*3.1. Inside–Out Method*

The Inside–Out method divides the mathematical model into an inside and outer loop for iterative calculation, and uses two sets of thermodynamic models, a strict thermodynamic model and an approximate thermodynamic model ( *K* value model and enthalpy value model). The simple approximate thermodynamic model was used for frequent inside loop calculations to reduce the time it takes to calculate the thermodynamic properties. The strict thermodynamic model was used for the outer loop calculations, and the parameters of the approximate thermodynamic model were corrected at the outer loop.

Defining the inside loop variables—relative volatility <sup>α</sup>*i,j*, stripping factors *Sb,j* of the reference components, liquid phase stripping factors *RL,j*, and the vapor phase stripping factors *RV,j*.

$$\mathbf{a}\_{i,j} = \mathbb{K}\_{i,j} / \mathbb{K}\_{b,j} \; \mathbb{S}\_{b,j} = \mathbb{K}\_{b,j} V\_j / L\_{j},\tag{6}$$

$$R\_{L,j} = 1 + lL\_j/L\_j \ R\_{V,j} = 1 + \mathcal{W}\_j/V\_j. \tag{7}$$

The inside loop uses the vapor and liquid phase flow rates of each component instead of the composition, for the calculations. Their relationship with the vapor and liquid phase composition and flow is as follows:

$$\mathbf{y}\_{i,\mathbf{j}} = \upsilon\_{i,\mathbf{j}} / V\_{\mathbf{j}} \ \mathbf{x}\_{i,\mathbf{j}} = \mathbf{l}\_{i,\mathbf{j}} / L\_{\mathbf{j}} \tag{8}$$

$$V\_j = \sum\_{i=1}^{c} v\_{i,j} \ L\_j = \sum\_{i=1}^{c} l\_{i,j} \tag{9}$$

According to the inside loop variables defined by the mathematical model of the Inside–Out method, the material balance Equation (M) and the phase equilibrium Equation (E) can be rewritten into the following relationship.

Material Balance:

$$(R\_{i,j-1} - (R\_{L,j} + a\_{i,j} S\_{b,j} R\_{V,j}) l\_{i,j} + (a\_{i,j+1} S\_{b,j+1}) l\_{i,j+1} = -f\_{i,j} + \sum\_{r} \upsilon\_{r,i} R\_{r,j}.\tag{10}$$

Phase balance:

$$w\_{i,j} = a\_{i,j} \mathbf{S}\_{b,j} l\_{i,j}.\tag{11}$$
