Download HEAT TRANSFER FROM IMPINGING FLAME JETS

HEAT TRANSFER FROM IMPINGING
FLAME JETS
TR diss A
1559
Theo van der Meer
, HEAT TRANSFER FROM IMPINGING
V
FLAME JETS
HEAT TRANSFER FROM IMPINGING
FLAME JETS
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus,
prof.dr. J . M . Dirken, in het openbaar
te verdedigen ten overstaan van een
commissie door het College van Dekanen
daartoe aangewezen, op
10 september 1987 te 14.00 uur
door
Theodorus Hendrikus van der Meer
geboren te Zoetermeer
natuurkundig ingenieur
TR diss
1559
Dit proefschrift is goedgekeurd door de promotor
prof.ir. C.J. Hoogendoorn
aan mijn ouders
aan Funny
CONTENTS
1 . INTRODUCTION
1 .1
Background
1.2
Aims of this study
1.3
Outline of the investigation
2. LITERATURE SURVEY
2.1
Hydrodynamics
2.1.1
Turbulent free jets
2.1.2
The stagnation flow region
2.1.2.1 A bluff body in a uniform cross flow
2.1.2.2 The stagnation flow region of an impinging
jet
2.1.3
The wall jet region
2.2
Heat transfer of impinging flows
2.2.1
Stagnation point heat transfer
Influence of the turbulent length scale on
stagnation stagnation point heat transfer
2.2.2
Heat transfer from cold impinging jets
2.2.2.1 The laminar impinging jet
2.2.2.2 The turbulent impinging jet
2.2.3
Heat transfer from flame jets
3. THEORY
3.1
3.2
3.2.1
3.2.2
3.2.3
3.2.4
3.3
The governing equations
Turbulence models
The k-£ model of turbulence
A low Reynolds number model
Drawbacks of the k-e model
The anisotropic model
The energy equation
11
12
13
17
18
20
21
25
28
29
29
35
36
37
40
50
53
55
56
58
60
62
67
4. THE NUMERICAL METHOD
4.1
The general finite difference equations
69
4.2
The hydrodynamic solver
73
4.3
The grid
74
4.4
The boundary conditions
76
4.5
Determination of the heat transfer coefficient
79
THE EXPERIMENTAL METHOD
5.1
Heat transfer from the isothermal jet
81
5.1.1
Experimental set-up
81
5.1.2
Temperature measurements with liquid crystals
83
5.2
Heat transfer from the flame jet
85
5.2.1
The experimental set-up
85
5.2.2
The Gardon heat flux transducer
88
5.3
The laser Doppler anemometer
91
5.3.1
The optical configuration
91
5.3.2
The electronic equipment
93
5.3.3
T h e seeding o f the flow
95
R E S U L T S OF T H E E X P E R I M E N T S
6.1
Introduction
97
6.2
Flow structure
97
6.2.1
Velocity and turbulence on the axis of the
6.2.2
The radial velocity gradient in the vicinity
free jet
97
of the stagnation point
1 07
6.2.3
The radial velocity profiles
110
6.2.3.1
The isothermal jet
110
H/d = 2
111
H/d = 6
113
The boundary layer thickness
116
The flame jet
117
H/d = 2
119
6.2.3.2
H/d = 6
119
6.2.4
Axial temperature distribution
122
6.3
Heat transfer
123
6.3.1
Stagnation point heat transfer
123
6.3.1.1
Isothermal jet
124
6.3.1.2
The flame jet
128
Radiation heat transfer
128
Convective heat transfer
130
6.3.2
Radial heat transfer distributions
137
6.3.2.1
The impinging isothermal jets
137
6.3.2.2
The impinging flame jets
141
Temperature distributions
141
The heat flux distributions
143
The Nusselt number distributions
145
7. RESULTS OF NUMERICAL SIMULATIONS
7.1
The laminar impinging jet
149
7.1.1
Comparison with literature data
154
7.2
The turbulent impinging jet
157
7.2.1
Comparison of results on different grids
159
7.2.2
7.2.2.1
Comparison of numerical with experimental
results
H/d = 6
162
162
7.2.2.2
H/d = 2
165
7.2.2.3
The stagnation point heat transfer
171
8. DISCUSSION AND CONCLUSIONS
8.1
The flow structure
173
8.2
Heat transfer
174
8.3
The simulated laminar impinging jet
175
8.4
The simulated turbulent impinging jet
176
8.5
Main conclusions
176
LIST OF PRINCIPLE SYMBOLS
179
LIST OF REFERENCES
183
SUMMARY
191
SAMENVATTING
1 93
CURRICULUM VITAE
195
NAWOORD
197
1. INTRODUCTION
1.1. Background
Heating, cooling and drying processes are often used in
industry. In most applications high heat transfer rates leading
to short processing times are required. The high heat transfer
rates are especially needed in circumstances where the energy
consumption of the process is relatively high. Obtaining short
processing times is often needed for reasons of product
quality. A very well-known technique for heating or cooling
purposes is the application of impinging jets. The high heat
transfer due to turbulent forced convection by impinging one or
more jets of hot air or one or more flames on the object to be
heated makes a relatively short exposure time possible. In the
metallurgical industry this technique is called rapid heating.
In cooling and drying a similar situation occurs; one or more
jets of cold (dry) air impinge to cool (dry) a product.
Rapid heating of products in furnaces is a common process
in, for instance, the glass and steel industry. To obtain a
uniform heat transfer rate to the object in most cases
radiative heat transfer is preferred over convective heat
transfer. Radiative heat transfer can be achieved by firing
gas, coal or oil in a radiation furnace. The walls of the
furnace are heated and in its turn the object is heated by
radiation heat transfer from these walls. Also often an
electrically heated wall is used. In this way the control of
the radiation temperature over the hot surface can easily be
obtained. When a short exposure time of the object to the high
temperatures is needed, it can be advantageous to enlarge heat
transfer by impinging gas flames directly on this object. For
this purpose high velocity burners are used, the major heat
transfer is by convection. There are several other advantages
of using these so-called impinging burners in rapid heating
furnaces:
- The furnace walls are less heated than in conventional radia-
11
tion furnaces, giving lower wall heat losses. Starting up and
cooling down periods are much shorter which also result in an
energy saving.
- Energy can be saved by switching on the burners only when
heat is demanded.
- Compared to heating electrically the primary fuel demand
is smaller.
- It is possible to heat locally.
The energy savings compared to a conventional radiation furnace
can be more than 50%. A major disadvantage of rapid heating
furnaces can be nonuniformity of the heat flux distribution.
With convective heat transfer it is much more difficult to
obtain uniform heating of an object than with radiation heat
transfer. It is possible that hot spots are created and
overheating at such spots (for instance, at a stagnation point)
can occur. For this reason it is important to know the heat
flux distribution of a flame jet impinging on an object.
1.2. Aims of this study
The main purpose of the investigation presented in this thesis
was to study the nonuniformity of the heat flux and to find out
the influence of turbulence on the heat transfer for a single
Ufflk/
1
I
''•Ml
or
Fig. 1.1. The impinging jet.
12
flame jet impinging on a flat plate. The flow configuration is
given in figure 1.1. The flame jets were produced by modified
commercial rapid heating tunnel burners. The highest heat
transfer rates applying impinging jets can be achieved for
distances between the nozzle exit and the plate of 2 to 12
nozzle diameters. In this region for a turbulent jet the shape
of the velocity profile and the turbulence intensity profile
change with the distance from the nozzle. The jet shape also
depends on the shape of the nozzle from which it originates.
For these reasons one simple expression for the heat transfer
to a plate on which the flame jet impinges cannot be given from
literature. In this study both heat transfer and flow structure
of impinging flame jets and of isothermal air jets from the
same burners are thoroughly examined.
1.3. Outline of the investigation
From literature much data on stagnation flows and impingement
heat transfer are available. In chapter 2 this literature is
discussed. Since the flow around bodies of revolution has its
analogies with the impinging jet on a flat plate these flows
are discussed in the first place. Here an important parameter
is defined: the gradient of the radial velocity near the
stagnation point just outside of the boundary layer: a R =
(3v/3r)r_,.0 A similar parameter can be defined in impinging jet
flow: the gradient of the maximum radial velocity near the
stagnation point: B = (3v x / 3 r ) r . These parameters appear to
depend on the shape of the body of revolution and the shape of
the impinging velocity profile, respectively. The influence of
the shape of the body or the shape of the impinging velocity
profile on the heat transfer at the stagnation point can be
expressed using the radial velocity gradients a R or g.
The governing equations for the flow and heat transfer are
presented in chapter 3. These are the continuity equation, the
Navier-Stokes equations, the energy equation and equations
forming a model to calculate the turbulent viscosity of the
13
flow (the k-e model). Since the turbulence in a stagnation flow
will be anisotropic due to the deceleration in axial direction
and acceleration in radial direction, the commonly used k-e
model has been extended with a third parameter, which takes the
anisotropy into account.
In chapter 4 the numerical technique used, the finite
volume method, is given. Together with the appropriate boundary
conditions we have all ingredients to be able to solve the
governing equations from chapter 3 numerically. The results
from these numerical calculations are discussed in chapter 7.
At first some experimental methods and set-ups for determining
heat transfer and flow characteristics are given in chapter 5.
The heat transfer measurements for the isothermal impinging
jets are performed with a liquid crystal technique. Also a
Gardon heat flux transducer is used for the determination of
the heat transfer distributions of the impinging flame jets.
Temperatures in the flame jets are measured with thin wire PtRh
thermocouples. At last in chapter 5 the laser Doppler
anemometer for velocity and turbulence intensity measurements
is discussed.
The next two chapters deal with the actual results from
our study. The experimental results in chapter 6 and the
numerical results in chapter 7. At first in chapter 6 results
of the flow structures of the impinging jets are given.
Important characteristics of the jets are:
- the axial velocity decay and the axial turbulence development
as a function of the distance from the burner. Comparisons
between isothermal jets and flame jets can give insight into
the effects of combustion on the turbulence.
- the radial velocity gradient near the stagnation point (6).
The impact velocity profile will have its influence on this
parameter. With (3 a first estimation of the heat transfer at
the stagnation point can be made.
- the radial velocity profiles close to the plate.
In the same chapter 6 the heat transfer results are discussed.
14
Stagnation point heat transfer coefficients, determined with
the radial velocity gradient Q, are compared with stagnation
point heat transfer coefficient, calculated from heat flux
measurements. The influence of turbulence is examined. Heat
transfer results from flame jets and from isothermal jets are
compared and described as much as possible in a similar way. At
last the radial heat transfer distributions of the impinging
isothermal jets and impinging flame jets are discussed in this
chapter.
Chapter 7 contains results on numerical calculations.
Laminar impinging jets with three different impact velocity
profiles (flat, parabolic and Gaussian) are simulated. With
these calculations the influence of the impact velocity profile
on the heat transfer distribution on the plate can be
determined. Besides this the computer code can be validated by
comparing the results with results from other investigators.
Results of simulations of turbulent impinging jets are also
given in this chapter. Calculations have been performed with
the standard k-e model of turbulence with modifications for low
Reynolds numbers and with a k-e model including a parameter for
the anisotropy of the turbulence. The computed flow fields and
heat transfer are compared with the measurements for validity
for the two H/d values: H/d = 2 and H/d = 6.
Finally, in chapter 8 the conclusions from this study and
their consequences for the practical use of impinging flame
jets are discussed.
15
2. LITERATURE SURVEY
2.1. Hydrodynamics
Extensive studies on the hydrodynamics of stagnation flows have
been done in the past. They will be reviewed in this chapter.
Before entering into details a brief description will be given
of the flow pattern of an impinging round jet on a flat plate.
This flow can be divided into three regions (see figure 2.1):
the free jet region, the stagnation flow region, and the wall
jet region.
stagnation
zone
U
wall let
*\
►
Fig. 2.1. Flow regions of a jet impinging on a flat
plate.
In the free jet region the flat plate has no perceptible
influence on the flow. According to Schrader (1961) this region
extends to a distance of 1.2 times the nozzle diameter (1.2d)
from the surface.
'-■
In the stagnation flow region the axial flow strongly
decelerates and the radial flow accelerates giving rise to an
increased pressure in this region. The characteristics''of the
17
stagnation flow region depend strongly on the dimensionless
nozzle to plate distance (H/d). It extends from 1.2d in axial
distance from the plate to about 1.1d in radial direction for
small nozzle to plate distances (H/d < 12).
In the wall jet region the fluid spreads out radially over
the surface in a decelerating flow.
In the following paragraphs the three regions will be
discussed in more detail.
2.1.1. Turbulent free jets
The free circular turbulent jet has been studied thoroughly in
the past. In this paragraph only a brief description will be
given of the results of these investigations. More detailed
information can be found in the handbooks of Rajaratnam (1976)
and Abramovich (1963).
The free circular turbulent jet can be divided into three
zones. Referring to figure 2.2 we have:
developed
zone
developing
zone
potential
core zone
F i g . 2 . 2 . The f r e e
18
jet.
1 . The potential core zone immediately downstream of the
nozzle. In this zone the potential core is the flow region
where the velocity remains constant and equal to the
velocity at the nozzle exit. Turbulence is being generated
by the large shear stresses at the jet boundary and diffuses
towards the axis. The length of the potential core depends
on the initial velocity profile and on the turbulence
intensity in the nozzle exit. According to Gauntner,
Livinggood and Hrycak (1970) the potential core length
varies from 4.7d to 7.7d.
2. The developing zone in which the axial velocity starts
decaying. The velocity profile develops into a profile which
is independent of the nozzle geometry.
3. The zone of fully developed flow, where the velocity profile
has reached its final shape. Tolmien (1948) and Gortler
(1942) calculated a radial velocity profile from boundary
layer type equations with the use of Prandtl's mixing length
theory. Reichardt (1942) performed measurements and found
that a Gaussian velocity distribution fitted his results
best.
It is shown by Rajaratnam (1976) that in the fully
developed jet flow the jet broadens linearly and the velocity
at the axis decays linearly. This has been justified by
experimental results. For the axial velocity decay Hinze and
v.d. Hegge-Zijnen
(1949) and
Schrader
(1961) give the
correlations:
Hinze and Zijnen:
u
6.39
— = —
uQ
x/d + 0.6
(x/d i 8)
(2.1)
Schrader:
u
8.0
— = —
uQ
x/d + 3.3
(x/d i 8)
(2.2)
If the jet has a different density than the surrounding fluid,
a correction is required. Based on the conservation of momentum
flux Thring and Newby (1953) find an equivalent nozzle-
19
diameter, :de, for non-constant density jets. Due to the high
rate of entrainment the density within the jet will approach
the density of the surroundings (Ps) within a short distance
from the nozzle. The momentum flux is:
ird2
G =—
ird*
P o V = — f " P s u o'
<2-3>
which leads to
d_ = d (^2)
Ps
(2.4)
The relationships for isodensity jets can be used for nonisodensity jets using this equivalent diameter. Chen and Rodi
(1978) come to the same equivalent diameter by dimensional
considerations. Due to density differences also a buoyancy
effect can occur. Chen and Rodi also give the limits within
which a hot round jet will be non-buoyant, being:
Fr - 2
(-°-)_4 d
Ps
< 0.5
(2.5)
Here Fr is a densimetric Froude number
Fr =
D u 3
1°^°g(Ps - P0)d
This densimetric Froude number in our experiments
enough to obey criterion (2.5). .
(2.6)
was high
2.1.2. The stagnation flow region
In the stagnation flow region the axial flow strongly
decelerates and the radial flow accelerates giving rise to an
increased pressure. The characteristics of this region depend
strongly on the dimensionless nozzle to plate distance H/d. The
limits of the stagnation flow region too are determined by H/d.
According to Schrader (1961) for nozzle to plate distances H/d
< 10 the stagnation flow region extends to 1.2d from the
20
impinged plate in axial direction and up to about 1.1d from the
stagnation point in radial direction. Before entering into
details of stagnation of an impinging jet, the simpler flow
around a bluff body will be discussed.
2.1.2.1. A bluff body in a uniform cross flow
The first solutions of the boundary layer equations for a twodimensional shear layer along a cylindrical body, which is
perpendicular to a uniform cross flow, were given by Blasius
(1908), Hiemenz (1911) and Howarth (1935) (see Schlichting,
1968). They supposed the flow outside of the boundary layer to
be a potential flow. The velocity along , the body can be
expressed as:
V(x) = v.] z + V3Z
+ V5Z
+ . . .
(2.7)
Here z is the coordinate along the surface of the body. The
velocity profile in the shear layer was calculated as a similar
polynomial in the coordinate perpendicular to the surface.
In the vicinity of the stagnation point the velocity decay
due to stagnation; and the acceleration of1 the fluid flow along
the surface just outside of the boundary-layer are given for
axisymmetric flow by:
U = - 2 aRy
and V = aRz
(2.8)
and for plane flow by:j • ■ ■
U = - axy and V = a x z'
'
(2.9)
,
Homann (1936) solved the boundary layer equations for the
case of axisymmetric flow with assumption (2.8)..
The constants a R and a x in equations* 2.8 and 2.9;' depend on-'the
shape and size of the body of impingement and on the. uniform
flow velocity.' For three different bluntr bodies it is found
from potential flow solutions (see Kays; 1966 and Kottke,
Blenke and Schmidt, 1977):
~ .
21
circular disc
a
R
= 4 U^/dTr
(2.10)
sphere
a
R
= 3 U„/d
(2.11)
cylinder
a
x = 4
(2.12)
Uoo/d
where d is the diameter of the body involved and U^ the uniform
flow velocity. More accurate experimentally determined values
of a R and a x are given by:
Kottke, Blenke and Schmidt (1977) for a disc:
a R = Ujö
(2.13)
Newman, Sparrow and Eckert (1972) for a sphere:
a R = 2.66 U^/d
(2.14)
and Hiemenz (1911) for a cylinder:
a x = 3.63 Ujd
(2.15)
Compared to the infinitely extended laminar flow around a
body, the turbulent flow is far more complex. Let us consider
the influence of turbulence.
Due to the experimentally found strong sensitivity of
stagnation point heat transfer of cylinders and spheres to
small changes in the intensity of free stream turbulence (see
Kestin and Maeder, 1957; Kestin, Maeder and Sogin, 1961;
Kestin, Maeder and Wang, 1961), Sutera, Maeder and Kestin
(1963) and Sutera (1965) did a theoretical investigation into
the vorticity amplification in stagnation point flow. In a
basically
two-dimensional
flow vorticity was distributed
periodically over the third dimension. The normal velocity far
from the stagnation surface had a periodic waviness along the
direction normal to the plane of the basic flow (see figure
2.3). They showed that such vorticity, having a sufficiently
large scale, can enter the boundary layer and significantly
alter the heat transfer at the wall. Vorticity with a scale
larger than the neutral wave length Xm±n = 21I/(aT}/v)i
1
22
K
or X •
'nun
Fig. 2.3. The distorted stagnation flow studied by
Sutera, Maeder and Kestin (1963).
2Tr/(ax/v)5 will be amplified. The distortion of the velocity
field seemed to be small. Figures 2.4 and 2.5 show the
distortions of velocity and temperature of the mean flow along
the surface compared to the undisturbed case for Pr = 0.74. The
shear stress increased by 4.85%, the temperature gradient by
26%. Experimental verification of this theory is presented by
Sadeh, Sutera and Maeder (1970). They conclude that turbulence,
and hence vorticity, is being amplified by the deceleration of
the stagnation flow and by stretching of fluid elements. The
amount of amplification seems to depend on the direction of the
vorticity
as was predicted by the theory. For natural
turbulence on a stagnation streamline the turbulence intensity
they found is given in figure 2.6. The dependence of
amplification on scale was also found to be in accordance with
the predictions of the vorticity amplification theory.
23
1—r
0.8
0.6
0.4
0.2
Fig. 2.4. The distorted stagnation velocity
Sutera, Maeder and Kestin (1963).
J
after
L
v '
Fig. 2.5. The distorted
temperature field on a
stagnation streamline after Sutera, Maeder
and Kestin (1963).
24
1
T
i
r
V7~
2.0
f.5
r.o
0.5
o
0
0.1
0.2
0.3
y
Fig. 2.6. The
turbulence
streamline
for
intensity
natural
on
a
stagnation
turbulence
(after
Sadeh, Sutera and Maeder, 1970).
2.1.2.2. The stagnation flow region of an impinging jet
In the case of a bluff body in a uniform cross stream we have
seen
that
the average
flow field
in the stagnation region is
dependent of the shape of the body. So, it can be expected that
in the case of an impinging jet on a flat plate, the average
flow
field
in
the
stagnation
region
depends
on
the
oncoming
velocity profile. For a free turbulent jet the velocity profile
changes from a flat profile into a fully developed one with a
Gaussian shape. Thus the character of the centreline
decay
in
the
stagnation
definitions
of a R
body
uniform
in
a
and a x
cross
region
also
changes. Similar
velocity
to the
(equations 2.8 and 2.9) for a bluff
flow,
this
same
parameter ■ can
bé
defined for an impinging jet:
axisymmetric flow
u = - 2 aRy
and
v m a x = aRr
(2.16)
plane flow
u = - axy
and
v m a x = axz
(2.17)
Here v
is the maximum velocity along the plate. The analogy
25
between a bluff body in a cross flow and an impinging jet only
exists in the direct vicinity of the stagnation point. Where
for a bluff body in a cross stream the value of a R is
determined by the shape of the body, this value for an
impinging jet is determined by the shape of the oncoming
velocity profile.
For an inviscid uniform impinging jet Strand (1964)
calculated for the velocity along the deflecting surface (H/d =
1 ):
V = 0.9032 ^ ° d
+ . . . .
(2.18)
Scholtz and Trass (1970) obtained a similar expression for an
inviscid parabolic impinging jet. They found for H/d = 0.25:
V = 2.322 ^ ° — + . . . .
d
(2.19)
From experiments it appears that the value of a R for a
disc with diameter d in a uniform
cross flow is the same as
for a uniform jet with diameter d impinging on a flat plate.
Schrader (1961) and Dosdogru (1974) found for a R in the case of
small nozzle to plate distances of a uniform turbulent
impinging jet (1 1 H/d S 10):
Schrader
Dosdogru
H un
a R = (1.04 - 0.034 -) -°d d
a R = (1.02 - 0.024 -) -°d d
(2.20)
(2.21)
Giralt, Chia and Trass (1977) correlate their radial
velocity gradient in the stagnation zone with the impact
velocity and the jet half radius at the beginning of the
impingement region, where the axial velocity in the impinging
jet becomes 98% of the axial velocity in the undisturbed jet.
The impact velocity from measurements by Giralt (1976) being:
26
H
u„
= u^ (1.004 - 0.003 -)
c
°
d
H
- S 5.5
d
H
(2.22)
H
u„ = u„ (1.35 - 0.066 -)
"°
d
5.5 < - S 10.0
d
(2.23)
7.37
u_
c = u„
° 0.67 + H/d
H
- > 10.0
d
(2.24)
The length scale at the beginning of the impingement region is
characterized by them as:
rni
H
-iz- = 0.493 + 0.006 d
H
1.2 s - S 6.8
d
r, 1
H
- i * = 0.069 (1 + -)
d
d
(2.25)
■ d
H
- > 6/8
d
(2.26)
For the value of a R can then be found:
aR = U 1 . ^ ° -
(2.27)
15
u-| is a function of H/d expressing the influence of the shape
of the velocity profile. For H/d = 1 .2, where u c = u 0 and r^j. =
id, they find:
a*RD = 0.916 ^2
d
(2.28)
For H/d > 10, however,
a
R
1.852 ^
(2.29)
di
2
Like in equations 2.18 and 2.19 one can see the strong
influence of the shape of the oncoming velocity profile on the
flow characteristics in the vicinity of the stagnation point.
The role of the turbulence in the flow field has been
visualized by Yokobori, Kasagi and Hirata (1977). They showed
that when the plate was positioned in the developing region of
27
the jet (4 < H/d < 10) large scale eddies existed in the
stagnation zone. The eddies seemed to be much larger than the
thickness of the laminar boundary layer, and appeared to be
generated by the large *shea_r at the jet boundary'upstream. For
H/d < 4 the stagnation zone looked laminar-like and for H/d >
12 the eddies seemed to be distorted and accompanied by small
scale turbulence.
2.1.3. The wall jet region
Where the velocity essentially is parallel to the plate the
wall;,jet region starts. Schrader (1961) gdves a correlation for
the radius r at which the velocity along the wall starts
decaying. This he defines as the beginning of the wall jet
. v
■
. 'i
region. The correlation fór r
is:
y
-2 = 1.09 (-) 0 - 034
d
H
'
'
(2.30)
For the maximum velocity in the wall jet he finds:
U,
J0..
:———:
L+ K,(H/d - 1 . 2 ) ( — - 1 )
1 +0.1.8. (H/d.- 1 .2);l ..2
, -.rgl
r -1.17
(— ) - ' • " '
(2.31)
r
g
with K1 = 1.10 and K 2 = 0.27 for H/d S 4.7
and K1 = 1.45 and K 2 = 0.09 for H/d > 4.7.
For the wall jet velocity profile Schrader found that already
at r/d 6 2 the profile was similar to the profile calculated by
Glauert (1956) for a fully developed wall jet (see figure 2.7).
The validity of Glauert's calculations is shown by Bakke (1957)
and Poreh, Tsuei and Cermak (1967) who did measurements for
larger distances from the stagnation point (r/d > 10). Because
our experiments 'are "restricted to small values of H/d (H/d i
12) and" r'/d (r/d s 5) no' details of their measurements are
given.
28
1.2
I
i
i
i
i
y-*
1
1
I
o r=S0 mm
■ I
r=60
r=80
A r=100
a r = 150
V
-
0
0.8 /
v
o*L,
a
0.4
£? o
0
V
O
O
i
i
0.4
i
i
l
0.8
I
1.2
-
o
I
1.6
2.0
y/y*
Fig. 2.7. Wall jet velocity profiles measured by
Schrader (nozzle diameter of 50 mm, H/d =•
2) and the profile calculated by Glauert
(a).
2.2.
Heat transfer of impinging flows
The heat transfer characteristics of impinging flows will be
discussed in the next three paragraphs. Of course the heat
transfer is determined by the hydrodynamics treated in the
previous paragraph. Firstly, results from literature, of heat
transfer at a stagnation point will be discussed, mainly for
cylinders in a uniform cross flow. The next paragraph concerns
local heat transfer distributions of impinging round jets with
almost constant fluid properties. In the last paragraph
impinging flame jets will be discussed where, due to the large
temperature differences, the fluid properties (like dynamic
viscosity, specific density and thermal conductivity) vary
strongly.
2.2.1. Stagnation point heat transfer
Heat transfer at a stagnation point of a body of revolution has
29
been studied extensively in the past. Knowledge of the heat
transfer at this point is of importance because here the heat
transfer will be at a maximum. From literature we know the
solutions in expansion series from Pohlhausen (1921), Eckert
(1942) and Merk (1958). Sibulkin (1952) solved the boundary
layer equations for laminar heat transfer to a body of
revolution near the forward stagnation point. His solution can
be regarded as the basis of all other experimental and
theoretical results. For the Nusselt number in the stagnation
point of a body of revolution he found:
Nu = 0.763 d (-)5 Pr 0 ' 4
v
(2.32)
In this equation 3 is equal to the velocity
outside the boundary layer:
gradient just
<2-33>
= <3r
— > y = 6,r+o
For a two-dimensional stagnation point flow Kays (1966) gives a
similar equation, which comes to:
Nu = 0.57 d (-)5 Pr 0 * 4
v
- -
(2.34)
For a sphere, cylinder and disc the values of 3 are known (see
paragraph 2.1.2), leading to the corresponding stagnation point
heat transfer results:
2.66
u
n c
r\ A
sphere
6 = aR =
Nu = 1.2 4 Re^^Pr"* 4
(2.35)
cylinder
3 = a„ = —
Nu = 1.09 Re 0 * 5 Pr 0 - 4
(2.36)
disc
3 = aR = —
Nu = 0.763 Re°- 5 Pr 0 - 4 (2.37)
For a uniform jet impinging on a flat plate equation 2.37 can
30
be applied. When an impinging jet has a nonuniform velocity
profile (e.g. parabolic or Gaussian) the value of 8 will in
general be higher leading to a higher heat transfer rate at the
stagnation point. This in analogy with the heat transfer to the
stagnation point of a cylinder or a sphere.
Much experimental and theoretical work has been done on
the heat transfer of a cylinder in a turbulent cross stream in
the past. From these studies much can be understood from the
influence of turbulence on stagnation point heat transfer. In
this paragraph a review of these studies will be given.
Kestin, Maeder and Sogin (1961) and Kestin, Maeder and
Wang (1961) showed that the influence of free stream turbulence
on the heat transfer rate on cylinders in cross flow was
important. From experiments on heat transfer to a plate at zero
incidence it was concluded that only in the presence of a
pressure gradient the free stream turbulence had large effects
on heat transfer coefficients. The biggest enhancement in heat
transfer occurred at relatively low turbulence levels. Kestin,
Maeder and Wang (1961) found that the local Nusselt number
increased by amounts of 25%-50% when the turbulence intensity
increased from 0.5% to 2%.
Sutera, Maeder and Kestin (1963) and Sutera (1965)
presented a mathematical model for a steady plane stagnation
point flow. They showed that probably the dominant mechanism of
heat
transfer
enhancement
by
turbulence
is
vorticity
amplification by stretching (see paragraph 2.1.2). Computations
done by them showed that a certain amount of vorticity in the
oncoming flow caused an increase of the wall shear stress of
4.85%, while the heat transfer was - augmented by 26% (at Pr =
0.74) .
Smith and Kuethe (1966) performed experiments in lowturbulence wind tunnels. They found that the influence of free
stream turbulence increased with increasing Reynolds number. At
high Reynolds numbers (Re > 105) a phenomenological theory for
stagnation point heat transfer on a cylinder agreed with their
31
experimental, results,. The. theoretical- curve they found for the
•heat transfer at the stagnation point on a cylinder is:
Nu
-—=
/Re
1 + 0.0277,Tu/Re
(2.38)
The assumption was made that the eddy viscosity is proportional
to the free stream turbulence and to the distance from the
wall;. From their theory; Smith and Kuethe then concluded that
Tu/Re; iwoyld be the single correlation parameter .to describe
stagnation point heat transfer. From, their experiments at Renumber.s lower than 10 5 they found that Tu/Re was not the only
parameter, instead there.- was another dependency on the Re­
number : -'!
.
- . ! ■ • ■
"
Nu
-—=
/Re
-j.
.
,.
" "
'"
c
1 + 0.0277 Tu/Re (1 - exp (- 2.9 10~bRe))
-
,
. . .
,
.
.
(2.39)
1
■ -i Many ..-investigato.rs , later used the parameter Tu/Re in their
correlations.. Some, of them used, the theoretical calculations ,of
Frossling (1940). as a -basis for their turbulent heat transfer
correlation. Frossling gives for the; laminar heat transfer ,at
the stagnation point on a cylinder:
Nu
. — - . = 0-34.45,.
/Re
(2.40)
which 'is the; same..as; .equation 2.36 with Pr =. 0.7 for air.
J<estin- and Wood (1971) presented their experimental
results-using the Smith-Kuethe parameter Tu/Re and the result
from-..Frossling. Thus they found- for the turbulent- heat transfer
at ; the stagnation' point of a..cylinder: (7.5 lO1* < Re, ' <
1 .25
105)
Nu
/Re
Tu/Re
0.945 +. 3.48 — — —
100
Tu/Re
- 3.99 (
,-);*
100
(2.41 )
:. Sikmanovic, Oka and-. Koncar-Dj'urdj evic (1974) found, thati.at a
relatively :low Re-number (Re ,= 19;000) the augmentation -of ;;the
32
heat transfer was absent for Tu < 2%. They found:
Nu
Tie
0.945 + 1 .94
Tu/Re
100
2.41
Tu/Re
(——-)•
100
(2.42)
This correlation comes very close to equation 2.39 for Re =
19,000.
Lowery and Vachon (1975) on the contrary did not find the
dependency on the Reynolds number as in equation 2.39. This is
not surprising because their Reynolds numbers vary from 1.10s
to 3.10s, where with the exponent in equation 2.39 the effect
hardly counts. Lowery and Vachon found that a turbulence
intensity of 14% gave a maximum increase of the laminar heat
transfer of 60%. Raising the turbulence intensity more did not
seem' to increase the local heat transfer' anymore, however, it
needed more data to justify this statement. They found the
correlation:
Nu
/Re
= 1.01
Tu/Re
+ 2.624 —
100
3.07
(
Tu/Re
. . ):
(2.43)
100
Fig. 2.8. Stagnation point heat transfer from cylinders.
33
The results expressed in equations 2.38, 2.41, 2.42 and 2.43
are gathered in figure 2.8.
Apart from the experimental results discussed so far, some
investigators studied theoretically the stagnation point heat
transfer on a cylinder in a crossflow. Already mentioned is the
phenomenological theory of Smith and Kuethe (1966).
Galloway (1973) formulated a roll cell model, which has
been simplified into an eddy viscosity. He used the findings of
Sadeh et al. (1970) who showed the formation of roll cells in a
two-dimensional stagnation flow. Galloway found a strong
amplification for high Prandtl number flows.
Traci and Wilcox (1975) used the Saffman turbulence model
in their partly analytic, partly numerical solution of
stagnation point heat transfer. They considered three regions:
the free stream flow, the still inviscid body distorted flow
and the viscous wall region flow. Solutions of the three
regions were matched to each other. In agreement with Sadeh et
al. (1970) they find amplification of turbulent energy in the
stagnation region, while their heat transfer calculations do
agree with the known experimental results.
Miyazaki and Sparrow (1977) constructed a model for the
eddy viscosity on the basis of measured turbulent velocity
fluctuations. It contained a single unknown parameter which was
determined from experimental heat transfer results. Their
numerical calculations showed that the Nusselt number increased
with the free stream turbulence but to a lesser extent as the
turbulence intensity increases. The effect of turbulence on the
friction factor was much less than on the heat transfer, which
also was shown by previous investigators.
Gorla and Nemeth (1982) constructed a mathematical model
in which the momentum eddy diffusivity depended on the free
stream turbulence and the length scale. Available experimental
data were used to find the eddy viscosity as a function of
Tu/Re.
A
34
more
detailed
numerical
study
on
heat
transfer
enhancement around the stagnation point of a cylinder was
performed by Hijikata, Yoshida and Mori (1982). They added an
extra equation to the k-e model of turbulence to take into
account the production of turbulent energy due to anisotropy
between the longitudinal and lateral Reynolds stress components
in the free stream (see paragraph 3.2.4). A reasonable
agreement with reported experimental data was found.
Influence of the turbulent length scale on stagnation point
heat transfer
Next to the influence of the turbulence intensity on the heat
transfer investigations were done to the role of the turbulent
length scale. For the definition of a characteristic scale most
investigators use the turbulent macroscale or integral scale
being the scale of the energy containing eddies.The macroscale
of turbulence can be found by integrating the area under the
space correlation function:
L v = ƒ R(x) dx
x
o
(2.44)
with
u(X) 2
Van der Hegge-Zijnen (1958) found a rapidly increasing
heat transfer with increasing macroscale. He suggested the
existence of an optimum value of the ratio between scale and
cylinder-diameter
which
corresponds
to
a
condition
of
resonance. Then, some frequency of turbulence coincides with
the frequency of the eddies shed by the cylinder. The work done
by Sutera, Maeder and Kestin (1963) and by Sutera (1965) has
already been mentioned before. From their mathematical model it
follows that wavelengths shorter than a so-called neutral
wavelength:
35
X
min =-2ir/(av)
(2.46)
cannot satisfy the governing equations. The physical signifi­
cance of this is that vorticity with a scale smaller than the
neutral scale is dissipated more rapidly due to viscous action
than it is amplified by stretching. Recent experiments varied
the length scales of the flow to measure its influence.
Sikmanovic, Oka and Djurdjevic (1974) found that the Nusselt
number slightly decreased with an increase of the turbulent
macroscale in the region Lx/d = 0.05 to Lx/d = 0.182. In the
region 0.015 < Lx/d < 0.095 Lowery and Vachon (1975) did not
find a noticeable effect of the macroscale of turbulence.
Neither did Katinas, Zhyugzhda, Zhukauskas and Shvegzhda (1976)
in the region Lx/d = 0.16 to 0.36. Yardi and Sukhatme (1978)
examined the effect of turbulent macroscale on the heat
transfer . explicitly. They varied the macroscale over the wide
range of Lx/d = 0.03.to Lx/d = 0.38. They found that the heat
transfer coefficient at the front stagnation point increases by
about 15% as Lx/d is reduced from 0.4 to 0.05. The effect seems
to diminish as Tu/Re is increased. At the value of the
parameter (Lx/d)/Re of about 10 the effect of the macroscale is
at a maximum.
More recently Gorla and Nemeth (1982) presented a mathema­
tical model to predict heat transfer from a cylinder in
crossflow. They used an eddy viscosity model in which Tu/Re and
(Lx/d)/Re are parameters. The dependence of the turbulent
viscosity on Tu/Re was determined by fitting the results to the
experimental data available. The measurements done by Yardi and
Sukhatme (1978) were used by them to find the expression in the
eddy viscosity for the length scale parameter.
2.2.2. Heat transfer from cold impinging jets
In the paragraphs 2.1.2 and 2.1.3 the flow structure in the
stagnation flow region and in the wall jet region of an
impinging jet has been described. The heat transfer as related
36
to this flow structure will be treated in the following
paragraph, where we restrict ourselves to flows of fluids with
constant fluid properties. Firstly, results from literature on
laminar impinging jets will be discussed, followed by results
on turbulent impinging jets.
2.2.2.1. The laminar impinging jet
Most of the results reported in literature on heat and mass
transfer from laminar impinging jets are from theoretical
studies, although some experimental works are also available.
From these studies influences of some parameters on the
transfer of mass and heat could be determined without the
existence of turbulence in the flow. The effect of the Reynolds
number on the Sherwood or Nusselt number in the stagnation
region (a), the influence of the velocity profile of the
impinging jet (b), the influence of the separation distance H/d
between nozzle and plate (c) and the dependency of the transfer
coefficient on the radial distance along the plate (d) will be
discussed. Because of the sparsity of results on axisymmetric
jets, also results on two-dimensional
(slot) jets are
considered.
a. As can be seen in paragraph 2.2.1 the Nusselt number at the
stagnation point of a body of revolution in a uniform flow
depends on Re 5 This same dependency has been derived by
Scholtz and Trass (1970)' for a parabolic impinging round jet
and by Sparrow and Lee (1975) for a nonuniform impinging
slot jet. In both studies a solution for the inviscid flow
field was obtained. This solution was employed as a boundary
condition for the viscous flow along the impingement
surface. Another way of predicting thé flow field and heat
transfer from a laminar impinging jet is by solving the full
Navier-Stokes equations with the appropriate boundary
conditions.
This is done using' a finite difference
representation of the equations by Saad (1975), also
37
published by Saad, Douglas and Mujumdar (1977) for an
impinging round jet and by Van Heiningen (1982), also
published by Van Heiningen, Mujumdar and Douglas (1976) for
an impinging slot jet. A conclusion for the round jet study
was that Nu ~ Re
for a parabolic velocity profile in the
range of 900 < Re < 1950. For a flat velocity profile in the
same Re-range they do find the 0.5 power of Re, as the
boundary layer theory predicts. For the slot jet Van
Heiningen et al. (1976) find for a flat velocity profile
again agreement with the boundary layer theory: Nu ~ Re • .
For a parabolic velocity profile, however, they find Nu ~
R e 0 , 6 , which differs from the similar axisymmetric case.
Finally, two references give experimental results on mass
transfer in the stagnation region. Scholtz and Trass (1970)
confirm their theoretical results and find experimentally
the Re • -dependency of the Sh-number in the stagnation
region of a nonuniform impinging round jet. Sparrow and Wong
(1975) experimentally confirm the results of Van Heiningen
et al. (1976) for a slot jet with a parabolic velocity
profile: Sh - Re 0 - 6 .
The influence of the velocity profile.
In paragraph 2.1.2 from equations 2.18 and 2.19 we can see
the influence of the shape of the impinging velocity profile
on the radial velocity gradient near the stagnation point.
According to the theory of Sibulkin (1952) this velocity
gradient (6) determines the heat transfer coefficient in the
stagnation point. In the case of a uniform flow over a body
of revolution the stagnation point heat transfer strongly
depends on the shape of the body (see paragraph 2.2.1). In
the same way the shape of the velocity profile of a
jet impinging on a flat plate will have its effect on the
stagnation point heat transfer. This influence has been
shown by several authors. From the boundary layer theory
Scholtz and Trass (1970) find for H/d = 0.5:
Sh
/Re
0.8242 Sc 0 * 3 6 1 + 0.1351 (-) 3 S c 0 - 3 8 6 R
0.0980(-)" S c 0 " 4 0 8 + . .'
R
(2.47)
This relation holds for a parabolic velocity profile. With
the inviscid solution of a uniform impinging jet from Strand
(1964) they calculate (H/d = 1.0):
Sh
/Re
0.3634 Sc 0 - 3 6 1 + 0.03441 (-)2 S c 0 - 3 8 6 R
0.002531 (-)" S c 0 ' 4 0 8 + . .
R
(2.48)
These two equations hold for 1 < Sc < 10.
The numerical computations done by Saad et al. (1977) also
show the importance of the velocity profile. Not only in the
stagnation region, but also in the wall jet region the heat
transfer from a parabolic impinging jet is higher than that
from a uniform impinging jet, according to their calcula­
tions .
c. The influence of the separation distance between nozzle and
plate has been studied by" Saad, Douglas and Mujumdar (1977).
They found from their numerical calculations on axisymmetric
impinging jets with a parabolic velocity profile in the
range 1 .5 < H/d < 12 a decrease in stagnation point heat
transfer of 15% with increasing H/d at Re = 450. At Re = 950
they found no perceptible decrease of the Nusselt number in
this range. At the same separation distances Sparrow and
Wong (1975) measured mass transfer from a laminar impinging
slot jet with the naphthalene sublimation technique. They
found no influence of the separation distance on the heat
transfer for H/d < 5 (277 < Re < 1700). At higher values of
H/d they find turbulence effects.
d. The radial variation of the transfer rate is given
Scholtz and Trass (1963) from their theory by:
by
39
Sh = 0.4264 R e 3 / 4 ( - ) " 5 / 4 g(Sc)
(2.49)
d
with
g(Sc) = 0.3733 S c 1 / 3
(2.50)
' for high Sc-numbers (Sc > 10).
They find agreement of this correlation with experiments
obtained with a liquid jet at a high Schmidt number (10004000). Later the same authors find agreement also for a
Schmidt number of 2.45 (at r/d > 1.5) (Scholtz and Trass,
1970). Also Kapur and Macleod (1974) found agreement between
their measurements and equation 2.49. They determined local
mass transfer coefficients by holographic interferometry.
Scholtz and Trass used for their solution of the boundary
layer equation for the mass concentration the analysis of
Glauert (1956). He obtained a solution of the boundary layer
equations for the motion of axisymmetric wall jets on the
basis of self-preservation of the form of the velocity
profile. The theory of Scholtz and Trass, therefore, cannot
predict the difference in wall jet heat transfer originating
from a parabolic or a uniform impinging jet as observed by
Saad et al. It gives a higher power of Re in the wall jet
region than is found in the stagnation point region.
However, the fully developed region may not yet be reached
in the calculations by Saad et al. Their results for the
wall jet region do not seem to agree with the theory of
Scholtz and Trass (1963) and the experiments by them and
Kapur and Macleod.
2.2.2.2. The turbulent impinging jet
In most practical applications of heat or mass transfer from
impinging jets the flow will be turbulent. Exact solutions of
the problem are then no longer possible. Because of the
practical importance of this flow many investigators performed
40
experiments and tried to correlate the heat or mass transfer
rate to the flow parameters. Also numerical studies with the
help of turbulence models were performed. In this paragraph
only
results from axisymmetric
impinging jets will be
discussed.
There are several ways in which the heat transfer can be
correlated to the flow parameters. One approach is correlating
the Nusselt number (ad/X) to the relevant parameters by the
Reynolds number in the nozzle Re = u Q d/v, the turbulence in
the nozzle exit (Tu = / u 0 ' 3 / u 0 ) , the separation distance
between nozzle and plate (H/d), the radial distance from the
stagnation point (r/d) and the fluid properties. Thus a
correlation would have the form:
H r
Nu = f(Re, Tu, -, -, Pr)
d d
(2.51)
Correlations in'this form have been used in the past. The
development of jet velocity, turbulence and jet velocity
profile with x/d is accounted for by a single parameter H/d in
this equation. The disadvantage of this method lies in the fact
that results on heat transfer from impinging jets with jets
from different orifices do not agree. Especially the turbulence
level at the jet origin and the initial velocity profile
influence the jet development and subsequently the transfer
rates.
Because of the complexity of a result in the form of
equation 2.51 the heat transfer at the stagnation point is
often separated from the radial dependency.
Another way to describe the stagnation point heat transfer
is to use local (impact) parameters of the flow. Parameters
which describe the free jet at the plane of impact when the
plate is not inserted. In this way a correlation can be found
of the form:
Nu > 5 = f(Re 5 ,Pr,Tu c Y )
(2.52)
41
Here the Reynolds number is based on the impact velocity; the
turbulence grade Tu c is based on the impact turbulence
intensity; y is a parameter" which'is a function of the shape of
the impact velocity profile. Now all parameters in equation
2.52 are a function of x/d.
A review will be given of the most important contributions
to literature on heat transfer from axisymmetric turbulent
impinging jets.
Smirnov, Verevochkin and Brdlick (1961) correlated their
heat transfer measurements together with results from Perry
(1954) and Schmidt, Schuring and Sellschopp (1930) into one
equation for the stagnation point heat transfer:
Nu = 0.034 d0-9 R e 1 / 3 p r 0 - 4 3 exp (-0.037 -)
d
(2.53)
The range of variables where this formula holds is: 0.5 < H/d <
10, 1600 < Re < 50,000 and 0.7 < Pr < 10. The dependence on the
non-dimensional nozzle diameter d (in mm) (which varied from
2.5 mm to 16.5 mm) in this correlation is rather surprising and
is not confirmed by later experimentalists,
Huang (1963) used the impact velocity measured by a
pressure probe on the spot of impingement to correlate the heat
transfer rate. He finds for the stagnation point (1 < H/d < 10
10 3 < Re < 10"):
Nu = 0.0233 R e c 0 - 8 7 p r 0 - 3 3
(2.54)
Surprisingly he did not find any other dependency on H/d than
that of the impact velocity alone.
It is difficult to verify these and former results because
little is known of the characteristics of the jets that were
used.
The first extensive experimentalists who studied the
influence of turbulence on the heat transfer were Gardon and
Gobonpue (1962) and Gardon and Akfirat (1965). They showed that
in contrast to a laminar impinging jet the stagnation point
42
Fig. 2.9. Radial heat transfer distribution for a
round impinging jet on a flat plate at
H/d = 2 (from Gardon and Akfirat, 1965).
r/d
Fig. 2.10. Heat transfer for a round impinging jet
on a flat plate for Re = 28,000 (from
Gardon and Cobonpue, 1962).
43
heat transfer from a turbulent impinging jet increases when H/d
increases from 0 to 5. This is due to an increasing turbulence
level on the axis of a jet in this range where the velocity
remains constant. Several peaks were found in the local radial
heat transfer distributions as can be seen in figures 2.9 and
2.10. At small separation distances (H/d < 4) the maximum heat
transfer rate was situated at r/d - 0.5. This can be explained
by the existence of a minimum of the boundary layer thickness
at this place as was predicted by Kezios (1956). At a higher
radial distance (r/d = 1.9) an outer peak was distinguished
which at low Reynolds numbers separated into two outer peaks at
r/d = 1.4 and' at r/d . = 2.5. Two reasons for the possible
existence of an outer peak were mentioned:
1) Penetration of turbulence into the boundary layer coming
from the mixing layer of the jet.
2) Transition from a-laminar to.a turbulent boundary layer.
At higher values of H/d the inner as well as the outer peaks
disappeared due to the higher turbulence level of the impinging
jet for higher H/d. Experiments with turbulence promoters in
the nozzle exit showed that turbulence indeed had an enormous
influence:- at H/d =. 2 -the stagnation point heat transfer was
augmented and the outer peaks disappeared. The results from
this study were confirmed by Schliinder and Gnielinski (1967).
Measurements of the turbulence intensity very close to the
impingement surface (0.15 mm) showed a qualitative agreement
between this turbulence intensity and the heat transfer
coefficients. From this could, be', concluded that the outer peak
in the radial heat transfer distributions at r/d ^ 1.9 is due
to turbulent eddies penetrating the- boundary layer.
From mass transfer measurements Jeschar
concluded that, for .1 ... s H/d. S 20 and for
Nusselt number can be correlated with (r/d
f(Re). For the stagnation point mass transfer
< H/d S 20, 8,000 < Re < '30,000, they found:
44
and Potke (1970)
5 s r/d é 40 the
+ 1)~1*1 pr0-42
in the ranges 10
Sh = 1.2 Re°- 7 (-)" 1 - 1 S c 0 ' 4 2
d
(2.55)
It was possible to find a correlation without Tu as a
parameter, because in this range of H/d the turbulence leyel
does not vary significantly anymore.
Nakatogawa, Nishiwaki, Hirata and Torii (1970) made an
attempt to correlate the heat transfer rate to local flow
parameters. Their starting point is the correlation for heat
transfer in a plane laminar stagnation point flow (see equation
2.34), however, they consider an axisymmetric flow. For the
velocity gradient near the stagnation point, the axial velocity
decay and the jet half width diameter, they use empirical
relations. In spite of some poor assumptions, the experimental
heat transfer results for small separation distances (H/d < 5)
agreed quite well with their predictions. For higher distances
H/d the experimental results were 1.25 to 1.5 times larger than
the predicted values probably due to turbulence effects which
were at H/d = 8 at the highest level. The dependency on the
shape of the velocity profile was not accounted for by them.
For the wall jet region theoretical solutions, obtained by
assuming a velocity profile according to the 1/7th power law,
agreed well with the experimental values.
The quantitative influence of turbulence has been studied
by Donaldson, Snedeker and Margolis (1971a). They applied a
correction factor to the laminar stagnation point heat transfer
which is a function of the free stream turbulence level. For
the theoretical description of the laminar heat transfer they,
used a correlation from Lees (1956):
C_
dv
j.
-E-^r { p u ( - ) r . n ) 2
, 2(PrT* ^ ^ d r ' r = ° J
(2.56)
This relation is similar to Sibulkin's equation for the
stagnation point heat transfer of a body of revolution
(equation 2.32). For the radial velocity gradient (dv/dr) r=0
experimental values were evaluated
from data given by
45
Donaldson, Snedeker and Margolis (1971b) who assumed:
dv
, 1 32P
i
<^>r=o = t" ^ > r / o > a
< 2 ' 57 >
Because extensive measurements were done on the flow structure
of the free jet, the ratio of the theoretical laminar heat
transfer could be determined as a function of the average
relative turbulent intensity in the free jet, defined by
k = -(ü71" + 2 V 7 1 ")^
(2.58)
In the range of 0.10 < k/u < 0.25 the ratio of turbulent to
laminar heat transfer varied from 1.4 to 2.2. Although very
much scatter was found they did not find any discernable effect
of the Reynolds number on this ratio.
For the average heat transfer coefficients Subba Raju
(1972) derived relations which fitted the experimental results
of different authors. In the range of parameters 1 < H/d < 10,
2.10" < Re < 4.10s, 0.7 < Pr < 8.0 and 1 < D/d < 60 he found:
Nu\(-) 0 - 5 = 1.54 Re 0 ' 5 Pr 1 / 3
d
- s 8
d
Nu Pr" 1 / 3 (-) 3 = 35.0 Re 0 * 5 + 0.28 Re°-8(
d
d
(2.59)
- è 8
d
(2.60)
This result gives an indication that for D/d s 8 the boundary
layer along the impingement surface is laminar (Nu - R e 0 * 5 ) ,
while it is turbulent (Nu = Re 0 - 8 ) for D/d a 8.
Kataoka and Mizushina (1974) investigated the local
enhancement of the heat transfer rate by free stream
turbulence. A minimum in heat transfer is found by them in the
stagnation point for H/d < 0.5 and a maximum at r/d = 0.6. Here
the large eddies coming from the mixing region penetrate the
boundary layer. The local skin friction showed a secondary peak
at r/d = 2.2. In contrast to this, the local Nusselt number had
46
8)
a secondary peak at r/d = 4 (for 6 < H/d < 8.5). It should be
noted that their measurements were performed at high Prandtl
numbers (2420 to 3300).
The necessity of using local parameters of the impinging
flow to correlate heat transfer was observed by Chia, Giralt
and Trass (1977). They adopted the already mentioned boundary
layer solutions obtained by Scholtz and Trass (1970) to the
velocity and length scales proposed by Giralt, Chia and Trass
(1977), discussed in paragraph 2.1.2. These scales are the
collision velocity at the stagnation point U c and the jet half
width radius at the beginning of the impingement region. The
result of this approach is a mass transfer rate for the
stagnation region without the influence of turbulence:
Sh,
1
V-,
r
n-5>i am = "2 V 1 a{r , (Sc) + — J d 2 ' ( S c ) (
)3 + . . . }
I
o
z
R e i u.b lam
v^
r ^
(2.61)
(
The functions c 0 '(Sc), d2'(Sc) etc. are tabulated by Scholtz
(1965). The coefficients V-j , V3 etc. are tabulated by Giralt et
al. (1977) for different nozzle to plate distances. The
coefficients V^ , V, etc. take into account the varying
impinging velocity profile.
The effect of turbulence is taken into account by:
Sh_-
Sh,
7itT=
(1 + Y
i » <7ie->lam
(2 62)
'
For Yf a form also used by Lowery and Vachon (1975) and
Galloway (1973) for heat transfer to cylinders in a cross flow
is assumed:
y±
= a S c 1 / 6 (TU;L Re^ - b)
(2.63)
Experimental results showed that beyond - H/d = 11.0 the
variation of Sh^/ZRe^ with r/r-j. is universal, although the
turbulence free mass transfer (Sh^//Re^ )- L a m is universal beyond
H/d = 8.0. This is attributed to a still increasing effect of
47
turbulence between H/d = 8.0 and H/d = 11.0, while the velocity
profile
does
not
change
shape
beyond
H/d
a
8.0.
For
the
enhancement factor Chia et al. (1977) found:
y±
= 0
Tuj/Re < 4.0
Yi = 0.0156 Sc1/,6 (Tu i /Re - 4)
= 0.468 S c 1 / 6
"yi
These
results
are
flow discussed
results
(2.64)
Tu ± /Re > 34.0
in
qualitative
obtained .for heat transfer
the
4<Tu i /Re < 34.0
agreement
from cylinders
with
results
in a uniform
cross
in .paragraph 2.2.1. It should be mentioned
found
by
Chia
et
al.
(1977)
are
that
based
measurements at a single Re-number (Re = 34,000) at Sc
on
=2.45.
However, the resultant mass transfer equations have been used
to predict literature data and
found
to be consistent over a
wide range of flow conditions.
' 'A
similar
transfer
approach
results
of
of
using
a, cylinder
the
in
stagnation
a
cross
point
flow
has
heat
been
undertaken by Den Ouden and Hoogendoorn (1974).' They influenced
the turbulence level at the nozzle exit by placing grids in the
nozzle,.. It was found that for small separation. distances (H/d <
4) the experiments could be correlated with equation
Nu'
'
Tu /Re
- — = 0.497 + 3.48
/Re
100
Almost
for
the same equation was found
cylinders
distances
(see
equation
(H/d > 4) apparently
Tu /Re 2
(
)
100
3.99
(2.65)
by Kestin and Wood
2.41).
At
higher
the influence
(1971)
separation
of the
changing
velocity profile was the cause that equation 2.65 did not hold
anymore.
"Special attention to the radial distribution of
transfer
coefficient
has
been
given
by
Vallis,
the heat
Patrick
Wragg (1978). They used an electrolytic mass transfer
and
technique
.in a Reynolds number range of 3.880 < Re < 23,000. Supposing a
Pr 1 "-dependency.they found for the stagnation point: .
48
Nu = 1 . 9 3
Re0-58 Pr1/3
(_)-°-74
d
This result differs very much
(1970)
discussed
earlier.
10 < - < 20
d
from
For
(2.66)
that of Jeschar and Pötke
the
fully
developed
wall
jet
region Vallis et al. (1978) found:
Nu = 0.078 R e 0 - 8 2 P r 1 / 3 (-)" 1 - 0 5
d
8 < r/d < 17
(2.67)
or
N u r = 0.11 R e r 0 * 8 2 P r 1 / 3
with
the
distance
from
(2.68)
the
stagnation
characteristic length scale in Re
A very detailed
heat
transfer
submerged,
in
the
impinging
has
r
as
a
and N u r .
the influence of
stagnation
jet
Hirata and Nishiwaki
field with
study of
streamline
turbulence
on
region of a two-dimensional,
been
done
by
Yokobori, Kasagi,
(1978). They observed the stagnation flow
the aid of a flow visualization technique. Results
are already discussed in paragraph 2.1.2. The large vortex-like
motions they observed in the region 4 < H/d
< 10 enhance heat
transfer considerably. By fixing a fine cylindrical rod at the
nozzle exit, it seemed possible to create artificially a pair
of
large
scale vortices on the impinging wall. Even when the
wall was
positioned
in the potential
core, the vortices were
observed. The heat transfer in this region was enhanced by the
artificial eddies to the same level as the maximum increase in
heat
transfer
eddies.
This
produced
study
by
thus
the
mixing
demonstrates
induced
that
large
heat
scale
transfer
predominantly is affected by large scale structures.
More
recently
Kataoka
et
al.
(1987)
also
studied
the
mechanism
of the enhancement of stagnation point heat transfer
by
scale
large
turbulent
structures.
They
demonstrated
the
existence of vortex rings at x/d = 1 as already shown by Yule
(1978) and
scale
Strange
structures
and
are
Crighton
produced
(1983). These coherent
due , to
the
instability
large
of
the
°4 9
laminar shear layer. At x/d = 2.2 two vortex rings pair into
one before breaking up into large scale eddies at the end of
the potential core region. Autocorrelation of centreline
velocity fluctuations indicate for x/d S 4 periodicity. With
the characteristic frequency fe and the centreline velocity u a
Strouhal number is defined as St = fgd/u. This number equals
about 0.6 for 1 < x/d < 2 and 0.3 for 2 < x/d < 4. For x/d > 4
the Strouhal number is defined with the frequency of large
scale eddies, determined from the integral time scale. This
resulted in a Strouhal number of about 2 at x/d = 6 decreasing
to about 1 at x/d = 10. Kataoka et al. correlated the heat
transfer enhancement with a surface renewal parameter being the
product of a turbulent Reynolds number Re^ = / u g ' 2 d/v and
this Strouhal number. The value of u s , a has been obtained from
measurements 5 mm upstream of the stagnation point. In this way
they also show that enhancement of stagnation point heat
transfer is mainly due to turbulent surface renewal by large
scale eddies.
2.2.3. Heat transfer from flame jets
Knowledge of stagnation point heat transfer (see paragraph
2.2.1) and of heat transfer from impinging jets (paragraph
2.2.2) can be used when heat transfer from flame jets is
studied. A large number of investigators have used the
theoretical solution of Sibulkin (1952) for the boundary layer
equations for heat transfer at the stagnation point of a body
of revolution as a starting point for the prediction of heat
transfer from flames. This theory leads for the heat flux
density at a stagnation point to:
q" = 0.763 (p f y f g) 0 - 5 (hf - h ) Pr" 0 - 6
(2.69)
where p f , yf and hf are the density, viscosity and enthalpy in
the flame just outside of the temperature boundary layer in the
stagnation point, h is the enthalpy of the gasses at the wall.
Fay and Ridell (1958) extended this theory by taking into
50
account the dissociation of air and recombination of radicals
in the boundary layer along a cooled object. Their theory can
be applied when the chemical reactions of the flame are still
present in the boundary layer along the impinged surface. This
resulted in:
q" = 0.763 (^W)0.1 ( P U | 3 ) 0 - 5 (hf - h„) Pr" 0 ' 6 .
PfUf
{1 + (Le 0 - 52 - 1) -ii£}
h
f
(2.70)
where Le = D/a is the Lewis number (D being the diffusion
coefficient) and h^_ D is the dissociation enthalpy.
Buhr, Haupt and Kremer (1976) found that for methane-air
flames without preheating the radical concentrations are low.
The heat coming free with the recombination of radicals was
then found to be negligible. A number of studies concentrated
on high temperature flames for which recombination of radicals
in the boundary layer of a cooled surface is an important
factor. Among these are studies from Conolly and Davies (1972),
and from Kilham and Purvis (1971 and 1978).
Beer
and Chigier
(1968) reported
results from an
experimental investigation of a flame impinging at an angle of
20° on the hearth of a furnace. Their results show that heat
transfer can be increased by a factor of 3 using direct
impingement. The contribution of convection to the total heat
transfer amounted 70%.
Milson and Chigier (1973) performed studies on methane and
methane-air flames impinging on a cold plate. Both flames had a
cool central core of unreacted gas giving rise to lower heat
fluxes near the stagnation point than at some distance from
this point (for 10 < H/d < 16). The heat transfer coefficient
in the wall jet region was higher than in the impingement
region due to the cool central core.
Horsley, Purvis and Tarig (1982) used impinging naturalgas-air flames from several types of burners. For the
51
stagnation point heat transfer they found that the results
showed differences depending upon the turbulence structure of
the free flames from the different burners. Yet all turbulent
flames considered gave stagnation point heat transfer in the
order of 1.2 to 1.6 times higher than calculated from
Silbulkin's theory. This is in agreement with the findings of
Giralt et al. (1977) for impinging isothermal jets discussed
earlier.
52
3. THEORY
3.1. The governing equations
The flow of the impinging jet can be described by the full
Navier-Stokes equations (the equations of motion) and the con­
tinuity equation. For a two-dimensional axisymmetric flow these
equations in cylindrical coordinates read (see Bird, Stewart
and Lightfoot, 1960):
- continuity equation:
3p
1 9
3
— +
(rpv) + — (pu) = 0
3t
r 3r
3x
(3.1 )
- equations of motion:
3u
„ 3u
.. 3u
13
~
3?^,
3p
p — + pu — + pv — = - { - — (rTr rx
- —
x ) + — ^ }
3t
3x
3r
r 3r
3x
3x
3v
: 8v
_ 3v
13
_
3T_V
p — + pu — + p v — = -{
(rt__)
+ —£*■
rr
3t
3x
3r
r 3r
3x
TOO
^}
r
(3.2)
3D
.3r
(3.3)
with u, v, T and p momentary values. The components of the
stress tensor T Q and TQ do not appear in these equations
because they are supposed to be zero due to the absence of a Eldependency in the problem. The non-zero stress-components are:
u{2
g {2
- g{2
3v
2
3r
3
v
2
r
3
->- ■+
(V.v)}
->--»■ ,
(V.v)}
3u
2
3r
3
(3.4)
(3:5)
■+ ->■
(V.v)}
(3.6)
53
3u
3v
u(— - — )
dr
3x
Assuming incompressible flow which means
equations of motion can be reduced to:
(3.7)
that V.v = 0,
the
3u
„ 3u . _ 3u
1 3
3u
9
9u
p — +pu —
pv — =
(pr — ) + — (u — ) + S~
u (3.8)
3t
3x
3r
r 3r
3r
3x
3x
3v
_ 9v
.. 3v
1 9
9v
9
3v
p — + pu — + pv — =
(pr — ) + — (u — ) + S~
v (3.9)
9t
9x
3r
r 9r
3r
3x
3x
with
13
3v
3
3ü
3p
S~
(ru — ) + — (w — ) - —
u = - _
r 3r
3x
3x
3x
9x
(3.10)
13
3v
3
3u
2uv
3p
S~
(ru — ) + — (U — ) - -r-a - —
v = - —
r 3r
3r
3x
3r
r
3r
(3.11)
If the flow under study is turbulent a time averaging of the
equations over a time larger than the biggest time scales of
the turbulence is appropriate. For this reason at first the
Reynolds decomposition of the variables will be executed: the
momentary value of a variable is the sum of the averaged value
and a fluctuating value
u = u + u'
v = v + v'
(3.1 2)
p = p + p'
Averaging of the equations for a stationary flow results in:
- the continuity equation:
13
3
- — (rpv) + — (pu) = 0
r 3r
3x
54
(3.13)
the equations of motion:
9u
9u
1 3
9u
9
9u
9
pu — + pv — =
(ur — ) + — (y — ) - { — pu
9x
3r
r 9r
3r
9x
9x
9x
1 9
+
pru'v') + S n
(3.14)
r 9r
3v
9v
1 9
9v
9
9v
9
pu — + pv — =
(ur — ) + — (u — ) - { — pu'v'
9x
9r
r 9r
9r
9x
9x
9x
+
1 3
Vfi'3
prv'2 - p -2
r 9r
r
} + Svv
(3.15)
with
19
9v
9
9u
9p
S u = - — (ru — ) + — (u — ) - —
u
r 3r
3x
3x
9x
9x
(3.16)
19
9v
3
3u
S„ = - — (ru — ) + — (u — )
v
r 9r
9r
9x
3r
(3.17)
2yv
9p
- —
r
3r
The terms between brackets in equations 3.14 and 3.15 are
called the Reynolds stresses. These express momentum transfer
by turbulent motion, and will be treated as turbulent
diffusion. Equations for the Reynolds stresses can be derived
from the Navier-Stokes equations but the resulting equations
contain higher order correlation terms which in their turn are
unknown. This is the closure problem of turbulence. In the next
paragraph it will be shown that by using turbulence models
estimates are found for the unknown Reynolds stresses.
3.2. Turbulence models
The analogy between the Reynolds stresses and the viscous
stresses is the basis for the Boussinesq hypothesis stating
(Hinze, 1975):
3uH
9u^
2
-pu i 'u i ' = u. ( i + — 1 ) - - p k ó ^
3
Z
:
9Xj
9Xi
3
(3.18)
55
Herein u^ is the turbulent viscosity
energy of turbulent fluctuations:
k = i (u ,a + v'2
and k is the kinetic
+ v6'2)
(3.19)
Assuming that the turbulent viscosity is a scalar implies
that nonisotropic effects of the turbulence cannot be taken
into account. The k-e
model of turbulence, discussed in the
next paragraph, makes this assumption. This model is very
widely used. From literature we learn that it also is applied
to nonisotropic flows, however, that is not fully justified.
Applying the Boussinesq hypothesis to the time averaged
Navier-Stokes equations (3.14 and 3.15) leads to:
3u
3u
1 3
9u
3
3u
pv — + pu — = - — (rue rf rf — ) + — (u-ff
—)
9r
3x
r 9r
9r
9x e £ t 3x
9
2
- — (p + - pk) + S„
9x
3
(3.20)
3v
3v
1 3
3v
3
3v
pv — + pu — = - — (rupeffr r — ) + —- (P-ff
—)
err
3r
3x
r 3r
3r
9x
3x
3
2
— (p + - pk) + S v
3r
(3.21 )
with the source terms:
S
3
3u
u =^<Meff^»
+
1 9
Ï J7
(rU
3v
eff ^ >
3
9u
1 3
3v
v
S v = — (ue erfrf —-) + - — (rp-rr
—-) - 2 eur rff —
etr
3x
3r
r 3r
3r
r2
(3.22)
(3.23)
and with
Ueff = U + U t
(3.24)
3.2.1. The k-e model of turbulence
The
56
value of the turbulent viscosity will be defined by
means
of a turbulence model. One of the most frequently used models
is the k-e model. The variables k and e represent indirectly
the characteristic velocity- and length scale of the turbulent
fluctuations. Their definitions are:
- kinetic energy of turbulent fluctuations:
k = i ui'ui'
(3.25)
dissipation of k:
v
3u-' 3u, '
i
i3x-
dx-
(3.26)
The macroscale of turbulence in terms of k and e can be defined
as:
3/2
L = '
C-D
/e
(3.27)
n k
If one assumes that, in analogy with the laminar viscosity, the
turbulent viscosity
is proportional to a characteristic
j.
velocity scale of the turbulent fluctuations (k2) and to a
characteristic length scale of the turbulent fluctuations (L)
one can define:
u t = Cypk2/e
(3.28)
Transport equations for k and e can be deduced from the
Navier-Stokes equations (see Tennekes and Lumley, 1972). How­
ever, the resulting equations contain a number of unknown
correlations between fluctuating quantities. Rodi (1980) shows
how these unknown terms in the equations can be modelled
resulting in the following 'convection-diffusion' equations for
k and e:
3k
3k
3
Ut. 3k
13
u<- 3k
pu — + pv — = — (-£ — ) +
(r -£ — ) + S k
3x
3r
3x 0^ 3x
r 3r
o^ 3r
(3.29)
3e
3e
3
u. 3e
13
u f 3e
pu — + pv — = — (—£ — ) +
(r —£ — ) + St F:
3x
3r
3x ak 3x
r 3r
a£ 3r
-
(3.30)
with
57
S k = Pk - pe
s
e = c1 l
p
(3.31 )
k - C 2P ^
, 3u
P k = Mufc{2 ( — )'
*■
3x
+
<3'32>
3v
< —)*
3r
+
v
(-)'
r
,
+
3u
3v
(— + — ) ' }
3r
3x
(3.33)
The five constants appearing in equations 3.28 to 3.33 are
partly determined from measurements on well-defined turbulent
flows and from computer optimizations. The set of constants
used in general is:
o k = 1, o e = 1.3, a
= 0.09, C1 = 1.44, C 2 = 1.92
3.2.2. A low Reynolds number model
For a wall, where turbulence will go to zero very near to it,
one needs special provisions in the model. It is customary to
use wall 'functions. Herewith the steep gradients of the
velocity along the wall and the variations of k and e near the
wall do not have to be calculated, but are supposed to agree
with universal profiles (see Launder and Spalding, 1972). For
the flow under consideration these wall functions cannot be
used. Especially in the stagnation point region the flow near
the wall cannot be compared to a fully developed boundary layer
flow for which the wall functions are valid. So the numerical
calculations have to be extended till very close to the wall in
order to determine the variations of u, v, k, e and T near the
wall properly.
Then another problem arises: the k-e model is only valid
at high Reynolds numbers. The fluid flow very close to the wall
will not be simulated correctly by the standard k-e model.
Several investigators have defined models for low Reynolds
number flow with which laminarization can be predicted. Source
terms for k and e are added to the equations or the constants
in the model are made functions of a turbulent Reynolds number
defined by
58
Re,. = —
ve
(3.34)
Patel, Rodi and Scheuerer (1981) give an evaluation of
turbulence models for near wall and low-Reynolds number flows.
The model developed by Chien (Chien, 1980) is used in this
study. It implies that the k-e model described in paragraph
3.2.1 is.changed in five respects:
- In the diffusion terms for k and e in equations 3.29 and 3.30
next to the turbulent viscosity the laminar viscosity is
added.
- The dissipation term C2pe /k in the e-equation (3.30) is
multiplied by a function fe, where
f_ = 1 - 0.22 exp {-(Jit)*}
e
6
(3.35)
By doing this the dissipation term fits experimental data of
decaying homogeneous grid turbulence at both low and high
Reynolds numbers.
- Since the diffusion of k very near a wall is finite an energy
dissipation near the wall is needed to balance this. An extra
dissipation term is added to the k-equation (3.29) which
becomes effective near the plate:
2 vk
(3.36)
(H - x)
where H-x is the distance from the plate.
A similar extra "dissipation" term is added to the e-equation
(3.30):
2 ve
(x - H)
exp {
C4 v*(H - x)
}
(3.37)
where v* is the friction velocity:
V ={ U ( ) x
* " ë = H}^
(3 38)
-
59
- The damping effect due to the presence of the wall is further
taken into account in the definition of v^.:
ka
v t = Cy — {1 - exp (- C 3 v* (H - x)/v}
(3.39)
The two additional constants C3 and C 4 are found by Chien by
means of computer optimization for channel flow. The set of
constants used in this model is:
C y = 0.0 9 ,Cj = 1 .35
C 2 = 1 .8
C 3 = 0.00115
C 4 = 0.5.
ok = 1
o£ = 1 .3
3.2.3. Drawbacks of the k-e model
The k-e model of turbulence has been widely used. It has
already been mentioned that for nonisotropic turbulent flows
the use of the model is not justified. Turbulence in a
stagnation region is nonisotropic, as can be seen from the
study of Hunt (1973). Yet several researchers simulate im­
pinging jet flows with k-e-alike models with varying success. A
'one-equation model is used by Wolfshtein (1969) (who obtained
agreement with experimental results) and by Bower et al. (1977)
(who found disagreement for small nozzle-to-plate distances).
Agarwal and Bower (1982) used a low-Reynolds number form of the
k-e model and showed that the k-e model leads to better results
than the zero- and one-equation models. Chieng and Launder
(1980), however, predicted with a low-Reynolds number form of
the k-e model heat transfer rates that were about five times
too large in the vicinity of a stagnation point. Amano and
Jensen (1982) used the k-e model to predict the flow and heat
transfer of an axisymmetric jet impinging on a flat plate. With
three different models for the calculation of k and e. at the
grid points nearest to the wall their results mutually differed
largely. For the stagnation point heat transfer they also found
an overprediction.
The equations for k and e in the k-e model of turbulence
60
consist of terms which are modelled from the exact equations
for k and e. It will be shown here that the modelling of one of
the terms cannot be correct. The exact equation for the kinetic
energy of turbulent fluctuations k derived from the NavierStokes equations reads (Tennekes and Lumley, 1972):
,
(iuW
3
1
=
I
, u-jP' ^ u » 3
u
in
- 2v u' i S i j ) - u'iu'jSij - 2v s i j S i j
IV
V
VI
(3.40)
where S ^ is the mean rate of strain:
3UJ
SÜ
x
3
= i (
3U^
- + —1)
3XJ
(3.41)
dx±
and s^-i is the fluctuating rate of strain:
, 3u'H
au'.
= i (
1 +
1)
(3.42)
Si1
3
* 3XJ
dx±
The rate of change of the kinetic energy of turbulent
fluctuations (I) thus is due to pressure gradient work (II),
transport by turbulent velocity fluctuations (III), transport
by viscous stresses (IV) and two kinds of deformation work (V
and VI). The deformation work - u'^u'^S^ also arises in the
energy equation for the mean flow, but with an opposite sign.
This term describes the exchange between kinetic energy of the
mean flow and kinetic energy of the turbulent fluctuations. It
can be noticed directly that a deceleration of the mean flow
leads
to an
increase
of kinetic
energy
of
turbulent
fluctuations while an acceleration of the mean flow causes a
decrease of the kinetic energy of turbulent fluctuations.
The flow of an axisymmetric jet impinging on a flat plate,
which is subject of study, has very large normal velocity
gradients near the stagnation point. The strongly decelerating
mean axial flow will give its energy to the kinetic energy of
61
turbulent fluctuations. The strong acceleration of the mean
radial flow extracts energy from the kinetic energy of
turbulent fluctuations'. In the k-e model of turbulence the
exchange term - u'^u'^S^-i between mean and fluctuating flow is
modelled by means of the Boussinesg hypothesis as mentioned
before:
Su.:
- U'JUV
= vf
(
3u.i
i + —1)
2
- - k 6±,
(3.43)
This hypothesis, however, is not valid when the turbulence is
anisotropic as is the case in a stagnation region, or more in
general in accelerating or decelerating flows. Equation 3.43
has proved to be a good approximation for the non-diagonal
components (i * j ) . For the diagonal components it turns out to
be incorrect as can be seen from the resulting modelled
equation for the turbulence energy in which - u'^u'-iS^ is
modelled by:
" u i u j S ij =
2 v
t S ij*
< 3"44>
All components on the right hand side of this equation are
positive, while the sign of the diagonal components on the left
hand side is determined by the sign of the mean rate of strain.
In this model accelerating and decelerating flows both produce
turbulent energy. This can be seen clearly in the production
term Pj, expressed in equation 3.33. If for many types of flows
the k-e model gives fairly good results, this is due to the
fact that the diagonal components of the Reynolds stresses in
these flows can be neglected when they are compared to the
non-diagonal components. However, this cannot be done in
stagnation flows.
3.2.4. The anisotropic model
The k-e model of turbulence as we have seen in the previous
paragraph assumes that the turbulence can be characterized by
62
one velocity scale. This, however, cannot be valid for a
stagnation point flow where the influence of the wall gives
rise to various velocity scales. Models are developed which
allow for these different velocity scales. These models use
transport equations for the individual stresses u^'iu1 and are
called Reynolds stress models. If for the normally impinging
jet the
assumption
is being made that the turbulent
fluctuations in radial and tangential direction are similar and
only the fluctuations in the direction normal to the plate
deviate, then a model can be developed with only one extra
equation. This being an equation for the difference in the
axial and radial velocity scales (u 12 - v ' a ) . Such a model has
been developed by Hijikata, Yoshida and Mori (1982). They
studied the flow around and the heat transfer to a cylinder in
a uniform air flow. The same type of model has been developed
for the flow under consideration and will be discussed in this
paragraph.
As distinct from a full Reynolds stress model where all
Reynolds stresses are calculated by their modelled equations,
here only two equations, namely for the normal stresses u , a and
v 1 2 , will be used. The component vg'3 is supposed to be equal
to v'2. The non-diagonal components of the Reynolds stress
tensor are approximated by the Boussinesq hypothesis (equation
3.18). So it is assumed that the plate causes an anisotropy
between the turbulent fluctuations parallel to it and the
turbulent fluctuations perpendicular on it.
The transport equations for u 1 2 and v'2 can be derived
from the Navier-Stokes equations
(see Hinze, 1980). For a
stationary axisymmetric flow with vg = 0 they read:
3u' 2
u
3x
3u'2
+ v
3r
3u'3
= vVu'2
3x
1 3
12
)
r 3r (rv'u
.
2 3u'p'
p
3x
diffusion
63
9u
9u
2 (u 12 — + u'v' — )
9x
9r
production
p' 9u'
p 9x
9u' 9u'
2 (
)
redistribution
dissipation
+ 2
u
9XJ
9v'2
9v'2
9u'v'2
+ v
= vV 2 v' 2 9x
9r
9x
1 9
(rv'3)
r 3r
9r
production
9v' 9v'
- 2v (
9xi dx±
(3.45)
3X:
p
9r
redistribution
2
9v'
vfi'2
+ -S—)
2 vfi
ö'
. r
99
r2
dissipation
The unknown correlations in the diffusion, redistribution and
dissipation terms have to be modelled.
The diffusion terms are modelled similar to the diffusion
of k.
■
.
The redistribution terms (or pressure strain) consist of
two parts; one part (11^ -j ) due to the interaction of the
fluctuating velocities; the second part (Ili 2 ) d u e to the
interaction between main strain and fluctuating velocities.
This can' be shown by diminishing the pressure fluctuation p'
via a Poisson equation (see Launder, 1976). Rotta (1972)
proposed for the first part:
64
e
i 1 = " C1 a - ^ i '
H
ij1
2
2
" T k)
' = k
One can see that IIi
1
(3.47)
3
is proportional to the anisotropy of the
turbulence. In isotropic turbulence u^' a equals 2/3 k. The term
can either act as a source or a sink term and thus redistribute
the energy among the components. Shir (1973) gave a correction
on n^ -j for wall effects. Near a wall the fluctuating velocity
normal to it is damped while the fluctuating velocity parallel
to the wall is enhanced relative to the fluctuating velocity in
free
shear
flows.
This
effect
is
included
in
the
following
correction for JIJ 1 :
k 3/2
e
n
C
u
i l' = 1 a' - < n'
l,I
I,g k
n
where
n
denotes
the
2
"
3
u
V
n
6
i '
l
direction
ni>
ni
normal
function k ' /(H-x) is introduced
T
(3.48)
e(H_x)
to
the
wall
and
the
to reduce the effect of the
wall with increasing distance from it.
The
part
in
the
redistribution
term
due
to
interaction
between main strain and fluctuating velocities can be modelled
by (see Noat et al., 1970, or Reynolds, 1970):
n
i,2 = " C 2 , g < P i - \
P
< 3 - 49 >
k>
In this equation P^ is the production of normal stress u^' s and
Pk
is the production
that IT^ 2 i
s
of k. From this equation
proportional
it can be seen
to the anisotropy of the production
2
of u^' . In isotropic turbulence P. equals 2/3 P k .
For the dissipation terms in equations 3.45 and 3.46 it is
assumed
that the turbulence
amounts
of
equal.
energy
This
dissipation
is locally
dissipated
assumption
can
in
each
be
made
isotropic, so that the
energy
component
because
the
takes place at the level of the smaller
are
energy
turbulent
eddies. So for the dissipation terms can be written:
2
Ei = - e
(3.50)
65
where e is defined by equation 3.26.
Now the new parameter characteristic for the anisotropy of
the turbulence is defined:
g = 'u~r7 - V^
(3.51 )
From this definition, the definition of k (k = 5 (u'J + v' 2 +
Vg' 2 ) and the assumption v 1 2 = Vg ' 2 one finds for the diagonal
components of the Reynolds stress tensor as a function of g and
k:
2
2
u'2 = _ g + - k
3
3
(3.52)
— 2
1
2
2
v' = vfl
= - - g +- k
ü '
3
3
(3.53)
From equations 3.45 to 3.53
transport equation for g:
one can deduce
the
resulting
9g
3g
3
Uf 3g
1 3
vt
3g
pu — + pv — = — (u + —-) — + — — r (u + —-) —
3x
3r
3x
o k 3x
r 3r
o k 3r
+
(1 " C2,g> Pg " PCi,g J g
2
2
k^
- 3 pC,
' (- k + - g)
1
'9
3
3
(H-x)
(3.54)
In this equation the diffusion term of g is presented in the
same way
way as the diffusion of k. P is the production term
reading:
2
8
3u
2
4
v
P g = P <- - 9 - - k) — + P (- g - - k) y
3
3
3x
3
3
r
3u
3v
+ 2 vt {( — ) 2 - ( — )2)
u
3r
3x
(3.55)
The thus defined turbulence parameter g can be used to model
the production term in the k-equation (term V in equation
66
3.40)
Pk = - u i U j S ± j
In the standard k-e model all Reynolds stress terms are
modelled by the Boussinesq hypothesis and equation (3.33)
results. In the present model only the non-diagonal terms of
the Reynolds stress tensor are modelled by the Boussinesq
hypothesis. Together with the continuity equation (equation
3.13) this results in:
3u
PKk = p (v,a - u'2) — + uzt
3x
3u
3v
< — + — )2
3r
3x
+ p (v 12 - v f l ' 2 ) -
(3.56)
r
The last term equals zero, so that
P
K
= -
3u
pg
3x
+u
c
3u
(
+
3r
3v
)'
3x
(3.57)
This completes the anisotropic model consisting of the
transport equations for k, e and g. This model does not have
the drawbacks mentioned in the previous paragraph. In our study
this model has been combined with the described model of Chien
for low Reynolds number flows. The constants used in this model
are partly suggested by Chien (1980) and by Hijikata et al.
(198 2) . They are:
Cy
=
0.09
C1
=
1.35
C2
C3
=
0.00115
C4
=
0.5
C
C1fg'
=
=
1,g
1.8
=
1
-8
°k = 1
°£
C
6
2,g = ° -
= 1
0.16
3.3. The energy equation
Since heat transfer to the plate is our primary concern, the
energy equation has to be solved. For a two-dimensional axisymmetric flow the energy equation reads:
67
3h
3h
3h
1 3
A
p — +pv — +pu — =
(r
3t
3r
3x
r 3r
C
3h
3
A
) + — (
3r
3x C
Jr
3h
)
3x
(3.58)
ïr
with h the momentary value of the enthalpy. Like the flow
equations the enthalpy equation is averaged over a time larger
than the biggest time scales of the turbulent flow. The
Reynolds decomposition applied to the enthalpy results in an
averaged (h) and a fluctuating value (h 1 ):
h = h + h'
(3.59)
Averaging equation 3.58 for a stationary flow then leads to:
3h
3h
1 3
A 3h
pv — + pu — = — — r (— — - pv'h' )
3r
3x
r 3r
C 3r
3
A 3h
+ — (
3x C n 3x
P u'h')
(3.60)
In analogy with the Boussinesq hypothesis for the Reynolds
stresses the heat transfer by turbulent fluctuations pui'h' is
supposed to be linear with the enthalpy gradient:
A^ 3h
Ui. 3h
- pui'h' = -£. (
) = —£_ (
)
C
p 3xi
°h,t 3 x i
(3.61)
This equation defines a turbulent diffusivity (At/pC
= afc)
being related to the turbulent viscosity by a turbulent Prandtl
number a n {.. Equation 3.61 in fact means that they are the same
turbulent eddies that transport momentum as well as heat. With
this hypothesis the averaged enthalpy equation becomes:
x
3h
3h
3 r u
pv — + pu — =
I (
3r
3x
r 3r
ohfl
3
+ —
3x
68
m- 3h
+ —^)—}
o h f t 3r
y
{(—— +
°h,l
Ui- 3h
—)—}
°h,t
3x
(3.62)
4. T H E N U M E R I C A L METHOD
In this chapter t h e numerical method will be discussed which
h a s been used t o compute t h e flow a n d h e a t transfer in laminar
and turbulent impinging j e t s . T h e governing e q u a t i o n s a r e
described in t h e previous c h a p t e r . R e f e r e n c e s to similar
c a l c u l a t i o n s k n o w n from literature c a n b e found in paragraphs
2.2.2 a n d 2.2.3. M o s t studies used a finite d i f f e r e n c e method
or the h e r e t o related finite v o l u m e m e t h o d . T h e last method is
a l s o applied in this study. F o r t h e m e r i t s and d e m e r i t s of t h e
three m e t h o d s w e w i l l suffice with a reference to Shih (1984)
w h o gives a d i s c u s s i o n on this s u b j e c t !
4.1. T h e g e n e r a l finite d i f f e r e n c e equations
Starting point f o r the n u m e r i c a l calculations is t h e general
finite d i f f e r e n c e e q u a t i o n . A l l transport equations which have
to be solved a r e w r i t t e n in the same form as a convectiondiffusion equation:
3
1 3
3
3<t>
— (pud)) +
(prvd>) = — (r\ eftft — ) +
3x
r 3r
3x
$<
3X
1 3
r ^
■-■■■
34>
<r
^
«
Sr"'
+S
»
(4
"1)
Table 4.1 gives the different values of <t>, T* e £f and S^ for
all equations that are solved in case of turbulent flow.
The finite volume method or also called the control volume
method we apply here, has been described in detail by Patankar
(1980). The solution domain is divided into a certain number of
adjoining control volumes. Each of these volumes is surrounding
a grid point. The convection-diffusion equation (4.1) is
integrated over each of these volumes. After applying the
divergence theorem this leads to balances for fluxes across the
surface areas of the control volumes. The most attractive
feature of this method is the integral conservation of the
quantity 4> over each control volume and thus over the complete
69
TABLE
4.1
DIFFUSIVITY AND SOURCE TERMS FOR THE EQUATIONS TO BE SOLVED
FOR THE TURBULENT IMPINGING JET
'
1
*
r
continuity
i
0
x-impuls
u
u
r-impuls
V
u
enthalpy
h
*
0
eff
3p
3
3x
3x
elr
3u
1 3
3x
r 3r
3p
1 3
3v
- — - + - — ( r ue er fr f — )
3r
r 3r
3r
eff
u
ut
°h,l
°h,t
u
t
u + —-
k
kinetic energy
S
*,eff
3v
ett
3r
3
+ —
3x
3v
(u
f — )
e tp r f
3x
EJ
2 UE
(x-H) 3
V
- (2 uz t + u)
T7
0
2 uk
R
°C
(x-H)2
°k
E
dissipation
E
u + —-
°E
1 ,£
k
k
2,£-'E,2
k
-C 4 Re
lt
k 3/2
anisotropy
g
u + —-
°k
3u
pk
-
- pg
pg
= p (- \ g
Vx
3u
♦ ut (—
3r
8
1
0.4
e
<! - C2,g> pg " PC1,g £ 9 "3 PC1,g £ <J k + 1 9>
3v
3v v
+ — ) J + 2 v*.
{-( — ) - +
t
3x
3r r
v „,
{-)')
r
3u
2
4
v
, 3u,,
,3v-:11
k) — + p (- g - - k) - + 2 u t { ( — ) ' - <T-> )
fc
3x
3
3
r
3r
3x
-<R 2 t / 6 )
3
(X-H)E
Fig. 4.1. A control volume around grid point P with
the four adjacent control volumes.
solution domain.
A control volume around grid point P is shown in figure
4.1. The neighbouring control volumes are surrounding the grid
points N, S, W and E. The interfaces of the adjacent control
volumes (n, s, e and w) are situated halfway the grid points.
The total fluxes (convection + diffusion) are defined by:
J
x
= Pu* "
r
r
= pvt - r ^
<D,eff
3<t>
g^
3<t>
j
e f f
With
these definitions
the
equation can be written as:
3
3x
x
1 3
r 3r
r
(4.3)
_
general
convection-diffusion
(4.4)
<•>
Integration of equation 4.4 over a control volume gives:
r
p <Je " J w>
+
' r n J n " r s J s> = r p 'v S *
dV
<4-5>
The quantities J e , J w , J and J are the integrated fluxes over
the control volume faces, e.g.:
71
J e = SJy. dr
(4.6)
or, if J x does not vary over this area:
J
e = «Pu* " r<t,,eff |^)e
Ar
(4
-7)
The source term S^ in equation .4.5 is linearized as follows:
S
(t> = S c
+ s
(4
p*P
-8)
so that its volume integral leads to:
IS^
dV = (Sc + Spd)p) ArAx
(4.9)
l
Now the discretization scheme determines the values of Je, J w ,
J n and Js. We have used the hybrid scheme which is a
combination of the central differencing scheme and the upwind
differencing
scheme.
For
the
diffusion
term
in
all
circumstances central differences are applied:
3d)
4>,eff g^>e
(r
Ar
Ar
= <r4>,eff>e J^~
(<t,
E " V
=
D e (<t)E - 4>p)
(4.10)
The convection term (pu<t>)„ Ar {= Fe((t>)e) is either determined
by central differences or by upwind differences. The decisive
criterion is defined by the cell-Peclet number being the ratio
between convection and diffusion through the concerning face of
the control volume. For instance, for the east area of the
volume:
pu6x
p
(4 11
e = ~J = (f>e
- >
e
<t>,eff
Depending on this number three different approximations of the
flux through the east area are made:
Pe s - 2
J
- 2 < Pe < 2
Je
Pe S 2
72
J
e
e
=
F
e*E
= J?F e (d)p + <t>E) ■
=
F
e*P
p
*E " *P
S
(«x)e
(4.12)
Similar expressions can be derived for the other three faces of
the control volume. Together these yield to the general finite
difference equation:
ap<t>p = ag^E
+ a
W*W
+ a ( ,
S * S + aN<')N
where
+ b
(4.13)
.
a p = a E + a w + a g + a N - SpAxAr
-.
(4.14)
b = ScAxAr
and the coefficients a E , a w , ag and a N are dependent on the
finite difference scheme.
4.2. The hydrodynamic solver
The momentum equations are particular cases of the general
differential equation 4.1. A difficulty, however, in solving
these equations lies in the unknown pressure field which
indirectly is given via the continuity equation. In this
paragraph it will be shown how the variables u, v and p can be
solved by an iterative method. For this method where the
primitive variables are calculated directly without eliminating
the pressure, it is recommended to use staggered grids for the
velocity components. Grid points for the velocities are defined
on the faces of the control volumes for the pressure and other
scalar variables. In this way it is prevented that an
unrealistic zig-zag pressure field can be interpreted like a
uniform pressure field by the momentum equations (see Patanker,
1980). In figure 4.2 a two-dimensional non-linear grid pattern
with staggered u and v locations is shown.
The solution of the velocity field is obtained by applying
the so-called SIMPLE procedure (Semi-Implicit Method for
Pressure Linked Equations). At first a pressure field is
estimated. With this estimation the momentum equations are
solved giving estimated velocities. The obtained velocity field
will not satisfy continuity. The continuity equation, turned
73
7 p-cell
2 v-cell
3 u-cell
Fig. 4.2. Control volumes in a staggered grid.
into
an
equation
for
the
pressure
correction,
leads
to
a
pressure correction. Next the velocities are corrected in such
a way that the momentum equations are satisfying. Then with the
help of this velocity field the other variables (h, k, E and g)
are
solved.
estimate
The
for
the
corrected
next
pressure
iteration
is
step.
procedure see Patankar and Spalding
regarded
For
as
details
a
of
new
this
(1972).
Lately other algorithms have been derived in order to try
to
improve
the
(see Patankar,
1984).
Both
rate
of
1980) and
convergence. Among
SIMPLEC
algorithms, when
we
these
(see Doornmaal
applied
are
SIMPLER
and Raithby,
them, did
not
give
significant improvements in our case.
4.3. The grid
The definition of the grid used to perform the calculations is
very
grid
important.
points
interested
than
in
Areas
areas
heat
with
large
gradients
with
small
gradients.
transfer
at
the
wall
we
points near the wall. If a low Reynolds number
a very fine mesh near the wall is needed
require
Since
need
74 •
many
more
are
grid
model is used,
to predict
the damping out of turbulence. For the laminar and
calculations different grids were used.
we
properly
turbulent
- the laminar case
This grid contained 20 points in radial direction and 30 points
in axial direction. Most of the grid lines were concentrated
near the axis of symmetry and near the plate. The grid was
defined by:
x(i) = 6D{A( nx " 1 ) 3 + B ( " x " 1 ) 3
nx - 1
nx - 1
+ C(" x " i)}
nx - 1
n_ - i ,3
n_ - i ,
r(i) = 3D{E(-^
) + F (-±
)}
nr - 1
nr - 1
(4.16)
(4.17)
with n x = 30, n r = 20, A = 0.94, B = 0.05, C = 0.01, E = 0.8
and F = 0.2 this resulted in a grid shown in figure 4.3.
*-x
Fig. 4.3. The grid pattern used for the laminar cal­
culations .
- the turbulent case
Since the gradients of velocity and of the turbulent parameters
near the plate are much bigger in turbulent than in laminar
flow, much more grid lines are required near the wall in the
turbulent case. At least a few grid points should lie within
the viscous sublayer of the wall jet. The x-coordinate of the
grid lines is defined by:
75
exp{(-i - n x )/a x ) - 1
'exp{(n - 1)/ax} - 1
x(i)
(4.18)
In radial direction within 0.5D from the axis; of symmetry an
equidistant grid is used with 1/5th part of the total number of
grid points in this direction. For the region 0.5D < r < 3D a
similar formula as equation 4.18 is used. The result is an
almost linear grid in the radial direction.
The total number of grid points was varied from 40 x 40 to
40 x 60. The largest occurring aspect ratio of the control
volumes for the finest mesh was 16. Much more grid points are
needed for this aspect ratio to be in the order of unity.
4.4. The boundary conditions
The problem to be solved is defined by its boundary conditions.
The boundary of the domain is divided into five regions as
given in figure 4.4.
JL
^rrftfffTK
I
-mi
w
i
L
;
E
-*-x,u
Fig. 4.4. The five regions for the boundary conditions.
For each of these regions the conditions were:
I. The solid wall:
<t> = 0 ( < t > = u , v , k , e , g )
h = 2
76
For the effective viscosity in a near wall control volume the
laminar viscosity is used.
II.
The a x i s of
symmetry:
-
3d>/3r = 0 (4) = u ,
-
v = 0
k,
e,
g,
h)
III. The burner outlet
- v = 0
- h =1
-
u
= u in
The velocity profile at the burner outlet could have
different shapes:
• a flat velocity profile in the turbulent as well as the
laminar case:
u
in = umax
- a parabolic velocity profile in the laminar case:
u
in = umax <1 "
(
^
) 2 }
(4
- 19 >
• a Gaussian profile which stretches itself over the regions
III and IV of the boundary (laminar case):
u
in = umax exP<
in
max
■■
<->'>
4b D
(4.20)
The value of b was chosen such that at r = -jD: u^ n = \
"
k
u
m ax
kk
= in
in
Measurements of the turbulent fluctuations gave:
u' 2 = 1 6 v "
(4.21 )
It is supposed that the tangential fluctuations are of the
same magnitude as the radial fluctuations: Vg' 3 = v 1 2 . The
kinetic energy of turbulent fluctuations at the burner
outlet then equals:
77
9
(4.22)
16
With a measured turbulence level of 7% (/u,2/u = 0.07) this
leads to the boundary condition for k at the burner outlet:
k
in = ^
• °- 49 U in 2
(4
9 = 9in
With equations 4.21 and 4.22 we
parameter at the burner outlet:
find
for
the anisotropy
9in = < u " " v ' 2 >in = J k i n
"
'23)
< 4 - 24 >
e
= e in
The dissipation of turbulent fluctuations at the outlet can
be calculated from
e
in = C u 3 / 4
k3/2
< 4 - 25 >
/!m
and from an expression for
Launder and Spalding (1972):
the
mixing
l m = 0.0375 D
length
given
by
(4.26)
IV. The region next to the burner outlet
In practice next to the burner outlet there will be a wall or a
free boundary. For the laminar case here a wall has been
chosen:
- u = 0
In the turbulent case for reasons of convergence it has been
found that this boundary should be treated as an inflow. In
this way a boundary with a free inflow is prevented.
The velocity of the inflowing mass was chosen such that
there was enough flow rate to provide the jet with sufficient
entrainment air, while the velocity itself was small enough to
leave the jet undisturbed:
78
r
-
u
The
- v
- h
- g
- k
= ~
u
<4-27)
in
other boundary conditions in this region were:
= 0
=1
= 0
and e were defined as small while Meff = u
V. The boundary with the free outflow
- 34>/3r = 0 (ct> = u , k, e , g, h )
- For the radial velocity the condition of continuity has been
applied. For every new iteration the velocities at the
boundary were calculated from the velocities at the next last
grid line.
4.5. Determination of the heat transfer coefficient
Since the first grid
lies well within the
the determination of
the laminar jet is
equals:
<'
=
X
point from the wall in the turbulent case
viscous sublayer of the wall jet (y+ < 5 ) ,
the Nusselt number for the turbulent and
similar. The heat transfer to the plate
<^>x = H
<4-28>
By definition this is also equal to:
qw" = a(TH - T»)
(4.29)
with a = heat transfer coefficient, T H = temperature of the
plate and T^, = temperature far from the plate.
In case of temperature independent fluid properties, this leads
to:
aD
D
3h
Nu = — =
■—- (— )'xV _= HH
A
(hH - h j 3x
(4.30)
79
From the numerical results this Nusselt number has been
calculated in two ways:
1 ) (9h/8x) x=H has been calculated from the enthalpy of the
first grid point from the wall and the wall enthalpy,
assuming a linear profile.
2) (3h/3x) x=H has been calculated from the enthalpies at the
two nearest wall grid points and the enthalpy at the wall
assuming a quadratic profile.
If the applied grid is fine enough, the two ways should give
the same answer.
80
5. THE EXPERIMENTAL METHODS
5.1. Heat transfer from the isothermal jet
5.1.1. Experimental set-up
For the determination of the heat transfer coefficients of an
isothermal jet impinging on a slightly heated plate an
experimental set-up using a liquid crystal technique has been
built as shown in figure 5.1.
-£- MD
©
1. thermostat bath
2. waterreservoir
3. copper plate
4. glass plate
5. liquid crystals
6. Isolation
7. burner
8. rotameter
9. calming vessel
10. pressure air line
1I®, £-
°l
©
©
©
:©
®
Fig. 5.1. Experimental set-up for the isothermal jet
heat transfer measurements.
Two thermostat baths provided a water flow of a constant
temperature. This was forced to flow through a channel between
a copper plate and a glass plate. These two plates were
separated by a small distance of 2.9 mm. The water flowed along
the copper plate into a reservoir behind this plate and from
that reservoir back into the first of the two thermostat baths.
In this way one side of the glass plate was kept to a nearly
constant temperature, provided the flow in the channel was
homogeneous and the heat transfer coefficient to the glass
plate very high. The other side of the glass plate was covered
with a thin layer of liquid crystals to measure its surface
temperature (see paragraph 5.1.2).
81
A jet of air issued by a burner impinged on the plate with
the liquid crystals on it. The air originated from compressed
air and was reduced in pressure by a reducing valve. A constant
flow was achieved by leading the flow through a large vessel.
To determine the flow rate the air passed through calibrated
flow meters. With this flow rate the Reynolds number (Reg) at
the exit of the burner could be determined. Finally the
temperature of the air coming from the burner was measured by a
Cu-constantan thermocouple.
In a steady state the heat flux from the water to the
glass plate equals the heat flux through the glass plate and
also equals the heat flux from the glass plate to the air jet,
or:
*w" = a wp <Tw - Ta> = V
d
g
(T
a " Tl > = a
(T
1 " T j > + q "rad
(5.1 )
where
a,,„ = heat t r a n s f e r
coefficient
from the water to the g l a s s
J
wp
plate. The Reynolds number of the channel flow varied from 570
to 870. a
was estimated to be 1000 W/m2K.
T
= the temperature of the water in the channel which was
measured by thermocouples at five locations and varied from
50°C to 80°C. T a = the temperature of the side of the glass
plate in contact with the water.
X = heat conductivity of the glass (1.0 W/mk).
d a = thickness of the glass plate (9.9 mm).
T-, = temperature measured by the liquid crystals.
T. = temperature of the air jet.
q"racj = heat flux density due to radiation from the glass plate
to the environment.
The thermal resistance of the thin layer of liquid
crystals (50 urn) is neglected since it is much smaller than the
uncertainty in 1/aw_. The heat flux by radiation equals:
q" rad = ea ( V - T j *) = a r a d {T± - T j )
(5.2)
With an emission coefficient of e = 0.9, a temperature of the
82
liquid crystal surface of 43.25°C and a jet temperature of
20°C, this gives
a
rad
= 5
*8
w
/m2K
(5
'3)
Eliminating T a from equation 5.1 gives for the heat transfer
coefficient from the plate to the impinging air jet:
1
1
/V
+
T
A
w - Tl
T
T
V g l" J
(5
«rad
-4)
This leads to:
T
a
—T
= 95.7 _w
1 - 5.8
T
l - Tj
W/m2K
With this equation the heat transfer
determined with an accuracy of about 5%.
(5.5)
coefficient
can
be
5.1.2. Temperature measurements with liquid crystals
In the transition from the liquid phase to the crystalline
phase (the so-called mesophase) cholesteric liquid crystals
have some peculiar optical properties. In the mesophase the
molecules have a specific order in which the forces between the
molecules are very weak. The molecular structure in this
temperature range is instable and can be influenced very easy,
changing the optical properties of the material. One of the
most striking optical effects is the spectral reflectivity of
light as a function of temperature. In the liquid phase as well
as in the crystalline phase the material transmits all wave­
lengths of the visible light while in the mesophase, when the
temperature rises, its colour changes from red to yellow, green
and violet. This phenomenon can be used to carry out
temperature measurements in a non-destructive way (see for
instance Fergason, 19 68 and den Ouden and Hoogendoorn, 1974).
Depending on the composition of a mixture of several
cholesterics all desired temperature-colour characteristics can
be obtained. One of the mixtures used in the experiments is
83
given in table 5.1.
TABLE 5.1
MIXTURE OF COMPONENTS USED IN THE EXPERIMENTS
parts in weight
10.4
3
33
24
35
component
butoxy benzoate
chloride
noanoate
oleate
oleyl carbonate
A calibration of this mixture gave a temperature-colour
characteristic of table 5.2.
TABLE 5.2
TEMPERATURE-COLOUR CHARACTERISTIC OF THE MIXTURE
FROM TABLE 5.1
colour
red
red-brown
light brown
yellow-brown
yellow
yellow-green
light green
blue green
light blue
blue
dark blue
violet
84
temperature in °C
42.8
42.95
43.1
43.2
43.25
43.3
43.4
43.65
43.85
44.15
44.6
44.8
From table 5.2 it can be seen that very 'accurate
temperature measurements are possible. Especially the colour
yellow at 43.25°C could clearly be detected and gave an
accurate measure of the temperature (± 0.1 °C).
The procedure of producing a layer of liquid crystals on
the glass plate was as follows: The different components were
brought to their liquid phase and mixed with a resin (neocryle
B-723) in toluene. The obtained solution was then sprayed onto
the glass plate and dried in a furnace for 24 hours. This
resulted in a thin layer of about 50 urn.
By varying the water temperature at the other side of the
plate, the colour yellow (representing 43.25°C) could be
detected at different spots. With the help of a coordinate
system engraved in the blackened copper plate behind the glass
plate a radial distribution of the heat transfer coefficient
could be measured.
5.2. Heat transfer from the flame jet
5.2.1. The experimental set-up
A small industrial rapid heating tunnel burner has been used to
obtain a premixed flame jet. In its original version the burner
had a tangential inlet of the gas-air mixture producing a swirl
acting as a flame holder. In case of the isothermal jet from
this burner the swirl still existed, however, the flame jet did
not show any swirl. To gnt rid of this unwanted difference
between isothermal jet and flame jet the tangential inlet
connection was replaced by an axial connection. A small disk
has been used as a flame holder.
Another alteration in the original burner design was made
for a more practical reason. In order to be able to make a
bigger version of the same burner, the shape of the inner
burner wall was altered in such a way that it could easily be
scaled. The final shape of the burner is shown in figure 5.2.
To determine the heat transfer coefficient from the flame
85
13.8mm
Alfi3
steel shield
flame holder
refractory material
inlet air/gas mixture
F i g . 5 . 2 . The b u r n e r .
jets to an isothermal plate the experimental set-up shown in
figure 5.3 has been built.
Volume rates of air and natural gas fed to the burner are
measured by rotameters before mixing these two flows in a
mixing chamber. The air is supplied by a compressed air line,
while the gas is available from bottled Groningen natural gas.
The stoichiometric mixture of gas and air after equalizing in a
vessel was led to the burner. A polished copper plate cooled by
water from a thermostat bath was placed on the top of the
burner. A Gardon heat flux transducer (Gardon, 1960) has been
included in the plate. Surface temperatures of the plate are
measured by eight thermocouples. The position of the burner.
relative to the heat flux transducer in the plate can be varied
by means of a 3-dimensional traversing mechanism in which the
burner is placed.
Temperatures of the flame jets are measured using
86
thin
ISOTHERMAL PLATE
THERMOSTAT
EQUALIZING CHAMBER
MIXING CHAMBER
X'
«
thermometer
manometer
rotameter
natural gas
"=j>i
_£.
air supply
Fig. 5.3. Experimental set-up for
transfer measurements.
*
flame
reducer
valve
jet
heat
87
10%RhPt-Pt and 30%RhPt-6%RhPt thermocouples with diameters
varying from 180 uin to 50 urn. The junctions are butt-welded and
to avoid contamination due to the high temperatures the wires
are coated with a mixture of beryllium oxide and yttrium
chloride as advised by Kent (1970). The thin wires are buttwelded to water-cooled supports of thicker wires of the same
material. The radiation loss of the thermocouple junctions
which becomes significant at temperatures over 1300°C is
estimated by extrapolating the results of measurements with
thermocouples
with
different
junction
diameters
to
a
thermocouple with an infinitesimal diameter, and checked by a
calibration.
5.2.2. The Gardon heat flux transducer
The heat flux transducer used in the experiments has first been
introduced by Gardon (1960). The design of this transducer for
our experiments is shown in figure 5.4.
cooling water
Isothermal plate
thermocouple
Isothermal surface
Fig. 5.4. The Gardon heat flux transducer.
88
A cylindrical copper body with cooling fins is cemented in
the water-cooled isothermal plate. On the cylinder axis of this
body a hole was drilled through. At one side this hole was
closed by a thin foil of constantari in such a way that the
surface of the copper body with foil was flushed with the
isothermal plate. In the middle of the constantari f bil a thin
copper wire is spot-welded. In this way a thermocouple is
constructed by which the temperature difference between the
centre of the foil and its edge, which is in contact with the
copper body, can be measured. The surface temperature of the
copper body,
which is flat with the isothermal plate, is
measured by a thermocouple and is kept equal to the plate
temperature by an extra cooling.
When the constantan is imposed to a heat flux, a
temperature gradient will exist over the radius of the foil.
The measured temperature difference will then be a measure of
the heat flux.
Gardon
(1960) made a theoretical analysis of the
performance of this heat flux transducer. For its sensitivity
he found:
—
e
= 2.29 103 —
R2
W/m2 mV
(5.6)
Measuring the emf. (e) of the thermocouple (in mV) from the
transducer with a foil with radius R = 2 mm and a thickness S =
0.2 mm gives a heat flux density equal to:
q" = 11.43
10"
e
W/m2
(5.7)
Because of the uncertainty in its dimensions the Gardon
heat flux transducer has to be calibrated. This has- been done
in a black body cavity (Hohlraum). In a cylindrical opening at
the bottom of this Hohlraum the transducer fitted exactly. The
configuration factor was equal to 1 . The surface of the
transducer
has been painted
black making the emission
coefficient very close to 1. The heat flux density absorbed by
89
its surface is equal to:
q" = O (Tg* - T0") + a c (Ts - T 0 )
(5.8)
The temperature T of the furnace cavity was measured by a
calibrated thermocouple. The temperature T Q of the heat flux
transducer was controlled by the water cooling and measured by
a thermocouple just beneath its surface.
The second term on the right hand side of equation 5.8
expresses the heat transfer to the surface by convection.
According to McAdams (1954) the heat transfer coefficient for
cooled square plates facing upward in air is:
a c = 0.12 (AT/L) 0 - 2 5
(5.9)
with L = side of the square plate. With this equation a c in 5.8
can be approximated by stating L = 2R.
In figure 5.5 the results from the calibration are given.
The sensitivity that is found appears to be linear for heat
flux densities from 30 kw/m2 to 150 kW/m2 with the relation:
q" = (13.0 ± 0.4) 10" e
W/m2
(5.10)
14012010080 60 40 20-
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
t[mV]
1.2
Fig. 5.5. Calibration curve of the heat flux transducer.
90
which comes reasonably near to Gardon's value (5.7). The
highest temperature that could be reached with the furnace was
1100°C corresponding with a heat flux density of 150 kw/ma.
Gardon found that these transducers had a linear relation
between q" and e up to temperature differences of 185°C. For
our transducer this corresponds with a heat flux density of 600
kW/m3. The result of the calibration expressed in equation 5.10
has been extrapolated up to this value.
5.3. The laser Doppler anemometer
Measurements of velocity and turbulence intensity have been
performed with a laser Doppler anemometer (LDA). Details of
this measurement technique can be found in textbooks (e.g.
Durst, Melling and Whitelaw, 1976; Watrasiewicz and Rudd, 1976;
Durrani and Greated, 1977). In this, paragraph the experimental
set-up that actually has been used will be discussed without
going into details on the technique itself. The experimental
set-up consists of two parts: an optical and an electronic
part.
5.3.1. The optical configuration
Figure 5.6 shows the optical configuration that has been used.
1
2
3
4
5
1/2-A plate
beam splitter
glass rod
Bragg cell
wedge
6 lens
7 diaphragm
8 collector lens
9 pinhole
10 photomultlpller
Fig. 5.6. The optics of the laser Doppler anemometer.
91
A laser beam passes a ^A-plate. With this retardation plate one
is able to rotate the polarization plane into the direction
suitable for th.e other optical components. Afterwards the beam
is split by a beam splitter into two parallel beams of equal
intensity. The beam separation is 50 mm. One of the beams
passes, a Bragg cell operating at a frequency of 40 MHz.
With this cell a positive or negative optical frequency shift
of the laser light can be obtained, which makes it possible to
detect the direction of the flow. Due to the acousto-optical
effect several beams with shifted frequencies leave the Bragg
cell. The cell can be adjusted in such a way that 85% of the
diffracted light is in the first order beam having a frequency
shift of 40 MHz relative to the original beam. Because the
first order beam leaves the Bragg cell at a certain angle, a
wedge is used to make it parallel to the original second beam
and to restore the separation distance of 50 mm. The second
beam is led through a glass rod to ensure that the shifted and
unshifted beams have an equal beam path length. All beams other
than the first order beam coming from the Bragg cell are
blocked. A lens focuses the two remaining beams. The
intersection of the two laser beams forms the so-called
measuring volume. Scattered light from particles moving through
the measuring volume is focused by a lens onto a pinhole. In
order to reach an optimal signal-to-noise ratio only 70% of the
width of the measuring volume (defined by the'1/e2-points of
the Gaussian intensity distribution of the laser beams) is
focused onto the pinhole. A photomultiplier is used to measure
the intensity of sthe light coming through the pinhoie.
The Doppler frequency
particles is equal to:
of
the scattered
light
from
the
2 U sin' (6/2"}
where 6 isiri the angle between the two beams and U is the
velocity component perpendicular to the optical axis.
92
Since the frequency (f) detected by the photomultiplier is the
sum of the preshift frequency (fD) and the Doppler frequency
(f^) the velocity component equals:
X
u =
— . (f - fn)
(5.12)
2 sin (0/2)
P
For the measurements two different lasers were used: a 5
mW He-Ne laser and a 50 mW Ar-ion laser. The higher output of
the Ar-ion laser more than compensates for the fact that the
optics have been optimized for the wave length of the He-Ne
laser. The characteristic properties of the set-up were:
He-Ne
Ar-ion
Width of the measuring volume
188 um 266
Length of the measuring volume
1.13 mm 3.21
90
Number of fringes in the measuring volume
97
60
Number of fringes observed through pinhole
68
Proportionality between velocity and
m/s
2.94
1 .925
frequency
MHz
um
mm
m/s
MHz
5.3.2. The electronic equipment
The electric signal from the photomultiplier has to be
processed
before
results
like
averaged
velocities
and
turbulence intensities can be calculated. Figure 5.7 gives an
outline of the electronic equipment.
The photomultiplier signal passes an amplifier (20 dB)
before it is downmixed to the desired frequency range. It means
that the preshift frequency, which optically is equal to 40
MHz, is electronically brought down to a lower value (in steps
variable from 0.01 to 20 MHz). Then the signal is analysed by a
signal processing counter (TSI-1980A). In this counter the
Doppler signal is band-pass filtered to remove high- and lowfrequency noise and it is amplified. A threshold detector
93
power supply I
photomultlplier
| pro-amplifier
mixer
|
HP • 1000
I
counter
micro-computer
terminal
graphic display
Fig. 5.7. The electronics of the laser Doppler anemo­
meter.
senses the presence or absence of a Doppler signal by the
amplitude of the incoming signal. A Schmitt Trigger converts
the analogue signal into a digital pulse train. A timing
circuitry, activated by the threshold detector, measures the
time needed for a fixed number of pulses (= periods of the
Doppler burst) to pass. This time is a measure for the velocity
of the particle passing the measuring volume. The individual
velocity measurements of the particles are read from the signal
processing
counter
into a home-made microcomputer
(see
Cornelissen,
1980).
Averaged
velocities
and
turbulence
intensities
are
calculated
from
5,000
(N)
individual
measurements by:
ü =
N
Z
j=1
i ,a =
N
I
j.i
,
3
N
3 -j = i
,u
>
(5.13)
:
3 j=1
in which the weight factor w^ can be:
- for arithmetic averaging: WJ = 1 ;
- for 1-D averaging (see Dimotakis, 1976): WJ = 1/|u,
94
(5.14)
Results of these calculations are routed to a terminal.
From the measurements also a probability density function is
made which is routed to a graphic display.
On basis of the observation of the probability density
function a measurement can be accepted or rejected. The
criterion for this decision is whether or not the probability
density function looks like a Gaussian distribution, which can
be expected. There are three possible ways of continuation:
1) If a measurement is accepted, the results are sent to a
HP1000 computer for storage and a new measurement can be
initiated.
2) If a measurement is rejected, the same measurement will be
repeated after eventual resetting : of the filters of the
signal processing counter, the amplification factor or the
preshift frequency.
3) If only a few of the 5,000 individual measurements appear
to be far outside of the expected velocity distribution, a
software filtering of the results can be applied, in order
to exclude these apparently wrong individual measurements.
Then the filtered results u, u'a and the probability den­
sity function are again calculated from the filtered data
and the results are stored by a HP1000 computer.
5.3.3. The seeding of the flow
The
laser
Doppler measurement
technique
requires
light
scattering particles that are small enough to follow the flow,
and big enough to scatter sufficient light for detection. At
high temperatures in flames one is limited in the choice of the
particles used to seed the flow. For the measurements on the
flame jets MgO is used.
According to Hjemfelt and Mockros (1966) MgO-particles
with a diameter smaller than 2.6 um follow the flow within 1%
if the frequency of the turbulent fluctuations does not exceed
1 kHz. Particles with diameters smaller than 0.8 um follow
frequencies below 10 kHz. In the course of the measurements
95
some rude estimates were made of the sizes of the MgO-particles
in the flow; Particles were made to collide on a small glass
plate which was viewed under a microscope. The collided
particles showed sizes of about 1 urn.
MgO has some disadvantages in its use as seeding material.
It is hygroscopic as well as electrostatic. For this reason the
powder- is firstly dried by keeping it for two or three days at
a tempera'ture of 120°C. Then it is thoroughly mixed with
Aerosol 972 and Al^On in percentages of weight: 94% MgO, 3%
A
"*"2°3' ^ % Aerosol. The powder is then put into a vessel with a
conical bottom (see figure 5.8). The mixture of gas and air
flows radially into this vessel. In this way a swirling flow is
created taking along particles from the bottom. .Only small
particles reach the outlet in the centre of the cover of the
vessel. In a second vessel big particles that possibly still
are carried along, are intercepted. This particle generator
•performed well for different flow rates by adjusting a valve in
a bypass of the two vessels.
As' it was not necessary to measure with MgO-particles for
the isothermal jet measurements, another particle generator was
used. In an atomizer the fluid DIOP (di-iso-octyl-phtalate) was
atomized. The signal-to-noise ratio achieved with these parti­
cles was much better than with MgO particles. For a description
of the atomizer see De Geus (1983).
Fig. 5.8.' The particle generator.
96
6. RESULTS OF THE EXPERIMENTS
6.1. Introduction
Experimental results of the flow structure and heat transfer
are given and discussed in this chapter. A small and a large
burner have been used from which both isothermal and flame jets
are measured at different Reynolds numbers. Comparisons will be
made between measurements of the flow structure and heat
transfer of the small and the large burner. For both cases
isothermal and flame jets are compared. To characterize the
four different jets, the following symbols are introduced which
will be used throughout this chapter:
si: small burner, isothermal jet
sf: small burner, flame jet
li: large burner, isothermal jet
If: large burner, flame jet.
In the next paragraph the axial flow and the radial flow
profiles, the stagnation point heat transfer, the radial
distribution of heat transfer and finally the correlation
between heat transfer and flow structure will be discussed.
6.2. Flow structure
6.2.1. Velocity and turbulence on the axis of the free jet
An important characterization of a turbulent jet is its axial
velocity decay. The shape o c the burner determines the velocity
profile and the turbulence profile at the exit of the burner.
These burner exit conditions of velocity and turbulence are
responsible for the way the jet spreads and decays in the
initial part of 8 or 10 burner diameters. The jets issuing from
the burners used in these experiments evidently differ from the
jets from smooth nozzles as discussed in paragraph 2.1.1. As is
mentioned earlier a jet with a high initial turbulence spreads
wider and decays faster than a jet with a low initial
turbulence.
97
i
i
»
!
|
i
1
i
i
f
1
1
1
I
-
1.0u/u0
>fw A
"
O.S ~
0.0
l
,
i
i
O
Re=3S80
Re=6130
&
Re=8S60
O
O
Re
^7
Re =14400
-
=10600
-
*~t~
.
.
.
.
1
.
,
,
i
■
•
.
■
I
I
I
.
15
10
20
x/d
F i g . 6 . 1 . A x i a l v e l o c i t y of
small burner.
30
-i
i
o
Tu(%)
20
1
1
1
i
1
1
9
v
r—i
jets
from
the
T
Re=3880
o
Re=6130
a
Re=8S60
o
v
1
isothermal
Re=10600
Re=14400
v
°
»
v
v
v
o
T
$ 8 6 £ 2 2 2
10
IS
10
I
I
L _
20
x/d
Fig. 6.2. Axial turbulence levels of isothermal jets
from the small burner.
98
The results on the axial velocity decay and the axial
turbulence development for the isothermal jet from the small
burner are given in figures 6.1 and 6.2. For the range of
Reynolds numbers studied the axial velocity decay is slightly
Re-dependent. At x/d > 10 the jets with the highest Re-numbers
(10,600 and 14,400) decay faster than the jets with the lower
Re-numbers (6,130 and 8,560). Precisely at that region (x/d >
10) turbulence is higher at high Re-numbers. From these
measurements a potential core length is hard to determine. A
95% criterion is used here. This defines the potential core
length as the distance from the burner exit to the place on the
axis where the velocity has decreased to 95% of its initial
value at the burner exit. This potential core length is about
4d for the isothermal jets from the small burner. The initial
turbulence level in the burner exit varies from 6% to 8%
without being correlated to Re (see figure 6.2). A turbulence
level of about 20% is reached after 10d. Here the jets appear
to be fully developed. Differences in turbulence intensity are
found for the various Re-numbers at x/d > 8. Higher turbulence
is generated at higher Re-numbers.
u/u0
1.0
0.8
0.6
Tu(%)
0.4
40
0.2
20
0.0
0
2
4
6
8
10
12
x/d
F i g . 6 . 3 . Axial v e l o c i t y and turbulence development
of isothermal j e t s from the l a r g e b u r n e r .
99
For the large burner the axial velocity and turbulence of
two isothermal jets together are shown in figure 6.3. There
seems to be a strong Reynolds dependency. The initial
turbulence levels are about equal, however, the jets at the two
Reynolds numbers are quite different. After the potential core
length of 3.3d (Re = 4,240) and 3.9d (Re = 9,500) the jet with
Re = 4,240 decays much faster than the jet with Re = 9,500.
This may be due to instabilities that can arise at this low Re­
number. The turbulence level of the low Re-number jet also
rises to a very large value, while the high Re-number jet comes
much closer to the expected values.
1.0
u/u0
0.8
-
// Re=9500
si Re = 10600
Tu(%)
0.6
30
0.4
20
0.2
10
2
_l
4
I
6
I
8
L
10
12
14
16
18
20
x/d
Fig. 6.4. Comparison of axial velocities and turbu­
lence levels of jets from the small and
large burner.
In figure 6.4 the axial velocity and turbulence intensity
of the jet from the small burner at Re = 10,600 and the jet
from the large burner at Re = 9,500 are compared to each other.
The jet from the large burner decays faster and shows at the
same time a higher turbulence, level. Also the initial
turbulence level of about 11% is a bit higher. The differences
which are found here might be due to small differences in
burner configurations (like wall roughness and position of the
blunt body flame stabilizer).
100
The characteristics of the flame jets from the two burners
are given in figures 6.5 up to 6.7. Figure 6.5 shows that the
velocity decay on the axis of flame jets is much faster than
the decay of the corresponding isothermal jets (see also figure
6.1). This is due to the density difference between the jet and
its surroundings. This effect of density can be taken into
account by introducing the effective diameter (see paragraph
2.1.1). For this reason a second horizontal axis is given in
figure 6.8 with the results from the flame jets. This axis
gives the axial distance non-dimensioned with an effective
diameter of 0.37d. This effective diameter is based on the fact
that the initial flame temperature is equal to the adiabatic
flame temperature of a stoichiometric mixture. For the gas used
in these experiments this temperature is 2133 K.
i
i
i
i
|
i
i
i
i
|
i
i
i
v
i
|
i
i
i
i
Re=1047
Fig. 6.5. Axial velocity decay of flame jets from the
small burner.
From figure 6.5 can be concluded that for a flame jet from
the small burner there does not seem to be a Reynolds
dependency of the axial velocity decay (as was the case for the
isothermal jet from this burner). It is remarkable that the
decay starts immediately at the burner exit. A potential core
does not seem to exist, however, this conclusion must not be
101
drawn too quickly. With the 0.95-criterion about 1.5d is found
for the potential core length. This matches to about 3.5
effective diameters, which corresponds well with the potential
core from the isothermal jet from this burner.
30
I
I
r-
Tu(%)
o
H
°
20
Re=1047
Re=1771
Re=2376
Re=2736
10
'0
5
10
15
20
x/d
Fig. 6.6. Axial turbulence levels of flame jets from
the small burner.
The development of the turbulence level on the axis is
given in figure 6.6. A very quick rise of the turbulence level
over a length of the first four jet diameters corresponds to
the sharp velocity decay in this region (see figure 6.5). The
initial turbulence of the flame jets from the small burner
varies from 6% to 12%. However, this does not give rise to
different
jet
characteristics
further
downstream.
The
turbulence levels in the fully developed jets are about 26%,
which already is reached after 5 jet diameters.
In figure 6.7 the velocity and turbulence on the axis of
the flame jets from the large burner are given. In contrast
with the results from the isothermal jets the flame jets at
three different Reynolds numbers give similar axial velocity
and turbulence developments. The 0.95-criterion here gives a
102
Re
1900
U/U0
2700
o
4226
o
a
Tu(%)
Fig. 6.7. Axial velocity and turbulence development
of flame jets from the large burner.
potential core of only 1d (2.7 d„ff).
However, the turbulence
"level at x/d = 1 has risen up to 20% already, so there seems to
be nearly no potential core at all. The initial turbulence
level of 16% is also very high. The flame jets from the large
burner are highly turbulent and develop very quickly. At x/d =
4 a fully developed turbulence level of about 29% is reached.
i
i
i
i
i
i
i
1.0
u/u0 ^ ^ V
0.6
\ \ \\
\ \
Tu(%)
30
- s-y^^^-T*
/■
/
/
0.4
i
sf
If
•*-^\
0.8
i
/
-
-.
\
v
/
/
\
V
/
20
^v
/
-^^^
0.2
- 10
1
1
1
1
1
1
1
1
I
6
8
10
12
14
16
18
20
x/d
_i
5
TO
15
20
25
30
35
40
45
50
*/de/f
Fig. 6.8. Comparison of the flame jets from the small
and large burner.
103
The results on the flame jets from the large and small burner
are compared to each other in figure 6.8. The lines drawn in
this figure and in the next three figures correspond to the
averaged values of the measurements at different Reynolds
numbers for the specific cases. Figure 6.8 shows that the flame
jets from the two burners behave similar to the isothermal jets
(see figure 6.4): A flame jet from the large burner also decays
faster and has a higher initial and axial turbulence level than
a flame jet from the small burner.
The comparisons between the flame jets and the isothermal
jets from the small burner and the large burner are shown
in figures 6.9 and 6.10 respectively. It should be noticed that
the effective diameter is being used in this comparison. Both
figures show the same remarkable result: Using the effective
diameter concept on the axial velocity decay as well as on the
axial turbulence development, the axial velocity of a flame jet
decays slower than that of an isothermal jet. On the other hand
the axial turbulence level of a flame jet increases faster with
axial distance than that of an isothermal jet. In the
isothermal jets the axial turbulence level starts increasing at
a place where the axial velocity starts decreasing. Both
effects occur at the axial position where the turbulence
1.0
u/u0
0.8
Tu(%)
0.6
30
0.4
20
0.2
10
_i
2
4
6
8
i
10
i
i _
12
14
16
18
20
Fig. 6.9. Comparison of the isothermal and flame jet
from the small burner.
104
i
i
i
i
i
1
r
// Re = 9500 If
Tu(%)
30
~~~ - .... - 20
- 10
J
2
4
I
6
I
I
L
a
10
12
J
14
16
i
18
20
Fig. 6.10. Comparison of the isothermal and flame jet
from the large burner.
originating from the jet boundary has penetrated to the axis.
In the flame jets, however, the turbulence level starts rising
before the velocity and also the temperature (see figure 6.29)
on the axis significantly decrease. It might be due to the
process of combustion taking place inside the burner.
This comparison between isothermal and flame jets from the
burners points at the existence of combustion . generated
turbulence which has no significant effect on the axial
velocity decay.
Figure 6.11 summarizes all results on the axial velocity
decay. In this figure also a comparison is made with some
results from literature (see paragraph 2.1.1) given by Hinze
(equation 2.2) and Schrader (equation 2.3). All jets show the
linear axial velocity decay which is characteristic for round
jets:
u
o
x/d
eff
+
b
Only the constants c and b in this equation are not equal for
all jets. These are calculated by linear regression and
together with the correlation coefficient given in table 6.1.
105
TABLE
6.1
THE CONSTANTS c AND
b FOR EQUATION
6.1
region
corr.coeff.
c
b
si
8.1
3.3
0.9994
x/d > 8
sf
7.4
-0.8
0.9988
x/d > 4
li
6.1
1 .7
0.995
x/d > 5
If
7.5
1 .3
0.9991
x/d > 3
7
I
I
I
I
I
I
I
I
I
I
-
6
u0/u
5
4
3
2
D
//
•
si
A
If
O
sf
Re = 9500
~
^
-
6.39
x/d+0.6
Hinze = —
-
«0
8.0
Schrader = —
1
i
-
x/d+3.3
«0
I
-
l
i
I
I
I
I
I
I
12
16
20
24
28
32
36
40
44
x/deff
Fig. 6.11. Comparisons of the axial velocity decays.
106
6.2.2. The radial velocity gradient
stagnation point
in the vicinity of the
An important parameter in the impinging jet flow is the radial
velocity gradient in the vicinity of the stagnation point:
3V
(—)y=6,r=o
<6-2)
As explained earlier in paragraph 2.2.1
this parameter
influences the stagnation point heat transfer strongly. In this
paragraph measurements of 3 will be presented and discussed.
It is assumed that the velocity just outside the boundary
layer in the vicinity of the stagnation point can be written as
V = 3r. If viscous effects are negligible it satisfies the
theorem of Bernoulli, which gives:
p + i p3 2 r 2 = constant = p s t
(6.3)
where p ^ is the static pressure in the stagnation point. From
measurements of the static pressure p distribution on the plate
the radial velocity gradient can be determined:
The tangent at r=o in the graph of (2{pst - p(r)}/p) 5 as a
function of r leads to the value of 3. For the density in
equation 6.4 in the case of an impinging flame jet the value is
used of the local density on the axis of the free flame jet at
the same axial distance from the burner exit (or: the density
at the impact temperature).
The figures 6.12 up to 6 .~1 5 contain the results from the
measurements of the radial velocity gradient. The expected
value of 3 at small burner to plate distances is 8 = u 0 / d - F o r
this reason the graphs present 3d/u_ as a function of H/d. It
can be seen from these figures that for small H/d-values where
the velocity profiles can be expected to be uniform, the
measurements correspond with the expected value. For higher
107
1
i
o Re 4240 -
1.4
o Re 9S00
M
"o 1.0
ÊË.
"o
1.2 -
\
\
r.oFr
-^
—
■
Giralt
\
Schrader
\ \.
~ - ,
0.8
o \
0.6
0.4
0.0
i
0.2
i
6
8
10
H/d
Fig. 6.12.
iRadial velocity gradient
g for the isothermal jet
from the small burner.
Fig. 6.13.
Radial velocity gradient
3 for the isothermal jet
from the large burner.
1.4
a Re = 1900
°
Re=2900
1.2
*
Re=4226
Giralt
Schrader
1.0
0.8
0.6
0.4
0.2
0.0
Fig. 6.14.
Radial velocity gradient
8 for the flame jet from
the small burner.
108
6
_i_
_j_
8
10
H/d
Fig. 6.15.
Radial velocity gradient
8 for the flame jet from
the large burner.
H/d-values B rises till a maximum will be reached. This maximum
lies at H/d = 4 for the impinging isothermal jets and at H/d =
3' for the impinging flame jets. The increase of 3 at small H/ddistances must be due for the major part to the changing impact
velocity profile. The influence of turbulence on 3 in this
region cannot be distinguished clearly. In the figures 6.12 and
6.14 the turbulence level on the axis of free jets from the
small burner has been given. The turbulence level in the
isothermal jets has hardly been increased (from 7% to 9%) when
3 reaches its maximum (see figure 6.12). The turbulence level
in the free flame jet on the other hand increases in the first
three diameters from 8% to 21%. From the figures 6.12 and 6.14
a correlation of 3 with the free stream turbulence level for
the region up to H/d = 4 cannot be found. This agrees with the
flow visualization studies done by Yokobori, Kasagi and Hirata
(1977). They found that for H/d < 4 the stagnation zone looked
laminar. The large scale turbulence generated in the mixing
layer of the jet has not yet been penetrated into the
stagnation region.
At small values of H/d the results in figures 6.12 up to
6.15 are compared with results from Schrader (1961) and Giralt
(1976). Schrader performed direct measurements of (9V/3r)r
and found a slightly decreasing 3 up to H/d = 8. Since he used
a jet issuing from a low-turbulence-level-nozzle the flat
velocity profile was maintained over a considerably axial
distance, which is the reason that his measurements do not show
an increasing 6 till H/d = 4. Giralt gives from static pressure
measurements a value of 3 at H/d É 1.2 : 3 = 0.916 uQ/d. For
almost all cases for H/d < 2 there is a good agreement with
values of 3 for the burners. They are equal or a little bit
higher than those measured by Giralt and Schrader.
From the figures it can be seen that the Reynolds
dependency of 3 is small as it is the case with the other flow
properties. Again the same exception has to be made for the
isothermal jets from the large burner.
109
Concerning
the
wall
jet
region
this
agrees
well
with
the
measurements done by Poreh, Tsuei and Cermak (1967). They found
a
turbulence
velocity
wall
intensity
relative
(/ v' 37vmax).100%
jet.
At
s/d
=
to
the
maximum
wall
jet
of about 30% to 40% all through
4
the
same
values
were
found
the
from
the
present measurements.
Examining
more
closely
the
profiles
of
the
turbulence
intensities at r/d = 1 and r/d = 2 maxima can be found at about
2 mm from the plate. The humps in the profiles at this distance
from
the
plate
are
probably
due
to
what
remains
of
the
turbulence generated in the mixing layer of the free jet before
impingement.
An important parameter~which characterizes the radial flow
along* the
plate
parameter
together
component
at
figure 6.18
v'maxa
value
of
is
the
the
maximum
with
the
location
radial
rms-value
of
velocity
vmax.
of
fluctuating
i/T' max
v
the
)
is plotted
seemed
the
to be almost equal to the maximum of the rms-
fluctuating
I
radial velocity
1
1
i
i
component
1
max
D
1
v
max
6
along
1
">
r=f
\ Re = 6400
0
4
/
3
/ /
2
3.0
2.5
5
,'
v
y
• i
T^L
i
V
1.5
6.18. Maximum
2.0
"max-)
1.5
*^
~*
1.0
"I
TV.
-*""
1/ '■'
-jjy
77
A' ^"
0.5
"max
^^^
11 (I/
r
1 '
2.0
velocity
""—
1
2.5
**—"-^
• ^ ^ " " " — ■ « ^ ^
1
I
3.0
along
fluctuating- component
'112
in
(for the Reynolds numbers at H/d = 2 ) . This value
w
Fig.
This
^^
'"
1
3.5
4.0
r/d
1.0
0.5,
0.0
the plate and its'
(si, H/d = *2) .
the
The maximum velocity in the boundary layer
plate: ( /"
max*
increases up tó a distance of r/d = 1 . This radial distance
corresponds with the edge of the stagnation zone. It has also
been found by Schrader (1961) and Hrycak (1981) that the
maximum of v m=v for small H/d-values is reached at r/d = 1 . The
IlldX
value of / v' - ' reaches its maximum value at 1.5 < r/d < 2.0,
which is remarkable because the average velocity in the
boundary layer has already decreased significantly at this
radial distance. The explanation probably is that the large
eddies generated in the mixing layer of the free jet, which are
deflected due to impingement penetrate into the wall jet only
outside of the stagnation region and gives rise to the increase
in turbulence.
H/d = 6
Figures 6.19 and 6.20 contain the measurement results for H/d =
6. Comparing the profiles of the average velocity at r/d = 1,
r/d = 2 and r/d = 4 we find that at r/d = 2 the flow is
constricted to the wall at the most. The variations in the
turbulence intensities existing at H/d = 2 are not present
here. Apart from small humps in the turbulence profiles within
6
y(mm)
5
4
3
2
1
0
0.5
1.0
1.5
Fig. 6.19. Radial velocity and turbulence intensity
profiles (si, H/d = 6, Re = 3300; for
legend see figure 6.16).
v 113
6
\I
y(mm)
..
tl
5
.
4
i»
r o
3
-
2
ft. \
K^
1
"
2
4
6
v(m/s)
.«oW/
0.5
» i
1.0
1.5
Vw*(m/s)
Fig. 6.20. Radial velocity and turbulence intensity
profiles (si, H/d = 6, Re = 6400; for
legend see figure 6.16).
half a millimetre from the plate at r/d é 1.0 the profiles look
rather flat and similar.
Examining this little hump in the turbulence intensity
close to the plate in the stagnation region we find that it
also is present in the measurements at H/d = 2. This could
agree with the theory of vortex stretching in the stagnation
zone of Sutera, Maeder and Kestin (1963) and the measurements
done by Sadeh, Sutera and Maeder (1970) on the turbulent
properties in the stagnation region of a circular cylinder.
However, these investigators found a more pronounced maximum.
On the other hand Gutmark and Wolfshtein (1978) did not find an
increase of the turbulence intensity in this region.
As for the measurements at H/d = 2 the values of
/ v' 2 /v max .100% for r/d = 4 vary between 30% and 40%.
Figure 6.21 shows a comparison of the results presented in
the figures 6.19 and 6.20. The profiles at r/d = 0.5, 1 and 2,
scaled to the average velocity at the burner outlet, from the
wall jets at Reynolds numbers 3600 and 6400, are compared to
each other. The differences of the dimensionless velocity
114
He=3600
Re=6400
r/d
r/d=O.S
y
(mm)
1
r/d=2
y
(mm)
J
I
O.1
I
I
0.2
y
(mm)
L.
0.1
0.1
0.2
0.2
,-*,
vl2/u0
Re-3600
Re=8400
e
y
(mm)
r/d=0.S
\\
\\
y
y
(mm)
(mm)
\\
4
2
r/d=2
-
*^
0.5
1.0
v/ua
0.5
1.0
v/u0
i
i
0.5
i
1.0
v/u„
Fig. 6.21. Comparison of radial velocity and turbu­
lence intensity profiles for two Reynolds
numbers.
115
v
max
6
Re=6400
5
Re = 3600
3.0
2.5
4
2.0
3
1.5
2,
1.0
1
0.5
nn
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
r/d
Fig. 6.22. Maximum velocity along the plate and its
fluctuating component (si, H/d = 6 ) .
profiles as well as the dimensionless turbulence intensity
profiles are small. There seems to be no Reynolds dependency of
the present results, as can be expected.
Figure 6.22 shows the maximum average velocity in the
boundary layer with the rms-value of its fluctuating component
as a function of r/d. The maximum of v m a x is reached at r/d =
1.5. The stagnation zone extends itself till this distance.
Again we see that the turbulence intensity does not vary much
over the entire range. This is due to the turbulence in the
free jet which at x/d = 6 is much more uniform than at x/d = 2.
i
However, the fact that the turbulence intensity does not
decrease 'in the same way as the average velocity does after r/d
= 1.5 points at an influence of the large eddies induced at the
edge of the jet.
The boundary layer thickness
Another characteristic parameter of the impinging jet flow is
the boundary layer thickness defined by the distance from the
plate to where the radial velocity reaches its maximum. In
116
" I —
S/d
I
J
1
H/d=6
0.20 - o Re=3600
a Re=6400
0.16 0.12
o
0.08
-
0.20
/ "
0.16
/
/
S/d
-
0.12
-
0.08
H/d=2
o Re=3600
□ Re=6400
/
-
0.04
0.04 -_a1
1
1
1
3
r/d
4
r/d
Fig. 6.23. The boundary layer thickness (si).
figure 6.23 the boundary layer thickness is given. In the
stagnation zone (0.25 < r/d < 1 for H/d = 2 and 0.25 < r/d <
1.5 for H/d = 6) the boundary layer thickness increases only
little. Where the wall jet becomes highly turbulent due to the
large eddies from the jet mixing layer, the boundary layer
thickness grows rapidly. From the measurements it can be seen
that the lowest Reynolds number gives the thickest boundary
layer.
6.2.3.2. The flame jet
Measurements.1; of the radial velocity profiles have also been
performed with the impinging flame jet from the small burner.
These measurements are difficult and from the scattering of the
data it is o'bvious that the accuracy of- these measurements is
worse than that of the isothermal flow measurements. The
temperature gradients induce refractive index gradients. This
caused the laser beams to deflect so the position in space of
the measuring volume could not accurately be determined. While
in the isothermal case measurements were possible up to a
distance of 0.2 mm from the plate, with the flame jet
117
r/d
5r
'
y(mm)
4
S
° 0.25
o 0.5 y(mm)
4
0d7S
- f\TVf\\
*
'„ u0
i.s
\v\N&
3
2
q
■
T^Sv
4-
*
°
-l / J X\
1-
8
12
16 20
24
6
8
10
4v^(m/s)
v(m/s)
Fig. 6.24. Radial velocity and turbulence intensity
profiles (sf, H/d = 2, Re = 1050).
6
y(mm)
5
3 2
1
8
12 16 20 24 28
v(m/s)
2
4
6
8
10 12
z
v' (m/s)
Fig. 6.25. Radial velocity and turbulence intensity
profiles (sf, H/d = 2, Re = 1900; for
legend see figure 6.24).
118
measurements could be made
plate. Besides deflections
gradients, the temperature
to jitter. Also the higher
higher Doppler frequencies
is a reason for the higher
up to a distance of 0.75 mm from the
of the beams due to the temperature
fluctuations caused the laser beams
range of velocities, resulting into
with a worse signal-to-noise ratio,
inaccuracy.
H/d = 2
Measurements
on
the
radial
velocities
and
turbulence
intensities for the short burner to plate distance are gathered
in figure 6.24 and 6.25. As in the isothermal case here also
different shapes of profiles can be distinguished. Comparing
the profiles at r/d = 0.75, 1.0 and 1.5 a strong constriction
can be noticed of the flow along the plate at r/d = 1.0. From
this distance on the strong mixing causes a flattening of the
velocity profile (see r/d = 1.5). The temperature decay causes
the strong velocity decay for higher r/d-values (r/d = 2 and
4). The profiles of the turbulence intensity show the same
tendencies as the velocity profiles: an increase of turbulence
intensity in the stagnation region till r/d = 0.75, then a
decrease of turbulence intensity due to the strongly decreasing
velocity.
Figure 6.26 gives the maximum average velocity and the
corresponding turbulence intensity as a function of r/d. In
this graph it can be seen more clearly that the maximum of v m a x
lies between r/d = 0.75 and r/d = 1.0. The maximum turbulence
intensity is only slightly lagging behind. The influence of the
mixing induced eddies is not as clear as for the isothermal
impinging jet.
H/d = 6
At the larger burner to plate distance (H/d = 6, figures 6.27
and 6.28) the flow develops as can be expected. Like in the
isothermal case turbulence in the stagnation region is rather
119
uniform. Figure 6.29, in which again the maximum velocity is
plotted, shows that the maximum v m a x lies close to r/d = 1.5.
The turbulence intensity already starts decaying before this
maximum is reached which is quite remarkable. However, any
influence of the free jet mixing layer is not expected, because
of the large H/deff value.
28
Fig. 6.26. Maximum velocity along the plate and its
fluctuating component (sf, H/d = 2 ) .
5
y(mm)
4
3
2
1
0
2
_i
i
i
u
4
6
8
10
v(m/s)
2
3
v'
2
4
(m/s)
Fig. 6.27. Radial velocity and turbulence intensity
profiles (sf, H/d = 6, Re = 1050).
120
S
y(mm)
4
5
y(mm)
4
2 -
8
12
16
v(m/s)
2
4
6
8
-<W*(m/s)
Fig. 6.28. Radial velocity and turbulence intensity
profiles (sf, H/d = 6, Re = 1900; for
legend see figure 6.27).
Fig. 6.29. Maximum velocity along,the plate and.its
fluctuating component (sf, H/d = 6 ) .
121
6.2.4. Axial temperature distribution
Next to the axial velocity decay the axial temperature
distribution of a flame jet is of importance. The hot flame jet
mixes with the environmental air at room temperature; so the
temperature of the jet decays. Due to the temperature dependent
density of the flue gasses the velocity decay of the flame jets
is a consequence of this temperature decay. The temperature
difference between flame jet and plate is also an important
parameter because it is the driving force for the heat transfer
to the plate.
Measurements of the temperature decay for the flame jets
studied are presented" in figure 6.30. The adiabatic flame
temperature of Groningen natural gas is 2133 K (see reference
"Basisgegevens over Gronings Aardgas", 1968). I,t is assumed
that all gas has been burned within the burner. Looking at the
axial temperature distribution this assumption is reasonable,
because the temperature at the burner outlet almost reaches the
2000
T(K)
1800
1600
1400
1200
1000
800
600
I
i
i
i
I
2
3
i
4
i
5
1
6
1
'
7
8
i
9
i
I
10
x/d
Fig. 6.30. Axial temperature
flame jets.
122
distributions
of
the
adiabatic flame temperature and the axial temperature decreases
rapidly after the flame leaves the burner. The temperature at
the burner outlet of three flame jets is only 50 K lower than
the adiabatic flame temperature. This difference can be attri­
buted to a slight deviation of the stoichiometric ratio or to
the radiation losses of the thermocouples which were used. The
radiation losses become larger when the convection to the
thermocouple decreases. This might be the reason that the exit
temperatures at the two lowest Reynolds numbers for jets from
the large burner (Re = 1900 and Re = 2900) are measured lower
than in the other three cases. The different heat losses from
the small and large burner might also contribute to a lower
exit temperature in case of the large burner. The difference in
heat loss is caused by the less effective thermal isolation of
the actual combustion chamber of the large burner. This has
been confirmed by the higher temperatures of the outer surface
of this burner.
6.3. Heat transfer
6.3.1. Stagnation point heat transfer
Heat transfer at the stagnation point has been measured for
both burners for the isothermal jet as well as for the flame
jet. When mentioning an isothermal impinging jet it is meant
that the jet is isothermal with the surrounding air and that
the temperature difference between the air and the plate is
below 30°C. The experimental set-ups and measuring techniques
are described in paragraphs 5.1 and 5.2.
From the measurements of the velocity gradient at the
stagnation point {3 = '5v max /3r) r _ 0 ) and the theory of Sibulkin
(1952) the stagnation point heat transfer can also be
calculated from:
Q
Nu R = 0.763 d (_)°-5Pr0-4
(6.5)
(see paragraph 2.2.1). Both results on stagnation point heat
123
i—!
1
■ I
i
i
i
i
1
1
1
1—
r
70Nu
6050 4030 r
— o — from heat flux measurements
20-
—••— from /3 measurements
~~ ""•• <Re=3300
10J
L _
I
I
1
2
3
4
-.
I
I
I
I
I
5
6
7
8
9
1
1
10
11
H/d
1
12
:.Flg. 6.31. Stagnation point heat transfer (si).
transfer from heat flux measurements and from velocity gradient
measurements are discussed in this paragraph.
6.3.1.1. Isothermal jet
.
Measurements on the stagnation point heat transfer for the
•isothermal impinging jet from the small burner are gathered in
figure 6.31. Heat transfer increases with H/d until at H/d = 5
a maximum is reached. There are several jet characteristics
which are. important, to ■ heat. transfer for those first five
.diameters, such as: the axial velocity, the.' turbulence
intensity on the axis and the shape of the velocity profile-. As
can be seen: from figures 6:1'.-and 6.2 the axial velocity'at x/d
.= •'5 has decreased 'to: 90% of its initial value' at x/d ■.=: 0
.(causing a decrease of the Nusselt '.number), while the
turbulence intensity has raised from 7% to 12% (possibly
causing an increase of the Nusselt number). Since the axial
velocity already decreased by 10%. the' velocity 'profile has
developed from an originally uniform profile to a more or less
Gaussian shaped profile (causing an increase of the: Nusselt
124
number). It is not possible to separate these three effects on
the heat transfer quantitatively.
The influence of the shape of the velocity profile and the
absolute value of the axial velocity can also be found in the
heat transfer results calculated with equation 6.5 with the
measured 3-values. As already stated in the discussion on the
results of the B-measurements
a separate influence of
turbulence on 3 has not been found. Concerning the measurements
at H/d < 4 this corresponded to flow visualization experiments
done by Yokobori et al. (1977). The same investigators Yokobori
et al. (1978) showed for a two-dimensional jet that the
influence of turbulence on the stagnation point heat transfer
was absent for H/d < 4. This fully agrees with our
measurements. The two ways of determining heat transfer give
corresponding results and since there was no influence of
turbulence on B in this region, this influence is also absent
in the direct measurements of heat transfer.
When the plate is positioned outside of the potential core
of the jet, the two ways of determining heat transfer start to
deviate. This is the region where the large eddies generated in
the mixing layer of the jet penetrate till the axis. The
turbulence in this region has a different character. While in
the potential core region the scale of turbulence is small
compared to the burner diameter, at axial distances higher than
the potential core length the scale of turbulence is of the
order of the burner diameter. For a jet with this large scale
turbulence we can understand that the Sibulkin formula does not
predict heat transfer anymore.
Figure 6.32 gives the same picture of the stagnation point
heat transfer measurements from the isothermal jet from the
large burner. It can be seen here that for the jet with Re =
4240 the heat transfer enhancement due to large scale
turbulence already starts at H/d = 2. This can be explained by
the shorter potential core length for this particular jet (see
figure 6.3).
125
80
Nu
70 -
Re=9500
60
SO
40
y
30
Re=4240
from heat flux measurements
from 0 measurements
20
10
_j
i_
1
2
3
4
5
6
7
8
9
H/d
10
Fig. 6.32. Stagnation point heat transfer (li)
Nu
from heat flux measurements
from |3 measurements
Sibulkin theory
100 -
50
20
2,000
5,000
10"
Re
Fig. 6.33. Stagnation point heat transfer at H/d = 2
(si + li).
126
The results show that heat transfer at the stagnation
point is laminar-like at small burner to plate distances. The
measurements from small and large burner at H/d = 2 are
gathered in figure 6.33.
From linear regression on results from heat flux measure­
ments of the four cases it is found that:
Nu = 0.57 R e 0 - 5 2 5
(6.6)
From linear regression on the four results from B-measurements
it is found that:
NUg = 0.60 R e 0 - 5 2 2
(6.7)
From figure 6.33 it is clear that the measurements at H/d = 2
are very close to the theoretical correlation for laminar heat
transfer from a uniform jet to the stagnation point of a flat
plate. Besides this it can be concluded that the results at H/d
= 2 from large and small burner match to each other.
The results presented in figures 6.31 and 6.32 are in
close agreement with the findings of Giralt, Chia and Trass
(1977), already mentioned in paragraph 2.2.2. With a similar
formula as equation 6.5 and measured stagnation pressure
profiles they found a "turbulence free" transfer coefficient
(see equation 2.6 1). They found enhancement of (in their case)
mass transfer by comparing the experimental results with this
"turbulence free" transfer coefficient.
Our results can also be interpreted in the same way: The
Nusselt number found with equation 6.5 is the laminar contribu­
tion. The difference of the Nusselt number from heat flux
measurements and from equation -6.5 gives the enhancement due to
turbulence. This can be expressed by a turbulence enhancement
factor defined by
Nu = (1 + Y)NUg
(6.8)
This turbulence enhancement factor is given in figure 6.34 as a
function of the turbulent Reynolds number Tu/Re for the
127
measurements on I both burners for several H/d-values. Different
symbols are used for values of y at H/d < 3, H/d = 4 and H/d a
6. It can be seen from the figure that for H/d 2 3 the value of
y is about zero even for values up to Tu/Re = 11. The results
from .Giralt et al. (1977) are also given in the figure. The
present results are in the same order of magnitude.
0.6
-
. H/d « 3
o H/d =4
° H/d >6
Gira,t
-
0.4
0.2
0.0
•0.2
J
I
4
8
I
. 12
I
I
16
20
1
24
I
I
L
28
32
36
40
TuJÏÏë .
.' Fig. 6.3 4. The turbulence
' ■
■
s i ) .'
enhancement
factor
(li' +
•'
6.3.1.2. The flame jet
With the heat flux transducer the total heat flux density for
flame jets can be measured, being the summation of convection
and
radiation. Before a discussion
of the heat . flux
measurements can be given, an estimation has to be made of the
radiation heat transfer from the hot burner to the cold plate.
Radiation heat transfer
It is supposed that the radiation of the flame itself is
negligible because of the very low emissivity of a hot gas
layer of small thickness. Then there is only left the radiative
heat transfer from the hot inner wall of the burner to the heat
flux transducer. '
,
Radiation1 from a cylindrical hole of finite depth- with
128
diffuse reflecting walls at constant temperature is analysed by
Lin and Sparrow (1965). They found an effective hemispherical
emissivity e n which gives the total amount of energy Q leaving
the mouth of the cavity in ratio to that emitted from a black
walled cavity:
Q
eh =
2
-..-■.
TTR
O(T.
=~rr)
(6-9)
where R is the radius of the cavity, o is the Bolzman factor
arid T w and T e are the cavity wall temperature and the
temperature of the environment.
Now the problem of radiation heat transfer from the burner
to the heat flux transducer is reduced to the problem of
radiation from a circular disk to a plane element dA2 / being
the surface of the heat flux transducer, parallel to the disk
(see figure 6.35). The fraction of energy leaving surface
m
Fig. 6.35. The burner exit A^ facing the heat flu?
transducer dAelement A-j that arrives at element dA2 is defined as the
geometric configuration factor F1^.2- Si'egel and Howell (1981)
give for the configuration of figure 6.35:
1+2
1 +.
id - ./(Z
2
dA-
V
2
- 4s )
(6.10)
with C = H/r, x, = R/r and Z = 1 + C 2 + S 2 .
The heat flux density absorbed by the heat flux transducer
can thus be estimated by:
u
1 + V
rad
c* h V '
XT * " T*)
/(Z 2 -' 4i;2)
(6.11)
129
where ec =
T =
Reasonable
ec =
eh =
the emissivity of polished constantan and
the heat flux transducer temperature.
estimates of the emissivities are:
0.84
0.7 (from Lin and Sparrow (1965) with e
=0.27)
A1 2 0 3
The most uncertain and also most determining factor is the
inner burner wall temperature.
Figure 6.36 gives
Q'Vad
calculated at three different burner wall temperatures: T w =
1900 K, 1800 K and 1700 K. Especially at small burner to plate
distances (H/d < 2) the contribution of radiation heat transfer
to the total heat transfer cannot be neglected.
I
I
I
1
1
1
100
"
f'rad
80
(kW/m2)
60
40
20
1
1
1
1
Tw=1900K
l'
1700K
-J'
-
IV,
- %
-
-
\l
-
- %
- %
- %
-
0
1
0
1
2
1
i^~-i
4
1—,—i
6
1
10
8
H/d
Fig. 6.36. Heat transfer by radiation.
Convective heat transfer
Measurements of the stagnation point heat flux distributions
were performed at two Reynolds numbers for the small burner and
at three Reynolds numbers for- the large burner. These
measurements are shown in figures 6.37 and 6.38. The dashed
lines give the convective heat transfer after a correction for
radiation calculated by equation 6.11 with T w = 1800 K. The
uncertainty of this correction is rather big because of the
unknown inner burner wall temperature. Very high convective
heat fluxes are reached at small burner to plate distances with
130
I
~T~
Q
(kW/m2)
o
o
--
600-
Re=2700
Re=1771
corrected for radiation
500
400
300
200
100
J
O
1
2
3
4
5
6
7
8
L_
9
10
H/d
Fig. 6.37. Stagnation point heat flux densities (sf)
q"
(kW/m2)
400
350
«e=4226
fte=2900
fle=7900
- corrected for radiation
300
250
200
150
100
50
0
a
9
10
H/d
Fig. 6.38. Stagnation point heat flux densities (If)
131
the small burner: 570 kW/m2 at H/d = 1.5. The radiation heat
flux for this H/d will be about 30 kW/m2. These high fluxes
give rise to another uncertainty. The heat flux transducer
could only be calibrated" up to heat fluxes of 150 kW/m2. Since
the response of the transducer was linear over a big range, the
calibration has been extrapolated.
A rapid decrease of the heat flux with increasing H/d is
caused by the axial developments of velocity as well as
temperature. Differences in results from large and small burner
are also due to the differences in axial velocity and
temperature development. As can be seen from the results in
figure 6.8 the jet from the large burner develops much faster
than the jet from the small burner. Figure 6.30 showed that the
*
temperature decay on the axis of the flame jet from the large
burner starts immediately at the burner outlet while this decay
on the axis of the flame jet from the small burner only starts
at x/d = 2. These differences give rise to the much faster
decay of q" with H/d for the large burner than for the small
burner.
The results from the flame jet can only be compared to
those of the isothermal jet when they are given in dimensionless numbers. In doing so the problem arises at which tempera­
ture the fluid properties has to be taken. In this respect the
method proposed by Eckert (1942) has been followed. He stated
that, in order to be able to compare the results with
isothermal measurements, the fluid properties should be taken
at a temperature (T*) where the, enthalpy across the boundary
layer has the average value. So the Nusselt number can be
calculated from:
q" C_(T*)d
Nu =
2
(6.12
h
< axis - hplt>*<T*>
The enthalpy on the axis ( n ax is' i s calculated from the
temperature on the axis in the free flame jet at x/d = H/d (the
impact temperature). The enthalpy near the plate is calculated
132
-\
1
1
r
H
I
I
T"
120
Nuu
100
80
60
A
ffe=4226l
o Re=2900 \ from heat flux measurements
°
Re=1900i
•
Re=4226]
• Re=2900} from/i measurements
•
Re=1900]
40
20
J
1
I
I
2
3
L_
_l
8
6
l_
9
10
H/d
Fig. 6.39. Stagnation point heat t r a n s f e r
(sf)
70
Nu
60
50
40
30
o
20
o
•
•
Re=270&]
} from heat flux measurements
Re=1771)
Re=2700]
from p measurements
Re=1771j
10
_i
1
2
3
4
5
i
6
i
7
i
8
i
9
H/d
Fig. 6.40. Stagnation point heat t r a n s f e r
i_
10
(If),
133
from the plate temperature.
A Nusselt number can also be calculated from the velocity
gradient at the stagnation point (see equation 6.5). For the
flame jet in a similar way this Nu-number can be calculated by:
NuR = 0.763 d (—)°-5Pr0.4
P
v(T*)
(6.13)
The results of the Nusselt numbers as a function of the
burner to plate distance are given in the figures 6.39 and
6.40. These Nusselt numbers are related to the convective heat
transfer results. So first the corrections for the radiation
heat flux were made.
From the plots it can be seen that between H/d = 2 and H/d
= 4 to 5 the Nusselt number rises. Similar to the isothermal
measurements this can be attributed to the changing velocity
profile in this region. However, looking at the axial velocity
and turbulence profiles, it is even more difficult as it is in
the isothermal case to separate the effects of the shape of the
velocity profile, the decreasing axial velocity and increasing
axial turbulence level. Looking back at figure 6.8 it can be
seen that the velocity in the region considered here decreases
from 0.9 u Q to 0.5 u Q , while the turbulence level increases
from 15% for the small burner and 20% for the large burner to
about 30%.
Comparing the results from the heat flux measurements with
the results from the 3-measurements it can be seen from figure
6.40 that for H/d £ 3 the agreement is rather good. At H/d = 3
the results from both measurements start to deviate. For H/d >
3 at the highest Re-number the difference between Nu and Nuo is
at its biggest. The measurements on the small burner show the
same effects less clearly. At Re = 1771 the Nusselt numbers
calculated in both ways do not deviate much for the whole H/d
region considered. However on the whole, figures 6.39 and 6.40
show the same trends as the results for the isothermal jets in
figures 6.31 and 6.32.
134
T
No
o
If 1
a
sf J
■
sf j
.
1
1 |
from heat
fluxes
from {'> measurements
Sibulkin
100 -
theory
-
50
1^^^
-
"o»
-
30
1
3.000
,
1
5.000
I
I
I
I
1
10.000
i
Re*
Fig. 6.41. Stagnation point heat transfer at H/d = 1
(If + sf).
To show that even for the flame jets with rather high
initial turbulence levels the influence of this turbulence on
the heat transfer at small burner to plate distances is not
big, all measurements at H/d = 1 are gathered in figure 6.41 .
They are given here as a function of the Re*-number, which is
defined as
u
Re*
od
v(T*)
(6.14)
where u_ is the velocity measured at x/d = 1 and v(T*) is the
viscosity
at
the
before
mentioned
"averaged
enthalpy
temperature". The results presented in this way agree very well
with the theoretical results of Sibulkin for a laminar stagna­
tion point flow, as was the case for the isothermal measure­
ments. Here also the results of both burners match each other.
The turbulence enhancement factor defined by equation 6.8
for the impinging flame jets is given in figure 6.42 as a
function of Tu/Re*. Again the same distinction is made between
135
T
•
o. e
);
I
I
H/d
<3
o H/d
=4
I
I
I
1
o H/d >6
1
a
I
Giralt
-
0.4
0.2
0.0
■0.2
4
J
I
I
I
I
I
I
L
8
12
16
20
24
28
32
36
40
TuVRë*
Fig. 6.42. The turbulence enhancement factor (If + sf).
the results for H/d i 3 and H/d a 6. Despite of the rather high
values of Tu/Re* (up to Tu/Re* = 28) the value of y is about
zero if H/d S 3. For H/d a 6 it is noticed that y * 0 and that
the turbulence enhancement factor is in reasonable agreement
with the findings of Giralt et al. (1977).
In paragraph 6.2.1. it was concluded that the flow
structures of the isothermal jets from the small burner
differed from the isothermal jets from the large burner (figure
6.4). A similar conclusion was drawn concerning the flame jets
from small and large burners (figure 6.8). Finally, the
comparisons between the isothermal jets and the flame jets from
the same burner (figures 6.9 and 6.10) showed that the flow
structure of the isothermal jets is completely different from
the flow structure of the flame jets. The heat transfer results
discussed in this paragraph, however, show that the stagnation
point heat transfer of all these different impinging jets
(flame jets and isothermal jets) can be described in a similar
way with 3 and y. This shows the wide applicability of this
description.
136
6.3.2. Radial heat transfer distribution
The heat transfer distribution as a function of the radial
position (r/d) on the plate has been measured for three flow
situations: the isothermal jets from the large as well as the
small burner (li and si) and the flame jets from the large
burner (If). The results will be discussed in this paragraph.
6.3.2.1. The impinging isothermal jets
With the liquid crystal technique described in chapter 5 radial
heat transfer distributions have been measured. Results of
these measurements for the small burner at different H/ddistances for two Reynolds numbers are given in figures 6.43
and 6.44. The low Reynolds number jet (Re = 3300) gives for all
H/d-distances a maximal heat transfer at the stagnation point.
For H/d = 2 and H/d = 3 the Nusselt number is constant over
the stagnation region (0 < r/d < 0.5) and monotonically
decreasing in the wall jet region (r/d > 0.5). For H/d a 5 the
heat transfer coefficient distribution expressed by Nu = f(r/d)
is bell shaped. The heat transfer for the high Reynolds number
jet (Re = 6300, figure 6.44) differs in one respect. For H/d =
2 and H/d = 3 secondary peaks can be noticed in the heat
transfer distributions at r/d = 2.
-i
1
1
1
1
Re=3300
H/d
* 2
a
30
3
S*
o 5
^^fc~
o 7
20
10
J
1
i
i
i
Ê
2
3
4
5
Fig. 6 . 4 3 . Radial heat t r a n s f e r d i s t r i b u t i o n
:
r/d
(si).
137
70
0\
1
1
1
1
1
1
2
3
4
5
1
r/d
Fig. 6.44. Radial heat transfer distribution (si).
These peaks can also be noticed in the results for Re =
4250 from the large burner (figure 6.45) and even more
pronounced for the higher Reynolds number jet from this burner
(Re = 9500, figure 6.46). The reason for the existence of these
secondary peaks in the heat transfer distributions around r/d =
2 can be the maximum of the turbulence intensity at almost the
same radial location in the wall jet (see figure 6.18). The
results from the large burner show the existence of a second
maximum. Especially the results from the high Reynolds number
(Re = 9500) at H/d = 2 and H/d = 3 show two maxima in the
radial Nusselt number distribution. The already mentioned
maximum at r/d = 2 and a maximum at r/d = 0.5. This has also
been found by other investigators (see e.g. Gardon and Akfirat
in figure 6.47) and is explained by the presence of a minimum
in the boundary layer thickness. It is not clear why this
maximum at r/d = 0.5 has not been found for the small burner.
A comparison of the present results at H/d = 2 has been
made with results from Gardon and Akfirat (1965) for the same
nozzle to plate distance. They presented results in the
Reynolds number range: 2800 < Re < 28,000. Figure 6.47 gives
138
Fig. 6.45. Radial heat transfer distribution (li)
Fig. 6.46. Radial heat transfer distribution (li).
139
100
80
60
40
20
O
6
4
2
0
2
4
6
r/d
Fig. 6.47. Comparison of radial heat transfer dis­
tribution with results from Gardon and
Akfirat (1965).
the comparison for the large as well as the small burner
results. Compared to our measurements the heat transfer results
of Gardon and Akfirat show higher values. This has also been
found by Hrycak (1983) who stated that the high values of the
results of Gardon and Akfirat are due to inaccuracies of the
heat flux transducers used by them.
The results compared
in figure 6.47
show another
difference: Gardon and Akfirat found three humps or peaks in
their radial heat transfer distribution, while we find only
two. The first peak found at r/d = 0.5 corresponds with the
peaks found by us for the impinging jets from the large burner.
Instead of the two outer peaks found by Gardon and Akfirat for
Re < 10,000 we found only one outer peak. The peak at r/d = 1.4
according to Gardon and Akfirat is due to a transition from a
laminar to a turbulent boundary layer. This peak vanishes for
Re > 10,000. Our measurements of the flow structure close to
the plate show that the wall jet also for r/d < 1.4 is
turbulent (see paragraph 6.2.3), so this transition does not
140
occur in our experiments. Due to the higher initial turbulence
of the jets from our burners the wall jets are fully turbulent
at lower Reynolds numbers. In this respect our heat transfer
results agree with the results of Gardon and Akfirat for Re >
10,000, where only one outer peak remains at the position r/d =
1.9.
:'•}■
"
6.3.2.2. The impinging flame jets
For the large burner at the Reynolds numbers of 1900, 2900 and
4226 at different burner to plate distances the maximum
temperatures near the wall, the temperature of the plate and
heat flux densities to the plate as a function of the radial
distance from the stagnation point are measured. From these
measurements with an equation similar to equation 6.12 in
paragraph 6.3.1.2, radial distributions of the Nusselt number
are calculated and presented in this paragraph. The Nusselt
number is calculated from
Nu =
q".c„(T*)d
2J
<hmax " h P lt> A<T*>
(6.15)
The fluid properties c n and A are determined at the temperature
(T*) at which the enthalpy of the gasses across the boundary
layer has the average value of (h ma ^ + h l t )/2.
Temperature distributions
With the thin wire PtRho6%-PtRho30% thermocouples temperature
profiles close to the wall have been measured." From these
measurements the maximum temperature close to the wall has been
found. Some of the results are presented in figures 6.48 and
6.49. Figure 6.48 gives the maximum of the' near wall
temperatures for Re = 2900 and 2 é H/d S 10. The temperature in
the stagnation region drops very rapidly with H/d. This agrees
well with the axial temperature development given in figure
6.30. In the wall jet region for r/d > 2, however, the maximum
141
H/d =2
Re = 2900
f500
T
Cc)
S
\
°loB
ofi.
4
• e
' a
*
.
" f900
2900
B
2
3
'
1000
800
ffe =
H/d
■■<
- 4226
öfl
o
e
0
10
8
4
^
8
- . ^ «t
0
D
1
i8 S
6
° 1
.
°
1
i
4
eo
r/d
r/d
F i g . 6.48 and 6 . 4 9 . Maximum boundary
ture (If).
layer
tempera­
H/d=2
a f900
o 2900
•■ 4226
r/d
r/d
Fig. 6.50 and 6.51. Radial distribution of the heat
flux density (If).
142
wall jet temperature is independent of H/d in the region 2 ë
H/d S 4. For r/d > 4 the maximum wall jet temperature is
independent of H/d in the region 2 s H/d s 6. Figure 6.49 gives
an example of the maximum temperatures close to the plate for
the three different Reynolds numbers at a fixed H/d value (H/d
= 2) . The maximum temperatures are independent of the Reynolds
number as was also the case with the axial temperature
developments of the free jets (see figure
6.30). Several
measurements have been carried
out twice to show the
reproducibility. Almost all reproduced measurements coincided
within 1 or 2%. Only a few deviated about 5%.
The heat flux distributions
For flames it is valuable for practical reasons to consider the
measured heat fluxes themselves. The heat flux results are
presented in figure 6.50 and 6.51. At Re = 2900 for 2 < H/d <
10 the radial heat flux densities are given in figure 6.50. The
figure shows that the heat flux density decreases very fast
when r/d decreases, especially at low H/d values. The
nonuniformity of the heat flux to the plate is enormous. Even
at H/d = 6 the heat flux drops a factor of 2 within two burner
diameters into the radial direction. It is clear that it is
only possible to reach a uniform heat flux distribution (for
instance, with an array of burners) at higher burner to plate
distances. Heat flux distributions at H/d = 2 for different Re­
numbers are given in figure 6.51. This figure shows once more
the large peak in the heat flux density at low H/d. In all
cases the plate temperature is in the range of 50-70°C, while
the maximum boundary layer temperatures range from 250-1500°C.
Comparing figures 6.48 and 6.50 it is obvious that the sharp
decrease of the heat flux density with r/d is to a large extent
caused by the sharp decreasing boundary layer temperature.
143
H/d=2
Re=
H/d=3
'm t900
90
4226
a 1900
a 2900
o 2900
*
fle=
Nu
60
30
r/d
r/d
H/d=4
a 1900
o 2900
». 4226
30
r/d
Fig. 6.52. Radial Nusselt number distributions (If)
144
The Nusselt number distributions
In figure
6.52
the radial Nusselt number distributions
calculated by equation 6.15, are presented. At H/d = 2 the
distributions in the stagnation region have a slight maximum at
r/d = 0.5. This has also been found in the results of the
isothermal impinging jet of the large burner at small H/ddistances. At higher H/d the Nusselt number in the stagnation
region monotonically decreases with r/d.
At the burner to plate distances H/d = 2 and H/d = 3 a
second maximum in the heat transfer distributions can be
distinguished at a radial position between 1.5d and 2.0d. Again
similar maxima have been found for the isothermal impinging
jet.
It can be concluded that the heat transfer distributions
of isothermal jets and flame jets compare very well, despite of
the large differences in heat flux densities and temperatures.
It was tried to correlate the radial Nusselt number
distributions in two ways:
1) For the stagnation region: Nu ~ R e 0 - 5 P r 0 , 4
2) For the wall jet region:
Nu ~ Re 0 , 8 Pr 0 - 4
Ad 1 ) It has already been shown for the stagnation point that
the Nusselt number is proportional to Re
for small
values of H/d (see figure 6.41). Correlating the results
from figure 6.52 with the square root of the Reynolds
number, it can be stated how far this proportion reaches.
The correlation is presented in figure 6.53. It shows
that the correlation holds for H/d = 2 and H/d = 3 in the
region 0 < r/d < 1 . At H/d = 4 and H/d = 6 the
correlation is bad. The conclusion is that not only at
the stagnation point, but also in the whole stagnation
region the heat transfer depends on Re^* 5 .
Ad 2) For the wall jet region the results are correlated with
R e 0 , 8 in figure 6.54. A power of 0.82 has been found by
Vallis, Patrick and Wragg (1978) (see equation 2.67) in
145
1.40
«89»
1.00
8
°§8
*
Re =
a 1900
Re =
a 1900
o 2900
a> 2900
» 4226
« 4226
ft
D
a
\
O
«
0
fc
B o o
0.60
0° °
6
,
□
0
H/d=2
0.20
,
H/d=3
i
i
—r
1
1
1.40
fle =
a »900
Re =
a 1900
o 2900
o 2900
•»
» 4226
1.00-
*»
* 4226
0»
o
« S 2o
o
n
■s g
S
»
B
0.60
°
o
S •
D 0
»
. •S ' "
° * a
H/d = e
H/d=4
1—
0.20
1
1—
r/d
Fig. 6.53. Correlation of Nusselt
tions with Re°- 5 Pr 0 - 4 .
number distribu­
the wall jet region of impinging isothermal jets.
However, this has been found for higher r/d-values than
the r/d-values considered in our study. Yet, figure 6.54
shows that at H/d = 2 this correlation with R e 0 , 8 is
possible for r/d > 3. For such a low burner to plate
distance the wall jet will be established earlier than in
the case of higher H/d-values. At H/d = 3 and H/d = 4 the
146
o.oa
□ 7900
o».m
a 7900
o 2900
"
0.06-
a 2900
*.%8
4226
« 4226
o g
8
fl
9
0.04-
H/d = 2
H/d=3
0.02
O
-
*
Re =
a 7900
0
o
C3
«a
0.06
fle=
a 7900
a
o
2900
o 2900
4226
.* 4226
o
□
*
o o o o o
' 8 „
Q
13
' * * *
0.04-
» ° o
®
. 1
*
0
a
*
i * ffl
^ t
6
2
i
H/d = 4
H/d = 6
0.02
6
0
r/d
r/d
Fig. 6.54. Correlation of Nusselt
tions with Re°- 8 Pr 0 - 4 .
number
distribu­
correlation evidently improves when r/d increases from 3
to a higher value. For the measurements at H/d = 6 no
correlation with Re
could be found in the r/d region
considered in our study.
This completes
distributions.
the discussion
on the radial
heat
transfer
147
7. RESULTS OF NUMERICAL SIMULATIONS
7.1. The laminar impinging jet
Numerical simulations were first performed on laminar impinging
jets. This was done in order to validate the computer code with
a known case. Also a study could be performed on the dependence
of the heat transfer on the shape of the impinging laminar
velocity profile. Constant fluid properties were assumed for
these calculations. The dependency of the heat transfer on H/d
was expected to be very small for laminar jets at Re-numbers
higher than 600 in the region 1 < H/d < 12. This in fact
has already
been found by Saad (1975). For this reason the
burner to plate distance was fixed at H/d = 6.
To show the importance of the shape of the velocity
profile on the heat transfer computed results on laminar jets
with a parabolic and a flat profile are compared in figure 7.1.
I
35
Nu
^^
\
I
I
\
I
I
Re=600
\
\
parabolic
N
25
"
flat
\
\
-
.-«•""""^"N.
^S^
15
—
S
\
"
^
^ \ ^
N.
\
^\.>v ^
_
\
>v
\ w
\
\
—
^ ^ s ^ ^ *"V
—
^^"»—^^_^- ^
5
I
0.5
I
I
1.0
1.5
I
2.0
I
2.5
3.0
r/d
Fig. 7.1. Radial heat transfer distributions
laminar impinging jets.
from
149
The Reynolds number defined on the averaged velocity at the
outlet is for both jets the same. This means that the velocity
on the jet axis of the parabolic jet is twice that of the flat
jet. Figure 7.1 shows that the heat transfer at the stagnation
point also differs by a factor of 2. However, not only at the
stagnation point, but for r/d È 2.5 at any radial position heat
transfer from the jet with the parabolic profile is higher than
heat transfer from the jet with the flat profile. The reason
for this is, that the flow from the parabolic jet is more
concentrated along the plate than the flow from the flat jet.
This can clearly be seen from the figures 7.2, 7.3 and 7.4
giving numerical results of radial velocity profiles at r/d =
0.486, the maximum radial velocity and the boundary layer
thickness as a function of radial distance. A comparison of the
radial velocity profiles along the wall at r/d = 0.486 in
figure 7.2 learns that the velocity gradient at the wall and
v
(m/s)
T
i
i
1
1
1
0.8
1
r
r/d = 0.486
parabolic
0.6
flat
0.4
0.2
'
0
2
4
6
Fig. 7.2. Radial velocity profiles
pinging jets (Re = 600).
8
of
y(mm)
laminar
im­
the peak velocity are much higher for the parabolic jet. The
maximum velocity of the wall jet from the impinging parabolic
jet is higher than that from the impinging flame jet at every
radial position considered (see figure 7.3). At the same time
the boundary layer thickness of the wall jet from the impinging
parabolic jet- (defined by the distance from the wall where the
150
i
v
max
(m/s)
0.8
1
1
l
1
parabolic
flat
'~~X
-
f
0.6
-
/
7
/
-
0.4
0.2
i
i
i
i
i
0.5
1.0
1.5
2.0
2.5
/
Fig. 7.3. Maximum wall jet velocities
impinging jets (Re = 600).
from
r/d
laminar
5
(mm)
_
-
2
jr
jT
""
,
1
«•«*
s*
j/r
_
^
^
y
^^
"^^V> v
—
s
«"^
J S ^
**
^^'
-
parabolic
y
,s'
-
flat
i
i
i
i
0.5
1.0
1.5
2.0
i
i
2.5
3.0
r/d
Fig. 7.4. Boundary layer thickness
pinging jet- (Re = 600).
of
laminar
im­
maximum velocity is present) is smallest in case of an
impinging parabolic jet (see figure 7.4). For r/d -► 0 we see
that the difference is a factor of 2. The boundary layer
thicknes of the impinging flat jet shows a minimum at r/d =
0.45. This corresponds with the maximal heat transfer at r/d =
0.40 (see figure 7.1). The comparison of the flow structure of
both impinging jets gives reasons for their differences in Jieat
transfer.
151
To find the influence of the shape of the velocity profile
the Reynolds number has to be defined on a different diameter
and velocity. We will take now Rei = u m , v di/v, with di being
2
max
2
2
the half width jet diameter (see also paragraph 2.1.2). In this
way also a Gaussian shaped velocity profile can be taken
respect.
Stagnation
point
heat
transfer
results
of
into
several
simulations are gathered in table 7.1 and figure 7.5.
TABLE
7.1
COMPARISON OF HEAT TRANSFER CALCULATED FROM 0 AND FROM
THE HEAT FLUXES
3
Rei =
Y =
0.661
2
u
max- i
1
V
3
d
u
max
di
6 i
di(-)2
2
Nui
=
2
V
a di
A
2
2
flat
583
53.3
1 .04
16.3
parabolic
849
240
1 .70
25.1
24.0
27.2
26.0
Gaussian
The
third
16.6
993
281
1 .70
1202
340
1 .70
29.9
28.9
1414
397
1 .69
32.3
32.9
1697
486
1 .72
35.7
35.3
1980
561
1 .71
38.4
39.6
2122
606
1 .72
39.9
40.4
2263
668
1 .78
41 .9
41 .4
2405
687
1 .72
42.5
41 .8
691
108
1 .88
23.8
21 .6
1037
162
1 .88
29.2
27.6
1383
221
-1 .92
34.1
32.9
column
in
table
7.1
gives
the
values
of
g
=
(8v/3r)/r->-0 determined from the calculated flow field. Then in
the fourth column the influence of the velocity profile on 3
can" be seen by the parameter y
152
= u m a x /(diB). with the formula
Nu 0.5
Nuos
SO
= 0.909
Re0°s5
0.5
30
0.5
flat
20
profile
parabolic
Gaussian
10
300
_L
500
_i_
1000
Re,0.5 5000
2000
Fig. 7.5. Stagnation point heat transfer for laminar
impinging jets.
of Sibulkin
(equation 2.32)
Nu = 0.763 L (-)2 Pr 0.4
the
stagnation
point
heat
(7.1 )
transfer
can
be
predicted.
The
results are given in column 5. The Prandtl number was Pr = 0.71
and the characteristic length scale L = dj.. Finally, in column
2
6
are
given
calculated
obtained
the
computed
values
of
the
Nusselt
number
from the computed temperature gradients at the wall
with
the
full
simulation
model. Comparing
columns
5
and 6 the conclusion is that the Sibulkin theory can be applied
to laminar
impinging
influences
g in the same manner as it influences the tempera­
ture
gradient.
concluded
From
jets. The shape of the velocity
the
results
in
table
7.1
can
profile
also
be
that from the three velocity profiles considered the
Gaussian shaped profile gives the highest stagnation point heat
transfer. Averaged values of y together with equation 7.1 give
the following relationships:
153
- flat velocity profile:
Y = 1.04
Nui = 0.675 Rej.0-5
2
(7.2)
2
- parabolic velocity profile:
Y = 1.72
Nui = 0.868 Rej.0"5
5
(7.3)
2
- Gaussian velocity profile:
y = 1.89
Nui = 0.909 Rej.0'5
3
(7.4)
2
The resulting stagnation point heat transfer found from 8
and from the temperature fields are compared once more in
figure 7.5. The results from the Gaussian jet and the parabolic
jet come very close due to the nearly similar profiles near the
jet axis.
7.1.1. Comparison with literature data
The results of the present laminar calculations can be compared
with the numerical results of Saad (1975) and the analytical
results of Scholtz and Trass (1970). The comparison is made in
the figures 7.6 and 7.7 between present results at H/d = 6,
results from Saad at H/d = 8 and results of Scholtz and Trass
at H/d = 1.0. In this region of nozzle to plate distances (1.0
< H/d < 8) Scholtz and Trass found only a slight influence of
H/d on heat transfer, so this comparison can be made.
Figure 7.6 gives the results from our simulations at Re =
1 ,000 and the results from Saad at Re = 950 both for a
parabolic velocity profile. The heat transfer results are
scaled with /Re, since a /Re-dependency at the stagnation point
has been found by Scholtz and Trass. Their results are also
shown in figure 7.6. At the stagnation point our result agrees
with the result of Scholtz and Trass, but deviates about 5%
with Saad's result. The little hump in the heat transfer just
next to the stagnation point has only been calculated by us.
Next to this hump our calculations show a faster decrease of
heat transfer at increasing r/d. In the wall jet region Saad's
154
i
i
I
i
Scholtz and Trass
1.4
Saad
(Re=950)
present ca/c. [Re = 1000)
1.0
0.8
0.6
0.4
0.2
0.5
1.0
1.5
2.0
2.5
r/d
Fig. 7.6. Radial
heat
transfer
distribution
for a
laminar impinging jet with parabolic velo­
city profile.
predictions are higher, however, up to r/d = 0.5 the difference
is only about 5%.
The same conclusion can be drawn from the comparison of
the results from the flat velocity profile in figure 7.7. Our
results deviate from Saad's results less than 5% for r/d 5 1.
The maximum in heat transfer found at r/d = 0.4 at Re = 600 is
not found by Saad at Re= 950. He found more or less a constant
heat transfer coefficient in the stagnation region. The results
from Scholtz and Trass also point at a maximum outside the
stagnation point. This also has been found experimentally by
them (Scholtz and Trass, 1970).
The stagnation point heat transfer results can also be
compared. Here Saad found Nu ~ R e 0 , 3 6 (950 < Re < 2,000) for a
parabolic velocity profile. This deviates from what Scholtz and
Trass found experimentally: Nu ~ Re ' . As can be seen from
155
I
I
Scholtz
Nu
-Me
0.6
and Trass
Saad
(Re=950)
— present
(Re=600)
calc.
0.S
0.4
0.3
0.2
0.1
0.5
1.0
1.5
2.0
2.5
r/d
Fig. 7.7. Radial heat transfer distribution for a
laminar impinging jet. with a flat velocityprofile.
figure 7.5 our results come very close to a proportionality of
Nu with R e 0 - 5 .
There are two possible reasons for the deviations of the
results of simulations done by Saad. In the first place Saad
used an upwind differencing scheme, while in our calculations a
hybrid differencing scheme is used (see paragraph 4.1). In the
stagnation region for the five gridlines nearest to the plate
the cell Péclet numbers were smaller than 2, so in this region
our code used the central differencing scheme. The use of
upwind differencing in this region may lead to "false diffu­
sion" (see Patankar, 1980), because here the flow changes
direction. In the second place the computational grids were not
the same. Saad used 21 x 24 gridlines, while we used 30 x 20
gridlines. His first gridline near the plate was at a distance
of 0.03d. Within this distance we had four gridlines with the
156
line nearest to the plate at 0.0026d. This means that the
temperature gradient at the wall could be determined with a
much higher accuracy.
With the computational results discussed in this paragraph
the computer code can be considered to be sufficiently
validated.
7.2. The turbulent impinging j et
Next to the numerical calculations of the laminar impinging
jets simulations have been performed for a turbulent impinging
jet. Two different models of turbulence have been used:
1 ) The k-e model with modification for low-Reynolds number
flows as proposed by Chien (see paragraph 3.2.2).
2) The anisotropic k-e model as modified by us with the lowReynolds number model as a basis (see paragraph 2.2.4).
All calculations have been performed at a Reynolds number of
6500 at H/d distances between 2 and 6. At H/d = 6 three
different grids have been applied on calculations with the
Chien model. At H/d = 6, 4 and 2 one of these grids has been
used on simulations with the anisotropic model.
Table 7.2 gives a summary of the cases that will be
discussed here. The number of gridlines that has been used is
equal to 40 x 40 or 40 x 60. The parameter a giving the degree
TABLE 7.2
THE NUMERICAL CASES DISCUSSED
Case
C1
C2
C3
A1
A2
H/d
turbulence
model
n
6
6
6
6
2
Chien
Chien
Chien
anisotropic
anisotropic
40
40
40
40
40
x
n
r
40
40
60
60
60
a
x
9
6
7.04588
7.04588
7.04588
157
of nonuniformity of the grid in the axial direction via
equation 4.18 varied between 6 and 9. The two grids of the
cases C1 and C2 are given in figure 7.8.
Case C2 had much more gridlines close to the plate than
case C1. The grid of C1 appeared to be too coarse to determine
heat transfer to the plate accurately. Heat transfer to the
wall has been calculated in two ways (with a linear or a
quadratic enthalpy profile near the wall, see paragraph 4.5).
*-X
Fig. 7.8. The two grids of case C1 (a) and case C2 (b)
158
This gave results that differed about 40% for grid C1 . It is
not clear which of the two methods gives the best results.
Evidently the number of gridpoints for the determination of the
temperature gradient at the wall was too small for this case.
The grid of case C2 gave much better results in this respect:
less than 1% difference in the two results of the Nusselt
number calculations. For the cases C3, A1 and A2 this
difference was always less than 5%.
7.2.1. Comparison of results on different grids
The numerical results of the four cases C1 , C2, C3 and A1
differ, which is shown in the following figures on the axial
velocity decay (figure 7.9), the axial development óf kinetic
energy of turbulent fluctuations in the stagnation region
(figure 7.10), the maximum velocity, in the boundary layer and
the kinetic energy of turbulent fluctuations at the same
location (figure 7.11), and the radial Nusselt number distribu­
tion (figure 7.12).
At first these four cases are compared in figures 7.9 and
7.10. The computational grids of C1 and C2 differ in this
respect, that in the axial direction C1 has more gridlines in
12.0
10.0
u
(m/s)
8.0
6.0
Fig. 7.9.
The axial velocity of
four cases.
4.0
2.0-
0.0
159
3.0
Fig. 7.10.
Kinetic energy of tur­
bulent fluctuations on
the axis in the stag­
nation region.
5.0
Fig. 7,11.
The maximum velocity
in the boundary layer
and the kinetic energy
of turbulent fluctua­
tions
at
the
same
location.
r/d
the free jet region and C2 more gridlines close to the wall.
Because of the fact that the C1 grid is finer in the free jet
region, the jet development should be simulated more accurately
with this grid. Case C1 results into less decay of the axial
velocity compared to case C2, accompanied by lower values of
the kinetic energy of turbulent fluctuations. Case C3 with more
160
120Nu
100-
80-
rC3
r*\ici
Fig. 7.12.
Radial Nusselt number
distributions.
\^
^i\V
60-
40-
20 -
0 0
i
1
1
2
3
r/d
gridlines in the radial direction gives even less velocity
decay and still lower values of k. The anisotropic model
finally results in the same axial velocity decay as case C3,
but the kinetic energy of turbulent fluctuations is lower with
a peak before stagnation closer to the plate.
Figure 7.11 shows a continuation of the already noted
differences. Case C2 has the lowest maximum velocities in the
boundary layer and the highest values of k. Cases A1 and C3
differ only slightly if it concerns the maximum velocity, but
the values of k for case A1 are significantly lower.
The differences in the flow give rise to different heat
transfer results, as can be seen from figure 7.12. The
deviation of C1 from C2 and C3 in this figure is rather big. As
already stated, C1 had too few gridlines near the plate to
predict the temperature gradient accurately. The Nusselt number
distributions of C2 and C3 are very close. The lower radial
velocities of C2 are compensated by the higher turbulence
level. The rather big difference between A1 and C3 must be
entirely due to the lower level of turbulence calculated with
the anisotropic model.
161
From these comparisons it can be concluded that the
calculated temperature gradient at the wall as well as the jet
development before impingement are dependent on the applied
grid. Case C1 certainly did not have enough gridlines close to
the plate, while case C2 did not have enough gridlines in the
region of jet development. A very fine grid will be necessary
to reach a high accuracy in both regions. It is not proven that
the grid of C3 and A1 is fine enough. Therefore, calculations
with much finer grids will be necessary. For the simulations
with H/d = 2 the jet development region is much smaller. It is
supposed that the applied grid is fine enough for this
situation.
7.2.2. Comparison of numerical with experimental results
In this paragraph the results of the simulations carried out
with the anisotropic model for burner to plate distances H/d =
6 and H/d = 2 will be discussed and compared to results from
measurements.
7.2.2.1. H/d = 6
The computational results presented in figures 7.9 and 7.10 are
not in good agreement with the experimental results presented
in paragraph 6.2.1. The axial velocity decay of the simulated
jet is much less than the decay measured on. the jets from the
burners, and consequently the turbulence on the axis of the
simulated jet is too low. The simulated jet has a longer
potential core length and looks much more like a jet from a
straight pipe with a well shaped nozzle (see figure 6.11). The
boundary conditions in the burner exit correspond with the
measurements on jets from the small burner: a flat velocity
profile and a turbulence level of 7%. This, however, does not
give rise to the same development of the jet in the first four
or five diameters from the burner exit, where the plate does
not yet have its influence on the flow. At x/d = 4 the velocity
162
on the axis of the simulated jet is still within 1% of the
velocity in the burner exit. At x/d = 5 this decay is only 4%.
The measurements on jets from the small burner gave a velocity
decay on the axis of 4% at x/d = 4, and 10% at x/d = 5 (see
figure 6.1 ). Apparently, the calculated jet does not mix with
the surroundings in the same degree as the experimental jet.
This cannot be due to the coarseness of the grid. As we have
seen before, a finer grid resulted in less decay of the axial
velocity.
There can be two reasons for the disagreement between
simulations and experiments. At first, the specification of the
profiles of u, k and z at the burner exit as flat profiles is
not sufficient to be able to predict the jet development in the
first few diameters. In this region the spread of a jet also
depends on the boundary layer of the flow in the nozzle.
According to studies performed by Yule (1978) and Strange and
Crighton (1983) the development of free jets is determined by
the presence at x/d = 1 of large scale axisymmetrically
coherent vortical structures due to the instability of the
laminar boundary layer. Within the first few diameters from the
nozzle two vortex rings firstly pair into one before breaking
up into several large scale eddies at the end of the potential
core region. If the boundary layer in the nozzle is turbulent
due to roughness of the nozzle inner wall, the coherent
structures do not exist. The present model of turbulence cannot
predict this behaviour of the flow.
A second reason for the disagreement between simulations
and experiments may be the height of the Reynolds number. The
convection, diffusion, production and dissipation of kinetic
energy of turbulent fluctuations can only be well predicted if
the Reynolds number is high enough. The low Reynolds number
modification of the k/e model takes into account the behaviour
of the turbulence in the low Reynolds number region close to
the wall. The modifications from Chien are only activated in
this region. It is not suitable, however, to predict the low
163
Reynolds number region at the boundary of the jet, where
turbulence is being generated. The Reynolds number of the
simulated jet may not be high enough to be sure that the k-e
model leads to good predictions in this region.
The maximum velocity in the boundary layer along the plate
together with the rms-value of the fluctuations of this
velocity are compared to the experimental results at Re = 6400
in figure 7.13. It is clear that due to the less developed
simulated jet the averaged velocities along the plate are
higher and the fluctuating velocities are lower than measured.
This is consistent with earlier experiences.
r/d
Fig. 7.13. Comparison of the maximum velocity in the
boundary
layer and the rms-value of its
fluctuating
component
with
results
from
measurements (H/d = 6 ) .
The final result of the simulations, being the heat
transfer distribution to the plate, is compared to the results
from measurements in figure 7.14. Although the anisotropic
model results in the lowest heat transfer (see figure 7.12),
164
I
i
Nu
simulations
Re = 6500
120
experiments
Re = 6300
80
40
O
0
1
1
2
3
r/d
Fig. 7.14. Comparison of experimental and numerical
results of radial Nusselt number distribu­
tion at H/d = 6 .
this is still too high compared to the measurements. A maximum
in the heat transfer is calculated at about r/d = 0.4. This
maximum is much less extreme than the maximum calculated with
the Chien model (C2 and C3 in figure 7.12). A similar maximum
has been found in our measurements (see figure 6.45 and 6.46)
and in the measurements of Gardon and Akfirat (see figure 2.9)
at r/d = 0.5, but in those cases at lower H/d-values. The
measurements of Gardon and Cobonpue (see figure 2.10) show that
at a Reynolds number of 28,000 this maximum is not present
anymore at H/d = 6. Again, this comparison and especially the
presence of the maximum in the simulated heat transfer
distribution, points at a
less developed
jet
in our
simulations.
7.2.2.2. H/d = 2
Results of simulations for a burner to plate spacing of H/d = 2
have only been obtained with the anisotropic model. At this low
165
x/d
x/d
r/d
r/d
Fig. 7.15. Contour lines of the turbulent parameters
for H/d = 2.
166
H/d-value the development of the free jet has less influence.
For this reason the simulations at H/d = 2 should be more
accurate.
To get a qualitative idea of the turbulent parameters of
the simulation the contours (lines of constant values) of k, e,
g and Ueff are given in figure 7.15. The highest turbulence
levels are reached in the mixing layer of the jet. Besides
this, also in the wall jet region just before r/d = 2 all
turbulent parameters have a maximum value. The kinetic energy
of turbulent fluctuations reaches a third maximum around r/d =
0.5 very close to the plate. This maximum in k leads to a
maximum in the turbulent viscosity at the same place, which can
hardly be seen from the figure.
Further results of the calculated flow are presented in
the form of radial velocity profiles and profiles of the
rms-value of the fluctuating component of the radial velocity
within the first 6 mm from the plate of impingement at six r e ­
locations
(figure 7.16). At four r/d-locations also the
measured profiles for Re = 6400, already discussed in paragraph
6.2.3.1, are given. At r/d = 0 a peak in the rms-value of the
fluctuating component is visible. This peak is in agreement
with the theory of Sutera (1965), the measurements of Sadeh,
Sutera and Maeder (1970) (see figure 2.6) and the recent
measurements of Van Fossen and Simoneau
(1987) on the
stagnation line of turbulent flow around a cylinder.
At the next four locations r/d = 0.25, 0.5, 1.0 and 2.0
the results from simulations and experiments can be compared.
The profiles of the averaged velocity agree within a few
percent. Values of the turbulent velocity are predicted too
low. This again has been caused by the fact that not enough
turbulence is being generated in the mixing layer of the jet.
The excellent agreement between experiments and simula­
tions can once again be noticed from figure 7.17. The maximum
velocity in the boundary layer is predicted within a few
percent. The predicted turbulence levels are too low for r/d >
167
y
(mm)
\
r/d=0.00
\ \
r/d =1.00
4-J
■
\
\
\
2-
1 '"V'
ï—i—i—i—i—i
i—i—i—i—i
i—r
(
i—i—i—i
i—i—i—i—i—i—i—i
0.25
I
1
1
1
1
1
1
1
1
1
1
1
1
1
w
S
y
(mm)
I
o.so
A'
4-\
\
V
2-
'|-^
o
i—r
f
2
4
v
i
"i
e
I—I—I—I—I—I—I—l
O
O.S
1.0
1.5
O
I
O
1
y
i
2
i
1
0.5
1
v'Um/s)
V
3.00
\
i
/==• } simulations
i
4
i
V
1
1.0
1
i—i
6
1 1
1.5
v'Um/s)
V
ï
r = , experiments
Fig. 7.16. Radial velocity profiles and profiles of the
rms-value of its fluctuating component at
H/d = 2 from simulations and experiments.
168
Fig. 7.17. Comparison of the maximum velocity in the
boundary layer and the rms-value of its
fluctuating component with results from
measurements.
0.5. The significant maximum of the turbulence level found in
the experiments at r/d = 2 is not found here, at least not at
the location of the maximum velocity. Fig. 7.15 shows that this
maximum is present in the simulations. From this figure
together with figure 7.16 it can be concluded that this maximum
lies somewhat further from the plate and is lower than in the
case of the experiments. In general, the experiments and
simulations of the flow field of the impinging jet at H/d = 2
agree excellent. The results affirm, that the disagreement found
for H/d = 6 is due to the erroneous predictions of the jet
development.
The heat transfer results of
fit the experimental results to a
from figure 7.18. The simulations
heat transfer distribution, which
the simulated jet at H/d = 2
lesser extent, as can be seen
give a maximum in the radial
was not measured for the jet
169
I
i
Nu
120
-----
i
simulations
Re = 6500
experiments
Re = 6300
Gardon and Akfirat
Re=7000
80
40
O
0
1
1
L
2
3
r/d
Fig. 7.18. Comparison of experimental and numerical
results of the radial Nusselt number.
with Re = 6300 from the small burner. Our measurements on jets
from the large burner and measurements performed by Gardon and
Akfirat also show a maximum heat transfer at r/d = 0.5. These
maxima, however, are less pronounced. They were attributed to a
minimum' in
the boundary
layer
thickness. The present
calculations do not show a minimum . in the boundary layer
thickness, but the maximum coincides with a maximum value of
the effective viscosity very close to the plate at this value
of r/d. .
Compared to the measurements the calculated heat transfer
in the wall jet region is too low. This is most probably due to
the lower turbulence predicted by .the simulations. The
deviations between results from simulations and from experi­
ments are mainly differences in turbulence or caused by these
differences in turbulence. In case of a burner to plate
distance H/d = 2 this difference in turbulence does not lead to
different averaged velocities. Thanks to the fact that the
170
averaged
velocities
transfer,
and
well,
prediction
the
that
to
a
large
these
of
extent
averaged
the
heat
determine
velocities
transfer
is
the
heat
are
predicted
in
reasonable
agreement with the measurements.
7.2.2.3. The stagnation point heat transfer
Like for the laminar jet, the stagnation
from
the
impinging
temperature
gradient
turbulent
jet
point heat
calculated
transfer
from
at the wall can be compared
the
to the heat
transfer calculated by means of the radial velocity gradient 6
(via equation 7 . 1 ) . This comparison has been made in table 7.3
for
the
cases
discussed
before. Only
case
C1
has
been
dis­
regarded because of the inaccurate result of its heat transfer
prediction.
The values
been determined from
of
8 for
the
calculation
of Nug
have
(8vmax/3r)r=0.
TABLE 7.3
COMPARISON OF Nu AND Nug
Nug
Nu
Case
A2
54.5
54.1
Exp H/d = 2
56.4
55.6
Exp H/d = 6
59.0
51 .2
C2
C3
A1
90.6
63.8
81 .3
63.3
80.4
66.1
For an impinging jet with a flat velocity profile it can
be expected that the value of 8 will be equal to u Q /d. The jet
with a Reynolds
number
of 6500 would
give 8 = u Q /d = 526 re­
sulting in a Nusselt number of Nug = 53.6. This agrees with the
calculations
done
gradient
the
at
for
wall
H/d
leads
=
2
to
(case
Nu
=
A2) .
54.5
The
and
temperature
the
velocity
171
gradient 3 to Nuo =54.1. At this short burner to plate
distance the impinging velocity profile is nearly flat and the
enhancement of heat transfer by turbulence is negligible as was
confirmed by the measurements. This has been predicted well by
the numerical model.
At H/d = 6 the experiments resulted in Nun = 51.2. This
experimental value can be explained as follows. The decrease of
3 due to the decayed axial velocity in the free jet at x/d = 6
is almost compensated by the increase of 3 due to the changed
velocity profile. As concluded before, the velocity decay of
the simulated jet is less than that of the experimental jet.
The relatively high value of the simulated Nup must be due to
the shape of the velocity profile, while the axial velocity has
hardly decreased.
The enhancement due to turbulence at H/d = 6 in the
experimental case is 15% and in the numerical case (A1 ) 22%.
This is rather high for the simulations considering the
predicted low level of turbulence in comparison to the
experiments.
From this closer look at the stagnation point heat
transfer it can be concluded that with 3 determined from the
simulated flow field a first estimate of the heat transfer at
the stagnation point can be made with Sibulkin's equation.
172
8. GENERAL DISCUSSION AND CONCLUSION
The results of our experiments and simulations have led to a
number of conclusions on heat transfer from impinging jets and
flame jets from rapid heating burners. In this chapter the main
conclusions from our study will be summarized and discussed.
8.1. The flow structure
The isothermal jets from the tested rapid heating burners
differ from the flow structure of jets issued by a long
straight pipe or by a well defined nozzle. The jets from the
two burners used differ also mutually. Compared to the well
defined jets (from pipe or nozzle) the jets from the burners
are much more turbulent. This is demonstrated by the high
turbulence level in the nozzle exit and the axial velocity
decay after a small potential core length. Of course there is a
reason for this high turbulence level. It is aimed to have a
good mixing of air and gas in the burner.
The flame jets from the burners show an even faster axial
development than the isothermal jets. The axial velocity decays
faster due to the axial temperature decay. However, if an
effective diameter is introduced which compensates for the
density difference between jet and environment, it has been
noticed that the flame jet axial velocity decays more slowly,
while its axial turbulence level increases faster, compared to
the isothermal jet from the same burner. This can point at
turbulence being generated by combustion, which does not effect
the mean flow.
The radial velocity gradient at the stagnation point (6)
determined from the static pressure distribution around the
stagnation point is about equal to uQ/d for burner to plate
distances up to 2d for the isothermal jet as well as for the
flame jet. Despite of the high initial turbulence levels of the
flame jets in particular, this agrees with well defined
impinging jets. At H/d = 2 the large eddies generated in the
173
mixing layer of the isothermal jet give rise to a maximum in
the turbulence level in the wall jet outside the stagnation
region (1.5 < r/d < 2.0). 'Maybe due to the strong temperature
decay in the wall jet this has not been found for the impinging
flame jet.
8.2. Heat transfer
Despite of the differences in flow structure of the jets from
the burners, the heat transfer from impinging isothermal jets
and flame jets from the rapid heating tunnel burners as used in
our study agree well with results from other investigators on
heat transfer from impinging jets. The heat transfer from the
impinging isothermal jet and the impinging flame jet at the
stagnation point can be described by an equation from Sibulkin
for laminar heat transfer at the stagnation point of a body of
revolution:
Nu = 0.763 d (-)0-5 Pr 0 - 4
v
with the radial velocity gradient 3 determined from static
pressure measurements in the vicinity of the stagnation point.
For small H/d-values (H/d s 3) this equation predicts heat
transfer well, despite of the high levels of turbulence (Tu/Re*
up to 28) that exist in the free jets at the same axial
distances x/d s 3. For higher values of H/d the Sibulkin
equation underestimates heat transfer and a turbulence enhance­
ment factor (y) has been defined by us. For the jets considered
by us this factor is between 1 and 1 .5 for the region 4 <
Tu/Re* < 32 (6 é H/d S 12). The impinging flame jet results
agree quantitatively with the results of the impinging iso­
thermal jets if the fluid properties are defined at the
temperature belonging to the averaged enthalpy over the boun­
dary layer. Then the stagnation point heat transfer for iso­
thermal and flame jets can be described with the same (3 and y.
The radial heat transfer distributions of the impinging
174
isothermal jets agree to a certain extent with results on
impinging jets from well defined nozzles reported in the
literature. The higher levels of turbulence of the jets from
the burners cause less pronounced peaks or humps in the radial
heat transfer distributions.
The radial heat transfer coefficient distributions of the
impinging flame jets compare well with the distributions of the
impinging isothermal jets. Of course, the distributions of heat
flux densities to the plate differ very much. The reason for
this is, that for a flame jet impinging on a watercooled plate
the heat flux density distribution follows both the heat
transfer coefficient and the wall jet temperature distribution.
Since the temperatures in the wall jet decrease very fast with
r/d, the heat flux density is very nonuniform with a maximum at
the stagnation point and a sharp decrease with increasing r/d.
For smaller temperature differences between the flame jet and
the surface of the object to be heated this nonuniformity will
be less. In a furnace with recirculating hot gasses the
temperature decay in the wall jet, and herewith the heat flux
decay, will also be less pronounced than in our study. However,
especially at the beginning of a heating process one has to be
aware of this nonuniform heat flux density distribution.
8.3. The simulated laminar impinging jet
Results
insight
transfer
defined
of simulations of laminar impinging jets have given
into the dependency of the stagnation point heat
on the impinging velocity profile. If Nu and Re are
on dj_, being the half width diameter, and on the
2
maximum velocity on the axis, the jet with the Gaussian
velocity profile results in the highest Nusselt number at the
stagnation point, compared to jets with a parabolic and a flat
profile.
An excellent agreement between heat transfer calculated
with Sibulkin's theory and heat transfer determined from the
temperature gradient at the wall is noted.
175
8.4. The simulated turbulent impinging jet
Simulations of the turbulent impinging jet at a Reynolds number
of 6500 and a burner to plate distance H/d = 6 were not
accurate enough in the prediction of the jet development. The
turbulence level of 7% in the burner outlet in the simulations
did not lead to the same amount of mixing with the surroundings
as in the experiments. This is most probably due to the
turbulence model used. This with a parameter for anisotropic
turbulence modified k-e model is not capable to predict the
large scale turbulent structures that originate in the mixing
layer of a free jet. The averaged velocities are simulated too
high; the fluctuating velocities are simulated too low. As a
result the simulated heat transfer is too high.
The simulations of an impinging jet at H/d = 2 agree much
better with experiments. The averaged velocities agree within a
few percent, the simulated fluctuating velocities are again too
low. The heat transfer results from experiments and simulations
deviated little, because of the differences in turbulence.
Future studies should concentrate on a better modelling of
the turbulence of a jet in its first five or six nozzle
diameters from the burner outlet.,— so that the present simula­
tions can be extended to impinging flame jets.
8.5. Main conclusions
The most important conclusions of our study are:
- Very simple static pressure measurements can give a first
estimate of the stagnation point heat transfer of an
impinging jet; also if this jet originates from a burner.
Together with temperature measurements in the free flame jet
the static pressure measurements can give a first estimate of
the stagnation point heat transfer from an impinging flame
jet.
- Heat transfer . from an impinging isothermal jet and from an
impinging flame jet can both be described with the radial
176
velocity gradient 8 and the turbulence enhancement factor y.
- Heat transfer at the stagnation point of an impinging
isothermal or flame jet is'only enhanced by turbulence if the
impinged surface is placed outside of the potential core of
the free jet. This holds even for very turbulent flame jets
with an initial turbulence level of 12%.
- The heat flux density distributions of the impinging flame
jets are very nonuniform for the high temperature differences
used in our study between the surface of the plate and the
impinging flame jet.
- The numerical simulations of an isothermal jet impinging on a
flat plate at a nozzle to plate distance of H/d = 2 done with
the k-e model of turbulence modified with a parameter for
anisotropic turbulence agree well with the experimental
results.
177
LIST OF PRINCIPLE SYMBOLS
aR
a
x
C
d
de
D
D
Fr
g
g
G
h
h'
h
H
J
k
k
Lx
nr
nx
Nu
p
p'
p
Pr
qw"
Re
Re
s.^
Sc
thermal diffusivity
velocity gradient in stagnation point
velocity gradient in stagnation point
specific heat at constant pressure
diameter of nozzle
diameter of body of revolution
equivalent diameter
diffusion coefficient
diameter of the plate (in eq. 2.5 and 2.60)
Froude number
parameter for anisotropy of turbulence
acceleration of the gravity (in eq. 2.6)
momentum flux
averaged enthalpy
fluctuating enthalpy
momentary enthalpy
nozzle to plate distance
flux (momentum or heat)
kinetic energy of turbulent fluctuations
mass transfer coefficient
macroscale of turbulence
■;
number of gridpoints in radial direction
number of gridpoints in axial direction
Nusselt number
averaged pressure
fluctuating pressure
momentary pressure
Prandtl number (v/a)
heat flux density to the plate
Reynolds number (u d/v)
turbulent Reynolds number (k3/ve)
fluctuating rate of strain
Schmidt number (v/D)
(m3/s
(1/s
(1/s
(J/kg.K
(m
(m
(m
3
(m /s
(m
(2
(m /sa
(m/s3
(kg.m/sa
(J/kg
(J/kg
(J/kg
(m
(m2/sa
(m/s
(m
(
(Pa
(Pa
(Pa
((W/ma
(((1/s
(r
17
Sh
SJJ
Si
t
T
T*
Tu
u
u'
u
v
v'
v
Vg'
V
>
x
y
z
a
ax
3
Y
r^ e £j
öjj
6X
Sherwood number (kcd/D)
mean rate of strain
general source term
time
temperature
temperature at averaged enthalpy
turbulence grade (/ u'a/u)
averaged axial velocity
fluctuating axial velocity
momentary axial velocity
averaged radial velocity
fluctuating radial velocity
momentary radial velocity
fluctuating tangential velocity
averaged radial velocity just outside the
boundary layer
axial distance
distance from the plate
distance along the surface from the stagnation
point
X
u
Ueff
heat transfer coefficient
parameter for nonlinearity of the grid
radial velocity gradient at stagnation point
turbulence enhancement factor
effective diffusivity
Kronecker delta
distance between two gridpoints in axial
direction
dissipation of kinetic energy of turbulent
fluctuations
thermal conductivity
dynamic viscosity
effective viscosity (u + ufc)
ut
turbulent viscosity
e
180
(-)
(1/s)
(s)
(K)
(K)
(-)
(m/s)
(m/s)
(m/s)
(m/s)
(m/s)
(m/s)
(m/s)
(m/s)
(m)
(m)
(m)
(W/m2.K)
(1/s)
(-)
(m)(m a /s 3 )
(W/mk)
(Pa.s)
(Pa.s)
(Pa.s)
v
p
a n -i
a^ ^
T
<t>
kinematic viscosity
density
laminar Prandtl number
turbulent Prandtl number
stress tensor
general parameter
(m 2 /s)
(kg/m 3 )
(-)
(-)
subscripts:
c
max
o
r
S
x
3
°°
referring to
of the plate
referring to
referring to
referring to
referring to
referring to
referring to
velocity has
referring to
the free jet value at the position
a maximum
the nozzle exit
the radial direction
the surroundings
the axial direction
the radial position on which the
reached half of its maximum
the undisturbed flow
181
REFERENCES
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Massachusetts (1963).
Agarwal, R.K. and Bower, W.W., Navier/Stokes computations of
turbulent compressible two-dimensional impinging jet field.
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Ali Khan, M.M., Kasagi, N., Hirata, M. and Nishiwaki, N.,
Proc. 7th Int. Heat Transfer Conf. , Munchen, FC63, 363-368
(1982) .
Amano, R.S. and Jensen, M.K., A numerical and experimental
investigation of turbulent heat transport of an axisymmetric
jet impinging on a flat plate. ASME-Report, 82-WA/HT-55
(1982).
Bakke, P., An experimental investigation of a wall jet. J.
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Beer, J.N. and Chigier, N.A., Impinging jet flames. Combustion
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measurements of two-dimensional turbulent jet impingement
flow
fields.
Proc.
Symp.
Turbulent
Shear
Flows,
Pennsylvania, J_, 3.1-3.8 (1977).
Buhr, E., Haupt, R. and Kremer, H., Konvektiver Warmeübergang
bei Verbrennung in der Grenzschicht. Westdeutscher Verlag
GmbH, Opladen (1976).
Chen, C.J. and Rodi, W., On decay of vertical buoyant jets in
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Toronto, MC 1 7, 97-102 (1978).
Chia, C.J., Giralt, F. and Olev Trass, Mass transfer in
axisymmetric turbulent impinging jets. Ind. Eng. Chem.
Fundam., 1_6, 28-35 (1977).
Chien, K.Y., Predictions of channel and boundary-layer flows
with a low-Reynolds-number two-equation model of turbulence,
AIAA paper 80-134 (1980).
Chieng, C.C. and Launder, B.E., On the calculation of turbulent
heat transport downstream of an abrupt pipe expansion.
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190
SUMMARY
Impinging flame jets are used in the glass and steel industry
for rapid heating purposes. Compared to a conventional radia­
tion furnace, in a rapid heating furnace, which applies imping­
ing flames, much higher heat flux densities can be obtained.
The dominant heat transfer mechanism in a rapid heating furnace
is convection. This study concentrates on the influence of
turbulence on heat transfer and on the nonuniformity of the
heat flux distribution to an object in such a furnace. In fact,
we study the heat transfer from a premixed flame jet impinging
perpendicularly on a flat plate. For this reason the flow
structure and heat transfer of impinging flame jets as well as
impinging isothermal jets from two rapid heating burners have
been measured.
The separation distance between burner and plate in this
study varied from 1 to 1 2 burner diameters. The Reynolds
numbers of the examined isothermal jets from the burners were
3,300 é Re s 10,000. The Reynolds numbers of the flame jets,
defined at the adiabatic flame temperature, were 1,700 S Re S
4,250.
Measurements of the flow field of free and impinging jets
were performed with a laser Doppler anemometer. Heat flux den­
sity distributions of isothermal jets impinging on a slightly
heated plate were measured with a liquid crystal technique.
Heat flux density distributions of the flame jets impinging on
a watercooled plate were measured with a Gardon heat flux
transducer.
Furthermore, static pressure measurements in the stagnation
region were performed, in order to find the value of the radial
velocity gradient just outside the boundary layer in the
vicinity of the stagnation point. With this parameter a first
estimate could be made of the heat transfer coefficient at the
stagnation point of the impinging jets. It is shown that heat
transfer from both isothermal and flame jets can be described
in the same way with this velocity gradient and a turbulence
191
enhancement factor. The results from flame jets agreed
quantitatively with the results from isothermal jets if the
fluid properties in the heat transfer correlation were taken at
a temperature belonging to the averaged enthalpy of the
boundary layer along the plate.
Radial heat transfer distributions of the impinging flame
jets were very nonuniform, mainly due to the large temperature
difference between the flame jet and the surface of the watercooled plate.
Using a numerical model impinging isothermal jets were
simulated. The two-dimensional Navier-Stokes equations, the
continuity equation and the energy equation have been solved by
the finite volume method. With simulations of laminar impinging
jets the effect of the impinging velocity profile on heat
transfer has been examined. Three different profiles (flat,
parabolic and Gaussian) were used. The Gaussian profile re­
sulted in the highest heat transfer at the stagnation point.
For simulations of a turbulent impinging jet (Re = 6500)
the turbulence of the flow was taken into account by a low
Reynolds number k-e model modified by us with a parameter for
anisotropic turbulence. At H/d = 6 the results showed that the
development of the free jet was not predicted well. The value
of the radial velocity gradient, however, again yielded to a
good first estimate of the heat transfer at the stagnation
point.
At H/d = 2 the results from our simulations agreed well
with the experimental results.
The most important conclusion from our study is, that from
isothermal measurements and from measurements of the radial
velocity gradient near the stagnation point a first approxima­
tion of the heat transfer from impinging premixed flame jets
can be made.
192
SAMENVATTING
In de glas- en staalindustrie wordt veelal gebruik gemaakt van
vlammen die loodrecht invallen op een te verhitten oppervlak.
Vergeleken met een conventionele stralingsoven kunnen in een
oven die gebruik maakt van deze gerichte vlammen, veel hogere
warmtestroomdichtheden bereikt worden. In een dergelijke oven
is convectie het belangrijkste warmteoverdrachtsmechanisme.
Deze studie betreft de invloed van turbulentie op de warmte­
overdracht en de niet-uniformiteit van de warmtestroomdicht­
heden naar een object in een dergelijke oven. In feite
bestuderen wij de warmteoverdracht naar een vlakke plaat die
loodrecht wordt aangestroomd door een yoorgemengde vlam. Hier­
toe zijn de eigenschappen van de stroming en de warmteover­
dracht van loodrecht treffende vlammen, maar ook van loodrecht
treffende isotherme stralen komende uit twee branders, experi­
menteel bestudeerd.
De afstand van de brander tot de plaat varieerde in deze
studie van 1 to 12 branderdiameters. De Reynolds getallen van
de onderzochte isotherme stralen uit de branders waren 3.300 s
Re é 10.000. De Reynolds getallen van de vlammen gedefinieerd
bij de adiabatische vlamtemperatuur waren 1.700 S Re s 4.250.
Metingen van het stromingsveld van de vrije en de lood­
recht treffende stralen werden met een laser Doppler snelheids­
meter uitgevoerd. Warmtestroomdichtheden van isotherme stralen
naar een weinig verwarmde plaat werden gemeten met een techniek
gebaseerd op het meten van temperatuur met vloeibare kristal­
len. Warmtestroomdichtheden van vlammen naar een loodrecht aangestroomde watergekoelde vlakke plaat werden gemeten met een
Gardon warmtestroommeter.
Bovendien werden statische drukmetingen in het stuwpuntsgebied uitgevoerd, teneinde de waarde van de radiële snelheids­
gradiënt in de nabijheid van het stuwpunt te vinden. Met deze
parameter kon een eerste schatting worden gemaakt van de
warmteoverdracht in het stuwpunt van loodrecht treffende stra­
len. Het is aangetoond dat de warmteoverdracht van zowel iso193
therme
stralen
als van
vlammen op dezelfde manier
beschreven
kan worden met deze snelheidsgradiënt en een parameter die ver­
hoging van de warmteoverdracht door turbulentie aangeeft. Kwan­
titatieve overeenstemming
werd gevonden,
indien de
stofeigen­
schappen in de warmteoverdrachtscorrelaties gedefinieerd werden
bij de temperatuur behorende bij de gemiddelde enthalpie in de
grenslaag langs de plaat.
De
radiele
niet-uniform,
verdeling
hetgeen
van
de
voornamelijk
grote temperatuurverschil
warmteoverdracht
werd
was
veroorzaakt
sterk
door
het
tussen de vlam en het oppervlak
van
de watergekoelde plaat.
Met een numeriek model zijn loodrecht treffende isotherme
stralen
gesimuleerd.
De tweedimensionale
lijkingen, de continuïteitsvergelijking
king
zijn opgelost
Navier-Stokes verge­
en de energievergelij­
met behulp van een eindige volumemethode.
Met simulaties van laminaire loodrecht treffende stralen is de
invloed van de vorm van het snelheidsprofiel op de warmteover­
dracht
bestudeerd. Er werden drie verschillende profielen ge­
bruikt (vlak, parabolisch en Gaussisch). Het Gaussische profiel
resulteerde in de hoogste warmteoverdracht in het stuwpunt.
Bij
straal
de
simulatie
van
een
loodrecht
treffende
turbulente
(Re = 6500) werd gebruik gemaakt van een laag Reynolds-
getal k-e
model, dat door ons is gemodificeerd met een para­
meter voor anisotrope turbulentie. Voor H/d = 6 wezen de resul­
taten uit dat de ontwikkeling van de vrije straal minder goed
werd
voorspeld. Wel bleek
snelheidsgradient
in het
weer
dat de waarde
stuwpunt
een
eerste
van de
goede
radiele
schatting
gaf van de warmteoverdracht in het,stuwpunt.
Voor H/d = 2 kwamen de resultaten van de berekeningen goed
overeen met de experimentele resultaten.
De belangrijkste conclusie die uit de resultaten van deze
studie^ getrokken
kan worden
metingen
en metingen
stuwpunt
de warmteoverdracht
is, dat met behulp van
van de radiele
van
isotherme
snelheidsgradiënt
loodrecht
treffende
mengde vlammen in eerste benadering bepaald kan worden.
1'94
in het
voorge-
CURRICULUM VITAE
13 juni 1951
geboren te Zoeterraeer
1957 - 1963
lagere school te Zoetermeer
1963 - 1967
St. Petrus Mulo te Zoetermeer
16 juni 1967
eindexamen Mulo-A en Mulo-B
1967 - 1970
St. Maartenscollege te Voorburg
12 juni 1970
eindexamen HBS-B
1970 - 1976
Technische Hogeschool Delft, afdeling Tech­
nische Natuurkunde :
1974 - 1976
4e- en 5e-jaars werk in de subgroep Warmte­
transport onder begeleiding van prof.ir.
C.J. Hoogendoorn
15 juni 1976
doctoraal examen natuurkundig ingenieur;
het afstudeeronderwerp was "Warmteover­
drachtscoëfficiënten voor zeer viskeuze
vloeistoffen in een statische menger"
augustus 1976 -
wetenschappelijk medewerker in de subgroep
Warmtetransport van de vakgroep Transport­
verschijnselen van de faculteit Technische
Natuurkunde aan de Technische Universiteit
Delft
195
NAWOORD
Dit proefschrift is tot stand gekomen met de hulp van veel
mensen. Allen die een bijdrage geleverd hebben, wil ik hierbij
mijn dank betuigen. Ik zal in dit nawoord volstaan met het
noemen van diegenen zonder wie het resultaat in deze vorm niet
voor u zou liggen.
Allereerst ben ik dank verschuldigd aan mijn promotor
prof.ir. C.J. Hoogendoorn voor zijn directe begeleiding. Hij
heeft mij in moeilijke momenten bijgestaan en gestimuleerd. Hij
heeft mij de ruimte gegeven dit proefschrift te schrijven.
Dr.inz. Cz.O. Popiel van de Poznan Universiteit in Polen
heeft gedurende het eerste jaar van mijn verblijf in de groep
Warmtetransport met mij op dit onderwerp samengewerkt. Hij
heeft mede richting gegeven aan dit onderzoek. Zijn bijdrage is
van onschatbare waarde.
Voorts is er een aantal studenten geweest die in het kader
van hun 4e- of 5e-jaars werk een deelonderzoek hebben gedaan.
Dit waren Pieter Broerse, Gerard Burger, Henk Buys, Hans
Dekker, Floris van Drunen, Jan Willem van Dijk, Aart de Geus,
Dolf van Hattem, Albert van der Heiden en Ad Voets.
Een speciaal woord van dank gaat uit naar twee personen
die bij de uitvoering van dit proefschrift een belangrijke rol
hebben gespeeld: Bram de Knegt en Riny Purmer. Bram heeft de
figuren verzorgd die zijn opgenomen in het proefschrift. Hij
stond altijd voor mij klaar. Riny heeft het typewerk verzorgd.
Zij heeft mij gewezen op vele fouten in het gebruik van de
Engelse taal. De opmaak van dit proefschrift is haar werk ge­
weest. De accuratesse van haar werk was verbluffend.
Een woord van dank gaat voorts uit naar die mensen van de
algemene dienst van de faculteit Technische Natuurkunde, die
hebben bijgedragen tot de totstandkoming van de experimentele
opstellingen. De contacten zijn altijd zeer goed geweest.
197