Optimization and design method of double-fluid porous heat exchanger based on three-period minimal surface

Abstract

The application discloses a kind of based on three periodic minimal surface double-fluid porous heat exchanger optimization and design method, comprising: step 1: the function representation of double-fluid porous heat exchanger;Step 2: establish optimization mathematical model;Step 3: optimization process and improvement optimization solver.The application is mainly aimed at the problem of enhancing the heat exchange efficiency of double-fluid heat exchanger, based on the porous structure of three periodic minimal surface has natural heat exchange advantage, for example, the specific surface area ratio is increased to increase the contact area of double-fluid, full connectivity and high smoothness can reduce flow resistance;In optimization model, the wall thickness between double-fluid is directly controlled by geometric parameter variable, to avoid adding complex wall thickness constraint;Secondly, the optimization solver of adding monotone programming operator is more suitable for the optimization of complex structure, and accelerates the convergence speed of optimization;In addition, the function representation of porous heat exchanger can guarantee the accuracy of model, good smoothness, connectivity and self-supporting are also conducive to additive manufacturing.

Description

Technical Field

[0001] This invention relates to the field of computer-aided design and is an optimization and design method for a two-fluid porous heat exchanger based on a three-period minimal surface, which is suitable for engineering applications such as cooling or heating fluids with low energy consumption. Background Technology

[0002] A heat exchanger is a device that transfers heat between two or more fluids to achieve the purpose of cooling or heating the fluids. As an essential piece of equipment for efficient heat transfer, heat exchangers have always been a crucial component in the development of advanced electronic equipment and large mechanical systems, especially in recent years with a significant increase in the demand for constructing highly efficient and low-energy-consumption heat exchangers. The performance improvement of a heat exchanger mainly depends on the convective heat transfer coefficient and the area of ​​the heat transfer surface. To achieve high heat exchange surface area density under the manufacturability limitations of traditional processes, conventional fluid heat exchangers rely on a few simple shapes in their design, such as multiple pipes or fins. However, as the heat load of related equipment gradually increases, these traditional heat exchangers are increasingly unable to meet the demand for more efficient heat exchangers, and the development of new heat exchangers faces significant technical challenges.

[0003] In the design of two-fluid heat exchangers, several solutions already exist. Industrially, computer-aided design (CAD) using geometry optimization software for heat exchanger design heavily relies on initial geometry and parameter selection. Compared to geometry optimization, topology optimization, as a structural optimization design method with higher degrees of freedom, can explore more complex geometries. Density-based methods are the most commonly used topology optimization methods. However, modeling and implementing the solid walls between different inflow channels is quite difficult in topology optimization. When Papazoglou et al. first applied density-based methods to the design of two-fluid heat exchangers based on the assumption of zero wall thickness, the designed heat exchanger's performance was limited by the zero wall thickness and did not conform to practical applications. Later, Tawk and Savier et al.'s density-based optimization method considered the design of solid walls, but some difficulties still exist: firstly, since there is no explicit separation between the solid and fluid phases, appropriate interpolation parameters or filters must be selected, intermediate densities must be used to represent the transition between different materials, and post-processing of the final design is required to obtain a model with clear boundaries. On the other hand, the channels are irregular during the optimization process, which cannot guarantee the continuity of the fluid channels. Therefore, the forced solid coating around each fluid element introduces new problems such as parameter selection. The irregularity of the solid wall makes the model have defects in manufacturability. Subsequently, Feppon et al. proposed a mesh evolution algorithm based on level sets, which has clear separation boundaries and high numerical accuracy. However, this method requires multiple constraints such as imposed thickness constraints and non-mixing constraints, which is not conducive to the numerical calculation and convergence of the optimization process. In addition, updating the mesh of the interface in each iteration requires more computation time.

[0004] With the development of additive manufacturing technology, it has become possible to design and manufacture fluid heat exchanger structures with more complex geometries. Minimal surfaces divide space into two unconnected parts; the high surface area-to-volume ratio increases the contact area between the two fluids, and the full connectivity and high smoothness reduce flow resistance, making them naturally suitable for fluid heat exchange design. Tri-periodic minimal surfaces can generate diverse designs by controlling parameters represented by functions, providing endless possibilities for structural design. However, due to the complexity of fluid heat exchanger design, finding effective representations, modeling, and optimization schemes for tri-periodic minimal surfaces in multi-fluid heat exchange problems is very difficult, and research on this structure in fluid heat exchange is scarce. In recent research, Kim and Yoo proposed a new method for describing compact heat exchangers, including a core structure based on tri-periodic minimal surfaces, Boolean operations, and additive manufacturing technology, which can effectively represent structures with high geometric complexity. Td et al. experimentally measured the thermal performance of a microarchitecture spiral lattice heat exchanger, concluding that its heat transfer efficiency is equivalent to that of a heat exchanger of 1 / 10th the size. However, neither of these studies provided an optimization scheme. Therefore, it is necessary to explore optimization and design methods applicable to two-fluid heat exchangers based on minimal surfaces. This patent provides a solution for this problem by using topology and geometry optimization to design an efficient two-fluid heat exchanger based on a three-period minimal surface.

[0005] This invention proposes an efficient optimization and design method and system for a two-fluid porous heat exchanger based on a three-period minimum surface. Compared with existing methods, the proposed method has the following advantages: First, the three-period minimum surface structure can easily distinguish between the two fluids and the solid wall using functional representation. The combination of topological parameters controlling the pore distribution and geometric parameters controlling the solid wall thickness, with thickness as a design variable, can prevent the two fluids from mixing without adding additional constraints. Second, the optimization solver incorporating a monotonic programming operator is more suitable for optimizing complex structures, accelerating the convergence speed to the constraint conditions and reducing the possibility of getting trapped in local minima during the optimization process. Furthermore, finite element calculations are performed within the design domain, eliminating the need to update the finite element mesh for each iteration, thus avoiding time-consuming meshing. Finally, models of arbitrary precision can be generated using the functional representation of the heat exchanger without requiring post-processing. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the existing methods and provide an optimization and design method and system for a two-fluid porous heat exchanger based on a three-period minimal surface.

[0007] The technical solution adopted in this invention:

[0008] An optimization and design method for a two-fluid porous heat exchanger based on a three-period minimum surface includes a functional representation module for the two-fluid porous heat exchanger, a module for establishing an optimization mathematical model, an optimization process, and an improved optimization algorithm module. The steps are as follows:

[0009] Step 1: The function representation module for the two-fluid porous heat exchanger is used to represent and distinguish the solid domain and the two fluid domains of the two-fluid porous heat exchanger using a single material function, and to pass the function representation of the porous heat exchanger to the optimized mathematical model;

[0010] The first step is to develop a precise mathematical representation of trigonometric functions based on a three-period minimum surface. Constructing an improved function representation:

[0011]

[0012] The topological parameter function P(r) > 0 is a continuously differentiable function. It controls the topology of the surface by controlling the distribution of holes in the surface, and maintains the topology during topological changes. The distance scale remains unchanged, and r = (x, y, z) is a vector in three-dimensional Euclidean space, where x, y, and z are the corresponding coordinates.

[0013] The second step involves improving the function surface from the first step. By offsetting to both sides, a porous solid with thickness is constructed. The two offset surfaces are represented as follows:

[0014]

[0015]

[0016] Among them, the geometric parameter function W(r) controls the offset distance, and thus controls the wall thickness of the porous solid; two offset surfaces and Divide space into fluid 1, that is... Fluid 2 and solid and Three unconnected regions; fluid 1 flows into the heat exchanger at a higher temperature, T in =1; Fluid 2 flows in at a lower temperature, T in =0;

[0017] The third step involves adding a heat exchanger shell to the porous solid using Boolean operations, thus constructing a complete porous heat exchanger based on a three-period minimum surface. The heat exchanger shell is represented by a directed distance field as a function Φ. shell If the value is greater than 0, then the two fluids, the solid, and the heat exchanger shell can be represented by the following functions:

[0018]

[0019]

[0020]

[0021] Where max(·,·) is the maximum value function. and These are the fluid 1 function, fluid 2 function, and solid function of a shell-and-shell heat exchanger. Based on these functions, the formula for distinguishing the three regions of fluid 1, fluid 2, and solid is defined as follows:

[0022]

[0023] Porous heat exchangers based on three-period minimal curved surfaces possess inherent heat exchange advantages. For example, the high specific surface area ratio increases the contact area between the two fluids, and full connectivity and high smoothness reduce flow resistance. The porous structure based on three-period minimal curved surfaces has also been proven to possess good mechanical properties, enabling it to resist external forces and thermal deformation to a certain extent.

[0024] Fourth, in order to distinguish the three regions by a single variable, a linear interpolation function is used to combine the directed distance field function into a material function for the design domain:

[0025]

[0026] Where H(x) is the Heaviside function, which is 0 when x is negative and 1 otherwise. ξ(r) is the material function, where ξ(r) = 1, 0.5, and 0 correspond to fluid 1, solid, and fluid 2, respectively.

[0027] Step 2: Establish an optimization mathematical model module to build a mathematical model for maximizing the total heat transfer problem under a fixed pressure constraint. Numerical calculations are performed based on the material function ξ(r) in the function representation module of the two-fluid porous heat exchanger. The calculated physical fields, objective function values, and constraint values ​​are then passed to the specific optimization process and solver. This includes the optimization model using total heat transfer as the objective function, pressure as the constraint, and the topological parameter function P(r) and geometric parameter function W(r) as design variables for optimization. The optimization focuses on two mass transfer problems (one for each fluid) and one global heat transfer problem under the assumptions of steady-state, constant, incompressible fluids and neglecting heat generation from viscous dissipation. The boundary conditions for the mass transfer problem include Dirichlet boundary conditions at the fluid inlet, outlet, and the remaining boundaries of the design domain, giving the inlet inflow velocity, the pressure-free condition at the outlet, and the no-slip condition for the fluid on the remaining boundaries. The boundary conditions for the heat transfer problem consist of the Dirichlet boundary conditions at the corresponding fluid inlet, giving the fluid inflow temperature. Homogeneous Riemann boundary conditions are applied to the remaining boundaries of the design domain, resulting in insulation outside the design domain. To simplify numerical implementation, all governing equations are presented in dimensionless form; therefore, physical units are omitted when referring to numerical values ​​used in our test cases. The optimized mathematical model for the two-fluid porous heat exchanger is established as follows:

[0028]

[0029] Make:

[0030]

[0031]

[0032]

[0033]

[0034]

[0035]

[0036] Where u = [u1, u2, u3] T Here, α represents the flow velocity, n represents the unit normal vector pointing out of the design domain, p represents the dynamic pressure, T represents the temperature, Re is the Reynolds number, Pe is the Péclet number, and α is the reverse osmosis rate. It is the fluid τ outlet boundary. τ represents the fluid inlet boundary, where the subscript τ = {1, 2} represents fluid 1 and fluid 2 respectively; Ω represents the design domain of the heat exchanger; J is the overall heat transfer objective function; G is the fluid pressure constraint, obtained by combining the geometric mean pressures g1 and g2 of the two fluids at the inlet using the KS function; σ is a constant parameter of the KS aggregation function; p max It is the maximum allowable pressure at the fluid inlet. Equation (1.10) is the variational form of the mass transfer and heat transfer equations. In the formula, a0(u,δu)=<u,δu> , b(δu,p)=-∫ Ω p·div(δu)dΩ, where <·,·> is the inner product of the two terms. The gradient operator is δu, δp, and δT, which are the test functions corresponding to flow velocity u, pressure p, and temperature T, respectively, belonging to the test spaces U0, M0, and T0. According to the variational form, the physical field (u, p, T) ∈ U×M×T can be directly obtained using the finite element method, where U, M, and T are the solution spaces. In finite element calculations, the directed distance field constructed by the function can effectively solve the physical field in the entire design domain without re-meshing, which is obviously more computationally efficient than continuously updating the subdomain interface mesh during optimization.

[0037] Step 3 divides the optimization process of the porous heat exchanger into three steps: pre-optimization, pore distribution optimization, and wall thickness distribution optimization. The design variables of the optimization model are iteratively updated using a GCMMA optimization solver with monotonic programming operators to obtain the optimized solution, including:

[0038] Step 3-1: The topological parameter function P(r) and the geometric parameter function W(r) can independently control the pore size and wall thickness of the porous structure. The radial basis interpolation method can transform the optimization of the parameter function into the optimization of the parameters at a finite number of interpolation base points.

[0039] Pre-optimization uses a small number of interpolation base points and constant geometric parameter functions as design variables. It involves initial adjustments to the porous structure, obtaining an optimized structure as a better initial structure for subsequent, more detailed adjustments. Selecting fewer interpolation base points in the pre-optimization process effectively reduces optimization complexity and the number of overall iterations. n0 interpolation base points are randomly selected in the solution domain. Then there is an interpolation form:

[0040]

[0041] Where n0 is the total number of control points within the design domain Ω, and M i (r) is the corresponding computable coefficient function. The topological parameter values ​​of the interpolation base points are used as design variables. The geometric parameter function W(r) = W0 is a constant function, and W0 is also a design variable. By using the constant wall thickness as a design variable, coupled optimization of the void distribution and wall thickness distribution is achieved, bringing the result closer to the global optimum.

[0042] Pore ​​distribution optimization uses topological parameter values ​​from a larger number of interpolation base points as design variables, allowing for further fine-tuning of the pores in the porous structure. In the solution domain, n is randomly selected. t interpolation base points Then there is an interpolation form:

[0043]

[0044] Where, n t N is the total number of interpolation base points within the design domain Ω. i (r) is the corresponding computable coefficient function. These are the topology parameter values ​​of the interpolation base points, used as design variables.

[0045] Wall thickness distribution optimization uses geometric parameter values ​​from multiple interpolation points as design variables to fine-tune the wall thickness of porous structures. Similarly, the interpolation form of the geometric parameter function is:

[0046]

[0047] in These are the geometric parameter values ​​of the interpolation base points, used as design variables.

[0048] Step 3-2: Use a monotonic programming operator to transform the original problem into an equivalent structural optimization problem. When the pressure constraint differs significantly from the initial pressure, due to the complexity of the two-fluid heat transfer problem, the optimization process is prone to getting trapped in local minima before the constraint conditions are met, and the convergence process is slow. To address this issue, a monotonic programming operator is added to the solver. The objective and constraints in the original problem can be represented by functions, as follows:

[0049]

[0050] Add the monotonic operator x = ln(1 + y1 / γ)·N norm Then, the original problem is transformed into an equivalent problem:

[0051] h γ (y)=h[ln(1+y1 / γ)·N norm ],

[0052]

[0053] Where X and Y γ These are functions h(x) and h, respectively. γThe domain of (y), low k and upp k These are the lower and upper bounds of the k-th design variable in the original problem, respectively; γ is the parameter of the monotonic programming operator, used to control the degree of change in transforming the original problem into an equivalent problem; N norm =[N norm,1 ,…,N norm,n ] T It is a normalized parameter vector used to ensure that the value of each design variable in the equivalent problem is between 0 and 1, thus avoiding numerical errors caused by the design variable range being too large or too small.

[0054] Step 3-3: Calculate the sensitivity of the equivalent optimization problem using the adjoint method, input it into the GCMMA solver to iteratively update the design variables, and converge to the optimal solution. The three optimization steps are performed sequentially, with each step optimized using the GCMMA solver. Although GCMMA has been proven to have theoretical global convergence, due to the high complexity of the mathematical model of the two-fluid heat transfer problem, a smaller number of internal iterations of GCMMA are used to ensure the speed and accuracy of numerical calculations. The sensitivity calculation formulas for the objective function and constraint functions with respect to the design variable D are as follows:

[0055]

[0056]

[0057]

[0058] Where H′ is the derivative of the Heaviside function, and the sensitivity of physical parameters u, p, T to design variables is calculated using the adjoint method. Topological parameters P(r) and geometric parameters W(r) are considered as design variables for optimization. and The value of is determined by the selection of different combinations of design variables in different optimization steps, which will lead to ... and The calculation results are different.

[0059] In the pre-optimization, the design variables are the topological parameter variables of the interpolation base points and the constant geometric parameter functions, D={P i ,W0|i=1,…,n0}. The sensitivity calculation results Inputting the data into the GCMMA optimization solver yields a preliminary optimized structure for the pre-optimization stage, which features smooth and continuous pores and wall thickness. This structure serves as the initial structure for subsequent pore distribution optimization and wall thickness distribution optimization.

[0060] In the optimization of hole distribution, the design variable is the topological parameter variable of the interpolation base point, D={Pi |i=1,…,n t The sensitivity calculation results in... Inputting the data into the GCMMA optimization solver yields an optimized structure with continuously varying pore sizes, which serves as the initial structure for optimizing the wall thickness distribution. Since the initial structure used in this optimization stage is the pre-optimized solution obtained during the initial optimization phase, and the geometric parameter function W(r) is fixed based on the pre-optimized solution W0, convergence in optimizing the pore distribution is easily achieved.

[0061] In wall thickness distribution optimization, the design variables are geometric parameter variables, D={W i |i=1,…,n t The sensitivity calculation results in... The input into the GCMMA optimization solver yielded a porous structure with both smooth pores and varying wall thickness. Since the topological parameter function P(r) is the optimal solution for the pore distribution optimization, and is fixed in this optimization stage, the convergence of the optimized geometric distribution is also well achieved.

[0062] The beneficial effects of this invention are as follows: The optimization and design system for a two-fluid porous heat exchanger based on a three-period minimum surface, belonging to the fields of computer-aided design and industrial design and manufacturing, is described above. The proposed porous heat exchanger can be represented in functional form and possesses good connectivity, mechanical properties, a high surface area-to-volume ratio, and smoothness. In the two-fluid heat transfer problem, a three-period minimum surface structure is applied to obtain an optimized porous heat exchanger with smooth pores and varying wall thickness. Compared with existing methods, the advantages of this invention are: First, the optimization process includes pre-optimization, pore distribution optimization, and wall thickness distribution optimization, simultaneously incorporating topological and geometric changes, increasing the design freedom. Furthermore, using wall thickness as a design variable prevents mixing of the two fluids without requiring additional constraints. Second, the improved solver incorporating monotonic programming operators accelerates the arrival at optimization constraints and reduces the possibility of getting trapped in local minima during optimization. In addition, finite element calculations are performed within the design domain, eliminating the need to update the finite element mesh for each iteration, thus avoiding time-consuming meshing. Finally, heat exchanger models of arbitrary precision can be directly generated using the function representation of the heat exchanger, and the good smoothness and connectivity are beneficial to additive manufacturing. Attached Figure Description

[0063] Figure 1 This is a flowchart illustrating the design and optimization process of a three-dimensional two-fluid porous heat exchanger based on a three-period minimal surface.

[0064] Figure 2 This is a schematic diagram comparing the heat exchange performance of a traditional plate-type two-fluid heat exchanger and a two-fluid porous heat exchanger based on a three-period minimal curved surface.

[0065] In the figure: a) Temperature distribution of a plate-shaped two-fluid heat exchanger in the XY and YZ planes; b) Temperature distribution of a uniform porous heat exchanger based on a three-period minimum surface in the XY and YZ planes; c) Temperature distribution of the optimized heat exchanger in the XY and YZ planes obtained by optimizing the uniform porous heat exchanger in b.

[0066] Figure 3 This is a comparison of the optimization results of a two-fluid porous heat exchanger based on a three-period minimal surface under two pressure constraints.

[0067] In the figure: a) the initial (before optimization) temperature distribution of the heat exchanger based on the three-period minimum surface in the YZ1 and YZ2 planes; b) the temperature distribution of the optimization result with pressure constraints lower than the initial pressure in the YZ1 and YZ2 planes; c) the temperature distribution of the optimization result with pressure constraints higher than the initial pressure in the YZ1 and YZ2 planes. Figure 4 This is an optimized result diagram of a two-fluid porous heat exchanger with a finned shell based on a three-period minimal surface.

[0068] In the figure: a) Temperature distribution of the initial (unoptimized) finned shell heat exchanger based on the three-period minimum surface in the YZ1 and YZ2 planes; b) Temperature distribution of the optimized finned shell heat exchanger based on the three-period minimum surface in the YZ1 and YZ2 planes.

[0069] Figure 5 This is an optimized result diagram of a U-shaped shell dual-fluid porous heat exchanger based on a three-period minimal surface.

[0070] In the figure: a) Temperature distribution of the initial (unoptimized) U-shaped shell heat exchanger based on the three-period minimum surface in the XZ1 and XZ2 planes; b) Temperature distribution of the optimized U-shaped shell heat exchanger based on the three-period minimum surface in the XZ1 and XZ2 planes. Detailed Implementation

[0071] The embodiments of the present invention include a function representation module for a two-fluid porous heat exchanger, a module for establishing an optimized mathematical model, a module for optimizing the process, and a module for improving the optimization algorithm.

[0072] Step 1: The function representation module for the two-fluid porous heat exchanger is used to represent and distinguish the solid domain and the two fluid domains of the two-fluid porous heat exchanger using a single material function. It passes the function representation of the heat exchanger to the optimized mathematical model, including:

[0073] First, we construct an improved functional representation of the three-period minimum surface: the three-period minimum surface can be precisely mathematically represented by trigonometric functions. Taking the G-triperium minimal surface as an example, its functional expression is:

[0074]

[0075] The positive and negative values ​​of its directed mileage field represent two disconnected spaces, and an improved functional representation is constructed based on this:

[0076]

[0077] The topological parameter function P(r) > 0 is a continuously differentiable function. It controls the topology of the surface by controlling the distribution of holes in the surface, and maintains the topology during topological changes. The distance scale of the directed distance field remains unchanged, and r = (x, y, z) is a vector in three-dimensional Euclidean space, where x, y, and z are the corresponding coordinates.

[0078] Then, the improved function surface By offsetting to both sides, a porous solid with thickness is constructed. The two offset surfaces are represented as follows:

[0079]

[0080]

[0081] The geometric parameter function W(r) controls the offset distance, which in turn controls the wall thickness of the porous solid. Two offset surfaces... and The space is divided into three regions: cold fluid, hot fluid, and solid. A solid is sandwiched between two curved surfaces, and two other unconnected spaces are connected to the hot fluid and cold fluid, respectively. In the experiment, the P(r) value of the G surface is in the range of [0.125, 0.5], which controls the pore distribution of the porous structure inside the heat exchanger, and the W(r) value of the G surface is in the range of [0.3, 1.1], which controls the wall thickness distribution.

[0082] Next, a thick outer shell is added to the constructed porous entity using Boolean operations, thus constructing a complete porous heat exchanger based on a three-period minimum surface. The complex and time-consuming Boolean operations are first converted into simple modifications to the directed distance field, using the Boolean union operation Ω of the two subdomains. i ∪Ω j It can be expressed as max(Φ) by the directed distance field function of the subdomain. i ,Φ j The outer shell is represented by a directed distance field as a function Φ. shell If the value is greater than 0, then functional representations for both fluids and solids can be constructed:

[0083]

[0084]

[0085]

[0086] Based on the aforementioned continuous functions, each subdomain can be represented by the directed distance field of the function. The formula for distinguishing the three regions of fluid 1, fluid 2, and solid, divided by the three-period minimal surface, is defined as follows:

[0087]

[0088] Finally, the three regions are each mapped to a single variable, and the distance field function is combined into a material function for the design domain using a linear interpolation function:

[0089]

[0090] Where H(x) is the Heaviside function, which is 0 when x is negative and 1 otherwise. ξ = 1, 0.5, 0 correspond to fluid 1, solid, and fluid 2, respectively. To calculate the derivative of the H(x) function during optimization, its normalized approximation form is generally used here:

[0091]

[0092] The parameter η∈(0,1) controls the shape of the Heaviside function, thereby controlling the degree of penalty for intermediate quantities. Since the topological parameter P(r) and the geometric parameter W(r) are offset surfaces... and The variables in the text are therefore obtained from the Boolean operation of the offset surface. and The material variable ξ obtained by linear interpolation is a function containing the topological parameter P(r) and the geometric parameter W(r).

[0093] Step 2: Establish an optimization mathematical model module to build a mathematical model for maximizing the total heat transfer problem under fixed pressure constraints. Numerical calculations are performed based on the material function of the two-fluid porous heat exchanger. The calculated physical fields, objective function values, and constraint values ​​are then passed to the specific optimization process and solver, including:

[0094] A suitable optimization model is selected: Based on the goal of obtaining a high-efficiency two-fluid porous heat exchanger with appropriate porosity and wall thickness distribution under given pressure constraints, the total heat transfer is used as the objective function, pressure as the constraint condition, and the topological parameter function P(r) and geometric parameter function W(r) as design variables for optimization. The optimization focuses on two mass transfer problems (one for each fluid) and one global heat transfer problem under the assumptions of steady-state, constant, incompressible fluids and neglecting heat generation from viscous dissipation. The boundary conditions for the mass transfer problem include Dirichlet boundary conditions at the fluid inlet, outlet, and the remaining boundaries of the design domain, given the inlet inflow velocity, the pressure-free condition at the outlet, and the no-slip condition for the fluid in the remaining boundaries. The boundary conditions for the heat transfer problem consist of the Dirichlet boundary conditions at the corresponding fluid inlet, giving the fluid inflow temperature. Homogeneous Riemann boundary conditions are applied to the remaining boundaries of the design domain, resulting in insulation outside the design domain. Based on the above description, the optimization model for the two-fluid heat transfer problem is established as follows:

[0095]

[0096] Make:

[0097]

[0098]

[0099]

[0100]

[0101]

[0102] Where u = [u1, u2, u3] T Let τ represent flow, p represent dynamic pressure, T represent temperature, Re be the Reynolds number, Pe be the Péclet number, and α be the reverse osmosis rate. The subscripts τ = {1, 2} represent fluid 1 and fluid 2, respectively. It is the fluid τ outlet boundary. τ is the fluid inlet boundary. J is the overall heat transfer objective function, representing the sum of the total heat released by fluid 1 and the total heat absorbed by fluid 2. G is the fluid pressure constraint, obtained by combining the geometric mean pressures g1 and g2 of the two fluids at the inlet using the KS function, where σ is a constant parameter of the KS aggregation function, and p max It is the maximum allowable pressure at the fluid inlet. (2.12) is the variational form of the mass and heat transfer equations, which can be directly obtained using the finite element method. The physical field (u,p,T)∈U×M×T is given by the variational form where <·,·> is the inner product of the two terms. b(v,q)=-∫ Ω q·div·vdΩ, solution space: U={u∈H1 (Ω)|u=u in on Γ in}、M=L2(Ω)、T={T∈H 1 (Ω)|T=T in on Γ in}, Test space: U0={δu∈H 1 (Ω) 3 |δu=0onΓ in}, M0=L2(Ω), T0={δT∈H 1 (Ω)|δT=0onΓ in}, H 1 It is a first-order Hilbert space.

[0103] Step 3: Optimize the process and improve the optimization solver module. The optimization process is divided into three steps: pre-optimization, pore distribution optimization, and wall thickness distribution optimization. A GCMMA optimization solver with monotonic programming operators is used to obtain a porous structure with efficient heat transfer that simultaneously features smooth pores and varying wall thickness, including:

[0104] First, the optimization of the topological parameter function P(r) and the geometric parameter function W(r) is transformed into the optimization of the parameters at a finite number of interpolation base points using the radial basis interpolation method.

[0105] Pre-optimization uses a small number of interpolation base points and constant geometric parameter functions as design variables. This initial adjustment of the porous structure yields a better initial structure for further, more detailed adjustments. Selecting fewer interpolation base points effectively reduces optimization complexity and the number of iterations. n0 (7–16 in experiments) interpolation base points are randomly selected in the solution domain. Then there is an interpolation form:

[0106]

[0107] Where n0 is the total number of control points within the design domain Ω, and M i (r) is the corresponding computable coefficient function. The topological parameter function value of the interpolation base point is used as a design variable. The geometric parameter function W(r) = W0 is a constant function, and W0 is a design variable. By using the constant wall thickness as a design variable, coupled optimization of the void distribution and wall thickness distribution is achieved, which is closer to the global optimum.

[0108] Pore ​​distribution optimization uses topological parameter values ​​from a larger number of interpolation base points as design variables, allowing for further fine-tuning of the pores in the porous structure. In the solution domain, n is randomly selected. t (200 interpolation base points in the experiment) Then there is an interpolation form:

[0109]

[0110] Where n t N is the total number of interpolation base points within the design domain Ω. i (r) is the corresponding computable coefficient function. These are the topology parameter function values ​​of the interpolation base points, used as design variables.

[0111] Wall thickness distribution optimization uses geometric parameter values ​​from multiple interpolation points as design variables to fine-tune the wall thickness of porous structures. Similarly, the interpolation form of the geometric parameter function is:

[0112] in These are the geometric parameter values ​​of the interpolation base points, used as design variables.

[0113] Then, using monotonic programming operators, the original problem is transformed into an equivalent structural optimization problem. The objective and constraints in the original problem can be represented by functions, in the following form:

[0114]

[0115] Add the monotonic operator x = ln(1 + y1 / γ )·N norm Then, the original problem is transformed into an equivalent problem:

[0116] h γ (y)=h[ln(1+y1 / γ)·N norm ],

[0117]

[0118] Where X and Y γ These are functions h(x) and h, respectively. γ The domain of (y), low k and upp k These are the lower and upper bounds of the k-th design variable in the original problem, respectively. γ is the parameter of the monotonic programming operator, used to control the degree of change in transforming the original problem into an equivalent problem. N norm =[N norm,1 ,…,N norm,n ] T This is a normalized parameter vector used to ensure that the value of each design variable in the equivalent problem is between 0 and 1, avoiding numerical errors caused by excessively large or small ranges of design variables. For the k-th design variable, we have:

[0119]

[0120] Next, the sensitivity of the equivalent optimization problem is calculated using the adjoint method, and then iteratively converged to the optimal solution in the GCMMA solver. In the experiment, the internal iteration count of GCMMA was set to 3 to balance the robustness and efficiency of the optimization algorithm. The three optimization steps were performed sequentially, with each step optimized using the GCMMA solver. The sensitivity calculation formulas for the objective function and constraint functions with respect to the design variable D are as follows:

[0121]

[0122]

[0123]

[0124] Where H′ is the derivative of the Heaviside function, and the sensitivity of physical parameters u, p, T to design variables is calculated using the adjoint method. Topological parameters P(r) and geometric parameters W(r) are considered as design variables for optimization. and The value of is determined by the selection of different combinations of design variables in different optimization steps, which will lead to ... and The calculation results differ. In pre-optimization, the design variables are the topological parameter variables of the interpolation base points and the constant geometric parameter functions, D={P i ,W0|i=1,…,n0}. The sensitivity calculation results Inputting the data into the GCMMA optimization solver yields a preliminary optimized structure with optimal pore size and overall wall thickness during the pre-optimization stage. This porous structure features smooth, continuous pores and wall thickness, serving as the initial structure for subsequent pore distribution optimization and wall thickness distribution optimization. In our experiments, the pre-optimization converged within 60 iterations. In the pore distribution optimization, the design variable is the topological parameter variable of the interpolation base point, D = {P}. i |i=1,…,n t The sensitivity calculation results in... Inputting the data into the GCMMA optimization solver yields an optimized structure with continuously varying pore sizes, which serves as the initial structure for wall thickness distribution optimization. Since the initial structure used in this optimization stage is the pre-optimized solution obtained during the initial optimization phase, and the geometric parameter function W(r) is fixed based on the pre-optimized solution W0, convergence in optimizing the pore distribution is easily achieved. In our experiments, the pore distribution optimization converged within 80 iterations. In the wall thickness distribution optimization, the design variable is the geometric parameter variable, D={W i |i=1,…,n t The sensitivity calculation results in... Inputting the data into the GCMMA optimization solver yielded a porous structure exhibiting both smooth pores and varying wall thickness. Since the topological parameter function P(r) is the optimal solution for the pore distribution optimization and is fixed at this optimization stage, the convergence of the optimized geometric distribution is also well achieved. In our experiments, the wall thickness distribution optimization converged within 80 iterations.

[0125] After optimization algorithms involving pre-optimization, pore distribution optimization, and wall thickness distribution optimization, the final optimized solution is a two-fluid porous heat exchanger with smooth pores and varying wall thickness, meeting the requirements and achieving high heat exchange efficiency. To the authors' knowledge, this is the first work to propose a function optimization and design approach for porous heat exchangers based on a three-period minimum surface for two-fluid heat transfer problems. The proposed porous heat exchanger based on a three-period minimum surface has structural advantages in heat transfer: a high specific surface area ratio increases the contact area between the two fluids; full connectivity and high smoothness reduce flow resistance; and good mechanical properties provide some resistance to external forces and thermal deformation. Compared with existing methods, the advantages of this invention are: First, the optimization process includes pre-optimization, pore distribution optimization, and wall thickness distribution optimization, simultaneously incorporating topological and geometric variations, increasing the design freedom; and using wall thickness as a design variable prevents mixing of the two fluids without requiring additional constraints. Second, using an improved solver incorporating monotonic programming operators accelerates the arrival at optimization constraints and reduces the possibility of getting trapped in local minima during the optimization process. Furthermore, finite element calculations are performed within the design domain, eliminating the need to update the finite element mesh with each iteration, thus avoiding time-consuming meshing. Finally, the functional representation of the heat exchanger can be used to directly generate models of arbitrary precision and different file types, and the good smoothness and connectivity of the heat exchange structure are beneficial for additive manufacturing.

Claims

1. An optimization and design method for a two-fluid porous heat exchanger based on a three-period minimum surface, characterized in that, The optimization and design method includes a functional representation module for two-fluid porous heat exchangers, a module for establishing an optimization mathematical model, an optimization process, and an improved optimization algorithm module. The steps are as follows: Step 1: The function representation module for the two-fluid porous heat exchanger is used to represent and distinguish the solid domain and the two fluid domains of the two-fluid porous heat exchanger using a single material function, and to pass the function representation of the porous heat exchanger to the optimized mathematical model; The first step is to develop a precise mathematical representation of trigonometric functions based on a three-period minimum surface. Constructing an improved function representation: The topological parameter function P(r) > 0 is a continuously differentiable function. It controls the topology of the surface by controlling the distribution of holes in the surface, and maintains the topology during topological changes. The distance scale remains unchanged, and r = (x, y, z) is a vector in three-dimensional Euclidean space, where x, y, and z are the corresponding coordinates. The second step involves improving the function surface from the first step. By offsetting to both sides, a porous solid with thickness is constructed. The two offset surfaces are represented as follows: Among them, the geometric parameter function W(r) controls the offset distance, and thus controls the wall thickness of the porous solid; two offset surfaces and Divide space into fluid 1, that is... Fluid 2 and solid and Three unconnected regions; fluid 1 flows into the heat exchanger at a higher temperature, T in =1; Fluid 2 flows in at a lower temperature, T in =0; The third step involves adding a heat exchanger shell to the porous solid using Boolean operations, thus constructing a complete porous heat exchanger based on a three-period minimum surface. The heat exchanger shell is represented by a directed distance field as a function Φ. shell If the value is greater than 0, then the two fluids, the solid, and the heat exchanger shell can be represented by the following functions: Where max(·,·) is the maximum value function. and Φ Solid (r) represent the fluid 1 function, fluid 2 function, and solid function of the shell-and-shell heat exchanger, respectively. Based on these functions, the formula for distinguishing the three regions of fluid 1, fluid 2, and solid is defined as follows: Fourth, in order to distinguish the three regions by a single variable, a linear interpolation function is used to combine the directed distance field function into a material function for the design domain: Where H(x) is the Heaviside function, which is 0 when x is negative and 1 otherwise. ξ(r) is the material function, where ξ(r) = 1, 0.5, and 0 correspond to fluid 1, solid, and fluid 2, respectively. Step 2: Establish an optimization mathematical model module to build a mathematical model for maximizing total heat transfer under a fixed pressure constraint. Numerical calculations are performed based on the material function ξ(r) in the function representation module of the two-fluid porous heat exchanger. Then, the calculated physical fields, objective function values, and constraint values ​​are passed to the specific optimization process and solver. The optimization model uses total heat transfer as the objective function, pressure as the constraint, and topological parameter function P(r) and geometric parameter function W(r) as design variables. The optimization mathematical model for the two-fluid porous heat exchanger is as follows: Make: Where u = [u1, u2, u3] T Here, represents the flow velocity, n represents the unit normal vector pointing out of the design domain, p represents the dynamic pressure, T represents the temperature, Re is the Reynolds number, Pe is the Péclet number, and α is the reverse osmosis rate. It is the fluid τ outlet boundary. τ represents the fluid inlet boundary, where the subscript τ = {1, 2} represents fluid 1 and fluid 2 respectively; Ω represents the design domain of the heat exchanger; J is the overall heat transfer objective function; G is the fluid pressure constraint, obtained by combining the geometric mean pressures g1 and g2 of the two fluids at the inlet using the KS function; σ is a constant parameter of the KS aggregation function; p max It is the maximum allowable pressure at the fluid inlet. Equation (1.10) is the variational form of the mass transfer and heat transfer equations. In the formula, a0(u, δu) =<u,δu> , b(δu, p)=-∫ Ω p·div(δu)dΩ, where <·,·> are the inner product of the two terms. It is a gradient operator, where δu, δp and δT are the test functions corresponding to flow velocity u, pressure p and temperature T respectively, and belong to the test spaces U0, M0 and T0 respectively. According to the variational form, the physical field (u, p, T) ∈ U×M×T can be directly obtained by the finite element method, where U, M and T are the solution spaces. Step 3: Optimize the process and improve the solver module. The optimization process is divided into pre-optimization, pore distribution optimization and wall thickness distribution optimization. A GCMMA optimization solver with monotonic programming operators is used to obtain a porous structure with efficient heat transfer that has both smooth pores and varying wall thickness. First, the topological parameter function P(r) and the geometric parameter function W(r) independently control the pore size and wall thickness of the porous structure. The radial basis interpolation method transforms the optimization of the parameter functions into the optimization of parameters at a finite number of interpolation base points. Pre-optimization uses the topological parameter values ​​of a small number of interpolation base points and the constant geometric parameter function as design variables, which is an initial adjustment of the porous structure to obtain an optimized structure as a better initial structure for more detailed adjustments. Pore distribution optimization uses the topological parameter values ​​of more interpolation base points as design variables, which is a further adjustment of the pore size of the porous structure. Wall thickness distribution optimization uses the geometric parameter values ​​of more interpolation base points as design variables, which is a further adjustment of the wall thickness of the porous structure. Then, all three optimization steps use the GCMMA optimization solver with monotonic programming operators to find the optimal solution of the established mathematical model, transforming the original problem into an equivalent structural optimization problem. The objective and constraints in the original problem are represented by functions: Add the monotonic operator x = ln(1 + y) 1 / γ )·N norm Then, the original problem is transformed into an equivalent problem: Where X and Y γ These are functions h(x) and h, respectively. γ The domain of (y), low k and upp k These are the lower and upper bounds of the k-th design variable in the original problem, respectively; γ is the parameter of the monotonic programming operator, used to control the degree of change in transforming the original problem into an equivalent problem; N norm =[N norm，1 ,…,N norm，n ] T It is a normalized parameter vector used to ensure that the value of each design variable in the equivalent problem is between 0 and 1, avoiding numerical errors caused by the design variable range being too large or too small; finally, the sensitivity of the equivalent optimization problem is calculated using the adjoint method, and the input is used to iteratively update the design variables in the GCMMA optimization solver until it converges to the optimization solution.

Images