# 数学代写|随机过程Stochastic Porcesses代考|STAT6540 Optimization through Sensitivity Analysis

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## 数学代写|随机过程Stochastic Porcesses代考|Optimization through Sensitivity Analysis

As aforementioned, the main manipulated variable for the flash drum is the operation temperature. The objective is to maximize the molar flow rate of methanol in the vapor stream. In this case, the maximum molar composition of water in the vapor stream acts as an inequality constraint. Because it is not possible to include constraints in the sensitivity analysis, the molar composition of water will be computed and the cases that accomplish the constraint will be considered as the candidates for the optimal solution.

To initiate the analysis, a new ID is created in the Sensitivity subfolder of the Model Analysis Tools folder. The default ID S-1 is used in this case (Figure 5.38). Now the information required in the Vary sheet must be completed. The manipulated variable is considered as the temperature of the flash, which is a block variable (Block-Var) defined as TEMP. The limits for the manipulated variable are set as $60^{\circ} \mathrm{C}$ and $100^{\circ} \mathrm{C}$. This range comprises the boiling temperatures of the pure components. The number of points to be analyzed is set as 100. All the mentioned information is shown in Figure 5.39. Next, the measured variables are defined. In this case, the objective function (the molar flow rate of methanol in the vapor stream) is the main measured variable. That flow rate is in the Streams category, with a type Mole-Flow (Figure 5.40). Nevertheless, the molar composition of water in the vapor stream is also defined as measured variable to evaluate the potential solutions that accomplish the inequality constraint. The molar composition is also in the category Streams, but its type is defined as Mole-Frac (Figure 5.41). In the Tabulate sheet, the Fill Variables button is pushed to indicate that both variables of the Define sheet are tabulated (Figure 5.42). Now, the Next button (or the Run button) is pushed to run the simulation. Figure $5.43$ shows the Results sheet. Methanol starts vaporizing at approximately $70.90^{\circ} \mathrm{C}$, with a molar composition of water in the vapor stream of $16.6 \mathrm{~mol} \%$. If the data shown in Figure $5.43$ are analyzed, it can be clearly observed that for temperatures higher than $72.52^{\circ} \mathrm{C}$, the composition of water in the vapor stream exceeds the upper bound of $20 \mathrm{~mol} \%$. Thus, the maximum flow rate of methanol, which can be obtained in the vapor stream without violating the inequality constraint, is $19.6842 \mathrm{kmol} / \mathrm{h}$ at $72.12^{\circ} \mathrm{C}$.

## 数学代写|随机过程Stochastic Porcesses代考|Optimization through Sequential Quadratic Programming

In this section, the optimization of the flash drum using the $\mathrm{SQP}$ method is presented. First, a new optimization routine is created in the Optimization subfolder of the Model Analysis Tools folder. The optimization routine is identified with the ID O-1, as shown in Figure 5.44. Now, the measured variables are defined. For our case of study, the variables involved in the optimization procedure are the following: the flash temperature, the molar flow rate of methanol in the vapor stream, and the molar composition of water in the same stream. The temperature is first defined, using the identifier TEMP. This variable is characterized in the Blocks category, with the type Block-Var (Figure 5.45). The molar flow rate of methanol is identified as FMEOH, falling in the category Streams, with a type Mole-Flow (Figure 5.46). Finally, mole composition of water is identified as $X_{\mathrm{H}_2 \mathrm{O}}$, being in the Streams category, with a type Mole-Frac (Figure 5.47). Now, the objective function and the constraints are loaded in the Objective \& Constraints sheet. The variable FMEOH is selected as the objective function, which will be maximized. Below the space where the objective function is defined, there is an area to select the constraints of the problem. Nevertheless, that space is blank (Figure 5.48). This is because the constraints are defined in other subfolder, which is called Constraint, and is also located in the Model Analysis Tools folder. If the Constraint subfolder is opened, a window where a new constraint must be defined will be opened.

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