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 parallelflow 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 thermalhydraulic 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, mm^{2}
 A _{in}

area of crosssection (mm^{2})
 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 m^{2} 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)
 ΔT

logmean 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 (–)
 Δp

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

RRBSTHX
 ETWC

STHXCPBETWC
 SWT

STHXCPBSWT
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 parallelflow 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 parallelflow STHXs have been proposed [5]. The famous one needed to mention is round rod baffle STHX (RRBSTHX) which is invented by Philips Company to solve the high pressure drop problem of segmental baffle STHX [6]. However, the RRBSTHX is vulnerable to flowinduced 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
In order to save energy and reduce cost, heat transfer enhancement technologies are always the hot point of energy utilization. Two new parallelflow STHXs with CPB and inserts (STHXCPBETWC and STHXCPBSWT) are proposed by Yu et al. [9]. The effects of geometrical parameters such as coil pitch, coil diameter, side length of equilateral crosssection, tape amplitude, tape pitch, and tape width on the thermalhydraulic performance are numerically explored with flow in turbulent regime.
For designers of STHX, it is meaningful to obtain the thermalhydraulic 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 quasiMonte 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 realworld applications with a significant reduction in cost and time. Different types of ANN are used to predict the thermalhydraulic 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 TiO_{2}–Al_{2}O_{3}/water nanofluid in mini double pipe heat exchanger [26]. The combination use of ANN and DOE can be found in some references [27–30]. 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 thermalhydraulic 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 STHXCPBETWC and STHXCPBSWT 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
2 Geometrical structure
For parallelflow 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
Figure 4
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 STHXCPBETWC, while T _{p}, Re, T _{a}, and T _{w} are used in the input layer for STHXCPBSWT. In the output layer, the Nu and f are used as the response parameters.
Figure 5
The train of ANN is very crucial. It is wellknown 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 wellestablished 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
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 STHXETWC and STHXSWT, 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
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
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
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
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
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
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
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 quasilevel 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
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
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
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
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
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:
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 STHXCPBETWC and STHXCPBSWT are shown in Figure 7. It includes inlet extended region, heat transfer region, and outlet extended region. Hybrid mesh is used.
Figure 7
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 thermalphysical 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:
Momentum equation:
Energy equation:
5.3 Boundary conditions
The 3D, doubleprecision, pressurebased 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 nearwall 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:
Uniform temperature 283.15 K is assigned for the velocity inlet boundary.
The surfaces of the baffles are set as adiabatic.
Pressure outlet boundary is used for the exit.
The four boundary walls of the unit model are set as symmetry boundary conditions.
The temperature of heated tube wall is set as a constant, 350 K.
5.4 Thermalhydraulic parameters
The Reynolds number Re can be obtained as:
The average heat transfer coefficient, h, and the average Nusselt number, Nu, are obtained as follows:
The friction factor is estimated by:
5.5 Validation and grid independence
For STHXCPBETWC and STHXCPBSWT, no experimental results can be found in open literatures. To validate the reliability of the numerical model, nonstaggered 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
Table 13
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 STHXCPBETWC and STHXCPBSWT obtained by TDM, UDM, and RSM are listed in Tables 14–21, respectively. Analysis of variance (ANOVA) of Nu_{ETWC} and f _{ETWC} obtained by TDML9 and TDML18 is listed in Tables 22–25, respectively. ANOVA of Nu_{SWT} and f _{SWT} obtained by TDML9 and TDML18 is listed in Tables 26–29, respectively. ANOVA of Nu_{ETWC}, f _{ETWC}, Nu_{SWT}, and f _{SWT} obtained by RSM is listed in Tables 30–33, respectively.
Table 14
Case no.  Nu_{ETWC}  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
Case no.  Nu_{ETWC}  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
Case no.  Nu_{ETWC}  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
Case no.  Level of parameter (actual value)  Response  

A  B  C  D  Nu_{SWT}  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
Case no.  Nu_{swt}  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
Case no.  Nu_{ETWC}  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
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
Case no.  Level of parameter (actual value)  Response  

A  B  C  D  Nu_{SWT}  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
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
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
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
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
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
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
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
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
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 
A^{2}  1  176.4078  176.4078  1.262127  0.2875 
B^{2}  1  2.456914  2.456914  0.017578  0.8972 
C^{2}  1  110.5712  110.5712  0.791093  0.3947 
D^{2}  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
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 
A^{2}  1  0.000123  0.000123  0.056797  0.8164 
B^{2}  1  0.000381  0.000381  0.176482  0.6833 
C^{2}  1  0.000307  0.000307  0.141838  0.7143 
D^{2}  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
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 
A^{2}  1  367.0631  367.063  10.140  0.0097 
B^{2}  1  1.880495  1.880  0.052  0.8243 
C^{2}  1  3.070484  3.070  0.085  0.7768 
D^{2}  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
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 
A^{2}  1  0.015292  0.015292  22.352  0.0008 
B^{2}  1  0.00025  0.000250  0.365  0.5590 
C^{2}  1  0.000218  0.000218  0.319  0.5848 
D^{2}  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 thermalhydraulic performance can be judged from the value of Variance (Var). For Nu_{ETWC}, 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 Nu_{SWT}, 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 Nu_{ETWC}, f _{ETWC}, Nu_{SWT}, 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 (CFDPre)/CFD% is shown in Figure 9–12. The sum absolute error is sum of the absolute of (CFDPre)/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
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
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
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
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
Figure 10
Figure 11
Figure 12
It is always attracting for designers to know the thermalhydraulic 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 Nu_{ETWC}, 55.99% for f _{ETWC}, 43.51% for Nu_{SWT}, 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 Nu_{ETWC}, 42.31% for f _{ETWC}, 51.26% for Nu_{SWT}, 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 Nu_{ETWC}, 45.30% for f _{ETWC}, 41.46% for Nu_{SWT}, 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 Nu_{ETWC}, 34.34% for f _{ETWC}, 90.93% for Nu_{SWT}, 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 Nu_{ETWC}, 13.89% for f _{ETWC}, 6.60% for Nu_{SWT}, 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 Nu_{ETWC}, from 13.89 to 5.3% for f _{ETWC}, from 6.60 to 4.01% for Nu_{SWT}, and from 10.55 to 8.60% for f _{SWT}. The sum absolute error is from 48.68 to 17.80% for Nu_{ETWC}, from 87.68 to 53.53% for f _{ETWC}, from 47.09 to 37.74% for Nu_{SWT}, 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 thermalhydraulic performance prediction of the STHXCPBETWC and STHXCPBSWT.
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 Nu_{ETWC}, 8.92% for f _{ETWC}, 6.82% for Nu_{SWT}, 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 Nu_{ETWC}, 100.87% versus 53.53% for f _{ETWC}, 120.41% versus 37.74% for Nu_{SWT}, 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, thermalhydraulic performance prediction of two new parallelflow 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:
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.
For RBF, it is not true that the more of train data, the more precision of the prediction.
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 thermalhydraulic performance of the STHXCPBETWC and STHXCPBSWT.
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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