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

Подождите немного. Документ загружается.

†

Packed-bed reactors (the packing disperses the ﬂuid laterally across the bed

resulting in a near uniform ﬂow).

7.1 DESIGN EQUATIONS AND AUXILIARY RELATIONS

The design equation of a plug-ﬂow reactor was derived in Chapter 4. The design

equation, written for the mth-independent reaction, is

¼ r



(7:1:1)

where Z

is the dimensionless extent, deﬁned by Eq. 2.7.2:

;

tot

)

(7:1:2)

and t is the dimensionless space time (dimensionless volume), deﬁned by

Eq. 4.4.8:

t ;

(7:1:3)

where t

is a conveniently selected characteristic reaction time, deﬁned by

Eq. 3.5.1, and C

is the concentration of a conveniently selected reference

Figure 7.1 Schematic description of plug-ﬂow reactor: (a) tubular reactor and ( b) packed-

bed reactor.

240 PLUG-FLOW REACTOR

stream, deﬁned by

;

tot

)

(7:1:4)

where (F

tot

)

and v

are, respectively, the molar ﬂow rate and the volumetric ﬂow

rate of the reference stream. As discussed in Chapter 4, to describe the operation of

a reactor with multiple reactions, we have to write Eq. 7.1.1 for each independent

chemical reaction. The solutions of the design equations (the relationships between

’s and t) provide the reaction operating curves that describe the progress of the

chemical reactions along the volume of the reactor.

To solve the design equations, we have to express the rates of all the chemical

reactions in terms of Z

’s and t. Below, we derive the auxiliary relations that

relate the species concentrations to Z

’s and t.

The volume-based rate of the ith chemical reaction (from Eq. 3.3.1 and

Eq. 3.3.6) is

¼ k

(u1)=u

’s) (7:1:5)

where k

) is the reaction rate constant at reference temperature T

, g

is the

dimensionless activation energy (g

¼ Ea

/RT

), and h

’s) is a function of the

species concentrations, given by the rate expression.

To express the rates of the chemical reactions in terms of the reaction extents, we

have to relate the species concentrations to Z

’s and t. For plug-ﬂow reactors, the

concentration of species j at a given point in the reactor is

(7:1:6)

where F

is the local molar ﬂow rate of species j, and v is the local volumetric ﬂow

rate. Using Eq. 2.7.7,

¼ (F

tot

)

tot

)

tot

)



)

()

(7:1:7)

where (F

tot

)

and y

are, respectively, the total molar ﬂow rate and the molar frac-

tion of species j at the reactor inlet. Substituting Eq. 7.1.7 into Eq. 7.1.6, the local

concentration of species j is

tot

)

tot

)

tot

)



)

()

(7:1:8)

7.1 DESIGN EQUATIONS AND AUXILIARY RELATIONS 241

When the inlet stream is selected as the reference stream, as done in most appli-

cations, (F

tot

)

¼ (F

tot

)

and y

¼ y

, and Eq. 7.1.8 reduces to

tot

)

(7:1:9)

For most liquid-phase reactions, the density of the reacting ﬂuid is almost con-

stant; hence, v ¼ v

, and Eq. 7.1.8 becomes

tot

)

tot

)

tot

)



)

()

(7:1:10)

When the inlet stream is selected as the reference stream, v

¼ v

, and Eq. 7.1.10

reduces to

¼ C

)

(7:1:11)

Equation 7.1.11 is used in most applications with liquid-phase reactions (when the

inlet stream is selected as the reference stream).

For gas-phase reactions, the volumetric ﬂow rate may vary along the reactor due

to changes in molar ﬂow rate, temperature, and pressure. Assuming ideal-gas beha-

vior, the local volumetric ﬂow rate is

v ¼ v

tot

)





(7:1:12)

Using Eq. 2.7.9 to express the total molar ﬂow rate in terms of the extents of the

independent reactions,

tot

¼ (F

tot

)

tot

)

tot

)

Equation 7.1.12 becomes

v ¼ v

tot

)

tot

)



(7:1:13)

where u is the dimensionless temperature, T/T

. Substituting Eq. 7.1.13 into Eq.

7.1.9, the local concentration is

¼ C

[(F

tot

)

=(F

tot

)

[(F

tot

)

=(F

tot

)



(7:1:14)

242 PLUG-FLOW REACTOR

When the inlet stream is selected as the reference stream, Eq. 7.1.14 reduces to

¼ C

)

1 þ





(7:1:15)

Equation 7.1.15 is used in most plug-ﬂow reactor applications with gas-phase reac-

tions (when the inlet stream is selected as the reference stream).

Since Eqs. 7.1.5 and 7.1.15 contain another dependent variable, u, the dimen-

sionless temperature, we should solve the energy balance equation simultaneously

with the design equations. For plug-ﬂow reactors with negligible viscous work, the

energy balance equation (derived in Section 5.2) is

CF(Z

, u)

HTN(u

 u) 

DHR

(7:1:16)

where HTN is the dimensionless local heat-transfer number, deﬁned by Eq. 5.2.22,

HTN ;



(7:1:17)

DHR

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

tion, deﬁned by Eq. 5.2.23,

DHR

;

)

(7:1:18)

and CF(Z

, u) is the correction factor of the heat capacity of the reacting ﬂuid,

determined by Eq. 5.2.54. The ﬁrst term inside the bracket of Eq. 7.1.16 is the

dimensionless local heat-transfer rate:

tot

)



¼ HTN(u

 u)(7:1:19)

The second term inside the bracket of Eq. 7.1.16 represents the heat generated (or

consumed) by the chemical reactions.

The speciﬁc molar speciﬁc heat capacity of the reference state,

, is deﬁned

differently for gas-phase and liquid-phase reactions. For gas-phase reactions, it is

deﬁned by Eq. 5.2.58,

;

)(7:1:20)

and for liquid-phase reactions, it is deﬁned by Eq. 5.2.60,

;



tot

)

(7:1:21)

7.1 DESIGN EQUATIONS AND AUXILIARY RELATIONS 243

To solve Eq. 7.1.16, we have to specify the value of HTN. However, its value

depends on the heat-transfer coefﬁcient, U, which depends on the ﬂow conditions

in the reactor, the properties of the ﬂuid, and the heat-transfer area per unit volume,

(S/V). These parameters are not known a priori. Therefore, we develop a procedure

to estimate the range of HTN. For isothermal operation (du/dt ¼ 0), we can deter-

mine the local HTN from Eq. 7.1.16 (taking the reactor temperature as the reference

temperature, u ¼ 1):

HTN

iso

(t) ¼

 1

DHR

(7:1:22)

Since dZ

/dt varies along the reactor, the value of HTN also varies from point to

point. We deﬁne an average HTN for isothermal operation by

HTN

ave

;

tot

HTN(t) dt (7:1:23)

where t

tot

is the total dimensionless space time of the reactor. Recall that for

adiabatic operation HTN ¼ 0. In practice in most cases, the heat-transfer number

would be

0 , HTN  HTN

ave

(7:1:24)

Equation 7.1.24 provides only an estimate on the range of the value of HTN. We

select a speciﬁc value after examining the reactor performance with different values

of HTN. When multiple reactions take place, it is important to examine the reactor

performance for different values of HTN, since it is difﬁcult to predict the effect of

the heat transfer on the relative rates of the individual reactions. Once the physical

reactor has been designed, it is necessary to verify that the ﬂow conditions in the

reactor actually provide the speciﬁed value of HTN.

For convenience, Tables A.3a and A.3b in Appendix A provide the design

equations and auxiliary relations used in the design of plug-ﬂow reactors.

Table A.4 provides the energy balance equation.

In the remainder of the chapter, we discuss how to apply the design equations

and the energy balance equation to determine various quantities related to the oper-

ations of plug-ﬂow reactors. In Section 7.2, we examine isothermal operations with

single reactions to illustrate how the rate expressions are incorporated into the

design equation and how rate expressions are determined. In Section 7.3, we

expand the analysis to isothermal operations with multiple chemical reactions. In

Section 7.4, we consider nonisothermal operations with multiple reactions. In all

these cases, we assume that the pressure drop along the reactor is negligible. In

Section 7.5, we consider the effect of pressure drop on the operations of plug-

ﬂow reactors with gas-phase reactions.

244 PLUG-FLOW REACTOR

7.2 ISOTHERMAL OPERATIONS WITH SINGLE REACTIONS

We start the analysis of plug-ﬂow reactors by considering isothermal operations

with single reactions. For isothermal operations, du/dt ¼ 0, and we have to

solve only the design equations. The energy balance equation provides the heating

(or cooling) load necessary to maintain isothermal conditions. Furthermore, for iso-

thermal operations, the reaction rate depends only on the species concentrations,

and Eq. 7.1.5 reduces to

r ¼ k(T

)h(C

’s) (7:2:1)

When a single chemical reaction takes place in the reactor, the operation is

described by one design equation, and Eq. 7.1.1 reduces to

¼ r



(7:2:2)

where Z is the dimensionless extent of the reaction and r is its rate. The solution of

the design equation [Z(t) versus t] provides the reaction operating curve that

describes the progress of the chemical reaction along the reactor. Once Z(t)is

known, we can apply Eq. 7.1.7 to obtain the species operating curves, indicating

the species molar ﬂow rates at any point along the reactor. Also, if one prefers to

express the design equation in terms of the reactor volume, rather than the dimen-

sionless space time t, using Eq. 7.1.3, the design equation becomes

tot

)

(7:2:3)

We have to solve Eq. 7.2.2 subject to the initial condition that at t ¼ 0, Z(0) is

speciﬁed (Z at the inlet of the reactor). In some cases, the design equation can be

solved by separating the variables and integrating

t ¼



out

(7:2:4)

where Z

and Z

out

are, respectively, the extent at the reactor inlet and outlet.

Equation 7.2.4 is the integral form of the design equation of plug-ﬂow reactors

with a single chemical reaction. Figure 7.2 shows graphically the relationship

among Z

, Z

out

, and t.

To solve Eq. 7.2.2 or 7.2.4, we have to express the reaction rate r in terms of the

dimensionless extent Z. To do so, we express the species concentrations in terms of

Z. Selecting the inlet stream as the reference stream, Z

¼ 0, and for liquid-phase

reactions, Eq. 7.1.11 reduces to

¼ C

( y

þ s

Z)(7:2:5)

7.2 ISOTHERMAL OPERATIONS WITH SINGLE REACTIONS 245

For gaseous reactions, assuming negligible pressure drop, Eq. 7.1.15 reduces to

¼ C

þ s

1 þ D Z



(7:2:6)

Note that for given inlet conditions, Eqs. 7.2.2 and 7.2.4 have three variables: the

dimensionless space time (or dimensionless reactor volume), t, the reaction extent

at the reactor outlet, Z

out

, and the reaction rate, r. We apply the design equation to

determine any one of these three variables when the other two are provided. In a

typical design problem, we have to determine the reactor volume needed to

obtain a speciﬁed extent (or conversion) for a given feed rate and given reaction

rate. A second application of the design equation is to determine the extent (or con-

version) at the reactor outlet for a given reactor volume and given reaction rate. The

third application is to determine the reaction rate when the extent (or conversion) at

the reactor outlet is provided for different feed rates.

Below, we analyze the operation of isothermal plug-ﬂow reactors with single

reactions for different types of chemical reactions. For convenience, we divide

the analysis into two sections: (i) design and (ii) determination of the rate

expression. In the former, we determine the size of the reactor for a known reaction

rate, speciﬁed feed rate, and speciﬁed extent (or conversion). In the second section,

we determine the rate expression and its parameters from reactor operating data.

7.2.1 Design

First, consider the application of the design equation when the reaction rate is pro-

vided in the form of an algebraic expression. We start with a ﬁrst-order, gaseous,

chemical reaction of the general form

A ! Products

whose rate expression is r ¼ kC

. For this reaction, s

¼ 21, and D is determined

by the stoichiometry. We select the inlet stream as the reference stream; hence,

Figure 7.2 Graphical presentation of the PFR design equation for a single reaction.

246 PLUG-FLOW REACTOR

, y

, and so on are speciﬁed. For a gas-phase reaction, the species concen-

tration is given by Eq. 7.2.6, and the reaction rate expression is

r ¼ kC

 Z

1 þ D Z

(7:2:7)

Substituting Eq. 7.2.7 into Eq. 7.2.2, the design equation becomes

¼ kC

 Z

1 þ D Z



(7:2:8)

Using Eq. 3.5.4, for ﬁrst-order reaction,

(Note this characteristic reaction time is the same as the one obtained by the pro-

cedure described in Section 3.5.) When only reactant A is fed ( y

¼ 1), the

design equation reduces to

1  Z

1 þ D Z

(7:2:9)

We solve this equation, subject to the initial condition that at t ¼ 0, Z ¼ 0. The

solution of this design equation for different values of D is shown in Figure 7.3.

Example 7.1 The ﬁrst-order, gas-phase, decomposition reaction

A ! B þC

is carried out in an isothermal, plug-ﬂow reactor. A stream of reactant A is fed to

the reactor at a rate of 10 mol/min. At the feed conditions, the concentration of A

Figure 7.3 Reaction operating curves for a single ﬁrst-order reaction of the form A !

Products.

7.2 ISOTHERMAL OPERATIONS WITH SINGLE REACTIONS 247

is C

¼ 0:04 mol=L. At the reactor operating temperature, the reaction rate con-

stant is k ¼ 0.1 s

a. Derive and plot the reaction and species curves.

b. Determine the volume of the reactor needed to achieve 80% conversion.

c. Determine the volume of the reactor needed to achieve 80% conversion if, by

mistake, we take D ¼ 0.

d. Determine the conversion if the volume calculated in (c) is used.

e. Determine the feed ﬂow rate to achieve 80% conversion in a reactor with the

volume calculated in (c).

Solution The stoichiometric coefﬁcients of the chemical reaction are

¼1 s

¼ 1 s

¼ 1 D ¼ 1

We select the inlet stream as the reference stream, and since only reactant A is

fed to the reactor, y

¼ 1, y

¼ y

¼ 0, C

¼ C

¼ 0:04 mol=L, and

10 mol=min 10 mol=min. The volumetric ﬂow rate of the reference stream is

tot

)

¼ 250 L=min

a. For a plug-ﬂow reactor with a single chemical reaction, the design equation is

¼ r



(a)

For gas-phase reactions, C

is expressed by Eq. 7.2.6, and the reaction rate is

r ¼ kC

1  Z

1 þ Z

(b)

Substituting (b) into (a), the design equation becomes

¼ kC

1  Z

1 þ Z



(c)

Selecting the characteristic reaction time to be

¼ 10 s ¼ 0:167 min (d)

and substituting (d) into (c), the design equation reduces to

1  Z

1 þ Z

(e)

We solve (e) numerically, subject to the initial condition that at t ¼ 0, Z ¼ 0.

248 PLUG-FLOW REACTOR

Figure E7.1.1 shows the reaction curve. Using Eq. 7.1.7, the species curves are

(t)

tot

)

¼ 1  Z(t)

(t)

tot

)

(t)

tot

)

¼ Z(t)

Figure E7.1.2 shows the species curves.

b. Using Eq. 2.6.5, the relation between the conversion and extent for a single

reaction,

Z ¼

(0)

¼

(1)

¼ f

For Z

out

¼ 0.8, from the reaction curve, t ¼ 2.42. Note in this case we can

determine t by separating the variables and integrating (e),

t ¼

0:8

1 þ Z

1  Z

dZ ¼ 2:42

Figure E7.1.1 Reaction operating curves.

Figure E7.1.2 Species operating curves.

7.2 ISOTHERMAL OPERATIONS WITH SINGLE REACTIONS 249