Inverse determination of steady-state heat transfer coefficient

Vol. 27, No. 8, pp. 1155-1164, 24)00 Copyright © 2000 ElsevierScience Ltd Printed in the USA. All rights reserved 0735-1933/00IS-see front matter

Int. Comm. H e a t M a s s Transfer,

Pergamon

PII: S0735-1933(00)00202-5

INVERSE D E T E R M I N A T I O N OF STEADY-STATE H E A T T R A N S F E R COEFFICIENT

S. Chantasiriwan Faculty of Engineering Thammasat University, Rangsit Campus Pathum Thani 12121, Thailand

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT An experimental setup for the inverse determination of steady-state heat transfer coefficient in a two-dimensional system is proposed. This setup requires temperature measurements at accessible and non-intrusive locations, but does not require heat flux measurement. The mathematical model of the system is solved numerically by using the boundary element method. It is shown that, as a result of uncertainty in temperature measurement, the estimated heat transfer coefficient will contain uncertainty. This uncertainty can be reduced by varying experimental parameters, some of which are discussed. © 2000 Elsevier Science Ltd

Introduction The Newton's law of cooling requires the knowledge of temperature and heat flux at the solidfluid interface for the determination of heat transfer coefficient. Direct techniques for measuring these quantities require either the insertion of sensors through the solid, which may presents technical difficulties, or the insertion of sensors through the fluid, which will disturb the flow field, causing experimental error. An indirect technique for determining heat transfer coefficient makes use of the analogy between convective mass transfer and convective heat transfer [1]. However, such a technique may result in considerable uncertainty in the value of determined heat transfer coefficient. There exists an alternative approach to determining heat transfer coefficient, which requires only temperature measurements. However, it also requires the solution of a corresponding inverse heat conduction problem. Although this approach is more computationally intensive than conventional approaches, it may have an advantage in a simpler setup and less expensive equipment. Previous works on the inverse problem of determining heat transfer coefficient have appeared in the literature [2-7]. In this paper, the problem of determining heat transfer coefficient in a two-dimensional system by 1155

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the inverse method is considered. In the spirit of the previous work by Martin and Dulikravich [6], a steady-state system in which part of the boundary is specified with known temperature distribution, part of the boundary is over-specified with both known temperature distribution and known heat flux distribution, and the other part of the boundary is unspecified will be investigated. However, the system under consideration is a little different from that considered by Martin and Dulikravich, and is designed to resemble the actual system found in an experiment. The numerical method used to solve the problem is the boundary element method. It will be shown that the solution will contain uncertainty as a result of uncertainty in temperature measurements. Factors influencing the solution will be discussed to provide guidance for an experimental design.

Boundary Element Formulation

F3 F

F2 FIG. 1 Two-dimensional source-less system

Consider the linear steady-state heat conduction in a source-less two-dimensional system shown in Fig. 1. The boundary F is divided into F~, F2, and F3. Let the temperature distributions on Fj and F2 be known. In addition, the heat flux distribution along F2 is also known. However, both the temperature and heat flux distributions on F3 and the heat flux distribution

on

F l are unknown. This system can be de-

scribed by the Laplace equation:

v2r(~)

=

o

(1~

T( Ye F, )

=

T0)(?)

(2)

T(;eF2)

=

T(2)(;)

(3)

=

q(~t(~)

(4~

and the boundary equations:

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Despite F3 being an unknown boundary, this system has a unique solution. This means that temperature and heat flux distributions on F3 can be inversely solved for if functions T(1)(~), T(2)(~), and q(2)(~-) are given. The heat transfer coefficient on F3 can then be calculated from the Newton's law of cooling. q(3)

(5) Instead of seeking the analytical solution, this paper will focus on the numerical solution based on the boundary element method, of which formulation for this problem is given by [8]

aT((,t)

=

a--if-,fq(~)G(F~- - ] d Y

S T(Y)~VG(Y - ()d~

P

r

(6)

where a is a coefficient that depends on the location of ~, and the fundamental solution G is lln(

1

/

Divide boundary F into elements, and make use of interpolating functions to approximate q and T. Equation (6) is now written in the following discretized form. m

E Rij qi

rn

-

Z S~ 7],.

i-I

=

0

for 1 _
(8)

i=1

which may be written as the following matrix equation: R~-

S~?

=

0

(9)

In order to determine heat flux and temperatures on F3, it is expedient to rewrite Eq. (9) in terms of heat flux and temperatures on F~, F2, and F3. R(H) ~(1) _}_ R(,2)~(2) + R03)~(3) _ S(,0 ~0) _ S(~2)~(2) -- 8(13) ~(3) R(20 ~0)

+

R(22)~(2) +

R(3~)q (~) + R(32)q p)

+

R(23)~(3)

_

R(33)q (3) -

S(21) f(l)

_

S(31)T (1) -

=

0

(10)

_

S(23) ~(3)

=

0

(ll)

S(32)T (2) -

S(33)T (3)

=

0

(12)

S(22) ~(2)

Rearrange the above equations so that known quantities appear on the right-hand side, and unknown quantities appear on the left-hand side. The result is

(13) Equation (13) can be solved if the number of unknowns on the left-hand side is less than or equal to the

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S. Chantasiriwan

Vol. 27, No. 8

total number of nodes. The result is expressions of heat flux and temperature on F3 in terms of temperatures on F~ and temperatures and heat flux on F2. ml

q}3)

=

Ti(3)

m2

m2

/=1

/=1

j=l

m I

rn 2

m 2

j=l

j=l

j=l

Z

aoT}')+Z b~Tj(2) + ~'~-'Cof4j~(2) ~

ZdiiT}')+ Ze!,~(2) + Z Z ,

=

(14)

ql 2)

(15)

Sample Experimental Setup Figure 2 shows a possible experimental setup for determining heat transfer coefficient in a forced convective flow over a flat plate. The solid is assumed to have a constant thermal conductivity over the range of temperatures experienced inside the solid during the experiment. Note that the dimension of the solid in the direction normal to the page is very long compared with its length L and width W. The left side and the right side of the solid are perfectly insulated. The bottom side is subjected to heat flux that varies only along the x-direction. In addition, the top side is exposed to air flow, of which velocity distribution does not vary in the direction normal to the page. As a result of these idealizations, the system can be considered to be two-dimensional.

I<

L

Insulation

l

Air flow with ambient temperature T~ ) Insulation

@ Solid of known thermal conductivity k

;

o x

-

~

O

O@O

o

O

o

o

Unknown heat flux qO)

FIG. 2 Experimental setup for determining heat transfer coefficient in a forced convective flow over a flat plate

Let Fi denote the bottom side, F2 denote the left and the right sides, and 1"3denote the top side. In solving a similar problem of determining heat transfer coefficient on 1"3, Martin and Dulikravich [6] considered two cases. First, heat flux and temperature on Ft and F2 were known. Second, heat flux and temperature on 1"~ were known, and only temperature on F2 was known. In the present study, a slightly different case will be considered. Heat flux on 1"2 is insulated, and temperatures on 1"1 and 1"2 are known. The

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reason for considering this case is that it seems relatively easy to be implemented in experiment as only temperature measurements on F~ and F2 are required. For this purpose, a certain number of temperature sensors are placed along F1 and F2. Figure 2 shows one possible arrangement of 19 sensors (denoted by little white circles). It should be noted that the division of boundary into elements will require more elements and, consequently, more boundary nodes than 19 sensors so that an acceptable solution will be obtained. In order to supply boundary temperatures at all nodes on F~ and F2, linear interpolations of measured temperatures may be performed. This experimental setup appears to offer practical advantages in that there is no disturbance of the flow field caused by intrusive sensors, and that the locations of sensors are quite accessible. Furthermore, no heat flux measurement is required although the heat flux on F2 must be negligible through the use of a good insulating material. More importantly, the knowledge of the heat flux applied at F~ is not necessary. Thus, the experimental will have the freedom to choose the heat source. However, the applied heat flux should not be rapidly varying along the x-direction lest many temperature sensors on F~ will be needed to gain a reasonably accurate knowledge of temperature distribution on F1.

Numerical Results and Discussion

3000 2500 h (3)

2000

(W/m2-°C) 1500 1000 500 0

w

0

0.05

0.1

0.15

0,2

x (m) FIG. 3 Distribution of exact (solid line) and estimated (white circles) heat transfer coefficient along 1"3

In order to test the algorithm for inverse determination of heat transfer coefficient, let's impose heat transfer coefficient at F3 and heat flux at F1 as follows. h(3)(x) =

2 0 0 0 - 250~rcos(5nx)tanh(0.5z)

(16)

2 + cos(5,=)

q°)(x)

=

20000

(17)

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For W 0.1 m, L = 0.2 m, T~, = 30 °C, and k = 50 W/m2-°C, the exact solution is TM

, 1, ,nr cos(5nx)cosh(0.5ny)q

r(x,y) =

L

j

Temperature measurements from the sensors located at specified locations

(18) on F 1

and F2 can be

assumed to correspond to Eq. (18), and used as input for the inverse algorithm. Figure 3 shows the result of the inverse determination of heat transfer coefficient at 5 locations along F~. Comparison between this result and the exact solution reveals a good agreement. Even when there are no experimental errors, the fact that temperature reading is displayed to only a few decimal points means that there is always uncertainty in an actual temperature measurement. For instance, a reading of 66.7 °C implies an uncertainty of 0.1 °C because the actual temperature is between 66.65 °C and 66.75 °C, In general, if all temperature sensors have an uncertainty of ~, or equivalently, ~ 0 ) _ 0.56"

<

7~0)

<

~0)+ 0.5~"

(19)

T,(2) - 0.5 a"

<

T,(2)

<

~(2) + 0.5 c

(20)

and heat flux on F2 is accurately known, it follows from Eqs. (14) and (15) that ~ ( 3 ) 0.5zXq~3)

<

ql 3)

<

~(')+ 0.5Aq} 3)

(21)

~(~)- 0.5AT, 0)

<

T (3)

<

T,(3) + 0.5AT~(3)

(22)

where mt

~i(3)

Ti(')

=

=

__

m2

m2

Z a , j V ) ') + l...,~'bif(2), , + Z c,).q{,2) j=l

/=1

j=l

ml

m2

m~

Z dog(') + Z e , i g (2) + ia..,, V / " 0"4 _(2) j=l

/=1

Aq}3) =

a¢ + •=

(24)

/=1

bo ~

(25)

e o. ~

(26)

=

d~i +

k / ; t3) =

(23)

=

The resulting value of estimated h}3) will range from (h} 3))m., and (h! 3))max with

(h{3))min

=

~(3)[

(h},i)m,.

=

¢(3t[

1 -- 0"5 Aq'3)/q,(3) 1 1+ 0.S AT,(3)/(~ (3) - T~) 1+ 0.5 Aq}3)/V,!3) 1 1 - 0.5 zXTi(3)/(g O) - T~ )J

(27)

(28)

For the same parameters as used to obtain the result shown in Fig. 3 and a uniform uncertainty in tem-

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perature measurement of 0.1 °C, Table 1 displays exact h (3) and the range of estimated h (3) at each of the 7 equally spaced locations on F3. Let's define uncertainty in estimated h}3) as Ah}3)

=

(h~3))max- (hf3))~.

(29)

TABLE 1 Table 1 Exact h (3) and Ran es of Estimated h (3) at 7 Locations on I'3 Exact h (3) Range ofestimated h (3) 427 395-470 0.033 481 388-528 0.067 656 530- 725 0.1 1000 921-1087 0.133 1573 1435- 1906 0.167 2314 2189- 2710 0.2 2720 2598-2916 In general, it is desirable to minimize this uncertainty. If the fluid conditions are kept constant in the sample experimental setup, it is interesting to explore how the experiment can be designed to minimize Ah~3) . Assume that, due to experimental constraints, the number of sensors is fixed at 19, and the length of the solid is fixed at 0.2 m. Then, factors that affect Ah! 3) are the thermal conductivity of the solid, the magnitude of the applied heat flux the arrangement of sensors, and the width of the solid. According to Martin and Dulikravich [6], decreasing thermal conductivity will yield a more desirable solution. This can be seen by inspection of Eqs. (6) and (23), which reveal that [a~[ is proportional to k. Since increasing k will leave both q}3) and T,(3) unchanged (as h~3) is fixed), it will cause an undesirable increase in the uncertainty of estimated heat transfer coefficient according to Eqs. (27) - (29). Similar outcome will result from decreasing applied heat flux q0). This is so because it will cause q(3) and Ti(3) to decrease without affecting [a,y[, [b~[, [d~[, or [e~[, leading to an increase in Ah}3). Although it may seem reasonable to design an experiment so that the thermal conductivity is as low as possible and the applied heat flux is as high as possible, doing so will raise temperature on F~ where sensors are located. Hence, there is in practice a lower limit to the former and an upper limit to the latter. Figure 4 shows three setups with different arrangements of 19 sensors. Table 2 compares uncertainty in estimated h} 3) of these setups. It can be seen that placing more sensors on F1 and fewer sensors on Fz results in smaller Ah}3)/h} 3) . For this sample problem, all arrangements shown in Fig. 4 yield comparable estimates of heat transfer coefficient on F3. However, if there are a rapid variation of temperature

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on F2 and a gradual variation of temperature on F~, arrangement (a) will be expected to produce a much better estimate than arrangement (c).

(a)

(b)

(c)

FIG. 4 Three different arrangements of 19 sensors (represented by white circles)

TABLE 2 Comparison of Ah}3) among Three Different Arrangements of 19 Sensors

Ah? /h?l x 0 0.033 0.067 0.1 0.133 0.167 0.2

(a)

(b) 0.115 0.213 0.284 0.166 0.311 0.306 0.173

(c) 0.112 0.201 0.274 0.166 0.296 0.287 0.171

0.095 0.150 0.218 0.151 0.230 0.228 0.135

TABLE 3 Comparison of Ah~3) among Three Cases with L = 0.2 m and Different W

Ah?l/h? x 0 0.033 0.067 0.1 0.133 0.167 0.2

W=0.1 m 0.115 0.213 0.284 0.166 0.311 0.306 0.173

W=0.15 m 0.115 0.216 0.240 0.128 0.267 0.319 0.191

W= 0.2 m 0.114 0.201 0.214 0.110 0.237 0.297 0.191

Finally, to compare the effects of solid width, the experimental setup shown in Fig. 1 will be used. The length is fixed at 0.2 m, but the width takes on the values of 0.1 m, 0.15 m, and 0.2 m. The numerical result is shown in Table 3. It is interesting to note that increasing width leads to smaller Ah}3)/h} 3) . Apparently, increasing F2, which is a known boundary (i.e. its heat flux and temperature are known), relative to F1 where heat flux is unknown is therefore beneficial as far as the inverse estimation is concerned. Martin and Dulikravich also reached similar conclusion in their previous related work [6]. Note that this conclusion is valid in this sample problem even though increasing the solid width results in more sensors

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being placed farther from F3.

Conclusion The inverse method of determining steady-state heat transfer coefficient offers some flexibility in designing an experiment. Instead of requiring direct measurements of heat flux and temperature at the surface where heat transfer coefficient is to be determined, the experiment can be designed so that temperature measurements are taken at locations where sensors can be conveniently placed and the sensors does not disturb the fluid flow field. A sample experimental setup is proposed in this paper to demonstrate the applicability of the inverse method. It is also shown that the estimated heat transfer coefficient contains uncertainty as a result of uncertainty in temperature measurements. However, by prudent experimental design, the uncertainty in the estimated heat transfer coefficient may be reduced. Although only a two-dimensional system is discussed in this paper, the inverse method will be much more useful in determining steady-state heat transfer coefficient in a three-dimensional system, which should be a subject for further study.

Acknowledgments The author would like to acknowledge the financial support from the Thailand Research Fund.

Nomenclature a o, b~, c o

coefficients relating heat flux at F3 to temperature at F~, temperature at F2, and heat flux at F2, respectively

co, do, eq

coefficients relating temperature at F~ to temperature at Fb temperature at F2, and heat flux at F2, respectively

G

fundamental solution

h (3)

heat transfer coefficient on F3

k

thermal conductivity

L

length of solid

m

the total number of boundary nodes

m b l'n2~ m 3

the number of boundary nodes on F~, F2, and F3

n

coordinate normal to and pointing outward from the boundary

q

heat flux

qO), q(2), q(3)

heat flux on FI, F2, and F3 position vector

R(U), S(u)

coefficient matrices

T

temperature

1164

S. Chantasiriwan

To

fluid temperature

T(~, T(z), 7~)

temperature on F~, F2, and I-'3

W

length of solid

Vol. 27, No. 8

uncertainty in temperature measurement position vector

References

1. W.M. Rohsenow, J. P. Hartnett, and Y. I. Cho, Handbook of Heat Transfer, Chap. 16, McGraw-Hill, New York (1998). 2. A.M. Osman and J. V. Beck, AIAA Journal of Thermophysics 3, 146 (1989). 3. A.M. Osman and J. V. Beck, Journal of Heat Transfer 112, 843 (1990). 4. D. Maillet and S. Degiovanni, Journal of Heat Transfer 113,549 (1991). 5. J. Xu and T. Chen, Heat Transfer Engineering 19, 45 (1998). 6. T.J. Martin and G. S. Dulikravich, Journal of Heat Transfer 120, 328 (1998).

7. S. Chantasiriwan, InternationalJournal ofHeatandMass Transfer 120, 1111 (1999). 8. P. K. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science, pp. 216-241, McGraw-Hill, London (1981).

Received September 2, 2000