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# CSTR Design for Reversible Reactions

By Ralph Levine |

Multiple CSTRs (continuous stirred-tank reactors) are advantageous in situations where the reaction is slow; two immiscible liquids are present and require higher agitation rates; or viscous liquids are present that require high agitation rates. Unlike in plugflow reactors, agitation is easily available in CSTRs. In this article, batch and plugflow reactors are analyzed and compared to multiple CSTRs.

The number of reactors required in a CSTR system is based on the conversion for each stage. When the final stage obtains the fraction of unconverted reactant that is equal to the desired final value from the plug-flow case, the CSTR system is complete.

The volumetric efficiency of multiple CSTRs is expressed as a function of conversion per stage and gives the total conversion required. In this article, we will apply this to reversible second-order reactions.

## 2nd-order, reversible reactions

The first case presented here is a kinetic process requiring a double component (2A) to be fed to a reactor, and producing two products (G and H). The design may be calculated for both CSTR and plug-flow reactors, determining the conversion in the first stage, the number of stages of equal volume, as well as the volumetric efficiency of the CSTR stages and the plugflow reactor.

The reactor design is developed by selecting a conversion in the first stage. Then, the second-stage conversion is equal to that of the first stage, since it requires an equal volume. This procedure is continued until the fraction of reactant exiting each reactor stage reaches the desired value in the last stage, or slightly less than the plugflow case, as illustrated in Figure 1.

The kinetic rate conversion of a reversible bi-molecular reaction at constant temperature and flowrate is represented by Equation (1). The reaction is illustrated below (nomenclature is defined on p. 49.

(1)

(2)

(2a)

Assume that G and H compounds are not present in the feed. Therefore, C_{G0}and C_{H0}are equal to zero, and the following expressions are true:

(2b)

(2c)

(2d)

The ratio may be different, as, for example, the concentration of G in the product may be five times that of H. However, we assume that the products are equal (C_{Gf}= C_{Hf}).

Expressing reaction rate. The rate equation can be modified to include conversion and equilibrium constant terms. Substituting Equations (1), and (2d) into Equation (1) give an expression for rate.

(3a)

(3b)

(3c)

At equilibrium, the net reaction rate equals zero.

(4a)

Using the quadratic equation, Equation (4a) is simplified to Equation (4c).

(4b)

(4c)

The quadratic equation can also be used to simplify Equation (3c), resulting in Equation (5a).

(5)

(5a)

The reaction rate expression can then be expressed as Equations (5b) and (5c).

(5b)

(5c)

Stirred reactor in a batch or plugflow reactor. The batch reactor case and the ideal continuous plugflow case are given in Equation (6).

The reaction time is t for the batch case, and V/v for the plugflow case.

(6)

Substituting Equation (5c) into (6) and rearranging, gives Equation (6a).

(6a)

(6b)

Volume of each CSTR stage. An expression for the first stage of a CSTR is given in Equation (7). The first stage conversion, X_{1}, occurs in each of the successive stages (X_{2}, X_{3}, and so on), and each has the same volume and reaction temperature.

(7)

Substituting Equation (5c) into (7) and rearranging gives Equation (7a).

(7a)

Equation (7a) has only one independent variable (V_{1}). If each stirred reactor stage is to be of equal volume and volumetric flowrate, then the result is a constant conversion per stage. That is, each stage, when at a fixed set of conditions, has the same conversion from each stage, expressed as:

X_{1}= X_{2}= X_{3}… = X_{n}

Number of stages. Conversion for stage 1 is expressed by equation (8).

(8)

The equilibrium conversion is based on time to reach a net reaction rate of zero, which may be calculated by Equation (4b) or (9).

(9)

Subtract Equation (9) from (8) and divide by (9) to obtain Equations (10a)and (10b).

(10)

(10a)

(10b)

The exit concentration, C_{A1}, can be calculated from Equation (10b). Also, the exit concentration from the second stage, C_{A2}, can be calculated from Equation (11a), based on each stage having the same volume and conditions.

(11a)

(12)

(12a)

(12b)

Total volume of all stages. Substitute Equation (7a) into (13a).

(13a)

(13b)

(13c)

By definition, C_{An}= C_{A0}(1 – X_{f}), where X_{f}is the total conversion in the nth stage or the final desired conversion of the plug-flow reactor. Bythe method used to obtain Equation (10), the following equation is similarly derived. Substitution of Equation (12b) into (13c) gives Equation (13d).

(13d)

Volumetric efficiency

Since V_{T}/v in Equation (13c) is residence time, as is V/v in Equation (6b), for CSTRs, these terms are equivalent. The volumetric flowrate is the same in all cases (a batch operation for one complete reaction cycle). Thus, the ratio of comparison should be V for plugflow or batch operation (reaction volume and time only) compared to V_{T}for multiple CSTRs. This ratio (V/V_{T}), volumetric efficiency is expressed as Equation (14), and is derived from Equations (6b) and (13d).

(14)

(14a)

Volumetric efficiency is independent of the initial or final concentration and velocity constant at constant temperature, as well as overall conversion. It is dependent on only the ratio of the first stage conversion compared to the equilibrium conversion. Calculations for Equation (14a) are shown in Table 1.

Table 1. Volumetric Efficiency for Equation (14a) | |||||

(14a) | |||||

X1/Xe | Xe | 1–(X1/Xe) | log[1–(X1/Xe)] | (V/VT)Xe | V/VT |

0.1 | 0.7 | 0.9 | 0.046 | 0.948 | 1.355 |

0.2 | 0.7 | 0.8 | 0.097 | 0.893 | 1.275 |

0.3 | 0.7 | 0.7 | 0.155 | 0.832 | 1.189 |

0.4 | 0.7 | 0.6 | 0.222 | 0.766 | 1.095 |

0.5 | 0.7 | 0.5 | 0.301 | 0.693 | 0.990 |

0.6 | 0.7 | 0.4 | 0.398 | 0.611 | 0.873 |

0.7 | 0.7 | 0.3 | 0.523 | 0.516 | 0.737 |

0.8 | 0.7 | 0.2 | 0.699 | 0.402 | 0.575 |

0.9 | 0.7 | 0.1 | 1.000 | 0.256 | 0.366 |

TABLE 1. For any ratio of conversion per stage to

equilibrium conversion, this table provides the

corresponding volumetric efficiency, based on Equation (14a)

## Reversible production of a dimer from two reactants

Another case exists, where two components are reversibly reacted to form a single product, a dimer, rather than two products (as shown in the reaction below). This case is similar to the previous case, but but with only one product, as shown below in Equation (15).

(15)

(16a)

(16b)

(17)

At equilibrium, the rate is zero.

(18a)

Using the quadratic equation, Equation (18a) becomes (18b).

(18b)

_{f}is found.

(19a)

Equation (19a) can be simplified to Equation (19b) using the quadratic equation.

(19b)

(19c)

(20a)

(20b)

(21)

(22)

(23)

Reversible production of a dimer from twin reactants

In another alternate but similar case, Equation (15) is modified for double components that are reversibly reacted to form a dimer, as shown in the reaction below. As in this last case, there is only one product.

(24)

(25)

(26a)

(26b)

At equilibrium, the rate is zero.

(27a)

Using the quadratic equation, Equation (27a) becomes (27b).

(27b)

(28a)

(28b)

For a plugflow reactor, the following expression is true.

(29)

(30)

The volumetric efficiency is found to be Equation (32).

(32)

The last two cases presented here are reversible and have only one product. The differences between these cases are the values calculated based on the quadratic equation for both Xe and Xf. All second order reactions that are reversible and produce one or two products require the quadratic equation for the calculation of Xe and Xf for each case. A summary of these equations is presented in the box above. â–

*Edited by Kate Torzewski*

## References

1. Levenspiel, O., “Chemical Reaction Engineering,” John Wiley & Sons, Inc., 1962.

2. Levine, R., Hydro. Proc., July 1967, pp. 158–160.

3. Levine, R. A New Design Approach for Backmixed Reactors — Part I, Chem. Eng. July 1, 1968, pp. 62–67.

4. Levine, R. A New Design Approach for Backmixed Reactors — Part II, Chem. Eng. July 29, 1968, pp. 145–150.

5. Levine, R. A New Design Approach for Backmixed Reactors — Part III, Chem. Eng. Aug 12, 1968, pp. 167–171.

6. Levine, R. CSTRs: Bound for Maximum Conversion, Chem. Eng. Jan. 2009, pp. 30–34.

## Summary of Equations

[6]

## Nomenclature

*C* Concentration, moles/unit volume

*k* Reaction rate constant

*K* Equilibrium constant

*M* Initial mole ratio of D/B

*n* Number of stages

*r* Reaction rate

*t* Reaction time

*V * Reactor volume

*v* Volumetric flowrate

*X* Conversion

### Subscripts

0 Initial conditions

1,2,3 First, second and third stages

*A* For component *A C* For component

*C*

DFor component

D

*D*

eEquilibrium conditions

e

*Overall or final conditions*

f

f

*Conditions for forward reaction*

F

F

*For component*

G

G

*G*

HFor component

H

*H*

jAny stage in the series of reactor stages

j

*n*The

*n*th stage

*For component*

P

P

*P*

RConditions for reverse reaction

R

*T*The total of all

*n*stages

## Author

Ralph Levine is a retired chemical engineer currently working as a consultant for plants, design or operations and R&D (578 Arbor Meadow Dr., Ballwin, Mo. 63021; Email: ralphle2000@yahoo.com). Levine earned a B.S.Ch.E. from the City University of New York, and did graduate work at Louisiana State University and the University of Delaware. Levine later served as an engineer for the U.S. Army Chemical Corps. He has worked for DuPont, Cities Service Co., and most recently, Columbian Chemical Co. Levine has filed several U.S. patents during his career, and is a published author, with his work featured in Chemical Engineering and Hydrocarbon Processing.

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