**2. Mathematical Model**

The schematic diagram of the reaction distillation column is shown in Figure 1. The whole column is composed of N stages. The condenser is regarded as the first stage and the reboiler is regarded as the last stage. There can be a liquid phase product, a vapor phase product, and a free water product at the top of the tower, and a liquid phase product at the bottom of the column. Chemical reactions can take place anywhere in the column. The general model of the jth theoretical stage is shown in Figure 2, where Fj is the feed flow rate of stage j, Lj is the liquid flow rate outputting stage j, and inputting stage j + 1, Vj is the vapor flow rate outputting stage j and inputting stage j − 1, Uj is the liquid side flow rate outputting stage j, Wj is the vapor side flow rate outputting stage j, Qj is the heat duty from stage j, Rr,j is the reaction extent of the rth reaction on stage j. The schematic representation from Figure 2 to represent all stages are shown in Figure 3.

**Figure 1.** Schematic representation of the reactive distillation column.

**Figure 2.** The jth equilibrium stage of the reactive distillation column.

The stages of the column are assumed to be in equilibrium conditions, the vapour and liquid phases leaving the stage are assumed to be in thermodynamic equilibrium, the stage pressure, temperature, flow, and composition are assumed to be constant at each stage. Five sets of equations are used to describe the equilibrium state of the stage—the material balances, the phase equilibrium, the molar fraction summation, the enthalpy balance, and the chemical equilibrium equations.

**Figure 3.** Schematic representation of input and output streams in all stages of a distillation column.

A set of highly nonlinear equations was obtained by modeling the steady state process of the reactive distillation column. The non-linearities were in the phase equilibrium, enthalpy balances, and chemical equilibrium equations. The solution of model equations is very difficult and the convergence depends strongly on the goodness of the initial guesses. After choosing appropriate iterative variables and providing the initial value of variables, the iterative calculation process can start. When the iterative calculation converges, the stage temperature, flow, and composition profiles can be obtained.

The mathematical model of reactive distillation [21]:

(1) Material balance Equation (M):

$$L\_{j-1}\mathbf{x}\_{i,j-1} - (V\_j + \mathcal{W}\_j)y\_{i,j} - (L\_j + \mathcal{U}\_j)\mathbf{x}\_{i,j} + V\_{j+1}y\_{i,j+1} + F\_jz\_{i,j} + \sum\_r \nu\_{r,i}\mathcal{R}\_{r,j} = 0,\tag{1}$$

(2) Phase equilibrium Equation (E):

$$y\_{i,j} - \mathbb{K}\_{i,j} \mathbf{x}\_{i,j} = \mathbf{0}\_{\prime} \tag{2}$$

(3) Molar fraction summation Equation (S):

$$\sum\_{i=1}^{c} y\_{i,j} - 1 = 0 \ \sum\_{i=1}^{c} x\_{i,j} - 1 = 0,\tag{3}$$

(4) Enthalpy balance Equation (H):

$$L\_{j-1}h\_{j-1} - (V\_j + \mathcal{W}\_j)H\_{\bar{j}} - (L\_j + \mathcal{U}\_j)h\_{\bar{j}} + V\_{\bar{j}+1}H\_{\bar{j}+1} + F\_{\bar{j}}H\_{\bar{F}\bar{j}} + Q\_{\bar{j}} - \Delta H\_{r,\bar{j}} = 0,\tag{4}$$

(5) Chemical reaction rate Equation (R):

$$r\_{\mathbf{j},\mathbf{r}} = f(T\_{\mathbf{j},\mathbf{r}}P\_{\mathbf{j},\mathbf{r}}x\_{\mathbf{i},\mathbf{j}\prime}y\_{\mathbf{i},\mathbf{j}}).\tag{5}$$
