*2.4. The Objective Function of the Process Optimization*

In the process of separating acetonitrile and water mixture by PSD, not only the separation requirements of product purity should be satisfied but also the economy of the process should be considered. TAC is a commonly used economic cost index for chemical process simulation and optimization, which is used in the evaluation of the economic rationality and feasibility of the process. The calculation of TAC is based on the following equations, which is taken from Luben and Chien [34]. Design variables, such as the tray numbers, the reflux ratios, and the feeding positions of the two columns, are optimized through the equations below, in which the minimum TAC and purity of acetonitrile are considered as the objective function and constraint condition, respectively.

Objective function:

$$\text{min } \text{TAC} = \text{C}\_{\text{operation}} + \text{C}\_{\text{crystal}} / 3 \,, \tag{1}$$

$$\text{TAC} = f(\mathbf{x}), \tag{2}$$

$$\mathcal{C}\_{\text{capital}} = \mathcal{C}\_{\text{exchanger}} + \mathcal{C}\_{\text{column}\nu} \tag{3}$$

$$C\_{\text{exchangger}} = 7296 A^{0.68} \,\text{,}\tag{4}$$

$$C\_{\text{column}} = 17640D^{1.066}L^{0.802},\tag{5}$$

$$\text{C}\_{\text{operation}} = \text{C}\_{\text{steam}} + \text{C}\_{\text{cooling water}} \tag{6}$$

$$L = 1.2 \times 0.61 \times (\text{N}\_{\text{T}} - 2),\tag{7}$$

$$A = \mathbb{Q}(\mathbb{K} \times \Delta T\_{\text{m}}).\tag{8}$$

Constraints:

$$w\_{\rm B1} < 0.001,\tag{9}$$

$$w\_{\rm B2} > 0.999.\tag{10}$$

where TAC is the total annual cost, *C*operation is the annual operation cost, and the annual operation time is 8000 h; *C*capital is the equipment investment cost, and the payback period is 3 years; *x* is the whole process operation variable, including the feeding positions of the LPC and HPC (NF1, NFR, and NF2), reflux ratios (RR1 and RR2), and tray numbers (NT1 and NT2); *C*exchanger and *C*column represent the heat exchanger (including condenser and reboiler) and the column equipment costs, respectively; *A* is the heat exchange area (m2); *L* and *D* represent the height and diameter of the column (m), respectively; The diameter of the column is calculated by the tool of "Tray Sizing" in Aspen Plus software; 0.61 m is the typical distance between the trays; *Q* and Δ*T*<sup>m</sup> are the heat duty (kW) and heat transfer temperature difference (K), respectively; K is the heat transfer coefficient, 0.852 kW/(K·m2) for the condenser and 0.568 kW/(K·m2) for the reboiler; *C*steam and *C*cooling water represent the cost of heating medium and cooling water, respectively; The LP steam is used in this work with the price of \$7.78 /GJ, and the price of the cooling water is \$4.43/GJ. Additionally, *w*B1 and *w*B2 are the purity of acetonitrile at the bottom streams of the LPC and HPC, respectively.

#### **3. Optimization of PSD**

#### *3.1. Optimization of PSD without Heat Integration*

#### 3.1.1. Process Optimization Sequence

The design variables of the PSD process include three feeding positions (NF1, NFR, and NF2), the reflux ratios (RR1 and RR2), and the tray numbers (NT1 and NT2) of the LPC and HPC. The variables are optimized through the sequential iteration method, which was commonly used to optimize the PSD process in the published literatures [17,35]. By editing the calculating formulas in Fortran, Aspen Plus software can automatically call the values of the corresponding variables to calculate TAC. The optimization sequence is illustrated in Figure 3. First, the feeding position is optimized as the innermost iteration, followed by the reflux ratio, and finally, the tray number is optimized. The objective function of the whole optimization process is to minimize TAC.

#### 3.1.2. The Optimization Results of PSD

According to the optimization sequence in Figure 2, three feeding positions (NF1, NFR, and NF2), the reflux ratios (RR1 and RR2), and the tray numbers (NT1 and NT2) of the LPC and HPC were optimized iteratively, and the optimum process parameters were obtained. The results can be seen in Table 4. That is, NT1 of the LPC is 21, NF1 and NFR are 19 and 16, respectively, and RR1 is 0.3; NT2 of the HPC is 25, NF2 is 16th, and RR2 is 0.5. The optimized flow diagram of PSD without heat integration for separating acetonitrile–water azeotrope is shown in Figure 3.

**Figure 2.** Optimization sequence of acetonitrile–water separation process by pressure swing distillation (PSD).

**Figure 3.** Optimized flow diagram of pressure swing distillation (PSD) without heat integration.

#### *3.2. Optimization of PSD with Full-Heat Integration*

PSD with full-heat integration is carried out by the huge temperature difference between the top stream of the HPC and the bottom stream of the LPC. That is, the top stream of the HPC can be used as the heating medium of the reboiler of the LPC. Full-heat integration can be carried out by adjusting the reflux ratios of the LPC and HPC.

#### 3.2.1. Process Optimization Sequence of PSD with Full-Heat Integration

The design variables are optimized according to the sequential iteration method. The optimization process is shown in Figure 4. First, the reflux ratio is optimized as the innermost iteration, then the feeding position, and finally, the tray number is optimized. The objective function of the whole optimization process is to minimize TAC.

**Figure 4.** Optimization sequence of acetonitrile–water separation process by pressure swing distillation (PSD) with full-heat integration.

#### 3.2.2. Optimization of the Reflux Ratios

In the whole process optimization, the reflux ratios of the two columns are used as the design variables, and the heat duty of the condenser of the HPC is equal to that of the reboiler of the LPC as the objective variable to achieve the full-heat integration design.

Figure 5 investigates the effect of RR1 in the LPC on RR2 in the HPC and heat duties of the condenser and reboiler (QCON2 and QREB2) in HPC. It can be seen that with the increase of RR1, RR2 increases gradually, but QCON2 and QREB2 of the HPC appear to have a minimum. When RR1 is 0.2 and RR2 is 0.7, the values of QREB2 and QCON2 are minimum.

**Figure 5.** Effect of the reflux ratio RR1 in the low pressure column (LPC).

The effect of the reflux ratios of the two columns on the TAC for the whole process is shown in Figure 6. It can also be seen that TAC is the smallest when RR1 is 0.2 and RR2 is 0.7. This finding is consistent with the conclusion in Figure 5. The optimum reflux ratios of the LPC and HPC are 0.2 and 0.7, respectively.

**Figure 6.** Effect of the reflux ratios of two columns on the total annual cost (TAC).

#### 3.2.3. Optimization of the Feeding Positions

The feeding position of the acetonitrile–water mixture (NF1), the feeding position of the recycle stream (NFR), and the feeding position of the HPC (NF2) affect the TAC to varying degrees, as shown in Figure 7. Taking the feeding position NF1 of acetonitrile–water mixture as an example, as the feeding position moves down, the TAC tends to decrease initially and then increase. The TAC is the smallest when feeding on the 20th tray. Similarly, the optimum feeding positions of NFR and NF2 are determined to be the 16th and 9th trays, respectively.

**Figure 7.** Effect of three feeding positions on the total annual cost (TAC).
