Method and system for optimizing a battery thermal management system coupled with a topological liquid cooling plate and phase change material

Abstract

The application discloses a kind of topological liquid cooling plate coupling phase change material battery thermal management system optimization method and system, it is related to battery thermal management technical field, topological liquid cooling plate coupling phase change material battery thermal management system optimization method mainly includes: using finite element analysis tool and numerical simulation tool to model composite battery thermal management system, obtain composite battery thermal management system model and liquid cooling plate two-dimensional topological model;Using the mathematical model of topological optimization to optimize the liquid cooling plate two-dimensional topological model, obtain the optimized liquid cooling plate two-dimensional topological model;According to composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topological model are optimized, obtain the multi-objective optimization design result of topological type liquid cooling plate and phase change material coupling composite cooling system.The topological liquid cooling plate coupling phase change material battery thermal management system optimization method and system provided by the application can improve the heat dissipation performance.

Description

Technical Field

[0001] This invention relates to the field of battery thermal management technology, and more specifically, to an optimization method and system for a thermal management system of a topological liquid-cooled plate coupled with a phase change material battery. Background Technology

[0002] Currently, lithium-ion batteries (LIBs) are widely used as the main power source for new energy vehicles due to their significant advantages such as high energy density, low self-discharge rate, and high average output voltage. The heat generated inside the battery is dissipated to the surrounding environment through mechanisms such as thermal conduction, thermal convection, and thermal radiation. However, under extreme conditions such as high-temperature operating environments or rapid charging and discharging, if the heat generated by the lithium-ion battery cannot be effectively released in a timely manner, the internal temperature of the battery will rise sharply. When the battery temperature exceeds 50°C, the anode active material may suffer structural damage, thus significantly accelerating the battery aging process. If the temperature continues to rise above the critical point, it is even more likely to trigger thermal runaway reactions in the battery, and may even lead to serious consequences such as explosions. Therefore, in-depth exploration and innovative design of novel battery thermal management systems (BTMS) have profound scientific significance and broad practical application value for improving the thermal management efficiency and safety of power batteries and the entire new energy vehicle.

[0003] Currently, mainstream BTMS technologies include air cooling, liquid cooling, phase change material cooling, and composite cooling. Research indicates that the optimal operating temperature range for lithium-ion batteries should be maintained between 20°C and 45°C, and the maximum temperature difference within the battery pack (ΔT) should be minimized. maxThe temperature should be strictly controlled below 5℃. Liquid cooling, due to its advantages such as high-efficiency heat dissipation, uniform temperature, small size, and light weight, has been widely used by major automakers. To increase the heat exchange area of ​​liquid cooling plates and improve cooling efficiency, many scholars have designed liquid cooling plate structures, including microchannel, fractal, biomimetic, and topology optimization designs. Among them, topology optimization design can break free from the limitations of size optimization, achieving higher design freedom, wider design space, and greater flexibility, and has been widely used in the cooling of square batteries. Wang et al. explored in depth the specific impact of various objective functions such as maximum heat exchange efficiency, maximum outlet enthalpy, and minimum temperature on the topology optimization channel morphology and its cooling performance, revealing significant differences in topology optimization results under different objective function guidance. Wu et al. used a multi-objective topology optimization strategy to innovatively design the cold plate and made a comprehensive comparison with the traditional seven-straight parallel channel cold plate. Under the condition of setting the inlet flow velocity to 0.4 m / s, the optimized cold plate is significantly better than the traditional design, specifically, the maximum average temperature of the LIB is reduced by 10.3%, the maximum average temperature difference between LIBs is reduced by 59.4%, and the pressure drop is also reduced by 23.9%. Zhan et al.'s research focused on the impact of inlet and outlet positions on battery heat dissipation efficiency during topology optimization. Through systematic analysis, they found that the cold plate with a parallel diagonal structure exhibited superior heat transfer efficiency, improving it by 6.1% compared to other structures, and also exhibiting the lowest pressure drop. With the rapid development of 3D printing technology, the fabrication of topology-optimized liquid cooling plates has become simpler and more efficient. Guo et al. successfully fabricated a topology-optimized cold plate using aluminum through 3D printing technology and verified the accuracy of the simulation results through experiments. The results showed that the maximum relative errors in temperature and pressure drop were controlled within 0.07% and 9%, respectively, verifying the reliability of the design method.

[0004] In the process of exploring the advantages of various cooling technologies and striving to overcome their limitations, composite cooling strategies have emerged as a cutting-edge research direction, specifically encompassing various combinations of air and liquid cooling, air and PCM cooling, and liquid and PCM cooling. Among the literature discussing hybrid battery thermal management systems, the combination of liquid and PCM cooling has attracted considerable attention due to its unique advantages, cleverly integrating the power-free characteristics of PCM cooling with the high efficiency of liquid cooling. Huang et al. proposed a novel PCM-water-jacketed liquid-cooled BTMS. Experimental results showed that when the battery spacing was set to 8mm and a 6-channel structure model was adopted, the system exhibited the best heat dissipation effect. Kong et al. further designed a composite PCM and liquid-cooled coupled BTMS and deeply analyzed the impact of key parameters such as battery spacing, distance from the battery to the cooling water pipe, number of channels, and refrigerant flow rate on system performance during cyclic charging and discharging. Through optimized design, the system successfully controlled the highest battery pack temperature to 41.1℃ at the end of 3C discharge, while maintaining the maximum temperature difference at 4℃, demonstrating excellent temperature control capabilities. Ping et al. established a thermal management method for prismatic batteries that couples PCM with liquid-cooled pipes. Their research shows that, while ensuring effective cooling capacity, appropriately reducing the coolant flow rate can significantly reduce the power consumption required by the liquid cooling system, achieving a good balance between energy efficiency and cost. Furthermore, Akbarzadeh et al. proposed an innovative liquid cooling plate structure design that incorporates PCM material, achieving significant reductions in energy consumption and weight of 30% and 36%, respectively, compared to traditional liquid cooling plates.

[0005] Currently, topology-optimized liquid cooling plates are typically used in prismatic batteries, with fewer applications in cylindrical batteries. Due to their circular structure, cylindrical batteries have a smaller direct contact area with the liquid cooling plate, and the use of topology-optimized liquid cooling plates alone in cylindrical batteries has certain limitations.

[0006] The above content is only used to help understand the technical solution of the present invention and does not represent an admission that the above content is prior art. Summary of the Invention

[0007] The purpose of this invention is to provide an optimization method and system for a thermal management system of a topological liquid cooling plate coupled with a phase change material battery, which can improve heat dissipation performance.

[0008] This invention provides an optimization method for a thermal management system of a battery with a topological liquid-cooled plate coupled to a phase change material, comprising the following steps: S1: Modeling the composite battery thermal management system using finite element analysis tools and numerical simulation tools to obtain a composite battery thermal management system model and a two-dimensional topological model of the liquid-cooled plate; S2: Optimizing the two-dimensional topological model of the liquid-cooled plate using a mathematical model of topology optimization to obtain an optimized two-dimensional topological model of the liquid-cooled plate; S3: Performing multi-objective optimization based on the composite battery thermal management system model and the optimized two-dimensional topological model of the liquid-cooled plate to obtain the multi-objective optimization design result of the composite cooling system coupled with the topological liquid-cooled plate and the phase change material.

[0009] This invention also provides an optimization system for a topological liquid-cooled plate coupled phase change material battery thermal management system. The system includes the following modules: a model building module configured to model the composite battery thermal management system using finite element analysis and numerical simulation tools to obtain a composite battery thermal management system model and a two-dimensional topological model of the liquid-cooled plate; a two-dimensional topology optimization module configured to optimize the two-dimensional topology model of the liquid-cooled plate using a mathematical model of topology optimization to obtain an optimized two-dimensional topology model of the liquid-cooled plate; and a multi-objective optimization module for the composite cooling system configured to perform multi-objective optimization based on the composite battery thermal management system model and the optimized two-dimensional topology model of the liquid-cooled plate to obtain the multi-objective optimization design result of the composite cooling system coupled with the topological liquid-cooled plate and the phase change material.

[0010] The optimization method and system for the thermal management system of a topological liquid-cooled plate coupled with a phase change material battery provided by this invention have the following beneficial effects:

[0011] This invention employs topology optimization for the structural design of the liquid cooling plate, with the optimization objectives being a dual goal of reducing average temperature and flow dissipation, and also incorporates a temperature root mean square error constraint; the heat source intensity is 5×10 5 W / m 3The optimal liquid cooling plate, optimized under a constraint of 1.4K, exhibits the best overall performance. The liquid cooling plate was 3D printed from aluminum alloy, with a flow channel thickness of 3mm and a plate thickness of 5mm. This invention first establishes the relationship between cooling system parameters and target values ​​using response surface methodology, then performs multi-objective optimization of the system parameters using a genetic algorithm. The optimal performance was achieved when the PCM thermal conductivity was 1.65 W / (mK), the latent heat of phase change was 268.82 J / g, the phase change temperature was 36.27℃, the inlet velocity of the liquid cooling plate was 0.04 m / s, and the inlet temperature was 34.9℃. At 6℃, the battery thermal management system achieves optimal heat dissipation, with the highest battery pack temperature at 41.05℃, the maximum temperature difference at 2.99℃, and the pressure drop at the inlet and outlet of the liquid cooling plate at 5.68Pa. This invention proposes an innovative heat dissipation solution for the special structure of cylindrical batteries: using PCM to encapsulate the cylindrical battery and attaching a liquid cooling plate to the outside of the PCM to improve heat dissipation performance. This design not only fully utilizes the advantages of PCM's flexible matching with batteries of different shapes and its zero-power consumption characteristics but also combines the high efficiency of liquid cooling, thereby achieving superior thermal management results. Attached Figure Description

[0012] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0013] Figure 1 This is a flowchart of the optimization method for the thermal management system of a topological liquid-cooled plate coupled with a phase change material battery provided by the present invention;

[0014] Figure 2 This is a schematic diagram of the composite battery thermal management system provided by the present invention;

[0015] in, Figure 2 (a) is a model of a composite battery thermal management system. Figure 2 (b) is its 1 / 4 symmetric model;

[0016] Figure 3 This is the two-dimensional topological model of the liquid cooling plate provided by the present invention;

[0017] Figure 4 This is a diagram of the liquid cooling plate topology under different constraints provided by the present invention;

[0018] Figure 5 This is a schematic diagram showing the changes in temperature difference, average temperature, and pressure drop of the liquid cooling plate under different constraints provided by the present invention;

[0019] Figure 6 These are (a) a three-dimensional model and (b) a physical image of the liquid cooling plate provided by this invention;

[0020] in, Figure 6 Image (a) is a 3D model of the liquid cooling plate. Figure 6 Image (b) is a photograph of the actual liquid cooling plate;

[0021] Figure 7 This is a comparison chart of the simulated average temperature of the liquid cooling plate and experimental data provided by the present invention;

[0022] Figure 8 This is a schematic diagram comparing the actual values ​​and predicted values ​​in the response surface regression model provided by this invention;

[0023] Figure 9 This is the Pareto optimal solution distribution diagram provided by the present invention;

[0024] Figure 10 This is a graph showing the variation of the highest temperature and temperature difference of the battery pack over time under charge-discharge cycle conditions provided by the present invention;

[0025] Figure 11 This invention describes the change in the liquid phase fraction of PCM over time under charge-discharge cycle conditions. Detailed Implementation

[0026] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0027] Figure 1 A schematic diagram of the optimization method for the thermal management system of a phase change material battery coupled with a topological liquid-cooled plate according to this embodiment is shown. In this embodiment, the optimization method for the thermal management system of a phase change material battery coupled with a topological liquid-cooled plate includes the following steps:

[0028] S1: The composite battery thermal management system is modeled using finite element analysis tools and numerical simulation tools to obtain the composite battery thermal management system model and the two-dimensional topology model of the liquid cooling plate;

[0029] As an exemplary embodiment, the finite element analysis tool and the numerical simulation tool are ANSYS Fluent and COMSOL Multiphysics, respectively;

[0030] S2: Optimize the two-dimensional topology model of the liquid cooling plate using a mathematical model for topology optimization to obtain an optimized two-dimensional topology model of the liquid cooling plate;

[0031] In one exemplary embodiment, step S2 specifically includes: optimizing the two-dimensional topology model of the liquid cooling plate using a mathematical model for topology optimization to obtain an optimized two-dimensional topology model of the liquid cooling plate, as shown in the formula:

[0032]

[0033] Findγ,

[0034]

[0035] in, Ω is the average temperature function of the cooling plate, i.e., the objective function for heat transfer; T is the temperature; ψ is the fluid dissipation power, i.e., the objective function for dissipation; μ is the fluid dynamic viscosity, u i Let u be the velocity component in the i-direction. j Let x be the velocity component in the j-direction. j Let x be the coordinate variable in the j-direction. i Let be the coordinate variable in the i-direction, α(x) be the reverse osmosis rate, Π be the bi-objective optimization function, and ω be the weighting factor of the multi-objective function. ψ0 is the initial value of the average temperature; ψ0 is the initial value of the fluid dissipation power; T stdev For temperature root mean square deviation constraints, A is the design domain area; T ave The mean temperature is γ; Find indicates solving, γ is the local density of the material; Minnimiza indicates minimizing. Let ρ be the velocity vector. f For fluid density, The gradient is represented by P, where P is pressure and α(γ) is reverse osmosis. f For the reverse osmosis rate of the fluid domain, α s Let C be the reverse osmosis rate in the solid domain, q be the reverse osmosis penalty factor, and C be the reverse osmosis rate. f Let λ be the specific heat capacity of the fluid, and λ(γ) be the thermal conductivity. As a heat source, λ s λ is the thermal conductivity of a solid. f V is the thermal conductivity of the fluid, p is the thermal conductivity penalty factor, and V f T is the average fluid volume fraction. i This represents the temperature mean square deviation constraint value.

[0036] S3: Based on the composite battery thermal management system model and the optimized two-dimensional topology model of the liquid cooling plate, multi-objective optimization is performed to obtain the multi-objective optimization design results of the composite cooling system coupled with the topology liquid cooling plate and the phase change material.

[0037] In one exemplary embodiment, step S3 specifically includes:

[0038] S31: Based on the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topology model, establish a response surface model;

[0039] In one exemplary embodiment, step S31 specifically includes: establishing a response surface model based on the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topology model, as shown in the formula:

[0040]

[0041] Where Y is the objective function, x i and xj Let βi and βj represent the i-th and j-th independent variables, respectively. i β ii and β ij The regression coefficients representing the intercept and x, respectively. i linear effect, x i The secondary effect and x i and x j The linear interaction effect between them, where k is the number of independent variables; T max The highest temperature of the battery thermal management system is given by ΔP, where ΔP is the voltage drop and ΔT is the voltage drop. max A is the maximum temperature difference, B is the thermal conductivity, C is the latent heat of phase change, D is the inlet water temperature, and E is the inlet flow rate.

[0042] S32: Based on the response surface model, a multi-objective optimization design is performed using a genetic algorithm to obtain the multi-objective optimization design results of the composite cooling system coupled with the topological liquid cooling plate and the phase change material;

[0043] In one exemplary embodiment, step S32 specifically includes: based on the response surface model, using a genetic algorithm to perform multi-objective optimization design, obtaining the multi-objective optimization design result of the composite cooling system coupled with the topological liquid cooling plate and the phase change material, as shown in the formula:

[0044]

[0045] Where x represents the set of independent variables, k PCM Where L is the thermal conductivity, L is the latent heat of phase change, and T is the thermal conductivity. m v is the phase transition temperature. in For the inlet water temperature, T in For the inlet flow rate, minT max Min represents minimizing the maximum temperature, minΔT max minΔP represents minimizing the maximum temperature difference, and minΔP represents minimizing the pressure drop.

[0046] This embodiment provides an optimization system for a topological liquid-cooled plate coupled phase change material battery thermal management system. The system includes the following modules: a model building module, configured to: model the composite battery thermal management system using finite element analysis tools and numerical simulation tools to obtain a composite battery thermal management system model and a two-dimensional topological model of the liquid-cooled plate; a two-dimensional topology optimization module for the liquid-cooled plate, configured to: optimize the two-dimensional topology model of the liquid-cooled plate using a mathematical model of topology optimization to obtain an optimized two-dimensional topology model of the liquid-cooled plate; and a multi-objective optimization module for the composite cooling system, configured to: perform multi-objective optimization based on the composite battery thermal management system model and the optimized two-dimensional topology model of the liquid-cooled plate to obtain the multi-objective optimization design result of the composite cooling system coupled with the topological liquid-cooled plate and the phase change material.

[0047] Specifically, the liquid-cooled plate two-dimensional topology optimization module is configured to: optimize the liquid-cooled plate two-dimensional topology model using a mathematical model for topology optimization, to obtain an optimized liquid-cooled plate two-dimensional topology model, as shown in the formula:

[0048]

[0049] Findγ,

[0050]

[0051] in, Ω is the average temperature function of the cooling plate, i.e., the objective function for heat transfer; T is the temperature; Ψ is the fluid dissipation power, i.e., the objective function for dissipation; μ is the fluid dynamic viscosity, u i Let u be the velocity component in the i-direction. j Let x be the velocity component in the j-direction. j Let x be the coordinate variable in the j-direction. i Let be the coordinate variable in the i-direction, α(x) be the reverse osmosis rate, Π be the bi-objective optimization function, and ω be the weighting factor of the multi-objective function. Ψ0 is the initial value of the average temperature; T is the initial value of the fluid dissipation power; stdev For temperature root mean square deviation constraints, A is the design domain area; T ave The mean temperature is γ; Find indicates solving, γ is the local density of the material; Minnimiza indicates minimizing. Let ρ be the velocity vector. f For fluid density, The gradient is represented by P, where P is pressure and α(γ) is reverse osmosis. f For the reverse osmosis rate of the fluid domain, α s Let C be the reverse osmosis rate in the solid domain, q be the reverse osmosis penalty factor, and C be the reverse osmosis rate. f Let λ be the specific heat capacity of the fluid, and λ(γ) be the thermal conductivity. As a heat source, λ s λ is the thermal conductivity of a solid. f V is the thermal conductivity of the fluid, p is the thermal conductivity penalty factor, and V f T is the average fluid volume fraction. i This represents the temperature mean square deviation constraint value.

[0052] Specifically, the multi-objective optimization module of the composite cooling system is configured as follows: based on the composite battery thermal management system model and the optimized two-dimensional topology model of the liquid cooling plate, a response surface model is established; based on the response surface model, a genetic algorithm is used to perform multi-objective optimization design to obtain the multi-objective optimization design result of the composite cooling system coupled with the topology liquid cooling plate and the phase change material.

[0053] Specifically, establishing the response surface model based on the composite battery thermal management system model and the optimized liquid-cooled plate two-dimensional topology model includes: establishing the response surface model based on the composite battery thermal management system model and the optimized liquid-cooled plate two-dimensional topology model, as shown in the formula:

[0054]

[0055] Where Y is the objective function, x i and x j Let βi and βj represent the i-th and j-th independent variables, respectively. i β ii and β ij The regression coefficients representing the intercept and x, respectively. i linear effect, x i The secondary effect and x i and x j The linear interaction effect between them, where k is the number of independent variables; T max The highest temperature of the battery thermal management system is given by ΔP, where ΔP is the voltage drop and ΔT is the voltage drop. max A is the maximum temperature difference, B is the thermal conductivity, C is the latent heat of phase change, D is the inlet water temperature, and E is the inlet flow rate.

[0056] Specifically, the step of using a genetic algorithm to perform multi-objective optimization design based on the response surface model to obtain the multi-objective optimization design result of the composite cooling system coupling the topological liquid cooling plate and the phase change material includes: using a genetic algorithm to perform multi-objective optimization design based on the response surface model to obtain the multi-objective optimization design result of the composite cooling system coupling the topological liquid cooling plate and the phase change material, as shown in the formula:

[0057]

[0058] Where x represents the set of independent variables, k PCM Where L is the thermal conductivity, L is the latent heat of phase change, and T is the thermal conductivity. m v is the phase transition temperature. in For the inlet water temperature, T in For the inlet flow rate, minT max Min represents minimizing the maximum temperature, minΔT max minΔP represents minimizing the maximum temperature difference, and minΔP represents minimizing the pressure drop.

[0059] In some embodiments, the above-described optimization method for the thermal management system of a topological liquid-cooled plate coupled with a phase change material battery can also be implemented in the following manner. In this embodiment, the optimization method for the thermal management system of a topological liquid-cooled plate coupled with a phase change material battery includes:

[0060] 1. Mathematical model theoretical analysis;

[0061] (1) Thermal effect model of lithium-ion battery;

[0062] Because the heat generation and heat transfer processes of batteries are quite complex, the following assumptions are made when establishing the thermal effect model of lithium-ion batteries:

[0063] 1) Assume that the physical properties of the 21700 battery are constant and do not change with temperature and SOC, and that the internal heat generation is uniform.

[0064] 2) Specific heat capacity and density are isotropic, while thermal conductivity is anisotropic, but constant in each direction.

[0065] 3) Since thermal radiation has a relatively small effect compared to other heat transfer methods, it is not considered in the calculation.

[0066] 4) Ignore the convective heat transfer of the electrolyte inside the battery and only consider the heat conduction method.

[0067] 5) During the discharge cycle, the current density is uniform throughout the battery.

[0068] Based on the above assumptions, the thermal conductivity control equation for a battery with an internal heat source is as follows:

[0069]

[0070] In the formula: ρ b —Battery density (kg / m³) 3 );c b —Specific heat capacity of battery (J / (kg K)); k r , k z The thermal conductivity of the battery in the radial, circumferential, and axial directions (W / (m K)); —Heat generation rate per unit volume (W / m³) 3 ).

[0071] The heat generated by a battery during charging and discharging includes reaction heat, Joule heat, polarization heat, and side reaction heat. Side reaction heat is negligible compared to the other heat sources and is generally ignored in calculations.

[0072] According to Bernardi's theory, the formula for calculating the battery heat generation rate is:

[0073]

[0074] In the formula, —Battery heat generation rate per unit volume (W / m) 3 V—Volume of a single battery cell (m³) 3T—Battery operating temperature (K); I—Battery charging and discharging current during operation (A) (negative value during discharging, positive value during charging); U ocv U—open circuit voltage (V) and operating voltage (V); —Entropy heat coefficient (mV / K).

[0075] In the formula, I(U) ocv -U) is irreversible heat, composed of Joule heat and polarization heat, therefore:

[0076] I(U ocv -U)=I 2 (R j +R p ) = I 2 R t #(3)

[0077] R j and R p R represents the Joule internal resistance and polarization impedance (Ω), respectively. t The equivalent internal resistance (Ω) can be measured experimentally.

[0078] Therefore, equation (3) becomes:

[0079]

[0080] This is called reversible heat. The entropy coefficient can be positive or negative during battery discharge. If I and Same symbols indicate exothermic reactions, while opposite symbols indicate endothermic reactions.

[0081] (2) Theoretical analysis of solid-liquid phase change cooling model;

[0082] To simplify calculations, this embodiment makes the following assumptions about PCM:

[0083] 1) The physical properties of the solid and liquid phases of the phase change material are constant and equal;

[0084] 2) Ignoring the flow of the phase change material after melting, set the viscosity to 10. 5 kg / (m·s);

[0085] 3) The effect of gravity is not considered;

[0086] 4) Since there is no relative slip between the fluid and the solid, the interface is set as a coupling wall;

[0087] 5) The PCM is fully bonded to the battery, and contact thermal resistance is not considered.

[0088] The enthalpy change method is used to simulate heat transfer in a PCM, and its energy equation is as follows:

[0089]

[0090] In the formula, ρ PCM k PCM c P，PCM —Density of PCM (kg / m³) 3 ), thermal conductivity (W / (m K)) and specific heat capacity (J / (kg K)); L—latent heat of phase change of PCM (J / kg); β—liquid fraction; T0—initial temperature (K); T m —The temperature at which PCM begins to melt (K); T l —Temperature (K) at which PCM is completely melted.

[0091] (3) Theoretical analysis of liquid cooling model;

[0092] In constructing a liquid cooling model, the first step is to determine the fluid flow state based on the Reynolds number (Re), as shown in formula (8). For fluids in a pipe, if the calculated Reynolds number is below 2300, the fluid state can be defined as laminar flow. In this embodiment, for cases with multiple coolant flow velocities, only the Reynolds number corresponding to the maximum flow velocity needs to be considered. The calculated Reynolds number at the maximum flow velocity is 847. Therefore, when analyzing fluid behavior, a laminar flow model should be selected as the theoretical basis.

[0093]

[0094] In the formula, ρ w ,μ w and u represent the density, kinematic viscosity, and velocity of water, respectively; D h This is the equivalent diameter of the cooling channel.

[0095] The coolant in the liquid cooling plate is water. To simplify the model, we assume the fluid is continuous and incompressible. Based on this, the continuity equation, momentum equation, and energy equation of the liquid cooling system are as follows:

[0096] Continuity equation:

[0097]

[0098] Momentum equation:

[0099]

[0100] Energy equation:

[0101]

[0102] In the formula, P represents pressure; λ w and C w These represent the thermal conductivity and specific heat capacity of water, respectively.

[0103] (4) Theoretical Analysis of Topology Optimization Model

[0104] In topology optimization, conjugate heat transfer is achieved through fluid-structure interaction. The following are assumptions regarding flow and heat transfer in the design domain:

[0105] 1) Fluids are incompressible;

[0106] 2) Assume the design domain is a porous medium, consisting of both fluid and solid phases;

[0107] 3) The heat flux density through the cooling plate is uniform;

[0108] 4) Since the Reynolds number of the fluid is below 2300, the fluid flow is considered laminar.

[0109] 5) The thermophysical parameters of batteries and liquids do not change with temperature, such as thermal conductivity, specific heat capacity, and dynamic viscosity.

[0110] Based on the above assumptions, during topology optimization, the fluid flow satisfies the Navier-Stokes equations, and the continuity and momentum equations are as follows:

[0111]

[0112] In the formula, Let P be the velocity, F be the frictional force during fluid flow, and ρ be the velocity. f μ and μ represent the fluid's pressure, density, and kinematic viscosity, respectively.

[0113] The energy equation for fluid heat transfer is:

[0114]

[0115] In the formula, λ f and C f These represent the thermal conductivity and specific heat capacity of the fluid, respectively. This represents the heat generated when the battery discharges.

[0116] The energy equation for the solid region is:

[0117]

[0118] In the formula, λ s This represents the thermal conductivity of a solid.

[0119] In this embodiment, a density-based method is used for topology optimization design. The design concept assumes the material in the design domain is a porous medium, and the design variable γ represents the local density of the material, which can control the changes in fluid and solid states. The value of γ is either 0 or 1 (0 represents solid, 1 represents fluid). To better allocate the solid and fluid domains within the design domain, Brinkman's penalty model is used to add frictional forces during fluid flow within the design domain.

[0120]

[0121] In the formula, α represents the local reverse osmosis rate of the porous medium, which depends on the design variable γ. In this embodiment, the following interpolation function is selected to calculate its value:

[0122]

[0123] Where q is the reverse osmosis penalty factor, mainly used to adjust the interpolation function; in this embodiment, it is set to 0.1 to reduce gray elements. α f and α s Representing the reverse osmosis rates in the fluid and solid domains, respectively, typically α f =0, α s It relates to the dimensionless number Da, and is calculated as follows:

[0124]

[0125] In the formula, Da is the Darcy number, which describes the relationship between viscous force and frictional force in porous media, and its value is Da = 1 × 10⁻⁶. -4 L is the characteristic length, which is set as the inlet flow channel.

[0126] The thermal properties of materials in porous media are also affected by the design variable γ. The following interpolation function is chosen for calculating the thermal conductivity:

[0127]

[0128] In the formula, p is the thermal conductivity penalty factor, which is taken as 0.01 in this embodiment, and λ s , λ f Thermal conductivity is divided into solid domain and fluid domain.

[0129] Based on the above interpolation function, equations (13) and (14) can be written as:

[0130]

[0131] In topology optimization, the input design variable γ is a scalar function with boundaries [0,1]. It can take any value, leading to ripple-shaped or checkerboard-like problems in the topology process. Previous research has shown that Helmholtz density filters are typically used to address these problems, avoiding grid dependence. The equation is as follows:

[0132]

[0133] In the formula, R min γ is the filter radius, whose value is derived from the grid. i and γ fi These are the input design variables and the filtered design variables, respectively.

[0134] Filters can make the design variable γ more continuous, but they also give it intermediate values ​​between 0 and 1. Therefore, this embodiment introduces the Heaviside function for projection to reduce the gray area between the fluid and solid.

[0135]

[0136] Where β represents the projection slope, which is taken as 8 in this embodiment, and γ β This represents the hyperbolic tangent projection point, which is set to 0.5 in this embodiment.

[0137] 2. Model building;

[0138] Composite battery thermal management system such as Figure 2 As shown, the system consists of nine 21700 type batteries, three phase change material (PCM) chambers, and three liquid cooling plates. The batteries are arranged in a 3×3 row, surrounded by a 1mm thick PCM. A topologically optimized liquid cooling plate is placed between each row of batteries, with a total thickness of 5mm, including a 3mm thick flow channel. The flow channels of the liquid cooling plate are obtained by stretching a topology-optimized two-dimensional model. Due to the symmetry of the model, a quarter of the model is used as the calculation model, with two symmetry planes: the side and the bottom. The ambient temperature is 35℃, and the convective heat transfer coefficient h = 5W / (m²). 2 The interfaces between the battery and the PCM, the PCM and the cooling plate casing, and the casing and the coolant are all fluid-structure interaction interfaces, set as coupled walls. The coolant inlet boundary condition is set as a velocity inlet, and the coolant outlet boundary condition is set as a pressure outlet. The solution type is selected as Pressure-Based, and the solidification-melting model is enabled, at which point the energy equation is automatically activated. A transient simulation method is used, selecting the SIMPLE algorithm for solving, with a time step of 1 second.

[0139] The topology optimization design of the liquid cooling plate was performed using COMSOL Multiphysics 6.1 software. To save computation time, the two-dimensional liquid cooling plate model is as follows: Figure 3 As shown, due to symmetry, a 1 / 2 model is used as the computational model. The gray area is the topology optimization design domain, and the size of the design domain matches the composite thermal management model: l = 69 mm, h = 70 mm, a = 8 mm, b = 5 mm. The initial fluid volume fraction is 0.5. To accommodate the heat generation rate of the battery at different discharge rates, a higher heat source intensity is applied within the design domain, with a value of 5 × 10⁻⁶. 5 W / m 3 The flow channel inlet is a fully developed velocity inlet with a Reynolds number of 100 and an inlet temperature maintained at 25°C; the outlet is a pressure outlet with a relative pressure of 0 Pa; and the wall surface employs a no-slip adiabatic boundary condition. Specific material thermophysical parameters are shown in Table 1.

[0140] Table 1: Material Thermophysical Properties

[0141] Material properties numerical values unit <![CDATA[Fluid density ρ f > 1000 <![CDATA[kg / m 3 ]]> <![CDATA[Fluid thermal conductivity k f > 0.6 W / (m K) <![CDATA[Specific heat capacity C of fluid pf > 4200 J / (kg K) <![CDATA[Solid density ρ s > 2719 <![CDATA[kg / m 3 ]]> <![CDATA[Solid thermal conductivity k s > 237 W / (m K) <![CDATA[Solid specific heat capacity C ps > 871 J / (kg K)

[0142] 3. Liquid cooling plate topology optimization design;

[0143] (1) Objective function and temperature root mean square error constraint;

[0144] Considering the significant heat generated during lithium-ion battery discharge, this embodiment optimizes the cooling plate from two aspects: heat dissipation and flow dissipation, i.e., maximizing heat transfer and minimizing flow resistance. Regarding heat transfer, the objective function is to minimize the average temperature of the cooling plate. For flow resistance, the goal is typically to minimize power consumption; generally, lower power consumption results in a smaller voltage drop across the cooling plate.

[0145] The average temperature function of the cooling plate is:

[0146]

[0147] Pressure drop is expressed as fluid dissipation power, which can be expressed as:

[0148]

[0149] Since the two physical quantities mentioned above are of different orders of magnitude, they need to be normalized. The final bi-objective optimization function can be defined as follows:

[0150]

[0151] In the formula, ω is the weighting factor of the multi-objective function, ω = 0.5. Let Ψ be the objective function for heat transfer, Ψ be the objective function for heat dissipation, and Ω be the design domain.

[0152] Considering the requirement for temperature uniformity during battery operation, a temperature root mean square deviation constraint is added, expressed as follows:

[0153]

[0154] Therefore, the mathematical model for topology optimization is:

[0155] Findγ

[0156]

[0157]

[0158] In the formula, V f The average fluid volume fraction is V. f =0.5; T i This is the temperature root mean square deviation constraint value, which is taken as a constant.

[0159] (2) Results of topology optimization for liquid cooling plate;

[0160] Based on the liquid cooling plate at a heat source intensity of 5×10 5 W / m 3 The mean squared temperature deviation (2.35K) was used as the constraint value, and the constraint size was set to 1.2~2K, with an interval of 0.2K. The liquid cooling plate topologies obtained under different mean squared temperature constraints are shown below. Figure 4 As shown, as the constraint decreases, the flow channels concentrate to the right, and the temperature is higher near the outlet side, resulting in uneven temperature distribution of the liquid cooling plate. Therefore, after increasing the constraint, in order to make the temperature distribution more uniform, more flow channels will be topologically generated in the high-temperature region to reduce the temperature and enhance temperature uniformity.

[0161] Figure 5 The figure compares the temperature difference, average temperature, and inlet / outlet pressure drop of five liquid cooling plates under different constraints. As shown, the temperature uniformity of the liquid cooling plate increases with decreasing constraint value. This is because more flow channels near the outlet side increase, lowering the temperature in the high-temperature region at the outlet and thus reducing the temperature difference. The temperature difference of liquid cooling plate C4 is 6.77K, which is 27.8% lower than that of the unconstrained liquid cooling plate. However, the average temperature of the liquid cooling plate does not decrease continuously with decreasing constraint value; it increases when the constraint value is less than 1.4K. According to the temperature root mean square error formula, enhancing temperature uniformity can be achieved by reducing the temperature in the high-temperature region and increasing the average temperature. Therefore, the average temperature of the liquid cooling plate first decreases and then increases with decreasing constraint value. In addition, the inlet / outlet pressure drop gradually increases with decreasing constraint value. Considering the average temperature, temperature root mean square error, and inlet / outlet pressure drop of the liquid cooling plate, liquid cooling plate C4 is selected for further research.

[0162] The C4 liquid cooling plate was 3D extruded using SolidWorks. The flow channel thickness was set to 3mm, and the liquid cooling plate thickness was 5mm. Because the designed flow channel was close to the top of the liquid cooling plate, a 1mm height was reserved on both the top and bottom sides of the liquid cooling plate for ease of machining. Therefore, the actual machined liquid cooling plate height was 72mm. The 3D model and physical sample of the liquid cooling plate are shown below. Figure 6 As shown, the liquid-cooled plate was 3D printed by the manufacturer using aluminum alloy. In the experiment, a heating plate with a power of 41W was used to heat the liquid-cooled plate. Water was used as the coolant with a flow rate of 0.013 kg / s. The initial temperature of the coolant and the environment was 26.4℃. Five T-type thermocouples were placed on the surface of the liquid-cooled plate, and their average value was used as the surface temperature of the liquid-cooled plate. The experiment was repeated three times. The simulation results and experimental data of the liquid-cooled plate surface temperature are shown below. Figure 7 As shown, the simulation results are close to the experimental data, with an error of less than 1%, indicating that the simulation model can predict the cooling effect of the liquid cooling plate well and has high accuracy and reliability.

[0163] 4. Multi-objective optimization design of a composite cooling system coupled with a topology-type liquid cooling plate and a phase change material;

[0164] (1) Establish a response surface model;

[0165] Box-Behnken design (BBD) is a type of response surface design that generates fewer design points, significantly reducing computational complexity. This embodiment utilizes a BBD-based response surface model to construct a regression response surface model between the objective function and independent variables. Based on the preliminary single-factor analysis results, the top five independent variables in terms of sensitivity are selected: the thermal conductivity of the PCM, latent heat of phase change, phase change temperature, and the inlet temperature and inlet velocity of the liquid cooler. As shown in Table 2, each independent variable is divided into three levels, and the value range of the variables is selected based on the ideal value range and preliminary experiments. The objective function is the highest temperature T of the battery pack. max Maximum temperature difference ΔT max And pressure drop ΔP. Objective function T max The functional relationship between ΔP and the independent variable can be described by equation (29), ΔT max The functional relationship between the independent variable and the function can be described by equation (30):

[0166]

[0167] In the formula, β0, β0, β ii and β ij The regression coefficients representing the intercept and x, respectively. i linear effect, x i The secondary effect and x i and x j The linear interaction effect between them.

[0168] Forty-six experimental groups were generated using BBD design. Based on the BBD results, the highest temperature TBTMS was established using RSM. max Maximum temperature difference ΔT max The surrogate regression model for the pressure drop ΔP is shown in Equation (31).

[0169]

[0170] The significance and fit of the regression model were assessed using analysis of variance (ANOVA), and the results are shown in Table 3. A higher F-value, a lower P-value, and a more significant correlation coefficient indicate that the response surface regression model has high statistical significance. 2 The closer the correlation is to 1, the more accurate it is. The results show the highest temperature T... max Maximum temperature difference ΔT max The p-values ​​for both the pressure drop ΔP and the pressure drop ΔP are less than 0.0001, and the regression model R... 2 All are greater than 0.98. Figure 8 The comparison between the actual and predicted values ​​in the response surface regression model shows that the actual and predicted values ​​agree well, indicating that the regression model can accurately predict T. max ΔT max And ΔP, and a regression model will be used for optimization design in the future.

[0171] Table 2: Range of Design Variable Values

[0172]

[0173] Table 3: Significance and Fit Tests of the Model

[0174] <![CDATA[R 2 ]]> p-value F value <![CDATA[T max ]]> 0.9878 <0.0001 101.57 <![CDATA[ΔT max ]]> 0.9808 <0.0001 63.95 ΔP 1.0000 <0.0001 5.59E+05

[0175] (2) Multi-objective optimization design based on genetic algorithm;

[0176] In battery thermal management design, the three key performance indicators that are of primary concern are the maximum temperature of the battery pack, the maximum temperature difference, and the pressure drop of the liquid cooling plate. There is a conflict between these three objectives: reducing the maximum temperature and decreasing the temperature difference may increase the pressure drop. Therefore, an optimization method that can consider these objectives simultaneously is needed. This embodiment will use a genetic algorithm (GA) to achieve multi-objective optimization of the maximum temperature, maximum temperature difference, and pressure drop of the battery pack. The genetic algorithm is a search algorithm that simulates the principles of natural selection and genetics. It searches for the optimal solution from a set of candidate solutions through iterative evolution. The genetic algorithm has the advantages of strong global search capability, good adaptability, and strong multi-objective optimization capability. It has significant advantages, especially in dealing with complex, nonlinear, and multi-objective problems. Compared with other algorithms, the parallelism, robustness, and global search capability of the genetic algorithm make it an ideal choice for the optimization of battery thermal management systems. The optimization objectives and design parameter ranges are shown in Equation (32):

[0177]

[0178] Figure 9 To calculate the Pareto optimal solution distribution and to find a balance between cooling performance and pressure drop, the red point in the graph is selected as the optimal solution. The optimized objective functions are T... max =41.05℃, ΔT max =2.99℃, ΔP = 5.68Pa, the corresponding design parameters are: k PCM =1.65W / (m K),L=268.82J / g,T m =36.27℃, v in =0.04m / s,T in =34.96℃. Compared to the initial operating condition, with a 13.89% reduction in voltage drop, the highest battery pack temperature at the end of discharge decreased by 13.04%, and the highest temperature difference decreased by 5.88%. Finally, the calculation results of this optimized design were verified through numerical simulation, and the results are shown in Table 4. The error between the optimized value and the simulation result is small, T max ΔT max The errors for ΔP and ΔP are 2.24%, 6.11%, and 2.26%, respectively.

[0179] Table 4: Comparison of Optimization Results and Simulation Results

[0180] <![CDATA[T max (℃)]]> <![CDATA[ΔT max (℃)]]> ΔP(Pa) Optimization results 41.05 2.99 5.68 Simulation results 40.14 3.19 5.82 error 2.24％ 6.11％ 2.26％

[0181] To investigate the cooling performance of the optimized battery thermal management system (BTMS) under dynamic conditions, a continuous charge-discharge cycle was simulated. Each cycle consisted of three cycles: a 5C discharge (720 s) and a 2C charge (1800 s). The changes in the battery pack's maximum temperature and temperature difference over time are shown below. Figure 10 As shown, during the charge-discharge cycles of the battery pack, the highest temperature of the battery pack remained below 45℃, with the highest temperatures of the three cycles being 40.14℃, 40.60℃, and 40.73℃, respectively. The temperature difference of the battery pack remained stable within 5℃, with a maximum temperature difference of 3.72℃. This was achieved through changes in the liquid phase fraction of the PCM, such as... Figure 11 As shown, the liquid phase fraction of PCM exhibits a fluctuating trend, with a maximum liquid phase fraction of 0.48 and a minimum liquid phase fraction of 0.1. This indicates that some of the latent heat of the phase change material was utilized, and the phase change material was cooled in time, thus restoring its latent heat and preventing heat accumulation.

[0182] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A method for optimizing a topological liquid-cooled board coupled phase change material battery thermal management system, the method comprising: determining a thermal management system design; determining a thermal management system performance; and optimizing the thermal management system design based on the thermal management system performance. The method comprises the following steps: S1: modeling the composite battery thermal management system by using a finite element analysis tool and a numerical simulation tool to obtain a composite battery thermal management system model and a liquid cooling plate two-dimensional topological model; S2: optimizing the liquid cooling plate two-dimensional topological model by using a mathematical model of topological optimization to obtain an optimized liquid cooling plate two-dimensional topological model; S3: performing multi-objective optimization according to the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topological model to obtain a multi-objective optimization design result of the composite cooling system coupled with the topological liquid cooling plate and the phase change material; Step S3 specifically comprises: S31: establishing a response surface model according to the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topological model; S32: performing multi-objective optimization design by using a genetic algorithm according to the response surface model to obtain a multi-objective optimization design result of the composite cooling system coupled with the topological liquid cooling plate and the phase change material; Step S31 specifically comprises: establishing a response surface model according to the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topological model, such as formula: ， ， ， wherein, is the objective function, and represent the i-th independent variable and the j-th independent variable, respectively, , , and represent the regression coefficient of the intercept, the linear effect of the i-th independent variable, the quadratic effect of the i-th independent variable, and and the linear interaction effect between the i-th independent variable and the j-th independent variable, is the number of independent variables; is the maximum temperature of the battery thermal management system, is the pressure drop, is the maximum temperature difference, is the thermal conductivity, is the latent heat of phase transition, is the phase transition temperature, is the inlet water temperature, is the inlet flow rate.

2. The method of claim 1, wherein, Step S2 specifically comprises: optimizing the liquid cooling plate two-dimensional topological model by using a mathematical model of topological optimization to obtain an optimized liquid cooling plate two-dimensional topological model, such as formula: ， ， ， ， ， ， ， wherein is the average temperature function of the cooling plate, i.e. the objective function of the heat transfer; is the design domain; is the temperature; is the fluid dissipation power, i.e. the objective function of the dissipation; is the fluid dynamic viscosity, is the velocity component in i-direction, is the velocity component in j-direction, is the j-direction coordinate variable, is the i-direction coordinate variable, is the inverse permeability; is the bi-objective optimization function, is the weight factor of the multi-objective function, is the average temperature initial value; is the fluid dissipation power initial value; is the temperature mean square error constraint, is the design domain area; is the average temperature; denotes the solving, is the local density of the material; denotes the minimization; is the velocity vector, is the fluid density, denotes the gradient, is the pressure; is the inverse permeability; is the inverse permeability of the fluid domain, is the inverse permeability of the solid domain, is the inverse permeability penalty factor, is the thermal conductivity, is the heat source, is the solid thermal conductivity, is the fluid thermal conductivity, is the thermal conductivity penalty factor, is the average fluid volume fraction, T i is the temperature mean square error constraint value.

3. The method of claim 1, wherein: Step S32 specifically comprises: performing multi-objective optimization design by using a genetic algorithm according to the response surface model to obtain a multi-objective optimization design result of the composite cooling system coupled with the topological liquid cooling plate and the phase change material, such as formula: ， wherein, represents a set of arguments, is a thermal conductivity, is a latent heat of phase change, is a phase change temperature, is an inlet water temperature, is an inlet flow rate, represents minimizing a maximum temperature, represents minimizing a maximum temperature difference, represents minimizing a pressure drop.

4. A topology liquid cold plate coupled phase change material battery thermal management system optimization system, characterized in that, The system comprises the following modules: A model construction module configured to model the composite battery thermal management system by using a finite element analysis tool and a numerical simulation tool to obtain a composite battery thermal management system model and a liquid cooling plate two-dimensional topological model; A liquid cooling plate two-dimensional topological optimization module configured to optimize the liquid cooling plate two-dimensional topological model by using a mathematical model of topological optimization to obtain an optimized liquid cooling plate two-dimensional topological model; A composite cooling system multi-objective optimization module configured to perform multi-objective optimization according to the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topological model to obtain a multi-objective optimization design result of the composite cooling system coupled with the topological liquid cooling plate and the phase change material; The composite cooling system multi-objective optimization module is specifically configured to: establish a response surface model according to the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topological model; perform multi-objective optimization design by using a genetic algorithm according to the response surface model to obtain a multi-objective optimization design result of the composite cooling system coupled with the topological liquid cooling plate and the phase change material; The composite cooling system multi-objective optimization module is specifically configured to: ， ， ， wherein, is the objective function, and represent the i-th independent variable and the j-th independent variable, respectively, , , and represent the regression coefficient of the intercept, the linear effect of the i-th independent variable, the quadratic effect of the i-th independent variable, and and the linear interaction effect between the i-th independent variable and the j-th independent variable, is the number of independent variables; is the maximum temperature of the battery thermal management system, is the pressure drop, is the maximum temperature difference, is the thermal conductivity, is the latent heat of phase change, is the phase change temperature, is the inlet water temperature, is the inlet flow rate.

5. The topological liquid cold plate coupled phase change material battery thermal management system optimization system of claim 4, wherein, establish a response surface model according to the composite battery thermal management system model and the optimized liquid cooling plate two-dimensional topological model, such as formula: The liquid cooling plate two-dimensional topological optimization module is specifically configured to optimize the liquid cooling plate two-dimensional topological model by using a mathematical model of topological optimization to obtain an optimized liquid cooling plate two-dimensional topological model, such as formula: ， ， ， ， ， ， ， wherein, is the average temperature function of the cooling plate, i.e. the objective function of heat transfer; is the design domain; is the temperature; is the fluid dissipation power, i.e. the objective function of dissipation; is the fluid dynamic viscosity, is the velocity component in i-direction, is the velocity component in j-direction, is the j-direction coordinate variable, is the i-direction coordinate variable, is the inverse permeability; is the bi-objective optimization function, is the weight factor of the multi-objective function, is the average temperature initial value; is the fluid dissipation power initial value; is the temperature mean square error constraint, is the design domain area; is the average temperature; denotes solving, is the local density of the material; denotes minimizing; is the velocity vector, is the fluid density, denotes gradient, is the pressure; is the inverse permeability; is the inverse permeability of the fluid domain, is the inverse permeability of the solid domain, is the inverse permeability penalty factor, is the thermal conductivity, is the heat source, is the solid thermal conductivity, is the fluid thermal conductivity, is the thermal conductivity penalty factor, is the average fluid volume fraction, T i is the temperature mean square error constraint value.

6. The topological liquid cold plate coupled phase change material battery thermal management system optimization system of claim 4, wherein, The multi-objective optimization design result of the composite cooling system coupled by the topological liquid cooling plate and the phase change material is obtained according to the response surface model and by using the genetic algorithm, and specifically includes: the multi-objective optimization design result of the composite cooling system coupled by the topological liquid cooling plate and the phase change material is obtained according to the response surface model and by using the genetic algorithm, as shown in the formula: ， wherein, represents a set of arguments, is a thermal conductivity, is a latent heat of phase change, is a phase change temperature, is an inlet water temperature, is an inlet flow rate, represents minimizing the maximum temperature, represents minimizing the maximum temperature difference, represents minimizing the pressure drop.

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