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 北京化工大学学报(自然科学版)  2018, Vol. 45 Issue (1): 8-12  DOI: 10.13543/j.bhxbzr.2018.01.002 0

### 引用本文

WEI LiShun, XU LanXi, LI DianQing. The influence of the gap width in a rotating liquid film reactor on the crystal size[J]. Journal of Beijing University of Chemical Technology (Natural Science), 2018, 45(1): 8-12. DOI: 10.13543/j.bhxbzr.2018.01.002.

### 文章历史

The influence of the gap width in a rotating liquid film reactor on the crystal size
WEI LiShun , XU LanXi , LI DianQing
Faculty of Science, Beijing University of Chemical Technology, Beijing 100029, China
Abstract: Under appropriate boundary conditions, the barium sulfate precipitation reaction was simulated by coupling the Navier-Stokes equation with convection diffusion reaction equations and the population balance equation. The volume mean diameter and size distribution of the crystals were evaluated for three different gaps with different rotor speeds. The reliability of the simulation method was verified by comparing the results with experimental data. The results show that the crystal size and crystal size distribution decreased with increasing gap width or rotor speed, and the partial elasticity of the crystal size as a function of rotor speed is greater than the partial elasticity of the crystal size as a function of the gap width.
Key words: rotating liquid film reactor    population balance    numerical simulation    crystal size distribution

1 反应器模型 1.1 反应器及其几何模型

 图 1 旋转液膜反应器几何模型 Fig.1 The geometric shape of the rotating liquid film reactor

1.2 数学模型

(1) 由于晶体颗粒粒径和体积分数均很小，假设整个流场都是均匀的液相；

(2) 不考虑颗粒间的碰撞、团聚、破碎。

 ${\rm{B}}{{\rm{a}}^{{\rm{2 + }}}}{\rm{ + SO}}_4^{2 - } = {\rm{BaS}}{{\rm{O}}_{\rm{4}}}$

 $\left\{ \begin{array}{l} \frac{{\partial \mathit{\boldsymbol{u}}}}{{\partial t}} + \mathit{\boldsymbol{u}} \cdot \nabla \mathit{\boldsymbol{u}} + \frac{1}{\rho }\nabla p - \nu \nabla \mathit{\boldsymbol{u}} = \mathit{\boldsymbol{F}}\\ \nabla \cdot \mathit{\boldsymbol{u}} = 0\\ \frac{{\partial {c_i}}}{{\partial t}} + \nabla \cdot (\mathit{\boldsymbol{u}}{c_i}) - \nabla \cdot ({d_i}\nabla ({c_i})) = {S_{{c_i}}}\\ \frac{{\partial {m_j}}}{{\partial t}} + \nabla \cdot (\mathit{\boldsymbol{u}}{m_j} - {\mathit{\Gamma} _{{\rm{eff}}}}\nabla {m_j}) = JL_{{\rm{min}}}^j + jG{m_{j - 1}} \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} {u_r}{|_{转子}} = {u_z}{|_{转子}} = 0\\ {u_r}{|_{定子}} = {u_\theta }{|_{定子}} = {u_z}{|_{定子}} = 0\\ {u_\theta }|转子 = \mathit{\Omega }({\mathit{R}_1} - \mathit{z}{\rm{cot}}\ \alpha )\\ {p_{\rm{i}}} + \frac{1}{2}\rho u_z^2 = 1\\ \frac{{\partial {p_{\rm{o}}}}}{{\partial r}} = \frac{{\rho u_\theta ^2}}{r}\\ {\mathit{c}_{{\rm{B}}{{\rm{a}}^{2 + }}}}{|_{入口}} = {c_{{\rm{SO}}_4^{2 - }}}{|_{入口}} = {c_0}\\ {m_j}{|_{入口}} = 0 \end{array} \right.$ (2)

 ${m_j} = \int_{{L_{{\rm{min}}}}}^{{L_{{\rm{max}}}}} {n{L^j}{\rm{d}}\mathit{L}, j = 0, 1, 2, 3, 4}$ (3)

 $S = \gamma \sqrt {\frac{{{\mathit{c}_{{\rm{B}}{{\rm{a}}^{2 + }}}}{c_{{\rm{SO}}_4^{2 - }}}}}{{{K_{{\rm{sp}}}}}}}$ (4)

 $J = \left\{ \begin{array}{l} 1.46 \times {10^{12}}{{\rm{e}}^{\frac{{ - 67.3}}{{{\rm{l}}{{\rm{n}}^2}\mathit{S}}}}}\;\;1\mathit{ < S < }1000\\ {10^{36}}{{\rm{e}}^{\frac{{ - 2686}}{{{\rm{l}}{{\rm{n}}^2}\mathit{S}}}}}\;\;\;\;\;\;\;\;\;\;\;\mathit{S} \ge {\rm{1000}} \end{array} \right.$ (5)

 $G = {k_{\rm{g}}}{\left( {S - 1} \right)^2}$ (6)

 ${S_{{c_{{\rm{BaS}}{{\rm{O}}_{\rm{4}}}}}}} = 3G{m_2}{K_{\rm{v}}}\frac{{{\rho _{\rm{s}}}}}{{{M_{\rm{s}}}}}$ (7)

2 模拟实验 2.1 模拟设置

 ${d_{43}} = \frac{{{m_4}}}{{{m_3}}}$ (8)

 $f\left( x \right) = \frac{1}{{\sqrt {2\pi } x{\rm{ln}}{\sigma _{\rm{g}}}}}{{\rm{e}}^{\left( { - \frac{{{\rm{l}}{{\rm{n}}^2}\left( {x/{{\bar x}_{\rm{g}}}} \right)}}{{{\rm{2l}}{{\rm{n}}^2}{\sigma _{\rm{g}}}}}} \right)}}$ (9)

2.2 模拟结果及分析 2.2.1 转速与间隙对晶体体积平均粒径的影响

 图 2 不同间隙下体积平均粒径随转速的变化 Fig.2 Variation of the volume mean diameter of the crystal with rotor speed for different gap widths
 图 3 不同转速下体积平均粒径随间隙的变化 Fig.3 Variation of the volume mean diameter of the crystal changing with gap width for different rotor speeds

2.2.2 间隙对晶体粒度分布的影响

 图 4 各转速下间隙对粒度分布的影响 Fig.4 The effect of gap width on the crystal size distribution for different rotor speeds

2.2.3 体积平均粒径对间隙和转速的偏弹性

 $\begin{array}{l} \frac{{Ef}}{{\mathit{E\Omega }}}\left| {_{_{\mathit{\Omega } = {\mathit{\Omega }_0}}}} \right. = \frac{{f(({\mathit{\Omega }_0} + \Delta \mathit{\Omega }), \mathit{d}){\rm{ - }}f({\mathit{\Omega }_0}, d)}}{{f({\mathit{\Omega }_0}, \mathit{d})}}\frac{{{\mathit{\Omega }_0}}}{{\Delta \mathit{\Omega }}}\\ \frac{{Ef}}{{\mathit{Ed}}}\left| {_{_{\mathit{d} = {\mathit{d}_0}}}} \right. = \frac{{f(\mathit{\Omega }, ({d_0} + \Delta d)) - f(\mathit{\Omega }, {d_0})}}{{f(\mathit{\Omega }, {d_0})}}\frac{{{d_0}}}{{\Delta \mathit{d}}} \end{array}$ (10)

2.3 实验对比

 图 5 0.3 mm间隙下体积平均粒径随转速变化模拟值和实验值对比 Fig.5 Volume mean diameter of the crystal obtained by simulation and experiment for different rotor speeds with a gap width of 0.3 mm
3 结论

(1) 在Ω∈[1000, 5000]或d∈[0.3, 0.5]内，增大反应器间隙宽度或转速使晶体体积平均粒径变小，粒度分布变窄，所以反应器参数设计应采用高转速或宽间隙。

(2) 体积平均粒径对转速的偏弹性大于对间隙的偏弹性，表明晶体粒径对转速更敏感。

(3) 实验值与模拟值较为相近，证明了本文模拟方法的可靠性，分析结果对反应器设计具有指导作用。

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