**1. Introduction**

Reactive distillation combines the chemical reaction and distillation separation process in a single unit operation [1]. As reactive distillation has the advantages of improving reaction conversion rate and selectivity and reducing costs, the research on design and optimization of the reactive distillation processes has increased rapidly in recent years [2]. Now reactive distillation technology has been successfully applied to the areas of esterification, etherification, hydrogenation, hydrolysis, isomerization, and saponification [3].

The production of methyl tert-butyl ether (MTBE), methyl tert-pentyl ether (TAME), and ethyl tert-butyl ether (ETBE), etc., adopts the reactive distillation technology, and the reaction conversion rate was significantly higher than the fixed-bed reactors [4]. Using the reactive distillation technology to produce methyl acetate can reduce investment and energy costs to one-fifth, and the reaction conversion rate is close to 100% [5]. Zhang et al. proposed a novel reaction and extractive distillation (RED) process and used it for the synthesis of isopropyl acetate (IPAc). The results showed that the purity of IPAc reached 99.5% [6].

As distillation and reaction take place simultaneously, there are complex interaction effects between them [7]. Therefore, it is very important to use mathematical methods to model and simulate the reactive

distillation process, and optimize the operating conditions and equipment parameters [8]. The traditional algorithms for simulating reactive distillation include the tridiagonal matrix method, the relaxation method, the simultaneous correction method (Newton–Raphson), and the homotopy method.

Zhang et al. proposed a mathematical model of the heterogeneous catalytic distillation process. For the boundary value problem solved by the model, the multi-target shooting method and the Newton–Raphson iteration were used. The calculation results were consistent with the experimental data [9]. The partial Newton method proposed by Zhou et al. used the results from a repeated relaxation method as the initial value of the iteration variable, and then used the Newton–Raphson method to solve the MESHR equation (material balance, phase equilibrium, molar fraction summation, enthalpy balance, chemical reaction rate) [10]. Qi et al. aimed at the equilibrium reactive distillation process and transformed the physical variables such as the stage composition, flow, and enthalpy, such that the transformed mathematical model of the reactive distillation was completely consistent with the ordinary distillation and the modified Newton–Raphson method was used to solve it [11]. Juha introduced the homotopy parameters in the phase equilibrium, the enthalpy equilibrium, and the chemical reaction equilibrium equations, to establish a homotopy model of reactive distillation, and successfully simulated an MTBE reactive distillation column [12]. Wang et al. used the relaxation method to calculate the methyl formate hydrolysis catalytic distillation column. The simulation results were compared with the experimental ones and showed grea<sup>t</sup> agreemen<sup>t</sup> [13]. Steffen and Silva divided the reactive distillation model into several smaller sets of equations based on the equation tearing method, and obtained a set of linear and nonlinear equations each, which simulated chemical reactions [14].

Boston and Sullivan first proposed the Inside–Out method to simulate a multi-component ordinary distillation process [15,16]. Compared to other algorithms, two thermodynamic property models are used. The approximate thermodynamic models are used for frequent inside loop calculations to solve the MESH equation (material balance, phase equilibrium, molar fraction summation, enthalpy balance). The strict thermodynamic models are used for outer loop calculations of the phase equilibrium constant K and the approximate model parameters. This saves a lot of time when calculating thermodynamic properties. The inside loop uses a stripping factor (Sb) as an iterative variable, which integrates the effects of the stage temperature and the vapor and liquid flow rates, reduces the number of iterative variables, and improves the stability of convergence.

Russell further improved the Inside–Out method by solving the sparse tridiagonal matrix and applying the improved Thomas Chase method, and applied it to the calculation of crude oil distillation [17]. The modified Inside–Out method was proposed by Saeger, who added a two-parameter model for liquid composition, representing the effect on phase equilibrium constant, improving the convergence stability of highly non-ideal systems, and improving the convergence rate [18]. Jelinek simplified the Inside–Out method proposed by Russell; the simplified method could solve the distillation process specified by any operation and did not use an approximate enthalpy model in the inside loop, reducing the complexity of the calculation process [19].

Basing the Inside–Out method on simulating ordinary distillation process, increased the chemical reaction rate equation and established a mathematical model of the reactivedistillation. In view of the obvious enhancement of the nonlinearity of the mathematical model, it was more di fficult to solve the problem than ordinary distillation [20]. This study used the Inside–Out method to solve the reactive distillation process, simplified the calculation of the *K* value of the approximate thermodynamic model, and proposed an initial value estimation method. This is an in-house model.
