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

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Differentiating with respect to time,

¼ M(t)



[T(0)  T

]

)

(9:3:18)

Noting that

¼

out



[T(t)  T

] ¼ h

where c

is the speciﬁc mass-based enthalpy of the liquid phase, we combine Eqs.

9.3.16 and 9.3.18:

M(t)



Q 



out

 h

) 

)

(9:3:19)

Using Eq. 9.3.5,

out

 h

) ¼

evap

)

evap

)

(9:3:20)

where (D

)

vap

is the speciﬁc molar-based enthalpy of evaporation species j. Also,

the rate heat transferred to the reactor is

Q ¼ U



(t)(T

 T)(9:3:21)

Combining Eqs. 9.3.19, 9.3.20, and 9.3.21, for reactors with negligible shaft work,

the energy balance equation becomes

M(t)





(t)(T

 T) 

M(t)



evap

)

evap

)

(9:3:22)

Equation 9.3.22 expresses the temperature changes of the reacting ﬂuid. To reduce it

to dimensionless form, we deﬁne dimensionless temperature by

u ¼

(9:3:23)

420 OTHER REACTOR CONFIGURATIONS

and divide both sides of Eq. 9.3.22 by (N

tot

)

HTN

(t)

(0)



 u) 

evap

)

vap

)



DHR

(9:3:24)

where DHR

is the dimensionless heat of reaction of the mth-independent reaction

deﬁned by Eq. 5.2.24, HTN is the heat-transfer number, deﬁned by Eq. 5.2.23,

is the speciﬁc molar heat capacity of the reference state, and CF is the correction

factor of the heat capacity. For liquid-phase reactions, the speciﬁc molar heat

capacity of the reference state is deﬁned by

M(0)



tot

)

(9:3:25)

and, using Eq. 9.3.10, the correction factor is

CF ¼

(t)

(0)

¼ 1 

evap

)

(t)

(9:3:26)

When the reactor temperature does not change, the rate of heat transferred is

tot

)



evap

)

evap

)

DHR

(9:3:27)

Example 9.4 The dehydration of two heavy organic species, reactants A and

B, is carried out in a distillation reactor where the water, product C, is being

removed continuously. The reactor is initially charged with a 60-L organic sol-

ution that contains 200 mol of reactant A and 300 mol of reactant B. The reactor

is operated isothermally at 2008C, and the following reactions take place in

the reactor:

Reaction 1: A (liquid) ! C (g) þ D (liquid)

Reaction 2: B (liquid) ! 2 C (g) þ E (liquid)

The rates of the chemical reactions are r

¼ k

and r

¼ k

a. Derive the design equations and plot the reaction operating curves.

b. Plot the species operating curves.

c. Determine the operating time needed to achieve 80% conversion of B and the

reactor volume at that time.

9.3 DISTILLATION REACTOR 421

d. Determine the heating load during the operation.

Data: At 2008C, k

¼ 0:05 min

1

¼ 0:1 min

1

¼ 30 kcal=mol DH

¼ 40 kcal=mol

¼ 18 g=mol r ¼ 800 g=L



¼ 0:9 cal=gK

1

Heat of vaporization of species C is 8300 cal/mol

Solution The two chemical reactions are independent and their stoichiometric

coefﬁcients are

¼1 s

¼ 0 s

¼ 1 s

¼ 0 D

¼ 1

¼ 0 s

¼1 s

¼ 2 s

¼ 0 s

¼ 1 D

¼ 2

We select the initial state as the reference state. Hence, V

¼ 60 L,

tot

)

¼ N

(0) þ N

(0) ¼ 500 mol

tot

)

¼ 8:33 mol=L

and y

(0) ¼ 0.4, y

(0) ¼ 0.6. For liquid-phase reactions, using Eq. 5.2.31, the

speciﬁc molar heat capacity of the reference state is

(60 L)(800 g=L)(0:9 cal=gK)

500 mol

¼ 86:4 cal=mol K

Since only species C is gaseous, Eq. 9.3.10 reduces to

(t)

(0)

¼ 1 

(t) þ 2Z

(t)] (a)

and, using Eq. 9.3.24, the correction factor of the heat capacity is

CF ¼

(t)

(0)

¼ 1 

þ 2Z

) ¼ 1  0:187(Z

þ 2Z

) (b)

The reference temperature is T

¼ 473 K, and the dimensionless heats of reac-

tion are

DHR

)

¼ 0:734 DHR

)

¼ 0:978

422 OTHER REACTOR CONFIGURATIONS

and the dimensionless speciﬁc molar enthalpy of vaporization of species C is

)

evap

8300

(86:4)(473)

¼ 0:203 (c)

a. We write Eq. 9.3.12 for each independent reaction:

¼ r

1 

þ 2Z

)





(d)

¼ r

1 

þ 2Z

)





(e)

We use Eq. 9.3.14 to express the reactant concentrations in terms of the

dimensionless extents:

(t) ¼ C

(0)  Z

(t)

1  (C

=r)MW

(t) þ2Z

(t)]

(f)

(t) ¼ C

(0)  Z

(t)

1  (C

=r)MW

(t) þ2Z

(t)]

(g)

We select Reaction 1 as the leading reaction and deﬁne a characteristic reac-

tion time by

¼ 20 min (h)

We substitute (f) and (g) in the rate expressions and the latter together with

(h) in the design equations and obtain

¼ (0:4  Z

(u1)=u

(i)

)

(0:6  Z

(u1)=u

(j)

b. For isothermal operation, u ¼ 1, and we solve (i) and ( j) subject to the initial

conditions that at t ¼ 0, Z

¼ Z

¼ 0. In this case, we obtain analytical sol-

utions:

(t) ¼ 0:41 e

t

ðÞ (k)

(t) ¼ 0:61 e

2t



(l)

9.3 DISTILLATION REACTOR 423

The reaction operating curves and the reactor volume curve are shown in

Figure E9.4.1.

c. Now that the extents of the independent reactions at any operating time are

known, we can determine the amount of each species formed or depleted

using relation Eq. 2.5.3:

(t)

tot

)

¼ 0:4  Z

(t) ¼ 0:4e

t

(t)

tot

)

¼ 0:6  Z

(t) ¼ 0:6e

2t

The species operating curves are shown in Figure E9.4.2.

d. For 80% conversion of B, N

¼ 0.2N

(0) ¼ 0.2[0.6(N

tot

)

] ¼ 0.12(N

tot

)

and, using Eq. 2.7.4, Z

¼ 0.48. From (l), this value of Z

is reached at

dimensionless operating time of t ¼ 0.805, and, at that time, Z

¼ 0.221.

The operating time is

t ¼ tt

¼ (0:805)(20 min ) ¼ 16:1 min

Figure E9.4.1 Reaction operating curves.

Figure E9.4.2 Species operating curves.

424 OTHER REACTOR CONFIGURATIONS

For t ¼ 0.805, V

(t)/V

(0) ¼ 0.785. The volume of the reactor when 80%

conversion of B is reached is

(t) ¼ (0:785)V

(0) ¼ 46:71 L

For isothermal operation, using Eq. 9.3.23

tot

)



)

evap

þ 2



þ DHR

(m)

which, upon integration, becomes

Q ¼ (N

tot

)

b(D

)

evap

þ 2Z

) þ DH

þ DH

c (n)

For 80% conversion of B, Z

¼ 0.221, Z

¼ 0.48, and Q ¼ 17.82  10

kcal.

9.4 RECYCLE REACTOR

A recycle reactor is a mathematical model describing a steady plug-ﬂow reactor

where a portion of the outlet is recycled to the inlet, as shown schematically in

Figure 9.5. Although this reactor conﬁguration is rarely used in practice, the recycle

reactor model enables us to examine the effect of mixing on the operations of con-

tinuous reactors. In some cases, the recycle reactor is one element of a complex

reactor model. Below, we analyze the operation of a recycle reactor with multiple

chemical reactions, derive its design equations, and discuss how to solve them.

To derive the design equation of a recycle reactor, we consider a differential

reactor element, dV, and write a species balance equation over it for species j:

¼ (r

) dV (9:4:1)

We follow the same procedure as the one used to derive the design equation of a

plug-ﬂow reactor (Section 4.4.2) and obtain

¼ r

(9:4:2)

Figure 9.5 Recycle reactor.

9.4 RECYCLE REACTOR 425

where

is the extent per unit time of the mth-independent reaction from the reac-

tor inlet to a given point in the reactor. This is the differential design equation for a

recycle reactor, written for the mth-independent chemical reaction. Note that Eq.

9.4.2 is identical to the design equation of a plug-ﬂow reactor. The main difference

between the recycle reactor and a plug-ﬂow reactor is in the way the volumetric

ﬂow rate and the concentrations of the species vary along the reactor.

Consequently, to solve the design equations, we have to express these quantities

in terms of the extents of the independent reactions. Since the reactor inlet is

affected by the outlet, let

out

denote the extent per unit time of the mth-indepen-

dent reaction in the entire reactor.

To reduce the design equation to a dimensionless form, we select the feed stream

to the system (stream 0) as the reference stream and deﬁne a dimensionless extent of

the mth-independent reaction by

;

tot

)

(9:4:3)

where (F

tot

)

¼ (v

) is the total molar ﬂow rate of the reference system, v

is its

volumetric ﬂow rate, and C

is the reference concentration. We deﬁne a dimension-

less space time by

t ;

(9:4:4)

Differentiating Eqs. 9.4.3 and 9.4.4,

¼ (F

tot

)

dV ¼ (v

) dt

and substituting these in Eq. 9.4.2, the design equation reduces to

¼ r



(9:4:5)

Equation 9.4.5 is the dimensionless design equation of a recycle reactor, written for

the mth-independent reaction. To describe the operation of a recycle reactor with

multiple chemical reactions, we have to write Eq. 9.4.5 for each of the independent

reactions.

To solve Eq. 9.4.5, we have to express the reaction rates in terms of the extents of

the independent reactions. We do so by expressing the local volumetric ﬂow rate

and the local molar ﬂow rates of all reactants in terms of Z

’s and calculating

the local species concentrations. Using Eqs. 2.7.8 and 2.7.10, the local molar

426 OTHER REACTOR CONFIGURATIONS

ﬂow rate of species j at any point in the reactor is

¼ F

)

(9:4:6)

and the local total molar ﬂow rate is

tot

¼ (F

tot

)

(9:4:7)

The local molar ﬂow rate of species j at the reactor inlet (point 1) is

¼ F

þ F

(9:4:8)

Using the deﬁnition of the recycle ratio,

R ;

the molar ﬂow rate at point 4 is related to the molar ﬂow rate at point 2 by

1 þ R

(9:4:9)

Writing Eq. 9.4.6 for the reactor outlet (point 2 where,

out

) and combining

it with Eqs. 9.4.8 and 9.4.9,

¼ (1 þ R) F

)

out

(9:4:10)

Substituting Eqs. 9.4.9 and 9.4.10 into Eq. 9.4.8, the molar ﬂow rate of species j at

the reactor inlet (point 1) is

¼ (1 þR)F

þ R

)

out

(9:4:11)

Substituting Eq. 9.4.11 into Eq. 9.4.6, and using the deﬁnition of dimensionless

extent (Eq. 9.4.3), the local molar ﬂow rate of species j is

¼ (F

tot

)

(1 þ R)y

þ R

)

out

)

(9:4:12)

9.4 RECYCLE REACTOR 427

To obtain the total local molar ﬂow rate, we sum Eq. 9.4.12 over all species and

obtain

tot

¼ (F

tot

)

1 þ R þ R

out

(9:4:13)

For the reactor inlet and outlet (points 1 and 2), Eq. 9.4.13 reduces, respectively, to

tot

)

¼ (F

tot

)

1 þ R þ R

out

(9:4:14)

tot

)

¼ (F

tot

)

1 þ R þ (1 þ R)

out

(9:4:15)

Using Eq. 9.4.12, the concentration of species j at any point in the reactor is

tot

)

(1 þ R)y

þ R

)

out

)

(9:4:16)

For liquid-phase reactions, the density is assumed constant; hence, v ¼ v

, v

, and v

¼ Rv

. Hence,

¼ v

þ v

¼ (1 þ R)v

(9:4:17)

Substituting Eq. 9.4.17 into Eq. 9.4.16, the local concentration for liquid-phase

reactions is

¼ C

1 þ R

)

out

1 þ R

)

(9:4:18)

Note that for R ¼ 0 (no recycle), Eq. 9.4.18 reduces to the local concentration in a

plug-ﬂow reactor with liquid-phase reactions (Eq. 7.1.11), and for R ! 1 (very

high recycle), Eq. 9.4.18 becomes

¼ C

)

out

which is the outlet concentration of a CSTR. Indeed, at very high recycles, the

recycle reactor behaves as a CSTR.

428 OTHER REACTOR CONFIGURATIONS

For gas-phase reactions, assuming ideal-gas behavior, the local volumetric ﬂow

rate at any point in the reactor is

v ¼ v

tot

)



(9:4:19)

Using Eq. 9.4.13,

v ¼ v

1 þ R þ R

out



(9:4:20)

Substituting Eq. 9.4.20 into Eq. 9.4.16, for isobaric operations, the local concen-

tration of species j is

¼ C

( y

)

þ R=(1 þ R)

)

out

þ 1=(1 þ R)

)

1 þ R=(1 þ R)

out

þ 1=(1 þ R)



(9:4:21)

Note that for R ¼ 0 (no recycle), Eq. 9.4.21 reduces to the local concentration in a

plug-ﬂow reactor, and for R ! 1 (very high recycle), it reduces to the concen-

tration of a CSTR.

For nonisothermal recycle reactors, we have to incorporate the energy balance

equation to express variation in the reactor temperature. The energy balance

equation is the same as that of a plug-ﬂow reactor,

CF(Z

, u)

HTN(u

 u) 

DHR

(9:4:22)

The only difference here is that the temperature at the reactor inlet depends on the

outlet temperature and the recycle. Hence, we have to solve Eq. 9.4.22 subject to

the initial condition that at t ¼ 0, u(0) ¼ u

(the dimensionless temperature at

point 1). To determine u

, we write an energy balance over the mixing point,

tot

)

¼ (F

tot

)

þ (F

tot

)

(9:4:23)

where

H is the speciﬁc molar enthalpy of the stream. Taking the feed stream to the

system as the reference stream, and assuming no phase change, the enthalpy of a

stream at temperature T is

tot

)

H ¼

) dT (9:4:24)

9.4 RECYCLE REACTOR 429