Application of improved decentralized fuzzy inference methods for estimating the thermal boundary condition of participating medium

International Journal of Thermal Sciences 149 (2020) 106216

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International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts

Application of improved decentralized fuzzy inference methods for estimating the thermal boundary condition of participating medium Shuangcheng Sun a, b, Guangjun Wang a, b, Hong Chen a, b, * a b

School of Energy and Power Engineering, Chongqing University, Chongqing, 400044, PR China Key Laboratory of Low-grade Energy Utilization Technologies and Systems, Chongqing University, Ministry of Education, Chongqing, 400044, PR China

A R T I C L E I N F O

A B S T R A C T

Keywords: Inverse heat transfer Coupled radiation and conduction Fuzzy inference method Heat flux estimation Participating medium

The decentralized fuzzy inference method (DFIM) was applied to estimate the time-dependent heat flux of 1D participating medium. The direct problem concerned on coupled radiation and conduction heat transfer in the medium was solved by the finite volume method and discrete ordinate method. The simulated boundary tem­ perature was served as input for the inverse analysis. The inverse problem was formulated as an optimization approach. Three improved decentralized fuzzy inference methods (IDFIMs) were developed to accelerate the convergence rate and enhance the estimation accuracy. Five kinds of time-dependent heat fluxes were considered to test the performance of the present inverse technique. No prior information on the functional forms of the unknown boundary conditions was needed for the inverse analysis. All retrieval results showed that the incident heat flux of participating medium can be accurately estimated by DFIMs. The proposed IDFIMs achieved better performance than the original DFIM in terms of computational accuracy and efficiency. Moreover, a comparison between the IDFIM and other optimization techniques was conducted. The proposed IDFIM was proved to be more efficient and accurate than conjugate gradient method, Levenberg-Marquardt method, stochastic particle swarm optimization algorithm and genetic algorithm.

1. Introduction Accurate knowledge of thermal boundary conditions in participating medium plays a crucial role in numerous engineering fields, such as continuous casting [1,2], glass industry [3], gas tungsten arc welding operation [4], tumor thermotherapy [5], industrial furnace [6], aero­ space engineering [7,8], automobile manufacture [9], material pro­ cessing [10], to name a few. However, the measurement of time-dependent boundary heat flux still remains a challenge in many practical applications, such as extreme thermal environment, high heating rate, external flow and surface condensation conditions [11,12]. During the last few decades, a great quantity of inverse techniques have been proposed and developed to solve inverse heat transfer problems, in which the surface heat flux is estimated by solving optimization tasks from the knowledge of temperature signals. Generally, the theoretical techniques for solving inverse heat transfer problems can be classified into two groups: stochastic intelligent algo­ rithm and gradient-based method. Stochastic intelligent algorithm is easy to implement and independent on initial guessed values. Mean­ while, many feasible solutions can be treated in parallel at each

iteration. Quite a few stochastic techniques including artificial neural network (ANN), genetic algorithm (GA), particle swarm optimization (PSO) algorithm and krill herd (KH) algorithm have been successfully applied to solve inverse heat transfer problems, through which the op­ tical and thermophysical properties of participating medium were recovered [13–25]. However, the stochastic intelligent algorithm is formulated using empiric variables and the searching process is proba­ bilistic. Hence, it is extremely difficult to solve large-scale estimation problems using heuristic algorithms without prior information, espe­ cially when the parameters to be estimated are closely related. Gradient-based method is an efficient and robust tool for solving optimization problems. Numerous estimation tasks have been solved using the gradient-based method due to its high computational effi­ ciency. For instance, Liu et al. [26] employed conjugate gradient method (CGM) to estimate the incident heat flux in a participating slab, in which the heat flux was considered as a constant value. Udayraj et al. [1] applied CGM to identify the time-dependent heat flux in continuous casting mold by solving an inverse heat conduction problem (IHCP). Yang et al. [9] used CGM to estimate the space- and time-dependent heat flux of the disc in a disc brake system from the knowledge of tempera­ ture measurements taken within the disc. Singh et al. [27] utilized

* Corresponding author. School of Energy and Power Engineering, Chongqing University 174, Shazheng Street, Shapingba District, Chongqing, 400044, PR China. E-mail address: [email protected] (H. Chen). https://doi.org/10.1016/j.ijthermalsci.2019.106216 Received 19 April 2019; Received in revised form 11 October 2019; Accepted 2 December 2019 Available online 9 December 2019 1290-0729/© 2019 Published by Elsevier Masson SAS.

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International Journal of Thermal Sciences 149 (2020) 106216

Nomenclature A B cp d e F h I K L m m0 n n N NN q r t T x

α β βe δ Δt

ε εrel

boundary emissivity scattering phase function absorption coefficient, m 1 scattering coefficient, m 1 thermal conductivity, W/(m⋅ K) density, kg/m3 standard deviation or Stefan-Boltzmann constant, W/(m2⋅ K4) sampling time, s a normal distribution random value scattering direction incident direction sensitivity coefficient ambient value blackbody calculate signal estimated value exact value input incident value estimation parameter number measurement time the maximum value measurement signal the minimum value output radiative heat transfer left boundary right boundary

εw Φ κa κs λ

fuzzy set of input fuzzy set of output specific heat at constant pressure, J/(kg⋅ K) inference result input signal objective function convective heat transfer coefficient, W/m2⋅ K radiative intensity, W/(m2⋅ sr) the number of fuzzy subset size of medium, m iteration number specified iteration number refractive index normal vector the number of discrete nodes the number of measurement signal heat flux, W/m2 domain range time, s temperature, K estimation parameter or coordinate, m weighting factor amplification factor extinction coefficient, m 1 a fluctuation sampling time step, s convergence accuracy relative error

ρ σ τ ς Ω Ω0 Ψ a b cal est exa i in j k max mea min o r w1 w2

Levenberg-Marquardt (L-M) method to determine the time-dependent heat flux based on single thermocouple data. Cui et al. [28] applied a modified L-M method to estimate the coefficients of time-dependent heat flux by solving IHCPs. Sun et al. [29] employed sequential quadratic programming (SQP) algorithm to identify the space- and time-dependent heat flux in 1D and 2D participating media. However, the process for retrieving the gradient distribution in gradient-based methods is time-consuming, especially for multi-dimensional transient heat transfer problems. Fuzzy inference method (FIM), with good anti-ill-posed character and low computational cost, is widely applied to solve numerous un­ certain problems, such as fuzzy control, fuzzy decision and fuzzy recognition [30]. The second and corresponding authors [31–33] pro­ posed a decentralized FIM (DFIM) to determine the boundary heat flux, in which a set of decentralized fuzzy inference units were constructed. The inference result of each fuzzy inference unit is weighted to generate compensation values for estimation parameters. Several kinds of spaceand time-dependent heat fluxes were accurately estimated by solving IHCPs based on DFIM. However, to the best knowledge of the authors, the application of DFIM for estimating the thermal boundary conditions of participating medium has not yet been reported. Thus, the purpose of this study is to introduce the DFIM to solve inverse coupled radiation and conduction heat transfer problems. The remainder of this work is organized as follows. The direct problem involving coupled radiation and conduction heat transfer in participating medium is investigated in Section 2. The principle of DFIM is presented in Section 3. Meanwhile, three improved DFIMs (IDFIMs) are developed in this section. In Section 4, the time-dependent heat flux exposed on 1D participating medium is estimated using the DFIM and IDFIMs. The main conclusions are pro­ vided in Section 5.

2. Direct problem The coupled radiation and conduction heat transfer in an absorbing, emitting, and scattering 1D slab is considered in this study. As shown in Fig. 1, the left boundary (x ¼ 0) and right boundary (x ¼ L) of the me­ dium are diffuse gray walls. Both of the boundaries are subjected to Cauchy boundary conditions with ambient temperature Ta and convective heat transfer coefficient hw. Meanwhile, the left boundary is exposed to a normally collimated laser irradiation. The energy equation governing the transient coupled radiation and conduction heat transfer in 1D participating medium can be given as [34]:

ρcP

∂T ∂2 T ¼λ 2 ∂t ∂x

∂qr ∂x

(1)

where ρ, cp and λ represent the density, specific heat and thermal con­ ductivity of the medium, respectively. T and t indicate the temperature and time, respectively. qr denotes the radiative heat flux. The initial condition for the energy equation is defined as: (2)

Tðx; tÞjt¼0 ¼ T0

where T0 indicates the initial temperature of the medium. The boundary conditions for the left and right boundaries can be expressed as: � ∂T � qrw1 λ �� ¼ qin þ hw ðTa Tw1 Þ (3) ∂x x¼0 �

qrw2 þ λ

∂T �� ¼ hw ðTa ∂x �x¼L

Tw2 Þ

(4)

where qin is the incident heat flux on the left boundary. The subscripts w1 and w2 represent the left and right boundaries of the medium, 2

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International Journal of Thermal Sciences 149 (2020) 106216

Fig. 1. Schematic of coupled radiation and conduction heat transfer in 1D participating medium.

respectively. The radiative source term in Eq. (1) and the radiative heat fluxes in Eqs. (3) and (4) can be obtained according to the following equations: � � Z ∂qr Iðx; ΩÞdΩ (5) ¼ κa 4π Ib ðxÞ ∂x 4π � 4 qrw1 ¼ εw1 n2 σTw1

Z

� 4 qrw2 ¼ εw2 n2 σTw2

Z

� (6)

Ið0; ΩÞjΩ⋅nw1 jdΩ Ω⋅nw1 >0

� IðL; ΩÞjΩ⋅nw2 jdΩ

Ω⋅nw2 >0

Table 1 Parameter settings for verification cases.

(7)

where κa denotes the absorption coefficient of the medium. n indicates the refractive index. Ib ¼ n2 σT4 =π represents the radiative intensity of blackbody at temperature T, σ is the Stefan-Boltzmann constant. ε in­ dicates the boundary emissivity. n is the normal vector on the boundary. The radiative intensity at the position x and direction Ω is denoted by I (x,Ω) which can be obtained by solving the following radiative transfer equation [34]: Z dIðx; ΩÞ κs ¼ βe Iðx; ΩÞ þ ka Ib ðxÞ þ Iðx; Ω’ÞΦðΩ’; ΩÞdΩ’ (8) dx 4 π 4π

Symbol

Unit

Case 1

Case 2

Size of medium Thermal conductivity Heat capacity Absorption coefficient Scattering coefficient Convective heat transfer coefficient Ambient temperature Laser power density Laser action time Boundary emissivity

L λ ρcp κa κs hw

m W/(m⋅K) J/(m3⋅K) m 1 m 1 W/ (m2⋅K) K W/m2 s –

0.01 0.7 2.2 � 106 1.0, 30.0 0 7.0

0.002 1.38 1.7 � 106 100 100 0

300 50,000 1.0 1.0

298 339,900 0.1 1.0

Ta qin tq

εw

3. Inverse model 3.1. Decentralized fuzzy inference method FIM is an effective and robust technique for solving optimization and control problems, which was proposed based on fuzzy set theory [38, 39]. In the conventional FIM, much measurement information is served as input for the inference system. Several output signals are generated by fuzzy inference unit. The inference rule in fuzzy inference unit is established based on the experience of experts. However, only one fuzzy inference unit is applied to conduct all inference processes (see Fig. 3). It is extremely difficult to solve multi-dimensional optimization problems using the conventional FIM, mainly due to the fact that an effective inference rule from input to output is too difficult to establish based on expert experience. In order to overcome the shortcomings of the conventional FIM, the DFIM is developed to solve multi-dimensional inverse heat conduction problems, in which an N-dimensional optimization task is decomposed into N one-dimensional inference models. The fuzzy inference rule for each optimization variable can be reasonably established based on expert experience. Moreover, a weighting mechanism is applied to ensure the rationality of output. The fuzzy inference system of the DFIM is shown in Fig. 4. Fuzzy inference unit plays a crucial role in the DFIM. There are three main parts in fuzzy inference unit, namely fuzzification, fuzzy inference and defuzzification. The fuzzification process converts the precise input into several fuzzy sets. For the estimation of boundary heat flux, the boundary temperature is employed as measurement signals. The input for the inverse analysis can be defined as:

where βe and κs represent the extinction and scattering coefficients of the medium, respectively. Φ(Ω0 ,Ω) denotes the scattering phase function, Ω0 and Ω indicate the incident and scattering directions, respectively. The boundary condition of opaque and diffuse surfaces for the radiative transfer equation can be expressed as: Z 1 εw Iw ¼ εw Ib;w þ Iðx; ΩÞjΩ⋅nw jdΩ (9)

π

Parameters

Ω⋅nw <0

The discrete ordinate method (DOM) and finite volume method (FVM) were employed to solve the radiative transfer equation and en­ ergy equation respectively, through which the coupled radiation and conduction heat transfer in participating medium can be solved. The detailed solution procedure is available in Refs. [34,35] and not repeated here. Two verification cases were conducted to verify the accuracy of the direct model. The parameter settings of the verification cases are illus­ trated in Table 1. The computational domain was discretized into 100 nodes and S4 quadrature scheme was employed in DOM. As shown in Fig. 2, the obtained results of the present FVM-DOM solution were in good agreement with those of Tan et al. [36] solved by ray tracing method and Andre et al. [37] solved by two-flux approximation method.

ek ¼ Tcal;k

Tmea;k

(10)

where e indicates the input signal for the inference system. The subscript 3

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International Journal of Thermal Sciences 149 (2020) 106216

which are denoted by fA1 ; A2 ; …AK g and fB1 ; B2 ; …BK g respectively. The membership functions are expressed by isosceles triangles. The membership values γ are shown in Fig. 5 when K ¼ 7. The Mamdani maximum-minimum inference engine is applied to determine the fuzzy set B of output variable uk, which can be expressed as [40]: � � γ B ðuk Þ ¼ maxKl¼1 min½γAl ðek Þ; γBl ðuk Þ� (11) where γAl ðek Þ denotes the membership value of input signal ek corre­ sponding to fuzzy set Al, which is determined based on Fig. 5(a). γ Bl ðuk Þ represents the membership value of output variable uk corresponding to fuzzy set Bl, which is determined based on Fig. 5(b). Since the inference result of Eq. (11) is a fuzzy variable, a defuzzi­ fication procedure is necessary to convert the fuzzy result into affirma­ tory output. The gravity method is employed as the defuzzifier, which can be written as [41]: R r0 γB ðuk Þuk du r dk ¼ R r00 (12) γ ðu Þdu r0 B k where d is the final inference result of the fuzzy inference unit. The results of fuzzy inference units are synthetically coordinated, after which the inference results for the update of estimation parameters can be obtained. The weighting factor of the kth measurement signal with respect to the jth estimation parameter is calculated using sensi­ tivity method, which can be written as: Ψ xj ðTk Þ

αk;j ¼ R t2 t1

(13)

Ψ xj ðTk Þ

where t1 and t2 denote the initial and final sampling time, respectively. xj indicates the jth estimation parameter. Tk is the simulated boundary temperature at the kth sampling time. Ψ xj ðTk Þ represents the sensitivity coefficient of Tk with respect to xj, which can be expressed by [42]: � � Tk xj þ δxj Tk xj þ δxj Ψ xj ðTk Þ ¼ (14) 2δxj where δ is a fluctuation which is taken as δ ¼ 0.005 in the present research. Thus, the estimation parameter can be updated based on the following equation: (15)

xmþ1 ¼ xmj þ αmj;k dkm j where the superscript m denotes the mth iteration.

Fig. 2. Verification of the present solution for solving transient coupled radi­ ation and conduction heat transfer problems. (a) Case 1, and (b) Case 2.

3.2. Improved decentralized fuzzy inference method In the original DFIM, the inference result is generated in the range of [ ro, ro]. The domain size of output has a significant influence on the estimation task. A small size conducts careful optimization works, while increases the computational time. In contrast, a large size accelerates the convergence rate, while an accurate solution is difficult to achieve. In the early optimization stage, the deviation between estimated and exact parameters is relatively large. Hence, a large step size is needed to reduce the computational time. However, a small step size is necessary in the later stage to ensure the estimation accuracy. Thus, a linearly decreasing output range is applied to synthetic the optimization process, which can be expressed as:

Fig. 3. Fuzzy inference system of the conventional FIM.

k denotes the kth measurement time. Tcal represents the simulated boundary temperature based on estimated boundary conditions. Tmea, the measurement signal for the inverse analysis, is calculated using the direct model based on exact boundary conditions. The ranges of input ek and inference result uk are denoted by [ ri, ri] and [ ro, ro]. The domains are linearly divided into K fuzzy subsets

ro ¼ ro;max

m � ðro;max mmax

ro;min Þ

(16)

where ro,max and ro,min represent the maximum and minimum values of the output range, respectively. The current and maximum iteration numbers are denoted by m and mmax, respectively. To ensure the 4

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International Journal of Thermal Sciences 149 (2020) 106216

Fig. 4. Fuzzy inference system of the DFIM.

introduced into the update of estimation parameters: (17)

¼ xmj þ βm αmj;k dkm xmþ1 j

where β is the amplification factor. In the present research, the ampli­ fication factor is generated by: 8β < max ðm m20 β ¼ : m

m0 Þ2 þ β0 ; β0 ;

m � m0

(18)

m > m0

where m0 indicates the specified iteration number to control the amplification factor. β0 denotes the minimum value of amplification factor. According to the above improved strategies, three IDFIMs are developed, namely DFIM with decreasing output domain (IDFIM1), DFIM with amplification factor (IDFIM2), and DFIM with iterationdependent output domain and amplification factor (IDFIM3). The orig­ inal and improved DFIMs are applied to estimate the thermal boundary condition of participating medium. The iteration stops until one of the following criteria is satisfied: (19)

m ¼ mmax NN X

Fobj ¼

ðTcal;l

Tmea;l Þ2 < ε

(20)

l¼1

where Fobj denotes the objective function of the optimization task. NN indicates the number of sampling time. ε is the specified convergence accuracy. The flowchart of the original and improved DFIMs is illustrated in Fig. 6. Since the tendency of Ψ xj ðTk Þ remains unchanged at each itera­ tion, the weighting factors are calculated before the iterative optimi­ zation in this study.

Fig. 5. Membership values of (a) fuzzy set A and (b) fuzzy set B when K ¼ 7.

stability of the improved model, the maximum output range is suggested to be less than 10. Since the inference result is limited to the output domain, the maximum output of the inference system is ro. However, the inference result is too small to achieve an effective update by Eq. (15) for largespace optimization tasks, especially in the early stage of the estima­ tion process. In order to make an intelligent utilization of the fuzzy inference result and accelerate the convergence rate, an amplification factor is

4. Results and discussion 4.1. Problem description In this study, the DFIM is applied to estimate the thermal boundary 5

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International Journal of Thermal Sciences 149 (2020) 106216

Fig. 6. Flowchart of original and improved DFIMs for estimating thermal boundary conditions of participating medium.

condition of 1D gray participating medium. As shown in Fig. 1, the left boundary of the medium was irradiated by an unknown heat source. The direct problem concerning on coupled radiation and conduction heat transfer was solved using the FVM-DOM solution. The simulated tem­ perature on the left boundary was served as input for the inverse anal­ ysis. The original and improved DFIMs were employed as optimization techniques, through which the time-dependent incident heat flux can be recovered. In order to evaluate the accuracy of estimation results, two estimation errors including relative error and relative root mean square error were considered:

εrel ¼

jxest

εRRMSE

xexa j � 100% xexa

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u �2 Nx � u1 X xest xexa � 100% ¼t Nx i¼1 xexa

where Texa is the boundary temperature calculated using the FVM-DOM solution based on the exact incident heat flux. σ indicates the standard deviation of measured boundary temperature. ς represents a random variable of normal distribution with zero mean and unit standard de­ viation in the interval [ 2.576, 2.576] for a 99% confidence. All cases were implemented using the Fortran code. The developed program is executed on an Intel Core(TM) i5-8400 PC. 4.2. Estimation of time-dependent heat flux of 1D participating medium The time-dependent heat flux of 1D gray participating medium is estimated using DFIMs in this section. The absorption and scattering coefficients were κa ¼ 1.0 m 1 and κs ¼ 1.0 m 1, respectively. The boundary emissivities of the medium were taken as εw ¼ 1.0. The other optical and thermophysical parameters were set as same as those for the verification case 1 in Section 2. To avoid inverse crime, the numbers of discrete nodes were set as N ¼ 100 and N ¼ 80 in the direct and inverse

(21)

(22)

where xest and xexa denote the estimated and exact heat fluxes, respec­ tively. The average relative error is denoted by εrel . Nx indicates the number of estimation parameters. Since measurement errors are inevitable in practical engineering applications, random standard deviations that follow Gaussian distri­ bution were considered to test the performance of the present inverse technique. Hence, the measurement signals can be given as: Tmea ¼ Texa þ σς

Table 2 Parameter settings of the original and improved DFIMs. Methods DFIM IDFIM1 IDFIM2 IDFIM3

(23)

6

Parameters ri

ro/ro,max

ro,min

K

β

3.0 3.0 3.0 3.0

3.0 9.0 3.0 9.0

– 3.0 – 3.0

7 7 7 7

1000.0 1000.0 1000.0 600.0

max

β0

m0

– – 300.0 200.0

– – 100 100

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International Journal of Thermal Sciences 149 (2020) 106216

models, respectively. The main parameter settings in DFIMs were listed in Table 2. The time step was taken as Δt ¼ 0.1 s. The number of the estimation parameters was Nx ¼ 150. All estimation tasks started with a uniform heat flux which was taken as q ¼ 10000 W/m2. The conver­ gence accuracy was set as ε ¼ 10 2 and the maximum iteration number was set as mmax ¼ 50000. To test the performance of DFIMs, a time-dependent heat flux with step changes was considered firstly. The incident heat flux was given by: 8 < 50000 0 < t < 5 qðtÞ ¼ 30000 5 � t < 10 (24) : 80000 10 � t < 15

Table 3 Retrieval results of different DFIMs.

The original and improved DFIMs were employed to estimate the incident heat flux from the knowledge of boundary temperature. The obtained results were compared with the actual heat flux distribution in Fig. 7. As shown, the estimation results of the original and improved DFIMs were in good agreement with the actual heat flux distribution, which demonstrated that the time-dependent heat flux on the left boundary of participating medium can be accurately estimated using DFIMs. Table 3 illustrates the relative errors and computational time of different fuzzy inference methods. It can be seen that the improved models were more efficient and accurate than the original DFIM. The IDFIM3 achieved the best performance in terms of estimation accuracy and convergence rate. To illustrate the optimization performance of the present technique, four kinds of time-dependent heat fluxes including sinusoidal, quadratic, linear and exponential forms were considered. The functional forms of the incident heat fluxes were expressed as: qðtÞ ¼ 30000 þ 10000 sinð0:2πtÞ; 0 � t � 15

(25)

qðtÞ ¼ 400t2 þ 4000t þ 25000; 0 � t � 15

(26)

8 20000; > > > > < 5000t þ 30000; qðtÞ ¼ 6000t 25000; > > > 4000t þ 65000; > : 17000;

(27)

� qðtÞ ¼

0�t�2 2
20000 expð 0:8tÞ; 20000 expð 6Þ þ 10 expðt 7:5Þ

1

0 < t � 7:5 7:5 � t � 15

Methods

εrel [%]

εRRSME [%]

Iteration number

Computational time [s]

DFIM IDFIM1 IDFIM2 IDFIM3

0.0530 0.0531 0.0433 0.0352

0.2735 0.2735 0.2069 0.1670

41434 23269 403 298

987.5 635.2 17.3 12.5

Fig. 8. Estimation results of sinusoidal heat flux.

(28)

The IDFIM3 was applied to estimate the above heat fluxes. The retrieval results are shown in Figs. 8 11. As illustrated, all the above time-dependent heat fluxes can be accurately estimated, which

Fig. 9. Estimation results of quadratic heat flux.

demonstrated the proposed inverse technique was robust and efficient for estimating the thermal boundary condition of participating medium. Moreover, the estimation results of IDFIM were compared with other optimization techniques such as CGM, L-M method, SQP algorithm, stochastic PSO (SPSO) algorithm and GA. The initial guess was taken as the same value in CGM, L-M method, SQP algorithm and IDFIM. The initial solutions were randomly generated in the search space [0, 100000] for the SPSO algorithm and GA. The convergence criteria for all inverse methods were set as the same conditions. The comparison of the estimation results obtained by different optimization techniques are shown in Table 4. It can be found that the time-dependent heat flux can be accurately estimated using the IDFIM, CGM, L-M method and SQP algorithm. Meanwhile, the proposed IDFIM achieved higher estimation accuracy in far less computational time. Although the incident heat flux

Fig. 7. Estimation results of the time-dependent heat flux with step changes using DFIMs. 7

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International Journal of Thermal Sciences 149 (2020) 106216

wide applicability of the proposed IDFIM. Accurate estimation of the incident heat flux on heating surface plays a crucial role in glass industry. Hence, the unknown heat flux on the glass surface was estimated here. The parameters of glass were set as same as those in Ref. [3] as shown in Table 6. The computational domain was discretized into 150 nodes. The sinusoidal heat flux expressed by Eq. (25) and the linear heat flux expressed by Eq. (27) were considered. The IDFIM3 was applied to estimate the time-dependent heat flux. As shown in Fig. 13, the incident heat fluxes can be accurately estimated using the IDFIM3. The maximum relative error was less than 0.03%, which demonstrated the time-dependent heat flux on the glass surface can be accurately estimated using the proposed IDFIM. Moreover, the influence of measurement errors on estimation results was investigated. Random errors were added to the measured boundary temperature. The linear heat flux expressed by Eq. (27) was considered to test the robustness of the IDFIM. As shown in Fig. 14, although the measurement error was increased to 1.0, the time-dependent heat flux can be accurately estimated by the IDFIM3, which illustrated the strong robustness of the proposed IDFIM. The optical and thermal properties of participating medium produce direct influence on the heat transfer process. However, deviations are inevitable in the measurement of medium properties. Hence, the influ­ ence of the random standard deviation in medium properties on the estimation result was investigated. The thermal conductivity, absorption coefficient and scattering coefficient of the medium were taken as λ ¼ 10.0 W/(m⋅K), κa ¼ 10.0 m 1 and κs ¼ 10.0 m 1, respectively. The

Fig. 10. Estimation results of linear heat flux.

Table 5 Parameter settings of optical and thermal parameters. Cases

Thermal conductivity [W/(m⋅K)]

Absorption coefficient [m 1]

Scattering coefficient [m 1]

Case 1 Case 2 Case 3

10.0 5.0 5.0

1.0 1000.0 1.0

1.0 1.0 1000.0

Fig. 11. Estimation results of exponential heat flux.

can also be estimated in parallel in SPSO algorithm and GA, the solution at each iteration presented stochastic characteristic, and thereby the relative error and computational time of SPSO algorithm and GA were much larger than gradient-based methods. Furthermore, the estimation tasks for different kinds of participating media were investigated. The optical and thermal properties of the medium including thermal conductivity, absorption coefficient and scattering coefficient are listed in Table 5. The quadratic heat flux expressed by Eq. (26) was considered. The estimation results of IDFIM3 are shown in Fig. 12. The time-dependent heat flux on the boundary can be accurately estimated for different kinds of participating media. The maximum relative error was less than 2.5%, which demonstrated the

Fig. 12. Estimation results for different participating media.

Table 4 Estimation results of incident heat flux using different optimization techniques. Methods IDFIM3 CGM L-M SQP SPSO GA

Sinusoidal

Quadratic

Linear

Exponential

εRRSME [%]

Time [s]

εRRSME [%]

Time [s]

εRRSME [%]

Time [s]

εRRSME [%]

Time [s]

0.0777 0.1120 0.1349 0.1016 6.7527 7.8693

9.71 1125.23 937.78 643.63 11345.40 18020.11

0.1072 0.1853 0.1793 0.1500 8.3461 10.1323

8.63 1113.61 916.53 601.51 15269.11 16900.43

0.1519 0.3062 0.2972 0.2137 9.1799 12.3645

9.83 1038.73 932.30 699.02 14506.05 13618.77

0.5906 0.9149 1.2331 0.8180 15.4597 18.3393

8.66 1014.90 960.15 657.34 11378.80 18751.89

8

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International Journal of Thermal Sciences 149 (2020) 106216

Table 6 Parameter settings in glass industry. Parameters Units Values

L m 0.15

cp

ρ kg/m 2219

3

J/(kg⋅K) 840

λ

κa

W/(m⋅K) 1.025

κs 1

m 1244

m 0

1

n

hw

– 1.4

W/(m2⋅K) 8

estimated average relative errors for each of 20 random samples for σ ¼ 0.5 are illustrated in Fig. 15. As shown, the measurement error of thermal conductivity produced significant influence on the estimation accuracy, whereas the measurement error of optical properties had no obvious influence on the estimation result. The main reason is that the measurement signals used for inverse analysis were highly sensitive to thermal conductivity but not sensitive to absorption and scattering coefficients. 5. Conclusions The DFIM was introduced to estimate the time-dependent heat flux of participating medium for the first time. The coupled radiation and conduction heat transfer in the medium was solved by a combination of FVM and DOM. The simulated time-resolved temperature on the left boundary was served as input for the inverse analysis. The DFIM was applied to estimate the incident heat flux from the knowledge of boundary temperature. Three IDFIMs were proposed to improve the optimization ability of the inverse technique, after which five kinds of time-dependent heat fluxes were accurately estimated. Moreover, the influence of measurement errors on the estimation result was investi­ gated. The following conclusions can be drawn: (1) The time-dependent heat flux can be accurately estimated using the DFIM even with noisy data. (2) The proposed IDFIMs achieved better performance than DFIM in terms of computational accuracy and convergence rate. (3) The IDFIM was proved to be more accurate and efficient than CGM, L-M method, SPSO algorithm and GA. (4) The random standard deviation in thermal conductivity produced obvious influence on the estimation result, whereas the deviation of absorption and scattering coefficients had few effects on the estimation result. The further research directions are to estimate the multi-dimensional time and space-dependent thermal boundary condition of participating medium and carry out the related experimental researches.

Fig. 13. Retrieval results of (a) incident heat fluxes and (b) relative errors.

Fig. 15. Effect of the random deviation in medium properties on the estima­ tion result.

Fig. 14. Retrieval results of incident heat flux for different measurement errors.

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International Journal of Thermal Sciences 149 (2020) 106216

Declaration of competing interest

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