Viscoelastic fluid heat flow coupling simulation method for micro-channel structure

Abstract

A viscoelastic fluid heat flow coupling simulation method oriented to a micro-channel structure comprises the steps that pretreatment is conducted, and a viscoelastic fluid constitutive model and material attributes are defined; then constructing a viscoelastic fluid heat flow coupling simplified mathematical model comprising a continuity equation, a simplified momentum equation, a simplified energy equation and a viscoelastic fluid constitutive equation based on logarithm reconstruction, and finally performing finite element simulation verification on the simplified mathematical model; according to the method, a mathematical model is simplified, a physical mechanism necessary for describing the flow and heat transfer coupling phenomenon of the low-Reynolds-number viscoelastic fluid in the micro-channel is reserved, and the numerical solution difficulty and the calculation cost of the model are reduced while the calculation precision is ensured; the method is suitable for efficiently and stably simulating the steady-state heat flow distribution of the viscoelastic fluid in the micro-channel device under the Reynolds number, and a new thought is provided for design and optimization of applications such as micro-channel heat management and micro-channel chips.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of micro-channel heat dissipation, and particularly relates to a viscoelastic fluid heat flow coupling simulation method for a micro-channel structure. BACKGROUND

[0002] With the rapid development of electronic information technology, electronic components represented by CPUs, GPUs and laser diodes are continuously evolving towards higher power density and smaller size, resulting in a sharp rise in heat flux density. The heat dissipation problem has become a key bottleneck restricting the performance, reliability and service life of the components. Traditional heat dissipation schemes such as air cooling, heat pipes and heat plates have gradually approached their limits in the face of extremely high heat flux density scenarios. Micro-channel technology is considered as a promising solution to the heat dissipation challenge of the next generation of high-power electronic devices due to its compact structure and high efficiency of heat exchange. However, due to the scale effect of micro-channels, Newtonian fluids such as water are usually in a laminar state under low Reynolds number conditions in the flow channel, and heat transfer mainly depends on the thermal conduction between fluid layers, which limits the heat exchange efficiency of micro-channels and poses a key challenge to improve their heat exchange capacity.

[0003] When non-Newtonian viscoelastic fluid flows through a micro-channel, the polymer chains will undergo compression and stretching. This characteristic enables viscoelastic fluid to generate elastic turbulence under low Reynolds number conditions, which effectively promotes the mixing between fluid layers and provides potential for significantly enhancing heat transfer. Studies have shown that under the same Reynolds number, the local heat transfer efficiency of viscoelastic fluid in elastic turbulence state can be up to three times higher than that of Newtonian fluid. The patent application entitled "Micro-scale heat transfer enhancement device using viscoelastic fluid pulsating flow resonance effect" (Publication No. CN112966420A) has verified the heat transfer enhancement effect of viscoelastic fluid in micro-channel experiments; in the numerical simulation of viscoelastic fluid, open-source simulation software OpenFOAM and commercial simulation software Fluent are commonly used for simulation. These simulation software are mainly applied to conventional flow channels with high flow rates, and the simulation calculation speed is relatively slow. There is still a lack of simplified calculation method for low Reynolds number viscoelastic fluid heat flow coupling in micro-channels. SUMMARY

[0004] In order to overcome the shortcomings of the prior art, the purpose of the present application is to provide a viscoelastic fluid heat flow coupling simulation method for a micro-channel structure, to establish a simplified mathematical model of viscoelastic fluid heat flow coupling, and to realize low Reynolds number viscoelastic fluid heat flow coupling calculation at micro scale.

[0005] In order to achieve the above-mentioned objectives, the technical scheme adopted by the present application is as follows:

[0006] A viscoelastic fluid heat flow coupling simulation method for micro-channel structure, first, a viscoelastic fluid heat flow coupling simplified mathematical model is established, the finite element method is used for solving, according to the control equation and boundary conditions, the control equation is transformed into weak form and spatially dispersed, forming a nonlinear algebraic equation group, Newton iteration method is used to solve the flow field variables.

[0007] A viscoelastic fluid heat flow coupling simulation method for micro-channel structure, comprising the following steps:

[0008] 1) Establish a geometric model, set material properties, and divide the grid:

[0009] 1.1) Establish a geometric model: according to the simulation requirements, establish a geometric model, divide the solid domain and fluid domain;

[0010] 1.2) Set material properties: according to the simulation requirements, select the viscoelastic fluid constitutive model, such as Oldroyd-B model, PPT model, power law model, Giesekus model, etc.; According to the material property definition, the density, solvent viscosity, solute viscosity, relaxation time, thermal conductivity, constant pressure heat capacity;

[0011] 1.3) Grid division: according to the simulation requirements, divide the grid of the geometric model;

[0012] 2) Set the model boundary conditions and finite element simulation parameters:

[0013] 2.1) Define the thermal boundary conditions and flow field boundary conditions: according to the design requirements, set the initial value and boundary conditions of the flow field and temperature field;

[0014] 2.2) Define the finite element simulation parameters: set the maximum number of iterations N max , set the maximum allowed error ε0 for terminating calculation;

[0015] 3) Establish a viscoelastic fluid heat flow coupling simplified mathematical model: for viscoelastic fluid flow, the dimensionless control equation is defined as:

[0016]

[0017] In the formula, u, ρ, t, p, η s , τ, T, k, c, Q are velocity, density, time, pressure, solvent viscosity, elastic stress, temperature, thermal conductivity, constant pressure heat capacity, heat source, all of which are dimensionless variables after processing;

[0018] The elastic stress τ needs to be expressed according to the properties of viscoelastic fluid, and the Oldroyd-B model is adopted, and the elastic stress τ satisfies:

[0019]

[0020] where η p is the solute viscosity, is the stress tensor Oldroyd-type inverse derivative, expressed as:

[0021]

[0022] and the Oldroyd-B model constitutive relation is:

[0023]

[0024] where C is the high polymer molecular deformation rate tensor, I is the unit tensor, and λ is the viscoelastic fluid relaxation time;

[0025] The logarithmic tensor reconstruction method is used to solve the constitutive equation, and the natural logarithm Θ of the deformation rate tensor is defined as:

[0026] Θ = ln(C) = R ln(Λ) R T

[0027] where R is the matrix composed of the characteristic vectors of C, and Λ is the diagonal matrix composed of the characteristic values of C;

[0028] In the simplified calculation of the low Reynolds number viscoelastic fluid heat flow coupling of the micro-channel, the time-varying term in the control equation is simplified; at the same time, the nonlinear convection term in the N-S equation is ignored;

[0029] The low Reynolds number viscoelastic fluid heat flow coupling simplified control equation is described as:

[0030]

[0031] In the formula, Ω is the skew-symmetric tensor, B is the diagonal tensor, and Θ is the natural logarithm of the deformation rate tensor;

[0032]

[0033] where ω ij and m ij are expressed as:

[0034]

[0035] 4) Construct a finite element solution model: solve the low Reynolds number viscoelastic fluid heat flow coupling simplified mathematical model by the finite element method, define the fluid domain discrete scheme according to the control equation and boundary conditions, convert the control equation into a weak form and perform spatial discretization, form a nonlinear algebraic equation system, and solve the flow field variables by the Newton iteration method, when the residual error ε is calculated, the maximum allowed error ε0 of the termination or the maximum iteration step number N maxWhen the time is up, the calculation terminates and the flow field results are output;

[0036] 5) Validation of simulation results of simplified mathematical model of thermal-fluid coupling of low Reynolds number viscoelastic fluid: The simulation results of simplified mathematical model of thermal-fluid coupling of low Reynolds number viscoelastic fluid are compared with the simulation results of unsimplified equation.

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] This invention provides a thermal-fluid coupling simulation method for low Reynolds number viscoelastic fluids. By simplifying the mathematical model, it performs thermal-fluid coupling calculations for viscoelastic fluids. Under low Reynolds number conditions in microchannels, it obtains relatively accurate analysis results while simplifying the calculation process, providing an efficient and feasible solution for the calculation of low Reynolds number viscoelastic fluids in microchannels. Attached Figure Description

[0039] Figure 1 This is a flowchart of an embodiment of the present invention.

[0040] Figure 2 This is a schematic diagram of boundary conditions in an embodiment of the present invention.

[0041] Figure 3 The results are finite element verification results for embodiments of the present invention. Detailed Implementation

[0042] The present invention will be further described below with reference to the embodiments and accompanying drawings.

[0043] Reference Figure 1 A viscoelastic fluid thermal-fluid coupling simulation method for microchannel structures includes the following steps:

[0044] 1) Establish the geometric model, set material properties, and generate the mesh:

[0045] 1.1) Establish a geometric model: Based on the simulation requirements, establish a geometric model and divide it into solid domain and fluid domain;

[0046] This embodiment describes the flow through a cylinder between two parallel plates. Due to the symmetrical structure, half of the geometric model is used for simulation. Figure 2 As shown, the geometric model has been dimensionless, with a flow channel width of 4L, an inlet flow development section length of 8L, a cylinder radius of L, a region near the cylinder length of 6L, an outlet section length of 11L, and L = 1.

[0047] 1.2) Set material properties: Depending on the specific case, adopt Oldroyd-B model, PPT model, power law model, Giesekus model, etc., and define density, solvent viscosity, solute viscosity, relaxation time, thermal conductivity, and constant pressure heat capacity according to material properties.

[0048] This embodiment uses the Oldroyd-B model; the material properties of the viscoelastic fluid are: density ρ = 1, solvent viscosity η = 1. s =β / Re, solute viscosity η p = (1-β) / Re, relaxation time λ = Wi*Re, thermal conductivity k = 1, constant pressure heat capacity c = Pr*Re; where β is the contribution of solvent viscosity to total viscosity, which is 0.59 in this embodiment; Re is the Reynolds number, which is 0.05 in this embodiment; Pr is the Prandtl number, which is 6.78 in this embodiment; Wi is the Wiesenberg number, which is 1 in this embodiment;

[0049] 1.3) Meshing: Mesh the geometric model according to simulation requirements;

[0050] 2) Set the model boundary conditions and finite element simulation parameters:

[0051] 2.1) Define thermal boundary conditions and flow field boundary conditions: Set the initial values ​​and boundary conditions for the flow field and temperature field according to design requirements;

[0052] The geometric model in this embodiment is as follows: Figure 2 As shown, half of the structure is simulated. Symmetrical conditions with zero normal flow and zero total tangential stress are used along the channel centerline. No-slip boundary conditions are applied to the channel walls and the cylindrical surface. The parabolic velocity inlet U... in The value is 1.5*(1-(y / 2)^2), and the inlet temperature T is... in =0, outlet pressure p out =0; The boundary heat source is distributed along the cylinder, with a size Q = 1, and the wall is adiabatic;

[0053] 2.2) Define finite element simulation parameters: Set the maximum number of iterations N max Set the maximum permissible error ε0 for the calculation termination;

[0054] In this embodiment, the maximum number of iteration steps N max Set the value to 100, and the maximum permissible error ε0 to 0.001;

[0055] 3) Establish a simplified mathematical model of thermo-fluid coupling for viscoelastic fluids: For incompressible viscoelastic fluid flow, the dimensionless governing equation can be defined as:

[0056]

[0057] In the formula, u, ρ, t, p, η s τ, T, k, c, and Q are velocity, density, time, pressure, solvent viscosity, elastic stress, temperature, thermal conductivity, constant pressure heat capacity, and heat source, all of which are dimensionless variables.

[0058] The elastic stress τ needs to be expressed by selecting the appropriate constitutive equation based on the properties of the viscoelastic fluid. Taking the Oldroyd-B model as an example, the elastic stress τ satisfies:

[0059]

[0060] Where η p For solute viscosity, The Oldroyd-type contravariant differential of the stress tensor is expressed as:

[0061]

[0062] Furthermore, the constitutive relation of the Oldroyd-B model is:

[0063]

[0064] Where C is the polymer molecular deformation rate tensor, I is the unit tensor, and λ is the viscoelastic fluid relaxation time;

[0065] To solve the high-dimensional Senberg number problem, the logarithmic tensor reconstruction method is used to solve the constitutive equations. The natural logarithm Θ of the deformation rate tensor is defined as:

[0066] Θ=ln(C)=Rln(Λ)R T

[0067] Where R is the matrix composed of the eigenvectors of C, and Λ is the diagonal matrix composed of the eigenvalues ​​of C;

[0068] For the thermal-fluid coupling problem of low Reynolds number viscoelastic fluids in microchannels, designers are more concerned with the steady-state temperature field distribution after the flow reaches thermal equilibrium over a sufficiently long period. Therefore, in the simplified calculation of thermal-fluid coupling of low Reynolds number viscoelastic fluids in microchannels, the time-varying terms in the governing equations can be omitted. Simplifications are made; meanwhile, due to the slow flow of viscoelastic fluids in microchannels at low Reynolds numbers, the nonlinear convection terms in the Navier-Stokes equations are simplified. It can be ignored;

[0069] In summary, the simplified governing equations for the thermal-fluid coupling of low Reynolds number viscoelastic fluids can be described as follows:

[0070]

[0071] In the formula, u, p, η s η p, λ, T, k, ρ, c, are velocity, pressure, solvent viscosity, solute viscosity, viscoelastic fluid relaxation time, temperature, thermal conductivity, density, constant pressure heat capacity, all are dimensionless variables after processing; C is the high polymer molecular deformation rate tensor, Ω is the anti-symmetric tensor, B is the diagonal tensor, Θ is the natural logarithm of the deformation rate tensor, R is the matrix composed of the characteristic vector of C, Λ is the diagonal matrix composed of the characteristic value of C, I is the unit tensor;

[0072]

[0073] where ω ij and m ij are expressed as:

[0074]

[0075] After the above key simplification steps, a viscoelastic fluid heat flow coupling simplified mathematical model is established, which includes continuity equation, simplified momentum equation, simplified energy equation and constitutive equation based on logarithmic reconstruction. The model retains the necessary physical mechanisms to describe the coupling of low Reynolds number viscoelastic fluid flow and heat transfer in microchannels, such as the elastic disturbance caused by the stretching and compression of polymer materials to the flow, thereby causing elastic turbulence of viscoelastic fluid in microchannels, while reducing the numerical solution difficulty and calculation cost of the model. The simplified model is particularly suitable for efficiently and stably simulating and predicting the steady-state heat flow distribution of viscoelastic fluid in microchannel devices at low Reynolds number, and provides an effective calculation tool basis for the design and optimization of microchannel thermal management, microfluidic chip and other applications.

[0076] 4) Build a finite element solution model: In this embodiment, a commercial finite element software COMSOL is used to simulate and calculate the low Reynolds number viscoelastic fluid heat flow; the low Reynolds number viscoelastic fluid heat flow coupling simplified mathematical model is solved by finite element method, the control equation and boundary conditions are defined, the control equation is converted into weak form and discretized in space, and a nonlinear algebraic equation system is formed. Newton iteration method is used to solve the flow field variables, and when the residual error ε reaches the maximum allowed error ε0 or reaches the maximum iteration step N max , the calculation is terminated and the flow field results are output;

[0077] 5) Verification of simulation results of low Reynolds number viscoelastic fluid heat flow coupling simplified mathematical model: compare the simulation results of low Reynolds number viscoelastic fluid heat flow coupling simplified mathematical model with the simulation results of unsimplified equation;

[0078] The simulation results of the first elastic principal stress, pressure and temperature before and after the mathematical model simplification in this embodiment are shown in Figure 3The simulation results show that the distribution of the first principal stress near the cylinder is basically the same before and after the simplification of the mathematical model, and the pressure and temperature distribution is the same; the simulation time of the simplified mathematical model is 6s, the range of the first principal stress component is-39.11-201.64, the range of the pressure is-11.31-1255.84, and the range of the temperature is 0.12-5.19; the simulation time of the non-simplified mathematical model is 8s, the range of the first principal stress component is-39.21-201.35, the range of the pressure is-11.31-1255.84, and the range of the temperature is 0.12-5.19;

[0079] The simulation results show that after the simplification of the mathematical model, the calculation time is reduced by 25%, the relative error of the first principal stress component is 0.079%, and the error of the pressure and temperature can be ignored; the simulation results show that the viscoelastic fluid heat flow coupling simulation method for micro-channel structure meets the simulation requirements of viscoelastic fluid flow in micro-channel under low Reynolds number, improves the simulation calculation speed, and has important significance for complex models and large-scale calculations.

Claims

1. A viscoelastic fluid thermal flow coupling simulation method for a microchannel structure, characterized in that: First, a simplified mathematical model of viscoelastic fluid heat flow coupling is established, which is solved by finite element method. According to the control equation and boundary conditions, the control equation is transformed into weak form and discretized in space to form a nonlinear algebraic equation group, and the Newton iteration method is used to solve the flow field variables.

2. The method of claim 1, wherein the method is a method of simulating thermal flow coupling of a viscoelastic fluid in a microfluidic structure. The method comprises the following steps: 3) establishing a simplified mathematical model of viscoelastic fluid heat flow coupling: for viscoelastic fluid flow, the dimensionless control equation is defined as: In the formula, u, p, t, p, h s , t, T, k, c, Q are velocity, density, time, pressure, solvent viscosity, elastic stress, temperature, thermal conductivity, constant-pressure heat capacity, heat source, all of which are dimensionless processed variables. The elastic stress τ needs to be expressed according to the constitutive equation of the viscoelastic fluid property, and the Oldroyd-B model is adopted, and the elastic stress τ satisfies: where η p is the solute viscosity, is the stress tensor Oldroyd type inverse derivative, expressed as: And the constitutive relationship of the Oldroyd-B model is: Where C is the deformation rate tensor of the polymer molecule, I is the unit tensor, and λ is the relaxation time of the viscoelastic fluid; The logarithmic tensor reconstruction method is used to solve the constitutive equation, and the natural logarithm Θ of the deformation rate tensor is defined as: Θ = ln(C) = R ln(A) R T Where R is a matrix composed of the characteristic vectors of C, and Λ is a diagonal matrix composed of the characteristic values of C; In the simplified calculation of the heat flow coupling of low Reynolds number viscoelastic fluid facing micro-channel, the time-varying term in the control equation is simplified; at the same time, the nonlinear convection term in N-S equation is ignored; The simplified control equation of low Reynolds number viscoelastic fluid heat flow coupling is described as: In the formula, Ω is an antisymmetric tensor, B is a diagonal tensor, and Θ is the natural logarithm of the deformation rate tensor; where ω ij and m ij is expressed as: 4) Constructing finite element solution model: solving the simplified mathematical model of low Reynolds number viscoelastic fluid thermal flow coupling by finite element method, defining the discrete scheme of fluid domain according to the control equation and boundary condition, converting the control equation into weak form and carrying out spatial discretization to form a nonlinear algebraic equation group, solving the flow field variables by Newton iteration method, when the residual error ε reaches the maximum allowed error ε0 or reaches the maximum iteration step N max , the calculation is terminated and the flow field results are output; 5) verification of the simulation results of the simplified mathematical model of low Reynolds number viscoelastic fluid heat flow coupling: compare the simulation results of the simplified mathematical model of low Reynolds number viscoelastic fluid heat flow coupling with the simulation results of the unsimplified equation.

3. The viscoelastic fluid heat flow coupling simulation method for a micro- channel structure according to claim 2, characterized in that, The method comprises the following steps: 1) establishing a geometric model, setting material properties, and dividing the grid: 1.1) establishing a geometric model: according to the simulation requirements, a geometric model is established, and a solid domain and a fluid domain are divided; 1.2) setting material properties: according to the simulation requirements, the viscoelastic fluid constitutive model is selected, including the Oldroyd-B model, the PPT model, the power law model and the Giesekus model; according to the material property definition, the density, solvent viscosity, solute viscosity, relaxation time, thermal conductivity and constant pressure heat capacity are defined; 1.3) dividing the grid: according to the simulation requirements, the geometric model is divided into a grid; 2) setting the model boundary conditions and finite element simulation parameters: 2.1) defining the thermal boundary conditions and the flow field boundary conditions: according to the design requirements, the initial values and boundary conditions of the flow field and temperature field are set; 2.2) Define finite element simulation parameters: Set the maximum number of iterations N max Set the maximum allowed error ε0for termination of the calculation.

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