RESEARCH ON HETEROGENEOUS SOLID STRESS MODEL BASED ON ARTIFICIAL NEURAL NETWORK
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摘要: 最小多尺度理论(Energy-Minimization Multi-Scale, EMMS)已经被引入多相质点网格法(multiphase particle-in-cell, MP-PIC)中, 建立了非均匀EMMS固相应力模型. 但现有的非均匀固相应力模型计算中, 中间步骤繁琐且花费时间长. 采用人工拟合的方式能获得非均匀固相应力表达式, 但需要人为确定拟合变量和拟合函数, 且针对于非均匀固相应力这种高度非线性函数所得到的拟合精度不高、用于MP-PIC模拟的结果相比原EMMS固相应力模型结果存在偏差. 针对上述问题, 本文提出通过机器学习的方法, 规避对固相体积分数的局部分布情况的表征, 并提出和建立能考虑颗粒浓度详细分布的人工神经网络(artificial neutral network, ANN)固相应力模型. 首先, 基于局部颗粒浓度和颗粒非均匀分布指数建立了双变量的ANN固相应力模型; 进一步将当前网格及其周边网格颗粒浓度组成的序列来详细表征颗粒浓度分布, 并建立相应的ANN固相应力模型. 然后, 将两种模型与EMMS固相应力模型进行了对比并测试了网格分辨率和粗化率对模型的影响. 研究表明: 基于ANN固相应力模型的模拟结果对EMMS固相应力模型结果有较高的还原度, 同时具有一定的网格分辨率无关性和粗化率无关性.
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关键词:
- 两相流 /
- 多相质点网格法 (MP-PIC) /
- 非均匀固相应力 /
- 机器学习 /
- 流态化
Abstract: The Energy-Minimization Multi-Scale (EMMS) theory has been introduced into the multiphase particle-in-cell (MP-PIC) method to establish the heterogeneous EMMS solid stress model to account for the effect of non-uniform solid distribution. However, the calculation process is very complex and also very time consuming for this heterogeneous solid stress model. The expression of the heterogeneous EMMS solid stress can be obtained by manual fitting method. However, the fitting variable describes heterogeneous solid distribution as well as the fitting function describe the shape of solid stress are required for manually fitting. Since the heterogeneous solid stress function is highly nonlinear in nature, the fitting precision is not high enough for the manually fitting model. And there is an obvious deviation between the fitting correlation and the original EMMS solid stress, because it is hard to find out an appropriate parameter to characterize the heterogeneous solid concentration distribution as well as to find out an appropriate fitting function. In order to solve the above problems, an artificial neutral network (ANN) based machine learning method was proposed to avoid the characterization of the local distribution of solid volume fraction. Subsequently an ANN solid stress model which accounts for the detailed distribution of particle concentration was proposed to improve the fitting accuracy. Firstly, a two-marker based ANN solid stress model was established based on local particle concentration and particle non-uniform distribution index. Further, particle concentrations in the current cell and its neighboring cells were arrayed to represent the particle concentration distribution, thus to establish the ANN solid stress model based on particle concentration distribution. Then, the two models are compared with the EMMS solid stress model, and the effects of grid resolution and coarse-graining ratio on the model are also tested. The simulation results predicted with ANN model agreed well with that of the EMMS solid stress model, and the dependence of simulation results on grid resolution and coarse-graining ratio was also reduced. -
图 1 反距离权重法计算网格颗粒非均匀分布指数的示意图(i、j、i + ii、j + jj均为网格格点的编号, 其中ii, jj为−1或1)
Figure 1. Demonstration for predicting the heterogeneity index of particle concentration distribution of a cell (i、j、i + ii and j + jj are the numbers of grid points, where ii, jj = −1 or 1)
图 2 隐藏层神经元数对R2决定系数和MAE损失函数的影响(横坐标神经元的个数代表两层隐藏层所包含的全部神经元个数)
Figure 2. The effect of cell number for hidden layer on R2 coefficient of determination and MAE loss function (The number of neurons on the horizontal axis represents the total number of neurons contained in the two hidden layers)
图 4 (a)人工拟合模型与EMMS固相应力模型的对比; (b) ANN model 1与EMMS固相应力模型的对比; (c) ANN model 2与EMMS固相应力模型的对比; (d)模型误差
Figure 4. (a) Comparison between the manually fitted model and EMMS solid stress model; (b) Comparison between the ANN model 1 and EMMS solid stress model; (c) Comparison between the ANN model 2 and model EMMS solid stress model;(d) Model errors
图 5 不同非均匀固相应力模型预测的时均轴向空隙率分布与瞬时固相流率
Figure 5. Time-average axial void distribution and instantaneous solid flow rate predicted by stress models with different heterogeneous solids
图 6 网格分辨率对时均轴向孔隙率分布的影响
Figure 6. Effect of grid resolution on time-average axial void distribution
图 7 网格分辨率对径向时均孔隙率分布的影响
Figure 7. Effect of grid resolution on time-average radial voidage distribution
图 8 粗化率对时均轴向孔隙率分布的影响(Grid: 20 × 200)
Figure 8. Effect of coarse-graining ratio on time-average axial voidage distribution (Grid: 20 × 200)
图 9 粗化率对径向孔隙率分布的影响
Figure 9. Effect of coarse-graining ratio on time-average radial voidage distribution
表 1 MP-PIC方法的控制方程和本构模型
Table 1. Governing equations and constitutive models of MP-PIC method
Description Equation Governing equations for gas phase $\begin{gathered}\dfrac{\partial }{\partial t}\left({\varepsilon }_{g}{\rho }_{g}\right) + \nabla \cdot \left({\varepsilon }_{g}{\rho }_{g}{u}_{g}\right) = 0\\ \dfrac{\partial }{\partial t}\left({\varepsilon }_{g}{\rho }_{g}{u}_{g}\right) + \nabla \cdot \left({\varepsilon }_{g}{\rho }_{g}{u}_{g}{u}_{g}\right) = -{\varepsilon }_{g}\nabla {p}_{g} + \nabla \cdot \left({\varepsilon }_{g}{\tau }_{g}\right) + {\varepsilon }_{g}{\rho }_{g}g-F\\ {\tau }_{g} = {\mu }_{g}\left[\nabla {u}_{g} + {\left(\nabla {u}_{g}\right)}^{{\rm{T}}}\right]-\dfrac{2}{3}{\mu }_{g}\left(\nabla \cdot {u}_{g}\right)I\end{gathered}$ Particle distribution function $\dfrac{\partial \phi }{\partial t} + \nabla \cdot \left({u}_{p}\varphi \right) + {\nabla }_{ {u}_{p} }\cdot \left({a}_{p}\phi \right) = 0$ Particle speed ${u}_{p} = \dfrac{{\rm{d}}{x}_{p} }{{\rm{d}}t}$ Particle acceleration ${a}_{p} = {\beta }_{p}\left({u}_{g,p}-{u}_{p}\right)-\dfrac{\nabla {p}_{gp} }{ {\rho }_{s} } + g-\dfrac{\nabla {p}_{s,p} }{ {\varepsilon }_{s}{\rho }_{s} }$ Gas-solid inter-phase momentum exchange rate $F = \displaystyle\sum _{k = 1}^{ {n}_{T} } \frac{ {n}_{p}^{k}{V}_{p}^{k} }{ {V}_{c} }{\beta }_{p}^{k}\left({u}_{g}^{k}-{u}_{p}^{k}\right)$ Homogeneous drag model ${\beta }_{0} = \dfrac{3}{4}{C}_{d0}\dfrac{ {\varepsilon }_{g}{\rho }_{g}\left|{u}_{gp}-{u}_{p}\right|}{ {d}_{p} }{\varepsilon }_{g}^{-2.7}$ Heterogeneous drag model[36] $\begin{gathered}{H}_{D} = a{\left(R{e}_{p} + b\right)}^{c}\\ {\beta }_{p} = {\beta }_{p0}{H}_{D}\end{gathered}$ Homogeneous solid force model ${p}_{s0} = \dfrac{ {p}_{s}^{*}{\varepsilon }_{s}^{\alpha } }{{\rm{max}}\left[{\varepsilon }_{s,cp}-{\varepsilon }_{s},\delta \left(1-{\varepsilon }_{s}\right)\right]}$ EMMS solid stress model ${p}_{s,EMMS} = \displaystyle\sum _{k = 1}^{ {n}_{T} } \left(\frac{ {f}_{k}{p}_{s}^{*}{\left({\varepsilon }_{sc}^{k}\right)}^{\alpha } }{{\rm{max}}\left[{\varepsilon }_{s,cp}-{\varepsilon }_{sc}^{k},\delta \left(1-{\varepsilon }_{sc}^{k}\right)\right]}\right)$ Artificial fitting model of EMMS solid stress $f\left({\varepsilon }_{s},{\delta }_{s}\right) = -3.273{\varepsilon }_{s}^{0.7115} + 1.370-{\delta }_{s}$
${ {p}_{s,Fit} }\left( { {\varepsilon }_{s} },{ {\delta }_{s} } \right)\text{=}\left\{\begin{aligned} & 1.164\varepsilon _{s}^{0.9862}\Biggr/\left[ { {\left( \frac{ { {\varepsilon }_{s} }+1}{0.2721} \right)}^{-0.7787} }-{ {\varepsilon }_{s} } \right],f\left( { {\varepsilon }_{s} },{ {\delta }_{s} } \right)\leqslant 0 \\ & { {{\rm{e}}}^{\left( -9.373\delta _{s}^{3}-0.4545{ {\delta }_{s} }-7.014 \right)\text{ } \cdot \text{ }{ {\left( { {\delta }_{s} }+3.319{ {\varepsilon }_{s} } \right)}^{-3.61} }+9.459} },f\left( { {\varepsilon }_{s} },{ {\delta }_{s} } \right) > 0 \\ \end{aligned} \right.$
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表 2 算例参数
Table 2. Simulation parameters
Schematic drawing Parameters Value 
Diameter, D/m 0.09 Height, H/m 10.50 Operating temperature, Top/K 298 Operating pressure, Pop/Pa 1.01 × 105 Particle diameter, dp/μm 54 Particle density, ρs/(kg·m−3) 970 Gas density, ρg/(kg·m−3) 1.185 Gas viscosity, μg/(Pa·s) 1.90 × 10-5 Superficial gas velocity, Ugo/(m·s−1) 1.52 Incipient average solid volume
fractions, εs00.058 Solid volume fraction at close pack, εs,cp 0.60 Solid stress parameter, ps*/Pa 100 Solid stress parameter, α 2 Particle–wall restitution coefficient, ew 0.90 Coarse-grained ratio, np 100,200,500,1000 Number of grids, I × J 20 × 200,20 × 400,30 × 500,40 × 600 Time step, Δt/s 5 × 10-4 Simulation time, t/s 25
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