1. 清华大学 热能工程系, 北京 100084 ;
2. 北京航天动力研究所, 北京 100076

Drop diameters during urea prilling
LIU Xiaodi1,2 , ZHU Yule1 , LV Junfu1 , MI Yan2 , GU Xueying2
1. Department of Thermal Engineering, Tsinghua University, Beijing 100084, China ;
2. Beijing Aerospace Propulsion Institute, Beijing 100076, China
Abstract:Urea is produced in a tower with falling liquid droplets to enhance the evaporation and drying of the urea particles. The physical process has a liquid column feeding an atomizer. This study analyzes the effect of the nozzle diameter, liquid velocity and air flow velocity on the urea maximum drop diameter. A model is developed to predict the critical heights of the liquid column and the maximum drop diameter. The model predictions agree well with the experimental data, indicating its ability to predict the maximum drop diameter. The atomizer nozzle diameter and the tangential velocity at the orifice strongly influence the drop size, while the liquid velocity at the orifice has little impact on the maximum drop diameter. The air flow velocity in the tower also affects the maximum drop diameter when the air flow velocity is greater than 1.5 m·s-1, but the effect is very small for the air flow velocity less than 1.5 m·s-1. Thus, the maximum drop diameter can be increased by increasing the atomizer nozzle diameter and the tangential velocity at the orifice. Secondary breakup will occur with large drops falling in the tower during cooling and solidification. The critical Weber number gives the maximum drop diameter of urea in the tower for granulation processing about 4 mm.
Key words: granulation processing tower     urea processing during solidification     liquid column     maximum drop diameter

1 尿素熔融液的液滴形成力学分析

 图 1 尿素液流出喷头的速度示意图

 图 2 尿素液柱受力情况分析

Fr1的方向与v1相反; Fr2的方向与v2相反,垂直向上,与Fb相同,和Fg相反; Fr3的方向与v3相反。在这些力当中,FgFr2FbFr3促使尿素液柱断裂,而Fr1FσFμ维持液柱形状不变。在此7个力的共同作用下,只有当促使液柱变形的力之和超过保持尿素液柱形状不变的力之和时,液柱才会出现变形和断裂[2],形成独立存在的尿素液柱,进而在表面张力及内部粘性力的共同作用下,形成近似于球形的液滴。发生断裂时对应的液柱长度即为最大液柱长度,亦即临界液柱长度。

 ${{F}_{\text{g}}}-{{F}_{\text{b}}}-{{F}_{\text{r2}}}+{{F}_{\text{r3}}}\ge {{F}_{\sigma }}+{{F}_{\text{a}}}+{{F}_{\text{r1}}}+{{F}_{\mu }}.$ (1)

 $\sqrt{{{\left( {{F}_{\text{g}}}-{{F}_{\text{b}}}+{{F}_{\text{r2}}} \right)}^{2}}+{{\left( {{F}_{\text{r3}}} \right)}^{2}}}\ge {{F}_{\sigma }}+{{F}_{\text{a}}}+{{F}_{\text{r1}}}+{{F}_{\mu }}.$ (2)

 ${{F}_{\text{g}}}=mg=\frac{\pi }{4}{{d}^{2}}l{{\rho }_{u}}g.$ (3)

 $\begin{matrix} 浮力 & {{F}_{\text{b}}}=\frac{\pi }{4}{{d}^{2}}l{{\rho }_{\text{a}}}g. \\ \end{matrix}$ (4)

 $\begin{matrix} 正面气流阻力 & {{F}_{\text{r1}}}={{C}_{\text{D1}}}\frac{\pi }{4}{{d}^{2}}\frac{1}{2}{{\rho }_{\text{a}}}v_{1}^{2} \\ \end{matrix}.$ (5)

 $\begin{matrix} 离心力 & {{F}_{\text{a}}}=\frac{\pi }{4}{{d}^{2}}l{{\rho }_{u}}\frac{v_{3}^{2}}{r} \\ \end{matrix}.$ (6)
 $\begin{matrix} 气流向上曳力 & {{F}_{\text{r2}}}={{C}_{\text{D2}}}dl\frac{1}{2}{{\rho }_{\text{a}}}v_{2}^{2} \\ \end{matrix}.$ (7)

 ${{C}_{\text{D2}}}=5{{\left( \frac{{{\mu }_{\text{a}}}{{v}_{2}}}{d} \right)}^{-\frac{1}{4}}}.$ (8)
 $\begin{matrix} 旋转气流阻力 & {{F}_{\text{r3}}}={{C}_{\text{D3}}}dl\frac{1}{2}{{\rho }_{\text{a}}}v_{3}^{2} \\ \end{matrix}.$ (9)

 ${{C}_{\text{D3}}}=5{{\left( \frac{{{\mu }_{a}}{{v}_{3}}}{d} \right)}^{-\frac{1}{4}}}.$ (10)
 $\begin{matrix} 表面张力 & {{F}_{\sigma }}=\pi d\sigma . \\ \end{matrix}$ (11)

 $\begin{matrix} {{F}_{\mu }}={{\mu }_{u}}\pi d{{v}_{1}}. \\ \end{matrix}$ (12)

 ${{l}_{\text{c}}}=\frac{4\pi \sigma -\frac{4}{d}\frac{v_{3}^{2}}{r}+4\pi {{\mu }_{u}}{{v}_{1}}+0.9\pi {{\rho }_{\text{a}}}v_{1}^{2}d}{\sqrt{{{\left( {{\rho }_{\text{u}}}-{{\rho }_{\text{a}}} \right)}^{2}}{{g}^{2}}{{\pi }^{2}}{{d}^{2}}-20\pi \left( {{\rho }_{\text{u}}}-{{\rho }_{\text{a}}} \right){{\rho }_{\text{a}}}gx_{2}^{\frac{7}{4}}\mu _{\text{a}}^{-\frac{1}{4}}{{d}^{\frac{5}{4}}}+\frac{100}{{{\pi }^{2}}}\rho _{\text{a}}^{2}\mu _{\text{a}}^{-\frac{1}{2}}{{d}^{\frac{1}{2}}}\left( v_{2}^{\frac{7}{2}}+v_{3}^{\frac{7}{2}} \right)}}.$ (13)

 ${{\phi }_{\max }}=\sqrt[3]{\frac{3}{2}{{d}^{2}}{{l}_{c}}}.$ (14)

 图 3 模型计算结果与实验测量结果的比较

 $\phi \propto {{d}^{0.4}}.$ (15)

 图 4 喷孔处切线速度和塔内通风上升速度对最大液柱长度和液滴直径的影响(d=1.5 mm,v1=3.2 m·s-1)

 图 5 喷孔出口速度对最大液柱长度和液滴直径的影响(v2=0.6 m·s-1,v3=5 m·s-1)

 图 6 喷孔直径对最大液柱长度和液滴直径的影响(v1=3.2 m·s-1,v2=0.6 m·s-1,v3=5 m·s-1)

 ${{l}_{\text{c}}}=\frac{4\pi \sigma }{\sqrt{{{\left( {{\rho }_{\text{u}}} \right)}^{2}}{{g}^{2}}{{\pi }^{2}}{{d}^{2}}+\frac{100}{{{\pi }^{2}}}{{\rho }_{\text{a}}}^{2}\mu _{\text{a}}^{-\frac{1}{2}}{{d}^{\frac{1}{2}}}{{v}_{3}}^{\frac{7}{2}}}}.$ (1)

2 液滴下落过程的二次分裂分析

 ${{{F}'}_{\text{g}}}=\frac{4}{3}\pi \phi _{\text{o}}^{3}{{\rho }_{\text{u}}}g.$ (17)

 ${{{F}'}_{\text{b}}}=\frac{4}{3}\pi \phi _{\text{o}}^{3}{{\rho }_{\text{a}}}g.$ (18)

 ${{{{F}'}}_{\text{r}}}={{C}_{\text{d}}}\frac{1}{4}\pi \phi _{\text{o}}^{2}\frac{1}{2}{{\rho }_{\text{a}}}{{v}^{2}}.$ (19)

 ${{{F}'}_{\sigma }}=\pi {{\phi }_{0}}\sigma .$ (20)

 ${{{F}'}_{\text{r}}}-{{{F}'}_{\text{g}}}-{{{F}'}_{\text{b}}}\ge {{{F}'}_{\sigma }}.$ (21)

 ${{{F}'}_{\text{r}}}\ge {{{F}'}_{\sigma }}.$ (22)

 ${{C}_{\text{d}}}\frac{1}{4}\pi \phi _{\text{o}}^{2}\frac{1}{2}{{\rho }_{\text{a}}}{{v}^{2}}\ge \pi {{\phi }_{\text{o}}}\sigma ,$ (23)

 $\frac{{{\phi }_{\text{o}}}{{\rho }_{\text{a}}}{{v}^{2}}}{\sigma }\ge \frac{8}{{{C}_{\text{d}}}}.$ (24)

3 尿素颗粒模型预测的实验验证

 图 7 尿素颗粒的粒度分布实验结果与模型计算结果对比

4 结论

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