Investigation on thermal-hydraulic performance prediction of a new parallel-flow shell and tube heat exchanger with different surrogate models

Xinghua Fu; Youqiang Wang; Chulin Yu; Haiqing Zhang; Jin Wang; Bingjun Gao

doi:10.1515/phys-2020-0218

Article Open Access

Investigation on thermal-hydraulic performance prediction of a new parallel-flow shell and tube heat exchanger with different surrogate models

Xinghua Fu , Youqiang Wang , Chulin Yu , Haiqing Zhang , Jin Wang and Bingjun Gao

Published/Copyright: December 31, 2020

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From the journal Open Physics Volume 18 Issue 1

Abstract

The thermal-hydraulic performance of a new parallel-flow shell and tube heat exchanger (STHX) with equilateral cross-sectioned wire coil (HCBetwc-STHX) is investigated in turbulent regime. Four different surrogate models are established to predict the thermal-hydraulic performance. Their merits and drawbacks are illustrated. The results show that the Nu_etwc/Nu_RRB and f _etwc/f _RRB are in the range of 1.1638–1.855 and 4.078–16.062, respectively. The precision of CFM is the lowest, whereas the precision of radial basis function + artificial neural network and Kriging model is the highest. A good balance can be achieved by response surface methodology between precision and cost. Finally, a general analysis procedure is presented for the predicting method of thermal-hydraulic performance of different STHX with relatively small cost and high precision.

Keywords: heat exchanger; thermal-hydraulic; prediction; surrogate model

Nomenclature

a: length (mm)
A _in: area of cross-section (mm²)
A _o: area of tube outer surface (mm²)
b: baffle width (mm)
c: length (mm)
C _p: specific heat at constant pressure (J kg⁻¹ K⁻¹)
d _c: coil diameter
D _h: hydraulic diameter (mm)
d _i: inner diameter of tube (mm)
d _o: outer diameter of tube (mm)
d _r: diameter of rod (mm)
f: average friction factor (–)
h: convection heat transfer coefficient (W m² K⁻¹)
K: total heat transfer coefficient (W m² K⁻¹)
L _b: baffle distance (mm)
l _et: side length of equilateral triangle
p: pressure (Pa)
p _c: coil pitch
Pr: Prandtl number (–)
P _t: tube pitch (mm)
R: radius (mm)
Re: Reynolds number (–)
T: temperature (K)
t: thickness (mm)
t _b: baffle thickness (mm)
V _in: inlet velocity (m/s)
Y: response
Δp: pressure drop (Pa)
ΔT: log-mean temperature difference (K)

Greek symbols

ρ: density (kg m⁻³)
μ: dynamic viscosity (kg ms⁻¹)
λ: thermal conductivity (W m⁻¹ K⁻¹)

Subscripts

etwc: HCBetwc-STHX
in: inlet
out: outlet
RRB: RRB-STHX

Abbreviations

w: wall
AAD: absolute average deviation
ANN: artificial neural network
ANOVA: analysis of variance
CCD: central composite design
HCB: hexagonal clamping baffle
KM: Kriging model
MAD: maximum absolute deviation
MLFF: multilayer feed forward
PEC: performance evaluation criterion
RBF: radial basis function
RRB: round rod baffle
RSM: response surface methodology
STHX: shell and tube heat exchanger

1 Introduction

With the development of computer science and computational fluid dynamics (CFD), more and more researchers are using CFD to develop new configuration of heat exchangers to meet the urging needs for saving energy [1,2,3].

To minimize the computational cost and time, surrogate models of heat exchangers are usually built based on some experimental design methods such as Taguchi method [4], response surface methodology (RSM) [5], and Latin hyper cubic method [6]. Once the surrogate models are built, the designers then can use them to predict the thermal-hydraulic performance of heat exchangers with different parameters.

Generally, four surrogate models can be used to predict the thermal-hydraulic performance of shell and tube heat exchanger (STHX). They are conventional fit model (CFM), RSM, artificial neural network (ANN) [7], and Kriging model (KM) [8]. Many examples can be found with the application of them.

CFM is the most widely used surrogate model. New correlations are proposed using CFM to estimate the values of the average Nusselt number Nu and friction factor f of a new smooth wavy fin-and-elliptical tube heat exchanger with three new types of vortex generators [9]. Fanning factor and Nusselt number correlations for the airfoil-printed circuit heat exchanger were obtained using CFM [10]. Heat transfer performances of molten salt in the shell side of a shell-and-tube heat exchanger are fitted with CFM [11]. The merit of CFM is that it can give an explicit form to designers. In addition, the model built by CFM may be embedded in some commercial software such as HTRI [12]. The drawbacks of CFM are also obvious. The precision of CFM strongly depends on the selected structure of function and the complexity of the investigated problem.

RSM can effectively show the relationships between the input variables and the output ones [13]. The heat and flow characteristics in a single-phase parallel-flow heat exchanger were examined numerically using RSM [14]. A second-order polynomial RSM was adopted to study the effect of fold baffle configuration parameters on the thermal-hydraulic performance [15]. The RSM and two phase mixture model were used to investigate the sensitivity of heat exchanger effectiveness in a double pipe heat exchanger filled with nanofluid [16]. The merit of RSM is that the analysis procedures are almost the same. The designer can complete the RSM analysis following simple step-by-step constructions. The drawback of RSM is that it may still not be able to reflect some complicated problems.

ANN can create a mapping between the input variables and output ones. Different network configurations were studied by the aid of searching a relatively better back propagation (BP) network for prediction of heat performance of STHX with segmental baffles or continuous helical baffles [17]. Applications of ANNs in flow and heat transfer problems in nuclear engineering are discussed [18]. An ANN model has been developed, which can predict the depth of a vertical ground heat exchanger using the soil thermal conductivity, grout thermal conductivity, inlet flow, inlet water temperature, underground water velocity, and heat flux as the input parameters [19]. The merit of ANN is that it can supply an easy modeling tool for engineers to obtain a quick preliminary assessment of heat transfer rate in response to the engineering modifications to the exchanger [20]. The drawback of ANN is that over-fitting problem may be encountered if the structure of ANN is not optimized or the train data are not enough [21].

KM is capable of modeling complex surfaces [22]. Shape optimization of a wire-wrapped fuel assembly in a liquid metal reactor has been carried out by combining a three-dimensional Reynolds-averaged Navier–Stokes analysis with the KM [23]. An improved algorithm combining a Kriging response surface and the multi-objective genetic algorithm for the optimization design of STHX with helical baffles is proposed [24]. The shape optimization of the plate-fin type heat sink with an air deflector is numerically performed with KM [25].

In the foregoing reports, a new parallel-flow STHX with hexagonal clamping baffle (HCB) and equilateral triangular cross-sectioned wire coil (HCBetwc-STHX) was proposed [26]. Taguchi method was adopted to investigate the influence of five geometric parameters such as baffle distance, baffle width, coil diameter, coil pitch, and the side length of equilateral triangle on heat transfer and pressure drop. Some useful conclusions are obtained. It is found that the coil pitch has a great influence, whereas the baffle width has a trifling effect. However, the effect of Re on thermal-hydraulic performance is still not investigated. In addition, the thermal-hydraulic performance prediction model was not built. In this paper, four different surrogate models of HCBetwc-STHX incorporating four factors (coil pitch, Re, coil diameter, and side length of equilateral triangle) are built. The precision of them has also been discussed. The work done in this paper can be regarded as the further research of the engineering application of HCBetwc-STHX.

2 Prediction method

2.1 CFM

The necessity of CFM is that the Nu is the function of Re and Pr, whereas the f is the function of Re. Various expressions of Nu of different heat exchangers are listed in Table 1.

Table 1

Various forms of CFM of different heat exchanger

Name	Working fluid	Expression
Rod baffle STHX [27]	Water	Nu = 0.0589 Re 0.815 × Pr 1 / 3 L b d e − 0.303 μ μ w 0.14
STHX without segmental baffle [28]	Molten salt	Nu = 0.04318 Re 0.7797 − 280 × Pr 0.4 1 + D e L 2 / 3 Pr f Pr w 0.25
STHX with fold helical baffle [29]	Water	Nu = c 1 Re m 1 Pr 1 / 3 c 1 = 0.346 + 0.06 β - 2.224 e − 0.14 β e + 8.033 × 10 − 4 β 2 + 3.678 e 2 m 1 = 0.339 − 0.008 β + 0.479 e − 0.006 β e + 4.411 × 10 − 5 β 2 + 0.235 e 2
Cross hollow twisted tape inserts [30]	Air	Nu = 0.3415 Re 0.5911 Pr 0.32 ( 0.9058 c 3 + 0.5439 c 2 − 1.345 c + 1.271 )

The mean error of CFM depends on situations. In some circumstances, the mean error is about 3% [29], whereas in other circumstances, the mean error can reach to 20% [30]. Some affecting factors can be concluded as follows:

The complexity of the response;
The form of CFM.

2.2 RSM

The RSM is proposed by Box and Wilson in the early 1950s. It has received considerable attention because of its good empirical performance in modeling. It can provide well-fitting models between input parameters and responses. The flow chart of RSM is shown in Figure 1.

Figure 1

Flow chart of RSM.

The second-order polynomial response surface mathematical equation is usually used to model the response as shown in equation (1):

(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; and Δ is the statistical error.

For RSM, there are two different sampling methods: Box-Behnken design (BBD) and central composite design (CCD). The BBD is a three-level design without any points at the vertices of a cubic region created by the upper and lower limits for each variable. The CCD includes a full or fractional factorial design with center points that are augmented with a group of axial points that allow estimation of the curvature in the resulting model.

2.3 ANN

There are many different ANNs such as multilayer feed forward ANN (MLFF + ANN), radial basis function ANN (RBF + ANN). Cong et al. described the merits and drawbacks of different ANNs. The accuracy of MLFF–ANN depends on the structure of ANN. Usually, the designer carry out trial and error to affirm the appropriate structure of ANN to reduce the over-fitting risk and therefore improve the generalization. Instead, the RBF–ANN has only three layers (input layer, hidden layer, and output layer) as shown in Figure 2. It has faster convergence, smaller extrapolation errors, and higher reliability than MLFF + ANN.

Figure 2

Agriculture of RBF–ANN.

It is reported that the mean error of ANN is about 2–13% for prediction of Nu [18]. The train of ANN is very crucial. In theory, the more train data, the more accuracy of the prediction of ANN. An important reason is that whether the train data of ANN are enough and representative. However, the train data are limited as the time and cost should be considered. For the RBF + ANN, the selection of the train set is very important for the accuracy. In this paper, we adopt the experiments designed by the CCD method of the RSM, because this method can provide a comprehensive sampling in the sample space. For the test of RBF + ANN, we adopt the experiments designed by the Taguchi method, because this method can provide typical sampling in the sample space.

2.4 KM

The KM is named by the professor Kriging. The formulation of the details of KM is omitted for brevity. Some details can be found in the literature [25].

Now, the KM can be easily accomplished with the help of Matlab Kriging toolbox. Usually, the KM is coupled with Latin Hypercube design (LHD). It is believed that the LHD gives one confidence because it can be infiltrating the design space well. In this paper, the LHD is not adopted as we want to test the applicability of KM whether the sampling points can be generated by the CCD.

3 Problem setup

The sketch of HCBetwc-STHX is illustrated in Figure 3. Four parameters named P _c, Re, l _et, and d _c are used as input parameters. Nu and f are used as the response parameters. The levels of the parameters are listed in Table 2.

Figure 3

Sketch of HCBetwc-STHX. (a) Front view and (b) left view.

Table 2

Factors and levels of HCBetwc-STHX

Factors (unit)	Level 1	Level 2	Level 3
Factors (unit)	−1	0	1
P _c (mm)	20	30	40
Re	14,465	21,698	28,931
l _et (mm)	2	3	4
d _c (mm)	13	14	15

4 Numerical model

4.1 Computational domain and boundary conditions

The computational domain includes the inlet extended block, heat transfer block, and outlet extended block. The inlet block and the outlet block are both 100 mm so as to avoid backflow. The large commercial CFD software Fluent is adopted. The details of computational domain, boundary conditions, and numerical methods of HCBetwc-STHX can be found in ref. [26]. To validate the reliability of the numerical model, non-staggered tubes supported by round rod baffle (RRB) are computed and compared with the results obtained by Dong et al. [27]. The working fluid, the boundary conditions, and the baffle distance of them are all the same. The grids adopted in this paper are the same with those in ref. [26]. Thus, the grid independency test and numerical model validation test can be deemed as satisfactory.

4.2 Thermal-hydraulic parameters

The Reynolds number, Re, can be obtained as follows:

(2) Re = ρ D h V in μ ,

(3) D h = 4 ( P t 2 − π d o 2 / 4 ) π d o .

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

(4) h = c p ρ V in A in ( T in − T out ) A 0 Δ T ,

(5) Nu = h D h λ ,

(6) Δ T = T in − T out ln T in − T w T out − T w .

The friction factor is estimated by:

(7) f = D h L 2 Δ p ρ V in 2 .

The Nu and f can reflect the heat and flow characteristics of the heat exchanger, whereas the performance evaluation criteria (PEC) can evaluate the overall thermal-hydraulic performance. PEC that define the performance benefits of an exchanger have enhanced structures that are applicable to single phase flow in tubes. It can be expressed as follows:

(8) PEC = Nu ( f ) 1 / 3 .

It is useful to determine the maximum absolute deviation (MAD) and absolute average deviation (AAD) observed for all models to give an indication of how accurate the model predictions can be. The MAD and AAD are used and defined as follows:

(9) MAD ( %) = max R pred − R CFD R CFD × 100 ,

(10) AAD (%) = 1 n ∑ i = 1 n R pred − R CFD R CFD × 100 .

4.3 Design of experiments

CCD is used to arrange the numerical experiments. The arrangement of experiments is presented in Table 3. A total of 25 experiments are adopted as the train set for RBF + ANN; nine experiments are adopted as the test set. The details are listed in Table 4. The parameters A, B, C, and D are P _c (coil pitch), Re, l _et (coil diameter), and d _c side (length of equilateral triangle), respectively.

Table 3

Numerical results of HCBetwc-STHX

Case no.	Parameters (level)				Response
Case no.	A	B	C	D	Nu_etwc	f _etwc	PEC_etwc	Nu_etwc/Nu_RRB	f _etwc/f _RRB	PEC_etwc/PEC_RRB
1	−1	−1	1	1	271.36	1.1199	261.31	1.7279	12.589	0.7428
2	−1	−1	−1	−1	205.55	0.5745	247.26	1.3089	6.458	0.7028
3	−1	1	−1	1	389.87	0.6366	453.21	1.4305	8.858	0.6913
4	−1	1	1	−1	440.53	0.8619	462.89	1.6164	11.995	0.7061
5	−1	1	−1	−1	351.49	0.5413	431.27	1.2896	7.533	0.6579
6	−1	1	1	1	497.50	1.0766	485.41	1.8254	14.982	0.7405
7	−1	−1	1	−1	245.19	0.9018	253.79	1.5613	10.137	0.7214
8	−1	0	0	0	324.36	0.7592	355.55	1.5050	9.723	0.7051
9	−1	−1	−1	1	220.80	0.6724	252.03	1.4060	7.559	0.7164
10	0	−1	0	0	213.29	0.5491	260.47	1.3582	6.172	0.7404
11	0	0	0	0	291.21	0.5273	360.46	1.3512	6.753	0.7149
12	0	0	0	1	298.36	0.5491	364.34	1.3844	7.033	0.7226
13	0	1	0	0	370.88	0.5190	461.51	1.3608	7.222	0.7040
14	0	0	0	−1	285.27	0.5114	356.72	1.3237	6.550	0.7074
15	0	0	1	0	316.33	0.6295	369.09	1.4678	8.063	0.7320
16	0	0	−1	0	268.32	0.4372	353.52	1.2450	5.600	0.7011
17	1	1	−1	−1	323.91	0.3660	452.84	1.1885	5.093	0.6908
18	1	1	1	1	396.75	0.5215	492.90	1.4557	7.257	0.7519
19	1	−1	1	−1	214.09	0.5141	267.25	1.3633	5.779	0.7597
20	1	−1	−1	1	201.29	0.4265	267.42	1.2818	4.794	0.7602
21	1	1	1	−1	372.46	0.4839	474.43	1.3666	6.733	0.7237
22	1	−1	1	1	224.57	0.5517	273.81	1.4300	6.202	0.7783
23	1	1	−1	1	343.66	0.3986	466.94	1.2609	5.547	0.7123
24	1	0	0	0	282.11	0.4441	369.77	1.3090	5.687	0.7333
25	1	−1	−1	−1	193.35	0.3915	264.29	1.2312	4.401	0.7513
Average	—	—	—	—	301.70	0.5986	362.34	1.4020	7.5488	0.7227

Note: The parameters A, B, C and D stand for the coil pitch, Reynolds number, length and diameter of the wire coil respectively.

Table 4

Details of nine experiments of test set

Case no.	Parameters (level)				Response
Case no.	A	B	C	D	Nu_etwc	f _etwc
26	−1	0	−1	−1	277.37	0.5517
27	−1	1	0	0	416.43	0.7477
28	−1	−1	1	1	271.36	1.1212
29	0	−1	0	1	216.77	0.5708
30	0	0	1	−1	307.12	0.6048
31	0	1	−1	0	337.93	0.4281
32	1	−1	1	0	219.46	0.5369
33	1	0	−1	1	269.50	0.4011
34	1	1	0	−1	349.92	0.4202

5 Results and discussions

According to the calculating formulas mentioned earlier, the results of HCBetwc-STHX obtained are listed in Tables 3 and 4. Analysis of variance (ANOVA) of Nu_etwc and f _etwc obtained by RSM is listed in Tables 5 and 6, respectively. The coefficients of the regression response surface models are listed in Table 7.

Table 5

ANOVA test result of Nu_etwc

Factors	DF	SS	MS	F	P
Model	14	1,54,362	11025.861	304.573	<0.0001
A	1	8644.056	8644.056	238.779	<0.0001
B	1	1,24,589	124588.957	3,441.581	<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	1,54,724.1

Standard deviation = 6.02.

R ² = 99.77%, R ² (Adjusted) = 99.44%.

Table 6

ANOVA test result of f _etwc

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%.

Table 7

The term coefficients of the regression response surface model for HCBetwc-STHX

Term coefficient	Nu_etwc	f _etwc	PEC_etwc
b0	150.75809	0.937654992	0.591455
b1	2.249	−0.002236847	0.004071
b2	3.62 × 10⁻⁴	−9.92196 × 10⁻⁶	−1.5 × 10⁻⁵
b3	−16.1873	0.051489677	−0.02166
b4	−12.49207	−0.081270354	0.022471
b1,2	−1.15 × 10⁻⁴	3.30979 × 10⁻⁸	−7.2 × 10⁻⁸
b1,3	−0.88264	−0.006542329	−0.00027
b1,4	−0.46444	−0.003019415	−0.00016
b2,3	1.42 × 10⁻³	−1.82406 × 10⁻⁷	8.5 × 10⁻⁷
b2,4	6.87 × 10⁻⁴	−7.25909 × 10⁻⁸	4.75 × 10⁻⁷
b3,4	2.28654	0.015453047	0.001569
b1,1	0.12005	0.000774895	3.44 × 10⁻⁵
b2,2	1.64 × 10⁻⁸	1.89374 × 10⁻¹⁰	1.23 × 10⁻¹⁰
b3,3	1.09802	0.009253198	0.000747
b4,4	0.58743	0.006138331	−0.00078

From Tables 5 and 6, it can be observed that the R ² (Adjusted) of Nu_etwc and f _etwc are close to 1.0. This indicates that the results of HCBetwc-STHX obtained by RSM are right.

It can be observed from Table 3 that the Nu_etwc/Nu_RRB is in the range of 1.1885–1.8254; the f _etwc/f _RRB is in the range of 4.401–14.982; and the PEC_etwc/PEC_RRB is in the range of 0.6579–0.7783. This means that the HCBetwc-STHX can enhance heat transfer rate compared with the RRB-STHX. However, this is achieved at the expense of large power consumption. As a result, the overall thermal-hydraulic performance of the HCBetwc-STHX is reduced compared with the RRB-STHX.

The Nu and f correlations are fitted for HCBetwc-STHX as shown in equations 11 and 12:

(11) Nu etwc = 0.10024 p c d e − 0.22945 Re 0.82669 × Pr 1 / 3 l et D h 0.26577 d c D h 0.58347 ,

(12) f etwc = 12.1764 p c d e − 0.8778 × Re − 0.08125 l et D h 0.63667 d c D h 1.17789 .

The adjusted residual square of the above fitted equations is 0.9813 and 0.9593, respectively. This indicates that the fitted formulas are acceptable.

After the results of Nu_etwc and f _etwc are obtained, the RBF–ANN is trained and tested. The predicted Nu_etwc and f _etwc with different surrogate models are shown in Figures 4 and 5, respectively.

Figure 4

Comparison of Nu_etwc of HCBetwc-STHX obtained with different surrogate models. (a) Comparison of Nu_etwc and (b) predicted error of Nu_etwc.

Figure 5

Comparison of f _etwc of HCBetwc-STHX obtained with different surrogate models. (a) Comparison of f _etwc and (b) predicted error of f _etwc.

To indicate the accuracy of different surrogate models, we use two indicators named MAD and AAD. The MAD is the maximum absolute deviation. The AAD is the absolute average deviation. The computed MAD and AAD with different surrogate models are shown in Tables 8 and 9, respectively.

Table 8

Comparison of error of Nu_etwc of HCBetwc-STHX with different surrogate models

Surrogate model	MAD (%)	AAD (%)
CFM	10.01	2.39
RSM	4.20	0.97
RBF + ANN	1.06	0.1
KM	1.84	0.24

Table 9

Comparison of error of f _etwc of HCBetwc-STHX with different surrogate models

Surrogate model	MAD (%)	AAD (%)
CFM	18.89	5.34
RSM	6.31	2.64
RBF + ANN	3.43	0.41
KM	1.68	0.21

It can be seen from Tables 8 and 9 that the MAD and AAD of Nu_etwc obtained with four different surrogate models follow the order: CFM > RSM > KM > RBF + ANN, whereas the MAD and AAD of f _etwc follow the order: CFM > RSM > RBF + ANN > KM. This indicates that: (1) the precision of CFM is the lowest; (2) the precision of RSM locates at the middle; and (3) the precision of RBF + ANN and KM is the highest.

From Figures 4 and 5, it can be observed that the MAD of RBF + ANN and KM are almost zero for the train set. This is quite different with that of CFM and RSM. As a result, the AAD of RBF + ANN and KM are also smaller compared with that of CFM and RSM.

For CFM, the precision is the lowest. This means that the selected form of CFM should be improved for the problem investigated. However, the improved form cannot be obtained easily as many attempts may be required.

For RSM, the precision is acceptable from the point of engineering application. The merit of RSM is that it has a good balance between time and precision. The designer need not do any further work to build other surrogate models as the designer can obtain the surrogate model easily with any RSM software.

For RBF + ANN and KM, a very high precision is achieved. The over-fitting problem is not obvious. This means that the train set based on the CCD of RSM is acceptable.

It is worth noting that this paper and ref. [26] give us a useful procedure to solve similar problems. The designer can use the Taguchi method to identify which factor is important with minimum expense (18 CFD runs for five factors). Then the sampling based on the CCD of RSM can be used for important factors (25 CFD runs for four factors). Finally, surrogate model with high precision can be established using RBF + ANN and KM.

6 Conclusions

In this study, the thermal-hydraulic performance of a new parallel-flow STHX HCBetwc-STHX is explored in turbulent regime. The efficiency and applicability of predicting of four different surrogate models are established and compared. Some main conclusions are drawn as follows:

The HCBetwc-STHX can enhance heat transfer rate than RRB-STHX. The Nu_SWT/Nu_RRB is in the range of 1.1885–1.8254; the f _SWT/f _RRB is in the range of 4.401–14.982; and the PEC_SWT/PEC_RRB is in the range of 0.6579–0.7783.
All of the four surrogate models proposed in this paper can be used to predict the thermal-hydraulic performance of HCBetwc-STHX with the MAD around 18%. The precision order of these four surrogate models follows the order: RBF–ANN ≈ KM > RSM > CFM.
The merits and drawbacks of these four surrogate models are illustrated.
A general analysis procedure is presented for the predicting method of thermal-hydraulic performance of STHX through the analysis of this paper. One can adopt similar procedure to explore similar problem.

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Acknowledgments

The authors wish to express their thanks for the National Science and Technology Major Project of China (No. 2010ZX06004) and Specialized Research Fund of postdoctoral program of DongFang Boiler Corporation.

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Received: 2020-09-21

Revised: 2020-11-15

Accepted: 2020-11-20

Published Online: 2020-12-31

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

Articles in the same Issue

https://doi.org/10.1515/phys-2020-0218

Keywords for this article

heat exchanger; thermal-hydraulic; prediction; surrogate model

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