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Methanol is produced by reacting carbon monoxide and hydrogen at \(644 \mathrm{K}\) over a \(\mathrm{ZnO}-\mathrm{Cr}_{2} \mathrm{O}_{3}\) catalyst. A mixture of \(\mathrm{CO}\) and \(\mathrm{H}_{2}\) in a ratio \(2 \mathrm{mol} \mathrm{H}_{2} / \mathrm{mol}\) CO is compressed and fed to the catalyst bed at \(644 \mathrm{K}\) and 34.5 MPa absolute. A single-pass conversion of 25\% is obtained. The space velocity, or ratio of the volumetric flow rate of the feed gas to the volume of the catalyst bed, is The product gases are passed through a condenser, in which the methanol is liquefied. (a) You are designing a reactor to produce \(54.5 \mathrm{kmol} \mathrm{CH}_{3} \mathrm{OH} / \mathrm{h}\). Estimate (i) the volumetric flow rate that the compressor must be capable of delivering if no gases are recycled, and (ii) the required volume of the catalyst bed. (Use Kay's rule for pressure-volume calculations.) (b) If (as is done in practice) the gases from the condenser are recycled to the reactor, the compressor is then required to deliver only the fresh feed. What volumetric flow rate must it deliver assuming that the methanol produced is completely recovered in the condenser? (In practice it is not; moreover, a purge stream must be taken off to prevent the buildup of impurities in the system.)

Step by step solution

01

(a) (i) Calculate the required feed of CO and H2

Using the stoichiometry of the reaction, for 54.5 kmol/h of CH3OH, the required feed of CO is also 54.5 kmol/h. As the ratio of H2:CO given is 2:1, the required feed of H2 is 2 * 54.5 = 109 kmol/h. The total molar flow rate of gases will be 54.5 (CO) + 109 (H2) = 163.5 kmol/h.

02

(a) (ii) Apply Kay's rule to find the volumetric flow rate

Applying Kay's rule, assuming unit volume at 0°C and 1 atm (1.013 bar), the total volume of feed gases at that condition will be 163.5 * 22.414 (molar volume at NTP) = 3664.3 m^3/h. Using the Ideal Gas Law, the volumetric flow rate for the feed gases can be found by correcting for actual temperature and pressure. At 644K (371°C) and 34.5 MPa (345 bar), this becomes 3664.3 * (273.15 + 371) / 273.15 * 1.013 / 345 = 44.95 m^3/h.

03

(a) (iii) Calculate the required volume of the catalyst bed

The space velocity (SV) given as ratio of volumetric flow rate to the volume of the catalyst bed, is defined as SV = V_feed/V_catalyst. The catalyst bed volume then can be expressed as V_catalyst = V_feed/SV. Given that SV is 2 h-1 and V_feed is 44.95 m^3/h, the volume of the catalyst bed required is V_catalyst = 44.95 / 2 = 22.475 m^3.

04

(b) Determine the fresh feed volumetric flow rate.

The fresh feed which is required to produce the methanol and is not recycled, includes the amount of gas which actually reacted in the reactor, which is 25% of the initial feed entering the reactor. Therefore, the fresh flow rate will be 0.25 * 163.5 (total molar flow rate as calculated earlier) = 40.875 kmol/h. Following the same procedure as before, the volumetric flow rate is 40.875 * 22.414 * (273.15 + 371) / 273.15 * 1.013 / 345 = 11.18 m^3/h.

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Key Concepts

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

Methanol Synthesis

Methanol (CH3OH) is a chemical with a wide array of uses, from fuel to feedstock for various chemical syntheses. To understand how methanol is made, one should look at the basic chemical reaction involving carbon monoxide (CO) and hydrogen (H2), generally over a catalyst like zinc oxide-chromium oxide (ZnO-Cr2O3). The reaction proceeds at elevated temperatures and pressures, yielding methanol through the following balanced equation:
CO + 2 H2 → CH3OH
For students grappling with the synthesis of methanol, remember it’s crucial to consider the molar ratios of reactants and the conditions required to favor the reaction, such as the 644 K temperature and 34.5 MPa pressure outlined in your textbook exercise.

Stoichiometry

Stoichiometry is the math behind chemistry. It's all about proportions and balancing act. When assessing the methanol synthesis, you may refer to the stoichiometric coefficients in the balanced reaction to understand the molar relationships between reactants and products. For the production of methanol, a 1:2 molar ratio of CO to H2 is needed, reflecting how many moles of each gas are necessary to form a mole of methanol. In practice, like in the textbook problem, we calculate the required feeds of CO and H2 based on the desired output of methanol—1 mole of CH3OH requires 1 mole of CO and 2 moles of H2. Thus, it's the stoichiometry that guides us in determining how much of each reactant we need to feed into the reactor to get our desired methanol output.

Catalyst Bed Design

In chemical engineering, the catalyst bed is where the magic happens. This is where reactant molecules meet and transform, with the help of a catalyst—our middleman. For a successful methanol synthesis process, the design of this catalyst bed is paramount. It needs to accommodate the correct volume of reactants and provide enough contact time for the reaction to occur. The space velocity, mentioned in the textbook problem, is a measure of how fast the gas flows through the bed relative to the catalyst's volume. It affects conversion rate, pressure drop, and ultimately the reactor's efficiency. Engineers must design the catalyst bed to ensure the right balance between size and flow rates to optimize production while minimizing costs and maximizing safety.

Kay's Rule

When dealing with gases under various conditions of temperature and pressure, one useful shortcut is Kay's rule. It's a thumb rule employed to quickly estimate the volume of a gas mixture at high pressures assuming ideal behavior. It asserts that the molar volume of a gas mixture can be approximated by the sum of the molar contributions of each individual gas, each adjusted for its mole fraction. This rule is applied, as demonstrated in the textbook solution, to estimate the volumetric flow rate of gases fed to the catalyst bed for methanol synthesis. Given the high-pressure conditions used in the methanol process, Kay's rule simplifies the step from knowing the molar flow rates to determining the actual gas volumes needed.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of chemical engineering and broader physical science, encapsulating the relationship between a gas's pressure (P), volume (V), temperature (T), and the amount of substance (n). The law is expressed in the equation PV=nRT, where R is the universal gas constant. This equation allows engineers to calculate any one of the variables if the others are known and is crucial for designing processes like methanol synthesis. In our textbook exercise, the Ideal Gas Law is used alongside Kay's rule to correct the volumetric flow rate for actual operating conditions of the process—after all, real-life systems rarely operate at standard conditions of temperature and pressure.