Modeling the rectification process on an analog computer
Abstract Generated abstract
This paper describes the formulation and analog-computer solution of differential mass-transfer equations for rectification of multicomponent mixtures. Using material-balance relations and equilibrium constants expressed through relative volatilities, the author derives equations for component changes in the enriching and stripping sections of a column and specifies boundary conditions for a column with a partial condenser. The method is applied to a three-component mixture on an EMU-10 analog computer, with variation of relative volatility along the column represented graphically by nonlinear blocks. A comparison with calculations assuming constant relative volatility indicates that the difference in results is insignificant for the case considered.
Full Text
Modeling the Process of Rectification on an Analog Computer
E. V. Makienko
(Presented by the scientific seminar of the Computing Laboratory of TPI)
For calculations of separation processes for multicomponent mixtures on analog computers, a mathematical description of the process by differential mass-transfer equations is used.
The basic equation of the process of mass transfer between the liquid and vapor phases, under the assumption that the vapor and liquid flows are constant over the height of the column, is written in the following form [2]:
\[ \frac{d l_i}{dH} = \beta_x \left( x_i - \frac{y_i}{K_i} \right), \tag{1} \]
where \(K_i\) is the equilibrium constant, \(x_i\) is the concentration of the component in the liquid, \(y_i\) is the concentration of the component in the vapor, and \(\beta_x\) is the mass-transfer coefficient.
The values of \(K_i\) are expressed through the relative volatilities of the components, taken with respect to the key component,
\[ \alpha_i = \frac{K_i}{K_b}, \]
where \(K_b\) is the equilibrium constant of the key component being separated. In any section of the column the condition must be satisfied
\[ \sum_{k=1}^{p} y_k = \sum_{k=1}^{p} K_k x_k = 1. \tag{2} \]
From (2) we obtain the equilibrium constant for the \(i\)-th component, expressed through the compositions in the section under consideration and the relative volatilities,
\[ K_i = \alpha_i \sum_{k=1}^{m} \frac{y_k}{\alpha_k}. \tag{3} \]
The material-balance equations for the \(i\)-th component are:
\[ v_i = l_i + d_i, \]
\[ v_i + l_i = \omega_i. \tag{4} \]
We obtain from (1), taking (3) and (4) into account, expressions for the change of the \(i\)-th component [1] in the enriching section
\[ \frac{d l_i}{dH} = \beta_x \left( \frac{l_i}{L} - \frac{l_i+d_i} {\alpha_i \displaystyle\sum_{k=1}^{p}\frac{l_k+d_k}{\alpha_k}} \right), \tag{5} \]
and in the stripping section
\[ \frac{d l_i}{dH} = \beta_x \left( \frac{l_i}{L} - \frac{l_i-\omega_i} {\alpha_i \displaystyle\sum_{k=1}^{p}\frac{l_k-\omega_k}{\alpha_k}} \right). \tag{6} \]
The calculation was carried out for a 3-component mixture of the following composition:
quantity of mixture entering the column:
\[ F=10\,\frac{\text{mol}}{\text{h}} \]
\[ f_a=1{,}5\,\frac{\text{mol}}{\text{h}}, \]
\[ f_b=4{,}25\,\frac{\text{mol}}{\text{h}}, \]
\[ f_c=4{,}25\,\frac{\text{mol}}{\text{h}}; \]
distribution of the components in the distillate:
\[ d_a=0{,}35\,\frac{\text{mol}}{\text{h}}, \]
\[ d_b=4{,}25\,\frac{\text{mol}}{\text{h}}, \]
\[ d_c=4{,}23\,\frac{\text{mol}}{\text{h}}; \]
distribution of the components in the bottom residue:
\[ \omega_a=1{,}146\,\frac{\text{mol}}{\text{h}}, \]
\[ \omega_b=0{,}002\,\frac{\text{mol}}{\text{h}}, \]
\[ \omega_c=0{,}03\,\frac{\text{mol}}{\text{h}}. \]
In the calculation, the change in the values of relative volatility over the height of the column was taken into account.
The dependences \(\alpha_i(t)\) and \(t(k_b)\), where
\[ K_b= \frac{\displaystyle\sum_{i=1}^{p}\frac{l_i+d_i}{\alpha_i}}{V}, \]
were specified graphically and reproduced on nonlinear blocks.
When integrating the systems of equations (5) and (6), it is necessary to specify boundary conditions. For a column with a partial condenser, the concentration of the first component in the liquid in the upper section of the column is taken equal to its concentration in the residual gas [1]:
\[ l_{di}=\frac{d_i L_d}{\alpha_i \displaystyle\sum_{k=1}^{p}\frac{d_k}{\alpha_k}} . \tag{7} \]
In the lower section of the column, the concentrations of the component in the vapor and in the liquid are assumed to be at equilibrium:
\[ v_{li}=\frac{\omega_i \alpha_i V_{\omega}}{\displaystyle\sum_{k=1}^{p}\omega_k \alpha_k} . \tag{8} \]
From the material-balance equation,
\[ l_i=v_i+\omega_i . \]
The calculation of the complete column is reduced to the simultaneous solution of equations (5) and (6) with initial conditions determined from (8) and (9), and with boundary conditions determined from the material-balance equation on the feed plate:
\[ l_{im}=l_{if}+l'_{in}, \]
where \(m\) is the section taken below the feed inlet;
\(n\) is the section taken above the feed inlet.
To determine the component, the equation of the overall material balance is used:
\[ l_p=L-\sum_{i=1}^{p-1} l_i, \]
where \(p\) is the number of components.
The solution was carried out on the “EMU-10” analog computer.
For comparison of the results, a solution of the systems of equations (5) and (6) was carried out for \(\alpha=\mathrm{const}\). The difference in the results proved to be insignificant.
REFERENCES
- B. G. Bergo. Separation of Multicomponent Mixtures. Khimiya Publishing House, 1965.
- V. V. Kafarov. Fundamentals of Mass Transfer. Vysshaya Shkola Publishing House, 1962.
- R. Aris. Optimal Design of Chemical Reactors, 1963.