Mann U. Principles of Chemical Reactor Analysis and Design: New Tools for Industrial Chemical Reactor Operations

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time, N

(t)=(N

tot

)

¼ 0:151 and N

(t)=(N

tot

)

¼ 0:388, which correspond

to 120.8 and 302.4 mol, respectively. Hence, the operating time of semibatch

is about 8 times longer than that of a batch operation, but more than twice the

amount of product V, and a smaller amount of undesirable product W are

generated.

d. The heating load curve is calculated by Eq. 9.1.50, and is shown in

Figure E9.1.5. For 90% conversion, t

¼ 4.4, and Q=[(N

tot

)

] ¼

Figure E9.1.3 Comparison of V production—isothermal operation.

Figure E9.1.4 Comparison of W production—isothermal operation.

Figure E9.1.2 Species operating curves—semibatch; isothermal operation.

390 OTHER REACTOR CONFIGURATIONS

20.0723, which corresponds to 24817.3 kcal. The instantaneous isothermal

HTN is calculated by Eq. 9.1.51.

e. For adiabatic semibatch operation, we have to solve design equations (g) and

(h) simultaneously with the energy balance equation. Using Eq. 9.1.35, the

speciﬁc molar heat capacity of the reference state is

(200 L)(1000 g=L)(1 cal=gK)

800 mol

¼ 250 cal=mol K

The reference temperature is T

¼ 333 K, and the dimensionless heats of

reactions are

DHR

)

¼0:0961 DHR

)

¼0:1441

The dimensionless activation energies of the two chemical reactions are

¼ 13:6 g

¼ 18:13

Figure E9.1.5 Heat-transfer curve—semibatch; isothermal operation.

Figure E9.1.6 Temperature curve—adiabatic operation.

9.1 SEMIBATCH REACTORS 391

For adiabatic operation, HTN ¼ 0. For uniform injection rate, using

Eq. 9.1.39, the correction factor of the heat capacity is

CF(Z

, u) ¼

(r)

Since T

inj

¼ T

inj

¼ 1), the energy balance equation (Eq. 9.1.38) reduces to



inj



(1  u)  DHR

 DHR



(s)

We solve (s) simultaneously with (g) and (h) for different values of operating

times, t

, subject to the initial conditions that at t ¼ 0, Z

¼ Z

¼ 0, and

u ¼ 1, and we determine Z

, Z

, and u for each t

. Once we obtain the reac-

tion operating curves (Z

and Z

versus t

), we use Eq. 9.1.15 to obtain the

species operating curves. Figure E9.1.7 shows the curve for product V and

compares it to that of an adiabatic batch reactor. Figure E9.1.8 shows the

curve for product W and compares it to that of an adiabatic batch reactor.

Figure E9.1.6 shows the temperature curve and compares it to that of a

Figure E9.1.7 Comparison of V production—adiabatic operation.

Figure E9.1.8 Comparison of W production—adiabatic operation.

392 OTHER REACTOR CONFIGURATIONS

batch reactor. Note that since the activation energy of the second reaction is

higher than that of the ﬁrst, and both reactions are exothermic, isothermal

operation is preferable.

For a batch reactor (constant volume), the design equations (Eq. 7.1.1) are

¼ r



(t)

¼ r



(u)

Using Eq. 6.1.5 to express the species concentrations, the reaction rates are

¼ k

(1  Z

 2Z

(u1)=u

(v)

¼ k

(1  Z

 2Z

)

(u1)=u

(w)

Substituting (v), (w), and (f) into (t) and (u), the design equations become

¼ (1  Z

 2Z

(u1)=u

(x)

)



(1  Z

 2Z

)

(u1)=u

(y)

For isothermal operation, u ¼ 1, and (x) and (y) are solved simultaneously.

For adiabatic operation, the energy balance equation is

¼DHR

 DHR

(z)

We solve (x), (y), and (z) for different values of operating times, t

, subject

to the initial conditions that t ¼ 0, Z

¼ Z

¼ 0, and u ¼ 1. The species

curves of the product V, and the undesirable product W, are compared in

the ﬁgures above.

Example 9.2 Valuable product V is produced in a semibatch reactor where the

following simultaneous, liquid-phase chemical reactions take place.

Reaction 1: A þ B ! V

Reaction 2: 2A ! W

The rate expressions are r

¼ k

, and r

¼ k

. The reactor is initially

charged with a 200-L solution of reactant B with a concentration of C

(0) ¼ 3

mol/L, and 200-L solution of reactant A (C

¼ 3 mol/L) is fed continuously

9.1 SEMIBATCH REACTORS 393

into the reactor at a constant rate. Both the injected stream temperature and the

initial reactor temperature are 608C.

a. Show qualitatively the advantage of semi-batch operation over batch

operation.

b. Derive the design equations, and plot the reaction and species curves for

isothermal semi-batch operation. Compare the product curves to those of

isothermal batch operation. What is the operating time for each mode of

operation for 80% conversion of reactant A? What is the amount of product

V and W generated in each mode?

c. Derive the design equations, and plot the reaction and species curves for

adiabatic semi-batch operation. Compare the product curves to those of

adiabatic batch operation with T(0) ¼ 608C, and T

inj

¼ 508C.

d. Derive the design and energy balance equations for batch operation.

Data:At608C, k

¼ 0:02 L=mol min

1

, k

¼ 0:04 L=mol min

1

¼9000 cal=mol DH

¼13,000 cal=mol

¼ 12,000 cal=mol E

¼ 20,000 cal=mol

The heat capacity of the solution is



¼ 0:9 cal g

1

, and its density is

0.85 kg/L.

Solution

a. To identify the preferable reactor operation mode, we write the ratio of the

formation rates of the desired and undesired products:

(T)C



exp 

 E



Hence, we would like to maintain high concentration of reactant B and low

concentration of reactant A. This is achieved by charging the reactor with

reactant B and then injecting reactant A slowly. Also, since E

, E

,itis

preferable to operate the reactor at a lower temperature.

b. The stoichiometric coefﬁcients of the chemical reactions are

¼1 s

¼ 1 s

¼ 0 D

¼1

¼2 s

¼ 0 s

¼ 1 D

¼1

Since each reaction has a species that does not appear in the other, the two

reactions are independent, and there is no dependent reaction. We select

the total amount introduced into the reactor during the operation as the

394 OTHER REACTOR CONFIGURATIONS

reference state. Hence, V

(0) ¼ 200 L, and, for constant injection rate,

inj

)

¼ v

¼ 200 L, and

¼ V

(0) þ (V

inj

)

¼ 200 þ 200 ¼ 400 L

Since only reactant B is charged initially, N

tot

(0) ¼ N

(0) ¼ 600 mol,

(0) ¼ 0, N

(0) ¼ 0, and N

(0) ¼ 0. Also, since only reactant A is

injected, (N

tot

)

inj

¼ (C

)

inj

)

¼ 600 mol, and the total molar content of

the reference state is

tot

)

¼ N

tot

(0) þ (N

inj

)

¼ 1200 mol

The reference concentration is

tot

)

1200 mol

400 L

¼ 3 mol=L

We write Eq. 9.1.20 for each independent reaction:

¼ r

1 þ





0  t  t

(a)

¼ r

1 þ





0  t  t

(b)

Using Eq. 9.1.21, the concentrations of the reactants in the reactor are

(t) ¼ C

þ t



(0:5)

þ (1)Z

(t) þ(2)Z

(t)



(c)

(t) ¼ C

þ t



[(0:5) þ (1)Z

(t)] (d)

The rates of the two reactions are

¼ k

þ t



(0:5)

 Z

 2Z



(0:5  Z

(u1)=u

(e)

¼ k

þ t



(0:5)

 Z

 2Z



(u1)=u

(f)

We select Reaction 1 as the leading reaction; hence, the characteristic reaction

time is

¼ 16:67 min (g)

9.1 SEMIBATCH REACTORS 395

Substituting (e), (f), and (g) into (a) and (b), the design equations reduce to

þt



(0:5)

Z

2Z



(0:5Z

(u1)=u

0 t t

(h)

)



þt



(0:5)

Z

2Z



(u1)=u

0 t t

(i)

For isothermal operation, u ¼ 1, and the design equations reduce to

þt



(0:5)

Z

2Z



(0:5Z

)0t t

(j)

)



þt



(0:5)

Z

2Z



0 t t

(k)

We solve ( j) and (k) for different values of operating times, t

, subject to the

initial conditions that at t ¼ 0, Z

¼ Z

¼ 0, and then determine the ﬁnal

extents for each t

. Figure E9.2.1 shows the reaction curves for isothermal

semibatch operation. Using Eq. 9.1.15, we determine the species curves,

shown in Figure E9.2.2. The curves of products V and W are compared to

those of isothermal batch operation (derived below) in Figures E9.2.3 and

E9.2.4, respectively.

For semibatch operation, 80% conversion of reactant A (N

tot

)

¼ 0.1), is reached at t

¼ 6.95, which corresponds to 116 min. At

that operating time, Z

¼ 0.180, Z

¼ 0.08, N

/(N

tot

)

¼ 0.20, and N

tot

)

¼ 0.10. Hence, 240 mol of product V and 120 mol of product W

were produced. For batch operation, 80% conversion is reached at

¼ 1.26, which corresponds to 21 min. At that operating time, N

tot

)

¼ 0.118, and N

/(N

tot

)

¼ 0.141. Hence, 141.6 mole of product V

and 169.2 mol of product W were produced.

Figure E9.2.1 Reaction operating curves; isothermal semibatch operation.

396 OTHER REACTOR CONFIGURATIONS

c. For nonisothermal semibatch operations, we have to solve design equations

(h) and (i) simultaneously with the energy balance equation. Using

Eq. 9.1.35,

(400 L)(850 g=L)(0 :9 cal=gK)

(1200 mol)

¼ 255 cal=mol K

Figure E9.2.2 Species operating curves; isothermal semibatch operation.

Figure E9.2.3 Comparison with batch reactor—product V (isothermal operation).

Figure E9.2.4 Comparison with batch reactor—product W (isothermal operation).

9.1 SEMIBATCH REACTORS 397

The reference temperature is T

¼ 333 K, and the dimensionless heats of

reactions are

DHR

)

¼0:106 DHR

)

¼0:153

The dimensionless activation energies of the two reactions are

¼ 18:14 g

¼ 30:21

For adiabatic operation, HTN ¼ 0. Using Eq. 9.1.39,

CF(Z

, u) ¼

1 þ



(l)

The energy balance equation (Eq. 9.1.37) reduces to

þ t



tot

)

inj

tot

)

inj

 u) þ DHR

þ DHR



(m)

In this case, u

inj

¼ 0.97. We solve (m) simultaneously with (h) and (i) for

different values of t

, subject to the initial conditions that at t ¼ 0,

¼ Z

¼ 0, u ¼ 1, and then determine the extents and u at each t

Figure E9.2.5 shows the reaction operating curves and Figure E9.2.6

shows the temperature curve. Using Eq. 9.1.15 we determine the

species curves. The curves of products V and W are shown in Figures

E9.2.7 and E9.2.8, respectively, and are compared to those of adiabatic

batch operation.

Figure E9.2.5 Reaction curves—semibatch; adiabatic operation.

398 OTHER REACTOR CONFIGURATIONS

d. For a constant-volume batch reactor, the design equations (Eq. 6.1.1) are

¼ r



(n)

¼ r



(o)

Figure E9.2.6 Temperature curve; adiabatic operation.

Figure E9.2.8 Comparison with batch reactor—W production; adiabatic operation.

Figure E9.2.7 Comparison with batch reactor—V production; adiabatic operation.

9.1 SEMIBATCH REACTORS 399