Chapter 2: Conversion and Reactor Sizing

Topics

  1. Conversion
  2. Design Equations
  3. Reactor Sizing
  4. Numerical Evaluation of Integrals
  5. Reactors in Series
  6. Space Time
  7. Useful Links

Conversion

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Consider the general equation

\( aA + bB \rightarrow cC + dD \)

The basis of calculation is always the limiting reactant. We will choose A as our basis of calculation and divide through by the stoichiometric coefficient to put everything on the basis of "per mole of A".

\( A + \frac{b}{a} B \rightarrow \frac{c}{a} C + \frac{d}{a} D \)

The conversion X of species A in a reaction is equal to the number of moles of A reacted per mole of A fed.

\( X = \frac{\text{moles reacted}}{\text{moles of } A \text{ fed}} \)

Batch Flow
\( X = \frac{(N_{A0} - N_{A})}{N_{A0}} \) \( X = \frac{(F_{A0} - F_{A})}{F_{A0}} \)

What is the maximum value of conversion?

    For irreversible reactions, the maximum value of conversion, X, is that for complete conversion, i.e. X=1.0.

    For reversible reactions, the maximum value of conversion, X, is the equilibrium conversion, i.e. X=Xe.


Batch: Moles A remaining, denoted as \( N_{A} \), is calculated by subtracting the moles of A reacted from the initial moles of A. The formula for this is:

\( N_{A} = N_{A0} - N_{A0} X \)

Alternatively, this can be expressed as: \( N_{A} = N_{A0} \) - (moles A initially) * (moles A reacted) / (moles A fed).


Flow: The rate of moles of A leaving, denoted as \( F_{A} \), is determined by subtracting the rate of moles of A reacted from the rate of moles of A fed. The formula is:

\( F_{A} = F_{A0} - F_{A0} X \)

This can also be represented as: \( F_{A} = F_{A0} \) - (rate of moles A fed) * (moles A reacted) / (moles A fed).


Design Equations

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The design equations presented in Chapter 1 can also be written in terms of conversion. The following design equations are for single reactions only. Design equations for multiple reactions will be discussed later.

Reactor Mole Balances in Terms of Conversion (Click on Reactor to see picture)


Reactor Differential Algebraic Integral
Batch

\( N_{A0} \frac{dX}{dt} = -r_{A} V \)

\( t = N_{A0} \int_{0}^{X} \frac{dX}{-r_{A} V} \)

Graph showing conversion (X) versus time (t) for a batch reactor, with the curve starting at the origin and increasing over time. Derive
CSTR

\( V = \frac{F_{A0} X}{-r_{A}} \)

Derive
PFR

\( F_{A0} \frac{dX}{dV} = -r_{A} \)

\( V = F_{A0} \int_{0}^{X} \frac{dX}{-r_{A}} \)

Derive
PBR

\( F_{A0} \frac{dX}{dW} = -r'_{A} \)

\( W = F_{A0} \int_{0}^{X} \frac{dX}{-r'_{A}} \)

Graph showing conversion (X) versus weight (W) for a PBR reactor, with the curve starting at the origin and increasing as weight increases. Derive

Reactor Sizing

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By sizing a chemical reactor we mean we're either detering the reactor volume to achieve a given conversion or determine the conversion that can be achieved in a given reactor type and size. Here we will assume that we will be given -rA= f(X) and FA0. In chapter 3 we show how to find -rA= f(X).

Given -rA as a function of conversion,-rA=f(X), one can size any type of reactor. We do this by constructing a Levenspiel plot. Here we plot either \(\frac{F_{A0}}{-r_{A}}\) or \(\frac{1}{-r_{A}}\) as a function of X. For \(\frac{F_{A0}}{-r_{A}}\) vs. X, the volume of a CSTR and the volume of a PFR can be represented as the shaded areas in the Levenspiel Plots shown below:

Graph titled 'Levenspiel Plot' showing a plot with the vertical axis representing F_A0 over -rA, and the horizontal axis representing conversion X. The curve starts near the origin and rises upward in a nonlinear manner, indicating a positive relationship between F_A0 over -rA0 and X.

Graph labeled 'CSTR' with a shaded rectangular area. The y-axis, labeled F_A0 over negative r_A, has a constant value across the width of the rectangle, indicating a horizontal line. The x-axis is labeled X and extends from 0 to X_1. The shaded rectangle represents the volume of the Continuous Stirred-Tank Reactor (CSTR), with the volume calculated as V = (F_A0 / -r_A) * X_1.

Graph labeled 'PFR' with a curve starting at the origin and increasing to the right, representing a Plug Flow Reactor (PFR). The y-axis is labeled F_A0 over negative r_A, and the x-axis extends to X_1. The area under the curve, shaded to indicate volume, represents the volume of the PFR. The formula for this volume is shown as V = integral from 0 to X_1 of (F_A0 / -r_A) dx.

Example Using the Ideal Gas Law to Calculate CA0 and FA0
Example Levenspiel plots in terms of concentrations

Numerical Evaluation of Integrals

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The integral to calculate the PFR volume can be evaluated using a method such as Simpson's One-Third Rule:

Graph illustrating numerical evaluation of integrals for calculating PFR volume. The y-axis is labeled 1 over negative r_A, and the x-axis ranges from 0 to X_2 with points at X_1 and 0. The curve increases non-linearly from the origin, with horizontal arrows at different y-values, labeled 1 over negative r_A(X_2), 1 over negative r_A(X_1), and 1 over negative r_A(0). Equal intervals on the x-axis are marked as ΔX, representing segments for integration. NOTE: The intervals \( \left( \Delta X \right) \) shown in the sketch are not drawn to scale. They should be equal.

\( \text{PFR: } V = \int_{0}^{X} F_{A0} \frac{dX}{-r_{A}} = \frac{\Delta X}{3} \left[ \frac{F_{A0}}{-r_{A}(X = 0)} + \frac{4 F_{A0}}{-r_{A}(X_{1})} + \frac{F_{A0}}{-r_{A}(X_{2})} \right] \)


Simpson's One-Third Rule (above) is one of the more common numerical methods. It uses three data points. Other numerical methods (see Appendix A) for evaluating integrals are:

  1. Trapezoidal Rule (uses two data points)
  2. Simpson's Three-Eighth's Rule (uses four data points)
  3. Five-Point Quadrature Formula

Reactors in Series

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Given -rA as a function of conversion, one can also design any sequence of reactors by defining the total conversion up to a point "i":


\( X_{i} = \frac{\text{moles of A reacted up to a point } i}{\text{moles of A fed to first reactor}} \)

Only valid if there are no side streams

Consider a PFR between two CSTRs
Diagram of reactors in series, illustrating a sequence of a Continuous Stirred-Tank Reactor (CSTR) followed by a Plug Flow Reactor (PFR) and another CSTR. Flow labeled F_A0 enters the first CSTR, achieving a conversion X_1 with flow F_A1, then proceeds to the PFR where it reaches conversion X_2 with flow F_A2, and finally enters the second CSTR to reach conversion X_3 with flow F_A3. Arrows indicate the flow direction between each reactor stage. Graph showing a Levenspiel plot with three shaded regions under a curve, representing the volumes of reactors in series. The y-axis is labeled F_A0 over negative r_A, and the x-axis is labeled X, extending to points X_1, X_2, and X_3. Each shaded area corresponds to a reactor volume: V_1 (blue), V_2 (red), and V_3 (green). Equations for each volume are provided on the right: V_1 = F_A0 X_1 / -r_A1, V_2 = integral from X_1 to X_2 of F_A0 over -r_A2 dX, and V_3 = F_A0 (X_3 - X_2) / -r_A3.




Example Use Levenspiel plots to calculate conversion from known reactor volumes
Critical One of the goals of the course is to have the readers further develop their critical thinking skills. One way to achieve this goal is through Socratic questioning. Throughout the course students will be asked to write questions on critical thinking drawing from information in the Preface section B2. Below are some examples of critical thinking questions (CTQ) that are either superficial or don't use R. W. Paul's Six Types of Socratic Questioning
Example Reactors in Series CSTR-PFR-CSTR
Critical What if ...?

Space Time

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The Space time, tau, is obtained by dividing the reactor volume by the volumetric flow rate entering the reactor:

\( \tau = \frac{V}{V_{0}} \)

Space time is the time necessary to process one volume of reactor fluid at the entrance conditions. This is the time it takes for the amount of fluid that takes up the entire volume of the reactor to either completely enter or completely exit the reactor.

Sample Industrial Space Times
  Reaction  
  Reactor    Temperature  Pressure atm    Space Time  
(1) C2H6 → C2H4 + H2
 PFR 860°C 2 1 s
(2) CH3CH2OH + HCH3COOH → CH3CH2COOCH3 + H2O
 CSTR 100°C 1 2 h
(3) Catalytic cracking PBR 490°C 20 1 s < τ < 400 s
(4) C6H5CH2CH3 → C6H5CH = CH2 + H2 PBR 600°C 1 0.2 s
(5) CO + H2O → CO2 + H2 PBR 300°C 26 4.5 s
(6) C6H6 + HNO3 → C6H5NO2 + H2O CSTR 50°C 1 20 min

Useful Links

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Self Test Numerical Evaluation of an Integral
Self Test Old exam questions
Self Test What's wrong with this solution?
Self Test What is more efficient?
Link Hippo Web Module
Link Access your learning style.
Link How the website can help your learning style.
Link More on learning styles.
Assess Objective Assessment of Chapter 2
 

* All chapter references are for the 1st Edition of the text Essentials of Chemical Reaction Engineering.

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