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

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)

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	Picture	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:

1. The complexity of the response;

2. 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.

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.

4 Numerical model

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.

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:

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.

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:

1. 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.

2. 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.

3. The merits and drawbacks of these four surrogate models are illustrated.

4. 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.