380

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Chemical Reaction Engineering Chap. 8

8.23 OPERATION OF A COOLED EXOTHERMIC CSTR

8.23.1 Concepts Demonstrated Material and energy balances on a CSTR with an exothermic reaction and coolingjacket.

8.23.2 Numerical Methods Utilized

Solution of a system of simultaneous nonlinear algebraic equations, and conversion of the system of equations into one equation to examine multiple steadystate solutions.

8.23.3 Problem Statement"

An irreversible exothermic reaction A ~ B is carried out in a perfectly mixed CSTR, as shown in Figure 8-14. The reaction is first order in reactant A and has

Fo CAO To

Figure 8-14 Cooled Exothermic CSTR

a heat of reaction given by A, which is based on reactantA. Negligible heat losses and constant densities can be assumed. A well-mixed cooling jacket surrounds the reactor to remove the heat of reaction. Cooling water is added to the jacket at a rate of Fj and at an inlet temperature of Tjo . The volume V of the contents of the reactor and the volume "J of water in the jacket are both constant. The reaction rate constant changes as function of the temperature according to the equation

k = ex exp(-EIRT) (8-137) The feed ft.ow rate F0 and the cooling water ft.ow rate Fj are constant. The jacket water is assumed to be completely mixed. Heat transferred from the reac

*This problem is adapted from Luyben8 with permission.

8.23 OPERATION OF A COOLED EXOTHERMIC CSTR 381

tor to the jacket can be calculated from

Q = UA(T-T) (8-138)

where Q is the heat transfer rate, U is the overall heat transfer coefficient, and A is the heat transfer area. Detailed data for the process from Luyben8 are shown in the Table 8-15.

Table 8-15 CSTR Parameter Valuesa

FO 40 U 150 btulh. oR F 40 ft3/h A 250 ft2 CAO 0.55 lb-mollft3 Tjo 530 oR V 48 ft3 To 530 oR

FJ 49.9 ft3/h A -30,000 btullb-mol

Cp 0.75 btuJlbm oR CJ l.0 btullbm oR a 7.08xl01O h-1 E 30,000 btullb-mol

P 50 lbn/ft3 Pj 62.3 lbm/ft3 R l.9872 btullb-mol. oR "J 12 ft3

aData are from Luyben8 with permission.

8.23.4 Solution (Partial)

(a) There are three balance equations that can be written for the reactor and the cooling jacket. These include the material balance on the reactor, the energy balance on the reactor, and the energy balance on the cooling jacket.

380 Chemical Reaction Engineering Chap. 8

8.23 OPERATION OF A COOLED EXOTHERMIC CSTR

8.23.1 Concepts Demonstrated

Material and energy balances on a CSTR with an exothermic reaction and coolingjacket.

8.23.2 Numerical Methods Utilized

Solution of a system of simultaneous nonlinear algebraic equations, and conversion of the system of equations into one equation to examine multiple steadystate solutions.

8.23.3 Problem Statement'

An irreversible exothermic reaction A ~ B is carried out in a perfectly mixed CSTR, as shown in Figure 8-14. The reaction is first order in reactantA and has

Fo ----,

,

CAO

To

Figure 8-14 Cooled Exothermic CSTR

a heat of reaction given by A, which is based on reactantA. Negligible heat losses and constant densities can be assumed. A well-mixed cooling jacket surrounds the reactor to remove the heat of reaction. Cooling water is added to the jacket at a .rate of Fj and at an inlet temperature of Tjo. The volume V of the contents of the reactor and the volume Vj ofwater in the jacket are both constant. The reaction rate constant changes as function of the temperature according to the equation

k = IX exp(-EIRT) (8-137) The feed flow rate F0 and the cooling water flow rate Fj are constant. The jacket water is assumed to be completely mixed. Heat transferred from the reac

*This problem is adapted from Luyben8 with permission.

---

z

382 le," V",-) Chemical Reaction Engineering Chap. 8 1> (J if. 1'. "', Mole balance on CSTR for reactant A

d.C F< ~ o (8-139)~klJ Energy balance on the reactor

') pCpCFoTo FT) - 'A,VkCA UA(T T j )::: 0 (8-140)

Energy balance on the coolingjacket

P C -F (T '0)))) T ,) + UA(T - T ,)) . ) = 0 (8-141)

(b) Introducing the numerical parameter values into Equations (8-139), (8140), C8-141), and (8-137) and entering the system of equations into the POLYl\1ATH Nonlinear Algebraic Equation Solver gives

Equations: f(CA)=40.*(0.55-L~)-48.*k*CA f('f)=50.*O.75*40.*(530.-'f)+30000.*48.*k*CA-150.*250.*(T-Tj)

f(Tj)=62.3*1.O*49.9*(530.-Tj)+150.*250.*(T-Tj)

k=7.08elO*exp(-30000./(1.9872*T))

Initial Conditions:

CA(0)=0.55

'f(0)=530

Tj(0)=530

A reasonable initial assumption is that there is no reaction; therefore, C Ao =

0.55, To ::: 530, and Tjo ::: 530. The solution obtained with these initial estimates is summarized in Table 8-16.

Table 8-16 Steady-State Operating Conditions for CSTR

Solution

Variable Value fO CA 0.52139 -4.441e-16 T 537.855 1.757e-9 T,J 537.253 -1.841e-9

k 0.0457263

The POLYl\1ATH data file for part (b) is found in the Simultaneous Algebraic Equation Solver Library located in directory CHAP8 with file named P8-23B.POL.

383 8.23 OPERATION OF A COOLED EXOTHERMIC CSTR

(c) Several different steady states may be possible with exothermic reactions in a CSTR. One possible method to determine these different steady states is to solve the system of nonlinear equations with different initial estimates of the final solution. While this approach is not very sophisticated, it can be of benefit in very complex systems of equations.

Another approach is to convert the system of equations into a single implicit and several explicit or auxiliary equations. (Incidentally, this is a good way to show that a particular system does not have a solution at all.) In this particular case, the material balance of Equation (8-139) can be solved for Ck

(8-142)

Also, the energy balance of Equation (8-141) on the cooling jacket can be solved forTj'

p C P T '0 + UATT= )))) (8-143)) (p.CP.+UA)) ) ) Thus the problem has been converted to a single nonlinear equation given

by Equation (8-140) and three explicit equations given by Equations (8-137), (8142), and (8-143). The equation sex for POLYMATH solution is

Equations: f{T)=50.*0.75*40.*{530.-T)+30000.*48.*k*CA-150.*250.*(T-Tj) k=7 . 08elO*exp {-30000./ (1. 9872 *T) } Tj=(62.3*1.0*49.9*530.+150.*250.*T}/(62.3*1.0*49.9+150.*250.) CA=40.*0.55/{40.+48.*k) Initial Conditions:

T(min)=500, T(max)=700

When f(T) is plotted versus T in the range 500 s T s 700, three solutions can be clearly identified, as shown in Figure 8-15. The first one is at low temperature as this is the solution that was initially identified. The second is at an intermediate temperature and the third is at a high temperature. The three resulting solutions are summarized in Table 8-17.

Table 11-17 Multiple Steady-State Solutions for CSTR

Solution No.

1 2 3

T 537.86 590.35 671.28

CA 0.5214 0.3302 0.03542

Tj 537.25 585.73 660.46

384 Chemical Reaction Engineering Chap. 8

1.800 Scale: 10-5

I(T) 1.200

0.600

0.000

-0.600

-1. 200 -i1--+----j---I---+----1~-+1---I--+-j---+------11 500.000 540.000 580.000 620.000 660.000 700.

T

Figure 8-15 Graphical Indication of Multiple Steady-State Solutions

The POLYMATH data file for part (c) is found in the Simultaneous Algebraic Equation Solver Library located in directory CHAP8 with file named P8-23C.POL.