Download .13.0 with Re,2/ = = k Mk C

P5-8-20
NUMERICAL ANALYSIS FOR HEAT TRANSFER ENHANCEMENT OF
A LITHIUM FLOW UNDER A TRANSVERSE MAGNETIC FIELD
Naotaka Umeda and Minoru Takahashi
Research Laboratory for Nuclear Reactors
Tokyo Institute of Technology
2-12-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan
Phone: 81-3-5734-2957, Fax: 81-3-5734-2959
E-mail: [email protected]
Abstract
Related to the first wall and blanket cooling for magnetically confined fusion reactors, heat transfer
characteristics of a lithium flow in a conducting rectangular channel in the presence of a transverse
magnetic field was investigated numerically. Special attention was paid to the effect of convective heat
transport by jets generated in sidelayers on heat transfer enhancement. The Navier-Stokes equations
including the Lorentz force term, the equation for scalar potential and the energy equation were solved
numerically to simulate the flow and heat transfer of the lithium flow with a heated wall parallel to the
direction of the magnetic field. The jets appeared adjacent to side walls parallel to the applied magnetic
field. The ratio of peak velocity in the jets to average velocity reached about six at the Hartmann number of
1,900.
Skin
friction
coefficient
agreed
well
with
the
simple
analytical
model:
C f = k p M 2 / 2 Re, with k p = 0.13. Temperature was lower in the center of a heating surface due to
the cooling effect of the sidelayer jet. The Nusselt number increased by 42%-50% at the Hartmann number
of 1,900 compared with hydrodynamic flow. It was confirmed that the jet in the sidelayer played an
important role in the enhancement of heat transfer in a lithium flow in a rectangular channel.
1
1. Introduction
Related to the first wall and blanket cooling for magnetically confined fusion reactors,
magnetohydrodynamics (MHD) and heat transfer characteristics in liquid metal flows have been studied
experimentally and numerically[1-18].
Previous studies on this problems have been reviewed by
Lielausis[19] Kirillov, et al.[20], and N.B.Morley, et al.[21]. Most of the experiments have been performed
using liquid metals other than lithium, such as mercury, sodium, potassium, NaK, and gallium, because the
characteristics of a lithium MHD flow can be simulated well using these liquid metals. As one of the quite
few experimental studies that directly dealt with a lithium MHD flow, heat transfer characteristics was
investigated by Takahashi et al.[22] for a lithium flow in a rectangular channel in the presence of a
transverse uniform magnetic field. It was found in the experiment that heat transfer coefficient increased
appreciably with an increase in the strength of an applied magnetic field. However, such MHD heat
transfer enhancement was not observed except under limited conditions in a similar lithium flow
experiment in a circular channel conducted by Takahashi et al.[23-25],.
According to most of the previous experimental studies on heat transfer characteristics of liquid metal
flows (e.g. [3]), it has been well known that heat transfer deteriorate under a magnetic field primarily due
to laminarization. In some cases, appearance of heat transfer enhancement was reported, e.,g., for a
mercury duct flow in the downstream from grids and obstacles[26,27]. A slight enhancement of heat
transfer was also observed in the mercury flow in a straight circular channel in the presence of a
longitudinal magnetic field up to 6T by Takahashi et al.[28]. However, no heat transfer enhancement
appeared observed in a similar mercury flow experiment using a helically coiled tube under an inclined
magnetic field up to 6T by Takahashi et al.[29].
Among the above mentioned experimental results, the heat transfer enhancement observed in a lithium
flow in a rectangular channel seems important, because the large increasing rates in heat transfer
coefficient with magnetic flux density suggests the existence of a certain heat transfer mechanism that is
contrary to our traditional understanding of heat transfer deterioration due to laminarization in a magnetic
field. One of the possible reasons for the heat transfer enhancement is a strong convection due to sidelayer
jets. Takahashi et al.[30] performed a numerical analysis to simulated the lithium flow, but did not
investigate the effect of the jets on the heat transfer enhancement observed in the lithium flow experiment.
2
The other possible reasons for the MHD heat transfer enhancement are an enhanced mixing by the
fluctuations of the sidelayer jets or the two-dimensional MHD turbulence[26,27]. The turbulence
characteristics were investigated experimentally and numerically for a mercury duct flow by Suzuki et al.
[31], Takahashi and Suzuki[32], and Yokoe and Takahashi[33]. However, the effect of MHD turbulence on
heat transfer enhancement has not been investigated in these studies.
In the present study, the effect of the sidelayer jets on the the enhancement of heat transfer is evaluated
by means of a numerical MHD analysis, and the analytical result is compared with the measured heat
transfer coefficient in the lithium MHD flow in a horizontal rectangular channel reported by Takahashi et
al.[22]. The basic equations and numerical techniques used in the present analysis are not new in
comparison with previous analyses [8-18] except that the electromagnetic field inside conducting walls are
numerically calculated together with those in the flowing fluid in the present analysis.
2. Analysis
2.1 Flow and coordinate systems
Numerical analysis were performed for a lithium flow in an electrically conducting rectangular channel.
The channel height was 5mm and 10mm, the channel width was 40mm, and the wall thickness was 5mm.
The bottom wall was heated uniformly under the conditions of Re =2,500-20,000 and the applied magnetic
flux density of 0-1.4T. The wall conductance ratio c is 0.25.
A schematic of the rectangular channel treated here and the cordinate system are shown in Fig.1. The
channel width is 2a , the channel height is 2b , and the wall thickness is t . The x -axis is perpendicular to
the top and bottom walls, the y -axis is parallel to the top and bottom walls, and the z -axis is in the
channel axis with the origin of the axes located at the center of the channel. A uniform magnetic field with
a magnetic flux density B applied to the lithium flow is in the direction of the y -axis, and the inside
surface of the bottom wall is heated uniformly with a heat flux q . The inside surfaces of the top and side
walls are adiabatic. The top and bottom walls which are parallel to the magnetic field direction are called
“the sidewalls” hereinafter.
2.2 Basic equations
3
Basic equations for a non-compressible, steady, laminar, and conductionless magnetohydrodynamic
lithium flow with no Joule heating are as follows:
Continuity equation
Ñ ×u = 0,
(1)
Momentum equation
ruÑu = -Ñp + J ´ B + mÑ 2 u ,
(2)
Conservation of charge
Ñ×J = 0,
(3)
Ohm’s law
J = s (- Ñf + u ´ B ) ,
(4)
Energy equation
rc u × ÑT = kÑ 2T ,
(5)
where u is the velocity vector,
p the pressure, B the magnetic flux density vector, J the electric
current density vector, T the temperature, r the fluid density,
m the fluid viscosity, s the fluid
electrical conductivity, c the specific heat, and k the thermal conductivity. It is assumed that the flow is
fully developed with uniform pressure and constant physical properties in the cross section of the channel.
Substitutions of Eq. (4) into Eqs.(2) and (3) lead to the following set of equations:
Continuity equation
¶u
=0,
¶z
Momentum equation
ru
Scalar potential equation
Energy equation
æ ¶ 2u ¶ 2u ö
¶u
¶p
¶f
=- sB 0
- suB0 2 + m ç 2 + 2 ÷ ,
ç ¶x
¶z
¶z
¶x
¶y ÷ø
è
¶ 2f
¶x
u
(6)
2
+
¶ 2f
¶y
2
= - B0
¶u
,
¶x
æ ¶ 2T ¶ 2T
¶T
= aç
+
ç ¶x 2 ¶y 2
¶z
è
(7)
(8)
ö
÷,
÷
ø
(9)
where a º k / rc is the thermal diffusivity, and the unknown variables are the velocity component in z direction u , the scalar potential f , the pressure p and the temperature T .
2.3 Boundary conditions
Boundary conditions are as follows:
(a) no slip at the inside surfaces of the walls,
u=0
at x = ±b and y = ± a , (10)
(b) electrical insulation at the outside surfaces of the walls, ¶f ¶x = 0 , at x = ± (b + t ) ,
4
(11)
¶f ¶y = 0 , at y = ± (a + t ) ,
(12)
(c) continuity of normal current at the inside surfaces of the walls,
s (¶f ¶x) f = s w (¶f ¶x) w , at x = ±b (13)
s (¶f ¶y ) f = s w (¶f ¶y ) w , at y = ± a
(14)
(d) uniform heat flux at the inside surface of the bottom wall,
q = - k ¶T ¶x at x = -b,
(15)
(e) thermally adiabatic at the inside surface of the walls,
¶T ¶x = 0 at x = b ,
¶T ¶y = 0 at y = ± a .
(16)
(17)
It is the characteristic feature of this analysis that the electromagnetic field was solved numerically not only
inside the channel but also inside the solid wall.
2.4 Solution procedure
Equations (6)-(9) are discretized into the following algebraic finite-difference equations using mesh
system shown in Fig.2.
ru in, j
u in, +j 1 - u in, j
Dz
ìï ¶p
æ u in-+11, j - 2u in, +j 1 + u in++11, j u in, +j 1-1 - 2u in, +j 1 + u in, +j 1+1 öüï
f in, +j 1 + f in-+11,j
÷
= í- sB 0
- sB0 u in, +j 1 + m ç
+
2
2
ç
÷ý
Dx
Dx
Dy
ïî ¶z
è
øïþ
(18)
f in1,+j1 - 2f in, +j 1 + f in++11,j
Dx 2
ui, j
Tin, j+1 - Tin, j
Dz
+
f in, +j -11 - 2f in, +j 1 + f in, +j +11
Dy 2
= - B0
u in++11, j - u in, +j 1
Dx
æ T n +1 - 2Tin, j+1 + T n +1 T n +1 - 2Tin, j+1 + T n +1 ö
i +1, j
i, j -1
i, j +1 ÷
ç i -1, j
=aç
+
÷
2
2
ç
÷
Dx
Dy
è
ø
(19)
(20)
The equivalent mesh sizes, 40x50 and 200x100, and non-equivalent mesh sizes, 79x118, were used.
Starting from initial values of a parabolic velocity profile, uniform profiles of scalar potential f =0
and temperature T = 350 o C , the equation of scalar potential, Eq.(19), was first solved using the Gauss-
5
Seidel method with 500 iterations. The momentum equation, Eq.(18), was then solved iteratively using
the same method until the velocity variation was less than 1% at all mesh points. The pressure gradient
was modified, and Eq.(18) was solved until the total flow rate agreed with the desired value. If the relative
axial gradients of f and u at all the mesh points were less than 10-4, the solutions were treated as a fully
developed flow and the calculation was proceeded to the solution of the energy equation, Eq.(20), and
otherwise it returned to the calculation of f and u .
Heat transfer coefficient at the inside surface of the bottom wall is defined by
h=
q
,
(Tw - Tb )
(21)
where q is the specified heat flux, Tw is the temperature at the center of the inside surface of the bottom
wall x = -b and y = 0 , and Tb is the mixed average lithium temperature given by
å å u (i, j )T (i, j )
Tb =
i
j
å å u (i, j )
.
(22)
i
The definition of the heat transfer is the same as that in the previous experimental study[22]. The Nusselt
number Nu is defined by
Nu = hd h / l ,
(23)
where d h is the thermally equivalent diameter which is equal to 8b , and l is the thermal concuctivity of
lithium.
3. Result and Discussion
The effect of mesh sizes on calculated velocity profiles in the planes of y =0 is shown in Fig.3(a).
Among the mesh numbers of 40, 100, 200 and 300, the difference in velocity profiles appeared near walls
only between the mesh numbers of 40 and 100. The effect of mesh sizes on calculated velocity profiles in
the planes of x =0 is shown in Fig.3(b). , Among the mesh numbers of 50, 100 and 200, the difference in
velocity profiles appeared near walls only between the mesh numbers of 50 and 100. Thus, the mesh
numbers of 200x100 were adopted for the present calculations.
6
Figures 4(a) and (b) show the calculated velocity profile at the Reynolds number Re=2,500 and the
Hartmann numbers M =135 and 1,900, respectively. It is found that there appear a core region with low
velocity and sidelayers with high velocity adjacent to the walls parallel to the direction of an applied
magnetic field. An example of the calculated electric current vectors in the channel cross section is shown
in Fig.5, which is reasonable from electromagnetic point of view.
The calculated results for skin friction coefficient C f are plotted against the interaction parameter
N º M 2 / Re with the parameter of the Reynolds number Re in the range of M from 271 to 1900 in
Fig.6. It is found that the present numerical results agree well with the solid line given by the Miyazaki et
al.’s simplified analytical model[34]: C f = k p M 2 / 2 Re , where k p º c /(1 + a / 3b + c) for a rectangular
channel, and k p =0.097 in the present case of a = 0.02 m, b = 0.005 , and c = 0.25 .
Figures 7(a) and (b) show the velocity profiles in the planes of y =0 and in the planes of x =0 with the
parameter of magnetic flux density, respectively. It is found that the velocities in the sidelayers increase
significantly and the peak positions approach the walls in the planes of y =0 with an increase in the
magnetic flux density. On the other hand, the flattened velocity profile known as the Hartmann flow in the
planes of x =0 does not change appreciably with an increase in the magnetic flux density from 0.2T to
1.4T. According to the resutl for the ratio of the peak velocity to the average velocity, the ratio increased
greatly with increasing the Hartmann number from 500 to 1,000, while the increasing rate decreased over
the Hartmann number of 1,000.
Figure
8
shows
the
result
for
the
contours
of
calculated
dimensionless
temperature
q (º (T - Tmin ) (Tmax - Tmin )) in the channel cross section. at Re =20000 and M =135. It is found that
heating wall temperature was higher in the region of y / a > 0.5 than in the region of y / a < 0.5 , which
implies that the convective heat transport by the jets in the sidelayers was stronger in the region
y / a < 0.5 than in the region
y / a > 0.5 . In other words, convective heat transport by the jets in the
sidelayers had appreciable influence on the heating surface temperature.
Figure 9 shows the effect of the Hartmann number M on the Nusselt number Nu with the parameter of
the Reynolds number. Since the condition of laminarization is M / Re > 1 / 225 , the result for the Nusselt
number is valid in the range of M > 11 at Re =2500, M > 44 at Re =10000, and M > 88 at Re =20000
7
in the present analysis with the assumption of a laminar flow. In the turbulent flow under a very weak
magnetic field, the Nusselt number of a liquid metal duct flow depends upon the Peclet number
Pe º Re× Pr , e.g. Nu = 7 + 0.025 Pe 0.8 , but in a laminar flow it becomes constant. It is expected that
under a strong magnetic field the Nusselt number may be primarily controlled by the Hartmann number
M through the change in a velocity profile near a heating wall, and that it does not depend on the
Reynolds number so much. It is found that the numerical result for the Nusselt number did not depend on
the Reynolds number so much, and that the Nusselt number increased by 42%-50% with increasing the
Hartmann number from 0 up to 1,900. One of the reason why the Nusselt number increased with the
Hartmann number may be the thinning of viscous sublayer near the heating surface. However, in the flows
of low Prandtl number fluids such as liquid metals with high thermal conductivity, the thickness of a
thermal boundary layer is much thicker than the viscous sublayer, and as a result the thickness of the
viscous sublayer does not have a large influence on the heat transfer coefficient. Since the sidelayer is
thinner than the thermal boundary layer as shown from the comparison of the velocity profiles in Fig.7(a)
and the temperature profile in Fig.8, the convective heat transfer by the sidelayer jets possibly has larger
influence on the heat transfer coefficient, or the Nusselt number. Therefore, the result in Fig.9
demonstrates that the application of a magnetic field enhances the convective cooling by the jet generated
in the sidelayer considerably.
The experimental result for the Nusselt number of a lithium flow in a fully developed region in the
channel of a =0.02m and b =0.005m obtained by Takahashi et al.[22] is shown in Fig.10. The flow was
treated as a fully developed one, since the temperature measuring position was at z = 929mm from the
leading edge of the magnet, and the length of 929mm, i.e. z / d e = 58 and z / d h = 23 , is long enough for
an entrance length. Due to some scattering in the experimental result, it is obvious that there were
disagreement in some dgree between the calculated Nusselt numbers in Fig.9 and the measured ones in
Fig.10, but the discrepancies were not large, and they agreed reasonably well with each other.
In order to compare the effect of the Hartmann number on the Nusselt number between the experiment
and analysis neatly, the Nusselt numbers are normalized by those obtained in the absence of a magnetic
field.The calculated normalized Nusselt numbers are compared with the measured ones in Fig.11. It is
found that the calculated Nusselt number increased with the Hartmann number qualitatively in a similar
way as those of the measured Nusselt number[22]. It is noteworthy that the calculated Nusselt numbers did
8
not depend on the Reynolds number so much, but that the dependence of the measured Nusselt numbers
on the Hartmann number changed significantly depending on the Reynolds number. For instances, the
Nusselt number increased more with the Hartmann number in the range of the higher Reynolds number:
Re ³ 7,400 , but it did not increase so much with the Hartmann number in the range of the low Reynolds
number Re £ 4,900 . Therefore, through the present numerical laminar flow analysis, heat transfer
enhancement could be explained qualitatively, but more sophisticated analytical model that takes into
account the other factors such as MHD turbulence may be required for quantitative explanation for this
dependence of the Nusselt number on the Reynolds number observed in the experimental study[22].
4. Conclusions
A laminar lithium flow in a conducting rectangular channel in the presence of a transverse magnetic
field was analyzed numerically, and the following conclusions were obtained:
(1) The jets appeared adjacent to side walls which were parallel to the direction of an applied magnetic
field. The ratio of peak velocity in the jets to average velocity increased with an increase in the
Hartmann number, and reached about six at the Hartmann number of 1,900.
(2) The skin friction coefficient agreed well with the result of the Miyazaki et al.’s simplified analytical
model[34]: C f = k p M 2 / 2 Re, with k p º c /(1 + a / 3b + c) = 0.097 for c =0.25.
(3) The temperature was lower in the center of a heating surface than near the sides on the inside surface of
the heating wall. This suggests that the sidelayer jet has a large cooling effect for the sidewall heating
through the convective heat transfer.
(4) The Nusselt number increased with an increase in the Hartmann number appreciably possibly due to
the cooling effect of the sidelayer jet. The Nusselt number increased by 42%-50% at the Hartmann
number of 1,900 compared with that at zero field. The Nusselt number depended on the Reynolds
number only slightly in the calculated result.
(5) The increase in the Nusselt number with the Hartmann number agreed qualitatively well with
experimental result obtained by Takahashi et al.[22]. The dependency of the Nusselt number on the
Reynolds number in the experimental result could not be simulated by the present analysis. This seems
to be one of impotant analytical work in the future.
9
Nomenclatures
B : Magnetic flux density, T
B : Magnetic flux density vector, T
C f : Skin friction coefficient, C f º (DP / Dz ) d e / 2 ru 2
c : Specific heat, J/kgK
c : Wall conductance ratio, c = s w t / sa
d e : Hydraulic diameter, d e = 2ab /( a + b) , m
d h : Thermal-hydraulic diameter, d h = 8b , m
h : Heat transfer coefficient, h = q (Tw - Tb ) , W/m2K
J : Electric current density vector, A/m2
k p : Constant, k p = c /(1 + a / 3b + c)
M : Hartmann number, M º de B s m
N : Interaction parameter, N º M 2 Re = sd e B 2 ru
Nu : Nusselt number, Nu º hd h l
p : Pressure, Pa
q : Heat flux, W/m2
Re : Reynolds number, Re º ude n
T : Temperature, K
Tb : Lithium bulk temparature,
Tb = å å u (i, j )T (i, j ) å å u (i, j ) , K
i j
i j
Tw : Bottom wall temperature, K
u : Axial velocity, m/s
u : Velocity vector, m/s
Greek symbols
a : Thermal diffusivity, m2/s
f : Scalar potential, V
10
l : Thermal conductivity of lithium, W/mK
m : Viscosity of lithium, Pa × s
n : Kinematic viscosity of lithium, m2/s
q : Dimensionless temperature, q º (T - Tmin ) (Tmax - Tmin )
r : Fluid density, kg/m3
s : Fluid electrical conductivity, W -1m -1
Suffices
f: Fluid
i: Mesh point in x -direction
j: Mesh point in y -direction
max: Maximum temperature
min: Minimum temperature
n: Step in z -direction
w: Wall
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[34]
K.Miyazaki, S.Inoue, N.Yamaoka, T.Horiba and K.Yokomizo, Magnetohydrodynamic pressure
drop of lithium flow in rectangular ducts, Fusion Tech. 10 (1986) 830-836.
Figures
Fig.1 Rectangular channel and coordinate system
Fig.2 Computational mesh
Fig.3 (a) Effect of mesh sizes on calculated velocity profiles in the plane y =0
Fig.3 (b) Effect of mesh sizes on calculated velocity profiles in the plane x =0
Fig.4(a) Calculated velocity profile at Re=2500 and M=135
Fig.4(b) Calculated velocity profile at Re=2500 and M=1900
Fig.5 Calculated electric current distribution at M=135 and Re=10000
Fig.6 Skin friction coefficient
Fig.7(a) Effect of magnetic field on velocity profile at Re=2500 in the plane y =0
Fig.7(b) Effect of magnetic field on velocity profile Re=2500 in the plane x =0
Fig.8 Contours of dimensionless temperature q at Re =20000 and M =135
Fig.9 Calculated result for Nusselt number
Fig.10 Measured Nusselt numbers in a lithium duct flow[22]
Fig.11 Comparison of calculated Nusselt numbers with measured ones
14
t
t
B
Li
2a
2b
t
x
z
q
y
Fig.1 Rectangular channel and coordinate system
B
x
u i+1,j
u i+1,j+1
T i+1,j
T i+1,j+1
fi,j
fi,j+ 1
u i,j
u i,j+1
T i,j
T i,j+1
y
Fig.2 Computational mesh
15
(m /s)
1
Mes h
N um ber
300
200
100
40
0.8
u
0.6
Size D x
0.066m m
0.1m m
0.2m m
0.5m m
R e= 2500
M= 1900
0.4
0.2
0
-1
-0.5
0
0.5
1
x/b
Fig.3 (a) Effect of mesh sizes on calculated velocity profiles in the plane y =0
(m/s)
0.1
Re=2500
M=1900
0.08
u
0.06
0.04
Mesh
Num ber Size D y
200
0.25m m
100
0.5 m m
50
1 mm
0.02
0
-1
-0.5
0
0.5
1
y/a
Fig.3 (b) Effect of mesh sizes on calculated velocity profiles in the plane x =0
16
0.3
0.2
u (m/s)
0.1
0
-1
y/a
-1
x/b
-1
Fig.4(a) Calculated velocity profile at Re=2500 and M=135
17
-1
0.8
0.6
u(m/s) 0.4
0.2
1
0-1
x/b
1
-1
Fig.4(b) Calculated velocity profile at Re=2500 and M=1900
18
y/a
B
x
j=10 4 (A /m 2 )
y
Fig.5 Calculated electric current distribution at M=135 and Re=10000
10
2
Cf
10 1
M=2.71x10 2 -1.90x10 3
Re
2.5x10 3
5.0
7.5
1.00x10 4
1.25
1.50
1.75
2.00
10 0
C f=k p M 2 /2Re
k p =c/(1+a/3b+c)
=0.097
c=0.25
10 -1 0
10
10 1
10 2
M 2 /Re
Fig.6 Skin friction coefficient
19
10 3
1
Plane y=0
Re=2500
0.8
u (m /s)
M =1.4T
0.6
0.6T
0.4
0T
0.2T
0.2
0
-1
-0.5
0
x/b
0.5
1
Fig.7(a) Effect of magnetic field on velocity profile at Re=2500 in the plane y =0
0.3
Plane x=0
Re=2500
M=0T
u (m/s)
0.2
0.1
1.0T
0.2T
1.4T
0.6T
0
-1
-0.5
0
y/a
0.5
1
Fig.7(b) Effect of magnetic field on velocity profile Re=2500 in the plane x =0
20
0.2
B
x
0.2
0.4
0.4
0.6
0.6
0.8
0.8
0.8
0.8
y
q
Fig.8 Contours of dimensionless temperature q at Re =20000 and M =135
18
Num erical analysis
Nu
16
Re
2.5x10 3
5.0
1.0x10 4
1.5
2.0
14
12
10
0
500
1000
1500
M
Fig.9 Calculated result for Nusselt number
21
50
a=0.02m , b=0.005m
40
Nu
30
Re e
3660
4900
7400
9900
14800
22100
z=0.929m ,
z/d e =58.1, z/d h =23.2
500
1000
20
10
0
1500
2000
M
Fig.10 Measured Nusselt numbers in a lithium duct flow[22]
N u B /N u B =0
2
N um erical analysis
1.5
1
R e e = 3.66x10 3
4.90
7.40
9.90
1.48x104
2.21
0.5
0
500
1000
1500
2000
M
Fig.11 Comparison of calculated Nusselt numbers with measured ones
22