Question

Compare Equations (6.22) and (6.29). Assuming a first-order reaction and the same operation conditions, which reactor needs less volume to obtain the same conversion, the CSTR or the PFR? Why?

Short answer

Based on our analysis, we can conclude that for a first-order reaction and under the same operation conditions, the Plug Flow Reactor (PFR) requires less volume to achieve the same conversion as the Continuous Stirred Tank Reactor (CSTR). This is attributed to the fact that the PFR has a concentration gradient along its length, thus utilizing the reactor volume more efficiently, whereas the CSTR operates at an average concentration throughout the reactor.

Step-by-step solution

Write down the given equations for CSTR and PFR

Equation (6.22) represents the design equation for a CSTR, and equation (6.29) represents the design equation for a PFR. Assuming the same operation conditions, these equations are as follows: CSTR (Equation 6.22): \[V_{CSTR} = \frac{-F_A0}{k}(X_f - X)\] PFR (Equation 6.29): \[V_{PFR} = \frac{-F_A0}{k}\ln\frac{1-X}{1-X_f}\] Where, \(V_{CSTR}\) & \(V_{PFR}\) are the volumes of the CSTR and PFR respectively \(F_A0\) is the molar flow rate of reactant A \(X_f\) is the feed conversion \(X\) is the desired conversion \(k\) is the reaction rate constant

Assume a first-order reaction and the same conversion in both reactors

Since we are assuming a first-order reaction and the same conversion in both reactors, we can rewrite both equations as: CSTR: \[V_{CSTR} = \frac{-F_A0}{k}(X)\] PFR: \[V_{PFR} = \frac{-F_A0}{k}\ln\frac{1-X}{1}\] Where, \(X\) is the desired conversion

Compare the volumes of CSTR and PFR for the same conversion

To compare the volumes, we will assess the relationship between the volumes of both reactors and determine which one is lower. Divide the volume of the CSTR by the volume of the PFR: \[\frac{V_{CSTR}}{V_{PFR}} = \frac{\frac{-F_A0}{k}(X)}{\frac{-F_A0}{k}\ln\frac{1-X}{1}} = \frac{X}{\ln(1-X)}\] Since \(\ln(1-X) < 0\) for \(0 0\), it follows that \(\frac{X}{\ln(1-X)}>0\). This implies that the CSTR will need more volume than the PFR to achieve the same conversion for a first-order reaction under the same operation conditions, i.e., \(V_{CSTR} > V_{PFR}\).

Conclusion

For a first-order reaction and under the same operation conditions, the Plug Flow Reactor (PFR) needs less volume to obtain the same conversion as compared to the Continuous Stirred Tank Reactor (CSTR). This is because the CSTR operates at an average concentration throughout the reactor, whereas the PFR has a concentration gradient along its length, allowing for more efficient utilization of the reactor volume.

Key concepts

CSTR

The Continuous Stirred Tank Reactor (CSTR) is a widely used type of chemical reactor characterized by a constant reactant concentration throughout its volume. It operates under steady-state conditions, and the reactants are continuously fed into the reactor while the products are simultaneously removed.

In a CSTR, mixing is complete and rapid, resulting in uniform reaction conditions. This homogeneous environment makes the CSTR an ideal model for reactions that require consistent temperatures and reactant concentrations. However, when it comes to volume efficiency for certain reactions, such as first-order reactions, CSTRs may not be as volume efficient as other reactor types, such as Plug Flow Reactors (PFRs).

PFR

A Plug Flow Reactor (PFR), in contrast to the CSTR, features a tubular design where reactants flow in one end and progress through the reactor in a 'plug' without back-mixing. The concentration of reactants thus changes along the length of the reactor.

Different from the CSTR, the PFR allows each 'slice' of the reactant plug to react at a rate corresponding to its point-specific concentration, usually resulting in higher conversion rates. This spatial variation enables the PFR to operate more volume efficiently for reactions such as first-order reactions. This is because the reaction can fully utilize the concentration gradient, leading to an overall reduced reactor volume for the same level of conversion.

First-Order Reaction

A first-order reaction is a type of chemical reaction where the reaction rate is directly proportional to the concentration of one reactant. This means that, in mathematical terms, the rate of the reaction, \(rate = k[A]\), where \(k\) is the rate constant and \[A\] is the concentration of the reactant.

First-order reactions are important in reactor design because they simplify the design equations and highlight the differences in performance between CSTRs and PFRs. As the reaction progresses, the concentration of the reactant decreases, diminishing the rate of reaction in a manner that is particularly well-accommodated by the concentration gradient in a PFR.

Reactor Volume Calculation

Calculating the volume of a chemical reactor such as a CSTR or PFR is pivotal for estimating the size and cost of the reactor needed for a particular chemical process. The volume calculation involves understanding the kinetics of the reaction and how the reactor type influences the overall conversion.

Using the design equations for each reactor type, we can determine the necessary volume for achieving a desired conversion level. For first-order reactions, the design equations generally show that less volume is required for a PFR than for a CSTR due to the more effective utilization of reactant concentration gradients in a PFR. This efficiency is quantified by comparing the reactor volumes derived from their respective equations and assessing which reactor gets to the same conversion with less volume.