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A Numerical Method on Inverse Determination of Heat Transfer Coefficient Based on Thermographic Temperature Measurement
Chinese Journal of Chemical Engineering, 16(6) 901ü908 (2008)
A Numerical Method on Inverse Determination of Heat Transfer Coefficient Based on Thermographic Temperature Measurement FAN Chunli (ֳҝॆ)*, SUN Fengrui (ൄמఫ) and YANG Li (ཷो)
Department of Power Engineering, Naval University of Engineering, Wuhan 430033, China Abstract The heat transfer coefficient in a multidimensional heat conduction problem is obtained from the solution of the inverse heat conduction problem based on the thermographic temperature measurement. The modified one-dimensional correction method (MODCM), along with the finite volume method, is employed for both twoand three-dimensional inverse problems. A series of numerical experiments are conducted in order to verify the effectiveness of the method. In addition, the effect of the temperature measurement error, the ending criterion of the iteration, etc. on the result of the inYerse problem is investigated. It is proved that the method is a simple, stable and accurate one that can solve successfully the inverse heat conduction problem. Keywords inverse heat conduction problem, heat transfer coefficient, finite volume method, modified one-dimensional correction method, measurement error
1
INTRODUCTION
A steady-state heat conduction problem in a flat plate with boundary conditions completely specified is a well-posed problem that can be solved by various analytical and numerical methods. However, when the boundary condition is to be determined from the temperature measurements of the plate surface, the problem is an ill-posed one known as the inverse heat conduction problem (IHCP) [1] which is often encountered in many fields such as chemical engineering and nuclear power engineering. The past three decades have been most active in the advancement of solution techniques for the IHCP, such as Tikhonov’s regularization procedure [2, 3], Alifanov’s iterative regularization techniques [57], Beck’s function estimation approach [8, 9] and their subsequent developments [10]. The aims of these methods are to obtain a solution that is accurate and not very sensitive to noise in input temperature data. Most of IHCPs that have been investigated so far are concerned with the estimation of boundary heat flux [10], boundary temperature [11] or boundary configuration [12, 13]. Another interesting problem that has not yet received as much attention is the estimation of heat transfer coefficient. Osman and Beck [14] treated the problem of estimating the time-dependent heat transfer coefficient in quenching a sphere as a nonlinear parameter estimation problem. Heat transfer coefficient was assumed to be a piecewise constant function of time. The unknown heat transfer coefficient parameters were estimated one by one using the sequential function specification method. Xu and Chen [15] studied the nonlinear problem of determining the heat transfer coefficient in two-phase flow in an inclined tube by building a steady-state two-dimensional heat conduction model. The advantage of the approach is that the solution is simple and needs little time for calculation. Martin and Dulikravich [16] developed an inverse boundary element method procedure for the determination of the unknown heat transfer coefficients on solid surfaces of
arbitrary shape. The procedure is non-iterative and cost effective, and has the advantage that it does not require any knowledge on or solution of the flow field. Chantasiriwan [17] presented an algorithm for estimating the time-dependent heat transfer coefficient for one-dimensional linear inverse heat conduction problem based on the sequential function specification method with the assumption of linear variation of the future boundary heat flux components. Based on the boundary element method Chantasiriwan [18] also solved numerically the problem of determining the heat transfer coefficient distribution in a two-dimensional flat plate by measuring the temperature distribution on plate surfaces. In this paper, a new numerical method for estimating the distribution of the heat transfer coefficient is presented by taking a multidimensional inverse heat conduction problem similar to the one concerned in Ref. [18] as a numerical example. The method employed is the modified one-dimensional correction method (MODCM) [19] which solves firstly the multidimensional inverse heat conduction problem by using a one-dimensional inverse function, and then the discrepancy of the estimated results caused by the one-dimensional simplification is corrected during an iterative process. The MODCM has the following main advantages when solving the inverse heat conduction problem: (1) the stability of the method can guarantee the convergence of the solution because the numerical method used in the algorithm is one of those for regular problem of differential equation; (2) the computational procedure is simple; and (3) it does not need an initial guess as most optimization methods do. Recent results by Fan et al. [19] showed that this method yielded good estimate of the distribution of effective thermal conductivity of a thin interlayer of a sandwich plate based on the temperature measurement of the outer inspection surface. Hence it is expected that the estimation of the heat transfer coefficient also performs well with the method, especially for the three-dimensional inverse problem about which little work has been done until now.
Received 2008-03-30, accepted 2008-09-20. * To whom correspondence should be addressed. E-mail: [email protected]
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2 MATHEMATICAL FORMULATION OF THE PROBLEM The problem to be considered is shown in Fig. 1. The flat plate is subjected to unknown heat transfer coefficient from the left surface. From the inspection surface, i.e. the right surface, heat is dissipated into the ambient by convection with a constant convective heat transfer coefficient hn. The other four surfaces are insulated. The temperature distribution of the inspection surface is measured by an infrared imager. The ambient temperature is Ta. The size of the flat plate is described by Lx, Ly and Lz in x, y and z directions, respectively, and the thermal conductivity of the plate is k. The measurement data along with the known geometrical and thermophysical data give rise to the inverse heat conduction problem, which can be mathematically described by the following steady-state governing equation: w 2T w 2T w 2T wx 2 wy 2 wz 2 and boundary conditions: wT wT wx x 0 wx x Lx wT wy
y 0
wT wy
0
(1)
0
(2)
0
(3)
y Ly
In the z direction, the boundary conditions on both the left surface and the right one are convective heat transfer. As illustrated in Fig.1, the bulk fluid temperature and the film coefficient on both sides of the plate are Tf, h(x,y) and Ta, hn, respectively.
one-dimensional correction method (ODCM) presented by Yang et al. [20] by correcting the iteration function, modifying the calculation of the correction terms, and giving the ending criterion of the iteration. In Ref. [19] the MODCM showed good performance on the identification of the conductivity distribution of the interlayer of a sandwich plate. The MODCM firstly calculates the parameters to be determined for the multidimensional inverse problem based on the one-dimensional inverse function. Then the discrepancy caused by the one-dimensional simplification is corrected by iteration to refine the solution. The expression of the iteration function is developed based on Taylor’s series expansion. For the inverse problem concerned in this paper, the heat transfer coefficient distribution h(x,y) is described by a matrix H formed by the heat transfer coefficient value of each discrete point hi , j (i 1 mx ; j 1 m y ) of the left surface where mx and my are the numbers of the node in x and y directions, respectively. hi,j is firstly calculated based on the discrete measured temperature value Toi , j (i 1 mx ; j 1 m y ) of the inspection surface, i.e. the element of the measured temperature matrix To, according to the one-dimensional inverse function h1D(To) which can be deduced easily from the one-dimensional heat conduction problem. Similar to Ref. [19], an iteration function can be derived based on Taylor’s series expansion when the matrixes formed by the elements hic, j (To ) and hicc, j (To ) are expressed as H c and H cc , respectively, for each iteration: H n 1 H n H cn * To T n H ccn * To T n * To T n 2! If the last term is neglected, Eq. (4) becomes H n 1
H n H cn * To T n
(4)
(5)
n
Figure 1 Three-dimensional inverse heat conduction problem to be solved for h(x,y)
The goal of the IHCP solution is to determine the heat transfer coefficient distribution h(x,y), the only unknown, on the left surface based on the thermographic temperature measurements of the inspection surface. 3 THE MODIFIED ONE-DIMENSIONAL CORRECTION METHOD (MODCM) 3.1
Iteration function of the MODCM The MODCM is developed on the basis of the
where H is the nth estimated result of the heat transfer coefficient after n 1 iterations; Tn is a matrix describing the temperature distribution on the inspection surface when the heat transfer coefficient on the left surface is Hn, which is obtained by solving Eq. (1); To is the original measured temperature distribution of the inspection surface; H c and H cc are the correction terms; “ ” in Eqs. (4) and (5) denotes the Hadamard product [21], i.e. the element by element product, of two matrixes with the same size; and superscript n is the iteration number. It should be mentioned that, in Eqs. (4) and (5), the matrixes H, H c , H cc , T and To have the same size mx u m y . 3.2
Obtaining the correction terms
The terms H c and H cc must be computed for each iteration for Eq. (4) or Eq. (5) to refine the identification result of the heat transfer coefficient distribution. The elements of the two matrixes are determined based on the following equations [20]: hic, j Toi , j 1/ Toci , j hi , j ° ® °¯hicc, j Toi , j ª¬ hic, j Toi , j º¼c
(6)
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For the MODCM, The term Toci , j (hi , j ) in Eq. (6) of each iteration is computed by perturbing the element hi,j of the heat transfer coefficient distribution matrix H obtained from the last iteration and computing the resulting change in temperature Toi,j of the corresponding point of the inspection surface from the solution of the multidimensional heat conduction problem in Eq. (1). Thereafter, Toci , j (hi , j ) is calculated by the ratio of perturbations of the two parameters: 'Toi , j / 'hi , j . Then the term hic, j (Toi , j ) can be computed according to Eq. (6) for every point to get H c . The term hicc, j (Toi , j ) can also be obtained by one more perturbation for calculating H cc . For most problems, Eq. (5) is good enough to obtain good identification results and the term H cc is not needed any more. 3.3 Ending criterion of the iteration For the MODCM on the identification of the heat transfer coefficient, the ending criterion of the iteration is given as J (H )
mx m y
¦¦
i 1 j 1
ª n 1 ¬ hi , j
2 hin, j ¼º
H
(7)
where İ is a small positive number. 3.4 Computational procedure of the MODCM The iterative computational procedure for the solution of this heat transfer coefficient identification problem using the MODCM can be summarized as follows. Compute the initial heat transfer coefficient distribution H1 based on the measured temperature distribution of the inspection surface To by using the one-dimensional inverse function h1D(To) to begin the iteration process. Suppose Hn is available. (i) Compute the temperature distribution of the inspection surface Tn based on the estimated heat transfer coefficient distribution of the left surface Hn according to the multidimensional heat transfer problem in Eq. (1) by the finite volume method. (ii) Check the ending criterion given by Eq. (7), and print the result if satisfied. (iii) Compute the correction terms H cn and H ccn in accordance with Eq. (6). (iv) Calculate the new heat transfer coefficient H n1 based on Eq. (4) or Eq. (5), and return to step (i). In this paper Eq. (5) is used.
with different heat transfer coefficient distributions. In all test cases considered here, we have chosen ˉ ˉ Tf 150°C, Ta 25°C and hn 10 W·m 2·K 1. The ˉ ˉ thermal conductivity k of the flat plate is 50 W·m 1·K 1. Eq. (1) is solved by the finite volume method. As in many IHCP studies [1012], in the present numerical experiments the measured temperature distribution To on the inspection surface is simulated by the solution of the multidimensional Eq. (1) based on the known heat transfer coefficient distribution of the left surface, to which errors can also be added to simulate the real thermographic measurement. Based on these simulated measurements the identification work is conducted and the results will be compared with the known exact heat transfer coefficient distributions to verify the effectiveness of the algorithm. In order for describing the precision level of the identification result, an average relative error (ARE) is also defined by comparing the estimated and the exact (known) values as
eARE
1 mx m y
mx m y
¦¦
i 1 j 1
hi , j hˆi , j u 100% hˆ
(8)
i, j
where hi,j is the estimated result; hˆi , j describes the known heat transfer coefficient distribution of the left plate surface in Fig. 1. The one-dimensional inverse function h1D(To) can be derived based on the one-dimensional heat transfer model illustrated in Fig. 2 with uniform heat transfer coefficients h and hn on both sides, respectively. In the figure To is the measured temperature of the inspection surface. The heat flux q transferred through the plate can be described as Tf Ta To Ta (9) q 1 G 1 1 h k hn hn where į is the thickness of the plate. From Eq. (9), the one-dimensional inverse function h1D(To) can be derived for this heat transfer coefficient identification problem as
h1D (To )
h(To )
G 1º ª Tf Ta « h T T k h » n ¼ ¬ n o a
1
(10)
4 NUMERICAL EXPERIMENTS AND DISCUSSION To illustrate the validity of the present inverse algorithm in identifying the heat transfer coefficient distribution h(x,y) of the left surface in Fig. 1 from the knowledge of temperature recordings of the inspection surface, numerical experiments are conducted as in many papers on inverse heat conduction problem [913]. Four examples are considered in this section, two of two-dimension and two of three-dimension,
Figure 2 One-dimensional heat transfer model of the flat plate
When solving Eq. (1) for the temperature distribution of the inspection surface based on the finite volume method, the relative error of the calculated
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temperature distributions used to judge whether the result Ti is grid-independent is defined for both the two- and three-dimensional problems as m 1 y Ti Tic u 100% 2 D ° ¦ m y i 1 Ti ° (11) egrid ® mx m y T T c i, j i, j ° 1 u 100% 3D ° m m ¦¦ T i, j ¯ x y i1j1 where Ti and Ti,j are the calculated temperature distributions when the discrete node number in the inspection surface is adopted as my and mx u m y for the twoand three-dimensional problems, respectively; Tic and Tic, j are the temperature distributions obtained for the grid meshing of 2m y and 2mx u 2m y .
4.1
Two-dimensional numerical test cases
If the dimension of the flat plate in Fig. 1 in x direction is very long compared with the dimensions in y and z directions and the heat transfer coefficient does not change in x direction either, the IHCP on determining the heat transfer coefficient of the left surface can be deemed as of two-dimension. For this two-dimensional problem, the dimensions of the flat plate in y and z directions are 0.5 m and 0.1 m respectively. As shown in Fig. 3, the corresponding grid meshing of the plate is adopted as mz 4 and m y 41 ( egrid 0.01% ). The functions of the heat transfer coefficient changing in y direction of the two test cases, A and B, are as follows:
Case A : h( y ) 50 800 y ; Case B : h( y )
0 İ y İ Ly
°40; 0 İ y İ 0.5Ly ® °¯20; 0.5 Ly y İ Ly
Figure 4 Temperature distributions of the inspection surface for test Case A and Case B ƻ Case A; Ƶ Case B
shown in Fig. 5 when the ending criterion is set as H 0.01 . The “exact” curves in the figures are plotted based on the known heat transfer coefficient distributions described in Eqs. (12) and (13). From the figures, one can see that the estimated and the exact curves have a good agreement. In the figures the initial identification results of the heat transfer coefficient are also shown which were obtained only by using the one-dimensional inverse Eq. (10) before the iteration begins. These two “initial” curves show that one-dimensional simplification will cause obvious discrepancy in the identification results; therefore, the iterative correcting process is necessary.
(12) (13)
(a) Case A
Figure 3 Schematic of grid meshing of the plate for the two-dimensional problem
Based on Eq. (1) with the third term deleted for this two-dimensional problem the temperature distributions on the inspection surface are calculated numerically by the finite volume method and plotted in Fig. 4. Based on these two distribution curves, i.e. the simulated measurement data, the identification work is conducted to certify the effectiveness of the inverse method. When no measurement error is considered, the identification results of the heat transfer coefficient distribution on the left surface of the flat plate are
(b) Case B Figure 5 Identification results of the heat transfer coefficient when no measurement error is considered for Case A and Case B ( H 0.01 ) ƽ estimated;ƻ initial; üü exact
In Table 1, the relationship between the ARE of the identification result and the ending criterion of the iteration process is studied systematically. The ARE increases with the increase of the ending criterion.
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Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008 Table 1 Relationship between the ARE of the identification result and the ending criterion İ
eARE (Case A)
eARE (Case B)
0.1
2.2%
5.4%
0.01
1.2%
4.5%
0.001
0.7%
2.9%
When H 0.01 , the AREs of the identification results are smaller than 5%; therefore H 0.01 is used in the following numerical examples in this subsection. In the practical work, errors always accompany the temperature measurement. In Fig. 6 the estimated heat transfer coefficient distributions are plotted for the two test cases when the measurement errors (e) are respectively +1.0°C, 1.0°C, +0.5°C and 0.5°C. It should be mentioned that uniform error is adopted here for every discrete measurement point because, for the thermographic temperature measurement, the effect of the surroundings, imager itself, etc. on the measurement result is nearly the same for every point in the inspection surface. From the figures one can see that the AREs of identification results increase with the increase of the measurement error. When the absolute value of measurement error is 1.0°C, the AREs of the identification results are smaller than 18.3%.
the analytical results for this one-dimensional problem are plotted in Fig. 7 when the measurement errors are +1.0°C and 1.0°C, respectively. From this figure, one can see that most of the errors reported in Fig. 6 and Table 2 are intrinsic for this heat transfer coefficient identification problem, but not resulted from the method employed. For the plate with thermal conductivity of ˉ ˉ 50 W·m 1·K 1, the temperature of the right surface is very close to that of the bulk fluid on the left side, especially when the heat transfer coefficient of the left surface is large; therefore it is much more difficult to increase or decrease 1.0°C of the right surface by changing the heat transfer coefficient of the left surface. This is the very reason why larger errors of the identification results appear in Fig. 7 and also in Fig. 6 at the part where the heat transfer coefficient is larger.
Figure 7 Relative error of the analytical results for the one-dimensional heat transfer coefficient identification problem e/°C: Ƶ 1.0;ƻ1.0 Table 2
(a) Case A e/°C:Ƹ1.0;ƽ 1.0;Ʒ0.5;ƻ 0.5; üü exact
4.2
(b) Case B e/°C:ƽ1.0;ƻ 1.0;Ƹ0.5;ͩ 0.5; üü exact Figure 6 Identification results of the heat transfer coefficient distribution with measurement error in consideration for Case A and Case B ( H 0.01 )
When the heat transfer coefficient does not change in y direction either, the identification problem becomes a one-dimensional one which can be solved analytically based on Eq. (10). The relative errors of
Relationship between the measurement error and the ARE of the identification result e/°C
eARE(Case A)/%
eARE(Case B)/%
0.5
8.6
6.7
0.5
7.4
6.3
1.0
18.3
9.0
1.0
13.2
8.5
2.0
33.1
14.8
2.0
23.8
12.7
Three-dimensional numerical test cases
In this subsection, the numerical method is tested for solving the three-dimensional IHCP for determining the heat transfer coefficient distribution on the left surface of the plate shown in Fig. 1. The dimensions of the plate used in this subsection is 1.0 m, 1.0 m and 0.1 m in x, y and z directions, respectively. The grid meshing of the plate for the finite volume method is mz 4 , mx 21 and m y 21 ( egrid 0.05% ). The heat transfer coefficient distributions to be determined for the two test Cases C and D are shown in Fig. 8. Based on the simulated temperature distributions of the inspection surface which are obtained according to Eq. (1), the heat transfer coefficient is estimated firstly based on Eq. (10) and plotted in Fig. 9. Then the
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(a) Case C (b) Case D Figure 8 Exact heat transfer coefficient distributions to be determined for Case C and Case D
Figure 9
(a) Case C (b) Case D Initial identification results of the heat transfer coefficient obtained by Eq. (10)
(a) Case C (b) Case D Figure 10 Final identification results of the heat transfer coefficient for Case C and Case D when no measurement is considered ( H 1.0 )
(a) Case C (b) Case D Figure 11 Identification results of the heat transfer coefficient for Case C and Case D when measurement error is in consideration ( H 0.01 ) upper: estimated ( e 1.0 °C); lower: estimated ( e 1.0 °C); middle: exact
discrepancy caused by using this one-dimensional inverse function is corrected by the iteration process according to Eq. (5). The final identification results of
the heat transfer coefficient are shown in Fig. 10 when the ending criterion is adopted as H 1.0 . The AREs of the results for Case C and Case D are 0.5% and
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
1.9%, respectively, when no measurement error is considered. The estimated and the exact coefficients have a good agreement. In this subsection, the effect of the thermographic temperature measurement error on the accuracy of the identification result of the heat transfer coefficient is also studied and reported in Table 3. The heat transfer coefficients estimated are also plotted in Fig. 11 when the measurement errors are +1.0°C and 1.0°C (the figure for Case D is plotted by rotating an angle for showing the result better). From the table, one can see that when the absolute value of the measurement error is smaller than 2.0°C, the AREs of the estimated results are not more than 23.7%. The results are good enough to be accepted as the solution of the IHCP. Table 3
Relationship between the measurement error and the ARE of the identification result e/°C
eARE (Case A)/%
eARE(Case B)/%
0.5
5.6
2.4
0.5
4.2
2.2
1.0
11.0
4.3
1.0
8.3
3.9
2.0
23.7
8.3
2.0
15.8
7.2
5 EFFECT OF TEMPERATURE MEASUREMENT ERROR
As stated in Ref. [20] for the ODCM, the numerical method used in the MODCM is one of those for regular problem of differential equation, and the stability of the method can guarantee the convergence of the solution. Therefore, the errors occurred in practical inspection work would only lead to errors in the identified results but will not make the method itself ill-posed. That is to say, as long as the ending criterion is small enough, the identification result obtained when the measurement error is considered is also a very accurate solution for this new simulated temperature measurement formed by the original temperature distribution and the measurement error added. Since the relatively larger error in the result caused by the measurement error is the intrinsic characteristic of this IHCP which can be seen from Fig. 7, more effort should be put in the practical inspection and identification work to decrease the measurement error of the infrared imager for obtaining a more accurate identification result. On the analysis and estimation of the thermographic measurement error one can refer to our previous work [22, 23]. 6
CONCLUSIONS
The MODCM along with the finite volume method is applied successfully in solving the IHCP for determining the heat transfer coefficient of one surface of a flat plate based on the thermographic temperature measurements of the opposite surface. A series of nu-
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merical experiments conducted, two-dimensional and three-dimensional, have certified the effectiveness of the method. Since the method is stable, the error of this method caused by a measurement error is a definite value and can be estimated and eliminated if only the measurement error can be estimated precisely in the practical inspection and identification work. This method is proved to be a simple, stable and accurate one which can be extended for solving other inverse problems. NOMENCLATURE H
h hn L m n T To T Ta Tf
matrix of heat transfer coefficient distribution of the left surface correction term correction term ˉ ˉ heat transfer coefficient, W·m 2·K 1 ˉ ˉ heat transfer coefficient of the inspection surface, W·m 2·K 1 dimension of the test piece number of discrete points iteration number matrix of temperature distribution on the inspection surface original measured temperature distribution on the inspection surface temperature, °C ambient temperature, °C temperature of fluid of left side of the plate, °C
ARE i,j o x,y,z
average relative error grid index original measured value in the x,y,z direction, respectively
Hc H cc
Subscripts
REFERENCES 1
2 3 4 5 6
7 8 9 10
11
12
13
Beck, J.V., Blackwell, B., “Inverse problems”, In: Minkowycz, W.J., Schneider, G.E., Pletcher, R.H., eds., Handbook of Numerical Heat Transfer, Wiley Interscience, New York (1988). Tikhonov, A.N., Arnold, K.J., Solution of Ill-Posed Problems, Winston & Sons, Washington (1977). Tikhonov, A.N., “Regularization of incorrectly posed problems”, Soviet Math. Dokl., 4, 16241627 (1963). Alifanov, O.M., Inverse Heat Transfer Problems, Springer-Verlag, New York (1994). Alifanov, O.M., “Solution of an inverse problem of heat-conduction by iterative methods”, J. Eng. Phys., 26, 471476 (1974). Alifanov, O.M., Mikhailov, V.V., “Solution of the overdetermined inverse problem of thermal conductivity involving inaccurate data”, High Temp., 23, 112117 (1985). Beck, J.V., Blackwell, B., Clair, C.R., Inverse Heat Conduction: Ill-Posed Problems, Wiley Interscience, New York (1985). Beck, J.V., “Calculation of surface heat flux from an internal temperature history”, ASME Paper, 62-HT-46 (1962). Ozisik, M.N., Oriande, H.R.B., Inverse Heat Transfer, Fundamentals and Applications, Taylor & Francis, New York (2000). Huang, C.H., Wang, S.P., “A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method”, Int. J. Heat Mass Transfer, 42, 33873403 (1999). Wu, S.K., Chu, H.S., “Inverse determination of surface temperature in thin-film/substrate systems with interface thermal resistance”, Int. J. Heat Mass Transfer, 47, 35073515 (2004). Fan, C.L., Sun, F.R., Yang, L., “An algorithm study on the identification of pipeline’s irregular inner boundary based on thermographic temperature measurement”, Meas. Sci. Technol., 18, 21702177 (2007). Fan, C.L., Sun, F.R., Yang, L., “A new computational scheme on quantitative inner pipe boundary identification based on the estimation of
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Fan, C. L., Sun, F.R., Yang, L., “A numerical method for determining thermal conductivity distribution of the interlayer of a sandwich plate based on thermographic temperature measurement”, J. Phys. D Appl. Phys., 41, 135501 (2008). Yang, L., Geng, W., Jiang, L., Zou, L., Hong, J., “Profile identification in dynamic infrared thermography: Adaptability and stability”, Int. J. Infrared Millimeter Waves, 20, 623634 (1999). Huang, R., “Some inequalities for the Hadamard product and the Fan product of matrices”, Linear Algebra its Appl., 428, 15511559 (2008). Yang, L., “Calculation and error analysis of temperature measurement using thermal imager”, Infrared Technol., 21, 2024 (1999). (in Chinese) Kou, W., Yang, L., “Error analysis of infrared measurement”, Infrared Technol., 23, 3235 (2001). (in Chinese)
A Numerical Method on Inverse Determination of Heat Transfer Coefficient Based on Thermographic Temperature Measurement FAN Chunli (ֳҝॆ)*, SUN Fengrui (ൄמఫ) and YANG Li (ཷो)
Department of Power Engineering, Naval University of Engineering, Wuhan 430033, China Abstract The heat transfer coefficient in a multidimensional heat conduction problem is obtained from the solution of the inverse heat conduction problem based on the thermographic temperature measurement. The modified one-dimensional correction method (MODCM), along with the finite volume method, is employed for both twoand three-dimensional inverse problems. A series of numerical experiments are conducted in order to verify the effectiveness of the method. In addition, the effect of the temperature measurement error, the ending criterion of the iteration, etc. on the result of the inYerse problem is investigated. It is proved that the method is a simple, stable and accurate one that can solve successfully the inverse heat conduction problem. Keywords inverse heat conduction problem, heat transfer coefficient, finite volume method, modified one-dimensional correction method, measurement error
1
INTRODUCTION
A steady-state heat conduction problem in a flat plate with boundary conditions completely specified is a well-posed problem that can be solved by various analytical and numerical methods. However, when the boundary condition is to be determined from the temperature measurements of the plate surface, the problem is an ill-posed one known as the inverse heat conduction problem (IHCP) [1] which is often encountered in many fields such as chemical engineering and nuclear power engineering. The past three decades have been most active in the advancement of solution techniques for the IHCP, such as Tikhonov’s regularization procedure [2, 3], Alifanov’s iterative regularization techniques [57], Beck’s function estimation approach [8, 9] and their subsequent developments [10]. The aims of these methods are to obtain a solution that is accurate and not very sensitive to noise in input temperature data. Most of IHCPs that have been investigated so far are concerned with the estimation of boundary heat flux [10], boundary temperature [11] or boundary configuration [12, 13]. Another interesting problem that has not yet received as much attention is the estimation of heat transfer coefficient. Osman and Beck [14] treated the problem of estimating the time-dependent heat transfer coefficient in quenching a sphere as a nonlinear parameter estimation problem. Heat transfer coefficient was assumed to be a piecewise constant function of time. The unknown heat transfer coefficient parameters were estimated one by one using the sequential function specification method. Xu and Chen [15] studied the nonlinear problem of determining the heat transfer coefficient in two-phase flow in an inclined tube by building a steady-state two-dimensional heat conduction model. The advantage of the approach is that the solution is simple and needs little time for calculation. Martin and Dulikravich [16] developed an inverse boundary element method procedure for the determination of the unknown heat transfer coefficients on solid surfaces of
arbitrary shape. The procedure is non-iterative and cost effective, and has the advantage that it does not require any knowledge on or solution of the flow field. Chantasiriwan [17] presented an algorithm for estimating the time-dependent heat transfer coefficient for one-dimensional linear inverse heat conduction problem based on the sequential function specification method with the assumption of linear variation of the future boundary heat flux components. Based on the boundary element method Chantasiriwan [18] also solved numerically the problem of determining the heat transfer coefficient distribution in a two-dimensional flat plate by measuring the temperature distribution on plate surfaces. In this paper, a new numerical method for estimating the distribution of the heat transfer coefficient is presented by taking a multidimensional inverse heat conduction problem similar to the one concerned in Ref. [18] as a numerical example. The method employed is the modified one-dimensional correction method (MODCM) [19] which solves firstly the multidimensional inverse heat conduction problem by using a one-dimensional inverse function, and then the discrepancy of the estimated results caused by the one-dimensional simplification is corrected during an iterative process. The MODCM has the following main advantages when solving the inverse heat conduction problem: (1) the stability of the method can guarantee the convergence of the solution because the numerical method used in the algorithm is one of those for regular problem of differential equation; (2) the computational procedure is simple; and (3) it does not need an initial guess as most optimization methods do. Recent results by Fan et al. [19] showed that this method yielded good estimate of the distribution of effective thermal conductivity of a thin interlayer of a sandwich plate based on the temperature measurement of the outer inspection surface. Hence it is expected that the estimation of the heat transfer coefficient also performs well with the method, especially for the three-dimensional inverse problem about which little work has been done until now.
Received 2008-03-30, accepted 2008-09-20. * To whom correspondence should be addressed. E-mail: [email protected]
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2 MATHEMATICAL FORMULATION OF THE PROBLEM The problem to be considered is shown in Fig. 1. The flat plate is subjected to unknown heat transfer coefficient from the left surface. From the inspection surface, i.e. the right surface, heat is dissipated into the ambient by convection with a constant convective heat transfer coefficient hn. The other four surfaces are insulated. The temperature distribution of the inspection surface is measured by an infrared imager. The ambient temperature is Ta. The size of the flat plate is described by Lx, Ly and Lz in x, y and z directions, respectively, and the thermal conductivity of the plate is k. The measurement data along with the known geometrical and thermophysical data give rise to the inverse heat conduction problem, which can be mathematically described by the following steady-state governing equation: w 2T w 2T w 2T wx 2 wy 2 wz 2 and boundary conditions: wT wT wx x 0 wx x Lx wT wy
y 0
wT wy
0
(1)
0
(2)
0
(3)
y Ly
In the z direction, the boundary conditions on both the left surface and the right one are convective heat transfer. As illustrated in Fig.1, the bulk fluid temperature and the film coefficient on both sides of the plate are Tf, h(x,y) and Ta, hn, respectively.
one-dimensional correction method (ODCM) presented by Yang et al. [20] by correcting the iteration function, modifying the calculation of the correction terms, and giving the ending criterion of the iteration. In Ref. [19] the MODCM showed good performance on the identification of the conductivity distribution of the interlayer of a sandwich plate. The MODCM firstly calculates the parameters to be determined for the multidimensional inverse problem based on the one-dimensional inverse function. Then the discrepancy caused by the one-dimensional simplification is corrected by iteration to refine the solution. The expression of the iteration function is developed based on Taylor’s series expansion. For the inverse problem concerned in this paper, the heat transfer coefficient distribution h(x,y) is described by a matrix H formed by the heat transfer coefficient value of each discrete point hi , j (i 1 mx ; j 1 m y ) of the left surface where mx and my are the numbers of the node in x and y directions, respectively. hi,j is firstly calculated based on the discrete measured temperature value Toi , j (i 1 mx ; j 1 m y ) of the inspection surface, i.e. the element of the measured temperature matrix To, according to the one-dimensional inverse function h1D(To) which can be deduced easily from the one-dimensional heat conduction problem. Similar to Ref. [19], an iteration function can be derived based on Taylor’s series expansion when the matrixes formed by the elements hic, j (To ) and hicc, j (To ) are expressed as H c and H cc , respectively, for each iteration: H n 1 H n H cn * To T n H ccn * To T n * To T n 2! If the last term is neglected, Eq. (4) becomes H n 1
H n H cn * To T n
(4)
(5)
n
Figure 1 Three-dimensional inverse heat conduction problem to be solved for h(x,y)
The goal of the IHCP solution is to determine the heat transfer coefficient distribution h(x,y), the only unknown, on the left surface based on the thermographic temperature measurements of the inspection surface. 3 THE MODIFIED ONE-DIMENSIONAL CORRECTION METHOD (MODCM) 3.1
Iteration function of the MODCM The MODCM is developed on the basis of the
where H is the nth estimated result of the heat transfer coefficient after n 1 iterations; Tn is a matrix describing the temperature distribution on the inspection surface when the heat transfer coefficient on the left surface is Hn, which is obtained by solving Eq. (1); To is the original measured temperature distribution of the inspection surface; H c and H cc are the correction terms; “ ” in Eqs. (4) and (5) denotes the Hadamard product [21], i.e. the element by element product, of two matrixes with the same size; and superscript n is the iteration number. It should be mentioned that, in Eqs. (4) and (5), the matrixes H, H c , H cc , T and To have the same size mx u m y . 3.2
Obtaining the correction terms
The terms H c and H cc must be computed for each iteration for Eq. (4) or Eq. (5) to refine the identification result of the heat transfer coefficient distribution. The elements of the two matrixes are determined based on the following equations [20]: hic, j Toi , j 1/ Toci , j hi , j ° ® °¯hicc, j Toi , j ª¬ hic, j Toi , j º¼c
(6)
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For the MODCM, The term Toci , j (hi , j ) in Eq. (6) of each iteration is computed by perturbing the element hi,j of the heat transfer coefficient distribution matrix H obtained from the last iteration and computing the resulting change in temperature Toi,j of the corresponding point of the inspection surface from the solution of the multidimensional heat conduction problem in Eq. (1). Thereafter, Toci , j (hi , j ) is calculated by the ratio of perturbations of the two parameters: 'Toi , j / 'hi , j . Then the term hic, j (Toi , j ) can be computed according to Eq. (6) for every point to get H c . The term hicc, j (Toi , j ) can also be obtained by one more perturbation for calculating H cc . For most problems, Eq. (5) is good enough to obtain good identification results and the term H cc is not needed any more. 3.3 Ending criterion of the iteration For the MODCM on the identification of the heat transfer coefficient, the ending criterion of the iteration is given as J (H )
mx m y
¦¦
i 1 j 1
ª n 1 ¬ hi , j
2 hin, j ¼º
H
(7)
where İ is a small positive number. 3.4 Computational procedure of the MODCM The iterative computational procedure for the solution of this heat transfer coefficient identification problem using the MODCM can be summarized as follows. Compute the initial heat transfer coefficient distribution H1 based on the measured temperature distribution of the inspection surface To by using the one-dimensional inverse function h1D(To) to begin the iteration process. Suppose Hn is available. (i) Compute the temperature distribution of the inspection surface Tn based on the estimated heat transfer coefficient distribution of the left surface Hn according to the multidimensional heat transfer problem in Eq. (1) by the finite volume method. (ii) Check the ending criterion given by Eq. (7), and print the result if satisfied. (iii) Compute the correction terms H cn and H ccn in accordance with Eq. (6). (iv) Calculate the new heat transfer coefficient H n1 based on Eq. (4) or Eq. (5), and return to step (i). In this paper Eq. (5) is used.
with different heat transfer coefficient distributions. In all test cases considered here, we have chosen ˉ ˉ Tf 150°C, Ta 25°C and hn 10 W·m 2·K 1. The ˉ ˉ thermal conductivity k of the flat plate is 50 W·m 1·K 1. Eq. (1) is solved by the finite volume method. As in many IHCP studies [1012], in the present numerical experiments the measured temperature distribution To on the inspection surface is simulated by the solution of the multidimensional Eq. (1) based on the known heat transfer coefficient distribution of the left surface, to which errors can also be added to simulate the real thermographic measurement. Based on these simulated measurements the identification work is conducted and the results will be compared with the known exact heat transfer coefficient distributions to verify the effectiveness of the algorithm. In order for describing the precision level of the identification result, an average relative error (ARE) is also defined by comparing the estimated and the exact (known) values as
eARE
1 mx m y
mx m y
¦¦
i 1 j 1
hi , j hˆi , j u 100% hˆ
(8)
i, j
where hi,j is the estimated result; hˆi , j describes the known heat transfer coefficient distribution of the left plate surface in Fig. 1. The one-dimensional inverse function h1D(To) can be derived based on the one-dimensional heat transfer model illustrated in Fig. 2 with uniform heat transfer coefficients h and hn on both sides, respectively. In the figure To is the measured temperature of the inspection surface. The heat flux q transferred through the plate can be described as Tf Ta To Ta (9) q 1 G 1 1 h k hn hn where į is the thickness of the plate. From Eq. (9), the one-dimensional inverse function h1D(To) can be derived for this heat transfer coefficient identification problem as
h1D (To )
h(To )
G 1º ª Tf Ta « h T T k h » n ¼ ¬ n o a
1
(10)
4 NUMERICAL EXPERIMENTS AND DISCUSSION To illustrate the validity of the present inverse algorithm in identifying the heat transfer coefficient distribution h(x,y) of the left surface in Fig. 1 from the knowledge of temperature recordings of the inspection surface, numerical experiments are conducted as in many papers on inverse heat conduction problem [913]. Four examples are considered in this section, two of two-dimension and two of three-dimension,
Figure 2 One-dimensional heat transfer model of the flat plate
When solving Eq. (1) for the temperature distribution of the inspection surface based on the finite volume method, the relative error of the calculated
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temperature distributions used to judge whether the result Ti is grid-independent is defined for both the two- and three-dimensional problems as m 1 y Ti Tic u 100% 2 D ° ¦ m y i 1 Ti ° (11) egrid ® mx m y T T c i, j i, j ° 1 u 100% 3D ° m m ¦¦ T i, j ¯ x y i1j1 where Ti and Ti,j are the calculated temperature distributions when the discrete node number in the inspection surface is adopted as my and mx u m y for the twoand three-dimensional problems, respectively; Tic and Tic, j are the temperature distributions obtained for the grid meshing of 2m y and 2mx u 2m y .
4.1
Two-dimensional numerical test cases
If the dimension of the flat plate in Fig. 1 in x direction is very long compared with the dimensions in y and z directions and the heat transfer coefficient does not change in x direction either, the IHCP on determining the heat transfer coefficient of the left surface can be deemed as of two-dimension. For this two-dimensional problem, the dimensions of the flat plate in y and z directions are 0.5 m and 0.1 m respectively. As shown in Fig. 3, the corresponding grid meshing of the plate is adopted as mz 4 and m y 41 ( egrid 0.01% ). The functions of the heat transfer coefficient changing in y direction of the two test cases, A and B, are as follows:
Case A : h( y ) 50 800 y ; Case B : h( y )
0 İ y İ Ly
°40; 0 İ y İ 0.5Ly ® °¯20; 0.5 Ly y İ Ly
Figure 4 Temperature distributions of the inspection surface for test Case A and Case B ƻ Case A; Ƶ Case B
shown in Fig. 5 when the ending criterion is set as H 0.01 . The “exact” curves in the figures are plotted based on the known heat transfer coefficient distributions described in Eqs. (12) and (13). From the figures, one can see that the estimated and the exact curves have a good agreement. In the figures the initial identification results of the heat transfer coefficient are also shown which were obtained only by using the one-dimensional inverse Eq. (10) before the iteration begins. These two “initial” curves show that one-dimensional simplification will cause obvious discrepancy in the identification results; therefore, the iterative correcting process is necessary.
(12) (13)
(a) Case A
Figure 3 Schematic of grid meshing of the plate for the two-dimensional problem
Based on Eq. (1) with the third term deleted for this two-dimensional problem the temperature distributions on the inspection surface are calculated numerically by the finite volume method and plotted in Fig. 4. Based on these two distribution curves, i.e. the simulated measurement data, the identification work is conducted to certify the effectiveness of the inverse method. When no measurement error is considered, the identification results of the heat transfer coefficient distribution on the left surface of the flat plate are
(b) Case B Figure 5 Identification results of the heat transfer coefficient when no measurement error is considered for Case A and Case B ( H 0.01 ) ƽ estimated;ƻ initial; üü exact
In Table 1, the relationship between the ARE of the identification result and the ending criterion of the iteration process is studied systematically. The ARE increases with the increase of the ending criterion.
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Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008 Table 1 Relationship between the ARE of the identification result and the ending criterion İ
eARE (Case A)
eARE (Case B)
0.1
2.2%
5.4%
0.01
1.2%
4.5%
0.001
0.7%
2.9%
When H 0.01 , the AREs of the identification results are smaller than 5%; therefore H 0.01 is used in the following numerical examples in this subsection. In the practical work, errors always accompany the temperature measurement. In Fig. 6 the estimated heat transfer coefficient distributions are plotted for the two test cases when the measurement errors (e) are respectively +1.0°C, 1.0°C, +0.5°C and 0.5°C. It should be mentioned that uniform error is adopted here for every discrete measurement point because, for the thermographic temperature measurement, the effect of the surroundings, imager itself, etc. on the measurement result is nearly the same for every point in the inspection surface. From the figures one can see that the AREs of identification results increase with the increase of the measurement error. When the absolute value of measurement error is 1.0°C, the AREs of the identification results are smaller than 18.3%.
the analytical results for this one-dimensional problem are plotted in Fig. 7 when the measurement errors are +1.0°C and 1.0°C, respectively. From this figure, one can see that most of the errors reported in Fig. 6 and Table 2 are intrinsic for this heat transfer coefficient identification problem, but not resulted from the method employed. For the plate with thermal conductivity of ˉ ˉ 50 W·m 1·K 1, the temperature of the right surface is very close to that of the bulk fluid on the left side, especially when the heat transfer coefficient of the left surface is large; therefore it is much more difficult to increase or decrease 1.0°C of the right surface by changing the heat transfer coefficient of the left surface. This is the very reason why larger errors of the identification results appear in Fig. 7 and also in Fig. 6 at the part where the heat transfer coefficient is larger.
Figure 7 Relative error of the analytical results for the one-dimensional heat transfer coefficient identification problem e/°C: Ƶ 1.0;ƻ1.0 Table 2
(a) Case A e/°C:Ƹ1.0;ƽ 1.0;Ʒ0.5;ƻ 0.5; üü exact
4.2
(b) Case B e/°C:ƽ1.0;ƻ 1.0;Ƹ0.5;ͩ 0.5; üü exact Figure 6 Identification results of the heat transfer coefficient distribution with measurement error in consideration for Case A and Case B ( H 0.01 )
When the heat transfer coefficient does not change in y direction either, the identification problem becomes a one-dimensional one which can be solved analytically based on Eq. (10). The relative errors of
Relationship between the measurement error and the ARE of the identification result e/°C
eARE(Case A)/%
eARE(Case B)/%
0.5
8.6
6.7
0.5
7.4
6.3
1.0
18.3
9.0
1.0
13.2
8.5
2.0
33.1
14.8
2.0
23.8
12.7
Three-dimensional numerical test cases
In this subsection, the numerical method is tested for solving the three-dimensional IHCP for determining the heat transfer coefficient distribution on the left surface of the plate shown in Fig. 1. The dimensions of the plate used in this subsection is 1.0 m, 1.0 m and 0.1 m in x, y and z directions, respectively. The grid meshing of the plate for the finite volume method is mz 4 , mx 21 and m y 21 ( egrid 0.05% ). The heat transfer coefficient distributions to be determined for the two test Cases C and D are shown in Fig. 8. Based on the simulated temperature distributions of the inspection surface which are obtained according to Eq. (1), the heat transfer coefficient is estimated firstly based on Eq. (10) and plotted in Fig. 9. Then the
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(a) Case C (b) Case D Figure 8 Exact heat transfer coefficient distributions to be determined for Case C and Case D
Figure 9
(a) Case C (b) Case D Initial identification results of the heat transfer coefficient obtained by Eq. (10)
(a) Case C (b) Case D Figure 10 Final identification results of the heat transfer coefficient for Case C and Case D when no measurement is considered ( H 1.0 )
(a) Case C (b) Case D Figure 11 Identification results of the heat transfer coefficient for Case C and Case D when measurement error is in consideration ( H 0.01 ) upper: estimated ( e 1.0 °C); lower: estimated ( e 1.0 °C); middle: exact
discrepancy caused by using this one-dimensional inverse function is corrected by the iteration process according to Eq. (5). The final identification results of
the heat transfer coefficient are shown in Fig. 10 when the ending criterion is adopted as H 1.0 . The AREs of the results for Case C and Case D are 0.5% and
Chin. J. Chem. Eng., Vol. 16, No. 6, December 2008
1.9%, respectively, when no measurement error is considered. The estimated and the exact coefficients have a good agreement. In this subsection, the effect of the thermographic temperature measurement error on the accuracy of the identification result of the heat transfer coefficient is also studied and reported in Table 3. The heat transfer coefficients estimated are also plotted in Fig. 11 when the measurement errors are +1.0°C and 1.0°C (the figure for Case D is plotted by rotating an angle for showing the result better). From the table, one can see that when the absolute value of the measurement error is smaller than 2.0°C, the AREs of the estimated results are not more than 23.7%. The results are good enough to be accepted as the solution of the IHCP. Table 3
Relationship between the measurement error and the ARE of the identification result e/°C
eARE (Case A)/%
eARE(Case B)/%
0.5
5.6
2.4
0.5
4.2
2.2
1.0
11.0
4.3
1.0
8.3
3.9
2.0
23.7
8.3
2.0
15.8
7.2
5 EFFECT OF TEMPERATURE MEASUREMENT ERROR
As stated in Ref. [20] for the ODCM, the numerical method used in the MODCM is one of those for regular problem of differential equation, and the stability of the method can guarantee the convergence of the solution. Therefore, the errors occurred in practical inspection work would only lead to errors in the identified results but will not make the method itself ill-posed. That is to say, as long as the ending criterion is small enough, the identification result obtained when the measurement error is considered is also a very accurate solution for this new simulated temperature measurement formed by the original temperature distribution and the measurement error added. Since the relatively larger error in the result caused by the measurement error is the intrinsic characteristic of this IHCP which can be seen from Fig. 7, more effort should be put in the practical inspection and identification work to decrease the measurement error of the infrared imager for obtaining a more accurate identification result. On the analysis and estimation of the thermographic measurement error one can refer to our previous work [22, 23]. 6
CONCLUSIONS
The MODCM along with the finite volume method is applied successfully in solving the IHCP for determining the heat transfer coefficient of one surface of a flat plate based on the thermographic temperature measurements of the opposite surface. A series of nu-
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merical experiments conducted, two-dimensional and three-dimensional, have certified the effectiveness of the method. Since the method is stable, the error of this method caused by a measurement error is a definite value and can be estimated and eliminated if only the measurement error can be estimated precisely in the practical inspection and identification work. This method is proved to be a simple, stable and accurate one which can be extended for solving other inverse problems. NOMENCLATURE H
h hn L m n T To T Ta Tf
matrix of heat transfer coefficient distribution of the left surface correction term correction term ˉ ˉ heat transfer coefficient, W·m 2·K 1 ˉ ˉ heat transfer coefficient of the inspection surface, W·m 2·K 1 dimension of the test piece number of discrete points iteration number matrix of temperature distribution on the inspection surface original measured temperature distribution on the inspection surface temperature, °C ambient temperature, °C temperature of fluid of left side of the plate, °C
ARE i,j o x,y,z
average relative error grid index original measured value in the x,y,z direction, respectively
Hc H cc
Subscripts
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2 3 4 5 6
7 8 9 10
11
12
13
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