西南大学学报 (自然科学版)  2020, Vol. 42 Issue (2): 118-128.  DOI: 10.13718/j.cnki.xdzk.2020.02.015 0
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1 数学模型

1.1 热风流场计算

 $\frac{{\partial {\rho _a}}}{{\partial t}} + \nabla \left( {{\rho _a}{\mathit{\boldsymbol{u}}_a}} \right) = 0$ (1)
 ${\rho _a}\frac{{\partial {\mathit{\boldsymbol{u}}_a}}}{{\partial t}} + {\rho _a}\left( {{\mathit{\boldsymbol{u}}_a} \cdot \nabla } \right){\mathit{\boldsymbol{u}}_a} = - \nabla p + \mu {\nabla ^2}{\mathit{\boldsymbol{u}}_a} - \mu \nabla \left( {\nabla {\mathit{\boldsymbol{u}}_a}} \right) + \mathit{\boldsymbol{F}}$ (2)

 $\varepsilon \frac{{\partial {\rho _a}}}{{\partial t}} + \nabla \left( {{\rho _a}{\mathit{\boldsymbol{u}}_a}} \right) = 0$ (3)
 $\frac{1}{\varepsilon }{\rho _a}\frac{{\partial {\mathit{\boldsymbol{u}}_a}}}{{\partial t}} + \frac{1}{{{\varepsilon ^2}}}{\rho _a}\left( {{\mathit{\boldsymbol{u}}_a} \cdot \nabla } \right){\mathit{\boldsymbol{u}}_a} = - \nabla p + \mu \frac{1}{\varepsilon }{\nabla ^2}{\mathit{\boldsymbol{u}}_a} - \mu \frac{1}{\varepsilon }\nabla \left( {\nabla {\mathit{\boldsymbol{u}}_a}} \right) + \mathit{\boldsymbol{F}}$ (4)

 $\frac{1}{\varepsilon }{\rho _a}\frac{{\partial {\mathit{\boldsymbol{u}}_a}}}{{\partial t}} + \frac{1}{{{\varepsilon ^2}}}{\rho _a}\left( {{\mathit{\boldsymbol{u}}_a} \cdot \nabla } \right){\mathit{\boldsymbol{u}}_a} = - \frac{\mu }{k}{\mathit{\boldsymbol{u}}_a}$ (5)

 $\frac{1}{\varepsilon }{\rho _a}\frac{{\partial {\mathit{\boldsymbol{u}}_a}}}{{\partial t}} + \frac{1}{{{\varepsilon ^2}}}{\rho _a}\left( {{\mathit{\boldsymbol{u}}_a} \cdot \nabla } \right){\mathit{\boldsymbol{u}}_a} = - \nabla p + \mu \frac{1}{\varepsilon }{\nabla ^2}{\mathit{\boldsymbol{u}}_a} - \mu \frac{1}{\varepsilon }\nabla \left( {\nabla {\mathit{\boldsymbol{u}}_a}} \right) - \frac{\mu }{k}{\mathit{\boldsymbol{u}}_a} + \mathit{\boldsymbol{F}}$ (6)

 ${\rho _a} = 1.293\frac{{p{T_0}}}{{{p_0}T}}$ (7)

 $\frac{\mu }{{{\mu _0}}} = {\left( {\frac{T}{{288.15}}} \right)^{1.5}}\frac{{288.15 + B}}{{T + B}}$ (8)

1.2 温度场计算

 ${\rho _a}{c_p}\frac{{\partial T}}{{\partial t}} + {\rho _a}{c_p}{\mathit{\boldsymbol{u}}_a}\nabla T = \nabla \left( {\lambda \nabla T} \right) + q$ (9)

 ${\left( {\rho {c_p}} \right)_{{\rm{eff}}}}\frac{{\partial T}}{{\partial t}} + {\rho _a}{c_p}{\mathit{\boldsymbol{u}}_a}\nabla T = \nabla \left( {{\lambda _{{\rm{eff}}}}\nabla T} \right) + \gamma I + q$ (10)

 ${\left( {\rho {c_p}} \right)_{{\rm{eff}}}} = \left( {1 - \varepsilon } \right){\left( {\rho {c_p}} \right)_s} + {S_l}\varepsilon {\left( {\rho {c_p}} \right)_l} + {S_g}\varepsilon {\left( {\rho {c_p}} \right)_g}$ (11)
 ${\lambda _{{\rm{eff}}}} = \left( {1 - \varepsilon } \right){\lambda _s} + \varepsilon {S_l}{\lambda _l} + \varepsilon {S_g}{\lambda _g}$ (12)

 ${S_l} + {S_g} = 1$ (13)
1.3 气相传递模型

 $\frac{{\partial {c_v}}}{{\partial t}} + \nabla \cdot \left( {D\nabla {c_v}} \right) + {\mathit{\boldsymbol{u}}_a} \cdot \nabla {c_v} = 0$ (14)

 $\frac{{\partial {c_v}}}{{\partial t}} + \nabla \cdot \left( {D\nabla {c_v}} \right) + {\mathit{\boldsymbol{u}}_{{\rm{tot}}}} \cdot \nabla {c_v} = R$ (15)

 $R = {K_{{\rm{evap}}}}\left( {{a_{\rm{w}}}{{\rm{c}}_{v\_{\rm{sat}}}} - {c_v}} \right)$ (16)

 ${\mathit{\boldsymbol{u}}_{{\rm{tot}}}} = {\mathit{\boldsymbol{u}}_a} + {\mathit{\boldsymbol{u}}_{{\rm{dif}}}}$ (17)

 $\mathit{\boldsymbol{J}} = - D\nabla {c_a}$ (18)

 ${\mathit{\boldsymbol{u}}_{{\rm{dif}}}} = \mathit{\boldsymbol{J}}\frac{M}{\rho }$ (19)

 ${\mathit{\boldsymbol{u}}_{{\rm{tot}}}} = {\mathit{\boldsymbol{u}}_a} + \frac{M}{\rho }{D_{{\rm{eff}}}}\nabla {c_v}$ (20)

1.4 液相传递模型

 $\frac{{\partial {c_l}}}{{\partial t}} + \nabla \cdot \left( {D\nabla {c_l}} \right) + {\mathit{\boldsymbol{u}}_l} \cdot \nabla {c_l} = - R$ (21)

 ${\mathit{\boldsymbol{u}}_l} = - \frac{k}{\mu }\nabla p$ (22)

2 模型验证

 图 1 干燥箱结构示意图

3 结果与分析 3.1 油菜籽的干燥特性曲线

 图 2 干燥特性曲线
3.2 干燥室内水蒸气分布

 图 3 干燥室内水蒸分布度云图
3.3 物料内部水分分布

 图 4 物料水分浓度云图(a 300 s；b 600 s；c 1 200 s)
3.4 干燥室内物料温度变化曲线

 图 5 干燥室内物料温度变化曲线
3.5 干燥室内热风风速分布

 图 6 风速云图

(a)，(b)，(c)，(d)分别为1，5，10，20 s时热风风速云图情况.由图(a)，图(b)可知1 s时刻干燥室中心区域流场有微量扰动；由图(b)，图(c)可知5 s时刻干燥室内热风流场开始趋于稳定；由图(c)，图(d)可知10 s和20 s时刻干燥室内部热风流场基本一致，这说明在很短时间内干燥室内部的热风流场就已经趋于稳态.在实际工程问题求解中，可先进行流场的稳态求解，然后再以该稳态解去计算能量与质量传递方程.该操作方法实质上是把流场与其它场的强耦合关系转化成了弱耦合关系，牺牲了一定的计算精度，但是该方法避开了流场的瞬态计算问题，极大地节约了计算成本，并且不会造成计算结果的较大偏差.因此工程领域在进行类似的分析计算时，可以先计算流场的稳态解，然后再计算其余物理场.

4 结论

1) 模拟值与实验值最大误差为13.3%，表明本模型可信；

2) 油菜籽在干燥过程中有干区、湿区、蒸发区之分，且干区与湿区被蒸发区隔开，蒸发区逐渐由物料外部向内部迁移；

3) 干燥室内水蒸气浓度总体上先增大后减小，干燥室中心区域水蒸气浓度比干燥室边缘区域水蒸气浓度高，表明干燥过程中干燥室内中心区域水蒸气无法及时排出，这不利于干燥过程进行，说明干燥室有待进一步优化；

4) 干燥初期，油菜籽平均温度上升迅速，表明油菜籽在干燥过程中存在着预热阶段；

5) 干燥室中心区域热风风速比边缘区域热风风速低，热风流场在极短时间内达到稳态，说明在工程计算时可采用流场的稳态解进行能量与质量传递方程的求解，并且不会对计算精度造成过大影响.

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A Multiphysics Coupling-Based Mathematical Model of Hot-Air Drying and Its Verification
HU Zhong-huan, YANG Ming-jin, YANG Zhuo-ran, YANG Ling
School of Engineering and Technology/Chongqing Key Laboratory of Agricultural Equipment for Hilly and Mountainous Regions, Southwest University, Chongqing 400715, China
Abstract: The mildew of grain during storage constitutes a major part of grain loss. The industry usually reduces the moisture content of the grain by drying, which reduces the loss caused by mildew. Among many drying methods, hot air drying has long occupied a large market share due to its advantages of simple operation, low cost, and low equipment requirements. Therefore, it is extremely necessary to optimize the hot air drying machines. During the optimization process, engineers often try to understand the distribution of physical fields such as temperature, wind speed and humidity in the drying machines. However, it is uneconomical to directly measure these physical fields from the drying machines, and there are great difficulties to measure them. Thus, based on the previous studies, this paper considers the influence of heat and mass transfer in the hot air drying process, and incorporates the fluid dynamics equations into the model framework, considering conservations of mass, energy and momentum. An analysis is carried out to establish mathematical equations based on the conservations, which fully describes the entire hot air drying process. In the text, four parts are used to introduce the whole mathematical model. (1) The hot air flow field adopts the fluid dynamics equation as the governing equation, which describes the transfer law of air outside and inside the material. (2) The temperature field is based on the law of conservation of energy, ignoring some minor thermal phenomena, and a heat exchange governing equation is constructed. (3) The gas phase transfer model of the moisture is based on the law of conservation of mass, and the phase change factor of moisture is introduced into the governing equation by means of source term. (4) The liquid phase transfer model of moisture is also based on the law of conservation of mass, whose governing equations have different signs on the source term than the gas phase transfer model. An experiment with rapeseed was carried out to verify the numerical simulation results of the model. The results showed that the maximum relative error between the model numerical simulation results and the real experimental results was 13. 3%, which indicated that the model could satisfactorily describe the hot air drying process. In addition, the numerical simulation results showed that a dry zone, a wet zone and an evaporation zone existed in the drying material during the drying process, the dry zone and the wet zone were separated by the evaporation zone, and the evaporation zone gradually migrated from the outside to the inside of the material. During the drying process, the concentration of water vapor in the drying chamber first increased and then decreased, and the concentration of water vapor was higher in the central area of the drying chamber than in the edge of the drying chamber. The average temperature of the material rose rapidly at the beginning of the experiment, and tended to be stable in the middle and late periods, which indicated that the material had a preheating time during the drying process, and the hot air temperature could be appropriately reduced in the middle and late drying period to reduce the energy consumption. The hot air flow field in the dry chamber reached a steady state in a very short time, which indicated that the steady solution of the hot air flow field could be directly used to calculate heat and mass transfer, for purpose of simulation cost saving.
Key words: hot air drying    heat and mass transfer    multiphysics coupling    porous medium