Single/Multi-Objective Optimization Design and Numerical Studies for Lead-to-Supercritical Carbon Dioxide Heat Exchanger Based on Genetic Algorithm

Abstract

Single-/multi-objective optimization based on genetic algorithm is employed in the present study to conduct an optimization design for the primary heat exchanger (HE) in a lead-cooled fast reactor (LFR), where the liquid lead and supercritical carbon dioxide (SCO₂) are the working fluids on the heat side and cold side of HE, respectively. A preliminary model of HE was first theoretically calculated by the subsection model based on equal heat transfer power, and an optimization design of HE was then performed based on genetic algorithm, where the entropy generation number and total pumping power were adopted as objective functions. Moreover, the numerical simulation based on Ansys-Fluent software was also performed to study the flow and heat transfer performances of working fluids in the optimized heat exchanger. The results show that the irreversible loss of HE is reduced by 25% after single-objective optimization. The heat transfer and hydraulic performance of optimized HE can be optimized together with multi-objective optimization based on a non-dominated sorting genetic algorithm II (NSGA-II). In addition, the field synergy angle of SCO₂ decreases, which indicates the improvement on the comprehensive performance of HE. The present work is helpful for the design of a primary heat exchanger in LFR.

1. Introduction

With the emphasis on environmental issues and carbon emission reduction, countries around the world are working to reduce the proportion of coal-fired cogeneration and actively develop new types of clean energy [1]. As a form of energy with low pollution, low cost and controllability, nuclear energy is receiving widespread attention to meet the needs of low-carbon energy conservation [2]. In order to achieve the sustainable development and improve the safety, reliability and economic competitiveness of nuclear energy, the Generation IV International Forum (GIF), organized by the US Department of Energy (DOE), is a collaboration with 12 other countries which has proposed six advanced Generation IV reactors after discussion and verification [3]. Among the six most potential advanced reactor types, the lead-cooled fast reactor (LFR) has drawn much attention due to its unique advantages, including high safety, high fuel utilization, excellent neutron physical, thermal-hydraulic characteristics and economic efficiency [4,5,6]. The lead-cooled fast reactor (LFR) adopts liquid lead or liquid lead-bismuth eutectic (LBE) as a core coolant. Due to higher core outlet temperature of the lead-based reactor (usually above 500 °C), the traditional steam Rankine cycle cannot meet the demand of a high power generation rate under high-temperature operation conditions [4]. The supercritical carbon dioxide (SCO₂) power cycles have been put forward as a promising candidate with the main advantages of being high thermal efficiencies, a simple and compact physical footprint and good operational flexibility [7,8,9]. Consequently, lead-cooled fast reactors coupled with a supercritical CO₂ power cycle are considered a highly competitive option and the most promising power conversion system for the miniaturization and modularization of advanced nuclear reactors.

As critical energy conversion equipment links the primary and secondary circuit in LFR, the primary heat exchanger (HE) is an important equipment which transfers heat from the core to the secondary heat transport system. When liquid lead and SCO₂ are employed as the working fluid in the primary heat exchanger (HE), the fluid flow and heat transfer characteristics are quite different from conventional coolants such as water and air. On the one hand, heavy metal fluids usually result in great loss of momentum and significant frictional pressure drop [10]. Moreover, high viscosity also means an extremely low Prandtl number (much lower than 1), which makes the flow more likely to show laminar flow characteristics and a thermal boundary layer which will be much thicker than the flow boundary layer, which means that the role of molecular heat conduction in the entire heat transfer process is more significant. As a result, a considerable part of heat transfer correlations concluded from the conventional coolants such as water are no longer applicable to liquid metals [3,11]. On the other hand, the supercritical carbon dioxide (SCO₂) power cycle utilizes the characteristics of SCO₂ near the quasi-critical region and realizes the efficient and compact design of the secondary loop, which makes SCO₂ a better choice of coolant [12,13]. The pressure of a secondary loop is usually higher than 20 MPa, and the temperature of the primary circuit using lead as coolant normally exceeds 500 °C which is limited by the high melting point of lead. The safety and convenient maintenance of HE under long-term operation is a crucial problem under such high-temperature and high-pressure conditions. In addition, as the key energy conversion device, the heat transfer efficiency of the primary heat exchanger determines the inlet temperature of the turbine and further influences the overall efficiency of the nuclear energy system. Meanwhile, the pressure loss of the fluids through the HE is also an important issue and should be considered since the total pressure loss has a significant impact on flow distribution and heat transfer performance. Therefore, it is crucially important to optimize the flow and heat transfer characteristics of lead-SCO₂ in the primary heat exchanger.

With a mature theory, sufficient experimental studies and extensive industrial application experience, the shell-and-tube heat exchanger (STHE) has been widely used in different fields [14,15]. Moreover, the shell-and-tube heat exchanger is a considerable choice of HE among the existing reactors including LFR [16,17]. Consequently, STHE is an available type of HE with liquid lead and SCO₂ as a working medium. In response to the dramatic changes in the physical properties of SCO₂ with temperature, the subsection model is used to simulate the variation in physical properties in the design of SCO₂ [18]. Some heat transfer enhancement methods used for the conventional fluids such as conventional segmental baffles may not be suitable choices, because this will lead to vast dead zones and increase the risk of liquid metal solidification [10]. Straight tube HE is recommended for the safety and miniaturization of the system design of LFR [19].

In addition to the design of a heat exchanger, the optimization of its performance is another important work. In recent years, the emergence of various intelligent optimization algorithms provides a powerful tool for the optimization of a heat exchanger. In the past decade, the genetic algorithm is definitely a competitive intelligent algorithm. It is widely used in the optimization of various thermal equipment and systems [20,21] and compared with other algorithms [22,23].

In the optimization process of the genetic algorithm, there is commonly single or multi-objective functions to evaluate the optimization results. Some researchers focus on the efficiency, pressure drop, heat transfer area and other thermodynamic parameters of HE [24], whilst others consider the cost and economy of HE [25], and some researchers take the entropy generation and exergy based on the second law of thermodynamics as the objective function. Furthermore, many researchers ensure comprehensive consideration on the design of HE through the combination of different perspectives [26,27]. Maida et al. [28] took the ecological function and cost as the objective function, used the non-dominated sorting genetic algorithm II (NSGA-II) to optimize the shell and tube heat exchanger and obtained the Pareto solution set. Song and Cui [29] carried out single/multi-objective optimization on the plate fin heat exchanger with the entropy production and operation cost as the objective functions, and optimized the comprehensive performance of the heat exchanger. The result of multi-objective optimization provides a flexible solution for practical production.

Specifically, some researchers also used genetic algorithm to optimize the design of a heat exchanger or energy system with SCO₂ or liquid metal as a working medium. Li et al. [30] carried out the multi-objective optimization of LFR system with thermoelectric conversion efficiency and a power generation cost as objective functions, improved the overall economy of the system and obtained the optimal geometric parameters of the intermediate heat exchanger. Fan et al. [31] proposed a CCHP system combined with a SCO₂ cycle, took a unit cost and efficiency as the objective functions and carried out multi-objective optimization using NSGA-II to obtain the best design performance of the system. The exergy efficiency is improved and the unit cost of the system is reduced after optimization. The temperature of the coal-fired power plant was optimized by Liu et al. [32] in order to improve the efficiency and the best temperatures of the cold end and hot end were obtained. Saeed and Kim [33] used the genetic algorithm to optimize the geometric structure of printed circuit heat exchanger with SCO₂ as a working medium and improve the hydraulic performance of the heat exchanger. To sum up, the genetic algorithm has been widely used in the design process of a heat exchanger, even with the working medium of SCO₂. However, for the design requirement of a primary heat exchanger in LFR where liquid lead and supercritical carbon dioxide (SCO₂) are employed as the working fluids on heat-side and cold-side of HE, respectively, only a few studies have been conducted.

In the present study, the genetic algorithm and NSGA-II are employed to carry out an optimization design for the primary heat exchanger (HE) in a lead-cooled fast reactor (LFR). A preliminary model of HE is first theoretically calculated by the subsection model based on equal heat transfer power, and then an optimization design of HE is conducted based on genetic algorithm with an entropy generation number and total pumping power as the objective functions. During the optimization process, the flow and heat transfer performance of the heat exchanger are evaluated. Then, numerical simulation based on Ansys-Fluent software is also conducted to study the flow and heat transfer performances of working fluids in the optimized heat exchanger.

2. Preliminary Design Based on Theoretical Calculation

2.1. Working Conditions of the Primary Heat Exchanger (HE)

According to the demand in [19],

Table 1. Main operated parameters of IHE.

Parameters	Value
Thermal power (MWt)	50
Operated pressure of primary loop (MPa)	0.1
Inlet and outlet temperature of core coolant (°C)	450/600
Operated pressure of secondary loop (MPa)	20
Inlet and outlet temperature of SCO2 side (°C)	360/560

lists the working conditions of the primary heat exchanger. The inlet and outlet temperatures of the lead-side are 450 °C and 600 °C, respectively, and depend on the parameters of the core and the properties of lead. The operating pressure of the SCO₂-side is 20 MPa which is mainly considered by the power generation efficiency of the secondary loop of a reactor.

2.2. Thermal-Hydraulic Calculation of HE

Heat transfer tubes are arranged as equilateral triangular layout in the present study, and the shell inner diameter is calculated by Equation (1) [34].

$$D_s=\left(1.1\sqrt{N_t}-1\right)\cdot s+3d_o$$

(1)

where N_t is the number of tubes, s is the tube pitch for the equilateral triangular tube arrangement, and d_o is the out diameter of tubes.

The heat transfer area of each (HEU) is given by:

$$A_j=\frac{{\dot{Q}}_j}{k_j\Delta t_{\mathrm{m},j}}$$

(2)

Here, k_j represents the overall heat transfer coefficient of the jth HEU, ${\dot{Q}}_j$ and Δt_m,j are the heat transfer rate and the LMTD of the jth HEU, respectively.

The overall heat transfer coefficient based on the external area of tubes is:

$$k_j=\frac{1}{\frac{d_o}{h_{t,j}{d}_i}+\frac{d_o}{2\lambda }\ln \frac{d_o}{d_i}+\frac{1}{h_{s,j}}}$$

(3)

where d_i and d_o are the inside and outside diameters of the tube, respectively; h_i,j and h_o,j are the heat transfer coefficients in the tube-side and shell-side, respectively; and λ is the heat conductivity of the tube-wall.

$$d_i=d_o-2t$$

(4)

$$t=\frac{P{d}_{\mathrm{o}}}{2\left(\left[\sigma \right]E+PY\right)}+C_1+C_2$$

(5)

where t is the tube thickness on the basis of national standard; P is the operating pressure of the tube-side; [σ] is the allowable stress of the material under the operating temperature; E is the welding joint coefficient; Y is the pressure coefficient related to the material; C₁ is the additional thickness due to thickness thinning; and C₂ is the additional thickness due to corrosion. All parameters may find a value on the basis of a national standard.

For the tube-side of each HEU, different correlations are proposed [35,36]. The heat transfer coefficient is calculated by Gnielinski correlation [36] in this study, which is widely used in the heat transfer calculation of SCO₂, and listed as follows

$$N{u}_{t,j}=\frac{\frac{f_j}{8}\times ({Re}_{t,j}-1000)\times {Pr}_j}{12.7\sqrt{\frac{f_j}{8}}\left({Pr}_{t,j}^{2/3}-1\right)+1}\left[1+{\left(\frac{d_i}{l_j}\right)}^{2/3}\right]$$

(6)

The correlation is valuable in the range of 0.5 < Pr < 2000, 2300 < Re < 5 × 10⁵, where

$$f_j={\left(1.81\times \lg {Re}_j-1.64\right)}^{-2}$$

(7)

For the shell-side of each unit, the heat transfer coefficient is calculated by the Gräber–Rieger correlation [37], as follows:

$$N{u}_{s,j}=0.25+6.2x+\left(0.032x-0.007\right)P{e}_{s,j}^{0.8-0.024x}$$

(8)

Here, x (x = s/d_o) is the ratio of the tube pitch to tube outside diameter, and Pe_j is the Peclet number of each unit.

$$P{e}_{o,j}={Re}_{o,j}{Pr}_{o,j}$$

(9)

$$R{e}_{s,j}=\frac{{\rho }_{s,j}{U}_{s,j}{D}_{es}}{{\mu }_{s,j}}$$

(10)

where d_e represents the hydraulic diameter of the shell side which can be expressed as follows:

$$D_{es}=4\frac{0.25\pi \left(D_s^2-Nt\cdot d_o^2\right)}{\pi \left(D_s+Nt\cdot d_o\right)}$$

(11)

The pressure drop of the tube-side can be expressed as follows by neglecting the pressure drop for the elbow of single tube pass.

$$\Delta P_t=\left(f_t\frac{l_j}{d_i}+1.5\right)\frac{{\rho }_{i,j}{U}_{i,j}^2}{2}$$

(12)

where f_t is the tube-side friction factor, and l_j is the length of tubes of the jth unit.

f_t is calculated by the Colebrook Equation [38] which is widely used in single-phase pressure drop calculation.

$$\{\begin{array}{l}{f}_{t,j}=\frac{64}{R{e}_{t,j}},Re\le 2000\\ \frac{1}{\sqrt{f_{t,j}}}=1.74-2\lg \left(\frac{2\epsilon }{d_i}+\frac{18.7}{R{e}_{t,j}\sqrt{f_{t,j}}}\right),Re>2000\end{array}$$

(13)

where ε is the roughness of the tube-wall.

The pressure drop of the shell-side can be calculated by the similar method such as the tube-side,

$$\Delta P_s=\left(f_s\frac{l_j}{d_e}+1.5\right)\frac{{\rho }_{s,j}{U}_{s,j}^2}{2}$$

(14)

f_s is also calculated by the Colebrook correlation [38].

2.3. Theoretical Calculation Based on Subsection Model

Supercritical carbon dioxide cannot be treated as a constant property’s fluid during the theoretical design process since the thermophysical properties vary greatly with temperature in a supercritical state. In order to simulate the variation of the properties of SCO₂, a subsection model based on equal heat transfer power is used for the theoretical calculation. Figure 1 shows the principle of the subsection model. The heat exchanger (HE) is initially divided into several heat exchanger units (HEUs) with the same heat flux where the properties of both hot and cold fluids are considered to be constants. The input parameters of the back HEU are given by the front one. In the subsection process, each unit is assumed to have undergone the same amount of heat exchange.

The thermodynamic model of HE introduced in Section 2.2 and the subsection model based on equal heat transfer power are written into a program by MATLAB language to carry out the design calculation of the preliminary HE. The working condition of the preliminary HE is listed in Table 1. In the program, the thermophysical properties of liquid lead are calculated according to correlations in

Table 2. Property correlation of liquid lead.

Property of Liquid Lead	Correlations
Density (kg·m−3)	ρ = 11367 − 1.1944·T
Viscosity (Pa·s)	μ = 4.55 × 10−4·exp(1069/T)
Thermal conductivity (W·m−1·K−1)	λ = 9.2 + 0.011·T
Heat capacity (J·kg−1·K−1)	Cp = 175.1 − 4.961 × 10−2·T + 1.985 × 10−5·T2 − 1.524 × 106·T−2 − 2.099 × 10−9·T3

, and the thermophysical properties of SCO₂ are from the NIST database. In addition, the flow velocity of liquid lead is limited to less than 1 m/s, the flow velocity of SCO₂ is lower than 5 m/s, and the pressure drop on both the tube-side and shell-side is below 50 kPa are restrictions in the design.

The preliminary designed results on HE are shown in

Table 3. Preliminary design results of HE.

Variables	Value
Tube—outside diameter (mm)	20
Tube—inside diameter (mm)	13
Length of tube (m)	6
Pitch of tubes (mm)	24
Shell-side diameter (m)	1.29
Number of tubes	2269
Mass flow of liquid lead and SCO2 (kg·s−1)	2303.3/205.2
Inlet/outlet temperatures of liquid lead (K)	873.15/723.15
Inlet/outlet temperatures of SCO2 (K)	633.15/831.40

.

To verify the accuracy of the preliminary design results of HE, the subsection model based on equal length was used. The total length of HE obtained from the design calculation is set as the input parameter in the subsection model based on equal length. This model assumes that the length of each HEU is the same and calculates the total heat transfer rate by the length of HE. Theoretically, the total heat transfer rate calculated by the subsection model based on equal length should be the same with the heat transfer rate in Table 1 since the thermodynamic model and boundary conditions are the same. The verification process of the preliminary HE is also written into a program by MATLAB language based on the subsection model based on equal length and the results are shown in

Table 4. Heat transfer rate validation result.

Variable	Total Heat Transfer Rate
Condition in Table 1	50 MWt
Verification result	48.97 MWt
Relative error	2.1%

.

It can be seen from Table 4 that the error between the verification result and the boundary condition in Table 1 is within 3%. This result verifies the preliminary design of HE based on the subsection model with equal heat transfer power so that the preliminary HE can be used for subsequent optimization design.

3. Optimization Design

3.1. Genetic Algorithm

There are different geometric factors which would have an impact on the performance of a heat exchanger such as the tube diameter and shell-side diameter. The effect of different factors are difficult to quantitatively analyze due to their completed change. Each factor is analyzed individually for the optimized thermal-hydraulic performance of a heat exchanger with the other factors constant and so much time will be used in the traditional optimization method. In order to solve this problem, the genetic algorithm is used in the optimization design. The genetic algorithm can estimate several factors at the same time to solve the optimal solution problem based on the biological evolution theory. Applying the genetic algorithm to the optimal design of HE can quantitatively analyze different variables at the same time and save a lot of time. Figure 2 illustrates the flow chart of genetic algorithm.

As shown in Figure 2, the parameters of each individual are transferred to the fitness function as input after the initial population is generated. The objective function of each HEU and the HE is calculated through the subsection model mentioned before and the thermal-hydraulic model of the heat exchanger. The conditions in the fitness function are set to eliminate unqualified individuals, which is realized in the algorithm through penalty function. After evaluating the fitness of each individual, the population is updated through selection, crossover and mutation until the optimal individual that meets the conditions is found or the maximum number of generations is reached.

Different related variables are associated with the fitness (objective function) to estimate the performance of HE. The first generation is randomly created and the next generation is generated by selection, crossover and mutation. Eventually, the heat exchanger with the best fitness will be proposed after the continuous alternation of the generation. The modified entropy generation number is the fitness in the present study. Entropy generation is caused by irreversible loss in the process of flow and heat transfer. It is used to evaluate the irreversible loss in the heat exchanger. The irreversible loss of the heat exchanger occurred in two parts: the irreversible loss from flow resistance and the irreversible loss from heat transfer with limited temperature difference. These two parts are expressed as [39]:

$${\dot{S}}_{gen,T}={\left(\dot{m}{c}_p\right)}_t\ln \left(\frac{T_{t,o}}{T_{t,i}}\right)+{\left(\dot{m}{c}_p\right)}_s\ln \left(\frac{T_{s,o}}{T_{s,i}}\right)$$

(15)

$${\dot{S}}_{gen,P}={\dot{m}}_t\frac{\Delta P_t}{{\rho }_t}\frac{\ln \left(T_{t,o}/T_{t,i}\right)}{T_{t,o}-T_{t,i}}+{\dot{m}}_s\frac{\Delta P_s}{{\rho }_s}\frac{\ln \left(T_{s,o}/T_{s,i}\right)}{T_{s,o}-T_{s,i}}$$

(16)

The total entropy generation rate is given by Equation (17).

$${\dot{S}}_{gen}={\dot{S}}_{gen,T}+{\dot{S}}_{gen,P}$$

(17)

Bejan [40] proposed the “entropy generation minimization” theory to optimize the performance of heat transfer equipment including the heat exchanger. The total entropy generation rate is nondimensionalized into an entropy generation number (EGN) by the larger heat capacity rate, and the expression is given by Equation (18).

$$N_s=\frac{{\dot{S}}_{gen}}{\dot{m}{c}_p}$$

(18)

where N_s is called Bejan’s definition of EGN.

In order to avoid the “entropy generation paradox” arising from the EGN calculation, Hesselgreaves [41] proposed the modified entropy generation number (MEGN) by drawing the inlet temperature of cold fluid as

$$N_{s1}=\frac{{\dot{S}}_{gen}{T}_{t,i}}{Q}$$

(19)

In the present study, five parameters were taken as the design variables which are the tube—outside diameter (d₀); pitch diameter ratio; ratio of shell—inside diameter to the shell—inside diameter calculated by Equation (11); the length of the tubes (L); and the layers of tubes. The ranges of these design variables are listed in

Table 5. Ranges of design variables.

Design Variable	Range
Tube—outside diameter (mm)	10–40
Pitch diameter ratio	1.25–2
Ratio of shell—inside diameter to the shell—inside diameter Calculated by Equation (1)	1–2
Length of tubes (m)	3–6
Layers of tubes	20–40

.

3.2. Single Objective Optimization of HE Based on Genetic Algorithm

Optimization design is carried out for HE based on the genetic algorithm with MEGN as the fitness to improve the thermal performance of HE. The optimization design method is written into a program by MATLAB language. During the optimization design process, the initial population is randomly generated according to the range set for every variable. The number of individuals in the initial population is 40, and the maximum number of generations is taken as 300. In the optimization design program, the heat transfer rate of HE is kept as the same value as 50 MWt. Additionally, the penalty function is used to eliminate those individuals who do not meet the design requirements in the process of population evolution.

For each generation in the evolution process, the best individual is selected based on the fitness of each individual. The parameters of the optimization model of HE and the comparison between the preliminary design and the optimized design are listed in

Table 6. Comparison between the preliminary HE and the optimized HE.

Variables	Preliminary Design	Optimized Design
Tube—outside diameter (mm)	20	10.2
Tube—inside diameter (mm)	13	8.8
Length of tube (m)	6	6
Pitch of tubes (mm)	24	12.9
Shell-side diameter (m)	1.29	0.987
Number of tubes	2269	4681
Mass flow of liquid lead and SCO2 (kg·s−1)	2303.3/205.2	2303.3/182.2
Inlet/outlet temperatures of liquid lead (K)	873.15/723.15	873.15/720.89
Inlet/outlet temperatures of SCO2 (K)	633.15/831.40	633.15/859.48
Entropy generation number	0.0854	0.0559

. The fitness of the best individual in each generation changes with the number of populations, as shown in Figure 3.

As shown in Figure 3, the optimal solution basically does not change after 120 generations, and the EGN achieves a 50% drop, from 0.11 for the first generation to approximately 0.055. In order to analyze the performance of the optimization model of HE with the decreasing MEGN, the effectiveness (ε), the number of transfer units (NTU), and the total pumping power of HE is calculated. ε [42] and NTU represent the thermal performance of HE, which can be expressed as:

$$\epsilon =\frac{1-\exp \left\{(-NTU)\left[1+\frac{{(\dot{m}c)}_{\min }}{{(\dot{m}c)}_{\max }}\right]\right\}}{1+\frac{{(\dot{m}c)}_{\min }}{{(\dot{m}c)}_{\max }}}$$

(20)

$$NTU=\frac{kA}{{(\dot{m}c)}_{\min }}$$

(21)

The total pumping power represents the flow resistance of both the tube-side and shell-side in HE. The total pumping power can be expressed as [43]:

$$\dot{W}=\frac{1}{1000\eta }\left(\frac{{\dot{m}}_s}{{\rho }_s}\Delta P_{s,sum}+\frac{{\dot{m}}_t}{{\rho }_t}\Delta P_{t,sum}\right)$$

(22)

where η is the pumping efficiency, and $\Delta P_{s,sum}$ and $\Delta P_{t,sum}$ are the sum of the pressure drop of shell-side and tube-side, respectively.

Figure 4 shows the variation in the effectiveness, number of transfer units (NTU) and total pump power of HE with MEGN in the optimization design.

It can be seen from Figure 4a,b that the effectiveness (ε) and NTU of HE decreases with EGN, where the enhancement of ε indicates an improvement on the heat transfer performance of HE. The result in Figure 4b means that the heat transfer capacity of HE is improved while more heat transfer area is needed for HE, probably for the better heat transfer based on Equation (20). When comparing the tube numbers in preliminary designed results with those in optimization designed results, as listed in Table 5 and Table 6, although the number of tubes of the optimized HE is more than twice that of the preliminary HE, the diameter of the tubes is reduced by nearly half. The heat transfer area of the optimized HE is increased by 5.2% compared with the preliminary HE through calculation. Although the heat exchange area is increased, the pressure bearing capacity of the optimized tubes is improved. In addition, the calculated effectiveness of the preliminary HE is 0.567 and NTU is 3.17. Compared with the preliminary design, the effectiveness and NTU of the optimized HE is increased by 5.2% and 75.9%, respectively. The improvement of the heat performance of HE is considerable so that the increase in heat transfer area is acceptable. Based on the calculation results illustrated in Figure 4c, the influence of MEGN on the total pumping power is not monotonous. In general, the curve of the total pumping power shows a down–up–down tendency with the increase in MEGN. In this case, it is necessary to evaluate whether the overall performance of the heat exchanger is improved. The overall performance evaluation factor ζ is used to estimate the enhanced heat transfer performance of optimized HE. The expression of ζ is:

$$\zeta =\frac{\left(Nu/N{u}_0\right)}{{\left(f/f_0\right)}^{1/3}}$$

(23)

where Nu and f are the Nusselt number and friction factor of HE, respectively. Nu₀ and f₀ represent the preliminary HE.

The factor ζ is used to evaluate the performance of the shell-side and tube-side, respectively, and the results are shown in

Table 7. Comparison of the performance of preliminary HE and optimized HE.

	ζs	ζt
Preliminary HE	1	1
Optimized HE	1.04	1.47

.

It can be seen from Table 7 that the performance of the shell-side of the heat exchanger is basically unchanged, but the performance of the tube-side is enhanced, which proves the improvement of the comprehensive performance of the optimized HE.

In addition, MEGN and the related thermal-hydraulic parameters are considerably optimized after only four generations compared with the first generation, though the total number of generations is 300. Therefore, the genetic algorithm can provide a better model HE and improve the efficient design of HE.

3.3. Multi-Objective Optimization of HE Based on NSGA-II

It can be seen from the results in Section 3.2 that optimizing HE with MEGN as a single objective function will greatly increase the total pumping power which indicates that the pressure drops of fluids on both sides of HE will increase. This is an undesirable result in the optimization design. In order to balance the heat transfer performance and hydraulic performance of HE, this section also takes the total pumping power as the objective function, and uses NSGA-II to carry out the multi-objective optimization of HE.

The program of NSGA-II is also written based on MATLAB language. The modified entropy generation number and total pumping power of the heat exchanger are the two objectives of optimization. The population is set to 80 and the genetic generation is 200 in the optimization. The offspring individuals are selected by tournament in the selection, and the polynomial mutation method is used during the mutation. The optimized variables and relative ranges are the same as those of single-objective optimization. Figure 5 illustrates the pareto front which results from optimization.

It can be seen from Figure 5 that the total pumping power decreases with the increase in the MEGN on the Pareto front. Select three representative points—A, B and C—as the analysis object, and pareto point A is similar to the model from single-objective optimization. The variable values of HE corresponding to pareto points A, B and C are listed in

Table 8. Variable values of pareto points A, B and C.

Variables	Point A	Point B	Point C
Outer diameter of tube/mm	10.1	13.4	20.8
Pitch diameter ratio	1.25	1.28	1.25
Length of tubes/m	6	6	6
Ratio of shell—inside diameter to the shell—inside diameter Calculated by Equation (1)	1	1	1
Layer of tubes	40	40	40
MEGN	0.0556	0.0601	0.079
Pumping power/kW	73.5	17.1	2.1
ε	0.598	0.587	0.565
NTU	5.651	4.492	3.127

.

It can be seen from Table 8 that the outer diameter of the heat exchange tube plays a major role in the multi-objective optimization to the objectives. The smaller outer diameter of the heat exchange tube will bring less irreversible loss to the heat exchanger. Although the other three design variables were almost the same for points A, B and C, they also provide an important reference significance for the design of HE. Compared to point C, the HE of point A shows better thermal performance where ε increases by 5.84% and NTU increases by 77.84%. However, with the decrease in MEGN of point A, the total pumping power of HE also greatly increases, which represents a large increase in the flow resistance of the fluid in the heat exchanger.

Furthermore, it can be found that the heat exchanger corresponding to point B has both good thermal performance and a relatively small flow resistance by comparing the variable values in the HE models corresponding to points A, B and C. In terms of thermal-hydraulic performance optimization, point A and point C are not the most suitable options, and point B is a better choice because it has the best comprehensive performance. The multi-objective results show that the multi-objective optimization design of HE aiming to reduce the modified entropy generation number and total pumping power can optimize both the thermal and hydraulic performance of HE. The results of multi-objective optimization provide various options of HE in practical application.

4. Numerical Simulation on Simplified Model of Preliminary Design HE and Optimized Design HE

Based on the preliminary design and optimization design results of HE, their simplified models are constructed. The coupled heat transfer simulation of the liquid lead and SCO₂ was carried out, and the influence of the optimization design on the thermal and hydraulic performance of HE was analyzed using the computational fluid dynamics software FLUENT.

4.1. Physical Model

It is difficult to directly use the prototype parameters for numerical simulation due to the large size of the preliminary model and the optimization model (point B) of HEs. Therefore, the simplified models of these two models are constructed based on the similarity principal method, which are called Model A and Model B. According to the similarity design method, the boundary conditions of the simplified model should remain the same as for the prototype, and several dimensionless criteria on the same point should also be equal. These dimensionless criteria reflect the influence of different parameters on the physical phenomena in HE. In the actual application, those parameters that are significantly related to the most concerned physical phenomena are preferentially guaranteed to be similar, while other weakly correlated parameters are ignored. In this study, attention is mainly focused on the heat transfer performance of HE so that the temperature distribution and velocity distribution in the simplified models should be consistent with those in the prototype. A simplified model of HE is designed based on the same MATLAB program in Section 2 and the information of Model A and Model B is listed in

Table 9. Information of the small-scale models of HE.

Variables	Model A	Model B
Tube—outside diameter (mm)	20	13.4
Tube—inside diameter (mm)	13	11.6
Length of tube (m)	5.9	5.9
Pitch of tubes (mm)	24	17.2
Shell-side diameter (m)	0.072	0.051
Number of tubes	7	7
Mass flow of liquid lead and SCO2 (kg·s−1)	7.106/0.633	3.445/0.277
Inlet temperatures of liquid lead (K)	873.15	873.15
Inlet temperatures of SCO2 (K)	633.15	633.15

.

It can be seen from Table 9 that the simplified models consist of seven heat exchange tubes, and the inlet and outlet temperatures on the shell-side and tube-side are consistent with the prototype. The mass flowrate of the fluid in the simplified model is proportionally reduced to ensure that the flow velocity of lead and SCO₂ is the same as that in the prototype. Because the velocity distribution and temperature distribution of the simplified model need to be similar to the prototype, the length of the heat exchange tube in the simplified model is almost the same as the prototype, which is not suitable for numerical simulation. After consideration, this study only takes five HEUs of model A and model B as the sample to simulate. The selection of HEUs is based on the relative change rate of MEGN of each HEU. The relative change rate of MEGN refers to the change in MEGN of each HEU in model B compared with that of each HEU in model A. A positive value means that the MEGN of the HEU decreases after optimization design, while a negative value is the opposite. The relative change rate of MEGN of each HEU is shown in Figure 6.

According to the results from Figure 6, five HEUs close to the outlet of SCO₂ are selected as the research object, because the MEGN of these heat exchanger units is reduced the most, which is more conducive to explain the impact of optimization design. The diagram of HE is shown in Figure 7. The five HEUs from Model A and Model B are named sample 1 and sample 2, respectively.

4.2. Boundary Conditions and Numerical Method

The physical model of HE is established by SOLIDWORKS, and the model is divided in ICEM. The regular tube side is divided into a hexahedral grid, while the shell side is divided into a tetrahedral grid. The final generated grid is shown in Figure 8. In addition, the boundary layer grids are generated on the outer surfaces of tubes. The physical properties of liquid lead are calculated based on the physical property relationship in Table 3 and compiled in FLUENT by user-defined function (UDF). The physical properties of SCO₂ were obtained from the REGPROP database by activating the NIST real gas model in FLUENT.

The velocity—inlet boundary condition was applied at the inlet of the lead and SCO₂, while the pressure—outlet was set as the outlet boundary condition at the outlet of lead and SCO₂. The coupled thermal boundary condition was applied for the inner and outer surfaces of heat transfer tubes which coupled the heat transfer between fluid and solid. The outside wall of HE was set as a zero heat flux boundary condition which assumes the outside-wall to be insulated.

The Reynolds number (Re) of liquid lead and SCO₂ are more than the magnitude of 10⁴ and 10⁵ in this study, respectively. Previous studies [10,44] have stated that the standard k-ε model is suitable for the flow of liquid metal and supercritical fluid under high Re. Therefore, the standard k-ε model is chosen as the turbulent model in the present study. The turbulent Prandtl number (Pr_t) of liquid lead cannot directly use the constant 0.85 set in FLUENT because the molecular Prandtl number of lead is so much lower that the Reynolds analogy is no longer applicable. In the present study, the Cheng model [45] was selected to be the turbulent Prandtl number model in the flow region of liquid lead, and the model was added into FLUENT through UDF. The turbulent Prandtl number in the region of SCO₂ still adopts the recommended value of 0.85 in FLUENT. The expression of the Cheng model is:

$$P{r}_t=\{\begin{array}{l}4.12,Pe\le 1000\\ \frac{0.01Pe}{{\left[0.018P{e}^{0.8}-\left(7-A\right)\right]}^{1.25}},1000<Pe\le 6000\end{array}$$

(24)

$$A=\{\begin{array}{l}5.4-0.9Pe,1000<Pe\le 2000\\ 3.6,2000<Pe\le 6000\end{array}$$

(25)

4.3. Grid Independence and Validation

Grid independence verification was carried out to eliminate the influence of the grid number on the simulation results. The average temperature on the outer surface of SCO₂ was taken as the monitoring object because it can reflect the heat transfer in HE. Taking sample 1 as an example, the grid models with 796,534, 2,544,917, 3,967,254, 5,386,445 and 7,508,074 cells were established for grid independence verification, and the results are shown in Figure 9.

It can be seen from Figure 9 that the average temperature on the outlet surfaces of SCO₂ changes less when the number of cells reaches 3,967,254; accordingly, the mesh that consists of 3,967,254 cells is chosen for the following study. The same procedure is carried out for sample 2, and the mesh number is approximately 4,788,641.

In addition to the grid independence, a suitable range of the y+ on the tube wall is also an important part to ensure the accuracy of the numerical simulation results. In this study, the flow patterns of liquid lead and SCO₂ are both turbulent flows, which the y+ within the range of 30–300 is suitable for such a zone. In the previous study, Heyong [46] verifies the turbulence model of SCO₂ in simulation and the results show that the error of calculation is less than 0.1 if y+ is in the range of 10–76. Therefore, y+ is controlled less than 76 in this study. The maximum value of y+ in the results is approximately 75.

In order to verify the accuracy of the design calculation, taking sample 2 as an example, the results of the design calculation are compared with the results of numerical simulation.

Table 10. The comparison between the design calculation and numerical simulation.

Results	Pressure Drop of Tube-Side/Pa	Pressure Drop of Shell-Side/Pa	Heat Transfer Rate/kW
Design calculation	1067.8	1283.1	9.35
Numerical simulation	1100.0	1117.3	9.01
Relative error/%	3.0	12.9	3.6

shows the pressure drop and heat transfer power of sample 2 in the design calculation and numerical simulation.

It can be seen from Table 10 that the pressure drops on the tube-side and heat transfer rate of sample 2 are in good agreement with the design calculation and numerical simulation. The error of the pressure drop on the shell-side is relatively larger, but it is still within an acceptable range.

4.4. Numerical Simulation Results

Based on the optimization design and the numerical model of HE, the flow and heat transfer characteristics were studied first. Figure 10 shows the flow velocity and temperature distribution of lead and SCO₂ in sample 2.

The flow velocity of lead is much less than the flow velocity of SCO₂ in sample 2 from the result in Figure 10a. It can be seen from the velocity contours of liquid lead and SCO₂ that the flow velocity of SCO₂ changes a little along the flow direction while the flow velocity of lead descends a little along the flow direction. It can be seen from Figure 10b that the temperature of SCO₂ in the heat exchange tube increases gradually along the flow direction, and the temperature in the center of the tube increases the slowest. The temperature of liquid lead decreases faster in the area close to the heat exchange tube. In addition, from the temperature contours of the four HEUs outlet sections, it can be found that the temperature gradient near the wall inside the tube and near the wall outside the tube is relatively large.

For a quantitative study, specific temperature distribution of SCO₂ and lead are collected. The axial temperature distribution of SCO₂ and liquid lead are shown in Figure 11. The temperature of SCO₂ increases 25.1 K in sample 2. The temperature of lead illustrates a uniform upward trend along the flow direction with a 18.2 K drop in temperature in sample 2. The temperature changing curves of lead and SCO₂ are approximately parallel, except for the area near the inlet of SCO₂.

The pressure distributions of lead and SCO₂ along the flow direction are shown in Figure 12. The operating pressure of liquid lead is 0.1 MPa and its pressure loss in sample 2 is 1.12 kPa. The operating pressure of SCO₂ is 20 MPa, and the pressure drop of it in sample 2 is 1.19 kPa. The flow velocity of liquid lead is much less than SCO₂, but the pressure drop is near to SCO₂, which is due to the high density and high viscosity of liquid lead.

In order to further illustrate the influence of MEGN on the thermal and hydraulic performance of HE and evaluate the comprehensive performance of it, the field synergy analysis was carried out for the heat transfer performance and flow resistance in sample 1 and sample 2. The field synergy angle commonly used in field synergy analysis is the angle between the velocity vector and the temperature gradient, as shown in the following equation:

$$\beta =\arccos \frac{\mathit{U}\cdot \nabla T}{\left|\mathit{U}\right|\cdot \left|\nabla T\right|}$$

(26)

where U is the velocity vector; and $\nabla T$ is the temperature gradient.

The cosine of the field synergy angle is called the field synergy number, which can also be used to evaluate the synergy of the velocity vector and temperature gradient. A larger field synergy number means a better thermal-hydraulic performance of HE. Figure 13 shows the average field synergy number in the heat exchange tubes.

It can be seen from Figure 13 that the average field synergy number of SCO₂ in the heat exchange tube of sample 2 is greater than that of sample 1 on the whole. Such results indicate that the synergy of the velocity vector and temperature gradient in sample 2 are better, i.e., that sample 2 has better comprehensive performance.

5. Conclusions

An optimization design for the primary heat exchanger (HE) in a lead-cooled fast reactor (LFR) was conducted based on the genetic algorithm and NSGA-II which takes the modified entropy generation number (MEGN) and total pumping power as the objective functions. The subsection model based on equal heat transfer power was employed in the theoretical calculation. Additionally, the numerical simulation based on Ansys-Fluent software was conducted to study the flow and heat transfer performances of the working fluids in the simplified models of preliminary HE and optimized HE. The results show that the MEGN of optimized HE is reduced by 25% through single-objective optimization. The optimized HE based on single-objective optimization shows higher effectiveness, a higher number of transfer units (NTU) and the total pumping power. Higher total pumping power indicates the high flow resistance of lead and supercritical carbon dioxide. The results of multi-objective optimization show that there is a confrontational relationship between the entropy generationand the total pumping power of HE. The pareto points generated by multi-objective optimization can provide various solutions for HE. Compared with the information of the preliminary HE, the irreversible loss of and the total pumping power can be reduced together. It is believed that the comprehensive performance of the simplified model of optimized HE is better than the simplified model of preliminary HE based on the numerical simulation and field synergy analysis.