## Study on the Relationship between the Theoretical Plate Number and the Mass Transfer Unit Number in Chemical Unit Operation

Li Xuehui, Chen Liang, Wu Zhengshun,

Abstract

Packed tower and plate tower were common equipment in the operation of chemical units. The calculation of the height of the packing layer in the packed tower and the determination of the number of theoretical plates in the tray tower were the key. The calculation of the height of the packing layer in the textbook was based on the mass transfer unit, and the determination of the theoretical plate in the plate tower was determined by the McCable-Thiele method, and the relationship between the number of mass transfer units and the theoretical plate was less. In this paper, The mass transfer unit number and the theoretical plate number were respectively calculated by the absorption factor method in detail, and the quantitative relationship between them was established. It could be used for the comparison of the two type towers, which had a certain guiding role for the analysis and design of the tower, and helped students to learn and understand the process of absorption and rectification unit operations.

Keywords： Chemical unit operation ; Theoretical plate ; Mass transfer unit number ; Absorption factor

Li Xuehui. Study on the Relationship between the Theoretical Plate Number and the Mass Transfer Unit Number in Chemical Unit Operation. University Chemistry[J], 2020, 35(2): 82-87 doi:10.3866/PKU.DXHX201905062

## 1 板式塔中理论板数的求取

$Y_{e}=m X_{e}+B$

$Y_{n}=m X_{n}+B$

Xn代入操作线方程则有：

$Y_{n+1}=\frac{L\left(Y_{n}-B\right)}{m V}+Y_{a}-\frac{L X_{a}}{V}$

$Y_{n+1}=A\left(Y_{n}-B\right)+Y_{a}-A m X_{a}=A Y_{n}-A\left(m X_{a}+B\right)+Y_{a}$

\begin{aligned}&Y_{a}^{*}=m X_{a}+B\end{aligned}

$Y_{n+1}=A Y_{n}-A Y_{a}^{*}+Y_{a}$

$Y_{2}=A Y_{a}-A Y_{a}^{*}+Y_{a}=Y_{a}(1+A)-A Y_{a}^{*}$

n = 2时有：

\begin{aligned}Y_{3} &=A Y_{2}-A Y_{a}^{*}+Y_{a}=A\left[Y_{a}(1+A)-A Y_{a}^{*}\right]-A Y_{a}^{*}+Y_{a} \\&=Y_{a}\left(1+A+A^{2}\right)-Y_{a}^{*}\left(A+A^{2}\right)\end{aligned}

$Y_{b}=Y_{a}\left(1+A+A^{2}+\ldots+A^{N}\right)-Y_{a}^{*}\left(A+A^{2}+\ldots+A^{N}\right)=Y_{a} \frac{1-A^{N+1}}{1-A}-Y_{a}^{*} A \frac{1-A^{N}}{1-A}$

$Y_{b}=A Y_{N}-Y_{a}^{*} A Y_{a}^{*}+Y_{a}$

$Y_{a}=Y_{b}-A\left(Y_{b}^{*}-Y_{a}^{*}\right)$

### 图1

$N=\frac{\ln \left[\left(Y_{b}-Y_{b}^{*}\right) /\left(Y_{a}-Y_{a}^{*}\right)\right]}{\ln (A)}$

## 2 填料塔中传质单元数的求取

### 图2

$Y_{e}=m X_{e}+B$

$Y=\frac{L}{G}\left(X-X_{a}\right)+Y_{a},得X=\frac{V}{L}\left(Y-Y_{a}\right)+X_{a}$

$\begin{array}{l}{N_{{\rm{OG}}}} = \int_{{Y_a}}^{{Y_b}} {\frac{{{\rm{d}}Y}}{{Y - {Y_e}}}} = \int_{{Y_a}}^{{Y_b}} {\frac{{{\rm{d}}Y}}{{Y - m \cdot \left[ {\frac{V}{L}\left( {Y - {Y_a}} \right) + {X_a}} \right] - B}}} \\ = \int_{{Y_a}}^{{Y_b}} {\frac{{{\rm{d}}Y}}{{Y\left( {1 - m \cdot \frac{V}{L}} \right) + \left[ {m \cdot \frac{V}{L}{Y_a} - \left( {m{X_a} + B} \right)} \right]}}} = \int_{{Y_a}}^{{Y_b}} {\frac{{{\rm{d}}Y}}{{Y\left( {1 - m \cdot \frac{V}{L}} \right) + \left[ {m \cdot \frac{V}{L}{Y_a} - Y_a^*} \right]}}} \\ = \frac{1}{{(1 - S)}}\ln \left( {\frac{{\left[ {(1 - S){Y_b} + S{y_a} - Y_a^*} \right]}}{{\left[ {(1 - S){Y_a} + S{Y_a} - Y_a^*} \right]}} = \frac{1}{{(1 - S)}}\ln \left( {\frac{{{Y_b} - S\left( {{Y_b} - {Y_a}} \right) - Y_a^*}}{{{Y_a} - Y_a^*}}} \right)} \right.\end{array}$

$\begin{array}{l}S = \frac{1}{A} = \frac{{mV}}{L} = \frac{{{\rm{相平衡线斜率}}}}{{{\rm{操作线斜率}}}} = \frac{{\left( {Y_b^* - Y_a^*} \right)/\left( {{X_b} - {X_a}} \right)}}{{\left( {{Y_b} - {Y_a}} \right)/\left( {{X_b} - {X_a}} \right)}} = \frac{{Y_b^* - Y_a^*}}{{{Y_b} - {Y_a}}}\\{\rm{即}}, {Y_b} - {Y_a} = A\left( {Y_b^* - Y_a^*} \right)\end{array}$

$N_{\mathrm{OG}}=\frac{A}{(A-1)} \ln \left(\frac{Y_{b}-Y_{b}^{*}}{Y_{a}-Y_{a}^{*}}\right)$

$H=H_{\mathrm{OG}} \times N_{\mathrm{OG}}=N_{\mathrm{G}} \times H_{\mathrm{G}}=N_{\mathrm{L}} \times H_{\mathrm{L}}$

$H_{\mathrm{OG}}=H_{\mathrm{G}}+\frac{m G}{L} H_{\mathrm{L}}$

$N_{\mathrm{OG}}=\int_{Y_{a}}^{Y_{b}} \frac{\mathrm{d} Y}{Y-Y_{e}}, \quad N_{\mathrm{G}}=\int_{Y_{a}}^{Y_{b}} \frac{\mathrm{d} Y}{Y-Y^{*}}, \quad N_{\mathrm{L}}=\int_{X_{a}}^{X_{b}} \frac{\mathrm{d} X}{X^{*}-X}$

$\frac{N_{\mathrm{OG}}}{N}=\frac{A \ln (A)}{A-1}$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

McCable, W. L.; Smith, J. C.; Harriott, P. Unit Operations of Chemical Engineering, 7th ed.; McGraw Hill Education: New York, 2014.

Green, D. W.; Perry, R. H. Perry's Chemical Engineers' Handbook, 8th ed.; McGraw-Hill: New York, 2008.

Judson, K. C. Separation Processes, 2nd ed.; McGraw-Hill: New York, 1980.

Seader, J. D.; Henley, E. J.; Roper, D. K. Separation Process Principles, 3rd ed.; John Wiley & Sons, Inc.: New York, 2010.

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