Download ENGR 7901 - Heat Transfer II 1 Introduction 2 The Flat Plate

ENGR 7901 - Heat Transfer II
External Flows
1
Introduction
In this chapter we will consider several fundamental flows, namely: the flat plate, the cylinder, the sphere, several other body shapes, and banks of tubes in cross flow. With the
exception of the flat plate, most models are either of empirical nature or combinations of
empirical and theoretical results.
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The Flat Plate
The most important fundamental result in convective heat transfer is the flat plate in a
streaming flow. Most students are aware of the classical boundary layer solution for skin
friction or drag prediction. We now examine the characteristics of a heated plate in a similar
flow field.
The Navier-Stokes and energy equations for this problem in their simplest form for steady
incompressible and constant property flows of a Newtonian fluid are:
∂u ∂v
+
=0
∂x ∂y
(1)
∂u
∂u
∂2u
+v
=ν 2
∂x
∂y
∂y
(2)
∂T
∂T
∂2T
+v
=α 2
∂x
∂y
∂y
(3)
u
u
These equations are subject to the following initial conditions and boundary condition:
u(0, y) = U∞
u(x, δ) = U∞
u(x, 0) = 0
(4)
T (0, y) = T∞
T (x, δT ) = T∞
T (x, 0) = Tw
(5)
and
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The solution for the above set of equations is not easy, but has been achieved using analytical
and numerical methods. For laminar boundary layer flows, 1000 < ReL < 500, 000, the
important parameters are the boundary layer thickness, the friction coefficient, and Nusselt
number.
The solution for the boundary layer thickness and skin friction coefficient as found in any
fluids text is:
5x
δ(x) =
1/2
(6)
Rex
and
0.664
Cf,x =
1/2
(7)
Rex
The average skin friction for the plate of length L found by integrating the shear stress over
the flow length is:
Cf =
1.328
1/2
(8)
ReL
In the case of the energy equation we have three special cases that we consider. These occur
when the hydrodynamic boundary layer thickness δ is thicker than, equal to, or less than
the thermal boundary layer thickness δT :
δ > δT
δ = δT
δ < δT
(9)
For the second case, the energy equation can be solved by analogy with the skin friction
problem. A particularly useful result is easily obtained when one recognizes that the momentum equation and the energy equation have a similar form. If we introduce θ = T − Tw ,
we obtain:
∂θ
∂θ
∂2θ
+v
=α 2
∂x
∂y
∂y
(10)
θ(0, y) = T∞ − Tw
θ(x, δT ) = T∞ − Tw
θ(x, 0) = 0
(11)
u
and
This equation is very similar in form to the momentum equation and its boundary and
initial conditions used to solve for the skin friction if we assume that ν = α. We may
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non-dimensionalize the velocity by defining u? = u/U∞ , and find the dimensionless velocity
gradient from the friction coefficient:
∂u? ρU∞
= 0.332Rex−1/2
∂y w
µ
(12)
Now we may also define a dimensionless temperature θ? = (T − Tw )/(T∞ − Tw ). In this form
the dimensionless temperature gradient is the same as the dimensionless velocity gradient
for the special case of ν = α. This gives:
∂θ? ρU∞
= 0.332Rex−1/2
∂y w
µ
(13)
Finally, we use Fourier’s law to obtain the heat flux at the wall:
∂T ρU∞
qw (x) = −kf
= −kf (T∞ − Tw )0.332Re−1/2
x
∂y w
µ
(14)
or if we define a local Nusselt number:
N ux =
hx x
qw (x)x
= 0.332Rex1/2 =
kf (Tw − T∞ )
kf
(15)
This simple result is only valid for the case when ν = α, i.e. the special case when δ = δT .
It has been observed theoretically and experimentally that
ν 1/3
δ
≈
= P r1/3
(16)
δT
α
where P r is referred to as the Prandtl number. It is a fluid property representing the ratio
of the kinematic viscosity and the thermal diffusivity. Fluids vary in their value of Prandtl
number. For example,
Oils
P r >> 1
Gases
Pr ∼ 1
Liquid Metals P r << 1
(17)
This observation leads to an approximate result valid for the range 0.1 < P r < 60:
N ux = 0.332Rex1/2 P r1/3
(18)
since the heat transfer coefficient is inversly proportional to the thermal boundary layer
1/2
thickness, i.e. h ∼ kf /δT ∼ kf P r1/3 /δ or in otherwords hx/kf ∼ Rex P r1/3 .
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This simple result provides much useful information. For example, we see that the heat
transfer coefficient is highest near the leading edge of the plate, i.e. h ∼ 1/x1/2 , and decreases
as we move further downstream. This behaviour is indicative of the boundary layer providing
greater resistance to heat transfer as it thickens.
The full analytical solution for the problem reveals that the Nusselt number takes the form:
N ux =
0.564(Rex P r)1/2
[1 + (1.664P r1/6 )9/2 ]
2/9
(19)
Besides the stipulated Reynolds number criterion for laminar flow, we must also ensure
P ex > 100. This criterion ensures that there is no upstream conduction within the fluid
film. The above result has three special limits:
1/2
P r → 0 N ux = 0.564Rex P r1/2
1/2
Pr = 1
N ux = 0.332Rex
1/2
P r → ∞ N ux = 0.339Rex P r1/3
(20)
For the isothermal surface, the mean Nusselt number is determined from:
1
h=
L
Z
L
hx dx
(21)
N uL = 2N ux=L
(22)
0
or
Thus the average or mean heat transfer coefficient is twice the local value at x = L. Note
this characteristic is only for the flat plate in laminar flow. Later results for mean Nusselt
numbers will yield a constant different from “2”.
An alternate form of Eq. (19) in your text is given by:
1/2
N ux = h
0.3387Rex P r1/3
i1/4
1 + (0.0468/P r)2/3
(23)
Both equations give essentially the same numerical results for h. Physically speaking, Eq.
(19) has more meaning, since the Peclet number in convection is more important than
Reynolds number or Prandtl number alone.
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2.1
Other Flat Plate Results
We now consider some variations on the theme, namely, the isoflux boundary condition,
turbulent flow, and delayed heating. The latter being important in the case of discrete heat
sources.
2.1.1
Isoflux Boundary Condition
In the case of an isoflux boundary condition in laminar flow, qw = Constant, we define the
Nusselt number as follows:
qw x
kf (Tw (x) − T∞ )
since the wall temperature now varies in this case.
(24)
N ux =
The following result for the local Nusselt number is obtained for the special case of 0.1 <
P r < 60:
1/3
N ux = 0.453Re1/2
x Pr
(25)
The analytical solution provides the full Prandtl number behaviour, which has been approximated by:
N ux = h
0.886(Rex P r)1/2
9/2
1 + (1.909P r1/6 )
i2/9
(26)
Eq. (26) is also subject to the same Reynolds number and Peclet number criteria as for the
isothermal wall case. The above result has three special limits:
1/2
P r → 0 N ux = 0.886Rex P r1/2
1/2
Pr = 1
N ux = 0.453Rex
1/2
P r → ∞ N ux = 0.464Rex P r1/3
(27)
For isoflux conditions there is no average Nusselt number, since the wall temperature varies
in this case. However, we may obtain an integrated average wall temperature:
T w − T∞
1
=
L
L
Z
(Tw (x) − T∞ )dx
(28)
0
Using the simple result for the local Nusselt number to obtain the wall temperature difference,
we can show that
5
T w − T∞ =
2
q L
3 w
1/2
0.453kf ReL P r1/3
(29)
or if we now define a new Nusselt number based on this mean wall temperature difference,
we get:
N uL =
qw L
1/2
= 0.679ReL P r1/3
kf (T w − T∞ )
(30)
This expression is only about 2 percent higher than the isothermal wall case. As a result,
many Nusselt numbers which were obtained for isothermal conditions are also valid for
isoflux conditions, provided one understands that the temperature difference involved is the
mean surface temperature difference and not the local temperature difference. In many
future results that we will present, it is implied that the surface condition is one of constant
temperature.
2.1.2
Turbulent Flow
For turbulent boundary layer flows, 500, 000 < ReL < 107 , the boundary layer thickness and
friction coefficient are:
0.38x
δ(x) =
(31)
1/5
Rex
0.059
Cf,x =
Cf =
(32)
1/5
Rex
0.074
(33)
1/5
ReL
The above results presume that turbulent flow prevails over the entire surface, which in
general it does not. If the boundary layer is composed of a combined laminar-turbulent flow,
i.e. ReL > 500, 000, the friction coefficient is computed from the integrated value:
1
Cf =
L
Z
xcr
Z
L
Cf,lam dx +
0
Cf,tur dx
(34)
xcr
which gives:
Cf =
0.074
1/5
ReL
−
1742
ReL
(35)
The subtractive term accounts for the fact that turbulent flow does not prevail fully over the
surface, and is the correction for such an assumption.
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The Nusselt number for turbulent boundary layer flow is also found by analogy with skin
friction. An approximate result that agrees very well with experiment is
N ux = 0.0296Rex4/5 P r1/3
(36)
Integrating the above result leads to
5
N uL = N ux=L
4
(37)
or
4/5
N uL = 0.037ReL P r1/3
(38)
Thus we see that the factor of “2” in laminar flow is not universal.
For a combined laminar/turbulent boundary layer, ReL > 500, 000, the following integrated
expression is useful:
1
h=
L
Z
xcr
Z
L
hx,lam dx +
0
hx,tur dx
(39)
xcr
which gives:
4/5
N uL = (0.037ReL − 871)P r1/3
(40)
Once again, the subtractive term accounts for the fact that turbulent flow does not prevail
fully over the surface, and is the correction for such an assumption.
2.1.3
Delayed Heating
In a number of applications, the start of the thermal boundary layer may be further downstream from the start of the hydrodynamic boundary layer. In these situations, we require
a modification to the theory discussed earlier for the flat plate having an isothermal surface.
One common application is in predicting the surface temperature for discrete heat sources.
It can be shown that the present theory need only be modified such that:
1/2
N ux =
0.332Rex P r1/3
[1 − (xo /x)3/4 ]
1/3
(41)
for laminar flow, and
4/5
N ux =
0.0296Rex P r1/3
[1 − (xo /x)9/10 ]
for turbulent flow.
7
1/9
(42)
These relationships are useful for assessing the effect of localized heating problems, when the
hydrodynamic boundary layer has already initiated upstream of the heat source. Integration
of the above results is possible (and tedious) in order to obtain mean heat transfer coefficients.
The results for the mean Nusselt number are:
x 34 23
L
o
1−
L − xo
L
(43)
x 109 89
L
o
1−
L − xo
L
(44)
N uL =
1/2
0.664ReL P r1/3
N uL =
4/5
0.037ReL P r1/3
for laminar flow, and
for turbulent flow.
Note however, that in all cases, x or L is measured from the leading edge of the surface where
the hydrodynamic boundary layer initiates and not the edge where the thermal boundary
layer initiates. However, in calculations involving h, we only use the heated surface area and
not the total surface area.
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The Cylinder
The circular cylinder in cross flow is another fundamentally important geometry and flow
configuration, due to its use in heat exchanger and pin fin applications. The heat transfer
coefficient varies around the perimeter of the cylinder in cross flow, with the greatest local
heat transfer coefficient at the leading stagnation point. At the rear of the cylinder, the heat
transfer coefficient is the lowest due to boundary layer separation in the wake of the flow.
In the vicinity of the forward stagnation point the Nusselt number is given by theory as:
1/2
N uD = 1.15ReD P r1.3
(45)
This result is exact for θ = 0, but holds approximately in the region of −15◦ < θ < 15◦ . It
is frequently used as a model for heat transfer coefficient on the leading edge of gas turbine
blades. Refer to figure 7.9 for a graphical presentation of local Nusselt number as a function
of Reynolds number, for gases.
The drag coefficient is already known through fluids mechanics courses and may be determined from the following expression which is valid for 0.1 < ReD < 250, 000:
CD =
10
2/3
ReD
8
+ 1.0
(46)
The mean heat transfer coefficient may be obtained from the following model proposed by
Hilpert:
1/3
N uD = CRem
DP r
(47)
For P r > 0.7, the values of m and C are tabulated:
ReD
0.4-4
4-40
40-4000
4000-40,000
40,000-400,000
C
0.989
0.911
0.683
0.193
0.027
m
0.330
0.385
0.466
0.618
0.805
A more general model for a cylinder in cross flow due to Churchill and Bernstein (1977) is:
"
5/8 #4/5
1/2
ReD
0.62ReD P r1/3
N uD = 0.3 +
1+
[1 + (0.4/P r)2/3 ]1/4
282, 000
(48)
This model is the recommended correlation, as it is based on a comprehensive set of data of
many researchers for a wide range of Prandtl numbers. Further, the model does not require
tabulated coefficients, and covers the range ReD > 0 (laminar and turbulent) provided that
P eD > 0.2.
In the case of non-circular cylinders, Hilpert also obtained data for several configurations,
including several orientations of a square cylinder and hexagonal cylinder, in addition to
a flat plate oriented normal to the flow stream. These data are modelled using Eq. (47)
with the values of C and m given in Table 7.3 of your text. In most cases they are valid in
the range of Reynolds number of 5000 < ReD < 105 . The length scale D is defined as the
maximum profile height.
4
The Sphere
The final fundamentally important geometry is the sphere. The sphere is modelled in a
similar manner as the cylinder. First, the drag coefficient from fluid mechanics is given for
the range 0 < ReD < 250, 000:
CD =
24
6
+
+ 0.4
ReD 1 + Re1/2
D
9
(49)
The mean heat transfer coefficient is obtained from the result proposed by Whitaker:
N uD = 2 +
1/2
(0.4ReD
+
2/3
0.06ReD )P r0.4
µ
µw
1/4
(50)
which is valid for 0.71 < P r < 380, 0 < ReD < 7.6 × 104 , and 1.0 < µ/µw < 3.2. The
viscosity ratio is the correction factor applied to the experimental results since the model
was derived using properties based on the stream temperature, and not the film temperature.
The constant factor of “2” is the exact result for pure conduction into a quiescent (stationary)
fluid medium. It is theoretically derived from results given in Chapter 3 of the text.
In the case of other three dimensional bodies, such as spheroids and cuboids, the following
model proposed by Yovanovich (which is not in your text), is:
N u√A
"
#
1/2
√
P
1/2
0.566
= 2 π + 0.15 √
P r1/3
Re√A + 0.35Re√
A
A
(51)
where A is the surface area and P is the maximum equitorial perimeter perpendicular to the
flow direction. Note the change in the characteristic length scale, it is now based on the square
root of the total body surface area. It is recommended for the range: 0 < Re√A < 2 × 105
and P r > 0.7. Further, the body aspect ratio C/B is limited to 0 < C/B < 5, where C is
the height of the body and B the length of the body in the flow direction.
5
Tube Banks
Flow through tube banks is quite important in heat transfer applications. Numerous applications such tube fin heat exchangers, pin fin heat sinks, and heating systems utilize arrays
of tubes (or cylinders) arranged in one of two common patterns: inline and staggered. An
inline tube bank contains tubes arranged on square centers such that all tubes longitudinally
and transversly are inline. Staggered tube arrays have every other row offset usually by one
half of the tube to tube spacing.
In such applications, the heat transfer coefficient and hence the Nusselt number degrades as
a result of the complex flow arrangements and warming of the fluid. Since the cylinders are
no longer isolated, the assumption of a constant free stream temperature can no longer be
made. Thus we expect that thermal performance will deteriorate with each added row in
the flow direction. Transversely, this effect is not experienced as each row will see its own
unique stream.
When analyzing tube banks, we must consider three geometric parameters in addition to the
tube diameter. These are: the longitudinal or flow direction tube pitch, SL , the transverse
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tube pitch, ST , and the diagonal tube pitch SD in the case of staggered tube banks.
The Nusselt number is obtained from a simple expression, namely:
1/3
N ud = 1.13C1 Rem
D,max P r
(52)
where we must satisfy the following conditions:
NL > 10
where NL is the number of tubes in the flow direction. Further, the usual Reynolds number
and Prandtl number conditions also apply:
2000 < ReD,max < 40, 000
and
P r ≥ 0.7
The constants C1 and m are given in Table 7.5 of the course text. The Reynolds number
requires careful attention, as it is now based on the maximum flow velocity through the tube
bank, and not the mean flow velocity.
ReD,max =
ρVmax D
µ
For inline tube banks the maximum velocity is:
Vmax =
ST
V∞
ST − D
(53)
For staggered tube banks the maximum velocity is defined as:
Vmax =
ST
V∞
2(SD − D)
(54)
if it occurs in the diagonal plane connecting tubes in adjacent rows, or as Eq. (53) if it
occurs in the transverse plane between adjacent rows. Refer to Fig. 7.11 in your text. Based
upon the tube arrangement, it will occur in the diagonal plane if:
SD <
ST + D
2
(55)
In general, if there are fewer than ten rows of tubes, i.e. NL < 10, the results for Eq. (52)
need further modification, by means of:
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N uD,NL <10 = C2 N uD
(56)
where C2 is tabulated in Table 7.6.
The data for tube banks are limited. Table 7.5 provides data for the range of 0.6 < SL /D <
3.0 and 1.25 < ST /D < 3.0 for inline tube banks, and 0.6 < SL /D < 3.0 and 1.25 < ST /D <
3.0 for staggered tube banks. Beyond the upper limits of ST /D, SL /D > 3.0, isolated tube
theory may be used, i.e if the spacing between tubes exceeds three diameters. In these cases,
there is sufficient fluid at free stream conditions to warrant isolation. Finally at values below
the lower limit, systems behave more like porous media as free space is limited. The above
results may be extrapolated with small error to systems with spacings of at least one tube
diameter, i.e ST /D, SL /D ∼ 1.0.
Now, having predicted the Nusselt number, one must still calculate the heat transfer rate!
If the free stream temperature is not constant, we need an alternate approach involving a
mean temperature difference. The appropriate temperature difference is referred to as the
Log Mean Temperature Difference or ∆TLM T D for short. It is defined as:
∆TLM T D =
(Ts − Ti ) − (Ts − To )
(Ts − Ti )
ln
Ts − To )
(57)
where the outlet temperature is given by:
(Ts − To )
πDN h
= exp −
Ts − Ti )
ρV∞ NT ST cp
(58)
Finally, the heat transfer rate is computed from
Q = h(N πDL)∆TLM T D
(59)
If this seems complicated (which it should), then it will become more apparent after Chapter
8 and Chapter 11 when concepts related to internal flows and heat exchanger theory are
covered.
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