91Ó°ÊÓ

A methanol-synthesis reactor is fed with a gas stream at \(220^{\circ} \mathrm{C}\) consisting of 5.0 mole\% methane, \(25.0 \%\) CO, \(5.0 \% \mathrm{CO}_{2},\) and the remainder hydrogen. The reactor and feed stream are at \(7.5 \mathrm{MPa}\). The primary reaction occurring in the reactor and its associated equilibrium constant are $$\begin{array}{l}\mathrm{CO}+2 \mathrm{H}_{2} \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH} \\\K=\frac{y_{\mathrm{CH}, \mathrm{OH}} y_{\mathrm{H}_{2}}}{y_{\mathrm{CO}} y_{H_{2}}^{2} P^{2}}=\exp \left(\begin{array}{c}21.225+\frac{9143.6}{T}-7.492 \ln T \\ +4.076 \times 10^{-3} T-7.161 \times 10^{-8} T^{2}\end{array}\right)\end{array}$$ where \(T\) is in kelvins. The product stream may be assumed to reach equilibrium at \(250^{\circ} \mathrm{C}\). (a) Determine the composition (mole fractions) of the product stream and the percentage conversions of CO and \(\mathrm{H}_{2}\). (b) Neglecting the effect of pressure on enthalpies, estimate the amount of heat (kJ/mol feed gas) that must be added to or removed from (state which) the reactor. (c) Calculate the extent of reaction and heat removal rate (kJ/mol feed) for reactor temperatures between \(200^{\circ} \mathrm{C}\) and \(400^{\circ} \mathrm{C}\) in \(50^{\circ} \mathrm{C}\) increments. Use these results to obtain an estimate of the adiabatic reaction temperature. (d) Determine the effect of pressure on the reaction by evaluating extent of conversion and rate of heat transfer at \(1 \mathrm{MPa}\) and \(15 \mathrm{MPa}\). (e) Considering the results of your calculations in Parts (c) and (d), propose an explanation for selection of the initial reaction conditions of \(250^{\circ} \mathrm{C}\) and \(7.5 \mathrm{MPa}\).

Step by step solution

01

Determine Product Stream Composition

First, it's necessary to determine the equilibrium constant with the given expression considering that T is in Kelvins (so we need to convert the 250°C to Kelvin by adding 273). Once we have the equilibrium constant, we can use the balanced chemical reaction and the mole percent given for each component of the mixture to set up a system of equations. This will allow us to solve for the mole fractions of the products using equilibrium expressions for the reaction.

02

Calculate CO and H2 Conversion Percentage

Having determined the mole fractions, we can use the initial and final quantities of CO and H2 to compute the percentage conversions of these molecules. The formula for conversion is ((initial quantity - final quantity) / initial quantity) * 100.

03

Estimate the Heat Amount

Neglecting the effect of pressure on enthalpies, we can now estimate the amount of heat (kJ/mol feed gas) that must be added to or removed from the reactor. We can use the heat of reaction (\( \Delta H \)) for the reaction and the stoichiometry to calculate this. If heat is received (\( \Delta H > 0 \)), then heat is added. If heat is released (\( \Delta H < 0 \)), then heat is removed from the reactor.

04

Compute Reaction Extent and Heat Removal Rate

At this stage, calculate the extent of reaction and heat removal rate for reactor temperatures between 200°C and 400°C with 50°C increments. This could be realized by repeating step 1 and 3 for each of these temperatures. This will assist in obtaining an estimate of the adiabatic reaction temperature, which is the temperature at which the heat removal rate equals zero.

05

Effect of Pressure on the Reaction

Now we must evaluate the effect of pressure on the reaction by recalculating the extent of conversion and rate of heat transfer at 1 MPa and 15 MPa, using the equilibrium constant expression. By comparing these results with those obtained at 7.5MPa, we'll be able to assess the effect of pressure on the given reaction.

06

Explain Initial Reaction Conditions

Given the calculations from part (c) and (d), the last step is to propose an explanation for why the initial reaction conditions of 250°C and 7.5MPa were chosen. This should involve analyzing the trade-offs involved in reaction rate, heat, conversion, and pressure.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilibrium Constant

In chemical reaction engineering, understanding the equilibrium constant is crucial for predicting how a reaction will behave under certain conditions. The equilibrium constant, denoted as \( K \), provides vital information about the concentrations of the reactants and products at equilibrium.
For this specific methanol-synthesis reaction, the equilibrium constant is calculated using a formula that involves temperature. It is given by:\[K = \frac{y_{\text{CH}_3\text{OH}} y_{\text{H}_2}}{y_{\text{CO}} y_{\text{H}_2}^2 P^2} = \exp \left( 21.225 + \frac{9143.6}{T} - 7.492 \ln T + 4.076 \times 10^{-3} T - 7.161 \times 10^{-8} T^2 \right)\]Here, \( T \) is the temperature in Kelvin, and \( P \) is the pressure. This equation shows that \( K \) depends on temperature and pressure, emphasizing their importance in determining equilibrium compositions.
Equilibrium is reached when the rate of the forward reaction equals the rate of the reverse reaction, and no further net change in concentration happens. By calculating \( K \), one can set up a system of equations using the balanced chemical reaction to solve for the mole fractions of each component in equilibrium, thus predicting the final product stream composition.

Conversion Percentage

Conversion percentage is a measure of how much of the reactant gets transformed into the desired product in a chemical reaction. It is calculated by comparing the initial and remaining concentrations of the reactant.
In our methanol synthesis problem, the conversion of carbon monoxide (CO) and hydrogen (H2) are of particular interest. The conversion is determined using the formula:

• \( \text{Conversion} = \left( \frac{\text{initial quantity} - \text{final quantity}}{\text{initial quantity}} \right) \times 100 \)

This tells us what percentage of the initial reactants have been converted into methanol.
High conversion percentages are generally desirable because they indicate that more reactant has been turned into product, making the process more efficient. However, achieving high conversions may depend on reaction conditions like temperature, pressure, and the use of catalysts. Ensuring the optimum conditions can minimize unreacted feedstock and maximize product yield, which is especially critical for large-scale industrial chemical processes.

Heat of Reaction

The heat of reaction, often symbolized as \( \Delta H \), is the heat energy exchanged during a chemical reaction at constant pressure. It is a fundamental concept in thermodynamics and critically important for chemical engineering.
For the methanol synthesis reaction, calculating the heat of reaction helps in designing the reactor and evaluating the energy necessary for sustaining the process.
- **Exothermic Reaction**: If \( \Delta H < 0 \), heat is released, and the process is exothermic.- **Endothermic Reaction**: If \( \Delta H > 0 \), heat is absorbed, making the process endothermic.In our exercise, we're tasked with estimating the heat needed to be added or removed from the reactor. By calculating \( \Delta H \) in kJ/mol of feed, we can infer whether energy needs to be supplied or dissipated. This allows for setting optimal energy management strategies, which is essential to maintain the desired reaction temperature and ensure process safety and efficiency.
Additionally, knowing the heat of reaction across a range of temperatures (200°C to 400°C) aids in determining the adiabatic reaction temperature, where no heat is lost to or gained from the surroundings, offering insights into the best operational temperature range.