Chapter 5: Isothermal Reactor Design: Conversion

Topics

1. Algorithm for Isothermal Reactor Design
2. Applications/Examples of CRE Algorithm
3. Reversible Reactions
4. ODE (Polymath) Solutions to CRE Problems
5. General Guidelines for California Problems
6. PBR with Pressure Drop
7. Engineering Analysis

Algorithm for Isothermal Reactor Design	top

French Menu Analogy                                          

Example: The elementary liquid phase reaction

is carried out isothermally in a CSTR. Pure A enters at a volumetric flow rate of 25 dm³/s and at a concentration of 0.2 mol/dm³.

What CSTR volume is necessary to achieve a 90% conversion when k = 10 dm³/(mol*s)?

Mole Balance
Rate Law
Stoichiometry	liquid phase (v = vo)
Combine
Evaluate	at X = 0.9,
	V = 1125 dm3
	Space Time

Here are some links to example problems. You could also use these problems as self tests.

Videos

The following humorous videos were made by Professor Lane's 2008 Chemical Reaction Engineering class at the University of Alabama, Tuscaloosa.

Applications/Examples of the CRE Algorithm	top

Gas Phase Elementary Reaction	Additional Information
	only A fed	P0 = 8.2 atm
	T0 = 500 K	CA0 = 0.2 mol/dm3
	k = 0.5 dm3/mol-s	vo = 2.5 dm3/s

Solve for X = 0.9

Applying the algorithm to the above reaction occurring in a Batch, CSTR, and PFR.

	Batch	CSTR	PFR
Mole Balance:
Rate Law:
Stoichiometry:	Gas: V = V0 (e.g., constant volume steel container)	Gas: T =T0, P =P0	Gas: T = T0, P = P0
		Per Mole of A:	Per Mole of A:
Combine:
Integrate
Evaluate
For X = 0.9:		V = 680.6 dm3	V = 90.7 dm3

Reversible Reactions	top

To determine the conversion or reactor volume for reversible reactions, one must first calculate the maximum conversion that can be achieved at the isothermal reaction temperature, which is the equilibrium conversion. (See Example 3-8 in the text for additional coverage of equilibrium conversion in isothermal reactor design.)

Equilibrium Conversion, X_e

From Appendix C:

Example: Determine X_e for a PFR with no pressure drop, P = P₀

Given that the system is gas phase and isothermal, determine the reactor volume when X = 0.8 X_e.

Reaction	Additional Information
	CA0 = 0.2 mol/dm3 KC = 100 dm3/mol	k = 2 dm3/mol-min FA0 = 5 mol/min

First calculate X_e:

X_e = 0.89

X = 0.8X_e = 0.711

One could then use Polymath to determine the volume of the PFR. The corresponding Polymath program is shown below.

ODE (Polymath) Solutions to CRE Problems	top

Algorithm Steps	Polymath Equations
Mole Balance	d(X)/d(V) = -rA/FA0
Rate Law	rA = -k*((CA**2)-(CB/KC))
Stoichiometry	CA = (CA0*(1-X))/(1+eps*X)
	CB = (CA0*X)/(2*(1+eps*X))
Parameter Evaluation	eps = -0.5	CA0 = 0.2	k = 2
	FA0 = 5	KC = 100
Initial and Final Values	X0 = 0	V0 = 0	Vf = 500

Polymath Screen Shots

Equations

Plot of X vs. V

Results in Tabular Form

A volume of 94 dm³ (rounding up from slightly more than 93 dm³) appears to be our answer.

General Guidelines for California Problems	top

Every state has an examination engineers must pass to become a registered professional engineer.  In the past there have typically been six problems in a three hour segment of the California Professional Engineers Exam. Consequently one should be able to work each problem in 30 minutes or less. Many of these problems involve an intermediate calculation to determine the final answer.

Some Hints:

group unknown parameters/values on the same side of the equation   example: [unknowns] = [knowns] look for a Case 1 and a Case 2 (usually two data points) to make intermediate calculations take ratios of Case 1 and Case 2 to cancel as many unknowns as possible carry all symbols to the end of the manipulation before evaluating, UNLESS THEY ARE ZERO

PBR with Pressure Drop	top

Note: Pressure drop does NOT affect liquid phase reactions

Sample Question:

Analyze the following second order gas phase reaction that occurs isothermally in a PBR: 

                                                               
Mole Balance

    

Must use the differential form of the mole balance to separate variables

Rate Law

Second order in A and irreversible:

Stoichiometry
  

Isothermal, T = T₀


Combine
                    

Need to find (P/P₀) as a function of W (or V if you have a PFR).

Pressure Drop in Packed Bed Reactors

Ergun Equation
Variable Gas Density
let
Catalyst Weight
where
let
then

English:
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Svenska:

	We will use this form for multiple reactions:
$$\frac{dp}{dW} = -\frac{\alpha}{2p}\frac{T}{T_o}\frac{F_T}{F_{To}}$$
$$ p=\frac{P}{P_o}$$
	We will use this form for single reactions:
	$\frac{dp}{dW} = -\frac{\alpha}{2p}\frac{T}{T_o}(1+\epsilon X)$
Isothermal Operation	$\frac{dp}{dW} = -\frac{\alpha}{2p} (1+\epsilon X) $
recall that	$\frac{dX}{dW} = \frac{k{C_{Ao}}^2(1-X)^2}{F_{Ao}(1+\epsilon X)^2} p^2$
notice that	$\frac{dX}{dW} = f(X,P) and \frac{dP}{dW} = f(X,P) or \frac{dp}{dW} = f(p,X)$
	The two expressions are coupled ordinary differential equations. We can solve them simultaneously using an ODE solver such as Polymath. For the special case of isothermal operation and epsilon = 0, we can obtain an analytical solution. Polymath will combine the mole balance, rate law and stoichiometry.

Analytical Solution, [e], PFR with

	$\frac{dp}{dW} = -\frac{\alpha}{2p} (1+\epsilon X)$
		CAUTION: Never use this form if

	$C_A = C_{A0}(1-X)p = C_{A0}(1-X)(1-\alpha W)^\frac{1}{2}$
Combine
Solve

Could now solve for X given W, or for W given X.

For gas phase reactions, as the pressure drop increases, the concentration decreases, resulting in a decreased rate of reaction, hence a lower conversion when compared to a reactor without a pressure drop. 

 

Here are some links to example problems dealing with packed bed reactors. You could also use these problems as self tests.

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Analysis using Simulation Tools

Consider the following gas phase reaction carried out isothermally in a packed bed reactor containing 100 kg of catalyst. Pure A is fed at a rate of 2.5 mol/s and with $ {C_{A0} = 0.2 \frac{mol}{dm^3}} $, and $ {\alpha = 0.0162kg^{-1}} $.

2AB

Mole Balance

$ {\frac{dX}{dW} = -r_A^{'} / F_{A0}} $

Rate Law

Elementary: $ {r_A^{'} = -kC_A^2} $

Stoichiometry

Gas with ${T = T_0}$

${A \rightarrow \frac{1}{2}B \\ C_A = \frac{F_A}{v} = \frac{C_{A0}(1-X)p}{1+\epsilon X} \\ \frac{dp}{dW} = -\alpha \frac{1+\epsilon X}{2p} \\ \epsilon = -1/2 \\ F_{A0} = 2.5 \\ C_{A0} = 0.2 \\ k = 5 \\ \alpha = 0.0162 \\ W_{final} = 100 \\ X(W = 0) = 0, p(W = 0) = 1 }$

The above equations can be easily combined with the help of simulation tools such as Polymath, Wolfram or Python.

Below is the result from Polymath

Here are links to Polymath, Wolfram, and Python code for the above example

Polymath Code Wolfram Code Python Code

Optimum Paritcle Diameter

Laminar Flow, Fix P₀, ρ₀,

ρ₀ = P₀(MW)/RT₀

ρ₀P₀∼P₀²

Increasing the particle diameter descreases the pressure drop and increases the rate and conversion.

However, there is a competing effect. The specific reaction rate decreases as the particle size increases, therefore so deos the conversion.

k ∼ 1/D_p

DP1 > DP2 k1 > k2 Higher k, higher conversion

The larger the particle, the more time it takes the reactant to get in and out of the catalyst particle. For a given catalyst weight, there is a greater external surgace area for smaller particles than larger particles. Therefore, there are more entry ways into the catalyst particle.

Later on in the course, we will learn that effectiveness factor decreases as the particle size increases  

Engineering Analysis - Critical Thinking and Creative Thinking	top

We want to learn how the various parameters (particle diameter, porosity, etc.) affect the pressure drop and hence conversion. We need to know how to respond to "What if" questions, such as:

"If we double the particle size, decrease the porosity by a factor of 3, and double the pipe size, what will happen to D P and X?"

(See Critical Thinking in Preface Table P1 and P2 on pages xvi and xvii.  e.g., Questions the probe consenquences)

To answer these questions we need to see how a varies with these parameters. 

Turbulent Flow                                    

 

Compare Case 1 and Case 2:

For example, Case 1 might be our current situation and Case 2 might be the parameters we want to change to.

For constant mass flow through the system= constant

 

Laminar Flow

Polymath Book Problem

The following is an example problem from the book. It is located on page 192 in Chapter 5. This is a problem done in polymath and the .pol file has been included for reference. The report and accompanying graphs generated in Polymath are also shown.

Here are more links to example problems dealing with packed bed reactors. Again, you could also use these problems as self tests.

 

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

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