Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Applications of the point-matching method in
coupled convection & conduction heat transfer
H. Barrow/ E. Buyruk,^ I. Owen*
"Dept. of Mechanical Engineering, The University of Liverpool, PO
Box 147, Liverpool L69 3BX, UK
*Dept. of Mechanical Engineering, The University ofCumhuriyet,
Sivas, Turkey
Abstract
In this paper, the point-matching method is used for the solution of conjugated
convection and conduction heat transfer, with special reference to the problem
of tubes in crossflow. In particular, the case of a non-uniformly fouled tube is
considered and the results are compared with those obtained previously using
a finite-difference scheme. In view of the variation of the thermal parameters
along the interface in the general case, the meaning of "average heat transfer
rate" for the surface is critically examined.
1 Introduction
Steady conjugated convection and conduction heat transfer occurs in many
engineering processes and the underlying principles are well-known and fully
documented. The essential feature of this coupling between the energy transfers
at a fluid-solid interface is the balance between the convected energy in the
fluid with the thermal conduction in the solid at all points along the interface.
This is the so-called Newton boundary condition and, of course, may occur in
conjunction with specified temperature (Dirichlet) and specified heat-flux
(Neuman) conditions over other pans of the boundary of the conduction
domain. The general two-dimensional situation is depicted diagrammatically
in Fig.l(a), while Fig.l(b) shows the more familiar one dimensional counterpart, viz. the plane wall, for which a simple analytical solution is available. A
typical two-dimensional engineering problem is also given in Fig.l(c) and
refers to a hot gas in flow across a tube with variable thickness fouling. This
case will be considered in detail later. Now the general problem is the
determination of the temperature field and hence heat transfer, and while this
may be effected analytically in many cases, it is frequently necessary to resort
to finite difference or finite element methods. The problem displayed in
Fig.l(c), for example, has been solved by a finite difference method by the
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
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Advanced Computational Methods in Heat Transfer
present authors, [1], using the bi-polar co-ordinate system. Mixed Dirichlet and
Newton boundary conditions were imposed in keeping with realistic conditions
for this situation. On the inner boundary it was assumed that the temperature
was known, and that the peripheral variation of the heat transfer coefficient h
was also known on the outer surface. It is clear that the finite difference
method suffices with simple geometries but that alternative methods may be
required when more irregular shapes are encountered. In this connection the
work of Cheng, Inaba&Gilpin, [2] is particularly valuable, since they employed
a point-matching method in their theoretical considerations of the energy
transfers in a variable thickness ice-layer on a tube in cross flow. The
essentials of the point- matching procedure are best understood by considering
first the simple heat transfer problem of the plane wall shown in Fig.l(b).
2 Point-matching method
With the mixed Dirichlet and Newton boundary conditions as indicated, and
in the notation shown in Fig.l(b), integration yields:
t=ax+b
(where a,b constants)
(1)
with, (i) t=tj at x=0
, the Dirichlet condition
and , (ii) -k(dt/dx), = h (t-t*J at x=l, The Newton condition
Here the matching aspect of the problem is basically the fitting of the
boundary conditions. In this very simple illustration this presents no difficulty
and results in the familiar linear relationship for the temperature in the solid.
In effect, the relative magnitudes of the internal and external resistances dictate
the value of the uniform temperature on the Newton boundary. However, had
h been a point function, then it would become necessary to effect this
matching at a sufficient number of variable temperature points along the
Newton boundary. Of course the relevant starting point is then the Laplacian
V4(x,y)=0 and the point matching involves the evaluation of the integration
constants of its general solution. This will become more evident as particular
problems are considered. The important point to make at this stage is that in
the point-matching method, the general solution for the temperature is first
obtained and then the constants are determined from a set of linear equations
which result from the imposition of the boundary conditions at a number of
points.
To investigate the method further the problem described in reference [1]
will be considered since the finite-difference results from that enquiry may
then serve as a yardstick for comparison. Furthermore, as shown in Fig.l(c),
realistic mixed boundary conditions have been employed and so the usefulness
of the method in these circumstances may also be studied. It is to be noted that
the work of Cheng et al. [2], pertain to the Dirichlet problem consistent with
known temperatures on the interfaces and the tube and that their objective was
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Advanced Computational Methods in Heat Transfer
35
the determination of the local external heat transfer coefficient.
In the present paper the converse problem is first explored, the variation
of h around the external interface being prescribed and the temperature
determined. The main steps in the calculation procedure are now outlined.
3 Crossflow with non-uniform fouling
In the notation listed in Fig.2. and using polar coordinates, then following
Ref.[2], it is easy to show that
\ % a* (R "-R -") Cos (n6)
where T=
t-t
* ; R=(r/rJ
t« E
This incorporates the Dirichlet boundary condition, viz. T=0. Now
Newton boundary, where h is known,
on the
Accordingly, equation (3) may be applied at selected points around
N. Furthermore in the present case, the normal gradient (3T/3n) may be related
analytically to the radial and peripheral temperature gradients through the
eccentric annular geometry. Finally combining equations (2) and (3) results in:
(4)
(Cos (n6) + 8 n - n s n n ) + kCos(-a) (^n_i+^_n_i) ^s (n6) ]
Equation (4) is now used at a number of peripheral locations and its solution
for bo and the values of a^ effected numerically. The temperature T is then
determined from equation (2). Typical temperature distributions are shown in
Fig.2 where it is seen that there is good agreement with the earlier prediction
using the finite difference formulation in bi-polar coordinates, as described in
Ref.[l]. If required, the total heat transfer may be calculated by
integration.This of course should equal the conjugated convection heat transfer
calculated using the local heat transfer coefficients and the corresponding
external temperature differences.
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
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Advanced Computational Methods in Heat Transfer
4 Average heat transfer coefficient, h
A useful engineering parameter is the average heat transfer coefficient h and
it is of paramount importance that its meaning is fully appreciated, h may be
defined in two ways:
H
ifhde
7W
;
H
*
fh(t-t.
7t(t-tJJ
the former being the true mathematical average value. The latter value is based
on overall heat transfer and average temperature difference. For the problem
studied in the previous section, these have been evaluated as 41.33 W/m K and
43.12 W/nfK respectively when the eccentricity is 0.002m. The interesting
point here is that these values are different, and this is in accord with the
simple analysis of average heat transfer coefficient presented earlier [3].
5 Newton boundary conditions
In the most general case, Newton boundary conditions with different
convective heat transfer coefficients over parts of the boundary occur. If the
same geometry as employed in Fig.l(c) is considered but now with an internal
fluid at temperature t% and convective coefficient h,, then the following
relationships corresponding to these given in the section dealing with crossflow
with non-uniform fouling pertain:
(5)
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Advanced Computational Methods in Heat Transfer
37
After some algebra, the following equations result. On the outer boundary :
a ( kCos(6-*)Cos(n6) (nR/'i) kSin(6-a) Sin(n0) (nR ")
"^
h^*r,R,h,
'
+R
/ kCos(6-*)Cos(n6) (nty"-*) kSin(6-a)Sin(n6) (nR -«)
"^
--^-
while for the inner boundary,
(9)
Of course if h, is very large, then t,->t, and equation (9) reduces to :
(10)
This corresponds exactly with the result for the internal Dirichlet condition of
the earlier analysis. Furthermore, if h, is very large, then from (8)
oo
l=b,,lnR2+£ a, (Rj»-Rj-») Cos (nO)
(11)
0
which (remembering that t^-^tj is in agreement with equation (2). In other
words, the result for the most general analysis is in accord with that for mixed
Dirichlet & Newton boundary conditions.
6 The concentric tube geometry
It is easy to demonstrate that the results for the eccentric annular geometry
concur with the simple axisymmetric case. Of special interest however is the
case of the axisymmetric geometry with h varying peripherally. This problem
is frequently encountered in the laboratory. The situation is shown
diagrammatically in Fig.3. Now the local heat transfer coefficient distribution
may be determined in different ways according to the accuracy required Most
simply,
27ir,(t-t
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
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Advanced Computational Methods in Heat Transfer
where t is the local temperature. This assumes that the surface heat flux is
uniform. Another method recognises the peripheral heat conduction but
assumes that t is a function of 0 only, when:
h=
h may then be determined numerically from the measured power input Q and
the curvature of the peripheral temperature profile. Finally the full calculation
using the previously described point-matching method may be employed.
Again, beginning with the Laplacian in polar coordinates, the boundary
conditions need to be imposed on both surfaces. The external temperatures are
determined from experiment for a given experimental power input. Initially the
inner value of temperature is unknown and so an arbitrary figure is chosen to
effect the calculation. Proceeding in the usual way, the values of h are
determined and calculated heat transfer is then adjusted until agreement with
the experimental power is achieved.
To demonstrate this procedure, a fictitious temperature profile has been
chosen for a concentric geometry and h has been calculated following the three
methods outlined above.These distributions of h are shown plotted in Fig.4.
The mathematical average values according to the three different methods of
calculation are 36.47 W/nrK , 36.45 W/nrK and 41.91 W/nrK, respectively.
The difference between the values according to equations (12) and (13) is
negligible an account of the small variation of the chosen surface temperature
distribution. However, the value obtained using the point-matching method is
significantly different indicating that caution needs to be exercised in
determining average heat transfer rates in experiments.
7 Discussion & Conclusions
Choosing turbulent forced-convection heat transfer in crossflow over a circular
tube as a model, the point-matching method in two dimensions has been used
to determine the temperature in the conjugated heat conduction domain.For the
special case of non-uniform fouling on the outer surface of the tube, the results
have been compared with those obtained using a finite-difference procedure of
an earlier study. In particular, two fouling geometries simulated by eccentric
annuli have been examined with mixed boundary conditions appropriate to a
real situation. In the point-matching method the polar co-ordinate system was
used, while in the finite-difference analysis, bipolar coordinates were chosen
to discretize the two-dimensional heat conduction domain.
Typical results for the temperature around the external surface of the
fouled-tube in crossflow are in good agreement and, as expected, the larger
difference occurs in the case of the geometry with the larger eccentricity. Of
special interest is the axisymmetric geometry with peripherally varying heat
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Advanced Computational Methods in Heat Transfer
39
transfer rate. Here the point-matching method has been used as a yard-stick to
investigate the accuracy of various approximate methods for calculating the
local heat transfer coefficients from experimentally measured surface
temperatures. This part of the investigation shows that considerable error may
be involved if the full two-dimensional nature of the heat conduction in the
wall is not modelled correctly.
The average heat transfer coefficient in this situation has also been
carefully examined. In this connection, distinction needs to be made between
the true mathematical average of the local values of heat transfer coefficient
along the interface, and an average value based on the quotient of the average
surface heat flux and average surface to fluid temperature difference. The
present numerical results confirm that, in general, there is a difference between
these two values. The difference, in turn, depends on the gradients of both heat
flux and temperature difference along the surface.
Of course, the point-matching method is not restricted to the solution
of the temperature field in heat conduction. It has been used by Cheng et
al.[4], to solve fully-developed laminar forced convection in an eccentric
annular channel with uniform heat sources and uniform heat fluxes at both
boundaries. There is, of course, no reason why other cases should not be
studied, and such an investigation might be a useful extension in the present
area of study.
References
1. Owen, I., Buyruk, E., & Barrow, H. Theoretical analysis of the thermal
resistance of non-uniform fouling on crossflow heat exchanger tubes, Applied
Thermal Engineering, 1996,16,43-50.
2. Cheng,K.C, Hideo Inaba, & Gilpin, R.R. An experimental investigation of
ice formation around an isothermally cooled cylinder in crossflow,
Trans.ASME, Journal of Heat Transfer, 1981,103,733-738.
3. Barrow,H. On average heat transfer coefficient, Int.J.Heat & Fluid Flow
1986,7,162-163.
4. Cheng,K.C. & Hwang,GJ. Laminar forced convection in eccentric annuli
AIChemE Journal,1968,14,510-512.
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
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Advanced Computational Methods in Heat Transfer
D
(b) 1 -d Case
(Plane Wall)
N
(ft _g
>>. T
( ^
1
X
eccentricity
N
(c) Typical 2-d example
(fouled tube in crossflow)
Figure 1. Examples in Conjugated Conduction&Convection Heat Transfer
with Mixed Boundary Conditions
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
Advanced Computational Methods in Heat Transfer
320"
" Point Matching
+ Bipolar f.d.,(1)
310,-f "
300
h-f(oc)
eccentricjty=1e-4m
rg=0.011 m
r, =0.008 m
cX
<D
OB
i_
0}
290
280
310
eccentricity=0.002 m
4-
300
-t '
E
.CD
4-.
290
280
0
20
40
60
80
100
Angle,o<
120
140
160
Figure 2. The Fouled Tube in Crossflow
180
41
Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
42
Advanced Computational Methods in Heat Transfer
Radius, r%
Thickness,&
Thermal Cond.k
Internal heat source
Q/unit length
Figure 3. Crossflow Over a Constant Thickness Tube
80
- Case 1,Equ.(12)
( *
3£
+ Case 2,Equ.(13)
*
^ From Expt. Temp. Values
for air, Re=9600
60
(
)
)
, *'
)K
40
- + Hh + Hr- -f ^^ 4- -K 4- -f- 4- Hf- -f Hh $ Hh 4- X
K
( )K
)
K ¥ ^
20
0
20
40
60
80
100
Angle Q
120
140
160
180
Figure 4. Heat Transfer Coefficient Distribution