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Decentralized fuzzy inference method for estimating thermal boundary condition of a heated cylinder normal to a laminar air stream
Computers and Mathematics with Applications 66 (2013) 1869–1878
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Decentralized fuzzy inference method for estimating thermal boundary condition of a heated cylinder normal to a laminar air stream Zhaoming Luo, Guangjun Wang, Hong Chen ∗ School of Power Engineering, Chongqing University, Chongqing 400044, China Key Laboratory of Ministry of Education of Low-grade Energy Utilization Technologies and Systems, Chongqing University, Chongqing 400044, China
article
info
Keywords: Fuzzy inference Thermal boundary behavior Inverse problem
abstract A decentralized fuzzy inference (DFI) method based on the fuzzy theory is proposed in this study for estimating the heat flux distribution of a heated cylinder experiencing conjugate heat transfer. A group of fuzzy inference units (FIUs) are designed. The deviations between the calculated and measured temperature at each measurement point are taken as the input parameters of FIUs, and the corresponding fuzzy inference components corresponding to the measured temperature are obtained by the FIUs. According to the importance of the various measured temperatures, the fuzzy inference components are then weighted and synthesized to gain the compensations of the guessed heat flux distribution. Ultimately, the prediction of the heat flux is accomplished. Numerical tests are performed to study the effect of initial guessed heat flux, measurement point numbers, measurement errors and the coupling of measurement point numbers and measurement errors on the estimated results. The results show that the DFI method can estimate the heat flux availably and possesses a better anti-ill-posed character and higher accuracy than the conjugate gradient method (CGM). The DFI method shows superiority. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction In direct heat transfer problems, the thermal parameters, boundary conditions and initial conditions of a heat transfer system are known and are used to establish the calculate the temperature field. Conversely, in inverse heat transfer problems (IHTPs), the unknowns are determined through a combination of the direct problem and inverse methods and using the measured temperature, known boundary conditions and initial conditions and thermophysical properties. IHTPs can be encountered in many natural sciences and engineering fields. The solving of inverse heat conduction problems has broad applications in practical engineering areas. These kinds of inverse techniques can be applied to estimating temperature, detecting heat flux, determining source terms, imaging thermophysical properties, identifying boundary conditions, estimating and designing configurations of bodies, etc. However, due to the ill-posed nature, IHTPs have become a kind of typical ill-posed problem. Scholars are active in research on the IHCP and have developed various valuable solving techniques. The traditional techniques for studying IHTPs are mainly the gradient-based methods, such as CGM [1,2] and the Levenberg–Marquardt (L–M) method [2]. However, difficulties still occur due to the ill-posed nature of the inverse problems when gradient-based optimization algorithms have been used [3,4]. The gradient-based methods require the evaluation of derivatives in the objective function and may also
∗
Corresponding author at: School of Power Engineering, Chongqing University, Chongqing 400044, China. Tel.: +86 23 65103512. E-mail addresses: [email protected] (Z.M. Luo), [email protected] (G.J. Wang), [email protected] (H. Chen).
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Nomenclature
(r , θ )
Cylindrical coordinates Temperature ° C Radius of cylinder m 1r Increment in radial coordinate m q Heat flux W/m2 e Temperature deviation ° C 1uk Fuzzy inference component M Number for nodes of angular coordinates N Number for nodes of radial coordinates L Number of measurement points X Sensitivity coefficient [−p, p], [−s, s] Universes of discourse
T r
Greek symbols
λ α 1θ σ ω ε µ
Thermal conductivity W/(mK) Weighting factor Increment in angular coordinate rad Standard deviation of measurement errors Random number Small positive number Degree of membership
Superscript 0 d
Initial guessed value Number of iteration
Subscript mea cal exa cen k i j
Measured value Calculated value Exact value Parameter at center of the cylinder Index of measurement points Index of radial coordinate node Index of angular coordinate node
need information on an initial solution vector for the searched parameters. In these approaches, the use of a bad starting point may result in the solution getting trapped in a local optimum [5]. In addition, the inverse results obtain by the L–M method and CGM may become worse when worse initial guessed values of the unknowns are given, or the measured information contains errors, or the measurement points are not enough [6,7]. In recent years, some intelligent optimization tools, such as the genetic algorithm (GA) [8–11] and particle swarm optimizer algorithm (PSOA) [12], are introduced to solve IHTPs. Although they can be used for solving IHTPs and own good adaptabilities, some deficiencies have been identified [13]. And when the number of optimization parameters is large, especially existing significant spatial distribution characteristics, the disadvantages of the GA could be very apparent [14]. Essentially, IHTPs are a kind of inference issue, an inverse inference from the measured information to the unknown input information. For IHTPs in industry, the direct evidence of the inference process is the measured information of the industrial equipment. Actually, the measured information is incomplete and inevitably contains interfering noise and measurement errors. Accordingly, IHTPs are a kind of uncertain inference issue. Fuzzy inference based on the fuzzy theory has a strong capacity of resisting disturbance to input information, good robustness and fault tolerance to the reasoning process, and can effectively utilize the imprecise, uncertain and incomplete information to infer and make strategic decisions. These characteristics of fuzzy inference can provide new ideas and methods for overcoming the ill-posed nature of inverse problems including IHTPs. The DFI method based on the fuzzy theory first proposed by Wang etc. [6] is established to solve a two-dimensional steady inverse heat conduction problem of estimating boundary condition, and is then applied to estimate heat-flux distribution at the metal–mold interface [7] and the temperature distribution of the furnace inner surface [14]. Advantages of this method have been shown in our research [6,7,14] compared with the L–M method and the CGM and the GA.
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Fig. 1. Diagram of half cylinder with computational mesh.
However, in our earlier research [6,7,14], the weighting matrices are determined relying on empirical correlations, and this is unrealistic for some complex heat transfer systems. To overcome the drawback of obtaining the weighting matrix, the authors improved the DFI method by taking advantage of the sensitivity coefficients to establish the weighting matrix [15]. In this paper, the DFI method advanced with the sensitivity coefficients is applied to solve an inverse conjugate heat transfer problem to estimate the unknown heat flux at the surface of a heated cylinder normal to a laminar air stream. In order to testify to the validity and superiority of this advanced DFI method, numerical tests are performed to study the effects of initial guessed heat flux, the measurement numbers and measurement errors on the predicted results, and comparisons with CGM are also conducted. This is the extended version of our earlier conference paper for FSKD 2012 [15]. Two main new contents: Sensitivity analysis in 5.1 and Test 4: Effect of coupling of measurement errors and measurement points number in 5.5, are added in this extended paper. The remainder of this work is organized as follows. Section 2 formulates the conjugate heat transfer problem. Section 3 provides detailed formulation on the DFI method. Section 4 states the computation procedure for solving the inverse problem using the DFI method. In Section 5 some numerical tests are performed for solving the inverse problem and the results are discussed. Section 6 concludes this paper. The numerical results show that the DFI method is less sensitive to initial guessed heat flux, measurement point numbers and measurement errors, and the DFI method has advantages compared with the CGM. 2. Physical model and problem statement A cylinder with an internal source in its center is considered in this paper. The air flows around it and conjugate heat transfer occurs between the cylinder surface and the air. This results in the spatially non-uniform heat flux, q(θ ). The cylinder is heated by an internal source which is assumed to maintain a constant temperature, Tcen , at the center of the cylinder. The radius of the cylinder is R, and the thermal conductivity is λk. We suppose that this cylinder is long enough so that the end effect can be neglected [16]. Due to the symmetric characteristics, only half of the cylinder is considered, as shown in Fig. 1. The corresponding governing equation and boundary conditions are stated as Eq. (1). 1 ∂ 2 T (r , θ ) ∂ 2 T (r , θ ) 1 ∂ T (r , θ ) + = 0 0 ≤ r ≤ R and 0 ≤ θ ≤ π + ∂r2 r ∂r ∂ r ∂θ 2 T (0, θ ) = Tcen r = 0 ∂ T (r , θ )/∂θ = 0 θ = 0
(1a) (1b) (1c)
∂ T (r , θ )/∂θ = 0 θ = π
(1d)
q(θ ) = −λ(∂ T (r , θ )/∂ r ) r = R.
(1e)
The cross section of the half cylinder is discretized uniformly. There are N nodes along the radial direction and M nodes along the circumferential direction, as shown in Fig. 1. Then the temperature of the cross section can be expressed as Ti,j = T (ri , θj ), where 1θ = π /(M − 1), 1r = R/(N − 1), θj = 1θ (j − 1), ri = 1r (i − 1), i = 1, 2, . . . , N; j = 1, 2, . . . , M. And the heat flux can be expressed as qN ,j . The heat balance method is employed to build the discretization equations of Eq. (1), and the Jacobi iteration method is applied to solve the direct problem expressed by Eq. (1) to obtain the temperature field of this two-dimensional heat transfer system. For the inverse problem of the above conjugate heat transfer process, the heat flux q(θ ) is unknown, and is difficult to be measured directly. q(θ ) should be determined through a combination of the direct problem and the DFI method and using the measured temperature, the known boundary conditions and thermophysical properties. In this paper, we locate L measurement points along r (N − 2). As there are M nodes (N − 2, j) (j = 1, 2, . . . , M ) along r (N − 2), we choose L(L ≤ M )
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Fig. 2. Decentralized fuzzy inference system.
Fig. 3. Membership functions of ek for fuzzy sets An .
nodes from these M ones as the measurement positions for simplification, and the corresponding temperatures are marked as Tk (k = 1, 2, . . . , L). 3. Decentralized fuzzy inference for heat flux 3.1. Decentralized fuzzy inference system The decentralized fuzzy inference system for estimating the heat flux is shown as Fig. 2. This system contains L onedimensional fuzzy inference units FIUk (k = 1, 2, . . . , L). The input information ek is the deviation of the temperature Tmea,k at the kth measurement point: ek = Tcal,k − Tmea,k
(k = 1, 2, . . . , L)
(2)
where Tmea,k is the measured temperature at the kth measurement point. Tcal,k is the calculated temperature corresponding to Tmea,k at the kth measurement point by solving the direct heat transfer problem using the guessed qN ,j . The subscripts cal and mea denote the calculated and measured temperature respectively. ek are regarded as the input information of the FIUk and the fuzzy inference components 1uk are obtained. Further, a weighting and synthesizing strategy is applied to 1uk using the weighting factors αj,k to generate the compensations 1qN ,j of the unknown heat flux qN ,j . Finally, the new guessed heat flux qdN+,j1 is obtained and the heat flux estimation is accomplished, and d denotes the iteration number. 3.2. Fuzzy inference units ek and 1uk are the input and output variables of FIUk. The universes of discourse of ek and 1uk are [−p, p] and [−s, s] separately. Fuzzy sets {A1 , A2 , . . . , A7 } and {B1 , B2 , . . . , B7 } are separately defined in the two universes of discourse for ek and 1uk , and the linguistic values of An and Bn (n = 1, 2, . . . , 7), separately, are NB (negative big), NM (negative medium), NS (negative small), ZO (zero), PS (positive small), PM (positive medium) and PB (positive big). µAn (ek ) and µBn (1uk ) are the triangular memberships of ek and 1uk for the fuzzy sets, and the triangular membership functions are adopted and shown in Figs. 3 and 4 respectively. The fuzzy inference rules of FIUk are determined according to the qualitative understanding of the heat transfer process. When the kth measurement position is considered only, if ek < 0, it indicates that the guessed heat flux qN ,j is higher than the exact value. The guessed heat flux should be decreased appropriately according to the values of ek , that is, the fuzzy inference components 1uk should be less than zero. And the bigger |ek | is, the more the guess is decreased. And the reverse
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Fig. 4. Membership functions of 1uk for fuzzy sets Bn .
is also true. Accordingly, the fuzzy rules are built as follows: If ek If ek If ek If ek If ek If ek If ek
is NB then 1uk is NB. is NM then 1uk is NM. is NS then 1uk is NS. is ZO then 1uk is ZO. is PS then 1uk is PS. is PM then 1uk is PM. is PB then 1uk is PB.
The membership function shown in Fig. 3 is employed to fuzzify temperature deviation ek . And the Mamdani max–min inference [17] combining with the fuzzy inference rules are used to gain the fuzzy set B by Eq. (3) in the range of the universe of discourse [−s, s]. 7
µB (wk ) = max{min[µAn (ek ), µBn (wkn )]}
(3)
n =1
where wk ∈ [−s, s]. Then the center of gravity method [18] is applied to obtain the fuzzy inference components 1uk of FIUk :
1uk =
q
µB (wk )wk dwk
−q
q
µB (wk )dwk .
(4)
−q
3.3. Weighting and synthesizing fuzzy inference components The fuzzy inference component 1uk is the compensation for heat flux of all nodes at the surface when only the kth measured information is considered. Actually, the heat flux of all nodes at the surface has a different effect on the temperature of all measurement positions. Consequently, the whole measured information should be considered when compensating for the unknown heat flux is performed. In this study, by weighting and synthesizing the fuzzy inference components 1uk (k = 1, 2, . . . , L), the compensations 1qN ,j (j = 1, 2, . . . , M ) are established by following Eq. (5):
1qN ,1 α1,1 1qN ,2 α2,1 . = . . . . . 1qN ,M αM ,1
α1,2 α2,2 .. .
αM ,2
··· ··· ··· ···
α1,L 1u1 α2,L 1u2 .. .. . .
αM ,L
(5)
1uL
where αj,k are the weighting factors. The weighting factors αj,k reflect the influence of the heat flux at node (N , j) on the kth temperature at the measurement point. The larger αj,k is the greater the influence is. The weighting matrix is determined relying on empirical correlations in our early researches [6,7], and this is unrealistic for some complex heat transfer systems. We improved the DFI method by taking advantage of the sensitivity coefficients to establish the weighting matrix to overcome the above drawback, as in Eq. (6):
αj,k = Xj,k
L
X j ,k
(6)
k=1
where Xj,k is the sensitivity coefficient of measured temperature Tmea,k for the heat flux qN ,j . It reflects the influence of qN ,j on the measured temperature Tmea,k , the bigger |Xj,k | is, the stronger the influence is. The sensitivity coefficients Xj,k are defined as: Xj,k = ∂ Tmea,k /∂ qN ,j .
(7)
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After weighting and synthesizing, the new guessed heat flux qdN+,j1 is updated as follows: qdN+,j1 = qdN ,j + 1qdN ,j (j = 1, 2, . . . , M ) where d denotes the iteration number. For the inverse process, the following Eq. (9) is taken as the stopping criteria: J =
(8)
L (Tcal,k − Tmea,k )2 ≤ ε
(9)
k=1
where ε is a specified small positive number. 4. Computation procedure The computational procedure for estimation is performed as follows. Step 1: Set a group of measurement points (nodes) and obtain the measured temperature Tmea,k ; Step 2: Give the initial guessed value q0N ,j to heat flux distribution qN ,j (j = 1, 2, . . . , M ) Step 3: Solve the direct problem expressed by Eq. (1) for Tcal,k , and examine whether the stopping criterion Eq. (9) is satisfied. Stop if the criterion is satisfied and take q0N ,j as the estimated results. Otherwise, go on; Step 4: Establish the inputs ek and outputs 1uk of the FIUs by performing the decentralized fuzzy inference; Step 5: Weight and synthesize 1uk to gain the compensations 1qN ,j for the guessed heat flux distribution; Step 6: Update the guessed heat flux using Eq. (9) and return to Step 3. 5. Numerical simulation and discussion In this paper, estimation of the heat flux distribution using the DFI method through numerical simulation is studied, and the estimated results are compared with the CGM. For details of the CGM refer to [19]. The material properties and the geometric parameters of the system are listed as follows: R = 0.05 m;
k = 14 W/(mK);
Tcen = 400 K;
M = N = 11;
p = 3 K;
s = 390 W/m2 .
We solve the direct problem expressed by Eq. (1) using the exact heat flux qexa (θ ) to obtain the exact measurement temperature Texa,k , and then get the measurement temperature Tmea,k by the following equation: Tmea,k = Texa,k + ωσ
(10)
where ω is a random number of normal distribution with zero mean and unit standard deviation over the range of −2.576 ≤ ω ≤ 2.567, and σ is the standard deviation of the measurement errors. The same stopping criterion, denoted by Eq. (9), is adopted for the DFI method and the CGM. According to the discrepancy principal [20], we use ε = Lσ 2 when measurement errors are present, and ε = 0 when the measurement errors are not present. In order to further inspect the accuracy of the inverse methods, we define the following average relative estimation errors E:
E=
M
|qexa,j − qest,j |/qexa,j /M .
(11)
j =1
5.1. Sensitivity analysis Fig. 5 shows the variations of Xj,k for temperature Tk (k = 1, 2, . . . , L) at measurement points. Fig. 5 shows that the sensitivity coefficients take on different varieties, and the positions that the sensitivity coefficients reach the max values are distinctly different for different nodes (N , j). That is, there are evidently different sensitive measurement regions for different qN ,j . And qN ,j , therefore, are not linear correlated, and can be estimated using the measured temperature. 5.2. Test 1: effect of initial guessed heat flux In order to study the effect of initial guessed heat flux on the inverse results, we choose that the standard deviation of the measurement errors is σ = 0, the measurement point number is L = M = 11. Two different initial values, q0N ,j = 0 W/m2
and 800 W/m2 , are given. The results are shown in Figs. 6 and 7, and the average relative estimation errors are shown in Table 1. When the initial guessed heat flux is 800 W/m2 , the inverse results gained by both the DFI method and CGM have excellent agreement with the exact results. However, the inverse results of CGM become worse when q0N ,j = 0 W/m2 is chosen, but the results of the DFI method still match the exact ones very well. And the average relative estimation errors E shown in Table 1 clearly reveals this conclusion as well. Comparing Figs. 6 and 7 and Table 1, we can conclude that the DFI method reduces the sensitivities of the prediction results on the initial guessed values.
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0.0
0.5
1.0
1.5
2.0
2.5
Fig. 5. Sensitivity coefficients Xj,k .
Fig. 6. Estimated results for initial value 800 W/m2 .
Fig. 7. Estimated results for initial value 0 W/m2 .
3.0
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DFI CGM
Test 1 q0N ,j = 800 W/m2
q0N ,j = 0 W/m2
4.04 3.16
4.04 5.25
Test 2
Test 3
Test 4
11.10 15.47
4.37 18.58
8.66 21.61
Fig. 8. Estimated results when σ = 0.15.
5.3. Test 2: effect of measurement errors In engineering applications, the measurement errors are inescapable, and they have a great influence on the performance of optimization methods. Accordingly, studying the performance of optimization methods when the errors are present is of great significance. We consider that the measurement point number is L = M = 11, the initial guessed values are q0N ,j = 800 W/m2 , and the standard deviation of the measurement errors is σ = 0.15. The predicted results are shown in Fig. 8 and Table 1. Both the inverse results of the DFI method and CGM take on deviations, but the errors of CGM are more obviously than that of the DFI method. The values of E for the DFI method and CGM are 11.10% and 15.47% respectively. The DFI method can still be used to predict the unknown. We can clearly see that the DFI method decreases the sensitivity of errors. 5.4. Test 3: effect of measurement points number In applications, we are not always allowed to locate too many measurement points. Using less measurement points is of much practical significance. In this test, we study the effect of measurement point number on the performance of the DFI method. We consider that σ = 0, the initial guessed values are q0N ,j = 800 W/m2 , and the nodes (N − 2, 2), (N − 2, 4), (N − 2, 6), (N − 2, 8) and (N − 2, 10) are chosen as the measurement points, that is L = 5. The estimated results are shown in Fig. 9 and Table 1. The values of E for DFI method and CGM are 4.37% and 18.58% respectively. We can clearly conclude that the results of CGM become worse, while the results of the DFI method still have excellent approximations with the exact values. Accordingly, we can conclude that the DFI method weakens the effects of the measurement point. 5.5. Test 4: effect of coupling of measurement errors and measurement points number The coupling effects of measurement errors and measurement point numbers on the results of both the DFI method and CGM are studied. When the initial guessed values are q0N ,j = 800 W/m2 , measurement point numbers L = 5 and measurement error is σ = 0.1. The estimated results are shown in Fig. 10 and Table 1. From Fig. 10 and Table 1, we can conclude that the estimated results of the DFI method still meet the exact values very well, but the results of the CGM obviously deviate from the exact ones. The values of E for DFI method and CGM are 8.66% and 21.61% respectively. We can conclude that the DFI method shows better anti-ill-posed character than the CGM. 6. Conclusions The DFI method is established in this study to predict the unknown heat flux distribution at the cylinder surface. The effects of the initial guessed heat flux distribution, the measurement point number, the measurement errors and the coupling
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Fig. 9. Estimated results when L = 5.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Fig. 10. Estimated results when L = 5 and σ = 0.1.
of measurement point number and measurement errors on the estimated results are studied and the comparisons between the DFI method and the CGM are also conducted. The numerical experiments show that the DFI method can estimate the heat flux availably. Compared with the CGM, the results also show that the DFI method reduces the dependence of the estimated results on the initial guessed value and the measurement point numbers, and weakens the effects of measurement errors on predicted results. The DFI method possesses better anti-ill-posed character and higher accuracy than the CGM. This paper presents the DFI method and its better results. These results indicate that DFI method and the inverse heat transfer scheme can be applied to identify the parameters of heat transfer systems, reconstruct the heat transfer environment, optimize and design heat transfer systems. However, limitations still exist. The comparison with other papers tackling the same topics is not done and should be done. In addition, the decentralized fuzzy inference system may be complex and difficult to implement. The above limitations will be tackled in our future research. Acknowledgments The authors are very grateful for the support of the National Nature Science Foundation of China (No. 51176211) and the Fundamental Research Fund for the Central Universities (No. CDJZR10140003). References [1] O.M. Alifanov, N.V. Kerov, Determination of thermal load parameters by solving the two-dimensional inverse heat-conduction problem, J. Eng. Phys. Thermophys. 41 (1981) 1049–1053.
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[2] B. Sawaf, M.N. Özisik, Determining the constant thermal conductivities of orthotropic materials by inverse analysis, Int. Commun. Heat Mass Transfer 22 (1995) 201–211. [3] M.M. Aral, Aquifer parameter prediction in leaky aquifers, J. Hydrol. 80 (1985) 19–44. [4] H. Karahan, M.T. Ayvaz, Groundwater parameter estimation by optimization and dual reciprocity finite differences method, J. Porous Media 8 (2005) 211–223. [5] M. Tamer Ayvaz, H. Karahan, M.M. Aral, Aquifer parameter and zone structure estimation using kernel-based fuzzy c-means clustering and genetic algorithm, J. Hydrol. 343 (2007) 240–253. [6] G.J. Wang, L.N. Zhu, H. Chen, A decentralized fuzzy inference method for solving two-dimensional steady inverse heat conduction problem of estimating boundary condition, Int. J. Heat Mass Transfer 54 (2011) 2782–2788. [7] L.N. Zhu, G.J. Wang, H. Chen, Z.M. Luo, Inverse estimation for heat flux distribution at the metal–mold interface using fuzzy inference, J. Heat TransferTrans. ASME 133 (2011) 081602. [8] M. Raudensky, J. Horsky, J. Krejsa, L. Slama, Usage of artificial intelligence methods in inverse problems for estimation of material parameters, Internat. J. Numer. Methods Heat Fluid Flow 6 (1996) 19–29. [9] G.R. Liu, J.J. Zhou, J.G. Wang, Coefficient identification in electronic system cooling simulation through genetic algorithm, Comput. Struct. 80 (2002) 23–30. [10] F.B. Liu, A modified genetic algorithm for solving the inverse heat transfer problem of estimating plan heat source, Int. J. Heat Mass Transfer 51 (2008) 3745–3752. [11] A. Adili, N. Hasni, C. Kerkeni, S.B. Nasrallah, An inverse problem based on genetic algorithm to estimate thermophysical properties of fouling, Int. J. Therm. Sci. 49 (2010) 889–900. [12] H. Qi, L.M. Ruan, H.H. Zhang, Y.M. Wang, H.P. Tan, Inverse radiation analysis of a one-dimensional participating slab by stochastic particle swarm optimizer algorithm, Int. J. Therm. Sci. 46 (2007) 649–661. [13] J. Kennedy, The particle swarm: social adaptation of knowledge, in: Proc. 1997 IEEE Int. Conference of Evolutionary Computation, ICEC’97, Indianapolis, IN, USA, 1997, pp. 303–308. [14] G.J. Wang, Z.M. Luo, L.N. Zhu, H. Chen, L.H. Zhang, Fuzzy estimation for temperature distribution of furnace inner surface, Int. J. Therm. Sci. 51 (2012) 84–90. [15] Z.M. Luo, H. Chen, G.J. Wang, Inverse estimation for the thermal boundary conditions of conjugate heat transfer using decentralized fuzzy inference method, in: 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2012, pp. 457–461. [16] J.H. Lin, C.K. Chen, Y.T. Yang, The inverse estimation of the thermal boundary behavior of a heated cylinder normal to a laminar air stream, Int. J. Heat Mass Transfer 43 (2000) 3991–4001. [17] J. Sargolzaei, M. Khoshnoodi, N. Saghatoleslami, M. Mousavi, Fuzzy inference system to modeling of crossflow milk ultrafiltration, Appl. Soft Comput. 8 (2008) 456–465. [18] E. Van Broekhoven, B. De Baets, Fast and accurate center of gravity defuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions, Fuzzy Sets and Systems 157 (2006) 904–918. [19] H.Y. Li, W.M. Yan, Identification of wall heat flux for turbulent forced convection by inverse analysis, Int. J. Heat Mass Transfer 46 (2003) 1041–1048. [20] O.M. Alifanov, Solution of an inverse problem of heat conduction by iteration methods, J. Eng. Phys. Thermophys. 26 (1974) 471–476.
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Decentralized fuzzy inference method for estimating thermal boundary condition of a heated cylinder normal to a laminar air stream Zhaoming Luo, Guangjun Wang, Hong Chen ∗ School of Power Engineering, Chongqing University, Chongqing 400044, China Key Laboratory of Ministry of Education of Low-grade Energy Utilization Technologies and Systems, Chongqing University, Chongqing 400044, China
article
info
Keywords: Fuzzy inference Thermal boundary behavior Inverse problem
abstract A decentralized fuzzy inference (DFI) method based on the fuzzy theory is proposed in this study for estimating the heat flux distribution of a heated cylinder experiencing conjugate heat transfer. A group of fuzzy inference units (FIUs) are designed. The deviations between the calculated and measured temperature at each measurement point are taken as the input parameters of FIUs, and the corresponding fuzzy inference components corresponding to the measured temperature are obtained by the FIUs. According to the importance of the various measured temperatures, the fuzzy inference components are then weighted and synthesized to gain the compensations of the guessed heat flux distribution. Ultimately, the prediction of the heat flux is accomplished. Numerical tests are performed to study the effect of initial guessed heat flux, measurement point numbers, measurement errors and the coupling of measurement point numbers and measurement errors on the estimated results. The results show that the DFI method can estimate the heat flux availably and possesses a better anti-ill-posed character and higher accuracy than the conjugate gradient method (CGM). The DFI method shows superiority. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction In direct heat transfer problems, the thermal parameters, boundary conditions and initial conditions of a heat transfer system are known and are used to establish the calculate the temperature field. Conversely, in inverse heat transfer problems (IHTPs), the unknowns are determined through a combination of the direct problem and inverse methods and using the measured temperature, known boundary conditions and initial conditions and thermophysical properties. IHTPs can be encountered in many natural sciences and engineering fields. The solving of inverse heat conduction problems has broad applications in practical engineering areas. These kinds of inverse techniques can be applied to estimating temperature, detecting heat flux, determining source terms, imaging thermophysical properties, identifying boundary conditions, estimating and designing configurations of bodies, etc. However, due to the ill-posed nature, IHTPs have become a kind of typical ill-posed problem. Scholars are active in research on the IHCP and have developed various valuable solving techniques. The traditional techniques for studying IHTPs are mainly the gradient-based methods, such as CGM [1,2] and the Levenberg–Marquardt (L–M) method [2]. However, difficulties still occur due to the ill-posed nature of the inverse problems when gradient-based optimization algorithms have been used [3,4]. The gradient-based methods require the evaluation of derivatives in the objective function and may also
∗
Corresponding author at: School of Power Engineering, Chongqing University, Chongqing 400044, China. Tel.: +86 23 65103512. E-mail addresses: [email protected] (Z.M. Luo), [email protected] (G.J. Wang), [email protected] (H. Chen).
0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.camwa.2013.07.019
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Nomenclature
(r , θ )
Cylindrical coordinates Temperature ° C Radius of cylinder m 1r Increment in radial coordinate m q Heat flux W/m2 e Temperature deviation ° C 1uk Fuzzy inference component M Number for nodes of angular coordinates N Number for nodes of radial coordinates L Number of measurement points X Sensitivity coefficient [−p, p], [−s, s] Universes of discourse
T r
Greek symbols
λ α 1θ σ ω ε µ
Thermal conductivity W/(mK) Weighting factor Increment in angular coordinate rad Standard deviation of measurement errors Random number Small positive number Degree of membership
Superscript 0 d
Initial guessed value Number of iteration
Subscript mea cal exa cen k i j
Measured value Calculated value Exact value Parameter at center of the cylinder Index of measurement points Index of radial coordinate node Index of angular coordinate node
need information on an initial solution vector for the searched parameters. In these approaches, the use of a bad starting point may result in the solution getting trapped in a local optimum [5]. In addition, the inverse results obtain by the L–M method and CGM may become worse when worse initial guessed values of the unknowns are given, or the measured information contains errors, or the measurement points are not enough [6,7]. In recent years, some intelligent optimization tools, such as the genetic algorithm (GA) [8–11] and particle swarm optimizer algorithm (PSOA) [12], are introduced to solve IHTPs. Although they can be used for solving IHTPs and own good adaptabilities, some deficiencies have been identified [13]. And when the number of optimization parameters is large, especially existing significant spatial distribution characteristics, the disadvantages of the GA could be very apparent [14]. Essentially, IHTPs are a kind of inference issue, an inverse inference from the measured information to the unknown input information. For IHTPs in industry, the direct evidence of the inference process is the measured information of the industrial equipment. Actually, the measured information is incomplete and inevitably contains interfering noise and measurement errors. Accordingly, IHTPs are a kind of uncertain inference issue. Fuzzy inference based on the fuzzy theory has a strong capacity of resisting disturbance to input information, good robustness and fault tolerance to the reasoning process, and can effectively utilize the imprecise, uncertain and incomplete information to infer and make strategic decisions. These characteristics of fuzzy inference can provide new ideas and methods for overcoming the ill-posed nature of inverse problems including IHTPs. The DFI method based on the fuzzy theory first proposed by Wang etc. [6] is established to solve a two-dimensional steady inverse heat conduction problem of estimating boundary condition, and is then applied to estimate heat-flux distribution at the metal–mold interface [7] and the temperature distribution of the furnace inner surface [14]. Advantages of this method have been shown in our research [6,7,14] compared with the L–M method and the CGM and the GA.
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Fig. 1. Diagram of half cylinder with computational mesh.
However, in our earlier research [6,7,14], the weighting matrices are determined relying on empirical correlations, and this is unrealistic for some complex heat transfer systems. To overcome the drawback of obtaining the weighting matrix, the authors improved the DFI method by taking advantage of the sensitivity coefficients to establish the weighting matrix [15]. In this paper, the DFI method advanced with the sensitivity coefficients is applied to solve an inverse conjugate heat transfer problem to estimate the unknown heat flux at the surface of a heated cylinder normal to a laminar air stream. In order to testify to the validity and superiority of this advanced DFI method, numerical tests are performed to study the effects of initial guessed heat flux, the measurement numbers and measurement errors on the predicted results, and comparisons with CGM are also conducted. This is the extended version of our earlier conference paper for FSKD 2012 [15]. Two main new contents: Sensitivity analysis in 5.1 and Test 4: Effect of coupling of measurement errors and measurement points number in 5.5, are added in this extended paper. The remainder of this work is organized as follows. Section 2 formulates the conjugate heat transfer problem. Section 3 provides detailed formulation on the DFI method. Section 4 states the computation procedure for solving the inverse problem using the DFI method. In Section 5 some numerical tests are performed for solving the inverse problem and the results are discussed. Section 6 concludes this paper. The numerical results show that the DFI method is less sensitive to initial guessed heat flux, measurement point numbers and measurement errors, and the DFI method has advantages compared with the CGM. 2. Physical model and problem statement A cylinder with an internal source in its center is considered in this paper. The air flows around it and conjugate heat transfer occurs between the cylinder surface and the air. This results in the spatially non-uniform heat flux, q(θ ). The cylinder is heated by an internal source which is assumed to maintain a constant temperature, Tcen , at the center of the cylinder. The radius of the cylinder is R, and the thermal conductivity is λk. We suppose that this cylinder is long enough so that the end effect can be neglected [16]. Due to the symmetric characteristics, only half of the cylinder is considered, as shown in Fig. 1. The corresponding governing equation and boundary conditions are stated as Eq. (1). 1 ∂ 2 T (r , θ ) ∂ 2 T (r , θ ) 1 ∂ T (r , θ ) + = 0 0 ≤ r ≤ R and 0 ≤ θ ≤ π + ∂r2 r ∂r ∂ r ∂θ 2 T (0, θ ) = Tcen r = 0 ∂ T (r , θ )/∂θ = 0 θ = 0
(1a) (1b) (1c)
∂ T (r , θ )/∂θ = 0 θ = π
(1d)
q(θ ) = −λ(∂ T (r , θ )/∂ r ) r = R.
(1e)
The cross section of the half cylinder is discretized uniformly. There are N nodes along the radial direction and M nodes along the circumferential direction, as shown in Fig. 1. Then the temperature of the cross section can be expressed as Ti,j = T (ri , θj ), where 1θ = π /(M − 1), 1r = R/(N − 1), θj = 1θ (j − 1), ri = 1r (i − 1), i = 1, 2, . . . , N; j = 1, 2, . . . , M. And the heat flux can be expressed as qN ,j . The heat balance method is employed to build the discretization equations of Eq. (1), and the Jacobi iteration method is applied to solve the direct problem expressed by Eq. (1) to obtain the temperature field of this two-dimensional heat transfer system. For the inverse problem of the above conjugate heat transfer process, the heat flux q(θ ) is unknown, and is difficult to be measured directly. q(θ ) should be determined through a combination of the direct problem and the DFI method and using the measured temperature, the known boundary conditions and thermophysical properties. In this paper, we locate L measurement points along r (N − 2). As there are M nodes (N − 2, j) (j = 1, 2, . . . , M ) along r (N − 2), we choose L(L ≤ M )
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Fig. 2. Decentralized fuzzy inference system.
Fig. 3. Membership functions of ek for fuzzy sets An .
nodes from these M ones as the measurement positions for simplification, and the corresponding temperatures are marked as Tk (k = 1, 2, . . . , L). 3. Decentralized fuzzy inference for heat flux 3.1. Decentralized fuzzy inference system The decentralized fuzzy inference system for estimating the heat flux is shown as Fig. 2. This system contains L onedimensional fuzzy inference units FIUk (k = 1, 2, . . . , L). The input information ek is the deviation of the temperature Tmea,k at the kth measurement point: ek = Tcal,k − Tmea,k
(k = 1, 2, . . . , L)
(2)
where Tmea,k is the measured temperature at the kth measurement point. Tcal,k is the calculated temperature corresponding to Tmea,k at the kth measurement point by solving the direct heat transfer problem using the guessed qN ,j . The subscripts cal and mea denote the calculated and measured temperature respectively. ek are regarded as the input information of the FIUk and the fuzzy inference components 1uk are obtained. Further, a weighting and synthesizing strategy is applied to 1uk using the weighting factors αj,k to generate the compensations 1qN ,j of the unknown heat flux qN ,j . Finally, the new guessed heat flux qdN+,j1 is obtained and the heat flux estimation is accomplished, and d denotes the iteration number. 3.2. Fuzzy inference units ek and 1uk are the input and output variables of FIUk. The universes of discourse of ek and 1uk are [−p, p] and [−s, s] separately. Fuzzy sets {A1 , A2 , . . . , A7 } and {B1 , B2 , . . . , B7 } are separately defined in the two universes of discourse for ek and 1uk , and the linguistic values of An and Bn (n = 1, 2, . . . , 7), separately, are NB (negative big), NM (negative medium), NS (negative small), ZO (zero), PS (positive small), PM (positive medium) and PB (positive big). µAn (ek ) and µBn (1uk ) are the triangular memberships of ek and 1uk for the fuzzy sets, and the triangular membership functions are adopted and shown in Figs. 3 and 4 respectively. The fuzzy inference rules of FIUk are determined according to the qualitative understanding of the heat transfer process. When the kth measurement position is considered only, if ek < 0, it indicates that the guessed heat flux qN ,j is higher than the exact value. The guessed heat flux should be decreased appropriately according to the values of ek , that is, the fuzzy inference components 1uk should be less than zero. And the bigger |ek | is, the more the guess is decreased. And the reverse
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Fig. 4. Membership functions of 1uk for fuzzy sets Bn .
is also true. Accordingly, the fuzzy rules are built as follows: If ek If ek If ek If ek If ek If ek If ek
is NB then 1uk is NB. is NM then 1uk is NM. is NS then 1uk is NS. is ZO then 1uk is ZO. is PS then 1uk is PS. is PM then 1uk is PM. is PB then 1uk is PB.
The membership function shown in Fig. 3 is employed to fuzzify temperature deviation ek . And the Mamdani max–min inference [17] combining with the fuzzy inference rules are used to gain the fuzzy set B by Eq. (3) in the range of the universe of discourse [−s, s]. 7
µB (wk ) = max{min[µAn (ek ), µBn (wkn )]}
(3)
n =1
where wk ∈ [−s, s]. Then the center of gravity method [18] is applied to obtain the fuzzy inference components 1uk of FIUk :
1uk =
q
µB (wk )wk dwk
−q
q
µB (wk )dwk .
(4)
−q
3.3. Weighting and synthesizing fuzzy inference components The fuzzy inference component 1uk is the compensation for heat flux of all nodes at the surface when only the kth measured information is considered. Actually, the heat flux of all nodes at the surface has a different effect on the temperature of all measurement positions. Consequently, the whole measured information should be considered when compensating for the unknown heat flux is performed. In this study, by weighting and synthesizing the fuzzy inference components 1uk (k = 1, 2, . . . , L), the compensations 1qN ,j (j = 1, 2, . . . , M ) are established by following Eq. (5):
1qN ,1 α1,1 1qN ,2 α2,1 . = . . . . . 1qN ,M αM ,1
α1,2 α2,2 .. .
αM ,2
··· ··· ··· ···
α1,L 1u1 α2,L 1u2 .. .. . .
αM ,L
(5)
1uL
where αj,k are the weighting factors. The weighting factors αj,k reflect the influence of the heat flux at node (N , j) on the kth temperature at the measurement point. The larger αj,k is the greater the influence is. The weighting matrix is determined relying on empirical correlations in our early researches [6,7], and this is unrealistic for some complex heat transfer systems. We improved the DFI method by taking advantage of the sensitivity coefficients to establish the weighting matrix to overcome the above drawback, as in Eq. (6):
αj,k = Xj,k
L
X j ,k
(6)
k=1
where Xj,k is the sensitivity coefficient of measured temperature Tmea,k for the heat flux qN ,j . It reflects the influence of qN ,j on the measured temperature Tmea,k , the bigger |Xj,k | is, the stronger the influence is. The sensitivity coefficients Xj,k are defined as: Xj,k = ∂ Tmea,k /∂ qN ,j .
(7)
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After weighting and synthesizing, the new guessed heat flux qdN+,j1 is updated as follows: qdN+,j1 = qdN ,j + 1qdN ,j (j = 1, 2, . . . , M ) where d denotes the iteration number. For the inverse process, the following Eq. (9) is taken as the stopping criteria: J =
(8)
L (Tcal,k − Tmea,k )2 ≤ ε
(9)
k=1
where ε is a specified small positive number. 4. Computation procedure The computational procedure for estimation is performed as follows. Step 1: Set a group of measurement points (nodes) and obtain the measured temperature Tmea,k ; Step 2: Give the initial guessed value q0N ,j to heat flux distribution qN ,j (j = 1, 2, . . . , M ) Step 3: Solve the direct problem expressed by Eq. (1) for Tcal,k , and examine whether the stopping criterion Eq. (9) is satisfied. Stop if the criterion is satisfied and take q0N ,j as the estimated results. Otherwise, go on; Step 4: Establish the inputs ek and outputs 1uk of the FIUs by performing the decentralized fuzzy inference; Step 5: Weight and synthesize 1uk to gain the compensations 1qN ,j for the guessed heat flux distribution; Step 6: Update the guessed heat flux using Eq. (9) and return to Step 3. 5. Numerical simulation and discussion In this paper, estimation of the heat flux distribution using the DFI method through numerical simulation is studied, and the estimated results are compared with the CGM. For details of the CGM refer to [19]. The material properties and the geometric parameters of the system are listed as follows: R = 0.05 m;
k = 14 W/(mK);
Tcen = 400 K;
M = N = 11;
p = 3 K;
s = 390 W/m2 .
We solve the direct problem expressed by Eq. (1) using the exact heat flux qexa (θ ) to obtain the exact measurement temperature Texa,k , and then get the measurement temperature Tmea,k by the following equation: Tmea,k = Texa,k + ωσ
(10)
where ω is a random number of normal distribution with zero mean and unit standard deviation over the range of −2.576 ≤ ω ≤ 2.567, and σ is the standard deviation of the measurement errors. The same stopping criterion, denoted by Eq. (9), is adopted for the DFI method and the CGM. According to the discrepancy principal [20], we use ε = Lσ 2 when measurement errors are present, and ε = 0 when the measurement errors are not present. In order to further inspect the accuracy of the inverse methods, we define the following average relative estimation errors E:
E=
M
|qexa,j − qest,j |/qexa,j /M .
(11)
j =1
5.1. Sensitivity analysis Fig. 5 shows the variations of Xj,k for temperature Tk (k = 1, 2, . . . , L) at measurement points. Fig. 5 shows that the sensitivity coefficients take on different varieties, and the positions that the sensitivity coefficients reach the max values are distinctly different for different nodes (N , j). That is, there are evidently different sensitive measurement regions for different qN ,j . And qN ,j , therefore, are not linear correlated, and can be estimated using the measured temperature. 5.2. Test 1: effect of initial guessed heat flux In order to study the effect of initial guessed heat flux on the inverse results, we choose that the standard deviation of the measurement errors is σ = 0, the measurement point number is L = M = 11. Two different initial values, q0N ,j = 0 W/m2
and 800 W/m2 , are given. The results are shown in Figs. 6 and 7, and the average relative estimation errors are shown in Table 1. When the initial guessed heat flux is 800 W/m2 , the inverse results gained by both the DFI method and CGM have excellent agreement with the exact results. However, the inverse results of CGM become worse when q0N ,j = 0 W/m2 is chosen, but the results of the DFI method still match the exact ones very well. And the average relative estimation errors E shown in Table 1 clearly reveals this conclusion as well. Comparing Figs. 6 and 7 and Table 1, we can conclude that the DFI method reduces the sensitivities of the prediction results on the initial guessed values.
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0.0
0.5
1.0
1.5
2.0
2.5
Fig. 5. Sensitivity coefficients Xj,k .
Fig. 6. Estimated results for initial value 800 W/m2 .
Fig. 7. Estimated results for initial value 0 W/m2 .
3.0
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Z.M. Luo et al. / Computers and Mathematics with Applications 66 (2013) 1869–1878 Table 1 Average relative estimation errors. E (%)
DFI CGM
Test 1 q0N ,j = 800 W/m2
q0N ,j = 0 W/m2
4.04 3.16
4.04 5.25
Test 2
Test 3
Test 4
11.10 15.47
4.37 18.58
8.66 21.61
Fig. 8. Estimated results when σ = 0.15.
5.3. Test 2: effect of measurement errors In engineering applications, the measurement errors are inescapable, and they have a great influence on the performance of optimization methods. Accordingly, studying the performance of optimization methods when the errors are present is of great significance. We consider that the measurement point number is L = M = 11, the initial guessed values are q0N ,j = 800 W/m2 , and the standard deviation of the measurement errors is σ = 0.15. The predicted results are shown in Fig. 8 and Table 1. Both the inverse results of the DFI method and CGM take on deviations, but the errors of CGM are more obviously than that of the DFI method. The values of E for the DFI method and CGM are 11.10% and 15.47% respectively. The DFI method can still be used to predict the unknown. We can clearly see that the DFI method decreases the sensitivity of errors. 5.4. Test 3: effect of measurement points number In applications, we are not always allowed to locate too many measurement points. Using less measurement points is of much practical significance. In this test, we study the effect of measurement point number on the performance of the DFI method. We consider that σ = 0, the initial guessed values are q0N ,j = 800 W/m2 , and the nodes (N − 2, 2), (N − 2, 4), (N − 2, 6), (N − 2, 8) and (N − 2, 10) are chosen as the measurement points, that is L = 5. The estimated results are shown in Fig. 9 and Table 1. The values of E for DFI method and CGM are 4.37% and 18.58% respectively. We can clearly conclude that the results of CGM become worse, while the results of the DFI method still have excellent approximations with the exact values. Accordingly, we can conclude that the DFI method weakens the effects of the measurement point. 5.5. Test 4: effect of coupling of measurement errors and measurement points number The coupling effects of measurement errors and measurement point numbers on the results of both the DFI method and CGM are studied. When the initial guessed values are q0N ,j = 800 W/m2 , measurement point numbers L = 5 and measurement error is σ = 0.1. The estimated results are shown in Fig. 10 and Table 1. From Fig. 10 and Table 1, we can conclude that the estimated results of the DFI method still meet the exact values very well, but the results of the CGM obviously deviate from the exact ones. The values of E for DFI method and CGM are 8.66% and 21.61% respectively. We can conclude that the DFI method shows better anti-ill-posed character than the CGM. 6. Conclusions The DFI method is established in this study to predict the unknown heat flux distribution at the cylinder surface. The effects of the initial guessed heat flux distribution, the measurement point number, the measurement errors and the coupling
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Fig. 9. Estimated results when L = 5.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Fig. 10. Estimated results when L = 5 and σ = 0.1.
of measurement point number and measurement errors on the estimated results are studied and the comparisons between the DFI method and the CGM are also conducted. The numerical experiments show that the DFI method can estimate the heat flux availably. Compared with the CGM, the results also show that the DFI method reduces the dependence of the estimated results on the initial guessed value and the measurement point numbers, and weakens the effects of measurement errors on predicted results. The DFI method possesses better anti-ill-posed character and higher accuracy than the CGM. This paper presents the DFI method and its better results. These results indicate that DFI method and the inverse heat transfer scheme can be applied to identify the parameters of heat transfer systems, reconstruct the heat transfer environment, optimize and design heat transfer systems. However, limitations still exist. The comparison with other papers tackling the same topics is not done and should be done. In addition, the decentralized fuzzy inference system may be complex and difficult to implement. The above limitations will be tackled in our future research. Acknowledgments The authors are very grateful for the support of the National Nature Science Foundation of China (No. 51176211) and the Fundamental Research Fund for the Central Universities (No. CDJZR10140003). References [1] O.M. Alifanov, N.V. Kerov, Determination of thermal load parameters by solving the two-dimensional inverse heat-conduction problem, J. Eng. Phys. Thermophys. 41 (1981) 1049–1053.
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