Step-by-step topological optimization design method for phase change heat accumulator structure based on molten salt

Abstract

The invention provides a phase change heat accumulator structure step-by-step topological optimization design method based on molten salt, which comprises the following steps: S01, defining a design domain and design parameters of a phase change heat accumulator fin, and determining a simulated physical model; s02, establishing a control equation, boundary conditions and initial conditions of the phase change heat accumulator, and determining a simulated mathematical model; s03, determining an objective function and constraint conditions based on the physical model and the mathematical model, and setting a projection slope beta and a penalty factor p; s04, based on the objective function, the constraint condition, the projection slope beta and the penalty factor p, optimizing and solving the mathematical model to obtain a temperature field of the design domain and a topological optimization model of the fins; and S05, observing whether the fin topological optimization model achieves an optimal convergence effect or not, and if not, gradually increasing the projection slope beta and the penalty factor p until the convergence effect is optimal and a preset design requirement is met.

Description

Technical Field

[0001] This invention belongs to the field of fin structure design technology, specifically relating to a step-by-step topology optimization design method for phase change heat storage structures based on molten salt. Background Technology

[0002] Solar energy is a clean, abundant, and resource-rich renewable energy source that has been widely used in power generation and building heating. However, solar energy has an inherent intermittent nature; due to the alternation of day and night and weather changes, its energy supply is difficult to maintain stably and reliably. Therefore, researching efficient and low-cost thermal storage systems to achieve continuous and stable operation of solar energy systems and fully unleash the enormous potential of solar energy is particularly important. Shell-and-tube phase change thermal storage systems can effectively solve the time-space mismatch problem in renewable energy utilization. However, due to the inherently low thermal conductivity of phase change materials, the thermal storage structure needs to be carefully designed to achieve the optimal heat transfer rate. Introducing finned structures is an effective method to improve the thermal efficiency of phase change thermal storage systems. However, existing fin designs often rely on empirical or semi-empirical methods, resulting in long design cycles, low design efficiency, and difficulty in obtaining optimal structural solutions. Although traditional straight-finned structures have the advantage of ease of manufacturing, they have significant limitations in terms of thermal conductivity and melting uniformity. Summary of the Invention

[0003] This invention provides a step-by-step topology optimization design method for phase change heat storage structures based on molten salt, which aims to...

[0004] To achieve the above objectives, the technical solution adopted by this invention is: to provide a step-by-step topology optimization design method for a phase change heat storage device based on molten salt, comprising the following steps:

[0005] S01, define the design domain and design parameters of the phase change heat accumulator fins, and determine the physical model for simulation;

[0006] S02, Establish the control equations, boundary conditions and initial conditions of the phase change heat storage device, and determine the mathematical model for simulation;

[0007] S03, Based on the physical model and the mathematical model, the objective function and constraints are determined, and the projection slope β and penalty factor p are set;

[0008] S04, Based on the objective function, the constraints, the projection slope β, and the penalty factor p, optimize and solve the mathematical model to obtain the temperature field of the design domain and the topology optimization model of the fin;

[0009] S05, observe whether the fin topology optimization model has achieved the best convergence effect. If it has not, gradually increase the projection slope β and the penalty factor p until the convergence effect is optimal and meets the predetermined design requirements. Then, output the fin structure using the topology optimization model at this time.

[0010] In one possible implementation of the step-by-step topology optimization design method for phase change heat storage structure based on molten salt provided by the present invention, in step S01, the design domain is a rectangular region, the design parameters include the width W and height H1 of the rectangular region, the height H2 of the fins, and the physical model includes the number of fins, the volume fraction of the fins, the material of the fins, and the molten salt-based phase change material.

[0011] In one possible implementation of the stepwise topology optimization design method for phase change heat storage structures based on molten salt provided by the present invention, in step S02, the governing equation includes:

[0012] The mass conservation equation during the melting process of the molten salt phase change material is as follows:

[0013]

[0014] The momentum conservation equations for the molten salt-based phase change material in the x and y directions are as follows:

[0015]

[0016] Where u and v represent the velocity components of the fluid in the x and y directions, respectively, t is time, ρ represents the fluid density, p represents the fluid pressure, μ represents the fluid dynamic viscosity, α represents the coefficient of thermal expansion, T is temperature, and T0 is the initial temperature; the fluid refers to the molten salt phase change material.

[0017] Heat transfer equation for fins:

[0018] ρ Al c is the density of the fin material. p,Al The specific heat capacity at constant pressure of the fin material is given by k, where T is the temperature of the fin, t is the time, and k is the constant pressure. Al It is the thermal conductivity of the fin material. For Hamiltonian operators;

[0019] During the phase transition, the energy conservation equation for the molten salt-based phase change material is:

[0020]

[0021] Where, ρ PCM c is the density of the molten salt-based phase change material. p,PCML(T) is the isobaric specific heat capacity of the molten salt-based phase change material, and k is the latent heat function. PCM The thermal conductivity of the molten salt-based phase change material is given by t; T is the temperature of the molten salt-based phase change material is given by t;

[0022] Define the latent heat function L(T):

[0023]

[0024] Among them, T m Let A1 and A2 be the phase transition temperatures of the material, A1 and A2 be the slopes, and T1 and T2 be the ranges controlling the phase transition temperatures.

[0025] This method describes the energy change during the phase transition process as a continuous change within a certain temperature range. This not only ensures the consistency between the phase transition potential per unit mass of the material and the actual value, but also significantly reduces the computational complexity and effectively reduces the computational time cost.

[0026] The sensible heat and latent heat of the molten salt-based phase change material are defined by a unified heat capacity expression and calculated as follows:

[0027]

[0028] Among them, c p,PCM ρ is the isobaric specific heat capacity of the molten salt-based phase change material. s c is the density of the solid molten salt-based phase change material. p,s ρ is the isobaric specific heat capacity of the solid molten salt-based phase change material, f is the liquid phase fraction, and ρ is the liquid phase fraction. l c is the density of the liquid molten salt-based phase change material. p,l The isobaric specific heat capacity of the liquid molten salt-based phase change material, α m The average coefficient of thermal expansion of the molten salt-based phase change material;

[0029] The liquid fraction f is defined as follows:

[0030]

[0031] Among them, T s,PCM The solid-state temperature of the molten salt-based phase change material is 414 K. l,PCM The liquid phase temperature of the molten salt-based phase change material is 416 K;

[0032] To effectively distinguish between the solid and liquid regions during the melting process, the variable viscosity coefficient method is used, with the melting temperature T as the reference value. m,PCM Using the dividing point as the boundary, the dynamic viscosity μ is expressed as a piecewise function, with the following specific form:

[0033]

[0034] Among them, T m,PCM T is the melting temperature of the molten salt-based phase change material. w The wall temperature;

[0035] Because the molten salt-based phase change material exhibits significant natural convection during melting, body forces are introduced into the momentum equation. To accurately simulate the effects of natural convection:

[0036]

[0037] In the formula, g is the acceleration due to gravity, with a value of 9.81 m / s². -2 γ is the volume expansion coefficient;

[0038] To ensure volume force The effect is applied only to the liquid region of the molten salt-based phase change material, introducing the volume expansion coefficient γ as a piecewise function, and using the material's melting temperature T as the basis for calculation. m,PCM As a dividing point:

[0039]

[0040] Among them, T m,PCM T is the melting temperature of the molten salt-based phase change material. w The wall temperature;

[0041] Thermal storage capacity Q is a key indicator for evaluating the performance of a thermal storage system, characterizing the system's thermal storage capacity per unit time; Q is composed of the sensible heat and latent heat of the molten salt-based phase change material, specifically defined as follows:

[0042]

[0043] Wherein, T0 represents the average temperature of the molten salt-based phase change material at the initial moment, T1 represents the average temperature when the molten salt-based phase change material reaches complete melting, and T1 is defined as the final heat storage temperature of the system, and L is the latent heat in the phase change process.

[0044] The heat storage power P of a phase change thermal storage system is defined as the heat capacity stored per unit time, which can be specifically expressed as:

[0045]

[0046] In one possible implementation of the stepwise topology optimization design method for phase change heat storage structures based on molten salt provided by the present invention, in step S02, the initial condition is: at the initial time t, the temperature in the design domain is T0 and uniformly distributed.

[0047] In one possible implementation of the stepwise topology optimization design method for phase change heat storage structures based on molten salt provided by this invention, the boundary conditions include:

[0048] The boundary conditions of the left-side pipe wall in the design domain are set as constant-temperature heat source boundary conditions, where the temperature T is equal to the temperature T of the heat transfer fluid. w This can be expressed as: T = T w =435K;

[0049] The boundary conditions for the upper and lower pipe walls and the right pipe wall of the design domain are set as adiabatic boundary conditions, and are described as follows:

[0050]

[0051] in, This represents the partial derivative of temperature T in the y-direction, i.e., the temperature gradient in the vertical direction. This represents the partial derivative of temperature T in the x-direction, i.e., the temperature gradient in the horizontal direction;

[0052] The coupling boundary condition between the fins and the molten salt-based phase change material is described as follows:

[0053]

[0054] Among them, T Al T is the surface temperature of the fins. PCM The temperature of the molten salt-based phase change material is k. Al k is the thermal conductivity of the fin. PCM The thermal conductivity of the molten salt-based phase change material is given. and The fin surface temperature T Al Temperature gradients in the x and y directions, and The temperature T of the molten salt-based phase change material PCM Temperature gradients in the x and y directions.

[0055] In one possible implementation of the stepwise topology optimization design method for phase change heat storage structures based on molten salt provided by this invention, in step S03, the optimization objective corresponding to the objective function is to maximize the average temperature Minimizez(T(x),x) of the design domain within a specified time t under the same heat source conditions, specifically expressed as:

[0056] Minimizez(T(x),x)=∫T(x)dx / V;

[0057] Where x represents the location of the design variable, V is the total volume of the design domain, and z is the average temperature within the design domain;

[0058] The constraints are as follows:

[0059]

[0060] Where, ρ s The density is the design variable, and Φ is a preset value for the proportion of the total volume occupied by the fins within the design domain. The design variable is defined as a continuous variable with a value range between 0 and 1, where 0 represents the molten salt-based phase change material and 1 represents the fins.

[0061] The first constraint specifies the physical model that the design variables must satisfy, meaning that temperature changes must strictly follow the aforementioned physical laws. In the second constraint, ρ... s The density represents the design variable, reflecting the proportion of different materials in each design variable, while the preset value of the total volume ratio of the fins within the design domain is a constraint that determines the total heat storage capacity of the system by limiting the amount of heat transfer material used.

[0062] In one possible implementation of the stepwise topology optimization design method for phase change heat storage structure based on molten salt provided by the present invention, in step S03, SIMP interpolation with a penalty factor p is used to interpolate the thermal conductivity of the molten salt-based phase change material and the fins. If the density, specific heat capacity, latent heat and permeability have little impact on the heat conduction process, linear interpolation is used.

[0063] The moving asymptote method is used as the optimization method to approximate the mathematical model, making its solution closer to the true value. To ensure the convergence of the optimization process, the maximum number of iterations is set to 500, and the convergence criterion is defined as a residual of less than 10. -9 ;

[0064] The design variables are density filtered using the Helmholtz partial differential equation, which takes the design variables as input parameters to obtain the filtered design variables.

[0065]

[0066] Where, r f The radius of the filter is r, which is taken as r in this study. f =0.0006, ρ f This refers to the density of the filtered design variables, which is also the density of the design variables used when solving the mathematical model. For the Hamiltonian operator, ρ s The density of the design variables before filtering;

[0067] A hyperbolic tangent projection function is used to eliminate grayscale cells introduced by the Helmholtz partial differential equation; the hyperbolic tangent projection function is:

[0068]

[0069] Where η represents the mapping threshold, which is taken as 0.5 in this paper, β is the projection slope, and ρ f For the filtered design variables, ρ p It is the design variable of the mapping. Through this mapping function, optimization results with clearer boundaries can be obtained while keeping the macroscopic material volume ratio unchanged.

[0070] In one possible implementation of the stepwise topology optimization design method for phase change heat storage structures based on molten salt provided by this invention, a permeability α(x) is defined to distinguish between the solid and liquid phases. If the fins are always solid, their flow velocity should be 0. Therefore, the solid region is characterized by defining α(x) >> 1, where the velocity obtained by solving the flow equation is very small, as specifically defined below:

[0071]

[0072] Using SIMP interpolation, the following physical quantities are redefined:

[0073]

[0074] ρ(x)=ρ PCM +ρ s (ρ HCM -ρ PCM );

[0075] c p (x)=c p,PCM +ρ s (c p,HCM -c p,PCM );

[0076] L(x)=(1-ρ s )L;

[0077] α(x)=α PCM (T)+ρ s (α HCM -α PCM (T));

[0078] Where, k PCM The thermal conductivity of the molten salt-based phase change material, k HCM Thermal conductivity of heat transfer material, ρ s To design the density of the variable, ρ PCM ρ is the density of the molten salt-based phase change material. HCM These represent the density of the heat transfer material, c p,PCM The isobaric heat capacity of the molten salt-based phase change material, c p,HCM The isobaric heat capacity of the heat transfer material, L is the enthalpy of phase change, α PCM The permeability of the molten salt-based phase change material, α HCM Permeability of heat transfer materials.

[0079] In one possible implementation of the step-by-step topology optimization design method for phase change heat storage structures based on molten salt provided by the present invention, the optimization analysis in step S05 includes the following steps:

[0080] S05.1, Determine the key parameters of the fin topology, the key parameters including volume fraction and filtration radius;

[0081] S05.2, set the projection slope β and penalty factor p to 1, and set the initial value to 0, and the result is generated as sol1;

[0082] S05.3, adjust the projection slope β and penalty factor p to 2, and use the parameters generated by sol1 as the initial values ​​to generate a new result record as sol2;

[0083] S05.4, adjust the projection slope β and penalty factor p to 4 and 3 respectively, use the parameters generated by sol2 as the initial values, generate a new result and record it as sol3;

[0084] In S05.5, the projection slope β and penalty factor p are adjusted to 8 and 4 respectively, the target result is generated and recorded as sol4.

[0085] The beneficial effects of the step-by-step topology optimization design method for phase change accumulator structure based on molten salt provided by this invention are as follows: Compared with the prior art, the step-by-step topology optimization design method for phase change accumulator structure based on molten salt provided by this invention results in a fin structure with higher thermal conductivity and better melting uniformity compared with the traditional rectangular fin structure. Attached Figure Description

[0086] Figure 1 The structural diagram of the fins generated after stepwise projection slope β and penalty factor p in the stepwise topology optimization design method for phase change heat storage structure based on molten salt provided in the embodiments of the present invention;

[0087] Figure 2 A comparison diagram of the fin structures generated by the stepwise topology optimization design method based on molten salt for phase change heat storage provided in the embodiments of the present invention and the direct topology optimization method (with no change in fin volume fraction);

[0088] Figure 3 A comparison of the liquid phase fraction at T=435K after the fins generated by the stepwise topology optimization design method based on molten salt and the direct topology optimization method (with no change in fin volume fraction) are applied to the molten salt phase change accumulator provided in the embodiments of the present invention.

[0089] Figure 4A comparison diagram of the fin structure and the rectangular fin structure generated by the step-by-step topology optimization design method for phase change heat accumulator structure based on molten salt provided in the embodiments of the present invention;

[0090] Figure 5 The finned structure, rectangular finned structure (without fin volume fraction) and finless structure (without fin volume fraction) generated by the step-by-step topology optimization design method for phase change accumulator based on molten salt provided in the embodiments of the present invention are compared with the liquid phase fraction at T=435K after being applied to a phase change accumulator based on molten salt.

[0091] Figure 6 A comparison of the total heat storage capacity at T=435K of the finned structure, rectangular finned structure (with no change in fin volume fraction), and finless structure (with no change in molten salt volume) generated by the step-by-step topology optimization design method for phase change accumulator based on molten salt provided in this embodiment of the invention after being applied to a phase change accumulator based on molten salt.

[0092] Figure 7 The temperature field comparison diagram shows the finned structure, rectangular finned structure (with unchanged fin volume fraction), and finless structure (with unchanged molten salt volume) generated by the step-by-step topology optimization design method for phase change accumulator based on molten salt provided in the embodiments of the present invention after being applied to a phase change accumulator based on molten salt and heated for the same time. Detailed Implementation

[0093] To make the technical problems, technical solutions, and beneficial effects to be solved by this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and are not intended to limit the scope of this application.

[0094] Please also refer to Figures 1 to 2 The step-by-step topology optimization design method for phase change heat storage structures based on molten salt, provided by this invention, is now described. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt includes the following steps:

[0095] S01, define the design domain and design parameters of the phase change heat accumulator fins, and determine the physical model for simulation;

[0096] S02, Establish the control equations, boundary conditions and initial conditions of the phase change heat storage device, and determine the mathematical model for simulation;

[0097] S03, Based on the physical model and the mathematical model, the objective function and constraints are determined, and the projection slope β and penalty factor p are set;

[0098] S04, Based on the objective function, the constraints, the projection slope β, and the penalty factor p, optimize and solve the mathematical model to obtain the temperature field of the design domain and the topology optimization model of the fin;

[0099] S05, observe whether the fin topology optimization model has achieved the best convergence effect. If it has not, gradually increase the projection slope β and the penalty factor p until the convergence effect is optimal and meets the predetermined design requirements. Then, output the fin structure using the topology optimization model at this time.

[0100] The predetermined design requirements stipulate that the final fin structure should have clear boundaries and be able to improve the melting rate of the phase change material.

[0101] The beneficial effects of the step-by-step topology optimization design method for phase change heat storage structure based on molten salt provided by this invention are as follows: Compared with the prior art, the step-by-step topology optimization design method for phase change heat storage structure based on molten salt provided by this invention results in a fin structure with higher heat transfer efficiency than the traditional rectangular fin structure.

[0102] like Figure 1 and Figure 2 As shown, in a specific embodiment of the stepwise topology optimization design method for phase change accumulator structure based on molten salt provided in this invention, in step S01, the design domain is a rectangular region, and the design parameters include the width W = 0.045m and height H1 = 0.1m of the rectangular region, the height H2 = 0.008m of the fins, and the physical model includes the number of fins, the volume fraction of the fins being 15%, the material being aluminum, and the phase change material being a molten salt-based phase change material.

[0103] like Figure 1 and Figure 2 As shown, in a specific embodiment of the stepwise topology optimization design method for phase change heat storage structures based on molten salt provided in this invention, the governing equation in step S02 includes:

[0104] The mass conservation equation during the melting process of the molten salt phase change material is as follows:

[0105]

[0106] Momentum conservation equations for molten salt-based phase change materials in the x and y directions:

[0107]

[0108] Where u and v represent the velocity components of the fluid in the x and y directions, respectively, t is time, ρ represents the fluid density, p represents the fluid pressure, μ represents the fluid dynamic viscosity, α represents the coefficient of thermal expansion, T is temperature, and T0 is the initial temperature; the fluid refers to the molten salt phase change material.

[0109] Heat transfer equation for fins:

[0110] ρ Al c is the density of the fin material. p,Al Here, T is the isobaric specific heat capacity of the fin material, t is the fin temperature, and k is the time. Al It refers to the thermal conductivity of the fin material. For Hamiltonian operators.

[0111] Specifically, when the fin material is aluminum, ρ Al c is the density of aluminum. p,Al K represents the specific heat capacity of aluminum under constant pressure. Al It refers to the thermal conductivity of aluminum.

[0112] The energy conservation equation for molten salt-based phase change materials during phase transition:

[0113]

[0114] Where, ρ PCM c is the density of the molten salt-based phase change material. p,PCM Let L(T) be the isobaric specific heat capacity of the molten salt-based phase change material, and k be the latent heat function. PCM t represents the thermal conductivity of the molten salt-based phase change material; T represents the temperature of the molten salt-based phase change material; and t represents time.

[0115] And define the latent heat function L(T):

[0116]

[0117] Among them, T m Let A1 and A2 be the phase transition temperatures of the material, A1 and A2 be the slopes, and T1 and T2 be the ranges controlling the phase transition temperatures.

[0118] This method describes the energy change during the phase transition process as a continuous change within a certain temperature range. This not only ensures the consistency between the phase transition potential per unit mass of the material and the actual value, but also significantly reduces the computational complexity and effectively reduces the computational time cost.

[0119] The sensible heat and latent heat of molten salt-based phase change materials are defined by a unified heat capacity expression and calculated as follows:

[0120]

[0121] Among them, c p,PCM ρ is the isobaric specific heat capacity of molten salt-based phase change materials. s For the density of solid molten salt-based phase change materials, c p,s ρ is the isobaric specific heat capacity of the solid molten salt-based phase change material, f is the liquid fraction, and ρ is the liquid phase fraction. lc is the density of the liquid molten salt-based phase change material. p,l The isobaric specific heat capacity of liquid molten salt-based phase change materials, α m The average coefficient of thermal expansion of molten salt-based phase change materials;

[0122] The liquid fraction f is defined as follows:

[0123]

[0124] Among them, T s,PCM The solid-state temperature of the molten salt-based phase change material is 414 K. l,PCM The liquid phase temperature of the molten salt-based phase change material is 416 K.

[0125] To effectively distinguish between the solid and liquid regions during the melting process, the variable viscosity coefficient method is used, with the melting temperature T as the reference value. m,PCM Using the dividing point as the boundary, the dynamic viscosity μ is expressed as a piecewise function, with the following specific form:

[0126]

[0127] Among them, T m,PCM T is the melting temperature of molten salt-based phase change materials. w This refers to the wall temperature.

[0128] Because molten salt-based phase change materials exhibit significant natural convection during melting, body forces are introduced into the momentum equation. To accurately simulate the effects of natural convection:

[0129]

[0130] In the formula, g is the acceleration due to gravity, with a value of 9.81 m / s². -2 γ is the volume expansion coefficient.

[0131] To ensure volume force This method applies only to the liquid region of molten salt-based phase change materials, introducing the volume expansion coefficient γ as a piecewise function, and using the material's melting temperature T as the basis for calculation. m,PCM As a dividing point:

[0132]

[0133] Among them, T m,PCM T is the melting temperature of molten salt-based phase change materials. w This refers to the wall temperature.

[0134] Thermal storage capacity Q is a key indicator for evaluating the performance of a thermal storage system, representing the system's thermal storage capacity per unit time. Q is composed of the sensible heat and latent heat of the molten salt-based phase change material, specifically defined as follows:

[0135]

[0136] Where T0 represents the average temperature of the molten salt-based phase change material at the initial moment, T1 represents the average temperature when the molten salt-based phase change material reaches complete melting, and T1 is defined as the final heat storage temperature of the system, and L is the latent heat in the phase change process.

[0137] The heat storage power P of a phase change thermal storage system is defined as the heat capacity stored per unit time, which can be specifically expressed as:

[0138]

[0139] like Figure 1 and Figure 2 As shown, in a specific embodiment of the stepwise topology optimization design method for phase change heat storage structure based on molten salt provided in this invention, in step S02, the initial condition is: at the initial time t, the temperature in the design domain is T0 and uniformly distributed.

[0140] like Figure 1 and Figure 2 As shown, in a specific embodiment of the step-by-step topology optimization design method for phase change heat storage structures based on molten salt provided in this invention, the boundary conditions include:

[0141] The boundary conditions on the left side of the design domain are set as isothermal heat source boundary conditions, with temperature T equal to the temperature T of the heat transfer fluid. w This can be expressed as: T = T w =435K.

[0142] The boundary conditions for the upper and lower pipe walls and the right-side pipe wall of the design domain are set as adiabatic boundary conditions, and are expressed as follows:

[0143]

[0144] in, This represents the partial derivative of temperature T in the y-direction, i.e., the temperature gradient in the vertical direction. This represents the partial derivative of temperature T in the x-direction, i.e., the temperature gradient in the horizontal direction.

[0145] The coupling boundary condition between the fins and the phase change material is described as follows:

[0146]

[0147] Among them, T Al T is the surface temperature of the fins. PCM K represents the phase change material temperature. Al k is the thermal conductivity of the fin. PCM The thermal conductivity of the phase change material, and The fin surface temperature T Al Temperature gradients in the x and y directions, and The phase change material temperature T PCM Temperature gradients in the x and y directions.

[0148] like Figure 1 and Figure 2 As shown, in a specific embodiment of the stepwise topology optimization design method for phase change heat storage structures based on molten salt provided in this invention, in step S03, the optimization objective corresponding to the objective function is to maximize the average temperature Minimizez(T(x),x) of the design domain within a specified time t under the same heat source conditions through optimization design. The specific equation is as follows:

[0149] Minimizez(T(x),x)=∫T(x)dx / V;

[0150] Where x represents the location of the design variable, V is the total volume of the design domain, and z is the average temperature of the material within the design domain.

[0151] x represents the spatial location or coordinates of the design variable within the optimization design domain. In numerical simulations and optimization, the design domain is typically discretized into a finite number of elements or nodes. x is often associated with these finite element nodes, each with one or more design variables describing the material properties or structural characteristics at that node. During topology optimization, the location of x (i.e., its corresponding finite element node) remains constant, but the values ​​of the design variables (such as density values) are dynamically adjusted according to the optimization algorithm. This adjustment aims to alter the material distribution and topological connections within the design domain to achieve optimization objectives (such as maximizing heat transfer capacity).

[0152] The constraints are:

[0153]

[0154] Where, ρ s The density is the design variable, and Φ is the preset value of the proportion of the total volume occupied by the fins in the design domain. The design variable is defined as a continuous variable with a value range between 0 and 1, where 0 represents molten salt-based phase change material and 1 represents fins.

[0155] The first constraint specifies the physical model that the design variables must satisfy, meaning that temperature changes must strictly follow the aforementioned physical laws. In the second constraint, ρ... s The density represents the design variable, reflecting the proportion of different materials in each design variable, while the preset value of the total volume ratio of fins within the Φ design domain is used as a constraint to determine the total heat storage capacity of the system by limiting the amount of heat transfer material used.

[0156] like Figure 1 and Figure 2 As shown in the embodiment of the stepwise topology optimization design method for phase change accumulator structure based on molten salt provided in this invention, in step S03, SIMP interpolation with a penalty factor p is used to interpolate the thermal conductivity of the molten salt-based phase change material and the fins. If density, specific heat capacity, latent heat, and permeability have little impact on the heat conduction process, linear interpolation is used. SIMP is a density-stiffness interpolation model.

[0157] The moving asymptote method is used as the optimization approach to approximate the mathematical model, making its solution closer to the true value. To ensure the convergence of the optimization process, the maximum number of iterations is set to 500, and the convergence criterion is defined as a residual of less than 10. -9 The Method of Moving Asymptotes (MMA) is an algorithm used to solve nonlinear optimization problems.

[0158] In topology optimization problems, design variables are the objects of optimization, each corresponding to a specific objective function and governing equation. During iterative calculations, the density values ​​of various design variables often differ significantly, and the density values ​​of design variables between adjacent nodes may fluctuate significantly. Since the density values ​​of design variables directly determine the values ​​of various parameters in the physical model, drastic changes in density values ​​can easily lead to oscillations in the numerical solution of the heat transfer problem, making convergence difficult. To address this issue, filtering methods are typically used to filter the optimization results. This method defines a filtering region radius centered on a node and uses the average value of the design variables of all nodes within the filtering radius as the node's variable value. This approach effectively smooths the density distribution between adjacent nodes, avoiding abrupt density differences, thus ensuring the stability of the heat transfer problem solution and improving convergence. This filtering process is achieved by solving the Helmholtz partial differential equation, which takes the design variables as input parameters to obtain the filtered design variables. The Helmholtz partial differential equation is:

[0159]

[0160] Where, r f The radius of the filter is r, which is taken as r in this study. f =0.0006, ρ f This refers to the density of the filtered design variables, which is also the density of the design variables used when solving the mathematical model. For the Hamiltonian operator, ρ s The density of the design variables before filtering.

[0161] The hyperbolic tangent projection function is used to eliminate the grayscale cells introduced by the Helmholtz partial differential equation; the hyperbolic tangent projection function is:

[0162]

[0163] Where η represents the threshold of the mapping, which is taken as 0.5 in this paper, β controls the steepness of the mapping, and ρ f It is the filtered design variable, ρ p It is a design variable with sharper boundaries after mapping. Through this mapping function, optimization results with clearer boundaries can be obtained while keeping the macroscopic material volume ratio unchanged.

[0164] like Figure 1 and Figure 2 As shown in a specific embodiment of the stepwise topology optimization design method for phase change accumulator structures based on molten salt provided in this invention, a permeability α(x) is defined to distinguish between the solid and liquid phases. If the fins are always solid, their flow velocity should be 0. Therefore, the solid region is characterized by defining α(x) >> 1, where the velocity obtained by solving the flow equation is very small, as specifically defined below:

[0165]

[0166] Using SIMP interpolation, the following physical quantities are redefined:

[0167]

[0168] ρ(x)=ρ PCM +ρ s (ρ HCM -ρ PCM );

[0169] c p (x)=c p,PCM +ρ s (c p,HCM -c p,PCM );

[0170] L(x)=(1-ρ s )L;

[0171] α(x)=α PCM (T)+ρ s (α HCM -α PCM (T));

[0172] Where, k PCM Thermal conductivity of molten salt-based phase change materials, k HCM Thermal conductivity of heat transfer material, ρ s To design the density of the variable, ρPCM ρ is the density of the molten salt-based phase change material. HCM These represent the density of the heat transfer material, c p,PCM The isobaric heat capacity of molten salt-based phase change materials, c p,HCM The isobaric heat capacity of the heat transfer material, L is the enthalpy of phase change, α PCM Permeability of molten salt-based phase change materials, α HCM Permeability of heat transfer materials.

[0173] like Figure 1 and Figure 2 As shown, in a specific embodiment of the step-by-step topology optimization design method for phase change heat storage structures based on molten salt provided in this invention, step S05 includes the following steps in the optimization analysis:

[0174] S05.1, Determine the key parameters of the topology optimization structure, including volume fraction and filter radius;

[0175] S05.2, set the projection slope β and penalty factor p to 1, and the initial value to 0, resulting in sol1; the topology-generated fin structure, such as Figure 1 As shown in (a);

[0176] In step S05.3, the projection slope β and penalty factor p are adjusted to 2, and the parameters generated by sol1 are used as initial values ​​to generate a new result, recorded as sol2; the topology-generated fin structure, such as... Figure 1 As shown in (b);

[0177] In S05.4, the projection slope β and penalty factor p are adjusted to 4 and 3 respectively. Using the parameters generated by sol2 as initial values, a new result is generated and recorded as sol3; the topology-generated fin structure is as follows: Figure 1 As shown in (c);

[0178] In S05.5, the projection slope β and penalty factor p are adjusted to 8 and 4 respectively, the target result is generated and recorded as sol4, and the topology-generated fin structure is as follows: Figure 1 As shown in (d).

[0179] It should be noted that the projection slope β controls the steepness of the projection function on both sides of the projection point. Therefore, similar to the penalty factor p and the filter radius, it also affects the number of intermediate densities and ultimately the topology optimization result. A larger projection slope β can yield results with almost no intermediate densities, which facilitates geometric reconstruction and actual fin fabrication. However, if it is too large, it will make the optimization problem difficult to solve. Therefore, the projection slope β is usually no more than 8.

[0180] Density-based topology optimization essentially involves rationally managing intermediate densities to minimize them while achieving a better objective function value. The penalty factor *p* determines the severity of the penalty applied to intermediate densities, while projection aims to reduce intermediate densities and obtain clear boundaries. Therefore, the penalty factor *p* and the projection slope *β* are crucial parameters influencing the topology optimization structure.

[0181] First, the projection slope β is set to 2, 4, 6, and 8 respectively. As the projection slope β increases, the number of intermediate density units at the fin edge decreases, and the boundary between the fin and the phase change material becomes clearer. This is because a larger projection slope β results in a steeper projection function, and a smaller transition region from 0 to 1 in the material density. When the projection slope β is 6, the fin edge is already sufficiently clear, with little difference in clarity compared to when the projection slope β is 8. It can also be observed that increasing the projection slope β also leads to more branched structures in the fin, enriching the fin details. This is because when the projection slope β is small, the projected structure still contains a large amount of intermediate density, mostly located at the fin edge and small branches. When density interpolation is applied later, the penalty effect filters out these intermediate densities, causing the fin details to disappear. However, when the projection slope β is large, these intermediate densities can be compressed towards the 0 and 1 sides before density interpolation penalty, reducing the intermediate density without filtering out fin details, thus retaining more fin branches after density interpolation.

[0182] Next, the penalty factor p was set to 1, 3, 5, 7, and 9 respectively. When the penalty factor p was 1, the material density in the entire computational domain was the intermediate density, and the density value was on the left wall (i.e., Figure 4The thermal conductivity is highest near the heating interface (marked in the image) and gradually decreases outwards along the width. This is because the interpolation function does not penalize the intermediate density for thermal conductivity; all physical properties are interpolated linearly. The intermediate density still has superior heat transfer performance, so the optimization algorithm cannot bring the intermediate density closer to the 0 and 1 extremes. This also means that if a material with similar physical properties to the intermediate density exists, its heat transfer effect will be better than that of the fin material. When the penalty factor p is 3, the interpolated thermal conductivity of the intermediate density unit decreases rapidly while the interpolated density and interpolated specific heat remain unchanged. Therefore, the intermediate density has poorer heat transfer performance compared to the fin material with a density of 1. The optimization algorithm makes the intermediate density unit gradually move closer to 0 and 1, resulting in a distinct fin structure. The fins exhibit a multi-level branching structure with numerous small branches on the fin trunk, but many branches are located in the intermediate density region, and the fin edges are also relatively blurred. When the penalty factor p continues to increase to 5, the intermediate density units decrease further, the fin branches decrease significantly, and all fins except the intermediate fins have only one level of branching. When the penalty factor p is 7, all the fine branching structures disappear, leaving only the fin trunk with first-order branching. At this point, the number of intermediate density units is greatly reduced, and the fin edges are more clearly defined. When the penalty factor p continues to increase to 9, the thermal conductivity of the intermediate density units is too poor, and the fin structure cannot be fully distributed within the design domain, ultimately resulting in a structure with only two fins and extremely uneven fin material distribution. Therefore, neither an excessively large nor an excessively small penalty factor p can yield effective topology optimization results.

[0183] Based on the characteristics of the phase transition process, after the phase change material melts, it enters the sensible heat storage stage, and its temperature rises rapidly. Empirically, using a large projection slope β and penalty factor p at the outset can cause the topology optimization to converge prematurely to a poor local minimum. The resulting topological patch structure means that while most of the phase change material has melted and its temperature has risen rapidly, a small portion melts very slowly, forming localized low-temperature regions. This is because using a large projection slope β and penalty factor p initially creates a large gradient in the material density distribution, making the objective function extremely sensitive to the material distribution. Even small changes in the material distribution in the early stages can lead to large fluctuations in the objective function.

[0184] To avoid this situation and ensure that the topology optimization converges to a better local minimum, a smaller projection slope β and penalty factor p are used initially, and gradually increased to obtain better sensitivity and stability. For phase change thermal storage topology optimization with a projection slope β of 8 and a penalty factor p of 4, this invention adopts a progressive continuous topology optimization scheme with constantly changing projection slope β and penalty factor p. Furthermore, the final temperature value of each step in the unsteady topology optimization needs to be carefully selected. This temperature value should be large enough to allow the phase change material to completely melt before the topology optimization converges, but not too large, lest the average temperature of the computational domain be too high at the end of the thermal storage process, approaching the temperature of the left wall, resulting in too small a change in the objective function obtained from each optimization step, leaving no room for further optimization.

[0185] like Figure 2 , 3 As shown, Figure 2 (1) A fin structure diagram generated by the step-by-step topology optimization design method for phase change heat storage devices based on molten salt provided in this application. Figure 2 (2) The fin structure diagram generated by the direct topology optimization method shows that the method provided in this application melts faster than the direct topology optimization method, with a melting time reduction of 29.52%. This can be attributed to the fact that, compared with the fin structure generated by the method provided in this application, the topological extension range of the fin structure generated by direct topology optimization is smaller and the branches are fewer, resulting in a smaller contact area and increased contact thermal resistance. Therefore, the method provided in this application can improve the melting speed, make the melting process more uniform, and achieve better convergence.

[0186] like Figure 5 As shown in the figure, the temperature and liquid fraction of the molten salt phase change material in the phase change thermal storage system change at a heating temperature of 435 K under three different structures (no fins, rectangular fins, and finned structure based on the method topology provided in this application) change over time. The introduction of the finned structure based on the method topology provided in this application significantly shortens the complete phase change time of the molten salt phase change material, from 7290 s for the no-fin structure to 2770 s, reducing the melting time by 62% and significantly improving the overall heat transfer performance of the system.

[0187] like Figure 6As shown in the figure, the total thermal storage capacity of the phase change thermal storage system varies under three different structures (no fins, rectangular fins, and finned structure based on the method topology provided in this application) at a heating temperature of 435K. Calculations show that the thermal storage capacities of the three structures are 456.94 kJ, 456.41 kJ, and 428.75 kJ, respectively, with average thermal storage power of 62.68 W, 84.99 W, and 154.78 W, respectively. The thermal storage capacity of the phase change thermal storage system with rectangular fins and the finned structure based on the method topology provided in this application increases by 35.59% and 146.93%, respectively.

[0188] like Figure 7 As shown in the figure, this is a comparison of the temperature fields of the phase change thermal storage system under three different structures (no fins, rectangular fins, and finned structure based on the method topology provided in this application) after heating for the same time of 2400s. It can be clearly seen that the phase change thermal storage system using the finned structure based on the method topology provided in this application has a higher average temperature.

[0189] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A stepwise topology optimization design method for a phase change heat storage device based on molten salt, characterized in that, include: S01, define the design domain and design parameters of the phase change heat accumulator fins, and determine the physical model for simulation; S02, Establish the control equations, boundary conditions and initial conditions of the phase change heat storage device, and determine the mathematical model for simulation; S03, Based on the physical model and the mathematical model, the objective function and constraints are determined, and the projection slope β and penalty factor p are set; S04, Based on the objective function, the constraints, the projection slope β, and the penalty factor p, optimize and solve the mathematical model to obtain the temperature field of the design domain and the topology optimization model of the fin; S05, observe whether the fin topology optimization model has achieved the best convergence effect. If it has not, gradually increase the projection slope β and the penalty factor p until the convergence effect is optimal and meets the predetermined design requirements. Then, output the fin structure using the topology optimization model at this time.

2. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 1, characterized in that, In step S01, the design domain is a rectangular area, the design parameters include the width W and height H1 of the rectangular area, the height H2 of the fins, and the physical model includes the number of fins, the volume fraction of the fins, the material of the fins, and the molten salt-based phase change material.

3. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 2, characterized in that, In step S02, the governing equations include: The mass conservation equation during the melting process of the molten salt phase change material is as follows: The momentum conservation equations for the molten salt-based phase change material in the x and y directions are as follows: Where u and v represent the velocity components of the fluid in the x and y directions, respectively, t is time, ρ represents the fluid density, p represents the fluid pressure, μ represents the fluid dynamic viscosity, α represents the coefficient of thermal expansion, T is temperature, and T0 is the initial temperature; the fluid refers to the molten salt phase change material after melting. Heat transfer equation for fins: ρ Al c is the density of the fin material. p,Al The specific heat capacity at constant pressure of the fin material is given by k, where T is the temperature of the fin, t is the time, and k is the constant pressure. Al It is the thermal conductivity of the fin material. For Hamiltonian operators; During the phase transition, the energy conservation equation for the molten salt-based phase change material is: Where, ρ PCM c is the density of the molten salt-based phase change material. p,PCM L(T) is the isobaric specific heat capacity of the molten salt-based phase change material, and k is the latent heat function. PCM The thermal conductivity of the molten salt-based phase change material is given by t; T is the temperature of the molten salt-based phase change material is given by t; Define the latent heat function L(T): Among them, T m Let A1 and A2 be the phase transition temperatures of the material, A1 and A2 be the slopes, and T1 and T2 be the ranges controlling the phase transition temperatures. The sensible heat and latent heat of the molten salt-based phase change material are defined by a unified heat capacity expression and calculated as follows: Among them, c p,PCM ρ is the isobaric specific heat capacity of the molten salt-based phase change material. s c is the density of the solid molten salt-based phase change material. p,s ρ is the isobaric specific heat capacity of the solid molten salt-based phase change material, f is the liquid phase fraction, and ρ is the liquid phase fraction. l c is the density of the liquid molten salt-based phase change material. p,l The isobaric specific heat capacity of the liquid molten salt-based phase change material, α m The average coefficient of thermal expansion of the molten salt-based phase change material; The liquid fraction f is defined as follows: Among them, T s,PCM T is the solid-state temperature of the molten salt-based phase change material. l,PCM This refers to the liquid phase temperature of molten salt-based phase change materials; To effectively distinguish between the solid and liquid regions during the melting process, the variable viscosity coefficient method is used, with the melting temperature T as the reference value. m,PCM Using the dividing point as the boundary, the dynamic viscosity μ is expressed as a piecewise function, with the following specific form: Among them, T m,PCM T is the melting temperature of the molten salt-based phase change material. w The wall temperature; Because the molten salt-based phase change material exhibits significant natural convection during melting, body forces are introduced into the momentum equation. To accurately simulate the effects of natural convection: In the formula, g is the acceleration due to gravity, and γ is the coefficient of volume expansion; To ensure volume force The effect is applied only to the liquid region of the molten salt-based phase change material, introducing the volume expansion coefficient γ as a piecewise function, and using the material's melting temperature T as the basis for calculation. m,PCM As a dividing point: Among them, T m,PCM T is the melting temperature of the molten salt-based phase change material. w The wall temperature; Thermal storage capacity Q is a key indicator for evaluating the performance of a thermal storage system, characterizing the system's thermal storage capacity per unit time; Q is composed of the sensible heat and latent heat of the molten salt-based phase change material, specifically defined as follows: Wherein, T0 represents the average temperature of the molten salt-based phase change material at the initial moment, T1 represents the average temperature when the molten salt-based phase change material reaches complete melting, and T1 is defined as the final heat storage temperature of the system, and L is the latent heat in the phase change process. The heat storage power P of a phase change thermal storage system is defined as the heat capacity stored per unit time, which can be specifically expressed as:

4. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 3, characterized in that, In step S02, the initial condition is: at the initial time t, the temperature in the design domain is T0 and uniformly distributed.

5. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 4, characterized in that, The boundary conditions include: The boundary conditions of the left-side pipe wall in the design domain are set as constant-temperature heat source boundary conditions, where the temperature T is equal to the temperature T of the heat transfer fluid. w This can be expressed as: T = T w ; The boundary conditions for the upper and lower pipe walls and the right pipe wall of the design domain are set as adiabatic boundary conditions, and are described as follows: in, This represents the partial derivative of temperature T in the y-direction, i.e., the temperature gradient in the vertical direction. This represents the partial derivative of temperature T in the x-direction, i.e., the temperature gradient in the horizontal direction; The coupling boundary condition between the fins and the molten salt-based phase change material is described as follows: Among them, T Al T is the surface temperature of the fins. PCM The temperature of the molten salt-based phase change material is k. Al k is the thermal conductivity of the fin. PCM The thermal conductivity of the molten salt-based phase change material is given. and The fin surface temperature T Al Temperature gradients in the x and y directions, and The temperature T of the molten salt-based phase change material PCM Temperature gradients in the x and y directions.

6. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 5, characterized in that, In step S03, the optimization objective corresponding to the objective function is to maximize the average temperature Minimizez(T(x),x) of the design domain within a specified time t under the same heat source conditions. The specific equation is as follows: Minimizez(T(x),x)=∫T(x)dx / V; Where x represents the location of the design variable, V is the total volume of the design domain, and z is the average temperature within the design domain; The constraints are as follows: Where, ρ s The density is the design variable, and Φ is a preset value for the proportion of the total volume occupied by the fins within the design domain. The design variable is defined as a continuous variable with a value range between 0 and 1, where 0 represents the molten salt-based phase change material and 1 represents the fins.

7. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 6, characterized in that, In step S03, the thermal conductivity of the molten salt-based phase change material and the fins is interpolated using SIMP interpolation with a penalty factor p, while density, specific heat capacity, latent heat and permeability are interpolated linearly. The moving asymptote method is used as the optimization method to approximate the mathematical model, making its solution closer to the true value. To ensure the convergence of the optimization process, the maximum number of iterations is set to 500, and the convergence criterion is defined as a residual of less than 10. -9 ; The design variables are density filtered using the Helmholtz partial differential equation, which takes the design variables as input parameters to obtain the filtered design variables. Where, r f Represents the filter radius, ρ f This refers to the density of the filtered design variables, which is also the density of the design variables used when solving the mathematical model. For the Hamiltonian operator, ρ s The density of the design variables before filtering; A hyperbolic tangent projection function is used to eliminate grayscale cells introduced by the Helmholtz partial differential equation; the hyperbolic tangent projection function is: Where η represents the mapping threshold, which is taken as 0.5 in this paper, β is the projection slope, and ρ f For the filtered design variables, ρ p These are the design variables for the mapping.

8. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 7, characterized in that, To distinguish between the solid and liquid phases, a permeability α(x) is defined. If the fins are always solid, their flow velocity should be 0. Therefore, the solid region is characterized by defining α(x) >> 1, where the velocity obtained by solving the flow equation is very small, as specifically defined below: Using SIMP interpolation, the following physical quantities are redefined: p(x)=p PCM +r s (r HCM -r PCM ); c p (x)=c p,PCM +ρ s (c p,HCM -c p,PCM ); L(x)=(1-ρ s )L; α(x)=α PCM (T)+p s (a HCM -a PCM (T)); Where, k PCM The thermal conductivity of the molten salt-based phase change material, k HCM Thermal conductivity of heat transfer material, ρ s To design the density of the variable, ρ PCM ρ is the density of the molten salt-based phase change material. HCM These represent the density of the heat transfer material, c p,PCM The isobaric heat capacity of the molten salt-based phase change material, c p,HCM The isobaric heat capacity of the heat transfer material, L is the enthalpy of phase change, α PCM The permeability of the molten salt-based phase change material, α HCM Permeability of heat transfer materials.

9. The step-by-step topology optimization design method for phase change heat storage structures based on molten salt as described in claim 8, characterized in that, In step S05, the optimization analysis includes the following steps: S05.1, Determine the key parameters of the fin topology, the key parameters including volume fraction and filtration radius; S05.2, set the projection slope β and penalty factor p to 1, and set the initial value to 0, and the result is generated as sol1; S05.3, adjust the projection slope β and penalty factor p to 2, and use the parameters generated by sol1 as the initial values ​​to generate a new result record as sol2; S05.4, adjust the projection slope β and penalty factor p to 4 and 3 respectively, use the parameters generated by sol2 as the initial values, generate a new result and record it as sol3; In S05.5, the projection slope β and penalty factor p are adjusted to 8 and 4 respectively, the target result is generated and recorded as sol4.

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