Methodology of surface heat flux estimation for 2D multi-layer mediums

International Journal of Heat and Mass Transfer 114 (2017) 675–687

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Methodology of surface heat flux estimation for 2D multi-layer mediums Jia-meng Tian, Bin Chen ⇑, Zhi-fu Zhou State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, China

a r t i c l e

i n f o

Article history: Received 24 February 2017 Received in revised form 10 June 2017 Accepted 12 June 2017

Keywords: Surface heat flux Inverse heat conduction problem 2D filter solution Optimal comparison criterion Spray cooling

a b s t r a c t Surface heat flux is an important parameter in various industrial applications. It is often estimated based on measured temperature by solving inverse heat conduction problems (IHCPs). In the present work, a filter solution to solve 1D single-layer IHCPs is applied to calculate the surface heat flux for 2D multilayer mediums. An optimal comparison criterion is implemented for 2D IHCPs to optimize the key regularization parameters. Afterward, the 2D filter solution is used for heat flux estimation with thin-film thermocouple (TFTC) and fine thermocouple (FTC) measurements during cryogen spray cooling. The accuracy of the estimated heat fluxes is tested with the measured temperature response to cryogen spray cooling. A small error (maximum value of 1.0740 °C) is observed between the temperature simulated based on estimated heat fluxes and the measured temperature. The maximum heat flux obtained by the 2D filter solution is 13.6% higher than that obtained by 1D method for TFTC measurement. This finding indicates that lateral heat transfer cannot be disregarded, especially when the heat conductivity coefficient of the material is large. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The heat flux characterizes the heat transfer capacity per unit area and is a significant index to evaluate the heat transfer performance of devices and facilities. Accurate estimation of surface heat flux profiles is important in various industrial applications, such as thermal protection of space shuttles [1], thermal management of electronic devices [2], metal heat treatment [3], maintenance of boilers [4] and nuclear reactors [5], spray cooling [6], and geophysics [7]. However, direct measurement of surface heat flux is difficult. By contrast, temperature measurement is easier. Thus, indirect estimation of surface heat flux by using surface or internal temperature has elicited much attention. Surface heat flux can generally be estimated by solving inverse heat conduction problems (IHCPs) according to the measured surface or internal temperature. The accuracy of surface heat flux estimation can be validated by comparing a hypothetical surface heat flux and one computed based on the temperature simulated with the hypothetical surface heat flux as a boundary condition. IHCPs are mathematically ill-posed, and a small error in temperature may significantly affect the accuracy of heat flux estimation [8,9]. To solve this kind of problem, many analytical and numerical techniques have been proposed; these techniques include sequential function specification (SFS) [10], transfer ⇑ Corresponding author. E-mail address: [email protected] (B. Chen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.053 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

function [11], conjugate gradient (CG) [12,13], singular value decomposition (SVD) [14,15], and Tikhonov regularization (TR) [16–18]. The SFS method proposed by Beck et al. [10] minimizes the effect of random errors by using future temperature data obtained with the least-squares method. However, the SFS method may cause uncertain heat flux fluctuation because of the inherent unstable nature of the algorithm when solving a multi-layer geometry [19]. The transfer function method regards heat flux as the input of a dynamic system and temperature history as the response; it constructs the relationship between the input and output by using Green’s function [20,21]. This method is simple, and its algorithm is stable. However, solving for the transfer function is difficult when dealing with a complex geometry. CG and SVD methods involve complicated algorithms and often cause inherent oscillations [22]. TR is usually regarded as the entire time domain method, which requires temperature data on all time steps and calculates the entire heat flux simultaneously [18]. All these methods have their advantages and disadvantages. Most of them are weak in terms of solving IHCPs with a complex geometry, and several (e.g., CG and SVD methods) involve complicated algorithms. Recently, a filter solution based on TR has attracted the interest of many researchers [9,23–27]. This solution was developed by minimizing the sum of the squares of the errors between estimated and measured temperatures with respect to the unknown heat fluxes and stabilized by Tikhonov regularization. This solution is expressed in a digital filter form, which allows for an almost real-time heat flux estimation, and has been applied in heat flux

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Nomenclature a c f F Hi HT Ht, Hs L mf, mp n Rq q q S t td T T W X Y

thermal diffusion coefficient (m2/s) specific heat capacity (kJ/kgK) filter coefficient filter coefficient matrix thickness of the ith layer (mm) depth of sensors (mm) temporal and spatial first order regularization matrix spray distance (mm) number of future and past time steps time step index sum of squares of the surface heat flux errors surface heat flux (kW/m2) surface heat flux matrix sum of squares of the temperature errors time (ms) time step (ms) temperature of substrates (°C) temperature matrix width of geometry (mm) sensitivity matrix experimental temperature (°C)

Greek symbols at, as temporal and spatial regularization parameter k heat conductivity coefficient (kW/mK)

measurement using a directional flame thermometer [28]. This method demonstrates superiority when solving IHCPs with a complex geometry, and its algorithm is simple. However, this method can only be used to solve 1D single-layer IHCPs directly. Little work has been conducted to solve multi-dimensional, multi-layer IHCPs. In this study, cryogen spray cooling (CSC) with a spray duration of several tens of milliseconds was selected as an example to investigate multi-dimensional, multi-layer IHCPs because such transient spray cooling often leads to ultra-fast surface temperature variation and a rapid change in time-dependent surface heat flux. In transient CSC, time-dependent surface heat flux can be estimated by solving IHCPs using the internal or surface temperature history of the solid substrate. Two typical measurements, namely, fine thermocouple (FTC) and thin-film thermocouple (TFTC), are commonly used to monitor internal and surface temperatures. For example, Anguilar et al. [29] utilized the SFS method to evaluate surface heat flux from internal temperature (
45 lm from the upper surface) measured by a type-T FTC (50 lm bead diameter) placed underneath a thin layer of aluminum foil (20 lm). The foil was positioned on the top of epoxy resin to provide rapid heat transfer from cooling cryogen droplets and mechanical support. Zhou et al. [19,30,31] measured time-dependent surface temperature by using a 2-lm type-T TFTC magnetically deposited onto an epoxy resin surface; the method accurately captured the temperature variation during CSC because of its ultra-fast thermal response (
1.2 ms). Afterward, surface heat flux was calculated with Duhamel theory. Although the temperature measured with TFTC is closer to the surface temperature than that measured with other methods, TFTC cannot be used to measure the temperature of metal materials because of electrical conductivity. Moreover, TFTC corrodes and oxidizes easily when it is exposed to hightemperature environments. Therefore, FTC measurement is widely used in many industries because of its reliability and stability. Unlike TFTC measurement with its single-layer geometry, FTC measurement consists of three layers, namely, aluminum, thermal

q

/

rY Dt Dx

e s

density (kg/m3) excessive temperature (°C) standard deviation of the random measurement errors (°C) spray duration (ms) interval between every two thermocouples (mm) uniform random temperature error (°C) response time of thermocouple

Subscripts c threshold value i layer index j sensor index k surface heat flux index 0 initial value int interface value min minimum value max maximum value MSE mean standard error Superscripts ^ estimated value

paste, and epoxy resin. For generality, multi-layer IHCPs need to be developed. Our recent work [19] compared 1D SFS, the transfer function, and the Duhamel theory method for TFTC and FTC measurements. The results indicated that the SFS method can be applied for TFTC and FTC measurements, but a noticeable discrepancy in the maximum surface heat flux was discovered. The transfer function method effectively inhibited noise and was suitable for TFTC and FTC measurements. The Duhamel theory method was insensitive to noise but unsuitable for FTC measurement. The Duhamel theory method was extended to the multi-layer case [19], in which surface heat flux was estimated based on the actual surface temperature calculated directly with traditional Duhamel theory from the measured internal temperature rather than the internal temperature. This method was validated in terms of its accuracy and applicability to TFTC and FTC measurements. We refer to this new method as Duhamel theory multi-layer method in this paper. Most of the abovementioned algorithms are based on 1D IHCP, which is based on the assumption that the lateral temperature distribution is uniform. In reality, the radial and temporal surface temperature variations during CSC result in significant nonuniformity of the surface heat flux [32,33]. Therefore, lateral heat transfer must be considered. Theoretically, surface heat flux distribution can be evaluated with a 2D IHCP model, especially when the heat conductivity coefficient is large. Therefore, a general 2D multi-layer IHCP needs to be developed. Najafi et al. [8] presented a filter solution for a 2D inverse heat conduction problem. However, the corresponding regularization and filter parameters that influence the accuracy of the estimated heat flux are given directly without any optimization. Also, the solution cannot be used for the evaluation of multi-layer IHCPs, which is essential for FTC measurement. In summary, a general 2D multi-layer IHCP is necessary. In the current work, a filter solution to solve 1D single-layer IHCPs was applied for a general 2D, multi-sensor, multi-layer surface heat flux estimation problem to consider lateral heat transfer.

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An optimal comparison criterion proposed by Beck [22] was implemented to optimize the regularization and filter parameters. Afterward, the filter solution and optimal comparison criterion were employed to estimate the surface heat flux in CSC, in which single-layer TFTC and three-layer FTC temperature measurements were conducted with six sensors placed laterally every 2 mm from the spray center. Six hypothetical triangular pulse heat fluxes were used to examine the accuracy and sensitivity of the measured temperature error.

The surface heat transfer of CSC was viewed as a 2D heat conduction problem in this work instead of a 1D heat conduction problem. The 2D model (W  H) for general situations with K heat fluxes, J temperature sensors, and I layers is illustrated in Fig. 1. The origin of the coordinate (0, 0) is located at the top left corner, and the thickness of the ith layer is Hi. The multiple surface heat fluxes are assumed to be uniform in a specific upper lateral surface area (e.g., 0  x  x1). The temperature sensors are placed at the lateral midpoint of each uniform surface heat flux range in the ith layer, and the interval between two thermocouples is Dx. All sensors are placed at an identical depth (y = HT) from the upper surface. The adiabatic boundary condition is considered on the three other sides (x = 0, x = W, and y = H). The initial temperature of the substrates is T0. The mathematical description of the direct heat conduction problem with K heat fluxes, J temperature sensors, and I layers is as follows:

ð0 6 x 6 W; 0 6 y 6 HÞ;

ð1Þ

where T is temperature, t is time, a is the thermal diffusivity coefficient (a = k/qc), and i denotes parameters in the ith layer. Without consideration of the thermal contact resistance, the boundary, interface and initial conditions with unchanged thermal properties can be expressed as

@T @T @T ð0; y; tÞ ¼ ðW; y; tÞ ¼ ðx; H; tÞ ¼ 0; @x @x @y

i X @T i Hl ; t x; ki @y l¼1

Tðx; y; 0Þ ¼ T 0

2. Mathematical description of IHCPs

@ 2 T i @ 2 T i 1 @T i þ 2 ¼ ai @t @x2 @y

8 q1 ðtÞ; 0 6 x 6 x1 > > > > < q 2 ðtÞ; x1 6 x 6 x2 @T 1 k ðx; 0; tÞ ¼ ; .. > @y > . > > : qK ðtÞ; xK1 6 x 6 xK

ð2Þ

!

i X @T iþ1 ¼ kiþ1 Hl ; t x; @y l¼1

ð3Þ

! ¼ qint;i ;

ð0 6 x 6 W; 0 6 y 6 HÞ:

ð4Þ ð5Þ

where qint,i denotes the heat flux at the ith interface between the ith layer and the (i + 1)th layer. In IHCPs, unknown heat fluxes are calculated with the known experimental temperature from the sensors, which can be described as

TðxTCj ; HT ; tÞ ¼ Y j ðtÞ j ¼ 1; 2; 3    ; J;

ð6Þ

where the subscript TCj denotes the lateral location of the jth sensor, Yj is the experimental temperature measured by sensors, and HT is the sensor location in the y direction. 3. Solutions for IHCPs and optimal comparison criterion The solution for 2D single-layer IHCPs was proposed by Najafi et al. [8]. The solution is derived by minimizing the sum of the squares of the errors between estimated and measured temperatures. This solution is expressed in a digital filter form, thus allowing for almost real-time heat flux estimation. Notably, the number of temperature sensors must be equal to or greater than the number of unknown heat fluxes. In this section, the solution for a 2D single-layer IHCP is introduced briefly. The solution is then applied to a 2D multi-layer IHCP. Afterward, the corresponding optimal comparison criterion is proposed to optimize the regularization and filter parameters. 3.1. Solution strategy for a 2D single-layer IHCP The discrete solution of temperature for direct heat conduction problems with K heat fluxes and J sensors described by Eqs. (1)–(5) can be presented in a matrix form as follows: [10]

T ¼ Xq;

Fig. 1. Geometry model of a 2D multi-layer IHCP.

ð7Þ

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where T is the real temperature matrix, X is the sensitivity matrix, and q denotes multiple heat fluxes described as follows:

2

Tð1Þ

3

2

T 1 ðnÞ

3

2

qð1Þ

3

2

q1 ðnÞ

3

7 6 7 7 7 6 6 6 7 6 7 7 7 6 6 6 6 qð2Þ 7 6 Tð2Þ 7 6 T 2 ðnÞ 7 6 q2 ðnÞ 7 7 6 7 7 7 6 6 6 7 6 7 7 7 6 6 6 7 6 7 7 7 6 6 6 6 .. 7 6 .. 7 6 .. 7 6 .. 7 6 . 7 6 . 7 6 . 7 6 . 7 7 6 7 7 7 6 6 6 7; qðnÞ ¼ 6 7; TðnÞ ¼ 6 7; q ¼ 6 7; T¼6 7 6 7 7 7 6 6 6 6 qðnÞ 7 6 TðnÞ 7 6 T j ðnÞ 7 6 q ðnÞ 7 7 6 7 7 7 6 6 6 k 7 6 7 7 7 6 6 6 7 6 7 7 7 6 6 6 6 . 7 6 . 7 6 . 7 6 . 7 6 .. 7 6 .. 7 6 .. 7 6 .. 7 7 6 7 7 7 6 6 6 5 4 5 5 5 4 4 4 T J ðnÞ

TðNÞ

6 6 6 6 6 Hs ¼ 6 6 6 6 4 ð8Þ

where N is the total time steps and n denotes the nth time step. Owing to the parabolic nature of the heat conduction problem, sensitivity matrix X is the lower triangular and has the Toeplitz structure [27].

2

að1Þ

6 6 6 að2Þ 6 6 6 6 X ¼ 6 að3Þ 6 6 6 . 6 . 6 . 4

0

0



að1Þ

0



að2Þ

að1Þ



.. .

.. .

..

0

3

7 7 0 7 7 7 .. 7 7 . 7; 7 7 7 7 0 7 5

.

ð9Þ

where

2

aJ1 ðnÞ ajk ðnÞ ¼

aJ2 ðnÞ

aJ3 ðnÞ



ð10Þ

aJK ðnÞ

@ðTðxTCj ; W T ; tÞ  T 0 Þ ¼ ujk ðnÞ; @½qk ð1Þ

ð11Þ

ujk ðnÞ ¼ /jk ðnÞ  /jk ðn  1Þ; n ¼ 2; 3;    ; N:

ð12Þ

ujk ð1Þ ¼ /jk ð1Þ

ð13Þ

Excessive temperature / can be presented as follows:

  /jk ðnÞ ¼ TðxTCj ; HT ; nt d Þ

qk ¼1; q1 ¼q2 ¼¼qK ¼0

 T0;

ð14Þ

where td is the time step. Given that the analytical solution is difficult to obtain directly, Eqs. (1)–(5) were numerically solved for the case of single-layer geometry with finite volume method (FVM). The sum of the squares of the errors between the estimated and measured temperatures plus one-order regularization, S, is

S ¼ ðY  TÞ0 ðY  TÞ þ at ½Ht q0 ½Ht q þ as ½Hs q0 ½Hs q;

0

3 2 0 0 1 60 0 07 7 6 6. .. .. 7 7 6. . .7 6. 7; I ¼ 6 7 60 0   I I 0 7 6 7 6 40 0   0 I I 5

I I .. .

  0   0 . . .. . .

0   0

0 0

Hs 0   0

0

0 Hs   0 .. .. . . .. . . . .

0 .. .

0

0   Hs 0

0

0   0 Hs

0

0   0

0

3

7 7 7 7 7 7; 07 7 7 05 0 .. .

0 Hs

3 0 07 7 .. 7 7 .7 7 0   1 0 0 7 7 7 0   0 1 0 5     .. .

0 1 .. .

0 0 .. .

0 0   0 2 1 1 6 0 1 6 6 . .. 6 . . 6 . Hs ¼ 6 6 0 0 6 6 4 0 0 0

0

0 0 .. .

0 0  

0

0 0

3:

0 07 7 .. .. 7 7 . .7 7   1 1 0 7 7 7   0 1 1 5

  0 .. . . ..

 

0

0 0 ð16Þ

b using the After minimizing S, estimated heat flux matrix q entire domain data yields [8]

b ¼ ½X0 X þ at H0t Ht þ as H0s Hs 1 X0 Y ¼ FY: q

ð17Þ

3.2. Solution strategy for a 2D multi-layer IHCP

aðNÞ aðN  1Þ aðN  2Þ    að1Þ 3 a11 ðnÞ a12 ðnÞ a13 ðnÞ    a1K ðnÞ 7 6 7 6 6 a21 ðnÞ a22 ðnÞ a23 ðnÞ    a2K ðnÞ 7 7 6 7 6 6 a ðnÞ a ðnÞ a ðnÞ    a ðnÞ 7 7; 31 32 33 3K aðnÞ ¼ 6 7 6 7 6 7 6 . . . . . 6 .. . . . . . 7 . . . 7 6 5 4

I 60 6 6 . 6 . 6 . Ht ¼ 6 60 6 6 40 2

qK ðnÞ

qðNÞ

2

ð15Þ

where Y is the measured temperature matrix with a form similar to that of real temperature matrix T. at and as are regularization parameters with respect to temporal and spatial terms, respectively. The superscript 0 denotes the transpose of a matrix. Ht and Hs are described as follows:

Generally, the IHCP solution for 1D multi-layer geometry is solved layer by layer, starting from the ith layer with the known experimental temperature. The heat flux is estimated at the interface with the (i  1)th layer. Thereafter, the interface heat flux between the (i  1)th and (i  2)th layers is calculated by using the interface heat flux as the input. Then, the surface heat flux is estimated [26]. Analytically, the IHCP solution for 2D multi-layer geometry can also be solved directly by numerical method for three-layer geometry rather than the layer-by-layer strategy. However, it cannot be used to analytically calculate sensitive matrix X. In our present work, we tried to elaborate a general method that can be used for an arbitrary 2D multi-layer geometry, in which matrix X can be calculated using both numerical and analytical solution. The solution for a 2D multi-layer IHCP is similar, except that the surface and interface heat fluxes are multiple. As shown in Fig. 1, the temperature at the ith layer is known. The interface b i between the ith and (i  1)th layers can be presented heat fluxes q in a similar form as that of the solution for 2D single-layer geometry.

b i ¼ ½X0i Xi þ at H0t Ht þ as H0s Hs 1 X0i Y ¼ Fi Y q

ð18Þ

Notably, Xi is different from X in the solution of single-layer IHCP. Xi is calculated by using (I  i + 1) layers below the ith layer. b i yields The estimated interface temperature T

b i ¼ Xi q b i: T

ð19Þ

b i ) are b i ) and temperature ( T Then, the interface heat fluxes ( q used as the known boundary condition to solve the solution for the (i  1)th layer, where the heat fluxes are unknown at y = H1 + b i1 between the (i  1)th H2 +  + Hi2. The interface heat fluxes q and (i  i  2)th layers can be expressed as

b i  Xi;u q b i1 ¼ ½X0i;d Xi;d þ at H0t Ht þ as H0s Hs 1 X0d ð T b iÞ q b i  Xi;u q b i Þ; ¼ Fi;d ð T

ð20Þ

where Xi,d and Xi,u have a similar form as the sensitivity matrix and X is for single-layer IHCP. However, excessive temperature / in X is different.

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For Xi,d, / yields

  /jk ðnÞ ¼ TðxTCj ; H1 þ H2 þ    þ Hi1 ; nt d Þ

qk ¼1;q1 ¼q2 ¼¼qK ¼0

 T0: ð21Þ

For Xi,u, / can be described as

  /jk ðnÞ ¼ ðTðxTCj ; H1 þ H2 þ    þ Hi2 ; nt d Þ

qk ¼1;q1 ¼q2 ¼¼qK ¼0

 T 0 Þ: ð22Þ

By substituting Eq. (18) into Eq. (20), the estimated interface heat flux is derived as

b i1 ¼ Fi;d ðXi  Xi;u ÞFi Y ¼ Fi1 Y: q

ð23Þ

b i2 ; q b i3 ;    q b 2 can be calculated by repeatInterface heat flux q b can be ing Eqs. (20)–(23). Thereafter, surface heat flux matrix q estimated. The solutions for 2D single-layer and multi-layer IHCPs are then obtained. 3.3. Filter solutions and optimal comparison criterion The solutions for single-layer and multi-layer 2D IHCPs (Eqs. (17) and (23)) require the entire domain temperature data, and this requirement results in a heavy computation load and is inappropriate for online monitoring of heat flux. Compared with the whole time domain method, the benefit of digital filter solution is not only its near real-time monitoring of heat flux, but also small memory and computational load, since it only requires temperature data from several previous time steps and a few future time steps [8,22,27]. In each experiment, we collect temperature at a time interval of 0.5 ms, i.e., 24,000 data for six-channel temperature measurement in the total computational time of 2 s will be acquired. If we use the whole time domain method (classic Tikhonov regularization method), the dimension of F matrix becomes 24,000  24,000, which requires large memory and heavy computation load. If we collect more temperature data in a larger time domain (such as 100 s or more), it is difficult to estimate transient heat flux using the whole time domain method (classic Tikhonov regularization method). Therefore, the filter solution rather than classic Tikhonov regularization method is chosen in our present work. The filter solution is not a new method; rather, it is a new lens through which many existing methods can be applied to the nearly real-time computation of heat fluxes. The filter solution indicates that the heat flux at each time step is only related to the temperature from several previous time steps and a few future time steps and is independent from the rest of the temperature data [8,9,27]. Several interesting characteristics can be found in Eqs. (17) and (23). The dimension of filter matrix F is (N  K)  (N  J), and the b and Y are (N  K)  1 and (N  J)  1. The structure dimension of q of F that disregards the first few and last several rows as well as the rest of the unaffected filter coefficients is shown as

2

f0 6 6 f1 6 6 6 . 6 .. 6 6 6 f F¼6 6 mp 6 6 .. 6 . 6 6 6 6 f N2 4 f N1

f 1

f 2



f 3N

f 2N

f0

f 1

f 2

f 3



..

..

.

..

..

..

f mp 1



f0



f mf þ1

.. .

.. .

.. .

..

.

.. .

f N3



.. .

f1

f0

f N2

f N3



f2

f1

.

.

.

.

f 1N

3

7 f 2N 7 7 7 .. 7 . 7 7 7 7 f mf 7; 7 7 .. 7 . 7 7 7 7 f 1 7 5 f0

where f denotes K  J block filter coefficients. mp and mf are the numbers of non-negligible filter coefficients before and after the current time step, respectively. Given that most filter coefficients can be disregarded except for those of the (mp + mf + 1) time step, the solutions for IHCPs (Eqs. (17) and (23)) can be simplified into a general filter solution as follows:

qk ðnÞ ¼

ðf k;m Y ðnmp 1ÞJþm Þ;

ð25Þ

m¼1

where fk,m denotes the filter coefficient in the mth column from one row of F associated with the kth unknown heat flux. As illustrated in Eq. (24), all the rows of the filter coefficients are identical but shift in time. Therefore, any one column can be selected, except for the first few and last several rows, to calculate the heat flux using partial temperature data rather than the entire domain data. Notably, the key parameters at, as, mp, and mf can significantly affect the accuracy of the estimated heat flux. These key parameters depend on the boundary conditions, material properties, sensor location, time step, and regularization [9,26,27], which should be determined for a specific IHCP. For 1D IHCPs, these parameters can be optimized by the optimal comparison criterion proposed by Beck et al. [22]. In this study, this method was applied to solve 2D single-layer and multi-layer IHCPs. The sum of the squares of the errors, Rq, between the estimated and real heat fluxes was used to examine the success of the calculation.

b  qÞ0 ð q b  qÞ ¼ ðFY  qÞ0 ðFY  qÞ Rq ¼ ð q

ð26Þ

A more efficient means is to minimize the expected value of Rq [22,27] as follows:

EðRq Þ ¼ ½ðFX  IÞq0 ½ðFX  IÞq þ r2Y trðF0 FÞ;

ð27Þ

where E is the expected value, tr denotes the sum of the diagonal elements, and rY is the standard deviation of the random measurement errors, which is assumed to be 0.01 °C given by Beck and Woodbury [22]. E(Rq) contains two components: Eq,bias, which can be written as

Eq;bias ¼ ½ðFX  IÞq0 ½ðFX  IÞq;

ð28Þ

and Eq,rand in the filter form

Eq;rand ¼ r2Y trðFT FÞ ¼ r2Y ðN  ðmp þ mf þ 1ÞÞ

þmf þ1Þ K ðmpX X k¼1

2

f k;ðj1ÞJþk :

ð29Þ

j¼1

The first term, Eq,bias, is the bias component and depends on the errorless heat flux; the second term, Eq,rand, is the random component and proportional to the variance of the temperature errors [22]. To examine the accuracy of the estimated heat fluxes, the mean standard error (MSE) between estimated and hypothetical heat fluxes was employed as follows [8,26]:

qMSE ¼

ð24Þ

ðmp þm f þ1ÞJ X

Nm Xf 1 2 ðb q ðnÞ  qðnÞÞ N  mf n¼1

!1=2 :

ð30Þ

4. CSC experiment To provide experimental data for estimating 2D surface heat flux, a transient spay cooling system was constructed, and corresponding surface temperature measurements were conducted. Cryogen R404A with a boiling temperature of 46.5 °C at 1 atm was atomized through a stainless straight-tube nozzle to cool

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Fig. 2. Schematics: (a) transient CSC experiment system and (b) TFTC and FTC measurement.

down the substrate. The surface temperature was measured by using six TFTCs and FTCs. As shown in Fig. 2(a), R404A (Dupont) reserved in a commercial container at saturation pressure (1.16 MPa, 22 °C) flowed through the high-pressure hose (length of 40 mm and inner diameter of 1.2 mm) to reach the nozzle. The hose was fixed on a 3D translational electric positioner (WN105TA300 M, Beijing Winner Optics Instruments Corp., China) with a space resolution of 8 mm. The tiny atomized droplets vertically impacted the epoxy resin surface. The spray duration was controlled with a fast-response solenoid (0090631-900, Parker, USA) that could open and close in 5 ms. Epoxy resin (50 mm  50 mm  5 mm) was used as the skin phantom because of its similar thermal property as the human skin. The heat conductivity coefficient (k), density (q) and specific heat capacity (c) of epoxy resin are 0.14 W/mK, 1930 kg/m3 and 1650 J/kgK, respectively [6]. Two different temperature measurements with TFTCs and FTCs were conducted. Schematics of TFTCs and FTCs were shown in

Fig. 2(b). In the TFTC measurement, the surface temperature of epoxy resin was measured by six type-T TFTCs. TFTCs have a small heat capacity, good contact with the underlying substrate, and fast response. The dynamic response is 1.2 ms, which is fast enough to monitor the rapid change in surface temperature during CSC [30]. The TFTC measurement involved only one material (epoxy resin), which can be regarded as single-layer geometry. In the FTC measurement, six standard FTCs of type-T with a 10 mm bead diameter (FLH1000-02/T, Captec, France) were assembled underneath a thin layer of aluminum foil (22 mm  22 mm  10 mm). A thin layer of thermal paste (OB-100, Omega, USA) with a thickness of 100 lm was placed between the aluminum foil and epoxy resin to ensure good thermal contact around the thermocouple and to provide mechanical support. In addition, the thermal paste can also be regarded as an insulation layer between thermocouples and aluminum foil. Therefore, the thermocouple signal will not be influenced by the high electrical conductivity of aluminum. According to Aguilar et al. [34], the response time (s) of thermocouple can

J.-m. Tian et al. / International Journal of Heat and Mass Transfer 114 (2017) 675–687

be estimated by the equation, s = hA/qcV, where h, A, q, c, and V are the surface heat transfer coefficient due to CSC (assumed to be 10,000 W/m2 K which is large enough for CSC), sensor’s surface area, density, specific heat, and volume, respectively. For bead diameter of 10 lm and foil thickness of 10 lm, the response time of thermocouple in the present work is 1.68 ms. In addition, the thermal diffusivity coefficient (a = k/qc) of aluminum foil and thermal paste is 9.65  105 m2/s and 2.93  106 m2/s (manufactory provided). Thus, diffusion time for aluminum and thermal paste is 1.03 ls and 3.33 ms. We can believe that the response time of FTC measurement is 3.33 ms, which is small enough to monitor the fast response of temperature during CSC. In fact, FTC measurement involves three materials (aluminum, thermal paste, and epoxy resin), and the sensor actually measures the internal temperature located in the second material (thermal paste); thus, it should be regarded as three-layer geometry. Consequently, TFTC and FTC measurements can be regarded as single-layer and three-layer IHCPs, respectively. The geometry of single-layer IHCP with TFTCs and three-layer IHCP with FTCs can be simplified as illustrated in Fig. 3 (a) and (b), respectively. For both TFTC and FTC measurements, the multiple surface heat fluxes were assumed to be uniform in a specific upper lateral surface area (eg. 0  x  x1) and different from each other. The other three sides were insulated. As shown

681

in Fig. 3(a), six TFTCs with a thickness of
2 lm were magnetically deposited directly onto the epoxy resin surface. Temperature was measured from the spray center (x = 0 mm, TC1) to the periphery of the spray (x = 10 mm, TC6), and the lateral distance between two TFTCs was Dx = 2 mm. The width of the geometry with TFTC measurement was W = 11 mm, which is almost half that of the spray diameter (see Fig. 3(a)) because the computational domain is symmetric. The thickness of the epoxy resin layer was H = 5 mm. Given that the physical properties of different TFTCs present many differences, the six TFTCs were calibrated separately. Table 1 presents the comparison of temperatures measured by a standard type-T thermocouple and TFTCs in the same location of the lowtemperature thermostat. The maximum temperature deviation is less than 1 °C, which verifies the accuracy of TFTCs. In this experiment, the solenoid valve was controlled by LabVIEW using a DAQ board (M-6251, NI, USA), and surface temperatures were measured simultaneously by a six-channel data acquisition program for TFTCs or FTCs. Our previous investigation [35] of R404A spray cooling revealed that spurt duration exerts a negligible effect on surface heat transfer if the spurt duration is longer than 50 ms at a spray distance (L) of 30 mm because of the heat transfer barrier caused by the increasing liquid film. Therefore, spray distance L and spurt duration Dt were set to 30 mm and 50 ms, respectively.

Fig. 3. Geometry models: (a) single-layer IHCP with TFTCs and (b) three-layer IHCP with FTCs.

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Table 1 Comparison of temperatures measured by a standard thermocouple and TFTCs. TC1/°C

TC2/°C

TC3/°C

TC4/°C

TC5/°C

TC6/°C

Standard TC/°C

Maximum deviation/°C

24.73 20.39 15.94 11.03 5.28 0.47 5.42 10.48 13.42 19.09 23.48 28.37 34.15

24.42 20.09 15.76 10.83 4.94 0.79 5.24 11.06 13.89 19.58 24.03 29.36 34.65

24.82 20.46 16.20 11.31 5.40 0.42 5.59 10.52 13.67 18.84 23.42 27.96 34.59

25.14 20.64 16.31 11.39 5.56 0.19 5.97 10.24 14.81 19.87 23.91 29.62 34.79

24.94 20.62 16.25 11.34 5.47 0.25 5.56 10.09 14.84 19.93 24.32 29.44 35.19

24.78 20.57 16.22 11.41 5.66 0.01 5.57 9.82 14.00 19.91 24.17 29.16 35.69

24.79 20.79 15.95 11.02 5.59 0.29 5.50 9.92 13.86 19.27 23.86 28.83 34.77

0.57 0.40 0.30 0.37 0.31 0.50 0.47 0.60 0.95 0.43 0.46 0.87 0.92

4.1. Measurement uncertainty The uncertainty of thermocouple location in depth direction is ±5 mm for FTCs. The maximum thickness uncertainty of TFTCs is ±0.05 mm, and maximum bead diameter uncertainty of FTCs is within 10% (provided by manufacturer). The uncertainty of the temperature measured by FTC is within ±0.5 °C (provided by manufacturer). The maximum temperature deviation is less than 1 °C for TFTCs. The signal from each TFTC and FTC is acquired at 100 kHz and converted to the temperature data after noise filtering with average 30 adjacent points using a DAQ board.

5. 2D heat flux estimation Two strategies can be used to examine the accuracy of heat flux estimation. One is based on the simulated temperature, and the other uses the experimental temperature. In the first method, the hypothetical heat fluxes are regarded as the real ones. The numerical temperature in measurement locations can be obtained by using these given heat fluxes as the surface boundary condition. Afterward, the new surface heat fluxes estimated based on the calculated temperature are compared with the assumed surface heat fluxes. Notably, the real surface heat fluxes are always unknown in an experiment, but the temperature is definite. An alternative method is to use the measured temperature to examine the accuracy of IHCP solutions. The surface heat fluxes are calculated from the experimental temperature data, and the temperatures calculated using the estimated surface heat fluxes are compared with the measured ones. In the present work, the solutions for 2D single-layer and three-layer IHCPs were examined with both simulated and experimental temperatures. 5.1. Heat flux estimation based on simulated temperature data The 2D single-layer IHCP with TFTC measurement was regarded as an example to provide an insight into the concept of the filter solution. Several rows of the filter matrix are plotted in Fig. 4(a) under the assumption that the total time steps are N = 1000, at = as = 107, and time step td = 0.5 ms. Most of the filter coefficients from the different rows are similar and shift with time, except for the 6th and 999th rows, because the Tikhonov regularization stabilization method provides a poor result at the beginning and end of time [27]. The 1st, 7th, 13th,    (n  6-5)th rows in the F matrix are associated with q1. The 2nd, 8th, 14th,    (n  6-4)th rows are associated with q2. Then, q3, q4, q5, and q6 can be written similarly. In a specific row of the F matrix, the corresponding temperature columns are associated with the temperature sensors and can be analogized by heat fluxes.

The 601st row was selected to investigate the characteristics of 2D IHCPs. The filter coefficients with respect to q1 are plotted in Fig. 4(b) The filter coefficients approach zero toward both ends. As illustrated in Fig. 4(c), mp and mf increase dramatically as fc increases and stabilize at their values (mp = 75, mf = 50) when fc = 0.0001. Therefore, the threshold value (fc) was set to 0.0001 to determine mp and mf. Filter coefficients equal to or less than 0.0001 can be ignored without exerting a considerable effect on heat flux estimation [26]. Through simple judgment, mp and mf were determined as 75 and 50, respectively. Fig. 4(b) shows that the filter coefficients exhibit similar trends with regard to the six sensors at different lateral locations; however, their values are different at the current time step. This difference can be clearly observed in Fig. 4(d). For q1, the largest filter coefficients are associated with the location of temperature sensor TC1 (x = 0 mm). The fitting curve proves that the filter coefficient exhibits exponential decay when the sensor location is far from the location of q1. This result is in accordance with the results of Najafi et al. [8]. Six triangular pulse heat fluxes (see Fig. 6) were utilized to examine the accuracy and sensitivity of the filter solutions to random errors for TFTC and FTC measurements. Random errors are not the deviation between standard and non-standard thermocouples; instead, these errors are caused by the noise in the experiment. Two 2D numerical modes for TFTC and FTC measurements (Fig. 3 (a) and (b)) were established and meshed in ANSYS. Six fluxes are at the top surface. A total of 400 time steps were set to td = 0.5 ms, and the forward-difference scheme was used to compute the temperature at the sensor locations. New heat fluxes for TFTC and FTC measurements can be estimated based on these temperature histories. The optimal comparison criterion implemented for 2D IHCPs was employed in this study to determine the optimal Tikhonov regularization parameters, namely, at and as. Fig. 5(a) illustrates the logarithmic components of the expected error provided by Eqs. (27)–(29) as a function of at or as for 2D single-layer IHCP with TFTC measurement. The three logarithmic components were stable on their respective constants when at or as was lower than 1011. The heat flux bias component decreased as at or as decreased in a range of 1011–107. However, the temperature of the random component presented a contrasting trend. As a result the sum of bias and rand components, E(Rq) reached the minimum value when at = as = 1010. The regularization parameters were selected as at = as = 1010, and the corresponding mp = 201 and mf = 21. The second case, 2D three-layer IHCP with FTC measurement (see Fig. 3(b)), was investigated similarly. Three components with respect to logarithmic Eq,bias, Eq,rand, and E(Rq) using the optimal comparison criterion are plotted in Fig. 5(b). Logarithmic random component Eq,rand increased as the regularization parameters at and as decreased. A similar trend is shown in Fig. 5(a), which

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Fig. 4. (a) filter coefficients at different rows of F as a function of time, (b) filter coefficients at row = 601 of F as a function of time, (c) mp and mf as a function of threshold value and (d) filter coefficients as a function of lateral location for single-layer IHCP with TFTCs: N = 1000, at = as = 107, td = 0.5 ms.

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14 log (Eq,bias )

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Fig. 5. Logarithmic components of the expected error for determining Tikhonov regularization parameters at or as: (a) single-layer IHCP with TFTCs and (b) three-layer IHCP with FTCs.

demonstrates that the random component increased with a small at and as. Logarithmic bias component Eq,bias showed a different characteristic from that of the 2D single-layer IHCP with TFTC measurement (shown in Fig. 5(a)); however, a trend similar to that in the case used by Najafi et al. [8] was observed. Eq,bias and E(Rq) reached their minimum value at at = as = 109. Eventually, the optimum regularization parameters were determined to be at = as = 109, and mp = 117 and mf = 140.

Fig. 6(a) compares the results of the hypothetical surface heat fluxes (solid lines) and the estimated ones (scatters) with the optimum regularization parameters for TFTC measurement. The filter solutions could accurately estimate all heat fluxes. However, the heat fluxes at the last 21 time steps could not be calculated because of the intrinsic nature of the filter solution. As shown in Fig. 6(b), good agreement was observed between the estimated heat fluxes (scatters) and the hypothetical ones (solid lines) for

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1200

1200

1000

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q (kW/m2)

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Fig. 6. Results of the comparison between estimated heat fluxes (scatters) and hypothetical ones (solid lines): (a) single-layer IHCP with TFTCs and (b) three-layer IHCP with FTCs.

FTC measurement. However, small deviations were noted at the descent stage after the maximum heat flux for q1 to q6. Furthermore, the heat fluxes at the last 140 time steps could not be calculated because future temperature data were required. The MSEs of the heat fluxes, qMSE, for TFTC and FTC measurement were computed and are shown in Table 2. The qMSE of q1 to q6 for TFTC measurement are 1.2478, 1.0399, 0.8319, 0.6239, 0.4160, and 0.2081 kW/m2, respectively. The values are small, which validates qMSE decreased as the given maximum heat flux decreased. This phenomenon indicates that accuracy increased when the maximum heat flux was small. Further investigation of the sensitivity of the filter solution to random temperature is essential because a small error of measured temperature would be amplified in IHCPs. Noted that it is difficult to add the real random errors or standard deviation errors (rY) to each temperature at every time step. Therefore, we applied the uniform temperature errors (e = 1 °C, large enough for the actual temperature measurement) adding to each temperature at every time step to simulate the random noise of temperature data alternatively. As illustrated in Table 2, the corresponding qMSE would increase but would still be acceptable. This result indicates that the accuracy and stability of the filter solution are satisfactory. The maximum MSEs of heat fluxes for 2D single-layer and multiple-layer IHCPs is 2.6021 kW/m2 and 6.4572 kW/m2. The mean relative error for FTC measurement is 1.7%, which is larger than that for TFTC (0.4%), because IHCP is solved layer by layer and the errors are accumulated. The filter solution and optimal comparison criterion show good performance with respect to the accuracy of heat flux estimation and the stability to measured temperature error, as evidenced by the cases of 2D single-layer and three-layer IHCPs with TFTC and FTC measurements. 5.2. Heat flux estimation based on experimental temperature data Fig. 7(a) shows the measured dynamic temperature at six lateral sensor locations with spray duration Dt = 50 ms and spray distance L = 30 mm using TFTC measurement during R404A spray

cooling. The temperature histories are similar, but differences exist in the specific values. Given that the response time of the solenoid electric valve was approximately 5 ms, 10 ms was required for the cryogen to flow inside the nozzle and for the droplets to reach the substrate. The temperature decreased dramatically in the first 20 ms when the cold droplets impacted the epoxy resin surface and then stabilized on the respective constant value because of the heat transfer barrier provided by the formed thin liquid cryogen film. It should be noted that cryogen spray cooling is a flash evaporative process, which is different from other spray cooling such as water spray. The droplet temperature was 59.4 °C [31] at spray center, which is 12.9 °C lower than the boiling point (46.5 °C) of R404A. Due to the heat transfer between subcooling droplet and surface, the minimum surface temperature (Ymin) at spray center was between boiling temperature (-46.5 °C) and droplet temperature (-59.4 °C). Ymin were 55.02 °C, 53.7 °C, 47.35 °C, 48.09 °C, 44.03 °C, and 36.48 °C from Y1 to Y6, respectively. The temperature increased gradually after the liquid film evaporated and the surface was exposed to warm air. The temperature at different lateral locations exhibited a large difference. The temperature obtained a high value when the location was far from the spray center (TC1 location, x = 0 mm). This result indicates that the heat transfer weakened at the spray periphery. Six surface heat fluxes ( b q 1 to b q 6 ) estimated based on the measured temperature (Fig. 7(a)) with at = as = 1010, mp = 201, mf = 21, and time step td = 0.5 ms are plotted in Fig. 7(b). The surface heat fluxes present a similar trend. q initially remained at 0 and then sharply increased to the maximum heat flux ( b q max ) due to the latent heat of vaporization of cryogen droplets. Afterward, q decreased to a steady value (approximately 100 kW/m2). At the end of the spray, q decreased to zero after the liquid film completely evaporated. The corresponding maximum heat fluxes b q max were 484.68, 445.43, 395.59, 370.25, 340.13, and 317.18 kW/m2. A large difference existed among the six heat fluxes, and the best cooling ability b q max was found at the spray center location (x = 0– 1 mm).

Table 2 Mean standard errors of heat fluxes for 2D single-layer IHCP with TFTCs and 2D three-layer IHCP with FTCs. q1,MSE

q2,MSE

q3,MSE

q4,MSE

q5,MSE

q6,MSE

TFTC

e = 0 °C e = 1 °C

1.2478 2.6021

1.0399 2.5319

0.8319 2.4768

0.6239 2.4383

0.4160 2.4172

0.2081 2.4141

FTC

e = 0 °C e = 1 °C

2.1230 6.4572

2.094 5.2969

1.002 4.0970

0.7780 2.4383

0.6082 2.4646

1.306 2.3026

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Fig. 7. (a) measured temperature and (b) estimated surface heat flux with TFTCs; (c) measured temperature and (d) estimated surface heat flux with FTCs: Dt = 50 ms and L = 30 mm during R404A spray cooling.

Table 3 Mean standard errors of temperature at the sensor locations for 2D single-layer IHCP with TFTCs and 2D three-layer IHCP with FTCs.

TFTC FTC

T1,MSE (°C)

T2,MSE (°C)

T3,MSE (°C)

T4,MSE (°C)

T5,MSE (°C)

T6,MSE (°C)

0.2424 1.0740

0.1987 0.7127

0.1812 0.6552

0.1800 0.7061

0.1777 0.5224

0.1504 0.1761

The dynamic internal temperature measured by FTCs with Dt = 50 ms and L = 30 mm is depicted in Fig. 7(c), which shows a larger delay response to CSC compared with TFTCs because of the heat transfer barrier of the thin layer of aluminum (10 mm). However, the overall trends of the six temperature profiles are similar to the results in Fig. 7(a). The minimum temperatures (Ymin) are 43.42 °C, 36.13 °C, 29.55 °C, 28.22 °C, 27.32 °C, and 22.45 °C from Y1 to Y6, respectively; these values are lower than those for TFTCs. Additionally, the measured temperature is lower at the spray center (TC1 location, x = 0 mm) than spray periphery. Fig. 7(d) presents the estimated heat fluxes calculated by the filter solution for 2D three-layer IHCP with FTC measurement. According to the preceding discussion, the optimum regularization parameters were set to at = as = 109 with mp = 117 and mf = 140. The time step was set to td = 0.5 ms. The estimated heat flux profiles in this figure are also similar to those in Fig. 7(b). However, a large difference was observed in heat fluxes at different lateral locations. The maximum heat fluxes b q max were 484.58, 351.46, 251.90, 214.94, 220.29, and 193.91 kW/m2, and the best cooling b q max was found at the spray center (x = 0–1 mm). The estimated heat fluxes during the actual R404A spray cooling using filter solutions could not be examined because the exact

heat fluxes on the surface are usually unknown. An alternative approach to calculate the error of simulated and measured temperatures was used in this study. The mean standard errors of temperature, TMSE, at the sensor locations were introduced as follows:

T MSE ¼

Nmf X 2 1 ð Tb ðnÞ  YðnÞÞ N  mf n¼1

!1=2 ;

ð31Þ

b is the simulated temperature at the sensor locawhere T tion obtained by solving the direct heat conduction problem (Eqs. (1)–(5)) based on the estimated heat fluxes (Fig. 7 (b) and (d)). The quantitative errors with respect to temperature for TFTC and FTC measurement are listed in Table 3. The maximum mean standard errors between predicted temperature from estimated heat flux and experimental measurement at the sensor locations for TFTC and FTC are 0.2424 °C and 1.0740 °C (see Table 3), respectively, which indirectly validates good precision of the estimated heat fluxes using the filter solution for 2D single-layer and three-layer geometries. The abovementioned relative errors of heat flux estimation are induced by the location errors of thermocouple sensors as well as the errors of simulated sensitive coefficient X.

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to b q 1 calculated by the 2D filter solution is larger than the 1D result [19] only around the maximum heat flux for TFTC measurement. The corresponding b q 1max is 13.6% higher (from 429.32 kW/m2 to 487.65 kW/m2). However, the heat fluxes that exclude b q 1max exhibited no differences mainly due to the small heat conductivity coefficient (k = 0.14 Wm1K1). These results demonstrate that lateral heat transfer should not be ignored, and the 2D filter solution is accurate in estimating heat flux with TFTC measurement. The high maximum heat flux obtained by the 2D method is reasonable because lateral heat transfer is considered to reach the same temperature in the spray center. Fig. 8(b) shows that the estimated heat flux, b q 1 , calculated by the 2D filter solution is larger than the heat flux estimated by the 1D Duhamel theory multi-layer method [19] with FTC measurement. This measurement is different from the TFTC measurement. The corresponding b q max is 484.58 kW/m2 for the 2D filter solution, which is almost 2.43 times larger than the 1D result (197.22 kW/m2) [19]. A probably explanation is that the heat conductivity coefficient of aluminum is k = 236 Wm1K1, which means that the lateral heat transfer at the surface is stronger than that in the TFTC measurement. Although the thermal paste also contributes the heat transfer in FTC, the heat conductivity coefficient of thermal paste (k
6 Wm1K1, provide by manufacturer) which is 38 times smaller than that of aluminum. Therefore, we can believe that the dominant contribution in terms of lateral heat transfer at surface is caused by aluminum. The 2D filter solution is accurate and suitable for heat flux estimation with TFTC and FTC measurements. This result proves that lateral heat transfer cannot be ignored, especially when the heat conductivity of the material is large. The trend of estimated heat flux b q 1 using TFTC and FTC measurement is similar (see Fig. 8(c)). However, the heat flux using FTC measurement is larger than that of TFTC measurement in the whole computation time mainly owing to the large heat conductivity coefficient of aluminum (k = 236 Wm1K1). Besides, the maximum heat flux corresponding time using FTCs is slightly lagged compared with TFTCs due to faster temperature response (s = 1.2 ms) of TFTC measurement. The computational time (
1.947 s) for TFTC measurement based on experimental temperature data is less than that for FTC measurement (
2.019 s) due to the less requirement of temperature and sensitivity coefficient data in mp + mf + 1 (223 for TFTC measurement, and 258 for FTC measurement) time steps, which indicates that a large mp and mf will increase the computational time. All computations in present work are done by using a personal computer (Intel(R) Core(TM) i74790 CPU, 3.6 GHz, 12 GB RAM).

250

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(c) Fig. 8. Results of the comparison of estimated heat flux b q 1 using 1D Duhamel theory multi-layer method [19] and 2D filter solution with (a) TFTCs and (b) FTCs; b 1 with TFTCs and FTCs. (c) comparison of estimated heat flux q

The 2D result was compared with that of the 1D method proposed in our previous investigation [19] because it shows better accuracy of heat flux estimation and stability to measured temperature errors compared with several different methods for 1D IHCPs. As shown in Fig. 8(a), the estimated heat flux with respect

The filter solution to solve 1D single-layer IHCPs was applied to solve a general 2D multi-layer IHCP for the estimation of surface heat flux. An optimal comparison criterion was investigated to optimize the key parameters, namely, at and as. The 2D multilayer filter solution and the implemented optimal comparison criterion were used to compare the heat flux estimation for TFTC and FTC measurements during CSC. Six hypothetical triangular heat fluxes and random temperature errors e = 1 °C were employed to analyze the accuracy and sensitivity of the filter solution for 2D single-layer and multi-layer IHCPs with TFTC and FTC measurements. The qMSE values for TFTC and FTC measurements with and without the random temperature errors were all within the acceptable range, which validates the good accuracy and stability of the filter solutions. Further examinations were conducted based on the measured temperature response to CSC. Given that surface heat fluxes are

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always unknown, the numerical temperature based on estimated heat fluxes was compared with the measured temperature. The results indicated that the maximum MSE of temperature TMSE was 1.0740 °C, which also reveals the accuracy of heat flux estimation using the filter solution for TFTC and FTC measurements. The simulated temperature with respect to the estimated heat flux showed good agreement with the measured temperature, which also indirectly validates the accuracy of the filter solution. The difference in the estimated heat flux in reference to q1 using the 2D filter solution and the 1D Duhamel theory multi-layer method [19] only occurred around b q max , which is 13.6% larger with the former method for TFTC measurement. However, the corresponding b q max using the 2D filter solution is almost 2.43 times larger than that using the 1D Duhamel theory multi-layer method [19] for FTC measurement. The increase in the b q max of FTC is larger than that of TFTC, which is reasonable because more energy is transferred laterally owing to the larger heat conductivity coefficient of the material. Therefore, the 2D filter solution is more accurate than the 1D method. Moreover, lateral heat transfer should not be ignored, especially when the heat conductivity coefficient of the material is large. Conflict of interest The authors declare that there is no conflict of interest. Acknowledgement This work was supported by the National Natural Science Foundation of China (51336006) and Fundamental Research Funds for the Central Universities. References [1] W.L. Ko, R.D. Quinn, L. Gong, Finite-element reentry heat-transfer analysis of space shuttle Orbiter, NASA Sti/recon Technical Report N, vol. 87, 1987. [2] J.P. Calame, High heat flux thermal management of microfabricated upper millimeter-wave vacuum electronic devices, in: Vacuum Electronics Conference, 2008, IVEC 2008, IEEE International, 2008, pp. 50–51. [3] P. Wikström, W. Blasiak, F. Berntsson, Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method, Appl. Therm. Eng. 27 (14–15) (2007) 2463–2472. [4] J. Taler, P. Duda, B. We˛glowski, W. Zima, S. Gra˛dziel, T. Sobota, D. Taler, Identification of local heat flux to membrane water-walls in steam boilers, Fuel 88 (2) (2009) 305–311. [5] C.-H. Lee, L.-W. Hourng, K.-W. Lin, Mathematical model predicting the critical heat flux of nuclear reactors, J. Comput. Sci. 8 (12) (2012) 1996. [6] J.M. Tian, B. Chen, D. Li, Z.F. Zhou, Transient spray cooling: similarity of dynamic heat flux for different cryogens, nozzles and substrates, Int. J. Heat Mass Transf. 108 (2017) 561–571. [7] I. Sumita, P. Olson, Rotating thermal convection experiments in a hemispherical shell with heterogeneous boundary heat flux: implications for the Earth’s core, J. Geophys. Res. 107 (107) (2002) 5–18, ETG 5-1-ETG. [8] H. Najafi, K.A. Woodbury, J.V. Beck, Real time solution for inverse heat conduction problems in a two-dimensional plate with multiple heat fluxes at the surface, Int. J. Heat Mass Transf. 91 (2015) 1148–1156. [9] J.V. Beck, Filter solutions for the nonlinear inverse heat conduction problem, Inverse Probl. Sci. Eng. 16 (1) (2008) 3–20. [10] J.V. Beck, B. Blackwell, C.R. St Clair, Inverse Heat Conduction, Ill-Posed Problems, Wiley, 1985.

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