Open Access Published by De Gruyter Open Access June 8, 2021

Thermal-hydraulic performance prediction of two new heat exchangers using RBF based on different DOE

Chulin Yu, Youqiang Wang, Haiqing Zhang, Bingjun Gao and Yin He
From the journal Open Physics

Abstract

Thermal performance prediction with high precision and low cost is always the need for designers of heat exchangers. Three typical design of experiments (DOE) known as Taguchi design method (TDM), Uniform design method (UDM), and Response surface method (RSM) are commonly used to reduce experimental cost. The radial basis function artificial neural network (RBF) based on different DOE is used to predict the thermal performance of two new parallel-flow shell and tube heat exchangers. The applicability and expense of ten different prediction methods (RBF + TDML9, RBF + TDML18, RBF + UDM, RBF + TDML9 + UDM, RBF + TDML18 + UDM, RBF + RSM, RBF + RSM + TDML9, RBF + RSM + TDML18, RBF + RSM + UDM, RSM) are discussed. The results show that the RBF + RSM is a very efficient method for the precise prediction of thermal-hydraulic performance: the minimum error is 2.17% for Nu and 5.30% for f. For RBF, it is not true that the more of train data, the more precision of the prediction. The parameter “spread” of RBF should be adjusted to optimize the prediction results. The prediction using RSM only can also obtain a good balance between precision and time cost with a maximum prediction error of 14.52%.

Nomenclature

A

area, mm2

A in

area of cross-section (mm2)

a

length (mm)

b

baffle width (mm)

c

length (mm)

C p

specific heat at constant pressure (J kg−1 K−1)

d c

coil diameter (mm)

d i

inner diameter of tube (mm)

d o

outer diameter of tube (mm)

d r

diameter of rod (mm)

D h

hydraulic diameter (mm)

f

average friction factor (−)

h

convection heat transfer coefficient (W m2 K−1)

L

length (mm)

L b

baffle distance (mm)

l et

length of equilateral triangular section (mm)

P c

coil pitch (mm)

p

pressure (Pa)

P t

tube pitch (mm)

Δ T

log-mean temperature difference (K)

T

temperature (K)

T a

tape amplitude (mm)

T w

tape width (mm)

T t

tape thickness (mm)

T p

tape pitch (mm)

t

thickness (mm)

u

velocity (m s−1)

V in

inlet velocity (m s−1)

t b

baffle thickness (mm)

Re

Reynolds number (–)

R

radius (mm)

Pr

Prandtl number (–)

Δ p

pressure drop (Pa)

Greek symbols

ρ

density (kg m−3)

ε

turbulence kinetic energy dissipation rate

k

turbulence kinetic energy

μ

dynamic viscosity (kg m−1 s−1)

λ

thermal conductivity (W m−1 K−1)

Subscripts

w

wall

in

inlet

out

outlet

rod

round rod

RRB

RRB-STHX

ETWC

STHX-CPBETWC

SWT

STHX-CPBSWT

Abbreviations

ANN

artificial neural network

ANOVA

analysis of variance

CCD

central composite design

CPB

clamping plate baffle

CFD

numerical result obtained by computation fluid mechanics

DOE

design of experiments

Pre

prediction result

RBF

radial basis function artificial neural network

RRB

round rod baffle

RSM

response surface methodology

SWT

sinusoidal wavy tape

STHX

shell and tube heat exchanger

TDM

Taguchi method

MLFF

multilayer feed forward artificial neural network

UDM

uniform design method

1 Introduction

Shell and tube heat exchangers (STHXs) are wildly used in various industries [1]. The greatest advantage of the STHXs over other types of heat exchangers is that they can withstand very high temperature and pressure simultaneously at a low cost [2]. As a result, it can be found that the STHXs are almost the only candidate for high pressure feed water heats in thermal power plants [3].

As a whole, the STHXs can be classified into three basic types according to the fluid flow path: cross flow, parallel flow, and helical flow [4]. Among the three types, the parallel-flow STHXs are more promising because they have the merits of both helical flow STHXs (low fouling and pressure drop, etc.) and cross flow STHXs (high heat transfer rate and low cost, etc.). In the past few decades, different parallel-flow STHXs have been proposed [5]. The famous one needed to mention is round rod baffle STHX (RRB-STHX) which is invented by Philips Company to solve the high pressure drop problem of segmental baffle STHX [6]. However, the RRB-STHX is vulnerable to flow-induced vibration failure due to the fact that the round rod and the tube are in point contact status [7]. Very large contact stress occurs when the flow impacts the tube. For this reason, clamping plate baffle (CPB) is proposed. Figure 1 illustrates one example of application of CPB in the 4th generation sodium cooled nuclear power station [8].

Figure 1 
               Picture of heat exchanger used in the 4th generation sodium cooled nuclear power station.

Figure 1

Picture of heat exchanger used in the 4th generation sodium cooled nuclear power station.

In order to save energy and reduce cost, heat transfer enhancement technologies are always the hot point of energy utilization. Two new parallel-flow STHXs with CPB and inserts (STHX-CPBETWC and STHX-CPBSWT) are proposed by Yu et al. [9]. The effects of geometrical parameters such as coil pitch, coil diameter, side length of equilateral cross-section, tape amplitude, tape pitch, and tape width on the thermal-hydraulic performance are numerically explored with flow in turbulent regime.

For designers of STHX, it is meaningful to obtain the thermal-hydraulic performance using design of experiments (DOE) to save time and costs. Taguchi design method (TDM), uniform design method (UDM), and response surface method (RSM) are three wildly used DOE. TDM developed by Genichi Taguchi is a statistical technique [10]. It can help researchers to learn the effect of many factors on object function with a small cost [11,12]. The uniform experimental design method (UDM) was proposed by Fang and Wang based on the quasi-Monte Carlo method, number theoretic method, and multivariate statistics [13]. Compared with the conventional statistical experimental design methods, the UDM can significantly reduce the number of experiments under the same number of factors and levels [14,15]. The RSM is an empirical modeling approach for determining the relationship between input parameters and responses with the various desired criteria [16]. One of the advantages of the RSM over the conventional experimental methods, in addition to reducing the experimental cost, is that it minimizes the variability around the target [17,18,19].

Artificial neural network (ANN) is finding increasing use as predictive tool in an extensive range of disciplines due to their strong adaptivity for solving complex and nonlinear systems [20]. Therefore, it has been applied to complex nonlinear engineering problems in different real-world applications with a significant reduction in cost and time. Different types of ANN are used to predict the thermal-hydraulic performance of STHXs [21]. It was reported that the multilayer feed forward artificial networks (MLFF) were the most widely used ANN for heat exchanger analysis due to their simplicity. However, the MLFF has some limitations in optimizing the network parameters [22]. Instead, radial basis function artificial neural network (RBF) has faster convergence, smaller extrapolation errors, and higher reliability than MLFF [23]. Radial Basis Function (RBF) neural network is a very important ANN. It is composed of three main categories of layers (input layer, some hidden layers, and output layer). The input layer in the RBF receives the input data without computing power and passes it through the hidden layer. The hidden layer performs some operations of the received data and extracted relevant information and sends them to the output layer [24]. Elsheikh et al. illustrated merits and debits of RBF [25]. Ghasemi et al. used RBF to predict and optimize energetic efficiency of TiO2–Al2O3/water nanofluid in mini double pipe heat exchanger [26]. The combination use of ANN and DOE can be found in some references [2730]. In most cases, the DOE was used to establish an appropriate structure of ANN. However, to the best of authors’ knowledge, the research on the train data design of ANN using DOE is rare. One may wonder if RBF and DOE can be used simultaneously to get a balance between precision and time cost.

Generally, the working flow chart of establishing RBF based on different experimental design methods to predict the thermal performance of STHX is shown in Figure 2. For STHX with the same parameters, the number of experiments needed to obtain the thermal-hydraulic performance numerically is quite different. As a result, the time needed to finish numerical calculation is also quite different. For example, for STHX with four design parameters (each with three levels), a minimum of 9, 10, and 25 experiments are needed according to TDM, UDM, and RSM. In present study, the STHX-CPBETWC and STHX-CPBSWT are taken as two objectives; the applicability and cost of prediction method using RBF based on TDM, UDM, and RSM are learned and discussed.

Figure 2 
               Working flow chart of prediction of thermal-hydraulic performance of STHX using RBF.

Figure 2

Working flow chart of prediction of thermal-hydraulic performance of STHX using RBF.

2 Geometrical structure

For parallel-flow STHX, the sketch of the tube bundle is illustrated in Figure 3. The sketches of ETWC and SWT are shown in Figure 4. The heat exchange tube inside the heat exchanger is arranged in a square form. When the medium flows, turbulence can be formed, which is favorable to the heat transfer process. The tube spacing of the heat exchange tube is Pt, and the outer diameter of the heat exchange tube is d 0. The clamping plate is used to fix the tubes. The insert element such as ETWC and SWT are placed in the space of the tubes to further enhance the heat transfer.

Figure 3 
               Sketch of tube bundle with CPB.

Figure 3

Sketch of tube bundle with CPB.

Figure 4 
               Sketch of two different inserts between tubes. (a) SWT, and (b) ETWC.

Figure 4

Sketch of two different inserts between tubes. (a) SWT, and (b) ETWC.

3 RBF

The construction of RBF involves three layers such as input layer, hidden layer, and output layer, which can be seen from Figure 5. The activation function used in hidden layer is RBF. In the software Matlab [31], two different kinds of RBF are presented (approximate RBF and exact RBF). In this paper, the exact RBF is selected as only one parameter “spread” can affect the precision and convergence of the prediction results. Four design parameters, named P c, Re, l et, and d c, are used in the input layer for STHX-CPBETWC, while T p, Re, T a, and T w are used in the input layer for STHX-CPBSWT. In the output layer, the Nu and f are used as the response parameters.

Figure 5 
               Architecture of RBF of STHX-CPBETWC and STHX-CPBSWT.

Figure 5

Architecture of RBF of STHX-CPBETWC and STHX-CPBSWT.

The train of ANN is very crucial. It is well-known that the more of train data, the higher of prediction precision. However, it is not practical to use a full factor test to train the ANN. Furthermore, there is not a well-established principle to decide how to train the ANN [14]. The basic method is trial and error. In this paper, three different experimental methods are used to generate the train data. Table 1 gives a detailed 9 training scheme of RBF. For example, for building method 1 and 2, the train data are generated by the TDML9 and TDML18, respectively. To test the performance of the trained RBF, the test data are generated by the TDM, UDM, and RSM. Two indicators, named the maximum absolute error and sum absolute error of the test data, are adopted. In addition, another prediction method using RSM only is also calculated and compared. It is worth noting that L9 is a part of L18 according to the orthogonal array of TDM. So, the test data of RBF is TDML18 + UDM + RSM.

Table 1

Different building methods of RBF based on different experimental design methods

Data Building methods of RBF
1 2 3 4 5 6 7 8 9
Train data TDML9 TDML18 UDM TDML9 + UDM TDML18 + UDM RSM RSM + TDML9 RSM + TDML18 RSM + UDM
Test data TDML18 + UDM + RSM

4 Experimental design method

4.1 TDM

TDM is also known as the orthogonal experimental method. Orthogonal experimental design is the study of multiple factors level and a design method. According to orthogonality from the selected part of the comprehensive experiment typical test points, the representative points with the “evenly dispersed, neat comparable to the characteristics of the Orthogonal DOE is the main method of fractional factorial design. It is an efficient, rapid, and economical experimental design method.

According to the theory of TDM, for STHX-ETWC and STHX-SWT, three levels of each factor are chosen as shown in Tables 2 and 3, respectively. Two different orthogonal arrays (L9 and L18) can be used to arrange experiments. The detailed arrangement is shown in Tables 4–7. The value in the bracket represents the actual value of selected factors. The other geometric parameters are the same with those listed in Table 8.

Table 2

The three levels of the selected factors used in TDM for STHX-CPBETWC

Code Factors (unit) Level 1 Level 2 Level 3
A P c (mm) 20 30 40
B Re 14,465 21,698 28,931
C l et (mm) 2 3 4
D d c (mm) 13 14 15
Table 3

The three levels of the selected factors used in TDM for STHX-CPBSWT

Code Factors (unit) Level 1 Level 2 Level 3
A T p (mm) 20 30 40
B Re 14,465 21,698 28,931
C T a (mm) 2 4 6
D T w (mm) 10 12 14
Table 4

The orthogonal array of experiments of L9 for STHX-CPBETWC

Case no. Control factors
A B C D
1 1 (20) 1 (14,465) 1 (2) 1 (13)
2 1 (20) 2 (21,698) 2 (3) 2 (14)
3 1 (20) 3 (28,931) 3 (4) 3 (15)
4 2 (30) 1 (14,465) 2 (3) 3 (15)
5 2 (30) 2 (21,698) 3 (4) 1 (13)
6 2 (30) 3 (28,931) 1 (2) 2 (14)
7 3 (40) 1 (14,465) 3 (4) 2 (14)
8 3 (40) 2 (21,698) 1 (2) 3 (15)
9 3 (40) 3 (28,931) 2 (3) 1 (13)
Table 5

The orthogonal array of experiments of L18 for STHX-CPBETWC

Case no. Control factors
A B C D
1 1 (20) 1 (14,465) 1 (2) 1 (13)
2 1 (20) 2 (21,698) 2 (3) 2 (14)
3 1 (20) 3 (28,931) 3 (4) 3 (15)
4 2 (30) 1 (14,465) 1 (2) 2 (14)
5 2 (30) 2 (21,698) 2 (3) 3 (15)
6 2 (30) 3 (28,931) 3 (4) 1 (13)
7 3 (40) 1 (14,465) 2 (3) 1 (13)
8 3 (40) 2 (21,698) 3 (4) 2 (14)
9 3 (40) 3 (28,931) 1 (2) 3 (15)
10 1 (20) 1 (14,465) 3 (4) 3 (15)
11 1 (20) 2 (21,698) 1 (2) 1 (13)
12 1 (20) 3 (28,931) 2 (3) 2 (14)
13 2 (30) 1 (14,465) 2 (3) 3 (15)
14 2 (30) 2 (21,698) 3 (4) 1 (13)
15 2 (30) 3 (28,931) 1 (2) 2 (14)
16 3 (40) 1 (14,465) 3 (4) 2 (14)
17 3 (40) 2 (21,698) 1 (2) 3 (15)
18 3 (40) 3 (28,931) 2 (3) 1 (13)
Table 6

The orthogonal array of experiments of L9 for STHX-CPBSWT

Case no. Control factors
A B C D
1 1 (20) 1 (14,465) 1 (2) 1 (10)
2 1 (20) 2 (21,698) 2 (4) 2 (12)
3 1 (20) 3 (28,931) 3 (6) 3 (14)
4 2 (30) 1 (14,465) 2 (4) 3 (14)
5 2 (30) 2 (21,698) 3 (6) 1 (10)
6 2 (30) 3 (28,931) 1 (2) 2 (12)
7 3 (40) 1 (14,465) 3 (6) 2 (12)
8 3 (40) 2 (21,698) 1 (2) 3 (14)
9 3 (40) 3 (28,931) 2 (4) 1 (10)
Table 7

The orthogonal array of experiments of L18 for STHX-CPBSWT

Case no. Control factors
A B C D
1 1 (20) 1 (14,465) 1 (2) 1 (10)
2 1 (20) 2 (21,698) 2 (4) 2 (12)
3 1 (20) 3 (28,931) 3 (6) 3 (14)
4 2 (30) 1 (14,465) 1 (2) 2 (12)
5 2 (30) 2 (21,698) 2 (4) 3 (14)
6 2 (30) 3 (28,931) 3 (6) 1 (10)
7 3 (40) 1 (14,465) 2 (4) 1 (10)
8 3 (40) 2 (21,698) 3 (6) 2 (12)
9 3 (40) 3 (28,931) 1 (2) 3 (14)
10 1 (20) 1 (14,465) 3 (6) 3 (14)
11 1 (20) 2 (21,698) 1 (2) 1 (10)
12 1 (20) 3 (28,931) 2 (4) 2 (12)
13 2 (30) 1 (14,465) 2 (4) 3 (14)
14 2 (30) 2 (21,698) 3 (6) 1 (10)
15 2 (30) 3 (28,931) 1 (2) 2 (12)
16 3 (40) 1 (14,465) 3 (6) 2 (12)
17 3 (40) 2 (21,698) 1 (2) 3 (14)
18 3 (40) 3 (28,931) 2 (4) 1 (10)
Table 8

The other geometric parameters of STHX-CPBETWC and STHX-CPBSWT

Parameter d o P t d r L b a b c tb
Value 25 32 6 400 10 20 10 3

4.2 UDM

According to the theory of UDM, for 4 factors and 10 levels of each factor, the array of experiments is shown in Tables 9 and 10. For the problem discussed in this paper, 5 levels of each factor are used. The quasi-level arrangement is adopted. The basic principle is shown in Figure 6. It illustrates that the level 6 should be substituted with level 1, the level 7 should be substituted with level 2, etc. when only 5 levels of each factor are considered.

Table 9

The array of experiments of U1110 for STHX-CPBETWC

Case no. Control factors
A B C D
1 1 (20) 2 (18,082) 5 (4) 7 (13.5)
2 2 (25) 4 (25,315) 10 (4) 3 (14)
3 3 (30) 6 (14,465) 4 (3.5) 10 (15)
4 4 (35) 8 (21,698) 9 (3.5) 6 (13)
5 5 (40) 10 (28,931) 3 (3) 2 (13.5)
6 6 (20) 1 (14,465) 8 (3) 9 (14.5)
7 7 (25) 3 (21,698) 2 (2.5) 5 (15)
8 8 (30) 5 (28,931) 7 (2.5) 1 (13)
9 9 (35) 7 (18,082) 1 (2) 8 (14)
10 10 (40) 9 (25,315) 6 (2) 4 (14.5)
Table 10

The array of experiments of U1110 for STHX-CPBSWT

Case no. Control factors
A B C D
1 1 (20) 2 (18,082) 5 (6) 7 (11)
2 2 (25) 4 (25,315) 10 (6) 3 (12)
3 3 (30) 6 (14,465) 4 (5) 10 (14)
4 4 (35) 8 (21,698) 9 (5) 6 (10)
5 5 (40) 10 (28,931) 3 (4) 2 (11)
6 6 (20) 1 (14,465) 8 (4) 9 (13)
7 7 (25) 3 (21,698) 2 (3) 5 (14)
8 8 (30) 5 (28,931) 7 (3) 1 (10)
9 9 (35) 7 (18,082) 1 (2) 8 (12)
10 10 (40) 9 (25,315) 6 (2) 4 (13)
Figure 6 
                  Sketch of quasi-level arrangement of U1110.

Figure 6

Sketch of quasi-level arrangement of U1110.

4.3 RSM

The RSM is an empirical modeling approach for determining the relationship between input parameters and responses with the various desired criteria. In this study, central composite design of RSM is used to arrange experiments. The factors and levels used in RSM are listed in Tables 11 and 12. The arrangement of experiments is shown in Tables 17 and 21.

Table 11

Factors and levels of STHX-CPBETWC used in RSM

Code Factor (unit) Level 1 Level 2 Level 3
−1 0 1
A P c (mm) 20 30 40
B Re 14,465 21,698 28,931
C l et (mm) 2 3 4
D d c (mm) 13 14 15
Table 12

Factors and levels of STHX-CPBSWT used in RSM

Code Factor (unit) Level 1 Level 2 Level 3
−1 0 1
A T p (mm) 20 30 40
B Re 14,465 21,698 28,931
C T a (mm) 2 4 6
D T w (mm) 10 12 14

It is worth nothing that the RSM can also predict the results by building response surface model. In this paper, the quadratic polynomial model is used to build the response surface model of Nu and f. It can be expressed as follows:

(1) Y = b 0 + I = 1 N ( b I X I ) + I = 1 N 1 J = I + 1 N b I , J X I X J + I = 1 N ( b I , J X I 2 ) + Δ
where Y is a response variable; X I and X J are the factors or variables; the symbols b 0, b I , b I , J , and b I , I are constants; N is the number of the factors or variables; Δ is the statistical error.

5 Numerical model

5.1 Computational domain

The computational domain and mesh scheme of the STHX-CPBETWC and STHX-CPBSWT are shown in Figure 7. It includes inlet extended region, heat transfer region, and outlet extended region. Hybrid mesh is used.

Figure 7 
                  Sketch of computational domain and boundary conditions.

Figure 7

Sketch of computational domain and boundary conditions.

To simplify the numerical simulation, the following assumptions are made: the working fluid is water which is continuous, Newtonian and isotropic; the gravity effect, thermal radiation, and viscous heating are negligible. The thermal-physical properties of water (density ρ = 999.7 kg m−3, thermal conductivity λ = 0.574 W m−1 K−1, specific heat c p = 4191 J kg−1 K−1, and μ = 0.001306 (Pa s)).

5.2 Governing equations

The steady dimensionless mathematic equations for continuity, momentum, and energy are as follows:

Continuity equation:

(2) ( ρ u j ) x j = 0

Momentum equation:

(3) ( ρ u i u j ) x j = p x i + x j μ eff u i x j + u j x i

Energy equation:

(4) ( ρ c p T u j ) x j = x j λ eff T x j

5.3 Boundary conditions

The 3-D, double-precision, pressure-based solver of Fluent is used [32]. SIMPLE algorithm is adopted; the second order upwind scheme is chosen to discrete momentum equation and energy equations; the second order difference scheme is used for the pressure [33]. The standard wall function method is adopted for the near-wall region. All equations take the convergent criterions of relative residual of 1 × 10−4 except energy taking 1 × 10−6.

The boundary conditions for computations shown in Figure 7 are described as follow:

  1. (1)

    Uniform temperature 283.15 K is assigned for the velocity inlet boundary.

  2. (2)

    The surfaces of the baffles are set as adiabatic.

  3. (3)

    Pressure outlet boundary is used for the exit.

  4. (4)

    The four boundary walls of the unit model are set as symmetry boundary conditions.

  5. (5)

    The temperature of heated tube wall is set as a constant, 350 K.

5.4 Thermal-hydraulic parameters

The Reynolds number Re can be obtained as:

(6) Re = ρ D h V in μ
(7) D h = 4 ( P t 2 π d 0 2 / 4 ) π d 0

The average heat transfer coefficient, h, and the average Nusselt number, Nu, are obtained as follows:

(8) h = c p ρ V in A in ( T in T out ) A 0 Δ T
(9) Nu = h D h λ
(10) Δ T = T in T out ln T in T w T out T w

The friction factor is estimated by:

(11) f = D h L 2 Δ p ρ V in 2

5.5 Validation and grid independence

For STHX-CPBETWC and STHX-CPBSWT, no experimental results can be found in open literatures. To validate the reliability of the numerical model, non-staggered tubes supported by RRB are computed and compared with the results obtained by Dong et al. [34]. The geometric parameters of the validation model are listed in Table 8. The computed results are plotted in Figure 8. It can be seen that the relative maximum deviation is within 7%, which shows the reliability of the numerical model adopted in this paper. In addition, mesh independency test is carried out to find the appropriate scheme (Table 13).

Figure 8 
                  The result of model validation.

Figure 8

The result of model validation.

Table 13

Geometric parameters of RRB-STHX for model validation (mm)

Parameter d o P t d r L b
Value 25 32 6 300

6 Results and discussions

According to the calculating formulas mentioned above, the results of STHX-CPBETWC and STHX-CPBSWT obtained by TDM, UDM, and RSM are listed in Tables 14–21, respectively. Analysis of variance (ANOVA) of NuETWC and f ETWC obtained by TDML9 and TDML18 is listed in Tables 22–25, respectively. ANOVA of NuSWT and f SWT obtained by TDML9 and TDML18 is listed in Tables 26–29, respectively. ANOVA of NuETWC, f ETWC, NuSWT, and f SWT obtained by RSM is listed in Tables 30–33, respectively.

Table 14

Numerical results of STHX-CPBETWC obtained by TDML9

Case no. NuETWC f ETWC
1 205.55 0.5745
2 324.36 0.7594
3 497.50 1.0761
4 216.77 0.5708
5 307.12 0.6048
6 337.93 0.4284
7 219.46 0.5369
8 269.50 0.4011
9 349.92 0.4210
Table 15

Numerical results of STHX-CPBETWC obtained by TDML18

Case no. NuETWC f ETWC
1 205.55 0.5745
2 324.36 0.7594
3 497.50 1.0761
4 198.91 0.4580
5 298.36 0.5491
6 393.04 0.5944
7 204.75 0.4493
8 301.94 0.5153
9 343.66 0.3986
10 271.36 1.1199
11 277.37 0.5513
12 416.43 0.7471
13 216.77 0.5708
14 307.12 0.6048
15 337.93 0.4284
16 219.46 0.5369
17 269.50 0.4011
18 349.92 0.4210
Table 16

Numerical results of STHX-CPBETWC obtained by UDM

Case no. NuETWC f ETWC
1 301.38 0.9191
2 375.60 0.7260
3 228.08 0.6319
4 293.59 0.5137
5 356.19 0.4408
6 240.56 0.8079
7 295.18 0.5613
8 349.40 0.4683
9 233.77 0.4107
10 304.35 0.3864
Table 17

Numerical results of STHX-CPBETWC obtained by RSM

Case no. Level of parameter (actual value) Response
A B C D NuSWT f SWT
1 −1 (20) −1 (14,465) 1 (4) 1 (15) 271.36 1.1199
2 −1 (20) −1 (14,465) −1 (2) −1 (13) 205.55 0.5745
3 −1 (20) 1 (28,931) −1 (2) 1 (15) 389.87 0.6366
4 −1 (20) 1 (28,931) 1 (4) −1 (13) 440.53 0.8619
5 −1 (20) 1 (28931) −1 (2) −1 (13) 351.49 0.5413
6 −1 (20) 1 (28,931) 1 (4) 1 (15) 497.50 1.0766
7 −1 (20) −1 (14,465) 1 (4) −1 (13) 245.19 0.9018
8 −1 (20) 0 (21,698) 0 (3) 0 (14) 324.36 0.7592
9 −1 (20) −1 (14,465) −1 (2) 1 (15) 220.80 0.6724
10 0 (30) −1 (14,465) 0 (3) 0 (14) 213.29 0.5491
11 0 (30) 0 (21,698) 0 (3) 0 (14) 291.21 0.5273
12 0 (30) 0 (21,698) 0 (3) 1 (15) 298.36 0.5491
13 0 (30) 1 (28,931) 0 (3) 0 (14) 370.88 0.5190
14 0 (30) 0 (21,698) 0 (3) −1 (13) 285.27 0.5114
15 0 (30) 0 (21,698) 1 (4) 0 (14) 316.33 0.6295
16 0 (30) 0 (21,698) −1 (2) 0 (14) 268.32 0.4372
17 1 (40) 1 (28,931) −1 (2) −1 (13) 323.91 0.3660
18 1 (40) 1 (28,931) 1 (4) 1 (15) 396.75 0.5215
19 1 (40) −1 (14,465) 1 (4) −1 (13) 214.09 0.5141
20 1 (40) −1 (14,465) −1 (2) 1 (15) 201.29 0.4265
21 1 (40) 1 (28,931) 1 (4) −1 (13) 372.46 0.4839
22 1 (40) −1 (14,465) 1 (4) 1 (15) 224.57 0.5517
23 1 (40) 1 (28,931) −1 (2) 1 (15) 343.66 0.3986
24 1 (40) 0 (21,698) 0 (3) 0 (14) 282.11 0.4441
25 1 (40) −1 (14,465) −1 (2) −1 (13) 193.35 0.3915
Table 18

Numerical results of STHX-CPBSWT obtained by TDML9

Case no. Nuswt f SWT
1 199.97 0.4811
2 248.20 0.9150
3 267.76 1.1615
4 198.37 0.4216
5 247.38 0.8089
6 258.91 0.9292
7 195.20 0.3899
8 213.24 0.5465
9 257.81 0.8687
Table 19

Numerical results of STHX-CPBSWT obtained by TDML18

Case no. NuETWC f ETWC
1 199.97 0.4811
2 349.02 0.9007
3 492.03 1.1542
4 198.37 0.4215
5 346.18 0.7891
6 467.53 0.9017
7 213.29 0.5465
8 361.65 0.8482
9 322.36 0.3638
10 267.76 1.1615
11 265.57 0.4624
12 450.76 0.8923
13 247.38 0.8089
14 362.06 0.9107
15 329.1 0.3967
16 257.54 0.8681
17 259.7 0.3716
18 366.15 0.5193
Table 20

Numerical results of STHX-CPBSWT obtained by UDM

Case no. Nu f SWT
1 309.57 1.0008
2 453.10 1.1159
3 270.25 1.0204
4 325.56 0.6854
5 376.14 0.5411
6 252.59 0.9535
7 315.12 0.6580
8 356.72 0.5004
9 228.19 0.3805
10 287.96 0.3626
Table 21

Numerical results of STHX-CPBSWT obtained by RSM

Case no. Level of parameter (actual value) Response
A B C D NuSWT f SWT
1 −1 (20) −1 (14,465) 1 (6) 1 (14) 267.76 1.1615
2 −1 (20) −1 (14,465) −1 (2) −1 (10) 199.97 0.4811
3 −1 (20) 1 (28,931) −1 (2) 1 (14) 345.22 0.4937
4 −1 (20) 1 (28,931) 1 (6) −1 (10) 454.54 0.9204
5 −1 (20) 1 (28,931) −1 (2) −1 (10) 332.73 0.4563
6 −1 (20) 1 (28,931) 1 (6) 1 (14) 492.03 1.1542
7 −1 (20) −1 (14,465) 1 (6) −1 (10) 251.76 0.9410
8 −1 (20) 0 (21,698) 0 (4) 0 (12) 349.02 0.9007
9 −1 (20) −1 (14,465) −1 (2) 1 (14) 205.28 0.5205
10 0 (30) −1 (14,465) 0 (4) 0 (12) 239.20 0.7451
11 0 (30) 0 (21,698) 0 (4) 0 (12) 332.01 0.7252
12 0 (30) 0 (21,698) 0 (4) 1 (14) 346.18 0.7891
13 0 (30) 1 (28,931) 0 (4) 0 (12) 426.05 0.7160
14 0 (30) 0 (21,698) 0 (4) −1 (10) 312.32 0.6487
15 0 (30) 0 (21,698) 1 (6) 0 (12) 390.02 1.0604
16 0 (30) 0 (21,698) −1 (2) 0 (12) 264.01 0.4033
17 1 (40) 1 (28,931) −1 (2) −1 (10) 317.19 0.3384
18 1 (40) 1 (28,931) 1 (6) 1 (14) 491.48 0.9357
19 1 (40) −1 (14,465) 1 (6) −1 (10) 241.37 0.7578
20 1 (40) −1 (14,465) −1 (2) 1 (14) 195.20 0.3899
21 1 (40) 1 (28,931) 1 (6) −1 (10) 431.03 0.7302
22 1 (40) −1 (14,465) 1 (6) 1 (14) 268.31 0.9675
23 1 (40) 1 (28,931) −1 (2) 1 (14) 322.36 0.3638
24 1 (40) 0 (21,698) 0 (4) 0 (12) 301.54 0.5710
25 1 (40) −1 (14,465) −1 (2) −1 (10) 193.35 0.3628
Table 22

ANOVA of NuETWC obtained by TDML9

Factors DF SS Var Rank
A 2 3.187 1.593 3
B 2 40.702 20.351 1
C 2 4.278 2.139 2
D 2 0.992 0.496 4
Error 0 0 0
Total 8 49.158 24.91
Table 23

ANOVA of f ETWC obtained by TDML9

Factors DF SS Var Rank
A 2 35.866 17.933 1
B 2 0.119 0.060 4
C 2 20.097 10.048 2
D 2 3.534 1.767 3
Error 0 0 0
Total 8 59.62 30.99
Table 24

ANOVA of NuETWC obtained by TDML18

Factors DF SS Var Rank
A 2 5.194 2.597 3
B 2 73.717 36.858 1
C 2 7.785 3.892 2
D 2 1.558 0.779 4
Error 9 0.38 0.04
Total 17 88.68 44.20
Table 25

ANOVA of f ETWC obtained by TDML18

Factors DF SS Var Rank
A 2 70.15 35.07 1
B 2 0.75 0.37 4
C 2 38.99 19.49 2
D 2 6.90 3.45 3
Error 9 0.91 0.10
Total 17 118.16 58.76
Table 26

ANOVA of NuSWT obtained by TDML9

Factors DF SS Var Rank
A 2 1.442 0.721 3
B 2 30.259 15.130 1
C 2 12.508 6.254 2
D 2 0.397 0.199 4
Error 0 0 0
Total 8 44.606 22.30
Table 27

ANOVA of f SWT obtained by TDML9

Factors DF SS Var Rank
A 2 15.04 7.52 2
B 2 1.64 0.82 4
C 2 84.71 42.35 1
D 2 2.40 1.20 3
Error 0 0 0
Total 8 103.79 52.70
Table 28

ANOVA of NuSWT obtained by TDML18

Factors DF SS Var Rank
A 2 2.379 1.190 3
B 2 70.746 35.373 1
C 2 25.124 12.562 2
D 2 0.771 0.385 4
Error 9 0.40 0.04
Total 17 99.75 49.73
Table 29

ANOVA of f SWT obtained by TDML18

Factors DF SS Var Rank
A 2 28.77 14.39 2
B 2 0.68 0.34 4
C 2 168.11 84.06 1
D 2 4.56 2.28 3
Error 9 0.80 0.09
Total 17 203.42 101.42
Table 30

ANOVA of NuSWT obtained by RSM

Factors DF SS MS F P
Model 14 192114.2 13722.44 98.1786 <0.0001
A 1 1034.828 1034.828 7.403783 0.0215
B 1 133546 133546 955.4682 <0.0001
C 1 46309.83 46309.83 331.3283 <0.0001
D 1 2212.355 2212.355 15.82852 0.0026
AB 1 80.65764 80.65764 0.577073 0.4650
AC 1 28.13198 28.13198 0.201273 0.6633
AD 1 33.42489 33.42489 0.239142 0.6354
BC 1 6247.454 6247.454 44.69804 <0.0001
BD 1 268.2549 268.2549 1.919257 0.1961
CD 1 841.5898 841.5898 6.021239 0.0340
A2 1 176.4078 176.4078 1.262127 0.2875
B2 1 2.456914 2.456914 0.017578 0.8972
C2 1 110.5712 110.5712 0.791093 0.3947
D2 1 48.30256 48.30256 0.345586 0.5697
Error 10 1397.702 139.7702
Total 24 193511.9

Standard deviation = 11.82.

R 2 = 99.28% R 2 (adjusted) = 98.27%.

Table 31

ANOVA of f SWT obtained by RSM

Factors DF SS MS F P
Model 14 1.55986 0.111419 51.55441 <0.0001
A 1 0.144421 0.144421 66.82493 <0.0001
B 1 0.002649 0.002649 1.225797 0.2941
C 1 1.290251 1.290251 597.011 <0.0001
D 1 0.072103 0.072103 33.36277 0.0002
AB 1 5.77 × 10−5 5.77 × 10−5 0.026698 0.8735
AC 1 0.005224 0.005224 2.417174 0.1511
AD 1 0.000252 0.000252 0.116477 0.7399
BC 1 1.33 × 10−5 1.33 × 10−5 0.006134 0.9391
BD 1 1.78 × 10−6 1.78 × 10−6 0.000825 0.9777
CD 1 0.034232 0.034232 15.83957 0.0026
A2 1 0.000123 0.000123 0.056797 0.8164
B2 1 0.000381 0.000381 0.176482 0.6833
C2 1 0.000307 0.000307 0.141838 0.7143
D2 1 0.001453 0.001453 0.672409 0.4313
Error 10 0.021612 0.002161
Total 24 1.581472

Standard deviation = 0.046.

R 2 = 98.63% R 2 (adjusted) = 96.72%.

Table 32

ANOVA of NuETWC obtained by RSM

Factors DF SS MS F P
Model 14 154362 11025.861 304.573 <0.0001
A 1 8644.056 8644.056 238.779 <0.0001
B 1 124589 124588.957 3441.581 <0.0001
C 1 12829.25 12829.248 354.389 <0.0001
D 1 2504.488 2504.488 69.183 <0.0001
AB 1 1105.535 1105.535 30.539 0.0003
AC 1 1246.482 1246.482 34.432 0.0002
AD 1 345.1229 345.123 9.533 0.0115
BC 1 1682.67 1682.670 46.481 <0.0001
BD 1 395.4134 395.413 10.923 0.0079
CD 1 83.65259 83.653 2.311 0.1594
A2 1 367.0631 367.063 10.140 0.0097
B2 1 1.880495 1.880 0.052 0.8243
C2 1 3.070484 3.070 0.085 0.7768
D2 1 0.878836 0.879 0.024 0.8793
Error 10 362.0108 36.201
Total 24 154724.1

Standard deviation = 6.02.

R 2 = 99.77% R 2 (adjusted) = 99.44%.

Table 33

ANOVA of f ETWC obtained by RSM

Factors DF SS MS F P
Model 14 0.961007 0.068643 100.331 <0.0001
A 1 0.515545 0.515545 753.535 <0.0001
B 1 0.004872 0.004872 7.120 0.0236
C 1 0.27287 0.272870 398.835 <0.0001
D 1 0.036134 0.036134 52.815 <0.0001
AB 1 9.17 × 10−5 0.000092 0.134 0.7219
AC 1 0.068483 0.068483 100.097 <0.0001
AD 1 0.014587 0.014587 21.321 0.0010
BC 1 2.79 × 10−5 0.000028 0.041 0.8441
BD 1 4.41 × 10−6 0.000004 0.006 0.9376
CD 1 0.003821 0.003821 5.585 0.0397
A2 1 0.015292 0.015292 22.352 0.0008
B2 1 0.00025 0.000250 0.365 0.5590
C2 1 0.000218 0.000218 0.319 0.5848
D2 1 9.6 × 10−5 0.000096 0.140 0.7158
Error 10 0.006842 0.000684
Total 24 0.967849

Standard deviation = 0.026.

R 2 = 99.29% R 2 (adjusted) = 98.30%.

From Tables 22–25, the effects of design factors on the thermal-hydraulic performance can be judged from the value of Variance (Var). For NuETWC, the effectiveness of different factors follows the order: B > C > A > D. For f ETWC, the effectiveness of different factors follows the order: A > C > D > B. This means that the Re has a significant effect on heat transfer rate, while the coil pitch has a significant effect on pressure drop. Similarly, for NuSWT, the effectiveness of different factors follows the order: B > C > A > D. For f SWT, the effectiveness of different factors follows the order: C > A > D > B. This means that the Re is the most significant factor for heat transfer enhancement, while the tape amplitude is the most significant factor for reducing pressure drop. For designers, it is very important to identify which factor is the control factor in the investigated range. The TDM just provides us a very efficient way. However, from ANOVA of NuETWC, f ETWC, NuSWT, and f SWT obtained by RSM, the effectiveness of every factor cannot be obtained directly. In addition, it is also not applicable to obtain ANOVA for UDM. As a result, the designer may face the diploma about which factor is the control factor.

After the results are obtained, the RBF based on 9 different experimental design methods is trained and tested. The comparison of error of Nu and f is shown in Tables 34–37, respectively. In the tables, the relative absolute error calculated by absolute of (CFD-Pre)/CFD% is shown in Figure 9–12. The sum absolute error is sum of the absolute of (CFD-Pre)/CFD%. In addition, for the column (time cost), it is assumed that the calculating time for all CFD runs are almost the same, that is 5 h per each.

Table 34

Comparison of error of NuETWC with different methods

Method Spread Max absolute error (%) Sum absolute error (%) Time cost (hour)
RBF + TDML9 33.39 404.47 45
RBF + TDML18 38.76 298.21 90
RBF + UDM 48.50 578.87 50
RBF + TDML9 + UDM 42.29 387.75 95
RBF + TDML18 + UDM 1 65.65 428.18 140
RBF + RSM 5.42 48.68 125
RBF + RSM + TDML9 9.80 40.02 170
RBF + RSM + TDML18 8.65 35.06 215
RBF + RSM + UDM 6.96 25.88 175
RSM / 2.88 34.16 125
RBF + RSM 2 5.67 30.88 125
RBF + RSM 20 2.17 17.80 125
Table 35

Comparison of error of f ETWC with different methods

Method Spread Max absolute error (%) Sum absolute error (%) Time cost (hour)
RBF + TDML9 55.99 565.78 45
RBF + TDML18 42.31 306.24 90
RBF + UDM 45.30 496.82 50
RBF + TDML9 + UDM 34.34 331.83 95
RBF + TDML18 + UDM 1 16.57 277.88 140
RBF + RSM 13.89 87.68 125
RBF + RSM + TDML9 12.28 68.51 170
RBF + RSM + TDML18 12.38 48.72 215
RBF + RSM + UDM 9.79 54.93 175
RSM / 8.92 100.87 125
RBF + RSM 2 9.37 71.43 125
RBF + RSM 20 5.30 53.53 125
Table 36

Comparison of error of NuSWT with different methods

Method Spread Max absolute error (%) Sum absolute error (%) Time cost (hour)
RBF + TDML9 43.51 501.77 45
RBF + TDML18 51.26 337.29 90
RBF + UDM 41.46 551.86 50
RBF + TDML9 + UDM 52.57 395.92 95
RBF + TDML18 + UDM 1 90.93 573.30 140
RBF + RSM 6.60 47.09 125
RBF + RSM + TDML9 6.14 41.26 170
RBF + RSM + TDML18 7.78 28.81 215
RBF + RSM + UDM 6.74 31.68 175
RSM / 6.82 120.41 125
RBF + RSM 2 4.01 37.74 125
RBF + RSM 20 5.60 38.21 125
Table 37

Comparison of error of f SWT with different methods

Method Spread Max absolute error (%) Sum absolute error (%) Time cost (hour)
RBF + TDML9 57.26 718.97 45
RBF + TDML18 54.72 405.92 90
RBF + UDM 155.50 1761.26 50
RBF + TDML9 + UDM 44.06 383.38 95
RBF + TDML18 + UDM 1 90.86 484.92 140
RBF + RSM 8.60 82.71 125
RBF + RSM + TDML9 12.60 91.11 170
RBF + RSM + TDML18 9.88 36.41 215
RBF + RSM + UDM 10.57 57.54 175
RSM / 14.52 214.42 125
RBF + RSM 2 8.97 78.68 125
RBF + RSM 20 10.55 85.08 125
Figure 9 
               Comparison of relative error of NuETWC obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

Figure 9

Comparison of relative error of NuETWC obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

Figure 10 
               Comparison of relative error of f
                  ETWC obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

Figure 10

Comparison of relative error of f ETWC obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

Figure 11 
               Comparison of relative error of NuSWT obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

Figure 11

Comparison of relative error of NuSWT obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

Figure 12 
               Comparison of relative error of f
                  SWT obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

Figure 12

Comparison of relative error of f SWT obtained by different methods. The corresponding experimental model names from (a) to (l) can be obtained from the figure.

It is always attracting for designers to know the thermal-hydraulic performance with the minimum expense. For the ten different prediction methods, the expense of RBF + TDML9 is the minimum as only 9 CFD runs are needed which is only one third of that of RBF + RSM. However, from Tables 34–37, it can be found that the max absolute error and sum absolute error of RBF + TDML9 are very large. The max absolute error is 33.39% for NuETWC, 55.99% for f ETWC, 43.51% for NuSWT, and 57.26% for f SWT. For RBF + TDML18, the time cost is doubled compared with RBF + TDML9. However, the max absolute error does not always decrease accordingly. The max absolute error is 38.76% for NuETWC, 42.31% for f ETWC, 51.26% for NuSWT, and 54.72% for f SWT. This indicates that the train data of RBF using TDM is not appropriate.

For RBF + UDM, the number of experiments is 10 which means the time cost of RBF + UDM is almost equal to that of RBF + TDML9. The max absolute error is 48.50% for NuETWC, 45.30% for f ETWC, 41.46% for NuSWT, and 155.50% for f SWT. It can be found that the max absolute error does not decrease accordingly compared with that of RBF + TDML9. The reason can be partly attributed to the scatter of design point in the design space of UDM.

For RBF + TDML9 + UDM and RBF + TDML18 + UDM, it can be found that the predicted max absolute error is also very large, up to 65.65% for NuETWC, 34.34% for f ETWC, 90.93% for NuSWT, and 90.86% for f SWT. Particularly, the time cost of RBF + TDML18 + UDM is three times that of RBF + TDML9. However, the predicted error becomes even larger. This seems contrary to the common sense of ANN that the more of train data, the more precision of the prediction. This indicates that the representativeness of training data is very important to the precision of RBF.

For RBF + RSM (spread = 1), it can be found that the predicted max absolute error is within the scope of engineering acceptance. The max absolute error is 5.42% for NuETWC, 13.89% for f ETWC, 6.60% for NuSWT, and 8.80% for f SWT. The time cost of RBF + RSM is smaller than that of RBF + TDML18 + UDM. For RBF + RSM + TDML9, RBF + RSM + TDML18, and RBF + RSM + UDM, a similar finding can be found that the prediction precision does not improve when the number of train data increases. Particularly, if the parameter “spread” is adjusted, a distinct prediction precision can be reached. The max absolute error is from 5.67 to 2.17% for NuETWC, from 13.89 to 5.3% for f ETWC, from 6.60 to 4.01% for NuSWT, and from 10.55 to 8.60% for f SWT. The sum absolute error is from 48.68 to 17.80% for NuETWC, from 87.68 to 53.53% for f ETWC, from 47.09 to 37.74% for NuSWT, and from 85.08 to 78.68% for f SWT. This fact suggests that RBF trained with RSM only can be an effective, efficient, and affordable tool for thermal-hydraulic performance prediction of the STHX-CPBETWC and STHX-CPBSWT.

For RSM, it can be found that the predicted max absolute error is also within the scope of engineering acceptance. The max absolute error is 2.88% for NuETWC, 8.92% for f ETWC, 6.82% for NuSWT, and 14.52% for f SWT. However, the sum absolute error of RSM is larger than that of RBF + RSM, 34.16% versus 17.80% for NuETWC, 100.87% versus 53.53% for f ETWC, 120.41% versus 37.74% for NuSWT, and 214.42% versus 78.68% for f SWT. The reason can be explained from Figures 9–12. The RBF + RSM can approximate the true value for train data, while there still exists relative larger error for train data obtained by RSM.

Selecting representative points in the sampling space is essential for different DOE. Not all sampling points are necessary. Some representative test points can be selected from the comprehensive test points.

TDM is also known as orthogonal experimental design method. Orthogonal design is a design method to study multifactors and multilevels. It selects representative points according to orthogonality. It has two characteristics in selecting representative points: uniform dispersion and neat comparison. Uniform dispersion makes the test sites evenly distributed within the test range, so that each test site has sufficient representativeness. Therefore, even if all the columns in the orthogonal table are filled, the satisfactory results can be obtained. However, in order to take care of the neat comparability, the test points of the orthogonal design cannot be fully evenly dispersed, and in order to achieve the neat comparability, the number of test points is relatively large. It must be tested at least once. For uniform design, especially in the case that the range of conditions varies greatly and multilevel tests are needed, uniform design can greatly reduce the number of tests. It only needs the same number of tests as the number of factor levels to achieve the test effect that orthogonal design can achieve at least one test. Uniform design only considers the full “uniform distribution” of the test sites within the test range, but does not consider the “neat comparable,” so the test results do not have the neat comparable of the orthogonal test results, and the test results are mostly processed by regression analysis method. Response surface analysis uses specific sampling method in the sample space. A multiple quadratic regression equation is used to fitting function relation between factors and response value. The optimal process parameters can be obtained through the analysis of regression equation, a statistical method to solve the problem of multivariate.

According to the different experimental design methods selected during the experiment, there are certain differences in the sample space obtained, resulting in a certain degree of difference in the data.

7 Conclusion

In this study, thermal-hydraulic performance prediction of two new parallel-flow STHXs using RBF based on different DOE is explored. The predicting efficiency and applicability of ten methods are discussed. Some main conclusions are drawn as follows:

  1. (1)

    The TDM not only can help us point out the significance of different factors, but also can save computing resource. However, thermal performance prediction using RBF + TDM is not applicable.

  2. (2)

    For RBF, it is not true that the more of train data, the more precision of the prediction.

  3. (3)

    The RBF + RSM behave best among the discussed ten prediction methods in terms of the accuracy of prediction. However, the parameter “spread” should be adjusted by trial and error to achieve a relative small error.

It must be emphasized that this does not mean the conclusions obtained above can be extrapolated to other heat exchangers. However, the authors encourage other researchers to carry out similar tests to further check the findings of this paper. After all, the RBF + RSM can present good performance in the task of prediction of thermal-hydraulic performance of the STHX-CPBETWC and STHX-CPBSWT.

Acknowledgments

The authors wish to express their thanks for the National Key R&D Program of China (No. 2018YFC0808600).

    Conflict of interest: Authors state no conflict of interest.

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Received: 2020-09-21
Revised: 2021-02-18
Accepted: 2021-02-24
Published Online: 2021-06-08

© 2021 Chulin Yu et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.