3.1.3. Improved *K*-Value Model

The *K*-value model was improved by simplifying the calculation of the weighting factor. This was derived from the molar fraction summation equation. The derivation process is shown below.

The relationship between the *K* value of the reference component *Kb,j* and the *K* value of the common component is:

$$
\ln \mathsf{K}\_{b,j} = \sum\_{i} \omega\_{i,j} \ln \mathsf{K}\_{i,j}.\tag{22}
$$

The weighting factors need to satisfy:

$$
\sum \omega\_{i,j} = 1.\tag{23}
$$

The stage temperature needs to satisfy the following relationship:

$$\Phi\left(T\_j\right) = \sum \mathbf{x}\_{i,j} - 1 = \sum \frac{y\_{i,j}}{K\_{i,j}} - 1 = 0. \tag{24}$$

The introduction of new variables *ui,j* was defined as:

$$
\mu\_{i,j} = \ln(\mathcal{K}\_{i,j} / \mathcal{K}\_{b,j}).\tag{25}
$$

The new relationship is:

$$\widetilde{\Phi}(T\_j) = \sum \frac{y\_{i,j}}{K\_{b,j}e^{\mu\_{i,j}}} - 1. \tag{26}$$

Since *Kb,j* was calculated by *Ki,j* through a weighting function, the trend with temperature changes was the same, so *uj,i* is a temperature-independent parameter, so at the stage temperature:

$$\mathcal{K}\_{b,j}(T\_j)e^{\mu\_{j,i}} = \mathcal{K}\_{j,i}(T\_j),\tag{27}$$

$$
\Phi(T\_j) = \overline{\Phi}(T\_j). \tag{28}
$$

The derivative of Φ *Tj* and Φ*Tj* at the stage temperature was equal to:

$$
\left.\frac{d\Phi}{dT}\right|\_{T\_j} = \left.\frac{d\overline{\Phi}}{dT}\right|\_{T\_j},\tag{29}
$$

$$\sum \frac{y\_{j,i}}{K\_{j,i}} \left(\frac{\partial \ln K\_{j,i}}{\partial T}\right)\Big|\_{\,^\*T\_j} = \left(\sum \frac{y\_{j,i}}{K\_{j,i}}\right) \frac{d \ln K\_{b,j}}{dT}\Big|\_{\,^\*T\_j} \tag{30}$$

$$\sum \frac{y\_{j,i}}{K\_{j,i}} \left(\frac{\partial \ln K\_{j,i}}{\partial T}\right)\Big|\_{\ T\_j} = \left(\sum \frac{y\_{j,i}}{K\_{j,i}}\right) \sum \omega\_{j,i} \frac{\partial \ln K\_{j,i}}{\partial T}\Big|\_{\ T\_j} \tag{31}$$

The weighting function used the new calculation method:

$$\omega\_{i,j} = \frac{y\_{i,j} / \mathcal{K}\_{i,j}}{\sum y\_{i,j} / \mathcal{K}\_{i,j}}.\tag{32}$$

#### *3.2. Initial Value Estimation Method*

The system of equations obtained from the modeling of the steady-state reactive distillation process was non-linear and was very difficult to solve. Therefore, it was important to provide good initial estimates, otherwise it would not be possible to reach a solution.

Before providing good initial estimates, all feed streams were mixed to perform chemical reaction equilibrium calculations to obtain a new set of compositions and flow rate, which were used to calculate the initial value of the variable.

Temperature: Calculate the dew and bubble point temperature by flash calculation, which was used as the initial temperature at the top and at the bottom of the tower. The initial temperature of the middle stages was obtained by linear interpolation.

Composition: The composition calculated by isothermal flash calculation was used as the initial value of the vapor-liquid composition of each stage.

Flow: According to the constant molar flow rate assumption and the total material balance calculation, the initial value of the vapor–liquid flow rate of each stage was obtained

Reaction extent: The reaction extent calculated by the chemical reaction equilibrium was taken as the maximum value of the reaction extent in the reaction section. According to the feeding condition of the reaction section, the position with the maximum reaction extent was selected. If there was only one feed in the reaction section, the position of the feed stage was selected. If there were multiple feeds in the reaction section, the position of the feed stage near the reboiler was chosen. If there was no feed in the reaction section, the position of the reaction stage closest to the feed was selected. The initial value of the reaction extent of the other stages were calculated according to the maximum value and a certain proportion of attenuation. The calculation is shown below.

Between the first reaction stage and the stage with the maximum reaction extent:

$$R\_{j,r} = R\_{\text{max}} \times \frac{j - j\_1 + 1}{\left(j\_m - j\_1 + 1\right)^2}. \tag{33}$$

Between the stage with the maximum reaction extent and the last reaction stage:

$$R\_{j,r} = R\_{\text{max}} \times \frac{j\_n - j + 1}{\left(j\_n - j\_m + 1\right)^2}. \tag{34}$$

where *R*max is the maximum value of the reaction extent in the reaction section, *jm* is the stage position where the maximum reaction extent occurs; *j*1 is the first stage position in the reaction section; and *jn* is the last stage position in the reaction section.

#### *3.3. Calculation Steps and Block Diagram*

The reactive distillation column adopts an improved Inside–Out method for calculation. The main working idea is that the the outer loop used a strict thermodynamic model to calculate the phase equilibrium constant and the vapor–liquid phase enthalpy difference, and the results were used to correct the approximate thermodynamic model parameters and update the reaction extents of each stage. The inside loop uses an approximate thermodynamic model to solve the MESHR equation to obtain the stage temperature, flow rate, and composition. After the inside loop calculation converges or reaches the number of iterations, the model returns to the outer loop to continue the calculation. Whentheinsideandouterloopconvergesatthesametime,thecalculationends.

The calculation steps are shown below:

*r*

## 1. Given initial value

(1) According to the feed, pressure, and operation specifications, the initial values of the temperature, flow rate, composition, and the reaction extent are provided (Section 3.2).
