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

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)

ΔT

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 (–)

Δp

pressure drop (Pa)

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.

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 [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 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.

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 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.

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.

4 Experimental design method

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.

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.

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)).

5.5 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.

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 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.

From Tables 22–25, the effects of design factors on the thermal-hydraulic 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 (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.

Figure 9

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

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.