Efficient Sensitivity Analysis for Enhanced Heat Transfer Performance of Heat Sink with Swirl Flow Structure under One-Side Heating

Abstract

Excellent heat transfer performance has increasingly become a key issue that needs to be solved urgently in the development process of large-scale fusion equipment. The study of heat transfer performance improvement to scientifically and reasonably determine the design parameters of the high heat flow (HHF) components of fusion reactors based on the efficient in-depth analysis of the heat transfer mechanism and its sensitive factors is of great significance. In this paper, a liquid-vapor two-phase flow model with subcooled boiling for a large length-diameter ratio swirl tube structure in the HHF calorimeter component is proposed to analyze the effects of key design parameters (such as inlet temperature of cooling water flow, swirl tube structure parameters, etc.) on its heat transfer performance. Then, considering the high computational cost of the liquid-vapor two-phase flow model, and in order to improve the efficiency of the sensitivity analysis of these design parameters, the polynomial response surface surrogate model of heat transfer performance function was constructed based on Latin hypercube sampling. On this basis, by combining the proposed surrogate model, the sensitivity index of each design parameter could be obtained efficiently using the Sobol global sensitivity analysis method. This method could greatly improve the calculation efficiency of the design parameter sensitivity analysis of HHF components in the fusion reactor, which provides vital guidance for the subsequent rapid design optimization of related components.

1. Introduction

Nuclear fusion can generate huge amounts of energy, which is undoubtedly the hope that mankind will finally solve the energy problem. Compared with other controlled nuclear fusions, Tokamak has many advantages, making future commercial reactor power generation more feasible. Excellent heat transfer performance has increasingly become a key issue that needs to be solved urgently in the development of Tokamak fusion reactors. The enhancement of heat transfer performance and the required values of heat transfer coefficient and critical heat flux (CHF) can be achieved using more than one solution, including swirl tubes, finned tubes, hypervapotrons, etc. Among these, the tube with twisted tape inserts has been found to be an efficient active cooling technology [1,2]. Twisted tape can not only increase heat transfer areas of the cooling tube but also induce bulk flows to form the swirling flow. The Heating Neutral Beams of International Thermonuclear Experimental Reactors (ITER-HNB system), which is the prototype of the whole ITER-HNB in Italy, The Neutral Beam Injection (NBI) system of the Large Helical Device in Japan, and the diverter of the Experimental Advanced Superconducting Tokamak (EAST) in China have all adopted a swirl tube as the main heat exchange structure for the internal high heat flow (HHF) component.

In the past few decades, a large number of studies on the heat transfer performance of the twisted tape structure have been carried out, the key factors affecting its heat transfer performance have been analyzed, and the empirical formulas with certain applicability have been given. Araki [3] performed heat transfer experiments on swirl tubes in the regions from non-boiling to high subcooled partial nucleate boiling and established heat transfer corrections for water under one-sided heating conditions for the subcooled boiling regime inside the swirl tubes. Marshall [4] developed a physical model by integrating the respective heat transfer correlations corrections in individual regimes for predicting heat transfer properties for a swirl tube with one-side heating. Yan [5,6] performed subcooled water flow boiling experiments in vertical circular tubes and proposed a modified Chen correlation to predict the subcooled heat transfer coefficient. Zhu [7] proposed a modified Liu and Winterton correlation for the entire subcooled boiling region to predict the heat transfer coefficients in swirl flow under high and non-uniform heat fluxes. Ji [8] explored the onset of nucleate boiling (ONB) of a one-side heated swirl tube and evaluated the prediction accuracy of the existing ONB correlations according to the experimental results. Tao [9] proposed a thermal dynamic simulation method based on the Eulerian multiphase method for swirl tubes under the subcooled boiling condition, considering the factors of the interphase transfer of energy and mass and the wall heat flux partitioning model. These studies show that whether the design parameters are reasonable or not has an important effect on the heat transfer performance of the HHF components of the fusion reactor [3,4,5,6,7,8,9,10,11,12,13]. The enhanced heat transfer performance optimization of HHF components of fusion devices is a multi-variable and multi-constraint optimization process, which has many design parameters, high CFD simulation cost, and low efficiency. It is of great significance to scientifically evaluate the sensitivity of enhanced heat transfer performance with the change of design variables and to classify the parameters according to the sensitivity analysis results for the design optimization of HHF components of fusion equipment, which can effectively reduce the complexity of design optimization.

At present, sensitivity analysis methods can be roughly divided into local sensitivity analysis (LSA) and global sensitivity analysis (GSA) [14,15]. LSA is the slope of the relationship between the statistical characteristics of output performance at the nominal point and the distribution parameters of input variables, and these mainly include Green’s function method [16], direct differentiation method [17], and finite difference method [18]. LSA cannot directly reflect the influence of input variables on the output response in the entire distribution domain, and it is difficult to effectively deal with nonlinear problems. The variation range of the input parameters of the GSA method can be extended to the entire definition domain, and the parameters can have different variation ranges and change at the same time. It is not limited by the linear model and can handle nonlinear problems. Common GSA methods include screening methods (such as the Morris method [19], Systematic Fractional Replicate Design method [20], etc.), Fourier Amplitude Sensitivity Test (FAST) and its extended methods (EFAST) [21], variance analysis sensitivity [22], moment independent sensitivity [23], random forest [24], etc. The screening method is a qualitative analysis method with a small amount of calculation. It is generally used to screen out the parameters that have a weak effect on the model output, to reduce the difficulty of further using other methods, or guide the reasonable selection of sensitivity analysis methods. FAST is a method for analyzing the sensitivity of uncertain systems. It can estimate the expectation and variance of monotone and non-monotone system responses and calculate the influence of a single input parameter on the variance, but it cannot analyze the high-order interaction between input parameters. EFAST can calculate the first-order sensitivity index, the overall sensitivity index, and the high-order interaction between parameters by analyzing the Fourier series spectrum curve. Compared with other methods, the Sobol method [25,26] is a GSA method based on variance decomposition, which has a clear physical meaning and clear calculation principle, is not limited by the model form (linear or nonlinear, monotonic or non-monotonic), can easily quantify the influence of input parameters, coupling between parameters and group input factors on the model output, and has good applicability to the “black box” problem. Therefore, it is more suitable for the sensitivity analysis of complex systems than other methods. However, the evaluation of the sensitivity index for the Sobol method often involves multi-dimensional integration of the model function. Considering the computational complexity, this is usually realized according to sampling analysis in practical application. For CFD simulation of the enhanced heat transfer performance, the calculation time of a single sampling analysis is long, the calculation cost of sensitivity analysis is huge, and it increases exponentially with the increasing CFD model complexity and design variables. Therefore, determining how to improve the solving efficiency of Sobol sensitivity analysis has always been the first problem to be solved in the design of complex engineering problems, such as the heat transfer performance design of large-scale fusion equipment.

Recently, in order to avoid the huge computational cost caused by the direct use of complex simulation models with high fidelity, the response surface method [27,28,29] has been widely used in the field of design optimization. It is a multivariable problem modeling and analysis technology based on the design of experiment (DOE) method. It uses approximate functions with simple and explicit expressions to approximate complex simulation models, simplify the calculation process, and shorten the calculation time on the premise of ensuring accuracy. According to the different forms of the approximation function, response surface methods can be divided into many kinds, such as artificial neural network (ANN), Kriging model, radial basis function (RBF), and polynomial response surface (PRS) model. Response surface methodology provides an effective way to improve the efficiency of the GSA of complex engineering problems [30].

This paper aims to analyze the heat transfer performance of the single-side heated swirl structure with a large length-diameter ratio arranged in the calorimeter in the negative ion NBI system of the China Fusion Experimental Rector (CFETR NNBI system), combining the RPI wall heat flux partitioning model [31,32,33] and subcooled boiling theory and establishing a liquid-vapor two-phase flow model. Then, in order to improve the efficiency of the sensitivity analysis of the heat transfer performance, the polynomial response surface surrogate model of the heat transfer performance function is constructed on the basis of Latin hypercube sampling. The root means square error (RMSE) and the complex correlation coefficient is used to test the error of the obtained surrogate models. On this basis, by applying the proposed surrogate model, the Sobol global sensitivity analysis of the heat transfer performance is carried out, and the sensitivity indexes of each design variable are obtained easily and efficiently. This study will provide important theoretical and application value for the enhanced heat transfer design of single-side heated pipeline components with a large length-diameter ratio in a nuclear fusion device.

2. Swirl Tube Bundles with a Large Length-Diameter Ratio for the CFETR NNBI Calorimeter

The detailed structural design of the CFETR NNBI calorimeter [13] is shown in Figure 1. The swirl tube bundles with a large length-diameter ratio are arranged in parallel. The swirl tube bundle is arranged in the front and back, and the front and back are staggered up and down. The staggered detailed size: 16.5 mm in front and back, 32 mm up and down, and the coincidence rate was 40%, each swirl tube was more than 3 m, the length-diameter ratio was 187.5, and the middle section design of every single tube was a U-shaped transition, which could effectively alleviate the thermal expansion and cold contraction. There are four different supports for the swirl tube model fixed on the support frame, the two ends of the single tube are fixed onto the support of the main pipeline shown as support 1 and support 2, and support 4 links the front and back swirl tubes, the middle section of the U-shaped tube is supported and defined as support 3. The material of the tube was CuCrZr, and the inner diameter of the cooling tubes was 16 mm, with a 2 mm thickness of the inserted stainless steel swirl tape. The high heat fluxes imposed on the inside part of the calorimeter are shown in Figure 2. Deionized water was chosen as the working coolant. The detailed physical parameters are shown in

Table 1. Material physical parameters.

Material Physical Parameter		Value (293 K)	Relational Expression Variation with Temperature
CuCrZr	Specific heat (J/kg·K)	390	$C_P=372.54105+0.0409T+4.2796\times {10}^{-5}{T}^2$
	Density (kg/m3)	8920	-
	Thermal conductivity (W/m·K)	390	$\lambda =421.47246-0.07225T+2.64976\times {10}^{-6}{T}^2$
304 stainless steel	Specific heat (J/kg·K)	470	-
	Density (kg/m3)	7850	-
	Thermal conductivity (W/m·K)	14.28	-
Water	Specific heat (J/kg·K)	4180	$C_p=\mathrm{10,608.87995}-55.7362T+0.15919T^2-1.49398\times {10}^{-4}{T}^3$
	Density (kg/m3)	998.5	$\rho =150.99277+7.54409T-0.02122T^2+1.82226\times {10}^{-5}{T}^3$
	Thermal conductivity (W/m·K)	0.599	$\lambda =-0.51402+0.00532T-3.35719\times {10}^{-6}{T}^2-6.23349\times {10}^9{T}^3$
	Viscosity coefficient (kg/m.s)	0.001006	$\mu =0.11157-9.51523\times {10}^{-4}T+2.7249\times {10}^{-6}{T}^2-2.61107\times {10}^{-9}{T}^3$

. The materials of the tube and swirl tape are considered homogeneous, uniform, and isotropic. Due to the complexity of the calorimeter, a basic large length-diameter ratio swirl tube modular in this HHF calorimeter component will be taken as a study object in the following sections, shown in Figure 2. Figure 2 shows a basic circuit formed by the front and back swirl pipes. The swirl structure is fixed inside the pipe, and the beam bombardment is on the pipe’s outer surface. The deposition power density on the front and back pipe surfaces is the same, and the heat-loaded area was calculated according to the actual beam deposition surface. According to the beam distribution, the whole tube bundle was roughly divided into two parts according to the beam distribution, and the middle part was connected and fixed with a U-shaped structure, which can alleviate the problem of deformation and reduce the temperature rise. Therefore, the heat density on the heat-loaded surface was also divided into two parts, the part near the beam inlet is defined as peak power density 2, which is slightly smaller than the part at the beam outlet (defined as peak power density 1), as shown in Figure 2. Correspondingly, the tape inserted inside was divided into four parts: swirl tape 1 inside (sw-1), swirl tape 2 inside (sw-2), swirl tape 3 inside (sw-3), and swirl tape 4 inside (sw-4).

3. Liquid-Vapor Two-Phase Flow Modeling with Subcooled Boiling for Heat Transfer Performance Analysis

In this section, the liquid-vapor two-phase flow model with subcooled boiling is established for the enhanced heat transfer performance analysis of the NNBI calorimeter swirl tube. The Eulerian multiphase model was chosen for the subcooled-type CHF simulation because of its excellent solving accuracy. The mass continuity equations and momentum equations are solved for the liquid phase and vapor phase, respectively. In order to ensure a fully developed profile of velocity magnitude and turbulence quantities at the inlet, a two-step approach was adopted. The Rensselaer Polytechnic Institute (RPI) model, which was proposed by Podowski and others [31,32,33] was adopted for the subcooled boiling flow.

(1)

Basic conservation equations

The continuity equation, momentum conservation equation, and energy conservation equation of fluid flow in the tube are, respectively:

$$\frac{\partial ({\alpha }_q{\rho }_q)}{\partial t}+\nabla ·({\alpha }_q{\rho }_q{\overrightarrow{v}}_q)=Sq+{\displaystyle \sum _{p=1}^n\left({\dot{m}}_{pq}-{\dot{m}}_{qp}\right)}$$

(1)

$$\begin{array}{l}\frac{\partial ({\alpha }_q{\rho }_q{\overrightarrow{v}}_q)}{\partial t}+\nabla ·({\alpha }_q{\rho }_q{\overrightarrow{v}}_q{\overrightarrow{v}}_q)+{\alpha }_q\nabla p-{\alpha }_q{\rho }_q{\overrightarrow{g}}_q=\nabla \cdot {\overline{\overline{\tau }}}_q+{\displaystyle \sum _{p=1}^n\left({\overrightarrow{R}}_{pq}+{\dot{m}}_{pq}{\overrightarrow{v}}_{pq}-{\dot{m}}_{qp}{\overrightarrow{v}}_{qp}\right)}\\ +({\overrightarrow{F}}_q+{\overrightarrow{F}}_{lift,q}+{\overrightarrow{F}}_{vm,q}+{\overrightarrow{F}}_{td,q}+{\overrightarrow{F}}_{wl,q})\end{array}$$

(2)

$$\frac{\partial }{\partial t}({\alpha }_q{\rho }_q{h}_q)+\nabla \cdot ({\alpha }_q{\rho }_q{h}_q{\overrightarrow{v}}_q)+{\alpha }_q\frac{\partial p}{\partial t}+\nabla \cdot {\overrightarrow{q}}_q={\overline{\overline{\tau }}}_q\cdot \nabla {\overrightarrow{v}}_q+{\displaystyle \sum _{p=1}^n\left(Q_{pq}+{\dot{m}}_{pq}{h}_{pq}-{\dot{m}}_{qp}{h}_{qp}\right)}+S_q$$

(3)

where the subscript $q$ represents the $q\mathrm{th}$ phase (liquid or gas), and the subscript $pq$ represents the value of interphase. On the right–hand side of Equation (1) is the mass transfer of interphase, ${\dot{m}}_{pq}$ characterizes the mass transfer from phase $p\mathrm{th}$ to phase $q\mathrm{th}$, and ${\dot{m}}_{qp}$ characterizes the mass transfer from phase $q\mathrm{th}$ to phase $p\mathrm{th}$, and the $S_q$ is the source term, ${\overrightarrow{v}}_q$ is the velocity of the phase $q$, and ${\alpha }_q$ is the phasic volume fraction. Additionally, ${\overrightarrow{R}}_{pq}$ is an interaction force between phases, $P$ is the pressure shared by all phases, ${\overrightarrow{F}}_q$ is an external body force, ${\overrightarrow{F}}_{td,q}$ is a turbulent dispersion force (in the case of turbulent flows only, Burns [34]), ${\overrightarrow{F}}_{lift,q}$ is a lift force [35], ${\overrightarrow{F}}_{vm,q}$ is a virtual mass force, ${\overrightarrow{F}}_{wl,q}$ is a wall lubrication force, and ${\overline{\overline{\tau }}}_q$ is the $q\mathrm{th}$ phase stress-strain tensor. Lastly, $h_q$ is the local enthalpy of the $q\mathrm{th}$ phase, $h_{pq}$ is the interphase enthalpy, $Q_{pq}$ is the intensity of heat exchange between $q\mathrm{th}$ and $p\mathrm{th}$ phases, ${\dot{m}}_{pq}{h}_{pq}$ and ${\dot{m}}_{qp}{h}_{qp}$ represents the interphase energy transfer caused by the interphase transfer, $q_q$ is the heat flux, and $N{u}_p$ is expressed by the Ranz–Marshall [36,37] relationship.

$$\begin{array}{l}{\overline{\overline{\tau }}}_q={\alpha }_q{\mu }_q\left(\nabla {\overrightarrow{v}}_q+\nabla {\overrightarrow{v}}_q^{\mathrm{T}}\right)+a_q\left({\lambda }_q^{}-\frac{2}{3}{\mu }_q\right)\nabla \cdot {\overrightarrow{v}}_q\overline{\overline{I}},Q_{pq}=h_{pq}(T_p-T_q),\begin{array}{c}\end{array}{h}_{pq}=\frac{6k_q{a}_q{a}_{p}N{u}_p}{d_p^2}\\ {\overrightarrow{F}}_{lift,q}=-C_1{\rho }_q{\alpha }_p\left({\overrightarrow{v}}_q-{\overrightarrow{v}}_p\right)\times (\nabla \times {\overrightarrow{v}}_q),\begin{array}{c}\end{array}{\overrightarrow{F}}_{vm,q}=0.5{\rho }_q{\alpha }_p\left(\frac{d_q{\overrightarrow{v}}_q}{dt}-\frac{d_p{\overrightarrow{v}}_p}{dt}\right)\\ {\overrightarrow{F}}_{td,q}=C_{td}{K}_{pq}\frac{D_p}{{\sigma }_{pq}}\left(\frac{\nabla {\alpha }_p}{{\alpha }_p}-\frac{\nabla {\alpha }_q}{{\alpha }_q}\right),{\overrightarrow{F}}_{wl,q}=C_{wl}{\rho }_q{\alpha }_p{\left|\left({\overrightarrow{v}}_q-{\overrightarrow{v}}_p\right)\left.{}_{\parallel }\right|\right.}^2{\overrightarrow{n}}_w,\begin{array}{c}\end{array}N{u}_p=2.0+0.6R{e}_p^{1/2}P{r}^{1/3}\\ {\overrightarrow{F}}_{lift,q}=-{\overrightarrow{F}}_{lift,p},{\overrightarrow{F}}_{td,q}=-{\overrightarrow{F}}_{td,p},{\overrightarrow{R}}_{pq}=-{\overrightarrow{R}}_{qp},{\overrightarrow{R}}_{qq}=0,Q_{pq}=-Q_{qp},Q_{qq}=0\end{array}$$

(4)

In Equation (4), $C_{wl}$ is the wall lubrication coefficient (Antal Model [32]), $C_1$ is the lift coefficient (Moraga [35]), ${\rho }_q$ is the primary phase density, ${\alpha }_p$ is the secondary phase volume fraction, $\left|\left({\overrightarrow{v}}_q-{\overrightarrow{v}}_p\right)\left.{}_{\parallel }\right|\right.$ is the phase relative velocity component tangential to the wall surface, ${\overrightarrow{n}}_w$ is the unit normal pointing away from the wall, and $C{p}_q,k_q,{\lambda }_q,{\mu }_q$ are the specific heat capacity, conductivity, shear, and bulk viscosity of the phase $q$. $C_{td}=1,{\sigma }_{pq}=0.9$, $K_{pq}$ is the covariance of the velocities of the continuous phase $q$ and the dispersed phase $p$, the dispersion scalar $D_p$ is estimated by the turbulent viscosity ${\mu }_{tq}$ of the continuous phase.

$$D_p=D_q=D_{tq}={\mu }_{tq}/{\rho }_q$$

(5)

(2)

Wall boiling model

The RPI model was adopted, and the total heat flux $q_{\mathrm{W}}^{″}$ from the wall to the liquid is partitioned into three parts, including the convective heat flux $q_{\mathrm{C}}^{″}$, the quenching heat flux $q_{\mathrm{Q}}^{″}$ (which represents the cyclic averaged transient energy transfer related to liquid filling the wall vicinity after bubble detachment), and the evaporative one $q_{\mathrm{E}}^{″}$.

$$q_{\mathrm{W}}^{″}=q_{\mathrm{C}}^{″}+q_{\mathrm{E}}^{″}+q_{\mathrm{Q}}^{″}$$

(6)

$$q_{\mathrm{C}}^{″}=h_c(T_w-T_l)(1-A_b), q_{\mathrm{E}}^{″}=\frac{\pi }{6}{d}_{vw}^{3}f{n}^{\prime }{\rho }_V{h}_{fv}, q_{\mathrm{Q}}^{″}=\frac{2}{\sqrt{\pi }}\sqrt{k_l{\rho }_l{C}_{pl}}f(T_w-T_l)$$

(7)

where $h_{\mathrm{c}}$ is the single-phase heat transfer coefficient (Kader, 1981 and Egorov, 2004) [38,39], $T_{\mathrm{w}}$ is the wall temperatures, $T_{\mathrm{l}}$ is the liquid temperatures, $A_b$ is the area covered by nucleating bubbles, $h_{fv}$ is the latent heat of evaporation, $d_{vw}$ is the bubble departure diameter, and $f$ is the bubble departure frequency. In Equation (7), $n^{\prime }$ is the active nucleate sites density, which indicates the number of nucleate sites per unit area and is related to surface roughness and wettability between the liquid phase and the wall [40]. Additionally, ${\rho }_l$ and $k_l$ is the liquid density and the conductivity within the control volume, ${\rho }_v$ is the vapor density within the control volume, and $C_{pl}$ is the specific heat of liquid at constant pressure.

$$\begin{array}{l}{A}_b=\min (1, K\frac{n^{\prime }\pi d_{vw2}}{4}), d_{vw}=\min [0.0014, 0.0006 \exp (-\Delta T_{sub}/45.0)], \Delta T_{sub}=T_{sat}-T_l\\ f=\sqrt{\frac{4g({\rho }_l-{\rho }_v)}{3d_{vw}{\rho }_l}}, K=4.8e^{(\frac{-J{a}_{sub}}{80})}, J{a}_{sub}=\frac{{\rho }_l{C}_{pl}△T_{sub}}{{\rho }_v{h}_{\mathit{fv}}}, n^{\prime }={[210(T_w-T_s)]}^{1.805}\end{array}$$

(8)

where $g$ is the gravity and $T_{\mathit{sat}}$ is the saturation temperature at the working pressure.

(3)

The turbulence model

Considering the flow characteristics of the two-phase fluid inside the swirl tube, a mixture realizable k-epsilon turbulence model was adopted. Compared with the standard k-epsilon model, the form of the epsilon equation is quite different [41] and considers the influence of vortex fluctuation, so it is more helpful to improve the accuracy of numerical simulation. The modeled transport equations in the realizable model are:

$$\frac{\partial ({\rho }_{m}k)}{\partial t}+\nabla \cdot ({\rho }_{m}k{\overrightarrow{v}}_m)=\nabla \cdot (({\mu }_{\mathrm{m}}+\frac{{\mu }_{t,\mathrm{m}}}{{\sigma }_k})\nabla k)+G_{k,m}-{\rho }_m\epsilon +S_{k,m}+G_{b,m}$$

(9)

$$\frac{\partial ({\rho }_m\epsilon )}{\partial t}+\nabla \cdot ({\rho }_m\epsilon {\overrightarrow{v}}_m)=\nabla \cdot (({\mu }_{\mathrm{m}}+\frac{{\mu }_{t,m}}{{\sigma }_{\epsilon }})\nabla \epsilon )+{\rho }_m{C}_{1}S\epsilon -C_2{\rho }_m\frac{{\epsilon }^2}{k+\sqrt{V_m\epsilon }}+S_{\epsilon ,m}$$

(10)

where $C_2$ is a constant, which was set as 1.9, $S_{k,m}$ and $S_{\epsilon ,m}$ are the user-defined source terms, $G_{k,m}$ is the generation of turbulence kinetic energy due to the mean velocity gradients, $G_{b,m}$ is the generation of turbulence kinetic energy due to buoyancy, ${\sigma }_k$ and ${\sigma }_{\epsilon }$ are the turbulent Prandtl numbers for $k$ and $\epsilon$, which were set as 1.0 and 1.2, respectively. In addition, ${\rho }_m$, ${\mu }_m$ and ${\overrightarrow{v}}_m$ are the mixture density, mixture molecular viscosity, and mixture velocity, respectively, and ${\mu }_{t,\mathrm{m}}$ is the turbulent viscosity of the mixture.

$$\begin{array}{l}{G}_{k,m}=-{\rho }_m\overline{v_i{v}_j}\frac{\partial v_j}{\partial x_i},G_{b,m}=\beta g_i\frac{{\mu }_{t,m}}{\mathrm{Pr}t}\frac{\partial T}{\partial x_i},C_1=\max [0.43,\frac{\eta }{\eta +5}]\\ \eta =Sk/\epsilon ,S=\sqrt{2S_{ij}{S}_{ij}},{\rho }_m={\displaystyle \sum _{p=1}^N{\alpha }_p{\rho }_p,}{\mu }_m={\displaystyle \sum _{p=1}^N{\alpha }_p{\mu }_p}{v}_{i,m}=\frac{{\displaystyle \sum _{p=1}^N{\alpha }_p{\rho }_p{v}_{i,p}}}{{\displaystyle \sum _{p=1}^N{\alpha }_p{\rho }_p}}\\ G_{k,m}={\mu }_{t,m}(\nabla {\overrightarrow{v}}_m+{\left(\nabla {\overrightarrow{v}}_m\right)}^T):\nabla {\overrightarrow{v}}_m,{\mu }_{t,m}={\rho }_m{C}_{\mu }\frac{k^2}{\epsilon }\end{array}$$

(11)

where ${\alpha }_p,{\rho }_p,{\mu }_p,v_{i,p}$ are the volume fraction, density, viscosity, and velocity of the $i\mathrm{th}$ phase, respectively, $C_{\mu }$ is a function of the mean strain and rotation rates, the angular velocity of the system rotation, and the turbulence fields, $\mathrm{Pr}t$ is the turbulent Prandtl number for energy, $g_i$ is the component of the gravitational vector in the i direction, and $\beta$ is the coefficient of thermal expansion.

$$\begin{gathered} C_{\mu }=\frac{1}{A_0+A_s{U}^{*}k/\epsilon }, U^{*}=\sqrt{S_{ij}{S}_{ij}+{\tilde{\Omega }}_{ij}{\tilde{\Omega }}_{ij}} \\ {\tilde{\Omega }}_{\mathit{ij}}={\Omega }_{\mathit{ij}}-2{\epsilon }_{\mathit{ijk}}{\omega }_k, {\Omega }_{\mathit{ij}}={\overline{\Omega }}_{\mathit{ij}}-{\epsilon }_{\mathit{ijk}}{\omega }_k, \beta =-\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial T}\right)}_p \end{gathered}$$

(12)

where ${\overline{\Omega }}_{\mathit{ij}}$ is the mean rate of rotation tensor viewed in a rotating reference frame with the angular velocity ${\omega }_k$.

$$A_0=4.04, A_s=\sqrt{6}\cos \varphi , \varphi =\frac{1}{3}\mathrm{across}\left(\sqrt{6}W\right),W=\frac{S_{ij}{S}_{ik}{S}_{kj}}{\tilde{S}}, S_{ij}=\frac{1}{2}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)$$

(13)

The whole calculation process is shown in Figure 3. (1) Firstly, the control equations are established, and the boundary conditions and initial conditions are determined according to the actual operation condition. (2) The model is meshed, and then the control equations, initial conditions, and boundary conditions are discretized. The control parameters in the solution process are set to detect the convergence of the results. (3) Using the Eulerian model of heterogeneous flow, taking the fluid as the continuous phase and the gas as the dispersed phase, the momentum and energy control equations are solved, and the relevant empirical formulas of the RPI-related boiling model are coupled to calculate the interphase mass and energy transfer. According to the empirical formula, it is judged whether the fluid temperature near the wall is boiling or not. If yes, the gas phase fraction generated by boiling should be calculated, and the relevant physical parameters need to be updated accordingly. Next, it is judged whether the calculation process has converged and, if yes, the calculation is completed, and the calculation result is output. If it does not converge, one should return to solve the governing equation again, and the solution iterates until the calculation parameters meet the convergence definition, and then the calculation of the current time step is completed, and the simulation of the next time step can be performed until the end of the simulation time.

4. Global Sensitivity Analysis for Enhanced Heat Transfer Performance

A large number of simulations and experimental tests have shown that different design parameters have a great effect on the heat transfer performance of the HHF components. To efficiently analyze the effect of various parameters on heat transfer performance, the Sobol sensitivity analysis method based on the polynomial response surface surrogate model was proposed.

4.1. Construction of the Polynomial Response Surface Surrogate Model

In this section, by taking the maximum temperature of the whole structure as the objective function, and the cooling water inlet flow rate, tape width, tape length, and twist length as the design variables, a polynomial response surface model is established for the subsequent sensitivity analysis of design variables. The polynomial response surface method [42,43,44] is essentially a multiple linear regression analysis model, which uses polynomials of different orders to express the relationship between input variables (design variables), $x={[x_1,x_2,\dots ,x_{N_D}]}^{\mathrm{T}}$, and output responses (target values), $y(x)$, in complex systems [42,43,44], that is:

$$y(x)=\tilde{y}(x)+\epsilon ={\displaystyle \sum _{i=0}^{N_P-1}{\beta }_i{f}_i(x)}+\epsilon$$

(14)

where, $\tilde{y}(x)$ is the approximate value of the output response (maximum temperature) $y(x)$, $f_i(x)$ is the polynomial function of the input variable (design variable) $x$, $(i=0,1,\cdots ,N_P-1)$ is the undetermined coefficient, $\epsilon$ is the error term, $N_D$ and $N_P$ are the number of design variables and the number of basis functions $f_i(x)$, respectively. The number of undetermined coefficients in the polynomial response surface method is closely related to the order of polynomials and the number of design variables.

To determine the coefficients ${\beta }_i$ of the polynomials, at least $N_S\ge N_P$ independent tests are required to obtain $N_S$ group sample data. The basis of the high-precision response surface model is an experiment of design, which is used to determine the location of sampling points in the sampling space of input variables to effectively include the information of the sampling space with as few sample points as possible to ensure the accuracy of the response surface. The Latin hypercube method is used for sampling within the value range of each design parameter. The Latin hypercube method adopts the principle of equal probability random orthogonal distribution. For any design variable $x_i(i=1,2,\cdots ,N_D)$, its value interval $\left[{\underset{¯}{x}}_i,{\overline{x}}_i\right]$ is divided into $N_S$ non-overlapping subintervals according to the principle of equal probability. A value is randomly selected and randomly sorted in each subinterval, and a sequence $S_i=\left(x_i^{(1)},x_i^{(2)},\cdots ,x_i^{(N_S)}\right)$ $(i=1,2,\cdots ,N_D)$ about $x_i$ is obtained. The column vectors after combining each $S_i$ is the $x_S^{(i)}={\left[x_1^{(i)},x_2^{(i)},\cdots ,x_{N_D}^{(i)}\right]}^{\mathrm{T}}(i=1,2,\cdots ,N_S)$. The corresponding output response (maximum temperature) $y(x_S^{(i)})$ of each $x_S^{(i)}={\left[x_1^{(i)},x_2^{(i)},\cdots ,x_{N_D}^{(i)}\right]}^{\mathrm{T}}(i=1,2,\cdots ,N_S)$ can be obtained using the simulation in Section 3.

For the above sample points, according to Equation (14), the error between the predicted value of the response surface and the true value can be expressed as [42]:

$$\mathsf{\epsilon }(\mathsf{\beta })=\tilde{y}-y_S=F_S\mathsf{\beta }-y_S$$

(15)

where

$$F_S=\left[\begin{array}{cccc}{f}_0(x_S^{(1)})& f_1(x_S^{(1)})& \cdots & f_{N_P-1}(x_S^{(1)})\\ f_0(x_S^{(2)})& f_1(x_S^{(2)})& \cdots & f_{N_P-1}(x_S^{(2)})\\ ⋮& ⋮& \ddots & ⋮\\ f_0(x_S^{(N_S)})& f_1(x_S^{(N_S)})& \cdots & f_{N_P-1}(x_S^{(N_S)})\end{array}\right],y_S=\left[\begin{array}{c}{y}_S^{(1)}\\ y_S^{(2)}\\ ⋮\\ y_S^{(N_S)}\end{array}\right],\mathsf{\beta }=\left[\begin{array}{c}{\beta }_0\\ {\beta }_1\\ ⋮\\ {\beta }_{N_P-1}\end{array}\right]$$

(16)

Due to $N_S\ge N_P$, Equation (15) has a unique least squares solution [42,43,44]. Minimize the sum of squares of errors:

$$S(\mathsf{\beta })={\left(F_S\mathsf{\beta }-y_S\right)}^{\mathrm{T}}\left(F_S\mathsf{\beta }-y_S\right)\to \min$$

(17)

The necessary condition for obtaining the minimum value of Equation (17) is:

$$\nabla S(\mathsf{\beta })=\nabla \left({\left(F_S\mathsf{\beta }-y_S\right)}^{\mathrm{T}}\left(F_S\mathsf{\beta }-y_S\right)\right)=2{\left(F_S\mathsf{\beta }-y_S\right)}^{\mathrm{T}}F=0$$

(18)

Thus, one can obtain [42,43,44]:

$$\mathsf{\beta }={\left(F_S^{\mathrm{T}}{F}_S\right)}^{-1}{F}_S^{\mathrm{T}}{y}_S$$

(19)

In the process of constructing an approximate model based on sample points, it is necessary to judge whether the approximate model meets the required accuracy requirements. If it meets the requirements, the approximate model can be used instead of the real physical model. Otherwise, it is necessary to add more sampling points or adjust the parameters of the approximate model to improve its predicted accuracy. In this paper, the mean relative error (MRE) and the complex correlation coefficient (R squared, $R^2$) are used to evaluate the accuracy of the approximate model [42,43,44].

$$MRE=\frac{1}{N_S}{\displaystyle \sum _{i=1}^{N_S}\left|\frac{y(x)-\tilde{y}(x)}{y(x)}\right|}, R^2=1-\frac{{\displaystyle \sum _{i=1}^{N_S}{\left(y(x)-\tilde{y}(x)\right)}^2}}{{\displaystyle \sum _{i=1}^{N_S}{\left(y(x)-\overline{y}\right)}^2}}$$

(20)

where $\overline{y}$ is the approximate mean value of sample points, $\overline{y}=\frac{1}{N_S}{\displaystyle \sum _{i=1}^{N_S}\tilde{y}(x)}$.

4.2. Sobol Sensitivity Algorithm Based on the Response Surface Surrogate Model

In this section, based on the response surface model of heat exchange performance, the Sobol method [25, 26,43,44] is further applied to obtain the sensitivity of design parameters to heat exchange performance. The Sobol method decomposes the variance of the output variables into sub-variance terms with different dimensions, which represent the contribution of each single input parameter and the coupling between these input parameters to the model output. By calculating the proportion of different sub-variance contributions in the total variance of the model output, sensitivity indices with different orders can be obtained to directly describe the effect of different input parameters and the coupling between these parameters on the model output.

It is assumed that the normalized design variable $x_i(0\le x_i\le 1,i=1,2,\cdots ,N_D)$ in $x={[x_1,x_2,\dots ,x_{N_D}]}^{\mathrm{T}}\in R^{N_D}$ is an independent random variable with consistent joint probability density, and the output $y=f\left(x\right)$ is square integrable, which can be decomposed into [25, 26,43,44]:

$$y=f(x)=f_0+{\displaystyle \sum _i{f}_i\left(x_i\right)}+{\displaystyle \sum _{1\le i<j\le N_D}{f}_{i,j}(x_i,x_j)}+\cdots +f_{1,2,3,\cdots ,N_D}(x_1,x_2,x_3,\cdots ,x_{N_D})$$

(21)

where $f_0$ is a constant, and the value of each item in Equation (21) can be calculated by the following integral function:

$$f_0={\displaystyle {\mathbf{\int }}_{R^{N_D}}f\left(x\right)}dx, f_i(x_i)={\displaystyle \int f(x){\displaystyle \prod _{s\ne i}d{x}_s}}-f_0, f_{i,j}(x_i,x_j)={\displaystyle \int f(x){\displaystyle \prod _{s\ne i,j}d{x}_s}}-f_0-f_i(x_i)-f_j(x_j)$$

(22)

And the like, other higher-order terms of Equation (21) can be obtained. Considering the orthogonality between the terms of Equation (21), by squaring both sides of Equation (21) and integrating them into the whole definition domain, one can obtain [25, 26,43,44]:

$${\displaystyle {\int }_{R^{N_D}}{f}^2(x)dx}-f_0^2={\displaystyle \sum _{s=1}^{N_D}{\displaystyle \sum _{1\le i_1<i_s\le N_D}{\displaystyle \int f_{i_1,i_2\cdots ,i_s}^2(x_{i_1},x_{i_2}\cdots ,x_{i_s})d{x}_{i_1}\cdots d{x}_{i_s}}}}$$

(23)

The variance $D$ and $D_{i_1,i_2,\cdots ,i_s}$ of $f\left(x\right)$ and $f_{i_1,i_2,\cdots ,i_s}(x_{i_1},x_{i_2}\cdots ,x_{i_s})$ are defined as:

$$D={\displaystyle {\int }_{I^k}{f}^2(x)dx}-f_0^2{D}_{i_1,i_2,\cdots ,i_s}={\displaystyle \int f_{i_1,i_2\cdots ,i_s}^2(x_{i_1},x_{i_2}\cdots ,x_{i_s})d{x}_{i_1}\cdots d{x}_{i_s}}$$

(24)

Thus,

$$D={\displaystyle \sum _{i=1}^{N_D}{D}_i}+{\displaystyle \sum _{1\le i<j\le N_D}{D}_{i,j}}+\cdots +D_{1,2,3,\cdots ,N_D}$$

(25)

Define the ratio of variance [25, 26,43,44]:

$$S_{i_1,i_2,\cdots ,i_s}=\frac{D_{i_1,i_2,\cdots ,i_s}}{D}\hspace{1em}(1\le i_1<\cdots <i_s\le N_D)$$

(26)

as the sensitivity of measuring the effect of input parameters, and it can be seen from Equation (25).

$${\displaystyle \sum _{i=1}^{N_D}{S}_i}+{\displaystyle \sum _{1\le i<j\le N_D}{S}_{i,j}}+\cdots +S_{1,2,\cdots ,N_D}=1$$

(27)

where $S_i$ is the “main effect” index or first-order sensitivity of the parameter $x_i$, which describes the contribution of random factors $x_i$ on the total variance of the output by oneself; $S_{i_1,i_2,\cdots ,i_s}$ is the $s\mathit{th}$ order sensitivity, which describes the contribution of the interaction of random factors $x_{i_1},x_{i_2}\cdots ,x_{i_s}$ to the total variance of the output.

$$S_i^{\mathrm{T}}=1-S_{~i}$$

(28)

where $S_{~i}$ refers to the influence of all factors except factor $x_i$ on the output.

The above Sobol sensitivity index can be obtained by the Monte Carlo integration method [25, 26,43,44]. Let matrix $A$ and matrix $B$ be $N_S\times N_D$ dimensional sampling arrays, and each row in the two matrices is a set of specific input combinations of the analysis function.

$$A=\left[\begin{array}{cccc}{x}_{1,1}& x_{1,2}& \cdots & x_{1,N_D}\\ x_{2,1}& x_{2,2}& \cdots & x_{2,N_D}\\ ⋮& ⋮& \ddots & ⋮\\ x_{N_S,1}& x_{N_S,2}& \cdots & x_{N_S,N_D}\end{array}\right]B=\left[\begin{array}{cccc}{x}_{1,1}^{\prime }& x_{1,2}^{\prime }& \cdots & x_{1,N_D}^{\prime }\\ x_{2,1}^{\prime }& x_{2,2}^{\prime }& \cdots & x_{2,N_D}^{\prime }\\ ⋮& ⋮& \ddots & ⋮\\ x_{N_S,1}^{\prime }& x_{N_S,2}^{\prime }& \cdots & x_{N_S,N_D}^{\prime }\end{array}\right]$$

(29)

The matrix $C_i$ is obtained by replacing the $i\mathit{th}$ column of the matrix $B$ with the $i\mathit{th}$ column of the matrix $A$; $C_{~i}$ is the matrix obtained by replacing the $i\mathit{th}$ column of the matrix $A$ with the $i\mathrm{th}$ column of the matrix $B$.

$$C_i=\left[\begin{array}{cc}\begin{array}{cccc}{x}_{1,1}^{\prime }& x_{1,2}^{\prime }& \cdots & x_{1,i-1}^{\prime }\\ x_{2,1}^{\prime }& x_{2,2}^{\prime }& \cdots & x_{2,i-1}^{\prime }\\ ⋮& ⋮& ⋮\\ x_{N_S,1}^{\prime }& x_{N_S,2}^{\prime }& \cdots & x_{N_S,i-1}^{\prime }\end{array}& \begin{array}{cccc}{x}_{1,i}& x_{1,i+1}^{\prime }& \cdots & x_{1,k}^{\prime }\\ x_{2,i}& x_{2,i+1}^{\prime }& \cdots & x_{2,k}^{\prime }\\ ⋮& ⋮& ⋮\\ x_{N_S,i}& x_{N_S,i+1}^{\prime }& \cdots & x_{N_S,N_D}^{\prime }\end{array}\end{array}\right]$$

$$C_{~i}=\left[\begin{array}{cc}\begin{array}{cccc}{x}_{1,1}& x_{1,2}& \cdots & x_{1,i-1}\\ x_{2,1}& x_{2,2}& \cdots & x_{2,i-1}\\ ⋮& ⋮& ⋮\\ x_{N_S,1}& x_{N_S,2}& \cdots & x_{N_S,i-1}\end{array}& \begin{array}{cccc}{x}_{1,i}^{\prime }& x_{1,i+1}& \cdots & x_{1,N_D}\\ x_{2,i}^{\prime }& x_{2,i+1}& \cdots & x_{2,N_D}\\ ⋮& ⋮& ⋮\\ x_{N_S,i}^{\prime }& x_{N_S,i+1}& \cdots & x_{N_S,N_D}\end{array}\end{array}\right]$$

(30)

Similarly, $C_{i,j}$, $C_{~i,~j}$, etc., can be defined. By introducing these input matrices into the polynomial surrogate model, the corresponding outputs can be obtained. Note that, $y_A$, $y_B$, and $y_C$ are the output vectors corresponding to the matrices $A$, $B$, and $C_i$, respectively. The following estimates can be obtained from the Monte Carlo algorithm:

$${\widehat{f}}_0^2=\frac{1}{N_S}{y}_A^{\mathrm{T}}{y}_B, \widehat{D}=\frac{1}{N_S}{y}_A^{\mathrm{T}}\left(y_A-y_B\right), {\widehat{D}}_i=\frac{1}{N_S}{y}_A^{\mathrm{T}}{y}_{C_i}-{\widehat{f}}_0^2, {\widehat{D}}_{~i}=\frac{1}{N_S}{y}_A^{\mathrm{T}}{y}_{C{~}_i}-{\widehat{f}}_0^2$$

(31)

Then the first-order sensitivity $S_i$ and total sensitivity $S_i^{\mathrm{T}}$ of the input $x_i$ can be estimated as:

$${\widehat{S}}_i=\frac{{\widehat{D}}_i}{\widehat{D}},\hspace{1em}{\widehat{S}}_i^{\mathrm{T}}=1-\frac{{\widehat{D}}_{~i}}{\widehat{D}}$$

(32)

Considering the high computational cost of the liquid-vapor two-phase flow model in Section 3, in order to realize the rapid evaluation of the parameter sensitivity on the enhanced heat transfer performance of the complex swirl flow structure and greatly improve the solution efficiency, this study proposed the use of the polynomial response surface to replace the actual complex CFD model to solve the Sobol sensitivity, which can easily and efficiently obtain the first-order and total global sensitivity indices of each design variable. The calculation flow of sensitivity analysis in this paper is shown in Figure 4.

5. Numerical Simulation and Analysis

5.1. Liquid-Vapor Two-Phase Flow Simulation and Enhanced Heat Transfer Performance Analysis

Based on the above liquid-vapor two-phase flow model and numerical solution method, the enhanced heat transfer performance of the swirl tube structure with a large length diameter ratio is presented in this section. As described in Section 2, the basic structural parameters for simulation analysis are shown in

Table 2. Base structural parameters for parametric analysis.

Parameter	Tape Width (m)	Twist Length (m)	Tape Length (m)	Diameter/Tape Height (m)	Inlet Temperature (K)
sw-1	0.002	0.056	1.44	0.016	305
sw-2	0.002	0.056	1.56
sw-3	0.002	0.056	1.56
sw-4	0.002	0.056	1.36

, including tape width, twist length (twist ratio), and tape length inserted in four different areas.

The basic operating conditions included an inlet pressure of 2.0 MPa, an inlet temperature of 305 K, an inlet velocity of 6 m/s, and a volume fraction of water vapor of 0. Additionally, the outlet of the tube was set as the pressure boundary condition, the peak power density 1 was 5.5 MW/m², the peak power density 2 was 5.2 MW/m², and the heating pulse length was 100s, except that the above heating surface was a constant heat source, other surfaces were set as thermal insulation surfaces. Based on the above liquid-vapor two-phase flow model and numerical solution method, this section analyzes the thermal-hydraulic performance of the swirl tube structure with a large length-diameter ratio. In the simulation, the fluid region and solid region mesh with Poly-Hexcore (as shown in Figure 5), the standard wall function for the near wall surface was adopted, and the five expansion layers were arranged in the fluid part of the fluid-solid coupling section. A grid independence test was carried out first for different grid levels. When the number of grids was higher than 4.56 million, the error of numerical simulation results was within 5%.

The velocity profile of the front swirl tube is shown in Figure 6, the temperature profiles of the front and back swirl tubes are shown in Figure 7, and the distribution of the void fraction inside the swirl tube under one-side heating is shown in Figure 8. From Figure 6, Figure 7 and Figure 8, one can find that the boundary layer was greatly disturbed by the addition of swirl tapes, the fluid velocity in the pipe where the swirl tape structure was arranged was effectively increased, and the maximum velocity of the fluid was up to 10.8 m/s. For the one-side heating tube, the maximum temperature appeared in the heat-loaded surface, responding to the peak power density 2 area in Figure 2, and the value was 606.7 K, which is far lower than the softening temperature of CuCrZr (823 K). The maximum temperature of the back swirl tube was 578.3 K, much lower than that of the front one, and the difference value was about 28.4 K. The total temperature rising of the cooling water was about 18.1 K. Along the direction of water flow, the water temperature rose gradually, and in the four straight pipe sections, the wall temperature of the heat-loaded surface was usually higher at the end of the swirl tape. Among them, the wall temperature at the inner side near the beam changed greatly, and the corresponding water temperature near the heat-loaded surface reached 578.3 K, far exceeding 485.55 K at the working pressure of 2 MPa. In addition, the maximum value of the corresponding heat transfer coefficient was about 41,100 W/(m²·K), and a large number of bubbles appeared in the front rows, while the maximum section void fraction was 0.039% and appeared at a distance of about 0.1 m from the entrance of the calorimeter structure. When the length of the swirl tape was shorter than the heat-loaded surface, bubbles appeared on the heat-loaded surface without the swirl tape structure, and the corresponding pressure drop of the whole circle was about 1.38 MPa.

5.2. Sensitivity Analysis of Design Parameters

Setting the parameters shown in

Table 3. Design parameters and their value ranges.

Parameter		Lower Bound	Upper Bound
sw-1	Tape width/P11	0.0015	0.0025
	Tape length/P12	1.4	1.6
	Twist length/P13	0.02	0.1
sw-2	Tape width/P21	0.0015	0.0025
	Tape length/P22	1.36	1.56
	Twist length/P23	0.02	0.1
sw-3	Tape width/P31	0.0015	0.0025
	Tape length/P32	1.36	1.56
	Twist length/P33	0.02	0.1
sw-4	Tape width/P41	0.0015	0.0025
	Tape length/P42	1.16	1.36
	Twist length/P43	0.02	0.1
Inlet temperature/P5		285	320

as the design variables and the maximum temperature of the structure as the objective function, 60 groups of design sample points were selected in the given design space, and the corresponding maximum temperatures were obtained by the CFD simulation proposed in Section 3. Then the polynomial response surface model between the design variables and the maximum temperature was established, and its accuracy was tested according to Equation (20). Based on the polynomial response surface model, the proposed Sobol method was used to solve the first-order sensitivity coefficient and the total sensitivity coefficient of each design variable.

The calculation results of the Sobol global sensitivity analysis have a great relationship with the sampling size in Equation (29). When the sampling size is small, it is difficult to reflect the real effect of parameter changes on the model output. Here the Latin hypercube sampling method was used to obtain the parameter samples. The first-order sensitivity of some design parameters to the maximum temperature under different sampling sizes is shown in Figure 9. The first-order sensitivity data tends to be stable with the increase in sampling size. The sensitivity analysis of the design parameters to the enhanced heat transfer performance was carried out based on the calculation results with a sampling number of 200,000. The calculation results are shown in

Table 4. Sensitivity indices of different design parameters.

Parameter		First Order Global Sensitivity	Total Global Sensitivity
sw-1	Tape width/P11	0.00486	0.01024
	Tape length/P12	0.42776	0.45768
	Twist length/P13	0.23414	0.25679
sw-2	Tape width/P21	0.00022	0.01162
	Tape length/P22	0.00175	0.02567
	Twist length/P23	0.00534	0.00851
sw-3	Tape width/P31	0.0186	0.03174
	Tape length/P32	0.03745	0.06278
	Twist length/P33	0.00474	0.04036
sw-4	Tape width/P41	0.00811	0.01948
	Tape length/P42	0.00405	0.02474
	Twist length/P43	0.0048	0.01612
Inlet temperature/P5		0.13035	0.15391

and Figure 10, respectively.

In Table 4 and Figure 10, the first-order global sensitivity reflects the influence of the variable itself on the maximum temperature, and the total global sensitivity not only reflects the influence of the variable itself on the maximum temperature but also reflects the influence of the interaction between these variable and other variables. If the first-order and the total global sensitivity of a variable differ greatly, there is an interaction between the variable with other variables. It is obvious that the sensitivity of the maximum temperature to variables P12, P13, P5, and P32 was relatively large, while the sensitivity of the maximum temperature to variables P11, P21, P22, P23, P31, P33, P41, P42, and P43 was relatively small. Therefore, in practical engineering, when designing process parameters, we should focus more on the influence of variables P12, P13, P5, and P32 on the maximum temperature. Next, taking variables P12 and P5 as an example, the influence of design variables on heat exchange performance is further shown.

(1)

The influence of tape length P12

According to the actual processing technology, the maximum length of the swirl tape inside the tubes is the length of the heat-loaded surface. If the length of the tape is too long and exceeds the length of the heat-loaded surface, the tape will deviate from the axis and bend, resulting in an uneven local flow. According to the change of heat-loaded surface, the length of the front and back and inner and outer four zones were slightly different. The maximum length of the tape length corresponds to the length of the heat-loaded surface: 1.6 m/1.56 m/1.56 m/1.36 m. In order to analyze the influence of different tape lengths on the heat transfer performance, three values of 1.44 m, 1.5 m, and 1.6 m for sw-1 were selected, and the other three tape lengths remained unchanged. The heat transfer performance of the swirl tube was more obvious than the smooth tube, as shown in Figure 11 and Figure 12. When the tape length of the sw-1 became larger and approached the length of the heat-loaded surface, the heat transfer performance became better at the position where the tape was added, and the overall surface temperature decreased (From 606.7 K to 567 K). Compared with the temperature field in Figure 7, the overheating and localized bubbles no longer occurred at the entrance of the structure, and the corresponding maximum bubble volume fraction decreased significantly (3.96 × 10⁻⁴, 2.8 × 10⁻⁴, 1.92 × 10⁻⁴). In addition, since only the length of the swirl tape in the tube was changed, the flow velocity in the tube was unchanged (still around 10.8 m/s).

(2)

The influence of inlet temperature P5

As an input condition, the inlet temperature was based on the reference ambient temperature, which changes mainly according to the different seasons during operation, and the specific change range is 293 K~318 K, while the other variables of the four swirl tapes remained unchanged, as shown in Table 2. The influence of the three different inlet temperatures of 298 K, 305 K, and 318 K on the heat transfer performance are shown in Figure 13. The inlet temperature directly affected the overall temperature increase in the swirl tube bundles and the amount of bubble volume fraction under the same heating conditions. It had little effect on the flow velocity (maintained at 10.8 m/s) and the temperature difference between the inlet and outlet (18.1 K).

6. Conclusions

In this paper, the liquid-vapor two-phase flow model with subcooled boiling for a large length-diameter ratio swirl tube structure in the HHF calorimeter component is proposed to analyze the effects of key design parameters and factors (such as the inlet temperature of cooling water flow, swirl tube structure parameters, etc.) on its heat transfer performance, and the numerical solution flow and corresponding simulation results were presented. In order to improve the sensitivity analysis efficiency, considering the high computational cost of the liquid-vapor two-phase flow model, the polynomial response surface surrogate model of heat transfer performance function was constructed on the basis of Latin hypercube sampling. On this basis, by using the proposed surrogate model, the sensitivity indexes of each parameter were obtained easily and efficiently according to the Sobol global sensitivity analysis method. The simulation results showed that the sensitivity of the maximum temperature to variables P12, P13, P32, and P5 was relatively large. Thus, in practical engineering, when designing process parameters, we should focus on the influence of these variables on the structural heat transfer performance.