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# 数学代写|随机过程Stochastic Porcesses代考|STAT6540 Optimization of a Tubular Reactor

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## 数学代写|随机过程Stochastic Porcesses代考|Description of the Problem

In this case of study, the esterification of lactic acid with methanol takes place in a tubular reactor. The reaction occurs as follows:
$$\mathrm{LACTAC}+\mathrm{MEOH} \leftrightarrow \mathrm{METLACT}+\mathrm{H}2 \mathrm{O}$$ where LACTAC represents the lactic acid; MEOH is the methanol; METLACT is the product of interest, methyl lactate; and $\mathrm{H}_2 \mathrm{O}$ represents water. The reaction is usually catalyzed by ion-exchange resins, but it has been reported that the reaction can be autocatalyzed using the acid for temperatures higher than approximately $360 \mathrm{~K}$ (Sanz et al., 2002). This approach is used in this example. The kinetic model reported by Sanz et al. (2002) is used to represent the reaction rate, but the effect of the reverse reaction is neglected. Thus, the reaction rate is expressed as follows: $$r=k_1 \exp \left(\frac{-E_A}{R T}\right) a{\mathrm{LACTAC}}^2 a_{\mathrm{MEOH}}$$

where $k_1$ is equal to $6.024 \times 10^8 \mathrm{~mol} \cdot \mathrm{min}^{-1}, E_{\mathrm{A}}$ is equal to $56.45 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}$, and $a_i$ is the activity of the component $i$.

In this case, $100 \mathrm{kmol} / \mathrm{h}$ of lactic acid is allowed to enter a tubular reactor. Methanol is also fed to the reactor, with a methanol/lactic acid mole ratio of 3:1. The reactor operates at $340 \mathrm{~K}$ and 1 bar. Esterification of lactic acid occurs inside the vessel; thus, the stream leaving the equipment contains methyl lactate, water, and unreacted methanol and lactic acid. The process is represented in Figure $5.54$.
In this case, the objective is to maximize the following function:
$$Z=F_{\text {METLACT }} C_{\text {METLACT }}-F_{0, \text { LACtAC }} C_{\text {LACTAC }}-\frac{C_{\text {REACT }}}{n}$$
where $F_{\text {METLACT }}$ is the mass flow rate of methyl lactate leaving the reactor (kg/year), $C_{\text {METLACT }}$ is the unitary sales cost of methyl lactate, $F_{0, \text { LACTAC }}$ is the mass flow rate of lactic acid entering the reactor ( $\mathrm{kg} /$ year), $C_{\text {LACTAC }}$ is the unitary cost of lactic acid, $C_{\text {REACT }}$ is the cost of the reactor, and $n$ is the recovery time of the investment (years). In this problem, $C_{\text {METLACT }}$ is assumed to be $12,700 \mathrm{USD} / \mathrm{kg}$, whereas $C_{\text {LACTAC }}$ is assumed to be $135 \mathrm{USD} / \mathrm{kg}$. These costs have been estimated on the basis of those reported by Sigma-Aldrich (www.sigmaaldrich.com). The cost of the reactor is approximated as follows:
$$C_{\text {REACT }}=4,000 L$$
where $L$ is the length of the reactor, in meters. Finally, $n$ has been assumed as 5 years.

## 数学代写|随机过程Stochastic Porcesses代考|Initial Simulation

In this section, the initial simulation of the reactor is described. Four components (lactic acid, methanol, methyl lactate, and water) are defined in the Components-Specifications sheet (Figure 5.55). The UNIFAC model is selected in the Methods-Specifications sheet (Figure 5.56) to represent the vapor-liquid equilibrium. This information is sufficient to run the properties test. Because the test is positive, we now move toward the simulation section, where the flowsheet is defined. An RPlug module is selected from the Reactors folder, representing the tubular reactor, and inserted in the flowsheet. The feed and product streams are also inserted (Figure 5.57). The conditions of the feed stream are now defined. Feed temperature and pressure are set equal to those of the reaction. As aforementioned, feed stream has $100 \mathrm{kmol} / \mathrm{h}$ of lactic acid and $300 \mathrm{kmol} / \mathrm{h}$ of methanol. The Input sheet for the feed stream can be viewed in Figure 5.58. As the next step, the characteristics of the reactor are loaded, in the module Setup sheet. First, the type of reactor is set as Reactor with specified temperature. It is assumed that the temperature of the reactor remains constant, equal to the feed temperature. This assumption is indicated as Operating condition (Figure 5.59). In the Configuration sheet, the length and diameter of the reactor are indicated. As a first approach, 5 and $0.1 \mathrm{~m}$ are selected for length and diameter, respectively. The reaction is assumed to occur in the liquid phase. Hence, the phase is indicated as the only valid phase in the reactor (Figure 5.60).

## 数学代写|随机过程Stochastic Porcesses代考|Description of the Problem

$$\mathrm{LACTAC}+\mathrm{MEOH} \leftrightarrow \text { METLACT }+\mathrm{H} 2 \mathrm{O}$$

$$Z=F_{\text {METLACT }} C_{\text {METLACT }}-F_{0, \text { LACtAC }} C_{\text {LACTAC }}-\frac{C_{\text {REACT }}}{n}$$
(年) 。在这个问题中， $C_{\text {METLACT }}$ 被假定为 $12,700 \mathrm{USD} / \mathrm{kg}$ ，然而 $C_{\text {LACTAC }}$ 被假定为 $135 \mathrm{USD} / \mathrm{kg}$. 这些成本是根据 Sigma-Aldrich (www.sigmaaldrich.com) 报告的成本估算的。反应器的成本近似如下:
$$C_{\text {REACT }}=4,000 L$$

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