• Author / Uploaded
• M.Effendy

• Categories
• Reactor Quimico
• Catálisis
• Ciencias fisicas
• Ciencia
• Ingeniería Química

Citation preview

•

• l\IlcGraw-HilL Book Company SI'«' York

SI. 1.011/.1

Sail FWIICil{(J

:{urk/a1ld

HlJmhurg Johall1U'sbu.,.g L"'id,," ,\[adrid ,\If'X;C'' ,\1(JIIlrNli Xl"" Dellt; Panama Pari\" S(lU POU/I) Tokm TOrrJlI/u Singa/mrt' Sw/lle\' . .

Rl1gu/ti

,

\.

•

•

•

NOTICE

The information presented in this book has been compiled from sources considered to be dependable and is accurate and reliable to the best of the author's knowledge and belief, but is not o,-uaranteed to be so. The author assumes no responsibility whatsoever with respect to any use which may be made of any of :he data or methods cesc-ibed in this book or any results· obtained by such t.se. Nothing in this b"ok is lO be construed as a recommendation of any practice in violation of any patent. ;aw, or regulation.

Library of Congress Cataloging i1l Publicat;"" Data

Tarhan, M. Orhan. Catalyti~ reactor design. Includes bibliographies and index . • I. Chemical reactors Design and construction. 2. Catalysis. 1. Title. TP157.T35 1983 660.2'995 82-18015 ISBN 0-07-062871-8 •

Copyright © 1983 by McGraw-HiII, lnc. All rights resen·ed Printed in the United States of America. Except as pelIuitted under the United States Copyright Act of 1976, no pan of this pUblication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system. wiThout the prior written permission of the publisher. 1234567890

898765-13

KGP/KGP

ISB l\ 0-07-062871-8

•

The edItors for this book were Diane Heiberg and Ceraldine Fahey, the designer was flliot 'Epstein. and the production supervisor was Sally Fliess. It was set in Baskerville by Techna Type. Printed and bound by The Kingsport Press.

•

•

CONTENTS

•

• ••

Preface

Chapter

"

•

••

Chapler

XIII

CLASSIFICATIC)N OF REACTORS AND PHASES (lIF DESIGN .

I

1.1 Classification of Reactors 1.2 Design Definitions

I 4

1.3 General Methods of Reactor Design 1.4 Phases of Reactor Design

4

t>.fodeling of Reactors Collection of Data Use of a Computer Mechanical Design 1.5 Operating Considel ations 1.6 Who Needs Reactor Design? References

6

•

-

;:,

7 8 9

9 10 10

OF NUMERICAL C()MPUTATION

13

2.1 Types of Differential Equations

13

•

Ordinan' Di fferential Equations Partial Diff~rential Equations 2.2 Principles of Algorithms Euler's Algorithm Runge-Kutw Algorithms 2.3 nth-Ord·er Ordinary Differential E X2, ... x,,, the step size h would be (2-6) •

and x, = Xf) + ih where i'= 0, I, ... n. How can the true solution )'(x) be approximated? Suppose the value )0' corresponding to the initial value Xo of x, is known and the next value )'1' corresponding to XI' is to be computed (see Fig. 2-2). One way of

•

•

16 CATALYTIC REACTOR DESIGN y ~Known

•

initiol volue of y

..---y(x,l True value of

XI

..---y(X)

Et Error

_1

- -b.x--

y, (

- Numerical value af y,

--h--

FIG. 2-2

Method of approximating a differential e«Juation.

getting from )'0 to )'1 is to compute the tangent at the initial point ),0, which is given by the derivative at Yo or the slope of the tangent at )'0:

dy

+ dx

= 'Yo

)'1

-

."=l .u

Ax = -1'(xd

Referring to Fig_ 2-2, what we are looking for is the true value Y(XI) on the true y(x) curve, but all we have from constructing the tangent at point )'0 is the numerical value YI' Now, according to Eq. (2-5), (dJldx)\:,,, = !(x;J, )'0) and we can write: (2-7)

which is the approximate value of )'1_ Here an ermr El is made. If one continues to approximate further values of)' by this method and is not careful, the error may accumulate excessively. The essential feature of all numerical calculations of differential equations is the estimation of this error El> since the true value Y(XI) is not known. The error E can be reduced if a smaller step h is selected. This can be seen from Fig. 2-2 and can be demonstrated by calculation. A simple method to estimate E is that of Euler. •

Euler's Algorithm Euler's integration algorithm is ),,, + 1

= Y"

+ !()'n,

xu) h

+

En

(2-8)

•

METHODS OF NUMERICAL COMPUTATION

17

This expression looks like a Taylor series, since f(y", x,,) is the derivative dy,/dx. Thus, a few more ternlS can be added by Taylor expansion: Y

t/+

J

= -

y + dy" . h + t/

dx

2

3

3

cly" . h + d y" . 1t +.. . . 2 3 21

dx

dx

31

(2 9)

-

The added terms in the brackets may be called truncation error. If we want to be really.accurate, we should use a large number of these terms. Ho~:ever, that would be too time-consuming and expensive, and we can compromise by using only the first term as a truncation error estimate, \\< hich is what Euler did. Such a procedure is called a one-term en-OT estimate. The derivative dy/dx can itself be expanded in a Taylor series:

h

dYn+J dx

(2-10)

2

2

From this expression d y/dx can be calculated:

cly" _ (dYn+ J/dx) - (dYn/dx) q

-

dx-

h

(2-11) ,

Equation (2-1 I) IS substituted into the first term in the brackets of Eq. (2-9) and one can write: to

-

h , " dx 21

(2-12)

where subscript e denotes "estimated." Equation (2-9) can then be written as

-

•,

""

=

d..,.

,

-

y. + I

=1'.

dy" + dx . h +

E

(2-13)

which is the essence of the Euler algorithm. Now, how can we convert this algorithm into a computer program? Let us asst.::ne that we have to integrate a single ODE: d~' . =x+J' -([x

(2-14)

with the initial value of y(xo) = y(O) = O. This initial value ODE has a known analytic solution: •

y=e'-x-I

(2-15)

In order to apply the Euler method to this ODE, we can start in Fig. _ 2-2 from the known point)o and decide up to what maximum x value, or XMAX, we will integrate. We can diagrammatically represent in Fig.

• CATALYTIC REACTOR DESIGN c

C

C

U L

E

R

•••• ~EAO 04TA ANO tNtTtALIZ~ • ~E4D CS.I00' XM4X. H .~ITE (6. 2001 X~AX.H

x = o. y = o• • TqUEY

=

E T HOD x. v. TRUEy •••• to!

O.

WRITE (6. 201)

v.

X.

TRUEv

c C

•••• EUL~R·S NSTEO : 00 3

N=l.

=

INTEGqATI~~ ••••

+ H/Z.)/H

(X~AX

V = V • H X

~=TH~O

NSTEP

.ex

FLOATlN)

+ YI •

H

TRUEY = EXP(X) - X - I . WRITE (6. 201) x. Y. TRUEY 3 CO'HINU:;

c C

•••• FOR~AT 100 ZOO 201

FO~MAT FOq~AT FOq~.T

1

FO~

J~PUT

AND OUTPUT

STATE~E~S ••••

(10X.~10.~.10X.FIO.6)

(7HIXMAX = .FIZ.6. sx. 5H H ~ • F12.5) (IH • 4HX = • FIO.b. 5X • • HY = • F16.6. 5X.

8HTRU~v

~

•

Fl~.61

c X.t.4Ax -,-

-

X

~

X X

X X X X

H

Y =

0.010000

y

0.020000 0.030aoo 0.0"0000 0.050000 0.060000

y

0.07000~

Y

;

o .oeoo ov

;

o.OQOOOJ

Y Y Y

-

X = X X X X

1.000000

0.0

X

X

=

--

--

= ;

0.100000 a.110000 0.120000 0.130000:

Y Y Y \'

y

Y Y

=

--

= --------

-

(I.olcoeo 0.0 0.0

TRUEY TRUEY

0.000100 0.000301

TQUEY

O.OC0604

TQUEY

0.001010

TI;:UEY

0.001520 0.00:21"35

TQUEY

0.002857

TQUEV

0.0036A5 0.00"622 0.005668

TRUEY

O.OO~~25

0.0")8093

T~UEY

""AUEY

'!J(UEY TRUEY

TRUEY TRUFV

--

= :

0.0

0.OOO:J50 a.~00201

O.OO;)4CO"4

0.000811 0.001271

O.OO193-b 0.002507 0.00 '2:86

--

0.00&174

=

0.005171 o .a06277 0.007496 0.006525

--

•

"

•

• •

• • • •

•

--"x --X --x -

O.57794~ TQUEY 0.5'39290 O .. ~ :':0000 TC:UEY Y O.-S;282fy Tc:;;utY : o .60S0C:;S 0.619951 y O.43~:>OOv TRUEY O.62~S35 0.635 7 09 X O.9~OOO:l Y O.t' 3Q270 TPUEY 0.~51696 O.97~OOO Y 0.6ving boundary conditions: y(a) = a

-v(b) = ~ •

(2-37)

as ca;~ be visualized in Fig. 2-9. The numerical solution of boundaryvalue ODEs is considerably more complicated than that of initial-value ODEs. Three types of algorithms are available, as described below.

•

•

METHODS OF NUMERICAL COMPUTATION 29 Y

10:

a

Yo .Q

b •

y..,'/3

•

{3

a

o

1 ft . , :

a

b

I

FIG. 2-9 Trial-and-error solution of a boundary-value problem. (From Car. nahan et al./ by pellnission from John WiIey.)

The Shooting Method The shooting method consists of a trial-and-error procedure in which a boundary point having the best-known conditions is selerted as the initial point. Any other missing initial conditions are assumed. The initial-value probl,em is then solved by one of the step-by-step procedures, such as the fourth-order Runge-Kutta algorithm (see Sec. 2.5). The initial conditions are adjusted and the problem is solved again. This process is repeated until the computed solution agrees with the known boundary conditions within specified tolerances.

.

The Finite Difference Method The finite difference method consists of approximating the derivatives in the ODEs by finite difference quotients. This results in a system of difference, or algebraic, equations that require considerable computer memory. However, present-day computers can easily handle such systems. Some ODEs can successfully be handled by means of the finite difference method, while others are too nonlinear and require the use of the method described below. The Combined Finite Difference and Quasilinearization Method The combined use of the finite difference method and quasilinearization 4 as described by E. Stanley Lee is recommended only in cases of comrlicat;:d and nonlinear ODEs that cannot be handled by the finite difference method alone. In most cases reactor designers do not have to use boundary-value ODEs, but there are cases when they are necessary, for example, when significant axial dispersion exists within the reactor. For integrating these equations, this book recommends either of the two methods or the combined method, as explained below.

•

•

30 CATAlYTIC REACTOR DESIGN

The types of boundary-value ODEs encountered in reactor design are of the general form: . . (2-38)

(2-39)

Z = dimensionless axial coordinate and independent variable

where

C

T

=

reduced molar concentration (= CICo), dimensionless dependent variable

TT = reduced absolute temperature (= TITo), dimensionless dependent variable

fh

f2 = functions of C and TT P~ = Peelet number for axial dispersion, gas-liquid phase T

Subscripts: mal denotes liquid-phase mass transfer ha2 denotes two-phase heat transfer

o

, •

denotes reactor inlet condition

The functions fl and i2 are highly nonlinear functions of catalyst surface temperatureTs and may also be non linear functions of catalyst surface concentration C.. depending on reaction order. In turn, in the case of first-order reactions Ts and Cs are related to bulk fluid temperature TT and concentration Cn respectively, through the relationships:

Cs = C Co/(l + Dam) T

Da,. = T1k1kca

(2-39a) (2-40)

Ts = To(TT + Dah)

(2-41 )

Dah = (- AH)ffiJa h To

(2-42)

where Da,. and Dah are Damkohler numbers for mass and heat, respectively. The reader is referred to Chap. 4 for more information on these surface-bulk relationships. .. It is believed today that catalyst surface concentrations and temperatures are practically identical to corresponding bulk fluid properties In industrial-scale trickle-bed reactors. Of course, the design equations are then considerably simplified. But there certainly are also cases in which this simplification does not apply.

•

MElHODS OF NUMERICAL COMPUTATION

31

The boundary conditions are: .

at bed inlet at bed outlet

Z=O

CT(O)

= 1

(2-43)

Z = ZT

dC,(Zr) dT,(ZR) = =0 dZ dZ

(2-44)

= T,(O)

In the following se\"eral cases, boundary-value ODEs of varying degreLS of nonlinearity will be considered. Case I: C, = C; T, = T

In a first-order rea((ion. by assuming that C, = C and T, and (2-39) can be written as: I dZC, . dZ 2 Pern.1

I Peh.2

~T, . 'dZ 2

-

dC, I dZ = hi exp (- E R(;T)C,C, =

=

T Eqs. (2-38)

13(C,.

T,)

(2-45) .

dT, -

dZ = - b',! exp (- EIRcT)C,Co = 14(Cp TT) (2-46)

b l and b2 are products of constants

where

E = activation energy, kJlkmol Rc = gas constant = 8.314 kJ/{kmol . K)

f3,14 = funaions of C and T, T

The combined finite difference and quasilinearization method must be used in this case because of the nonlinearitv• of the ODEs. In order to apply the finite differena approximation metlwd the catalyst bed length ZT is subdivided into M equal increments of length az as diagrammatically shown in Fig. 2-10. Since ZT is not really known yet, it must be guessed; it is preferable to overestimate so that we do not run the risk of underestimating and repeating the whole calculation. Let C(l) and T,(l) denote the values of er and Tn respective\:", at positions I!1Z; i.hf"n the derivatives of Egs. (2-45) and (2-46) can be replaced by the following difference quotients: 2

d C, I -7? = ? [Ca + 1) - 2 C,(l) + C,(l dZ.lZ,

~tlZ"l 1 I: 1

.

2 2

3

5• 6 ;

4

3 4

5

9'

7

M-2 M-l

M increments

FIG. 2-10

Diagram of reactor length increments.

M M M+l

I»)

(2-47)

•

32 CATALYTIC REACTOR DESIGN

(2-48)

(2-49)

dZ dT

TX

dZ

=

I [T (I + 1\ - T (I)]

tlZ

TX

(2-50)

)TX

in whic!; I = 1,2,3,. ., (M - 1). The subscript x denotes an unknown value in a forthcoming iteration, while nonsubscripted TT values are known eililer from the initial value or from the previous iteration. Equations (2-49) and (2-50) are an old version of the finite difference approximation of the first denvative. It was used in this section because 4 E. Stanley Lee used it in his work. H,)wever. it is more accurate to write: •

dC,

dZ dT,x

dZ

=--

(2-51 )

=--

(2-52)

This is done in the development of partial differential equations in Sec. 2.7. The difference in accuracy would probably be insignificant in reactor design. Let us now replace the derivatives in Eqs. (2-45) and (2-46) by the proper difference quotients: \.

Pe

1

tlZ2 [C,(I + 1) - 2C,(I) + CT(I - 1)]

mal

I

- tlZ [C,(l

+ 1) -

C,(l)]

= bl exp (- EIRcT,(l)T,,) . C(l) . Co = f3(C n TT)

1 Peha2 t1Z2 [Til

+

1)

2T,(l)

+

TT(l -

1 - tlZ [TT(l =

+

(2-53)

1)]

1) - TT(/)]

- b2 exp (- EIRcTT(I)To)CT(l) . Co = f4(C" •

n

(2-54)

Equation (2-53) is perfectly linear in C and does not need to be quasilinearized. However, Eq. (2-54) is not fully linear in TT; it contains a TT term in the exponential. Thus only Eq. (2-54) has to be quasilinT

•

• METHODS OF NUMERICAl COMPUTATION 33

earized. The function f4(C" TT) can be quasilinearized by using the following equation for TT only: . J(T,x) = J(T,)

dJ(TT) dT (Tn:

+

-

(2-55)

TT)

r

TLus, Eq. (2-54) can be quasilinearized as follows: •

J(T,x)

=

b2 exp (E/RcToT,(/»C,(I)Co + l(df(T,)/dTr)1 . [Trx(I) - Ir(/)]

-

(2-56)

Here the derivative df(Tr)ldT, can be calculated by using well-kno.m formulas: dJ4/dT,

= b2

C,(I) Co [d exp (- E/RcToTr(/»/dT,] •

Let u

=

exp [E/RcToRT{l)]. Then du-l/dT,

=

u-'\duldTr)

=

(l/exp (2E/RcToT,»(duldTTI

Substitution of the above derivatives into Eq. (2-56) gives J(T.,)

= -

E(T",(I) - T,(l)) T oRc(T,(/»2 - I

b2CoC.(I) exp ( - E/RcToTl 1)

(2-57)

Equation (2-53) is rearranged and Eq. (2-54) is combined with Eq. (2-57) and rearranged to give CT(I

+ +

+ 1)[(l/Pema lllZ

Cr (..')£( -

2

(l/LlZ)]

) -

9

2/Pemal~Z-)

Cll - 1)(lIPema1 LlZ

Trx(l + 1)[(lIPemaI LlZ -l

•

2

Trx(/)[( -2/Peha2tlZ2)

)

+ 2

)

(l/LlZ) - blCo exp (- EIRcToT,fJ))] = 0

(2-58)

(lIAZ)]

-

+ (I1LlZ) -

2

(b2COC (/»E exp (-EIRcToT,{l) T

2

/ToRcTT(I)] + TTiI - 1)( I1Pe ha2 Z ) =

b2COCT(l) exp [.- E/RcToT,(/)] [( - EIRcToT,(l)2) -

_ (2-09)

I]

It should be noted that in Eq. (2-58) the C, terms have no subscript x because that equation is already linear in C and all C terms in it are considered "known" or calculable bv the method of Thomas. which is . ' explained below. The C term in the right-hand side of Eq. (2-53) is assumed to be in the "main" increment I and not in incremem.5 I + 1 or I - 1 and is so subscripted. All the C, and TT terms in Eq. (2-58) and T

T

.

T

•

-

34 CATALYTIC REACTOR DESIGN

Eq. (2-59), which do not originate from Eqs. (2-47) to (2-50), are subscripted I. . Equations (2-58) and (2-59) can be simplified and rearranged by defining the following coefficients: 2

l

Cll = (Pemal • I1Z r

CI2 = -2' Cll + (1/I1Z) -b l C o CCl Cl3 = Cl! - (1/I1Z) ,

Cl4 = 0

C21 = (Peha2' I1Zr

l

C22 = -2' C21 + (lII1Z) + CC~ . CCI

(2-60)

C23 = C21 - (lh~Z)

C24 = CCI . CC3 where CC 1 = e -EIRGTOT,(l)

-----;; - 1

Substituting these coefficients in Eqs. (2-58) and (2-59) Cl I . C,(I - 1) + CI2 . C,(I) + C13 . CT(1 + 1)

Cl4

(2-61)

C21 . T",(I - 1) + C22 . T",(I) + C23 . T,/I + 1) = C24

(2-62)

=

-

Note that the terms are ordered from left to right in the order of I - 1, I, I + l. For I = I, the term containing (I - 1) would be zero. Because a subscript is not pennitted to be zero in Fortran IV, instead of setting I = 1,2, 3, ... , M, the values of I will be set I = 2,3,4 .... , (M + I), as shown in Fig. 2-10. Then, at the reactor inlet Z = 0

at reactor outlet

Z =

ZT

I = I

I = M +

I C,(i\t + 1) T,{M

+

= C,(M)

1) = TT(M)

(2-63) When the value of I is varied, the following types of equations are obtained: • C1I . C T(I) CII . C,(l - 1)

• CI2 . CT(2)

+ + CI2 . Cll)

+ Cl3 . CT(3) + Cl3 . ClI + I)

Cl4

(2-64)

= Cl4

(2-65)

=

•

METHODS OF NUMERICAL COMPUTATION 35

+ Cl2 . C,(M) + C13 . Cr(M + 1) C21 . T",(l) + C22 . T;x(2) + C23 . Tr.•(3) C21 . T.)/ - 1) + C22 . Trx(l) + C23 . TTX(l + 1) C21 . Tn:(M - 1) + C22 . TrAA1) + C23 . Tn;(M + 1)

C 11 . Cr(M - 1)

= CH

(2-66)

= C24

(2-67)

= C24

(2-68)

= C24

(2-69)

Since C. ~ 1) = 1, and Trx(l) = i., the terms C ll'Cr(l) and C21· T,x(l) become known quantities and can be transferred to the right-hand side of Eqs. l2-64) and (2-67). Also, because of the boundary conditions of Eq. (2-63), Eqs. (2-66) and (2-69) can be simplified and the following system of six equations is obtained: C12 . Cr(2)

+

= C14 - Cll

C13 . C,(3)

(2-70) ell· er(I -

Cll . e,(M -

+ CI2 . cm + CI3 . e,(I + I) + (C12 + C13)C,(M) C22 Trx(2) + C23 . T ,,(3)

1)

1) = C14

(2-71)

= C14

(2-72)

= C21 - C21

(2-73) C21 . T,x(I - 1) C21 . T",(M -

1)

+ C22 . Trx(I) + C23 . + (C22 + C23)Trx (M)

Trx{I

+ 1)

= e24

(2-74) •

=

C24

(2-75)

It can be seen that the first and last equations of each set of M-I

,-

1

equations contain only two unknowns, and all other equations in the two sets contain three unknowns. These equations have a tridiagonal coefficient 56 matrix and may be solved by using Thomas' melhod. • This method and a 2 subroutine TRIDAG to implement it are presented by Carnahan et al. This procedure is applied alternately to the mass and heat equations until Cr and 1~ values for any Z do not change any more than a desired amount from one iteration to the next. There are some problems here: the bl and b2 "constants" are not really constant. Also, these quantities cannot be calculated from increment !:J.Z to increment AZ because all increments are calculated simultaneously by the Thomas method and not in steps beginning with the reactor inlet. The only time new coefficient values are calculated is between iterative steps and then they can be picked up by the computer program for each A2 or each I value.

•

•

2: Cr ;I!: C; Tr ;I!: T In case of a first-order'reaction, Eqs. (2-38) and (2-39) can be written as •

(2-76)

•

36 CATALYTIC REACTOR DESIGN

(2-77)

where Cs is replaced by its value from Eq. (2-39a), while Ts is left as is. The following equations result: (2-78)

(2-79)

Equation (2-78) is linear in Cr and Eq. (2-79) is linear in T r. The Ts in Eq. (2-79) can be related to Tr during local iterations. Both equations can be treated 0~11y by finite difference approximation. No quasilinearization is needed. The simplifying coefficients are defined as follows:

•

C 11 = r Cl2 = -2' CII + (1/.6.Z) - bl CC l 2 (Pe ma l.6.Z •

Cl3 = CII - (1I.6.Z) Cl4

=

0

C21 = (Peha2.6.Z2rl C22 = -2' C21

+

(2-80)

(1/.6.Z)

C23 = C21 - (1/.6.Z) .

.

'.

C24 = - b2Cr(f) . CC

Co

where CC =

I

e-EIRcT,

+

D

am

In Case 2, the overall iteration is not adequate to calculate new coefficient values. The values of TlJ), Cif) and functions of other parameters related to reaction kinetics must be calculated for each I value before integrating each ODE. This must be done by an efficient local iteration. It is best to recalculate the temperature Ts(f) by this local iteration and to compute other parameters from that recalculated temperature value. A Newton-Raphson iteration procedure required only three steps to arrive within ±O.OOI of the true figure, while a Picard iteration needed four steps for the same ·purpose. Iteration tpchniques are discussed in .. App. A.

• MElHODS OF NUMERICAL COMPUTATION 37

=

=

Case 3: Cs C; Ts T; Reaction Is Second-Order Equations (2-38) and (2-39) c~m be written as: •

(2-81 )

(2-82)

Again, both finite difference approximation and quasilinearization are applied to both equations in the same way they were used in Case 1. The simplifying coefficients are defined as: CII = (Pemal.::lZ2rl C12 = -2' Cll + (lI.::lZ) - 2 - blCCl

CI3

=

CII - (1/.::lZ)

. Cl4 = -bl - CT(I) . CCI 2 l C21 = (Peha2 .::lZ

(2-83)

r

C22

=

C23

= C21 -

C24 =

where CCI =

-2' C21 -b~·

C~

+

(lI.::lZ) - CC2 - C,(l) . CCI . b2IT,(l)

(If.::lZ)

C'(l) . CCI(CC2 + I)

C,(I)

e-E1Rr,ToT.(/)

CC2 = ElRcToT,(l) •

The rest of the plrOcedure is again quite similar to that of Case 1. A word of caution is absolutely necessary before leaving. the subject of boundary-value ODEs: the foregoing algorithms do work with reaction kinetics of reasonable nonlinearity. However, if the nonlinearity exceeds certain limits, the computer program might not work. The only possible remedy I know of in such cases is to try to decrease the nonlinearity of the kinetic equations by juggling parameters.

2.7 ALGORITHMS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS PDEs describing NI]'.;~.F reactors are of the parabolic type, i.e., d'f

-

dZ

= A

.

a r., 2

iJR-

I aY

+ -

R aR

+ C

(2-84)

•

38 CATALYTIC REACTOR DESIGN

where Z is the axial coordinate, R the radial coordinate, and Y a dependent variable such as concentration or temperature. A conventional approach to the numerical solution of PDEs is to replace the spatial derivatives in both the axial and radial coordinates with finite difference approximations. The resulting system of nonlinear algebraic equations is then solved by an iterative procedure such as the Newton-Raphson algorithm. Typically, the equations are coupled tridiagonal systems, which can be solved in a circular iterative fashion by Thomas' method. A generalized program for PDEs based on this api proach has been reported by M. G. Zellner et. al. Alternatively, the radial derivatives of the PDEs can be replaced with finite difference approximations, while the axial derivatives are retained. This approach leads to a system of coupled non linear ODEs consisting of subsets of ODEs for each PDE. The ODEs are initial-value in the axial position if only first-order axial flm" terms appear in the original PDEs. The entire system of oeEs is then integrated numerically with respect to axial position, starting at initial conditions corresponding to those at the inlet to the reactor. The dependent variables Y, i.e., the reactor temperature and concentrations, can be visualized as evolving along lines of constant radial position. Therefore this approach is often termed the method of lilles, since the ODEs contain difference terms. The method has also been termed the state-7)G1iable algorithm. The method of lines will now be illustrated in detail as applied to the solution of the PDEs that model the XINAF reactor. As we will see later in Chap. 5, mese PDEs are of the form: aCT

az aT

T

az

•

q

1 _a-_c-,;., + I aCT =-2 aR R aR

(2-85)

1 aT + ---'R aR

(2-86)

1

T

=-

and have the following auxiliary conditions: Bed inlet

(2-87)

CT(R, 0) = T,(R, 0) = I

°

(2-88)

Bed centerIine

aCr(O, Z)/aR = aT,(O, Z)/aR =

Bed wall

aC,(RT,Z)/aR = 0

(2-89)

aTr(RnZ)/aR = (BiUT, - T(RT,Z»

(2-90)

.

The radial derivatives in Eqs. (2-85) and (2-86) are replaced bv standard second-order central difference approximations:

ac, +-.-....: 1

R

aR

-

+ C ri-l - ~~----~----~ + C..tri+l - 9C· .... 1"/

I1R~

il1R

211R

(2-91 )

•

METHODS OF NUMERICAl. COMPUTATION 39

- 2Tri aR 2

+ T ri - I

+

1 T ri +1 - T ri - I iaR . 2aR

(2-92)

These approximations are written for the ith radial grid point, where r = iaR, i = 0, 1,2, ... , N" and aR is the radial increment and is assumed constant in the subsequent development. If Eqs. (2-91) and (2-92) are substituted in Eqs. (2-85) and (2-86), two ODEs result for each radial grid point:

dC ri dZ dT" - 1 . dZ Peh

-2Cri + Cri - 1 aR 2 Tn+1 -

2Tri

+

+ T ri _ , +

1 . Cri + 1 - Cri - 1 iAR 2aR

, ri 1 ri 1 . T + - T iaR 2/lR

_

f

R I

I

+f R 2

I

(2-93)

(2-94)

where i = 0, I, 2, ... , N r Similarly Eqs. (2-87) to (2-90) are approximated as: (2-95) (2-96) (2-97) (2-98)

\

Equations (2-95) to (2-98) are used to eliminate the fictitious L-0undary points Cr_1> Tr_ h CrNr + I, and TrNr+ 1 from Eqs. (2-93) and (2-94). The initial conditions for Eqs. (2-93) and (2-94)

C

Tl

=

I

T rz = 1

correspond to the inlet conditions for the reactor, Eq. (2-87). Equations (2-91) to (2-100) constitute an initial-value problem in ODEs with, in this case, a total of 2(Nr + 1) ODEs (two equations for each of the radial grid points i = 0, 1, 2, ... , N r). In other words, the computational problem is now to integrate the two derivative vectors dCri(Z)1 dZ and dT ri(Z)/dZ that define the concentration and temperature distributions within the reactor. Any numerical integration algorithm can in principle be used to solve this initial-value problem. This book will use the fourth-order Rung~-Kutta integration algorithm, which was develop~d in Sec. ~.3 for initial-value ODEs. The implementation of this method presents two problems: .first, at the centerline the expression (lIiAR)(C ri + I - C'i- d/2t1R becomes 010, or

•

40 CATALYTIC REACTOR DESIGN

indeterminate; second, some of the subscripts become zero, something that the computer does not accept. Consequently, measures must be taken to solve these two problems. In order to avoid the first problem, the PDEs for i = 0 must be modified. The terms (lIR)(iJC/iJR) and (lIR)(iJTjiJR) in Eqs. (2-85) and (2-86) can be modified by applying l'Hospital's rule to them (i.e., differentiation of the numerator and denominator with respect to R). lim R~O

1

ac,

R

aR

1

" iJT, a-T, iJR = -aR~2

R

-

(2-101) (2-102)

Substituti'ln of the rt:lationships in Eqs. (2-85) and (2-86) gives:

ac,

az

=

aT, =

az

" 2 a-c, 2

1 . Pem

•

f,(R,)

(2-103)

f

(2-104)

aR

" 1 . 2 a-T, Pe" aR 2

+

2(R I )

The finite difference analog for these new equations are:

2 . C'i+1 - 2C ri + C,i-I _ f'R )

oCr =

az

Pem

iJTr _

2 . T'i+' -2T'i + T,i-, Pe" . !1R2 +

az -

..

!1R2

1\

f

(2-105)

I

(2-106)

2(Rd

Equations (2-95) and (2-96) can be substituted into Eqs. (2-lO5) and (2-106), respectively, to obtain: ..

ac,

az

=

aT, =

az

4 . C,i+ I

Pem

-

!1R2

CT ;

_

4 . T'i+l - TTi + Peh !1R2

f·

(R ) I

f

=

0)

(2-107)

(for i

=

0)

(2-108)

I

(R ) 2

(for i

I

Thus the first problem is resolved. The second problem can also be resolved by setting i = 1, 2, 3,4, 5,6 instead of i = 0, 1, 2, 3,4, 5. Then NT = 6. In this way i will never become zero or negative. In order to modify Eqs. (2-93) and (2-94) for this change, the fJra~tions lIi!1R will be replaced by l/(i - 1)!1R. The set of equations to be integrated by FUNCTION RUNGE 'can then be set up as shown in Fig. 2-11. Under certain conditions computer programs that integrate partial differential equations by the foregoing method can develop serious instability problems. In its mild form, instability becomes evident when temperature or concentration profiles begin to swing. Of course, in its

•

METHODS OF NUMERICAl

41

C

•••• CALL ON ThE FOURTH-ORDER RUNGE-KUTTA FUNCTION •••• 2 K=RUNGE (12,Y,F,Z,H) . C IF Kc1 COMPUTE DERIVATIVE VALUES IF{K.NE.1) GO TO J DO 10 I= 1, NR IF{I.NE.1)GO TO 11 P (1.1 ) .. (4./PEM)· (CR (1+1 )-CR Cl) )/(DEL'l'AR**2)- (DP*R1(1 )/(CO"U» J:o'CI .2 )&(4 ./PEH)· (TR (1+1) -'l'R (I) )/(DELTAR"2)+",HR(I )"uf 1 /( CP"U.RHOC*'l·O) GO TO 10 11 1F(I.EQ.NR) GO TO 12 WALL=(TR(I+1}-TR(I-l»/(2.·D~LTAR)

F (I .1 )=( l./FEM)" ( (CR(I+1) -2. "CR (I )+CR( 1-1) )/(DE.L'l"R"2) 1 ... (1. I{ (I-1) "DBL'l'AR) ). (CR(I+1 )-CR( 1-1) )/( 2. "ufU.'l'AR) ) -(DP*Rl(I)/(co*u» 2 F(I.2)=(1./PEH)·«TR{1+1)-2."TR(I}+TR{I-l»/{DELTAR**~J

1 + ( 1. 1« 1-1 ) "DELl'AR )) *WALL) 2 +(DHR(I)*DF/(CP*U*RriOC*TO» GO TO 10 12 WALL=BIW*(TW-TR(NR» 1

F(1.1)=(2./PE~)*(CR(I-1)-CR(I»/(DBLTAR*·2)

-(DP*Rl(I)/{Co·u» F(1,2)=(2.IPEH)*(TR(I-l)-TR(I»/(DELTAR··2) 1 +(l./{{I-l)*DELTAR)*WALL) 2 +(DHR(I)*DF/{CP*U*RhOC.TO» 10 CONTINUE GO TO 2 •.•• IF TR EXCEEDS TR";AX, TERb:INATE INTl:.GRAHOh •••• C ) IF(TR{l).LE.TR~~X) GO TO 4 GO TO 5 (5 is a WRITE s'tatement)

FIG. 2-11 The essentials of the computer progr am to solve partial differential • equations.

\.

more serious form, it makes the program unworkable. Instability is usually caused by too large a I:!.ZII:!.R2 ratio. For linear PDEs there is a classical stability condition that requires (I:!.ZII:!.R2)(l/Pe) to be -

-

~

-- 0.6

-

'-'

-~ 0.4

-4

.o

'-'

.:: 0.2

1 0.2

2 0.4

0.6

0.8

1.0

Fraction poisoned a

FIG. 3-4 Selectivity of catalyst pore poisoning. (Reproduced with the pe, lIIission of the Academic Press and Dr. Ahlbol n Wheeler. IS)

ied and known before any reactor design can begin. There is no amount of a priori knmdedge that can make up for the lack of such information.

3.2 CATALYTIC REACTION RATE MODELS There are two well-known models for:! catalytic reaction rate. The first one is that of Langmuir-HinslzelwoodlHuugen-Watsoll (L-HIH-W).17 The other is tile power laU'. The L-H/H-W model gives for a reaction of the type A + B

,-

-

) D

(3-20)

the reaction rate: r

k K 1K 2C ....C S = ----------~~~~----~ [1 + K1C ... + K 2C S + K 3CDf

(3-21)

where CA, CB, and Cv are the molar concentrations of species A, B, and D, respectively. The constants KJ> K 2 , and K3 are adsorption equilibrium constants and k is the reaction rate constant. In Eq. (3-21) there are too many coefficients that are difficult to determine. For the same reaction [Eq. (3-20)] the power law is: (3-22)

The power law has only one temperature-dependent coefficienrk. One usually operates at a narrow temperature range of interest. It is at best \'ery difficult to prove \~hether the one or the other model is closer to reality. I n general it does not make much sense to retain a complicated model if the existing data can be made to fit the simpler power law just as well.

•

•

56 CATALYTIC REACTOR DESIGN

•

r

FIG. 3-5 Normal reaction.

The factor kin Eq. (3-22) is called the reaction rate constant and by the well-known Arrhenius equation:

i~

given

k = A exp( - EIRT)

where E = activation energy, kJ/kmol (see Fig. 3-1) . R = gas constant = 8.314 kJ/(kmol . K) T = absolute temperature, K A = frequency factor or preexponential factor (the dimension of this parameter depends on the order of the reaction)

,.

,,

,

,

A catalyst affects both the activation energy and the frequency factor of the reaction rate constant. :\[ezaki and Kittrell have shown that it is possible to distinguish between various reaction models by using linear regression techniques. From the viewpoint of the reactor designer, there can be no justification in using anything but the power law unless the other model has been significantly differentiated by a better curve fit. In Eq. (3-22) r is the reaction rate, which is usually expressed as a der;'-ative such as r = -dC...ldt, which is the consumption of the species A during the differential time dt. It also may be expressed as the rate of formation of a species, such as r = dCr/dt. In this case the sign in front of the derivative is positive. The reaction rate constant k is related to temperature by the Arrhenius equation [Eq. (3-22a)].

Reaction Order The order of reaction is the value of the exponent of the concentration term of a rate expression., For example, scme reaction rates are independent of any concentration. Then Eq. (3-22) can be simplified to •

r = k

(3-23 )

Such reactions are termed zero-order, since Eq. (3-23) can also be written as

r = kC~C~C~ = k(l)(l)(l)

•

CATALYTIC KINETIC MOORS 57

If a reaction rate is proportional to a single concentration (el III and is thus described by an equation of the form r = kC A

(3-24)

the reaction is said to be first-order with respect to reactant A. Some reactions may be second-order: r = k C~

(3-26)

There are also reactions of fractional order, such as the of toluene to benzene and methane:

h~'drodealkylation

(3-27)

•

r=

o.s

k Cto" C H2

(3-28)

The reaction order has no relationship to stoichiometry. The only time the order coincides with stoichiometry is when the reaction occurs "as . " wntten. Not mal and Autocatalytic Reactions

A reaction is called normal if its rate falls off with conversion

XA

(Fig.

3-5). There are some reactions for which the rate may go up with increasing conversion XA and then fall off after having reached a maximum. 18 This is typical of autocatalytic reactioni (Fig. 3-6). Carbon monoxide oxidation and many aromatic oxidation reactions are autocatalytic:

co

+

'h

O

2

Pt, Pd, Rd, Ru.

CO

2

(3-29)

In an autocatalytic reaction, the rate at the start is low because little product is present. It increases to a maximum as more product is formed MOl I

,

I

r

I I I

•

I I I

,XA

FIG. 3-6

Autocatalytic reaction.

•

•

CATALYTIC RE.A.CTOR DESIGN •

and then drops again to a low value as reactant is consumed. The rate expression is usually of the form -

k

rA =

c;. . C;;rodud

(3-30)

Here at the right-hand side of the maximum the order is negative.

3.;i DETERMINATION OF REACTION RATE The reactor designer needs an accurate expression for the catalytic reaction rate which has been determined for the catalyst to be used. Although in most cases someone else will make this determination. it is important that the reactor designer knows how it was done. The expression needed is of the general type: Rate = r = where A,B C

f

(CA,CB,Cn , T,A,EL

(3-31)

reactants 3 molar concentration, kmoVm D = reaction product T = absolute temperature, K A = frequency factor E = activation energy, kJ/kmol = =

•

This expression represents the intrinsic reaction rate. There are a number of methods for determining catalytic reaction rates. Some are batch, some are semibatch, and some are continuous. The Differentk! Flow Reactor This method of measuring the reaction rate uses a long, thin tube which has been packed with a small amount of catalyst. It contains thermocouples to measure the temperature before and after the catalyst bed. The tube must be kept absolutely isothermal by a liquid at constant temperature surrounding the tube. High heat transfer is essential for isothermality (Fig. 3-7). A series of experiments is carried out in this tube at absolutely isothermal conditions while trying to keep the conversion Iow, so that it may be considered differential conversion. The concentration CA of a key reactant is analyzed before and after conversion, and actual moi:'r con-+>====~~c=======-

CA.· c~

Calal ysl

__ CA Ca ~

Co T

T. FIG. 3·7 Schematic representation of a differential flow reactor.

•

•

CATALYTIC KINETIC MODELS 59

Slope is rate rA

I

I I I I

FIG. 3-8 Reaction rate as slope of conversion vs. space time curve.

centrations at reaction conditions are calculated. From tions, one calculates conversion XA: or

x ... =

the~e

1 - (CA/CAD)

I + (eAC ...IC AU )

concentra-

(3-32)

where C...o is the molar concentration of reactant A at the inlet to the reactor and eA is the fractional change in volume of the system during complete conversion of reactant A and is defined by CA

\.

•

=

v,

x .... = 1 -

l"x.\ =0

(3-33)

In Eq. (3-33) v' is the volume of the reactants and V;.,~ I means the value of V' when x ... = L Now we can plot x.'\ versus V/F, where F is the volumetric feed rate in cubic meters per second. The V/F ratio is de filled by V. - = T = space time (3-34) F We can graphically obtain the derivatives of the curve in Eg. 3-8. Each derivative, or slope at a tangent, is the reaction rate rA at the corresponding space time and concentration. We can plot these rate data versus C .. (Fig. 3-9). If the points fall on a straight line, we can say that rA is directly proportional to CA, or first-order with respect to CA' If the points f;ln on a straight line but the straight line is parallel to the abscissa, we can tell that TA is independent of CA, or zero-order. If the points fall on a curve, then we may have a case of second-order or fractional-order reaction. The shortest way of determining the reaction order a in such a case is to consider, as in Fig. 3-10, r A = k C~

In r A = In k

(3-35)

+ a In

CA

(3-36)

• 60 CATALYTIC REACTOR DESIGN

•

Zero order

- - Ifsl order CA

FIG. 3-9 Detellllination of reaction order• •

and to plot In TA versus In GA' The sbpe of the line represents the reaction order 0:. Of course this method will work for fractional-order reactions and for second-order reactions of the type TA = kG! but will not work for second-order reactions of the type (3-25)

Here we can write: (3-37)

and plot In TA against In GAGB as shown in Fig. 3-11. If the reaction is of the type of Eq. (3-25), the plot will be a straight line. The main drawback of the differential reactor is that it must work with small conversions, and consequently extremely good analyses of inlet and outlet streams are required. These are not always available .. One of the best procedures commonly used in s~arching for a rate equation is differential analysis, which uses a differential reactor and is more convenient with complex rate expressions. The design equation for plug-flow reactors (3-38)

can be rearranged to dx A dV/FAo

(3-39)

· One makes a series of runs in the packed bed of Fig. 3-7, using fixed feed concentration GAO byt varying FAO such that a wide range of values for VIFAO and X A is obtained. · One plots

XA

vs. VIF AO for all runs (Fig. 3-8).

· One fits the best curve to the XA vs. VIFAO data, making it pass through the origin.

•

•

CATALYTIC KINETIC MODElS 61

· The rate of reaction at any value of XA is the slope of the curve at that value. For each XA value one can find a rate value and a CA value. · Now all one has w do is to plot 3-9 and 3-10, respectively.

TA

vs. CA or In A vs. In CA as in Figs.

Differential reactors are discussed in detail by Levenspiel,18 Smith,7 li and Hougen and Watson.

The Integral Flow Reactor Integral

Analysi~

The same packed laboratory reactor shown in Fig. 3-7 can be used as an integral flow reactor. However, more catalyst can be packed into it, so that 20% conversion is obtained. Again, the reactor is operated under isothermal conditions. Another commonly used procedure in searching for a rate equation is integral anal)'5is which uses the integral flow reactor. A specific mechanism with its corresponding rate equation is put to the test by integrating the rate equatiun for the reactor flow conditions. For this: · A series of runs is made in a packed bed, with fixed feed concentration CA!) but with varying catalyst volume \f and/or feed rate FAo, in such a manner that a wide range of V/FAil vs. x" data are obtained. · A rate equation is selected for testing and used in integrating the plugflow performance equation to give V

--

x .•

dxA

(3-40) •

\, ,

· For each experimental run, the left and the right sides of Eq. (3-38) are evaluated numerically. · The results are plotted one against the other and tested for linearity. Integral analysis provides a straightforward and fast method for test-

•

a slope 1

•

.l"k h FIG. 3-10

CA

Detexulination of reaction order and reaction rate constant k for

rates depending on a single cC)Dcentration CA'

•

62 CATALYTIC REACTOR DESIGN

1-

>

.4, CA CB

Dtte.mination of reaction order and reaction rate constant k for rates depending on two concentrations, CA and CB'

FIG. 3-11

,

..

ing simple rat~ expressions. However, with increasing complexity, the integrated forms of the rate expressions be:ome too messy. In the tubular packed integral reactor, ideally the gas flow, as represented by either the feed rate FA or the Reynolds number, should be constant and the reactor length should be varied. However, this is frequently not done, because it is inconvenient to re pack a laboratory re19 actor for every data point. Instead, a fixed length of packed bed is used while attempting to maintain the pressure, the temperature, and the reactants ratio constant and varyin~ only the flow rate. After such a series of experiments is completed, further series are run by varying 7 the flow rate at other levels of constant reactant ratio. This type of reactor must be kept absolutely isothermal or the data obtained from it will be worthless. According to J. J. Carberry,5.19 a tube diameter equivalent to 5 to 6 particle diameters are the maximum to have in the tube to be certain that isothermality is maintained. However, this conflict" with the general packed-bed rule that a minimum of8 to 15 particle diameters is necessary to minimize wall effects. This means that no matter what is done some error will be made. The Continuous Stirred-Tank Reactor (CSTR)

The CSTR can be used in the laboratory as a tool for reaction rate determination. It is normally used for homogeneous reactions. If it is perfectly mixed, the reaction rate can be obtained b~- materials balance (Fig. 3-12): •

,

_ rA =

(3-41 )

In order to adapt the CSTR to heterogeneous catalysts, Carberry developed his basket-type experimental CSTR.20 In this reactor, catalyst

•

CATALYTIC KINETIC MODRS 63

pellets are placed in wire baskets, which rotate around a vertical axis in a cylinder full of the reactant gases, The performance equation becomes: F

x"

F "I

- rA

-= ••

(3-42)

from which (3-43)

Here, each run gives directly a \'alue for the rate at the composition of the exit fluid. There is an alternate design for a similar reactor with a fixed catalyst and recirculated gases. However this alternate design costs considerably more ~han the Carberry• reactor.

The Recycle Reactor The recycle reactor can also be used in the laboratory as a tool for reaction rate determination. It uses a plug-flow reactor and converts it to a backmixed reactor by adding a recycle (Fig. 3-13). Again a specific kmetic equation is tested bj' inserting it into the performance equation for recycle reactors -

v

= (R

+

1)

FAO

Rx:\(R+Il

(3-44)

- rA •

.

The equation is integrated and a plot of the left-hand and right-hand :>ides of Eq. (3-44) then tests the linearity. At high recycle ratio R the reactor approaches a CSTR. The most critical part of the reactor is the pump. According to J. J. Carberry,2! the best pump is the all-glass pump developed by M. Boudart. The CSTR and the recycle reactor do not give the best yields, but at this stage of learning, no one cares about yields it is the rate constants that are desired. Laboratory reactors will be discussed in greater detail in Chap. 10 . •

•

- - -

- --

..

FIG. 3-12 Schematic representation of a continuous stirred-tank reactor (CSTR).

•

64 CATALYTIC REACTOR DESIGN x •

q

~25

fiG. 3-13 Schematic

of a Iecycle

3.4

EFFECTS OF PORE STRU~TURE AND TPANSPORT PHENOMENA ON REACTION KINETICS

,

--

,

Most solid catalysts are porous pellets, some having internal surface areas 2 as large as 300 m /g or more. This sort of porous structure strongly affects the kinetics of chemical reactions, not only with respect to reaction rate but also by favoring certain products over others. We say that pore structure affects the selectivity of the reaction. A typical heterogeneous catalyst pellet is a small solid (spherical, cylindrical, or irregular in shape) containing a multitude of pores. In Fig. 3-14 such a pellet is shown with exagerated pores. Gases or liquids which surround the pellet diffuse into the pores, where they react on catalyt!cally active sites; the products of the reaction diffuse back to the bulk phase and are transported by the bulk phase. Because the actual catalytic reaction takes place 011 the active sites of the catalyst surface, which are mostly inside the pores, both the temperature and the concentratio~s may vary between the bulk phase and the pores. All properties related to the space inside }he pellets are called intraphase properties, while all properties related to the space between pellets are called interphase properties. The Global Rate of Reaction

The global rate of reaction is the rate that can be measured. It includes also the mass transfer. The simple case is a nonporous solid catalyst plate, as shown in Fig. 3-15. A reactant of initial concentration Co may approach the solid. The solid is assumed to be covered by a fictitious film E)f thickness S, which represents a sort of resistance to the reaction on the surface. The reaqant will penetrate the film and will be adsorbed on the surface; its surface concentration will then be Cs. The rate of • surface reaction will be: _ (3-45)

•

• CATAlYTIC I(lNETIC MODElS 65

where k = reaction rate constant (units vary according to a) a = reaction order . kg = mass-transfer coefficient, m1s a = the ratio of surface available per unit volume of reactor space, 2 3 m /m Equation (3-45) simply states that the number of moles of reactant consumed by the surface reaction is equal ~o the number of moles transferred through the film to the catalyst surface. The coefficient kg is a phenomenological coefficient, which is also related to the diffusivity D of the reactant by: (3-46) With increasing flow or with increasir,g Reynolds number Re:

•

5 )0 Let us solve Eq. (3-45) for Cs. If a C, can be extracted from it:

1, Eq. (3-45) is first-order. Then

=

(3-47) Substituting C, from Eq. (3-47) into the rate equation

~ •

kCo

=

I

,

~ = kC,

+

(k/kga)

gives: (3-48)

•

A

B

FIG. 3-14 Schematic representation of a porous catalyst pellet.

-

'---__ • Co

-------------.-----Cs

8 Film

Ihi~kness

~ Solid plote FIG. 3-15 Schematic representation of a nonporous catalyst.

•

•

66 CATALYTIC REACTOR DESIGN

In Eq. (3-48), the expression kl[ I + (klkga)] is the experimental value of the rate constant. Thus. this' whole expression, not just k, is the rate constant. At low temperatures kga > k; (klkga) ~ 0; ~R = kC o and kexpcrimcntai

==

kjflt.rinsk

At higher temperatures kga < k; kI[ 1 + (klk"a)] (k • kga/k) = kga; 5R = kgG Co and

=

(k . kga)/(k[!a + k)

)

To express the relationship between the experimentally measurable global reaction and the intrinsic reaction rate we will define an external effectiveness factor

5 21 1lx • ;

(' .K --

~\'

k CIJ

(3-49)

where I

"

(3-50)

The dimension of k for first-order reaction is seconds-I. Here the ratio klkga is dimensionless and is called the Damkohler number (Da). It expresses the relationship of the chemical reaction to the mass transfer. The external effectiveness factor 11x should not be confused with the intraphase effectiveness factor TJ. If 11x is plotted against the Damkohler numbel' ;)0 a double logarithmic chart, the curve on Fig. 3-16 is obtained. The factor 11x can also be computed in cases of reaction orders different from 1 and plotted on the same Fig. 3-16. We see that the diffusion intrusion is the more serious, 22 the higher the order of reaction. If we investigate the Arrhenius relationship of the experimentally measured reaction rate (3-51 )

by plotting In kx against IIT in Fig. 3-17, we see that at low temperatures (high liT), the slope of the curve is - ElRc, where E is the activation energy of the catalytic intrinsic reaction. At intermediate T we have a mixed phase with a flatter curve. At very high temperatures (low liT) most of the reaction will o€cur in the bulk gas phase rather than on the surface, and we will be measuring the intrinsic reaction rate in bulk gas phase. Then E will be the activation energy of the thermal reaction and kexperimental = kintrinsic. thermal' This means that if the measured activation en-

•

CATAlYllC KINETIC MODELS 67 •

10 •

0.Q1

0.1

10

1.0

Do

Plot of the isothellllal extelllal catalytic effectiveness factor for reaction order a as a function of the Damkohler number Da. (From Cassiere and Carben'Y,22 by pel mission from the American Society for Engineering E iucation.)

FIG. 3-16

erg)' E is found is involved.

to

be low in studying a reaction, considerable diffusion k

•,

•,

(3-52)

Equation (3-52) works with all reaction orders. In a pure diffusion region, all reactions appear to be first-order. By obtaining 1), from observable rate data, it is possible to calculate the real order of the reaction. Let us consider again a flat catalyst plate (Fig. 3-18). However, let this be a porous plate instead of the nonporous one of Fig. 3-15. With the porous plate wc' en study the intra phase aspect of the heterogeneous reaction rate. In order to correlate the observable reaction rate constant kexperimemal v.;ith kimrinsio we define an effectiveness facto!- 1)

1)

= ----

k.C.

which under isothermal conditions tends

-

lli

= ----~

C,

(3-53)

to

1.0

(3-54)

Here dl is a differentia!' distance and L is a characteristic dimt:nsion of the particle.: • volume of particle - 1 L = - external surface area a

.

• 68 CATAlYTIC REACTOR DESIGN ___ k, =kjnlrinsit •

Thermal reaction .l., k ~

Bulk phase

I

I I

t

I I I I

I

I

I I

Mixed phose

Intrinsic (catalytic!

I I

I

I

I

I

I

Slope:-t

I t

I I

liT

fiG. 3-17 Arrhenius plot for the experimentally measured reaction rate constant

k. in a nonporous catalyst.

Thus

At nonisothermal conditions 1]nonisothennal

~ 1.0

The rate of diffusion of the reactant equals that of its consumption by the reaction; therefore we can write d

2

L'

D df = kCU

(3-55)

For first-order reaction, a = 1 and we have: -'

,

D

tf~ dl

= kC

(3-56)

..

We may use the dimensionless forms of some quantities:

f

1

= C

z =L

Cs

(3-57)

With these quantities, Eq. (3-56) becomes or

-

(3-58)

The value of the quanti,ty in parentheses on the right-hand side of Eq. 23 (3-58) is equated to qJ2, qJ ~eing the well-known Thiele modulus. Analytical solution of Eq. (3-58) by a complicated and lengthy procedure gives: 1];

=

tanh qJ

qJ

(3-59)

•

CATALYTIC KINETIC MODElS 69

where tanh 'I' is the hyperbolic tangent of quantity 'I' is given by the relationship: orat:JrY reactors deviate from this fixed ,"alue. Thus, axial dispersion in fixed beds can be completely neglected in reactor design. It will be seen in Chaps. 5 and 7 that axial dispersion is completely omitted from design calculations. The two types of expressions of axial dispersion of ma~s are related lo ll as follows . : (4-9)

where n is the number of equivalent CSTRs and ZT is the reactor bed lerigth in meters .

•

"

Interphase Mass Transfer The concentration of the key reactant on the external catalyst surface in a fixed-bed gas reactor is related to the concentration of the same reactant in the bulk gas phase by the equation: kga (C - C.)

= 1l1k1C; = ~Ii.

(3-45)

where kf{ = interphase mass-transfer coefficient. m/s a = ratio of external particle-surface-area/particle-volume, m-I C, = concentrativn of key reactant on exterior particle surface, kmol!m~ ••

111 =

effectiveness factor of reaction I

kl = reaction rate constant of reaction I J

- reaction order

• 84 CATAlYTIC· REACTOR

The left-hand side ofEq. (3-45) expresses the interphase mass transfer and the right-hand side expresses the depletion of the key reactant by the global reaction gt as observed from the catalyst surface. The left- . hand side of Eq. (3-45) is the product of a driving force, C - Cs> by a constant value kga.Since the constant a can be easily measured for any catalyst, we will be mainly interested in kg. Some aspects of the relationship represented by Eq. (3-45) have already been discussed in Sec. 3.4, including the derivation of the Damkohler number from it. In Sec. 5.3 we will use this relationship in the design of the NINAF reactor in the form C

=

C,

[(1}1

k/kga)

+

1]

=

C, [Dam

+

(4-10)

I]

This is the desired relationship between the bulk gas concentration C and the c::mcentration C, of the key reactant on the catalyst surface. This relationship is further improved by making it dimensionless. Let us divide both sides of Eq. (4-10) by the inlet concentration Co and let us set: C,Ir:o = Cn = reduced concenn-ation on catalyst surface CICo = C r = reduced concentration in the bulk gas phase

Thus, we can write: (4-11 )

\

where Da., = Damkohler number for mass. Dam relates the effective reaction rate to the interphase mass transfer. In the following we will be mainly interested in the interphase mass transfer coefficient kl{ and in its calculation. Ergun 12 and CarberryI3.14 reviewed past studies on kg and Carberry arrived at the following correlation, based on a "developing boundary layer concept":

JD where

JD JH

= =

= (kglu)e Sc~' =

J factor J factor

1.15 Re-I,..

=

JH

= (hlpcpu)

Pr'"

(4-12)

for mass for heat

u = superficial velocity, m1s e = void fraction, dimensionless Sc = Schmidt number = • = viscosity, Pa . s Dim =

~pDI""

•

dimensionless

molecular diffusion coefficient for multiple component mixtures, m 2/s

p =

gas density, kg/m

3

diffusing species in

•

•

TRANSPORT PHENOMENA IN FIXED-BED GAS RE,J\CTORS 85

Re

=

Reynolds number

=

dpG/~,

dimensionless

G = mass velocity of gas, kg/(m2 . s)

dp = catalyst particle diameter, m h = interphase heat transfer coefficient, kj/(m cp

=

heat capacity, kJ/(kg . K)

Pr

=

Prandtl number

=

Cp~Ag,

2

•

s . K)

dimensionless

Ag = thermal conductivity of gas, kj/(s' m . K)

Equation (4-12) shows that the mass-transfer coefficient R.I( correlates not only with the flow characteristics of the gas, as expressed by the Reynolds number, but also with the heat-transfer coefficient h. The applicability of this correlation has some limits, however: 1. It is applicable below the Reynolds number, corresponding to turbulent boundary-layer development, which probably has a value of , "C>, "" 'I w"?','e / ' --'; ',':;::;1 / ' " ,,' li' ", n-" se\'eral hundred 13 ~.,

•

I

""'ul~-:""'~'

\\'~""_'-,j",,,\

',_

\"";:)

''''",

,-

'~

'.

2. It is applicable in the region where radiant heat transfer is negligible.

v

Radiant heat transfer would at least do away with the correlation between mass and heat transfer. The effect of radiant heat will be discussed further below.

Interphase Heat Tt ansport

Heat transport phenomena in the void spaces of the catalyst bed include the radial dispersion of heat, the transfer of heat between the catalyst surface and the bulk gas, and radiant heat transfer in the bed.

Radial Dispersion of Heat

The general backmixing phenomenon in fixed-bed reactors, which we already discussed in relation to mass dispersion, also causes some radial dispersion of heat. This heat dispersion is of no practical consequence in isothermal and adiabatic gas reactors. However, it must be definitely taken into account in sidewall-cooled NINAF reactors. The radial dispe~~~()n of heat is eA-pressed by a radial Peelet number for heat flow: •

(4-13)

where Kr is the radial dispersion coefficient for heat in square meters per second. The value of Peh lies between 5 and 10. In Chap. 5 we used a value of Peh = 7.

• 86 CATALYTIC REACTOR DESIGN

Interphase Heat Transfer The temperature of a key reactant on the external catalyst surface in a fixed-bed gas reactor is related to the temperature in the bulk gas phase by the equation: ha (Ts - T) = (- tlH)

where

(4-14)

9{

h = interphase heat tr~nsfer coefficient, kJ/(m ( - JlH) - reaction heat, kJ Ikmol

2

•

T

=

temperature of the gas in the bulk phase, K

T,

=

temperature on catalyst surface, K

s . K)

a = surface area-to-volume ratio for the catalyst, m-I

Equation (4-14) can be rearranged into T, = T

+

[(-tlH)~Klha]

(4-15)

which can be made dimensionles" by dividing both sides by .. he inlet temperature Tn: (TjTfI ) = (TlTn)

+ [(-tlHYRIT"ha]

(4-16)

If we set (TJTfI) equal to T", the reduced surfare temperature of catalyst,

and (T1To) equal we can wr~te

to

TT' the reduced temperuture in the bulk gas phase,

•

.

(- tlH)~R •

,

•

T"

=

Tr +

ahTo

( 4-17)

The term (- tlH)Yl..lahTo is another dimensionless ~roup relating the heat produced by reaction (numerator) to the interphase heat transfer (denominator) and is called the Damkiihla 1wmberfor heat: ( 4-18)

-

Equation (4-17) will be used in the NINAF reactor design problem in Chap.5. The interphase heat transfer coefficient can be calculated from the correlation of theJD andJH factors in Eq. (4-12). As mentioned abm'e, Eq. (4-12) is not applicable to turbulent flow conditions or to high tem- . l5 peratures at which radiant heat transfer becomes significant. Martin • used a different approach to obtain h. He had observed that for low Peclet numbers most or the experimentally obtained particle-to-fluid heat transfer coefficients in packed beds were several orders of magnitude below the values predicted for a single spherical panicle in cross flO\,'. Martin attempted to establish a relationship between the heat trans-

•

TRANSPORT PHENOMENA IN FIXED-BED GAS REACTORS 87

fer coefficient of the single particle and that of the packed bed and to calculate the latter from the fprmer. In the range Pe > 100 the N usselt number of a packed bed can be calculated from the following equations: Nu

=

[1

+ 1.5(1-

E)] . r\u"

(4-19)

(4-20)

NUl' = 2 + F· (Pe/E)'" IY',·

where Nl1 is the !\usselt number for the packed bed = lld/A and Nul' is the Nusselt number for a single sphere and equals ht/VA, A being thermal conductivity. In the laminar region the factor F has values between 0.6 and 0.664 as found by different workers. For intermediate turbulence l6 le\'els. F can be calculated from Gnielinski's correlation : F = 0.664

I +

0.0557 (Re/E )11:1 Pr"1 I + 2.44(Pr"I - 1)(Re/Er"I (PI'

> 0.6)

(4-21 )

Here hp is the interphase heat transfer coefficient for a single sphere and h :s thp interphase heat transfer coefficient for a fixed bed in kilojoules per square meter per second per keh'in. Calculations with the above equations show that for the same superficial velocity and for intermediate Reynolds numbers. the heat transfer coefficient of a p?cked bed of spheres is about 3 times that of a single sphere. Equatic0 O.1I0001~2

2.016

11.314 a.0831"

n.000026

50. 5.73

D.le

4'3'31".0 E+06 6 0.003

3.142

16.043

FIG. 5-11 The RDMOTI computer program for the design of an adiabatic fixed-bed reactor for the hydrodealkylation of toluene to benzene (continued) . •

•

122

TOlUE~E

HYOPOO£AlKYLATION TO

XO=O.OOOO TO= 873.2 K FYO= .02512 KHOt,S .

RSTAR=

.0831'

THAX=

BAQ.H··~IKHOl.K

OH= ~9974.0 KJ'KMOL . E= 1~~11~.0 KJ'KHOl HUI= .0000200 HU2= .~O~0?24 EPS= .3~O OP = .00300 M MW1= ~?141 HW2= 78.114 lBEll= 6

REACTOP LENGTH. M .500 1.000 1.500

2.000 2.31!0 END OF BED. BEU "0= 1

'306.2 I( RC = 6.314 KJ'KMOl.1( P= 50.0 BAR A~S H= .050 AC=3.142 ... ··2 A= .5730000E+01 MU3= .0000182 MU4= .0000260 PAS 2.016

"PFO B£O CONVERSION .040 .063 .131 .16S .220

QUfNCH =

~. 3~O

3.1100 1t.300 '.800 '5.300 '.600 6.250 £NO OF IlHl. BED NO= "

e7~.6 6~7.0

&'34.'1

,

BAR

"9.'3'3 '37 4'3.'3f> 49.95

"9.

.01'350 1".700

ENt' OF 81"0. IlEO ,.0= 3

P"fSSU~E.

"'l.g4

FIXED ftEn REACTO!> LENG TH. {,. 7'50 7.250 7.750 11.2'50 !I.750 Q.2'j0

1

TEMPfRAT~:.K

FIXE!:

IlfACTOR LfNGTH. H 2.1100

SEN1EN~

. •

Tf""'(PATtJRf. K

p,~:SSUQ'~ ~ 8~ ,~.

~75.3

c.q. "C

877." 879.5

4'l.7Q 4 0 .17

~A1.6

.~46

~·:r.7

Itn.7e; 4 0 • 7 ;>1

.5E~

88S.'!

4 0 .72

.57'3 .596 .EL!

8~~.n

4G.70

~90

"-9.

.62'3 . f 46

.66< .E-7Q .6~5

.711 .727 .741

.1

~

A

P'32.3

4C?.66

89".'t

4 0 .6 = specifir he::t of gas, KJlkg (includes all gases) p = specific gravity of gas, kg/m:l (includes all reacting gases) KT = radial thermal diffusivity if' packed bed, m:!ls When divided by

n - 7tH:

~----~~

+

z

27rr~TAz cpup

-Kr(aTlar)

.

uAr

-

r

- Kr(r

+ Ar)(aTfar) u

TI,]/Az = (K'/u) .. [(r

[TI,+.1: -

and rearrangeJ, Eq. 15-48) gives:

r~r

(5-49) r+ar

+ Ar)(aTfiJr)L+~r - "(aTlo7')lr]lrAr

+ When Az ) 0 and flr = 0, [TI,+, [(r

TU/Ilz

(-AHhRlcpup

(5-50)

•

) aTliJz

+ Ilr)(aTfar)lr.ur - r(iJTfi1r)irJlrllr ) (1/r) .. i1(raTfiJr)/iJr

The derivative of a product can be expressed as follows: d(roTlor)lar = ro2Tfal

+

(aTfiJr)(orlar)

Thus Eq. (5-50) becomes: 2

(dTliJz) = (K/u)[(rPTlar ) + (aT/ror)]

+ (- AH)~R./cpup

(5-51)

Equations (5 .. 46) and (5-51) are the two PDEs that describe the NINAF reactor. Together with the proper rate equations and auxiliary condition expressions, they can be conveniently solved by the computer technique developed in Chap. 2 for PDEs. However, it is more desirable to transform Eqs. (5-46) and (5-51) into forms that contain dimensionless variable~, not because the mathematical solution will be any easier it will not but because the res,!iting equations contain a number of dimensionless groups which are more generally applicable and thereby make designing considerably easier. •

•

•

134 CATALYTIC REACTOR DESIGN

Let us use the following dimensionless va.·iables: •

C, = CICo

reduced concentration of key reactant 3

Co =

molar concentration of key reactant at inlet, kmol/m

T, = TITo

reduced gas temperature

To =

absolute temperature of reactant at inlet, K

Z = zldp

reduced axial coordinate

dp

catalyst particle diameter (for catalyst particles of cylindrical shapes, dp represents the diameter of the sphere having the same surface as the cylinder)

=

reduced radial coordinate With these dimensionless variables Eq. (5-46) becomes:

Again, the derivative of a product can be written as [a{R(Jc/aR)JIaR = R . a«(JC/aR)I(JR

+

(aC/aR) =

(Ra~C/aR2)

+ (aC/aR)

Thus, considering that dpulD, is called the radial Ph/et number Jor mass (Pem ), we obtain (see Chap. 4) aCT

az

2

1 a c, 1 = Pe m aR 2 +

R.

aCT iJR

dp,j(

- Coli

(5-52)

Similarly, Eq. (5-51) becomes, upon substitution \rith dimensionless variables,

aT,

az

=

I _.

(5-53)

where K/wlp has been replaced by the di,nensionless group called radial Ptclet number Jor heat, wqich is designated by the symbol Pe". In Eqs. (5-52) and (5-~3) the left-hand terms represent the axial gradients, the middle terms the radial gradients, and the right-hand terms the specie~ generation terms, which include the apparent reaction rate ~R..

•

PROCESS DESIGN OF FIXED-BED GAS REACTORS 135 •

Finally, we need the auxiliary values of the PDEs (5-52) and (5-53), in order to integrate them. The auxiliary values are: Bed inlet

Z = 1

Centerline Reactor wall

aC,(R T , Z) .

aR

C,(R, C) = I

T,(R, 0) = I

(5-54)

aC,(O,Z) = 0 aR

T,(O,Z) --.:...;.......:.-.:. = 0 aR

(5-55) (5-56)

= U

iJT,(R T , Z)

(5-57)

aR

The boundary conditions, Eq. (5-54), mean that at :he reactor inlet both reduced concentration and reduced temperature are unit\·. The boundary condition, Eq. (5-55), states that at centerLle or reactor axis, the reduced concentrations and reduced temperatares do not change with radial coordinates. The boundary condition, Eg. (5-56). !>tates that no mass transfer can take place through the reactor wall. The boundary condition, Eq. (5-57), states that the heat flux at the wall. as expressed by the modified version of the Fourier law, is equal to the product of the wall heat transfer coefficient It", by the temperature driving force between reduced bulk gas temperature T, and the reduce(! inside wall temperature Tu" The reactor wall expression can be rearranged as:

-aT, iJR(T, -

T.J

=

Bi".

(5-58)

The Biu. numbers were discussed in Chap. 4. The calculation of flu presents some difficulties. Reliable data are not l6 available. Froment gives a plot of hu. d/Ag vs. Re with a number of scattered lines obtained by various workers. Obviously the heat transfer coefficient at the wall is one of the most important data and one should be determined carefully if it turns out that a new pf()('ess might be run in ;\I;\AF reactors. Howner, if such information cannot be obtained. a conservative way to design a l'\Il'\AF reactor would be to use a rather lower h". value corresponding to a given Reynolds number. If it turns out that more heat transfer is obtained at the wall during the startup tests of che reactors than the selected h". value would provide, there is nothing to fear since the reactors would then be more stable than intended. The system (If Egs. (5-.52), (5-53), and (4-1), the kinetic equation of the type (5-1), (3-45), (4-11), (4-12), (4-11), (4-17), and (3-59), and auxiliary conditions given by Eqs. (5-54), (5-55), (5-56), and (5-57) adequately describe a single first-order reaction occurring in a I'\I:,\AF reactor.

•

136 CATALYTIC REACTOR DESIGN •

•

However, it must be modified to handle a system of several first-order reactions by replacing the ( -I1H R I) expression in Eq. (5-53) by I( - aH ;Ri) in order to take account of all heat developed by all reactions taking place in the reactor. In the case of two consecutive reactions Ri = TJ1k l Cs

(5-59)

R2 = TJ2k2Cn

where CB is the concentration of the product of reaction I on the catalyst surface. The v,lue of CB can be calculated for a "macromicropore model," accurding to Carberry.17 In .;olving this system of equations, we must use the computer program developed in Chap. 2 to solve PDEs. However, we cannot start at the inlet of the reactor and apply the PDE technique to compute TT and C step by step for each Z, because at the inlet of the reactor we start with bulk g-as conditions er and Tr and have to calculate catalyst surface conditiom C" and T" using Eqs. (4-26) and (4-28), which relate bulk conditions to cataly~t surface conditions. Such calculations car only be done by iteration, as will be seen in great detail in the example of the oxidation of naphthalene to phthalic anhydride, which will be discussed next. T

Example of NINAF Reactor Design: Air Oxidation of Naphthalene to Phthalic Anhydride This example was chosen because it has been widely discussed by l8 19 Carberry l3.14 and others • and sufficient published data are available to design a reactor. \Vhen this oxidation reaction is conducted in an excess of oxygen, the reaction system may be summarized by the following simplified equations: CIOHS Naphthalene

*,

+ 4.502 -~) C 6 HiCOhO +

2C0 2

+ 2H 20

•

(5-60)

Phthalic anhydride

C 6 HiCOhO + 7.502

k. .)

8C0 2 + 2H 20

(5-60a)

Here, kl and k2 are the rate constants of the two consecutive reactions. Input data for designing a NINAF reactor for this reaction system are given in Table 5-2. Kinetic studies reported by DeMaria '8 for a catalyst with small fluidizable particles suggest that all reactions at:e pseudo-firstorder in naphthalene and phthalic anhydride and zero-order in oxygen. Oxygen is always present in vast stoichiometric excess. The oxygen/ naphthalene ratio was sdected slightly outside the explosive region. The excess of air constitutes both an excess of reactant and an inert diluent and hence forms a beneficial heat sink. On the other hand, it also (on-

-

•

PROCESS DESIGN OF FIXED-BED Gf\S REACTORS 137 •

•

stitutes a factor for increasing the number of reactor tubes, since the reactor diameter cannot be enlarged without badly affecting heat transfer. The sources of the data used in this example are indicated in Table 5-2. Some data were recalculated. For most of the data the correctness of information given in the literature was relied on. The computer program RDMOT2 developed to solve this type of problem is explained in detail in the following, with the assistance of Figs. 5-16 to 5-20. Fir~t, all the input data are read in and also printed in order to define the problem. In the unlikely event that the temperature would run away, TR~IAX = 2.0T" is also read in as a limit. A fixed step size is read in. Next, dependent variable Y and its derivatives F are dimensioned in the form of two-dimensional arrays. There are two types of PDEs and each has six I values. An equivalence statement establishes corresFondence between e" T" and their Y equivalents. Some information such as the highest I value, NR, and the frequency of printing, IFREQ, is supplied in form of data statements. Next Z, Y(I,I), Y(I,2), and ICOUNT are initialized. A DO loop assigns I values from 1 to 6 to all the \,'s. Then nine quantities, p, cp' Re, Pr, Sc, kl(, It, BiOI, and ~T are calculated. 0:ow, dummy variables CSI and TSI are initialized in order to com-

plHe the catalyst-surface concentration CS(I) and catalyst-surface

•

r~m­

perature TS(I). As a starting point, CSI is assumed to be the same as the concentration on the axis and inlet of the reactor, namely CSI = CS( 1). This statement calls on the function CS(I) which is set up as one of the several separate functions. On Fig. 5-17, the sequence of computatiori which gives CSt.!) is shown by an arrow diagram. We will come back to this sequence of computation. The FUNCTION RUNGE is called. This function is set up to process a one-dimensional array of differential equations. Since there-is one material balance ODE and one heat balance ODE, each with six I values, N = 12. The FUNCTION RUNGE works exactly as discussed in the adiabatic reactor examples. A DO loop assigns values between 1 and 6 to I. For I = I or on the reactor axis, special ODEs apply. For I = 2, 3,4,5, the regular ODEs apply and for I = 6 or at the reactor wall, the ODEs include the boundary conditions. The calculation of I-dependent quantities, such as the temperaturecorrected RHOC, Sc, Kg, HTC, and Bi"" also have to be located inside the DO loop in which the value of I is defined. Following the FUNCl'ION RUNGE a temperature limit is set: if the bulk temperature exceeds-2.0 times the inlet temperature, the integration is terminated and the program writes its final statements and stops. Otherwise, it continues. Here, the axis points are taken as the temper-

•

138 CATALYTIC REACTOR DESIGN TABLE 5-2

Input Data for the Computer Program RDMOT2 to Design a NINAF Reactor for the Oxidation of Naphthalene to Phthalic Anhydride Data

Text symbol

a Al

A

Al

Dimensions

Value

1200.

m

5.809 x 10

13 •

-I

C.lculated from 14

-I

14

s

A2

2.222 x 10'

s

Bi"

BIW

Note 1

T)imensionless

cp

CP

30.334

kJl(mol'K)

-

D

.,

•

10-' •

source (ref. no.)

-I

A2

('

14

m'/s

14

Dim

DIM

2.05 10-'

m2Js

14

DglD ...

DBDA

0.93

Dimensionless

14

dp

DP

0.001

m

14

DT

DT

0.05

m

20

El

El

158990.

kJlkmol

14

Eo-

E2

83680.

kJ/kmol

14

H

H

0.1

Dimensionless

kl

Kl

7.0

-I S

14

0.035

-I s

14

k2 •,

Computer sym601

K2

•

MI

M''\' I

128.174

M..

MW2

28.860

Pe rn

PEM

10.0

Dimensionless

14

Peh

PEH

7.0

Dimensionless

14

D

He

RC

8.316

kJ/(kmol·K)

To

TO

643.2

K

T rm:u

TRMAX

2.0

Dimensionless

14

ature sensing points, since the peak is expected to show up there. Each time that a certain number of integration steps are compiled, the results are printed. -The number of integration steps between each printing is designated as-IFREQ. Various quantities needed for the calculation of the ODEs are computed as separate functions. The relationships between these functions is shown in Fig. 5-17. There are some problems in these relationships which require a detailed explanation. Thus, starting with the bulk gas

•

PROCESS DESIGN OF AXED-BED GAS REAol

Value

(ref. no.)

Dimensions

Tu.

TW

0.983

Dimensionltss

u

U

8.0

m/s

yo

YO

Note 2

Dimensionless

VIS

kg/m's

14

LG

0.0000305 5 103.853 x

kJ!(s·m· K)

14

f1H,

DHl

-1.8 x 106

kJ/kmol

Note 3

f1H,-

DH2

- 1.82 x

kJlkmol

Note 3

At;

1O~

KOTEl: Heat transfer across the reactor walls of Kli'\AF reactors is reponed'o to be in the order of 200 tu 400 Btu/(h·ft 2 .oF). which corresponds to 1.1356 - 2.2712 kjl(s·m 2 'K). The author selected the upper range and calculated but •

Bi. = h" Pe,/pcpu = (2.27)(7 .)/{0.56094)( 1.0246)(8.) = 3.456 Too Iowa Bi", would make the NINAF reactor approach an adiabatic react" ... This is exactly what would happen if the above calculated Biw figure were used. We have used Bi,.. = 92.0 to demonstrate a strong wall effect. Heat transfer in the process side is reponed to bew 32 Btul(h·ft'·oF) or 0.1817 kJ/s·m 2 ·K), which is 6.46 times lower than the value calculated by fOllllula. We prefer to use the . experimental data. '.

NOTE 2: One must be very careful in selecting the naphthalene mole fraction in air because of the explosi\ity of air-naphthalene mixtures: •

Lower explosive limit: 0.9 mol

~

naphthalene in air

Upper explosive limit: 5.9 mol \k' naphthalene in air Thus )'0 = 0.0075 (¥.\k' naphthalene) is probably the highest safe concentration. However, it is also kno,,"n that some industrial reactors are being operated e,'en in the explosive range.'" NOTE 3: Literature data on I1HI and I1H2 are spread oyer an order of magnitude. Data of Coleman and Pilcher}1 G. C. Parks et al.." L. M. Elkin.'" and a pri"ate source were compared and the following data were adopted: -~, = 0.18 X 10' kJfkmol and -;;;H. = 0.182 x 10' kJfkmol.

.

concentration CR(I) and bulk gas temperature TR(I), it is not possible to calculate the catalyst surface concentration CS(I) and temperature TS(I) explicitly from the differential equations. The reason is that the terms 1'/1> kl> RI> and I/lJi'R, all of which depend on temperature and concentration, must be evaluated as functions of CS(I) and TS(I). This means that CS(I) and TS(I) appear implicitly in the system of equations and must therefore be computed by iteration. Thus, CS(I) and TS(I) are assumed; 1'/1, k l, RI, and I/lH·R are computed based on these as-

•

140 CATAlYTIC REACTOR DESIGN •

NO YES YES

YES YES

'- YES

NO

FIG. 5-'16 Block diagram of the RDMOT2 computer proglam.

sumed values; new values of CS(I) and TS(I) are computed from the set of equations; and the entire calculation is then repeated until convergence on CS(I) and TS(I) is obtained. Specifically, th.e call to RI (I) in calculating F(I, 1) initiates the iterative calculation of the catalyst surface concentration and temperature:

I RI

I +-.-

EFFI . . . -

PHI!

(I-1,)/(2.*DELTAR)) 2 - :DP.Rl{l)/(CO.ll)) 1 (1,21 = (1./PEBl * «TB (1+ 11 -2 •• TR (11 +rB (1-1) )1 (OELT1R**2) + 1 (1./( (I-11*OELTAR)) .WUL) 2 + (DHR (1) *DPI (CP*u *RH:JC*TO)I GO TO 10 12 l!ALL~BliI" (TlI-TB (IIR)) F (I, 1) = (2. /PE!!) • (Cl! (1 -1) -CR (I) ) I (DELTAR**Z)

•

-(D~*Rl(I)/(CO*U))

1

l" (1,2) = (Z./pEH) *«TR (1-11 -TB (I) ) I (DELTA n**z) + 1 (1./( (I- l l*OELTAII»*1I1LLI 2 +(DHR(I) *OP/(CP*0*1I110C*'1'0) 10 CONTIIIUE GO TO 2 3ZZ=Z"DP

c

••.•• IF

TB EXCEEDS TP.IIAX, T..lR'IIIIAT.1'! INrEGRATION •••• Il"(TB(l).LE.TRlIll.l, GO T~ 4 GO TO 5 4 ICOUNT " lCOUNT + 1 IF(CS(H.U.O.Ml) GO TO 5 IF (ZZ. GE. 1.5) GO TO 5 Il"IICO{JIIT.IIE.IFREQ) GO TO 2

C

C

C

•••• PRINT SOLGTIOM ••••

C

WRITE (6,Z02) Z, ZZ WRITE (6,203) (CR(I) ,I=l,NR) WRIrE {6,2041 (T!!(I), I~l,!lRI 202 FO~'AT(lHO,2X,4HZ = ,E10.3,J~,SBZZ = , El0. 3/1H 1 l1H R~O ,1111 R:1 ,11H , F=2 2 118 R=J ,11H &=4 ,1111 R=5 ) 203 FOBIlAT (18 ,68CR(1)=,6El1.3) 204 fOR!lAT (1l! ,6RTR(I}=,6E11.J) ICOUIIT ~ 0 GO TO 2 •

\

C

C C C

THE CALL ro Rl(I) IR CALCULATING F(I,l) INITIATES THE ITERATIVE CALCUL1TIOII OF THE CATALYST SURFACE CONCENTRATIOII AND TEftPERATURE, CSI 1IID TSI

C

5 WRlrE(6,202) Z, zz IIRITE(6,203) (:::R III ,I=l, IIR) IlRITE(6,ZOIl) (TR(I) ,1:1,118) STOP END C

FIG. 5-19 The RDMOT2 computer pLOglam for the design of a NINAF reactor for the oxidation of naphthalene to phthalic anhydride (continued) .

•

144

•

•

fUBCTIOK RUHG~ (I,T,',Z,m IlIf EGER BU ICE

•

DII1BIISIOII PHI (50) ,SA'lr:t(50) ,T(II) ,F(R) DATA 11/0/

c

11;11+1 GO TO (1,2,3,4,5), 11

C

•

•

PASS 1

1 RUNGE = 1 RETURII

C

PASS 2 2 DO 22 J=l,N SAYEl (JI =1 (J) PHI (JI;1' (JI 22 T(J)=SlVEl(J) +0.5 • z = z + 0.5 H

ROIIGE ; 1 RETURB

C

*

PASS 3 J=l,1I = PBI (J) • 2.0 SAY El (3) + O. 5

] DO 3] PHI CJI 33 T (JI = Rll!!GE .. 1

H. F(JI

*

*

f' (J) H •

F (JI

BBTDall

C

" 44

C

5 55

PASS ~ DO 44 J=l,N PHI (J) = PHI (JI + 2.0 * F(JI T(J) .. Sl'lET(J) + H • F(J) Z = Z + 0.5 • H ROIGE '" 1 BETURI PASS 5 DO 55 J ; 1,. T IJ) = SUET (J) + (PHI (J) + l' (Jl). H/6.0 11=0 RIJ'GI'! .. 0 RETIJII

!ID

C

FD.cn O. a1

(11

C C C

BElL 1t1 • THE FOLLOWIIIG STlrEIIENT Wl. LL ITERATIVELI CALCULAT & CSI UD TSI, 'lIA THI'! CALL TO CS (I) CSI=CS (I) R1 = E1'F1 (I)*K1(1) *CSI RETURN

ElID

C

c

FUlIC1'IO!l R2 (I)

ilEAL 1t2 a2 = E1"F2 (I) *K2 (1) *CB(I) RETURN

EIID

FIG: 5-19 The RDMOT2 computer program for the design of a NINAF reactor for the oxidation of napht}!aJene to phthalic anhydride (continued) • ••

145

• C

YUICTION EYY1!I)

•

C

PHI " PRIl (1) ilff1 = TUR (PHII /PRI RE'J'URt! ElIll

•

•

C FO NCT 1011

~F1'2

(I)

C

PHI " PHI2 (1) El'F2 " TANH(PBI)/PHI R1WJRK

END

C

FONcnON PHll (1) C

C C

REA L r; 1 CO""CM A,Al,A2.BI~.CO.D.DBDA,DR1#nh2~DP.E1#E2,ffTC,XG,RG.TO, 1 CSI,TSI,CR,TR llI!ElISI~H CR(61, TR(6) PHI1 = llP /6. *SQRT (lt1 (I) Ill) SET PRI1 ro A IIIHIIIU! VALOE TO AVOID A NUI!ERICAL PROBLEII IN COIIPUTIN;; EFl", SPlIIP!'! . IF(PHI1.LT.1.0E-10) PHI1 " 1.0E-l0 IlETUllll END

C

-•

F!JlICTIOII PRI2 (1)

RElL 1':2 COl!lIOll I, A " 12, BI~, CO, 0, DB 01., OH' ,DR2,DP, El, E2 ,aTe.!,(G ,RG. TO. 1 CSI.TSI,CR,TR DIIIEIISIOll CR(6), rR(6) PRI2 '" OP/6.*SQRT(1'(2 (I) Ill) IF(PHI2.LT.l.0E-10) PRI2 " 1.0E-l0 RETURlI EIID

C

REAL l'OliCTIOII 1t1 (l) C

C

NOTE - 1(1 IS 1I0W

BASED Oll THE CATALYST S(JBl'ACI! TI!!lPERAT(JRI!, TSI

C

•

COft!OH 1.1',12,BIII,CO,D,DBDA,DH1,DR2,DP,E',E2,HTC,r;C,RG,TO.

,

.

eSI,1sI,CR,TR

DIIIElISIOR CR(6). TR(6)

1(1 = 11*EIP(-ElI(RG*TSI)) RETUIlll EIID

•

•

FIG. 5-19 The RDMOT2'computer program for th~ design of a NINAF reactor for the oxidation of naphthalene to phthalic anhydride (continued).

146

•

•

C

RBAL PURCTIOR

~2(~

C

C.

ROTE -

~2

COftftOR

a,Al,A2,815.CO,D,DBDA.DH1,DH2,DP,El,E2,8TC,~G,RG,TO,

IS IOW BASED 01 THE CATALYST SURFACE TEftPER1TDRE,

•

TSI

C

1

CSI, TSI,CR, Tl! . D1ftfRS10I CB(6J, TR(61 K2 ; l2*ElP(-E2/(RG*TS111 RETURN ERD

C PUNCTIO~

CS (I)

C

C C C C C

CS IS THE ClTALYST SURFACE CONCENTRATION OF KEY REACTANT COBPU:ED VIA BASS AND HEAT TRAHSFER COEFFICIERTS CSI AND rSI ARE ClLCULATFO ITERATIVELY VIA EQUATIONS CSI ; CO*CR/(1.0+(Effl*tO/(KG*A)I J (~\ ISI = TO*TR+DHR/{RTC*'l (S)

C

C C C C

COI!I!CII I, A1, A2, BIII,CO, D, DBDA, OH', DR2,DP,E 1 ,E2 ,HTC ,KG, RG ,TO, 1 CSI,TSI,CR,T' DIIIENSIOR 0::2(61, TR(61 REAL tt1,~:; SET TBE NlJftSEB Of' ITER1TI)NS, HITER IHTER = 3 SET THE INITIAL VALUES OF CSl AND TS1 TO THE BULK GAS CONCENTRATION AND TE~PERATURE, CSI = CO*cR(I) TS1 = TO-TB (1) CCIIPUTE CSI UD TS1 VH EQIJATIOns A AND B

C

DO 1 L= " HITER CSI = CO*CR(II/(l.+{Eff' (I)*1C1(I)/(ICG*A»)) 1 TSI = TO*TR (I) +(DBR (11 / (IITC.AI) CS ; CSI RETURl! EIID

•

•

C FUNCTIOI! CB(I) C C C C

CB IS tHE CATALYST SURFACE CONCENTRATIOR OF THE PRODUCT OF RElCnOI 1 A,Al,A2,81ft.:O,D.DBDA,DH1,DH2,DP,E1,£2,HTC,ICG,RG.TO, 1 CSI,TSI,CR,TR DIKENSIOR CRCE), IR(6) REAL IC " tt2 RK1K2 = IC1 (I)/I(2 (IJ S = Sl!ftPII (11 ROTE - :B IS ROi BASED OR CSI CSCO = CSI/CO CB = CO. ( (S- DBDA) I (S-1.) ) * (RK 11\'2/ (RIC 1tt2*DBDA -1.) ) * ( (CS CO" S) -CSC 0) If(CB. LT. D.) C8=0. BETURII EIID CO~ftOI

C

•

FIG. 5-19 The RDMOT2 computer program for the design of a NIN AF reactor

for the oxidation of naphtlralene to phthalic anhydride (continued). ..

147

•

c

FUlIcnOIl SIIIIPII (rl

c c

•

SIIIIPII IS THE PACTOB 5 POB THE IIACBO-IIICRO PORE ftODEt

c

COIIIIOII

•

1.11,12,8I",=O,D,D8Dl,DH1,OH2,OP,El,E2.HTC,~G,RG,TO,

CSI.TSI.CR,TA· OIII!1I51011 CR (6), TR (6) RElL 11111,11112 SIIIIPII=D8Dl*!!1I1 (I). (PBI2 (11 "2j.EFF2 (II / 1 (11112 (1)* (PHI 1 (11 "2)-.eFFl (1)) RETORI

1

ERD

C

RElL FUIiCTIOII 11111 (I)

c C

11111 IS 1 PACTOR OS!D III '.!'RE C1LCULATIOI OF 5 FOR THE RACRO-RICRD

poa E

C

COIIIICII 1

1I01)!",

t,11,12.8I~.CO,D,D8DA,DR1.0R2,DP,El,E2,BTC,KG,RG.TO,

CSI,TSI,CR,TR DIIIERSIOH CB(6), TB(6) PHI = PHI1 (I) ~1I1= 1 • • (PHI.TANH(PHI)/BI") EET{lRR END

C

REAL FUICTION 11112(11 C

C C

11112 IS 1 FACTOR ijSED IN THE CALCULATION OF S FOR THE IIlCRO-IIICRO

PORe IIODEL COII!!!'R A, A " A2, BIll, CO, D. DBDl. DH 1 ,DH2,DP,E 1 ,E2 ,HTC, KG, RG, TO. 1 CSI,TSI,CR,TR DI!lERSION Cl! (6). rR (6) PHI = PHI2 (I) RII2= 1 •• (PHI-TUH (PHI) /BIII) RETUR'i END •

c FtlllCTION DRR C C

C

c C

(1) •

DRR IS THE Stlll OF TRE PRODUCTS OF THE HEATS OP REACTIOR AND R1T!S OF REACTION COIIIIOll A,Al,12.BI~,CO,D.DBDA.DH1,DH2,DP,E1,E2,HTC,~G,RG,TO. 1 CSI,TSI.CR,TR DI~~!!SIO!! CR (6), TR (6) RElL It 1 COIIPUTE Bl 11 Dlll*R1 + DH2*S2 LOCALLY TO "DID A RECORSI'! CALL TO FOlICTIOII CS (CSI IS USED III PLACE OF cs (1) ) RH = EFF1 (l) -It 1 (I) .CSI DHR - DH1-R1I • DH2-R2(I) RETtlRlI • ERD

FIG. 5-19 The RDMOT2 computer progtam for the design of a NINAF reactor for the oxidation of naphthalene to phthalic anhydride (continued).

• 1

DESIGI Ol' UUI' GAS REACTOR o AIB OXID1TIOI Ol' IlPHrB1LEIE ro PHTHALIC AIHYDRIDE o ro= 643.2 K 0= 8.000 ft/s TBftlX= 2.000 A= 1200.0 1/ft tOs 0.250B-02 CP=30.334 KJ/KftOL ftll=128.174 ftI2=26.860 PEft=10.000 El= 0.159E+06 Al=0.5809E+14 Dr= 0.050 ft R=0.100 PER= 7.000 E2= 0.8378+05 12=0.22228+06 OBOl t O.930 TI=0.983 OP=0.001" 8II=0.920£+02 "VIS=0.2908-04 PAS 081=0.180£+07 KJ/KftOL Le= 0.3952-04 KJ/S.ft.K BG=9.316 KJ/KftOL.K 082=0.1922+07 U/KIIOL D= 0.100 E-04 11**2/5 D 111=0. 205!!-04 11**2/5 o Z = 0.250£+02 ZZ = 0.2502-01 R=q 8;0 R=1 R=2 R=l R=5 C8 (11 = 0.9768+00 0.976&+00 0.976£+00 0.976 £+00 0.977£+00 O.986E.00 Ta (I) = 0.101£+01 0.101E+Ol 0.100£+ 0 1 0.983E+00 0.101E+01 0.101&+01 o Z = 0.500£+02 ZZ = 0.500E-Ol 8=3 R=4 8=0 R= 1 8=2 R=5 CB (I) = 0.947£+00 0.947E+00 0.J47£+0~ 0.9461':+00 0.953£+00 J.970&'00 TR(I)= 0.101&+01 0.101E+0· 0.101E+0: 0.101&+01 O.I~I1':.Ol O. './83 £+00 0 7 ~ 0.750&+02 ZZ = 0.750l-01 8=0 B=1 8=2 R=J R=4 R=5 CR{I) = 0.913&+00 0.913£+00 0.91'"£+00 J.9171'+00 0.929 E+OO 0.953E+OO TR(l)- 0.102£+01 0.102&+01 O. i02E+Ol 0.1021':t01 0.10lEtOl 0.9831':tOO o z = 0.100R+03 ZZ = 0.100&+00 8=0 8=1 B=2 R=J P=5 R=" CB(I) = 0.871£+00 0.871£+00 0.873£+00 0.880 E+OO 0.902E+00 0.934E+00 TR(I)= 0.1032+01 0.1032+01 0.1032+J1 0.102E+Ol O.101E+Ol 0.983E.00 o z = 0.125£+03 ZZ = 0.125£+00 B=O B=1 8=2 B=3 !I=4 R-5 CR (I) = 0.817&+00 0.8178+00 0.821&+00 0.836 £+00 0.873E+OO 0.915E+00 TB (I) = 0.104£+01 0.104£+01 0.104£+01 0.103E+Ol O.lOlE+Ol 0.983£+00 o z ~ 0.150£+03 ZZ = 0.150£+00 8=0 8=1 R=2 8=3 R='I R=5 CB (I) = 0.740£+00 0.742£+00 0.752£+00 0.782£+00 0.8111£+00 0.893E+00 TB(I) = 0.106£+01 0.106E+Ol 0.106E+01 0.10';E+Ol 0.102£+ 01 0.983£+00 o z = 0.175&+03 ZZ = 0.175E+00 . 8=0 B=1 8=2 B=3 B=4 !!=5 cau) = 0.617£+00 0.623£+00 0.649£+00 0.711E+00 0.804E+00 0.869£+00 TR(I) = 0.109£+01 0.109£+01 0.108£+01 0.106E+Ol 0.103E+01 0.983£+00 o z = 0.200E+03 ZZ = 0.200£+00 8=0 8=1 R=2 P. =3 R=4 R=5 CII (I) = 0.411£+00 0.425£+00 0.490£+00 0.607£+00 0.759E+00 0.643E+00 Ta(I) = 0.114£+01 0.114£+01 0.112£+01 0.106£+01 0.103£+01 0.983£+00 o z = 0.225E+03 ZZ = 0.225£+00 8=0 B=l 11=2 B=3 1l=4 R=5 CB(I) = 0.206£+00 0.219£+00 0.275£+00 0.447£+00 0.701E+OO 0.812E+00 T8(I) = 0.120£+01 0.119£+01 0.117£+01 0.112E+Ol 0.105£+01 0.983E+00 o z = 0.250£+03 ZZ = 0.250E+00 B=O 8=1 B=2 B=3 8=4 B=5 CR (X) = 0.913£-01 0.995£-01 0.137&+00 0.269£+00 0.621&+00 0.775£+00 TR(I)= 0.123E+Ol 0.122£+01 0.121£+01 0.117£+01 0.106£+01 0.983E+00 o Z = 0.275&+03 ZZ = 0.275£+00 a=o B=l 1=2 11=3 R=4 B=5 CII (I) = 0.392£-01 0.439£-01 0.660£-01 0.153 E+OO O.515E+00 0.729&+00 TB(I) = 0.124£+01 0.124E+Ol 0.123£+01 0.119£+01 0.10BE+Ol 0.983E+00 o z = 0.300£+03 Z~ = O.JOOE+OO a=o 8=1 R=2 R=3 11=4 R=5 CR (I) = 0.168£-01 0.194£-01 0.326£-01 0.920E-Ol 0.336£+00 0.669E+00 TB (1) = 0.125£+01 0.125E+01 0.124£+01 0.121£+01 0.1111\+ 01 0.984 &+00 o Z = 0.325&+03 ZZ = 0.325£+00 B=O R=l R=2 B=3 8=5 R=II CR (X) = 0.727£-02 0.881E-02 0.170B-Ol 0.576£-01 0.262£+00 0.598£+00 TB (I) = 0.125£+01 0.125£+01 0.124£+01 0.121£+01 O. 114E+ 01 0.964 E+OO o Z = 0.350£+03 ZZ = 0.350B+00 R=O R=l B=2 R=3 R=4 8=5 CR (I) = 0.323£-02 0.416E-02 0.944E-02 0.367E-Ol O.17I1E+OO 0.522E+00 TB [Il = 0.125£+01 0.125E+Ol 0.124£+01 0.121£+01 0.115£+ 01 0.9811E+00 o z = 0.374£+0] ZZ = 0.374E+00 R=O B=l g=2 B=3 B=4 R=5 CR (I) = 0.149£-02 0.207£-02 0.553£-02 0.241£-01 0.122&+00 0.447&+00 TB (1) = 0.125£+01 0.125£+01 0.124£+01 0.121&+01 0.115£+01 0.9811 !+OO o Z = 0.388R+03 ZZ = • ~0.388E+00 B:3 B=O 11=1 R=2 B=4 8=5 O. 998E' OJ CB (L) = 0.146E-02 0.42'1£-02 0.195£-01 0.104£+00 0.410£+00 TB(l)= 0.125E+Ol 0.125£+01 0.124&+01 0.121E+Ol 0.115£+01 0.984£+00

-

,.

•

FIG. 5-20

Printout of RDMOT2 computer run.

149

"

•

150

-

CATAlYTIC REACTOR DESIGN

The computation of CB, the concentration of the product of reaction 1 (phthalic anhydride) on the catalyst surface, was carried out according to themacromicropore 1flOd.f'/17 by the expression: Si

CB = Co

- (D B ID,4) Si 1

~,

C

kl/k z (k/k~)(DBID/\)

.

C --

1

-

o

-

where

DIl ml If't 11~ DA

ml -

=

1112

If'i 111

1 + «1f'1 tanh 1f'1)/Bim )

lnz = 1

DA and DB

o

+

[(If'z tanh If'z)/Bi m ]

micropore (Knudsen) diffusivities of naphthalene and phthalic anhydride, resp~ctively

CBO = initial concentration of phthalic anhydlide at reactor If'I and If'z = Thiele moduli for macropores

111 and 112 = macropore effectiveness factors

.

,

The derivation of this kinetic relationship will not be treated in this book. It can be found in !Jart in a paper by Carberry.li In our example CBO = O. Other quantities were taken from Carberry's paper. In this example we will calculate the design of a single reactor tube. A number of these tubes will have to be grouped together into a heatexchanger bundle in order to meet a desired production capacity. The feed to the reactor is determined by the mole fract:Jn Ju of naphthalene in air and by the fluid velocity u in the bed. Naphthalene contents of 0.25 to 1.0% in air have been used, corresponding to Yo values of 0.0025 to 0.01. At higher concentrations the mixture becomes explosive and the excess of oxygen is depleted. Various quantities must be calculated by the computer before the call for CS(J). Co = (Yo/22.414)(273.2/To) = 12.189 YolTo

The average molecular weight of the reacting mixture is: •

•

The dimensionless reactor radius is RU'

=

DY/2dp- The gas density P is

calculated at the inlet and, later, in the bed by Po

=

273.2 Mo/(22.414 To) - 12.189 Mo/To

-

pr~OCESS

•

DESIGN OF FIXED-BED GAS REACTORS

151 •

The various dimensionless groups must now be calculated: (Mo/22.414){273.2ITo)(u/~) =

Re = Pr

12.189 Mo uf(TolL)

= CpIL/Ac;

Se = IL/P D I., tl5

1J

k/i

=

1.15 uRe-

It

=

I. 15 u Re - 05 P ("Pr - '"

Bi m

=

k/'1'a

ill' =

Rj5

Sc-

Some of these calculations must be repeated inside the DO loop which calculates the derivatives F{I,I) and F{L2) because the expressions contain the variable I. Figure 5-18 lists definitions of symbols used in the source program RDMOT2. The complete RDl\fOT2 source program and its output printout are given in Figs. 5-19 and 5-20, respectively.

NOMENCLATURE A Frequency factor A, C

.

.

.)

Cross-sectional area of reactor bed, m-

.,

Molar concentration of reactant or product, kmol/m"' Molar concentration of toluene, kmollm

•

3 3

CB

Molar concentration of hydrogen, kmol/m 3 tration of phthalic anhydride, kmol/rn

Cpi

Molar heat capacity of each i reacting species, kJ/(kmol· K)

E

Activation energy, kJ/kmol

F

Molar flow, kmolls

k

Reaction rate constant

mi

Molar flow of each i reacting species, kmol/s

N

N umber of moles passing from the reactor per unit time

p

Partial pressure of reactant or product, bars

P

Total pressure, bars

l',

Reaction rate, mole converted per unit time per unit surface area of catalyst,. kmol/(s'm2)

T,_

Intrinsic reactiort r:'ote, kmol/(s'm3) ..

,

or molar concen-

Reaction rate expressed as partial pressure converted per second 3 per unit volume, p/(s·m )

•

152 CATAlYTIC REACTOR DESIGN

•

• •

Intrinsic reaction rate, moles converted per unit time per unit weight of catalyst, kmol/(s'kg) •

R

Gas constant = 8.3l4 kJ/(kmol'K)

R

Gas constant = 0.08314 (ms·bar)/(kmol·K)

,

3

E/(l - E) substituting this value of V,. in Eq. (6-8):

4 Vp E D = ----'-t (1 - E)Sp

(6-10)

-

In order to determine AL in Eq. (6-3), Turpin et al. determined what they call the liquid saturation or the fraction of the voids fIlled with liquid er/£. The m-:>ss velocity G, based on the total cross-sectional area of the unpacked reactor, was correlated with EdE as follows: For upflow For downflow

(6-11)

EL

= -0.017 + 0.132(Gr/Gd

24

(6-12)

with 1.0 :5 (GdGc)°·24 :5 6.0 for both equations, which are said to be only ±25% accurate. The pressure loss due to the acceleration of the liquid does not have to be considered for operating pressures below 3.4 atm gauge (3.36 bars gauge). Above this pressure, the acceleration terms can be calculated from inlet and outlet quantities. A trial-and-error procedure must be used in the solution of Eq. (6-3). Tlie frictional pressure drop (AP)GLF can be obtained directly from Eq. (6-4) and Eq. (6-5) or (6-6). However, a trial-and-error solution is

•

...

,

•

•

GAS-UQUID-PHASE FIXED-BED REACTORS 165 • .

required to obtain the total pressure drop - (P 2 - PI)' The procecure is to assume an average column pressure, which in effect fixes P 2 • Then •

This quantity is used in Eq. (6-3) and the procedure is repeated until the calculated average column pressure is sufficiently close to the assumed average colUlnn pressure. Turpin's correlations were made with the well-known water-air system. For other gases, the viscosity was not expected to change appreciably, but for liquids other than water a viscosity correction factor 9 (J.LuiJ.Ldo. was used. Here subscripts wand L denote water and other liquid, respectively. This factor is used to multiply the Re~167/Re~;67 term in Eqs. (6-5) and (6-6). Pressure drop in packed beds in COUIltercurrent flow is determined by means of the generalized pressure drop correlation first established by Sherwood et al.,32 which can be found in its improyed present form 33 in a brochure by ~he Norton CO. Unless extremely large particles of catalyst are used, the design of countercurrent gas-liquid fixed-bed reactors does not make much sense, since with the usual sizes of catalysts • the packing factor F would be very high and flooding would occur at quite low rates of gas and liqllid flow. Thus, industrial-size countercurrent gas-liquid fixed-bed rClctors are not practical.

"

Axial Dispersion and Hoidup in Trickle-Bed Reactors In contrast to fixed-bed gas reactors, axial dispersion in gas-liquid-phase fixed-bed reactors is not insignificant. In studies made with laboratory and pilot-plant reactors an app':"~ciable amount of liquid holdup was measured in the porous catalyst bed. Even the gas flow appeared to have some axial dispersion. However, in large industrial trickle-bed reactors both the liquid and the gas flow were found to be plug flow. The studies of Lapidus,21 Schiesser and Lapidus,16 Hochman and 21l Effron,19 Sater and Levenspiei,34 and Glaser and Lichtenstein are of great interest in this field. These workers used various tracers in gas and liquid phases to detect and measure the residence time distribution of the fluid in laboratory-scale packed beds. They did this by two methods: •

1. The step function input: In this method the tracer is fed at tull concentration during a sufficiently long period of time to reach a steady concentration in the c;ffluent equal to that in the input. Then the tracer addition is suddenly discontinued and the decrease in tracer concentration in the effluent was measured (Fig. 6-5).

•

•

166 CATALYT1C REACTOR DESIGN 1.0

r====:::-----

O.S

•

~Porous

S 0.6 ......

u

•

spheres

Nonporous spheres--

0.4

0.2

o'-- --'-- --'-- -:::::: =o

10

20 30 Time,s

40

Comparison of the step response for porous and nonporous packings. C :=: tracer concentration in effluent; Co :=: tracer concentration at inlet. (From Schiesser and Lapidus, 16 by from the American Institute of ChemFIG. 6-5

ical Engineers.) .

2. The pulse function input: In this method the packing is hrought into equilibrium with a tracer-free flow. Then suddenly, a tracer conc:entration is added to the inlet of the packing, maintained constant for a short time, and then again suddenly discontinued. Again the tracer wncentration in the effluent is monitored (Fig. 6-6). 21

'.

•

Figure 6-7 shows three types of time-of-contact curves obtainable by a step function input. If there is absolutely no backmixing in the reactor, the response' would be a steplike line labeled A plug flow. If there is complete backmixing, as in a CSTR, curve B would be obtained. In real21 life cases, type C curves are obtained. Lapidus found that in concurrent flow over nonporous packing the time-of-coT1tact or residence-time curve approached the plug-flow regime and that the effects of minor deviations from plug flow would not be important unless chemical conversions of • 95% or greater were 8:esired. By contrast, residence time experiments with porous packing produced distorted curves, the distortion being a consequence of mass transfer of tracer into and from the internal voids, or pores, of the packing. Pulse-type input curves are affected only to a minor degree by this inass transfer and thus produce results almost 21 identical to those of a nonporous packing. Lapidus and Schiesser and Lapidus 16 separated the contributions of pore diffusion and backmixing 16 in bulk voids. As can be seen from the plot of Schiesser and Lapidus in Fig. 6-8, diffusion of liquids into pores adds a long tail to the step response curve. Lapidus found that only 40 to 50% of the pore \'olume is filled with liquid. A corresponding fraction of the packing is thus unavailable for liquid-phase catalytic reaction, which indicates poor overall liquid-catalyst conta~ting. The reactor moJel which emerges from all this is one in which a r~actant molecule enters a catalyst bed either in the gas phaSe Oi' the liquid phase, moves along with the bulk flow to a particular catalyst particle, diffuses into a catalyst pore to an active site,

•

•

GA5-UQUID-PHASE RXED-BED REACTORS

167

•

becomes adsorbed at the site. and reacts with another adsorbed molecule or all by itself (depending upon the particular chemical reaction). The reaction product another molecule becomes desorbed and diffuses all the \vay back to the bulk flow. By that time the bulk flow may have moved, say. some 20 min downstream and the produ"Ct molecule coming out of the pore joins fresh reactant molecules in the main bulk flow with this much delay. This is the essence of axial dispersion and its net effect is a decrease in overall conversion. The task of expressing this axial dispersion in a manner useful for design purposes has been carried out by DeMaria and White/ Hochman and Effron.19 Sater and Levenspiel,34 and others. The axial dispersion term in the chemical reaction equation has been expressed as a Peck, number, as discussed in Sec. 4.3. The Peelet numbers proposed by various workers are listed below: 1. In the gas phase of a two-phase syster1:

Sater and Levenspiel

34

PeG

=

(udplDak

3.4 Rec067

=

(6-13)

DeMaria and White'

(6-14)

Hochman and Effron 19 PeG with

(j

=

=

1.8

Re,~o.,

10 -fUN)S

RCL

(6-15)

+40%

2. In the liyuid phase of a two-phase system: Sater and Levenspiel

34

PeL

=

3

711

Re~'s

10'U)()$

7.58 10- ReZ

(6-16)

:l

• '-

,

Hochman and Effron 19 PeL = 0.034

-,'" o

0.100

ReG

(6-17)

alumina spheres

-

U 0.050

rous alumina spheres

::>

OO~

10

20

30

40

50

Time, S

Comparison elf pulse respor.se for porous and nonporous packing. C = tracer concentration.. in effluent; V = volumetric flow rate of liquid; Q = moles tIacer injected in pulse input. (From Schiesser and Lapidus,t6 by pennission from the American Institute of Chemical Engineers.) FIG. 6-6

..

•

•

168 CATALYTIC REACTOR DESIGN •

•

...

•I

--

-.-'"'"

, I

\

\ \ \ \

t

\

I

t

Perfect mIXing

,,

•,t

c:

o u

•

Plug flow

\

•t

'" -e -c.,u:

A \

t

o .-

"\

\

I

,• o

•

•

B

'..,.-C

" Intermediate

8 Time t

Thlee types of experimental time-of-contact curves. (From Leon Lapidus, «Flow Distribution and Diffusion in Fixed-Bed Two-Phase Reactors, Ind. Eng. Chem., 49:1001 (1957), by pe/mission from the American Chemical Society.) FIG. 6-7

I ' ,

where u Da

=

=

axial velocity, m/s 2

axial dispersion coefficient, m /s

dp = nominal catalyst particle diameter (diameter of a sphere having the same surface area as the actual particle) Subscripts Land G denote liquid phase and gas phase, respectively. The fact that these workers came out with such a variety of results is quite disconcerting and confusing to the designer. It seems that everyone has been missing something important. Fortunately for the reactor designer, most design problems encountered involve industrial-size reactors, which are of the plug-flow type. In the case of laboratory or pilotscale reactors, the designer can act conservatively without committing a significant economic sin. He can simply select the Peelet number correlation which gives the highest backmixing (the poorest yield), hence the smallest Pe. If the actual pilot plant then produces a higher yield, no one will complain. 35 Mears has proposed a minimum reactor length z required for freedom from significant axial dispersion effects in trickle-bed reactors. For less than 90% conversion 'lnd first-order reactions, Mears' condition is expressed as: •

Z

200:

Co

- > 1n ---"d, Bo CJ

(6-18)

•

GAS-llQUID-PHASE FIXED-BED REACTORS 169 ,

where

0:

•

= reaction order

Z = length of reactor bed, m , d, = equivalent = (dp Lp

sph~rical

diameter of catalyst particle, m

+ tfpI2)0.5

dp = diameter of catalyst particle. m

Lp = length ('f ~talyst particle. m Bo = Bodenste;n number = ri, uliJ"n dimensionless u = superficial vebcily. :11/s

Da,

=

2

axial eddy diffusivity, m /s

C = concentration of reactant, kmoL'm Subscripts: 0 = initial value;

f

=

3

final value

The Bodenstein number is a Peelet number based on the particle diameter d, instead of dp. Mears used the correlations of Hochman and l9 Effron to calculate it. 1.0 = - - - - - - - - 0,8 0,6

OA 0.2 1.0

,

,

,

0

u "u

0.8

, o

10, the diffusion region 1

At 10- < Sh < 10, the transition region 1

At Sh < 10- , the kinetic region This model was checked experimentally for catalytic hydrogenation of cydohexene in methanol and I-propanol as solvent in presence of a Pt 51 black catalyst. A more realistic model of a vertical column of spheres was developed 50 by the same authors. This mathematical model is very similar to the 38 experimental model of Satterfield et al. The foregoing information on mass-transfer relationships in tricklebee: reactors should enlighten the reader in general terms. In specific terms, mass transfer will have to be expressed by means of an effectiveness factor for the particular liquid-phase reaction. That effectiveness factor is used in the design equations. How this factor is calculated will be discussed in detail in the next chapter. NOMENCLATURE a Catalyst particle surface-area/volume ratio, m-I -

9

A

Average area, m- [Eq. (6-3)]

Bo

Bodenstein number = d,ulD a , dimensionless 3

C

Molar concentration of reactant, kmoVm

C*

~oncentration

Cl C 2 C3

Constants of Eq. (6-3)

Cj

Molar concentration in the liquid at the gas-liquid interface, 3 kmol/m

Cp

Specific heat, kJ/(kg . K)

C;

Specific heat of saturated air, kJ/(kg . K)

Cs

Molar concentration oii. the c;;.~alysL surface, kmoVm

dp

Catalyst panicle diameter, m

d,

Equivalent spherical diameter of catalyst particle, m

D

Diffusivity in liquid phase, m%

Do

in the gas phase that would be in equilibrium 3 with the liquid at the catalyst interface, kmoVm

2

Axial dispersion coefficient for mass, m /s

3

•

182 CATAlYTIC REACTOR DESIGN •

Axial eddy diffusivity, m% •

Equivalent diameter of reactor, m Activation energy for the group Kvdvc Activation energy of gas-liquid reaction based on gas-phase concentrations, kJ7kmol Err

Activation energy of gas-liquid reaction based on liquidphase concentrations, kJ/kmol

f

Frictional factor, dimensionless

F

Feed rate of gas at reaction temperature and pressure, 3

m /s dim~nsionless

Fr

Froude number,

g

Gravitational constant,

m!S2

Force-to-mass conversion factor

=

:Mass flow rate in gas phase, J.: g/(m2 . s) Mass flow rate in liquid phase, kg/(m2 . s) Gallileo number, dimensionless 2 Equivalent heat transfer coefficient, kJ/(s . m . K) Gas-phase reaction rate constant Two-phase reaction rate constant based on concentration in the gas phase .

.

•

Two-phase reaction rate constant based on concentration in the liquid phase •

Constant of Eq. (6-3), depending on variation of pressure with the longitudinal coordinate

-

K

Vapor-liquid equilibrium constant

K,

Overall effective thermal conductivity, kJl(s . m . K)

KLS

Overall mass-transfer coefficient from gas-liquid interface to [he catalyst surface

Lp

Length of catalyst particle, m

P

Pressure: PI

Pe

Peelet number, dimensionless

Q

Heat transfer rate, kJ/s

TO

Radius of reactor tube, m

=

inlet pressure and P 2 = outlet pressure

Hydraulic radius of reactor, m •

•

GAS-UQUIO-PHASE FlXED-BED REACTORS 183

Re

•

Reynolds number, dimensionless 2

Sp

Total area of solids or catalysts, m

t

Temperature, DC

tb

Outlet mixing cup temperature, DC

u

Superficial velocity based on empty reactor cylinder, mls

v

Molar volume

Vc Vp

Volume of single catalyst particle, m

3

Total volume of catalyst particles, m

3

V"

Total volume of voids in the catalyst bed, m

W

Weight flow rate

x

Conversion, dimensionless

y

Mole fraction in gas phase, dimensionless

z

Axial coordinate, m

Z

Reactor length, m

a

Reaction order

3

Dimensionless constant in the equation of Ergun and Orn• mgor R?dial fraction of gas mass velocity External mass transfer resistance factor = C/C; or Radial fraction of liquid mass velocity (tlP)

I1tlm

Pressure drop per unit length. Palm Log mean temperature difference, DC .

Fractional void volume, dimensionless Ec

Gas-phase fraction. dimensionless

EL

Liquid-phase fraction or liquid holdup, dimensionless

Eo

Operating holdup, dimensionless

tp

Catalyst-particle-phase fraction, dimensionless

Es

Static holdup. dimensionless Effectiveness factor. dimensic:1kss Molecular conductivity of gas. kJl(s . m . K) •

Molecular copductivity of liquid. kJl(s . m . K) Viscosity. , . Pa . s Viscosity of water, Pa . s

•

CATAlVfIC REACTOR DESIGN •

•

Fl~id density, kg/ms

p

.

Standard deviation

(J'

•

,.

Empirical factor to correct for ":ortuosity" and for nonuniformity of catalyst pore cross section

Subscripts f Final F

Frictional

G

Gas phase

L

Liquid phase

LG

Two-phase

o

Initial or inlet

T

Total

w

Wall

REFERENCES 1.

J. Klassen and R. S. Kirk, "Kinetics of the Liquid-Phase Oxidation of Ethanol," AIChE J. 1:488 195 (1955). (Cited as example of countercurrent trickle-bed process.)

2. L. Philip Reiss, "Cocurrent Gas Liquid Contacting in Packed Columns," Ind. Eng. Chem. Process Design DeveWp. 6:486 199 (1967). ,

3. M. Orhan Tarhan, Vaporizing and Pretreating Aromatic Hydrocarbons Feedstocks without Polymerization, U .S. Patent 3,448,039, 1969. 4. Paul B. Venuto and P. S. Landis, "Organic Catalysis over Crystalline Aluminosilicates," Advan. Catalysis 18:259-371 (1968) . 5.

•

J. C. Charpentier, C. Post, and P. LeGoff, "Ecoulement ruisselant de liquide dans une colonne it garnissage. Determination des vitesses et de- debits relatifs des films, des filets et des gouttes," Chim. llld. Genie Chim. 100(5):653-4)65 (1968).

6.

J.

C. Charpentier, C. Post, W. Van Swaaij, and P. LeGoff, "Etude de la

retention de Iiquide dans une colon ne it garnissage arrose it co-courant et a contre-courant de gaz-liquide. Representation de sa texture par un modele de film~, filet~, et gouttes," Chim. lnd. Genie Chim. 99(6):803-826 (1968). 7. F. DeMaria and R. R. White, "Transient Response Study of Gas Flowing through Irrigated Packing," AIChE J. 6:473 181 (1960). 8.

E. Dutkai and E. Ruckenstein, "Liquid Distribution in Packed Columns,"

Chem. Eng. Sei. 23:1365-1373 (1968).

9. C. J. Hoohendoorn and J. Lips, "Axial Mixing of Liquids in Gas-Liquid Flow through Packed Beds," Can.]. Chem. Eng. 125-131 (June 1965).

• GAS-UQUI

FIXED-BED REACTORS 185 •

•

10. G. J. Jameson, "A Model for Liquid Distribution in Packed Columns and Trickle-Bed Reactors," Trq.ns. Inst. Chem. Engr. 44:TI98-T206 (1~6). 11. L. Musil, C. Prost, and P. LeGoff, "Hydrodynamique des colonnes noyees. a' garnissage avec contre-courant de gaz-liquide. Comparaison avec les colonnes arosees," Chim. Ind. Genie Chim. 100(5):674-682 (1968). 12. K. E. Porter, "Liquid Flow in Packed Columns. Part I: The Rivulet Model," Tran. There are no standard published methods. Many oil cO!llpanies have developed and are using their own proprietary design procedures. In the following we will attempt to develop a procedure based on the present-day best published knowledge of tricklebed reactor design.

7.1 GENERAL PROCEDURE

The first thing the designer should do is study the reacting system with all its kinetic and mass-transfer peculiarities, because this infOllIlation will determine the kind of design model selected. There is a choice between a plug-flow and a partially backmixed reactor. In either case the reactor will be concurrent downflow and adiabatic. The use of criteria discussed in Sec. 6.2 will determine whether the reactor is plug-flow or not. We will select a process and design one reactor of each type in this book. A hydrocarbon oil desulJurization reactor is selected as the example. The following assumptions can be made: · The reaction occurs between the liquid phase and the catalyst surface. · The reaction is adiabatic. •

• The key reactants are sulfur compounds dissolved in the oil feed and a hydrogen-containi~g gas. · The liquid volume in the reactor remains constant.

•

•

f90 'CATALYnC REACTOR DESIGN •

Although OUt main purpose is to design a reactor for the conversion of sulfur compounds, we must take account of all the side reactions occUlling• in the reactor because they all contribute to heat evolution and make up the composition of the reacting mass. Because the number of individual reactions is too large, it is customary to lu~p them into groups of reactions. We may write one equation for each of the following lumped • reactIon groups: · Hydrodesulfurization · Hydrodeoxygenation · Hydrodenitrogenation · Hydrocracking · Saturative hydrogenation

,

, "

'

Because economics dictates the recycling of the unused hydrogen, the recycled gases normally contain certain impurities. Hydrogen sulfide, the product of hydrodesulfurization, can be easily removed from the recycle gas.] Thus the major impurity will be methane, a product of the hydrocracking side reaction. It is customary to remove at least part ef 2 this methane from the effluent gas and to produce hydrogen from it. Other impurities are water and ammonia. Ammonia is normally removf'd together with the hydrogen sulfide in a monoethanolamine absorber, and water vapor is removed from the desulfurized gas in a molecular sieve or equivalent dryer. In order to develop the design equation, reference is made to Fig. 7-1, which represents a cylindrical packed bed of catalyst. In this bed a Gas in{C (0) =Co T(O)=T v Liquid in z=O

r

Area

'If

2

0 T /4 f, Z - •

Pa eked bed of ealalysl

~'

V.L

,• ,

•Z • Z + f,z

,...

ZT

•

C{z) Cdzl} T{z) TL(z)

V

•

z=z T Gas out ,

•

LIQ UI d out

FIG. 7-1

reactor.

Schematic representation of a typical downflow gas-liquid fixed-bed

•

•

PROCESS DESIGN OF TRICKLE-BED REACTORS

191

differential height !:!.Z is considered for which a materials balance and a heat balance ",ill be obtained. The design will be determined for the liquid-phase reaction, and gas concentrations will be calculated from the liquid-phase reaction at all points. Let us desip"nate: ~

CH = molar hydrogen cOf'centration in bulk phase, kmol/m

3

CL = molar conu'ntrarion of key reactant in liquid phase or molar

concensation of hydrocarbons containing an atom of sulfur, 3 km:JUm ., A, = cross-sectional area of the catalyst bed = 71'DT/4, m.)

z = axial coordinate in the reactor

ZT -.: total reactor :ength, m Let us introduce the concept ..lf phase fraction, which is the fraction of the reactor bed volume occupied by anyone phase. The total volume V of the packed catalyst beds consists of the volume taken by the catalyst particles Fp (which includes both the volume of the solids and that of the pores). the gas volume Vc, and the liquid volume V L • The following equations can be written:

,

Fp V +

and

I

and

F(; - cc V

Ifweset

•

Vp F

cp

VL V

CL

it follows that (7-1)

where

cc = CL

=

cp =

also

c

=

gas-phase fraction liquid-phase fraction or liquid holdup catalyst particle phase fraction void fraction

=

Cc

+

CL

While tp and hence £ are constant along the bed length, lOG and tL may vary somewhat in s()me cases and appreciably in others. \Ve will compute CL according to Satterfield's Eq. (6-19): (6-19)

•

•

192 CATALYTIC REACTOR DESIGN ,

Plug-Flow Trickle-Bed Reactor The case of the plug-flow industrial size reactor will be considered first. Let us ,make a materials balance for the liquid phase of the differential reactor volume increment:

(Molar inflow axially) - (molar outflow axially) - (molar disappearance by reaction)

=

°

Let us calculate the value of each set of parentheses in general terms: Molar inflow axially

=

Molar outflow axially = Molar disappearance by reaction =

I 4' DT UL EL CL = 1T

2

~ D~ UL EL CJ-.l= -E

L

'iT -

4

2 DT

az: Cr :-;, CrhY' a!1d ..C rhc : --

dp

~R2

Cox(l llL

(7-19)

•

•

196 CATALYTIC REACTOR DESIGN

•

•

dCrN - dp !K3 ·dZ C NO UL

(7-20)

dCrby _ dp 9l. 4 dZ ChyO UL

(7-21)

• •

dCrbc

dZ

-

dp ~R5 ChcO

(7-22)

UL

Because the major reactant feed components are rather high-boiling liquids, there is no reason to be concerned with the mass transfer of these components from one phase to the other. However, ifit is necessary to calculate the mass transfer of H2 from the gas phase to the liquid phase and the mass transfer of reaction products such as H 2 S, CH 4 , H 20, and NH:\ from the liquirl phase to the gas phase, the following equations can be written: kKHa (CUB kKH?S a(CLH,s -

CuH,s)

=:0

L

(7-23)

(H 2-consuming reactions)

!RI

(7-24)

:H • -

CUCH.) =

!R.4

(7-25)

kKHP a(CI.Hp -

CuH,p) =

!R.2

(7-26)

kK:-;H, a(CL:-;H, -

Cu:-;H,) =

!R.3

(7-27)

kt;

In Eq. (7-39) everything except the Peelet number is relatively easily o~tainable. The Peelet number Peha2 is a function of gas-phase and liquidphase axial dispersion coefficients, which have to be determined. The reactions, which take place in the catalyst partide. release the exothermic heat right inside the particle. This heat is propagated across the exterior surface of the particle to the bulk liquid and from there to the gas stream. As in the case of the NINAF reactors, the temperature inside the particle is assumed to be constant. Now let us summarize the relationships that can be used to describe the trickle-bed reactor system with axial dispersion: The five reaction rates given by Eqs. (7-28) to (7-32) are of course the same as in the plug-flow case. The design equations are Eq. (7-39) as the heat baiance equation and dZ

-

(7-40)

, .

200 CATALYTIC REACTOR DESIGN •

•

d2Crox I dC,ox 2 dZ 'Pe"",1 dZ 0

dp 9(2 UL Coxo

(7-41)

•

1 ~CTN - dp ~R3 dC T.'X 2 Pern.... dZ Ut. CNO dZ " my _ d" ~R4 I d-C dC m, -2 Ut. C hvO dZ rlZ Pe"",l

(7-42)

0

o

(7-43) .

•

dC me d7.

--

'1

I 0

Pc "",I

d-C me 1Z2

d" !K" ill. C hd •

(7-44)

The following boundary conditions must be written:

z=o Crln = l.U

Inlet

.

•

".

=

C"IL

Crox

= Le

dC T - 0.0 Z = ZT dZ

Outlet

L

CT

1.0

--

1.0

C,N --

1.0

T T = 1.0

dTT - 0.0 dZ

It is not possible to write boundary conditions for each ODE. One can design a reactor for the conversion of a specific fraction of the suI fur , the oxygen, or the nitrogen or for a certain amount of hydrocracking or saturation, but not for all these specific conversious at the same time. If it is decided to design for su!fur conversion, one has to accept whatever one obtains from the other reactions.

7.2 DESIGN EXAMPLES FOR PETROLEUM HYDRODESULFURIZATION Let us now calculate two specific examples for the petroleum oil hydrodesulfurization reactor that was discussed generally in the previous section. Selected design parameters related to both examples are listed below: o

Desired final fractional sulfur conversion

Xv =

0.65

· Feedstock boiling in the 260 to 365°C range and containing 4% sulfur, no oxygen, 0.35% nitrogen, and olefins corresponding to a bromine number of 5.0 · Temperature 375°C = 648.2 K •

o

o

Pressure

=

55.0 bars

H2/oil ratio

3

=

*

0.5 m H 2 (NTP)/kg oil

· Catalyst: cobalt molybdate on alumina, 5-mm square cylinder pellets of PE = 950 kg/m3

•

PROCESS DESIGN OF TRICKLE-BED REACTORS •

· Kinetics: key reaction is first-order, kinetic data are given in Eqs. (7-28) to (7-32); 111 = 0.4; weight .hourly space velocity (WHSV) = 2000 3 kg/(m 'h) · Void space in catalyst bed

E

= 0.36

· Oil density at ambient temperature, PLO, is given as 35.3° API or 141/ (35.5 + 131 5) = 843.3 kg/m3 · Thermal expansion of the oil, 1.49 at 375°C, ~iyes density at reaction ~emperature PL = (843.3/1.49) = 569.3 kg/m' -

CpL

=

3.3095 kJl(kg' K)

Most industrial-s:ze reactors are of the plug-flow type. The general design contiguration of such a trickle-bed reactor is shown in Fig. 7-2. The reacting gas G enters the reactor vessel from the top. The reacting liquid L also enters from the top but is distributed by means of a liquid distributor over the catalyst bed As will be seen further below, the reactor will need twc catalyst beds with a gas quench in between as a means of interbed cooling. The effluent gas leaves the vessel at the bottom. The reacted liquid accumulates at the bottom and is rt:leased from the reactor by means of a leyel controller. Design parameters specific to the plug-flow reactor are given below: The plant capacity is 250,000 tlyr. The approximate reactor volume can be calculated from the weight ,

G L ...r--Bed 1

Quench

v-Bed 2 •

G• - - -- - - -

-- -

,Le ,, , L

FIG. 7-2

Trickle-bed reactor with interbed quenching.

•

CATALYTIC REACTOR DESIGN ,

hourly space velocity. Assuming an on-stream time of 310 days/yr, the hourly capacity is ' ,

,

250,000 kg/yr (310 days/yr)(24 h/day)

=

33 602 k /h , g

=

9.33393 kg/s

Then the approximate reactor volume is 33,602 kg/h, = 16.801 m''\ 2000 kg/(h'm") In order to determine the reactor's hr:/Sitt1diallleta mlio, we must assure that the reactor will indeed operate as a trickle-bed reactor. For this, we may refer to Fig. 6-4 and calculate the gas/liquid ratio as follows. We 3 note that 0.5 Nm Hikg oil is equivalent tu (0.5)(2.016)/(22.4 m~/kmol) which equals 0.045 kg H2/kg oil. Then, sillce 5000 lb oiU(h·ft2) appears to be safely in the gas-continuous region up to 250 Ib gas/(h'ft~), the hydrogen rate, assuming pure hydrogen is used, is [5000 Ib/(h·[(2)] . (0.0451bllb) = 225 lb H 2 /(h·fe). Indeed, 225 Ib/(h'fe) is in a safe region in Fig. 6-4 and the process is certain to be trickle-bed. The minimum reactor cross-sectional area is: 9

9

(33,602 kg/h) . (0.0929 m-/fe) = 1.3782 m 2 [5000 lb/(h'fe)] . (0.453 kgnb)

•

•

Since we already know the volume, the reactor height will be l6.80 1 3 2 m /1.3782 m = 12.19 m and the reactor diameter 1.324 m, which are quite reasonable values. A lower height/diameter ratio would also be permissible since the process would then be slower. An economic study should be made of the cost of the reactor at different height/diameter ratios. At large diameters and high pressures, the reactor wall must be thicker and hence more expensive. On the other hand, the cost of the reactor is also a function of it:> external surface area, which for perfect cylinders is given by the relationship: Surface area 4 -----= Volume H

H D

+ 0.5

where H is height and D is diameter. This relationship gives a minimum .surface/volume ratio for H = D. Because in this example we do not want to go into cost problems, we will simply select a somewhat larger diameter, namely Dr= 1'.5 m. The corresponding cross-sectional area A, 2 is 1.7671 m and the approximate reactor height is 9.5 m. The sulfur content of the feedstock is gi,'en as 4.0 wt% of sulfur. This corresponds to 4.0/(32.064)(100 kg) = 0.12475 kg-atoms SOIIOO kg of SO-containing material.

•

PROCESS DESIGN OF TRICKLE-BED REACTORS 203

When designing a reactor that is actually going to be built, the designer should use actual molecular weight data obtained in a reliable laboratory on properly sampled representative feedstocks. However. for the purpo~e of this book we can look up the molecular weight of oil from page 6 21 of Maxwell's book from which we obtain a value of 670 for an oil with a mean boiling point of 594°F (322.2°C) and API gravity of 35.3°. Then 100 kg of oil contu.ins 100/670 = 0.14925 kmol. Thus the mole :ractio'l of S-containing oil is: 0.12475 mol SIlOO kg 0 _ = .8358 0.1492;) mol/lOO kg

= •)'"

This means that 83.58% of all molecules contain one atom of sulfur. Lf't us calculate C /.!,: (569.3 kg1m:{) = 0.7102 kmol S/m:! (32.064)(100 kg)

= 14.0)

Also, 0, ,,·t 'k oxygen = 0)16 kmolll 00 kg of oxygen-containing material. The gas oil selected contains no oxygen. However, for reasons of completeness, equations for oxygenated compounds will still be developed: Co,,, == __ O-"'-,P..:c'-_ kmol/m:1 (16) (lOO)

The

\

of

is best expressed as a bromine number. A bromine numbc" uf 1.0 means that 1.0 g of bromine is added to the double bonds contained in 100 g of feed stock. The molar concentration of double bonds in the feed can be expressed as follows: content

11llsaturaies

-

ChcO

=

Br. - no. PL (0.001) k (159.832)(0.1 g)

=

0.1781 kmol unsaturates/m

:l

The contt'lIt of h.• drocrackable compounds is difficult to define. Only the reaction results show the extent of hydrocracking. The basis of feed concentration is the entire feed: 569.3 k g / m : ' , == 670 = 0.8497 kmol/m

The nitrorzclI colI(t'lltratioll is 0.35 wt % N in feed. The molar c0ncen,. tration of nitrogen-conpining compounds is: \0.35) I'L ,~ C,," = k /k = 0.14224 kmol N compounds/m(100)(14.008 'g g-atom) The molar COllfelltration of suIfuT in liquid phase will then be

-

• •

s GLO = (4.0)(569.3)/(100)(32.064) = 0.7102 kmollm initially and GL = 0.7102(1 - x) at any point. in the reactor after that. Since the liquid volume is assumed to remain constant throughout the reactor, conver-' si on x may be defined in tenus of concentrations:

Similarly Xj

= 1 -

erN

X~

= 1 -

Grhy

X;

=

I - Grhe

One kilo mole of gaseous process mixture at 375°C and 55 bar occupies the volume of 3

(22.414 m NTP)(648.2 R)( 1.0] 325 bar) (55 bar)(273.2 K)

=

0.97972 m3 .

S

Furthermore, 1 m of reactor void space will contain at reaction con3 ditions 110.97972 = 1.0207 kmollm , which is also the initial hydrogen concentration GHO ' The Hloil mol£ ratio can be calculated from the H/oil ratio: S

(0.5 m H 2 )(670 kg/kmol). = 14.946 kmol H 2/kmol oil 3 (22.414 m NTP/kmol) (1 kg od) '-

-',

For simplicity's sake in calculations, the reactor is a~svmed to use pure 3 hydrogen, although this is not common. Thus GHO = 1.0342 kmol/m • The densit), of oil is given as 35.3° API. This corresponds to 141.5/(35.3 s 3 + 131.5) = 0.8483 g/cm or 848.3 kg/m . The thermal expansion of this 6 oil is 1.49 at 375°C according to page 143 of Maxwel!'s data book. Thus the density at reaction temperature is 848.3/1.49 = 569.3 kg/ms = PLo This figure may be used as a constant because it does not vary significantly in the temperature range of the reactor. Howeyer the gas demit)· does vary with conversion and should be expressed by the function: p = (273.2 P122.711 T)(2.016(1. -

+ 34.032 where

QI

= (4.0%

QIXl

Q1X 1 -

+ 18.016 Q2X2 + •

Q2X2 -

Q4X4 - Q;x;) 16.043 Q.~X3 + 5.6773 Q4x.d QgXg -

•

S/IOO)(670·kg oillkmol)/(32.066 kg .s/kmol). (14.946 kmol H2 available/kmol oil)

If all the sulfur would react, it would involve QI fraction of the initial

.

---,-. '>,.--'-

~---~>----~­

_.. .

.

PROCESS DESIGN OF TRICKLE-BED REACTORS

""

205 •

.

•

•

•

H 2 • Similarly. if all oxygen in the oil would react with H 2 • it would involve . Q2 fraction of the initial H 2 , ete.

Q2

= (% 0:/100)(670 kg oillkmol)(l.O kmol H:/katom 0)/(16.0 kg ~

O/katom)( 14.946 kmol H2 ayailable/kmol oil) Q:l

=

(0.35 % 1\;/100)(670 kg oil/kmol)( 1.5 kmol H 2/katom f\)/( 14.0 11 kg N/katom)(l4.946 kmol H2 available/kmol oil)

We will assume that every hydrocarbon mole is hydrocrackable.

Q4

l

(kmol H:/kmol oil)(22.4l4 m:l/kmol)/(0.5 m: H/kg oil)(670 kg/kmol)

=

= 0.0669

Q'>

Br no.l159.832)(0.01)(670 kg/mol)(22A14 m:' /kmol)/(14.946 kmol H2 available/kmol oil)

=

Reynolds m'mbers are calculated as follows:

and Here dp is the diameter of a sphere that has the same outside surface as the catalyst pellet we use. The surface of the square cylinder pellet is 271"1"2 + (21")(271"1") = 671"1"2. When r = 0.0025 m, the surface area is 2 0.00011781 m "'hich equals the surface of the sphere, 471"r~. Then r., = (0.000117811471")0.5 = 0.0030618, and thus dj, = 0.00612 m. Viscosity of hydrogen at 375°C = 0.015 cp = 0.000 IS g/(cm's) = 0.000015 kg/(m·s). (See Chemical Engineers' Handbook/ Fp. 3-21.) Viscosity of oil = 0.0021 g/(cm's) = 0.00021 g/(m·s). (See page 163 of 6 MaxweIl's book. ) •

•

~,~

,

~

,

~

GL

=

(9.3339 kg oills)/( l. 767 m

2

)

=

5.2819 kg/(m2·s)

G = (S.2819 kg/m2·s)(0.045 kg H2/kg oil)

=

0.2377 kg/(m2·s)

Rec = (0.00612 m)(0.2377 kg/m2·s)/[0.00001S kg/(m's)] ReL =

=

•

96.98

(0.00612 m)(S.2819 kg/m2·s)/[0.00021 kg/Cm's)] = 153.93

In order to calculate the linear velocities u(; and must first calculate the phase fractions E(; and El.'

UL

in the reactor, we 3

Liquid flow = (9.3339 kg/s)/(569.3 kg/m:l) = 0.016395 m /s Gas flow •

3

= (9.3339 kg/s)(O.S m /kg)(648.2 K)(1.01325 bars)/ 273.2 K . S5.0 bars = 0.20399 m%

The void fraction is ·assumed to be E Equation (6-16) is used to estimate El.' -

=

0.36. Thus

EL

+ Ec = 0.36.

-1/3

.

•

206 .CATALYTIC REACTOR DESIGN

,

where Re

=

153.93

d

=

dp

= 0.00612 m

J.LL = 0.00021 kg/(m's) PL = 569.3 kg/ms

g Thus

2

9.807 m/s

=

= 0.208

EL

= O. ~0833

Eo

= 0.36 - 0.208 = 0.152

Now we can calculate

UG

and

UL

at the inlet: 3

UG

=

_ UL -

0.20399 m /s ="'_ / (1.767 m 2)(0.152) 0./;)95 m s 3

0.016395 m /s _ 00 6 / (1.767 m2)(O.208) - . 44 m s

We can also calculate the residence times of the gas and the IUfuid in the bed: =

OG

e L

3

(16.801 m )(0.152) = 12. 19 s 0.20399 m3/s 5 3

= (16.801 m )(0.208) = 213 15

0.016395 m3/s

.

s

•

= 3.5525

. mm

The specifu; heat of the oil is assumed to be constant and equal to 3.3095 kJ/(kg· K). However, the specific heat of the gas is a function of gas composition and will change with conversion. We need numerical values of the specific heats of water, H 2 , H 2S, NHs, and CH 4 : cp (H 2 0) = 34.401 + 0.000628 T + 1.34

X

10- P If(Z.GT.Q98J.O) GO TO 0 If (ICOUNT.NE.IFREQ) GO TO 2 WRITE(6,200) ZZ,Y(1),Y(ZJ,Y(3J,I(4J.T 200 fOB:!A! (18 ,lP6El0.3) ICO O!iT = J GO TO 2

C

.....

• • •• E NJ 0 F BE 0 ......

" WRITE (6,2\)0) 2Z,1(1) ,Y(2) ,1(3) ,1{1I) ,T WRITE (o,;:Ol) I 201 FOoMA: (lA ,SX,11AilND Of B£ll .12) C •••• CO~P;)TE GAS C(,WOSITICN AND QUENCH AT !lED OUTLET •••• EMF =(H~D*O.5/22.4'Q)*(1.-Ql*(1.-CR) - ;)3* {1.-CllNl 1 Q4*(1.-CRHY) - :)5*(1.-CRHCii H2SMF - FEE)*S*(1.-CR)/32J6.6 AI!~I' = FEED*N1TliO*(1.-CRN)/1400.8 CH4~F = FEED*(1.-CEHY)/~WL SU~ = B"F + H2S~f • A~l!f + CH4~1' QUENCH; (FEED*CPL/~i/L) + (SU!I*CP») * (TR*TO - TO) / (29. 895 * 1 (TO-298.2» WHTE (6. 202) QUE Nca 202 FOHHT(1HO, 5I,9HQUEIICR = ,1'10.5,78 KIIOL/S) WRITE (6. 230)

FIG. 7-5

Fortran program RDMOT4 for designing a plug.flow industrial·size

trickle-bed reactor for petroleum oil hydrodesulfurization (continued). 214

•

230 FOBft1T(1aO.1SX.29HGAS CG"PCSITIOR. "OL FBACTIOR/1BO.9X. 1 8HHIDaOGER,SX,lBB2S.7X.lBMHl.7X,3BCHfll BID = B"F /50" H2S = B2SIIF/SOII A" = AII"F /SO" CHII = CHilli F/StJ" iRITE{6,2311 HID.82S. All. CHII

231 FOI8AT(1HO, 8I,F8.5,2X,F8.S,2I,F8.S,2X,F8.51 SlYEZ '" Z SAVEtl: Y(lI SlYET2= 1(2) 51 VE13= I PI SAVEIQ= 1(41 51 VE IS = 1. 0 CORTI!IUE USE THE TOP -5" IF THE REV BED SHOULD ST1~T AT TB=1. USE THE BOTTO" "S" IP THE HEll BED SHOULD SlABT 11 la=IBPtJT VALOE. •••• END OF REACTOR •••• IIElTE (6, 200) ZZ, 1(1). 1(2), Y (31, Y (11). T IlRITE (6, 2031 FOBIIAT(1H ,5X.14*(I.-CRH(I»))

•

•••• CALC~LATIOS OF THE S?FClf~C HEAT CP OF ~EACTING GAS ~IX ••• , c;'il2 - 27.698 + (0.003189 .. TO .. TR(I») CPH2S = 10.1243 + (3.015J6 * TO * TR(I» (PCR" = 22.343 .. (0.04312 lO • TIl(II) Ci'N33 ~ 28_0328 .. (D.02i36 * ~o *TR(!) CP = CPH2 *(1.-(;)I*(1.-Cll(I)j) - (Ql*(1.-GIN(I»)) 1 (Q4*(1.-CRilY(II» - (QS*(I.-CRHC(I»)) .. (CPH2S.~1.(1.-CR(IlI/2.)

*

2 .. (CPNi!3*;;3*(1.-CBN(I»/3.) C

._ ••

•

(CPC~4.-.14.(I.-CRHY{I»)

OF SDi'3fiI'ICIIL GAS VELOCITY 0 ••• , (GL*T)*1.01325/(273.2*E?SC*P) )*(HOIL-( (CNO. p_-CRN (1)/3» CA~CJLATIOa

;) = 1 (C'lCO* (1_-CRHC

(1»»

•

/RHCL)

C

sa =DP*EPSL/(TO*«EPSG*RHO*C?*U) .. (EPSL*RHOL*CPL*UL»I SlG~AD=

(DH"Rl(I)/DA~I(IJJ"

1

\,

".

(DH'+*R4(I)/DB4(I» C22 = -(2.*C21)" (1./ilELTA2) C211~-!lB' S IG1lAD AA(I) = C21 E(l) ~ C22 C (1) = C23 18 D (1) ~ C2'!

. C

..

(DH3*R3(I)/DAI'I3(I))

..

(DHS.RS(I)/DA~5(I»

•••• SET BOUNDARY VALUES •••• 0(2) = 0(2) - C21 B (11)

~

C23 + C22

CALL TRIDAG (2,.;."'!,AA,B,C,.D,.'IRX) TRX(",) = TRX (M)

IiRITE (6,202) (TRqI), :-41, 'l01, 202 FQR:lAT (10HO TRX(I) =.lP13E12.1) ro 19 I = 2, M1 19 IF(TRX(I).Ll.1_)TRX(I)=1. C

c

40)

.... '1::57.... _ :l0 30 1=2. ~ IF(AB5(!RX(I)-TR(I)}.GT. J.OJ1) TR (Il =':ilX (I)

ITER=.T?UF.. •

Fortran program RDMOT3 for designing a backmixed laboratorysize trickle·bed reactor for-petroleum oil hydrodesulfurization (continued).

FIG. 7-9

229

, 30 CONTINUE IF (ITER.lIID.1I0IT"R.Li:.20) GO TO 20 WUTE(6, 220) 220 FORIU.T (lilO, 13X, 11311S ~ A C 'I 0 R O E S I G N

RES U L 'I S)

C

(6, 2(13) (ZZ(l), I~41, 401, 40) (10HO ZZ (I) =, lPl0E12. 3) (6, 198) (CR(I). 1=41. 401, 40) (6. 199) (CRII(I). I=Ql, 401, qO) WaIn: (6, 200) (CRIlT(I), I=ql, 401, 40) WHITE (6, 201) (CRHC(I), 1=41, 401, ~C) WRITE (6, 202) (TRX(I), 1=41, 401, 110) 'IS (l! 1) = '!'S (!\) WHITE (6, 2(4) ('IS (I) , 1=41, 401, 40) 2011 FORIIAT(10ilO TS(l) =, 11>10E12.3) CS (Ill) = CS (11) WRITE (6, 2051 (CS(I), 1=41, 4Cl, 40) 205 FOR 11 AT (10HO CS(I) =, lPl0E12.3) IIR1TE(6, 230) 230 FORIIAT(lHO,1SX,291l0AS CCIIPOSITIOII MCL-FRACTIOH/1HO,10T, 1 5HZZ (I) , 4T, 8HHYDPOGEII, SX, 3HH2S, 71, 3HNH3, 7 X, 3HCH4) DO 231 1=111, 401, qv HMI' = (YEEo*O.5/22.q"')*(1.-~1*(1.-CIi(:I11 - Q3*(1.-CRII(III 1 Q4*p.-CRHI(I)) - QS*p.-CIiUC(I») H2S'iF= P?:Eo*S* (l.-CR(I) )/3200.6 AIIIIF = PEEo*/lITBO*p.-CRN(l») 114CO.8 CH4l!F= FEED*(l.-CRHY(II)/IIWL SU~ = H~F + H2SKF + A~~F + Caq'F IlYO = HIIP/Sill! 1l2S = H2SIIF/3U!! All = All:! P/SUII CH4 = CB4I1P/SD~ WRITZ (6, 232) ZZ(I). HTD, 82S, A~, CH4 231 CONTINUE 232 FOR/IAT (1HJ, 10X, F6.4,' 4X, F6.5, 41, F6.S, 41, F6.S. qX. FG.S) STOP END SUBROUTINE TRIOAG SUBROUTINE FOS SOLVING A SISTE" OF LINEAR SIIIULTANEODS EQUATIONS HAVIHG A TRIDIAGO!lAL COEFFICIENT !!ATRIX. THE 01"01

'1. 15r,r-(l.l_,,_B.A9L~:)J_

1:005£+02

1.20n'02

1.405P.'02

1.605E+02

1.805E+02

2.005£+02

O,i,1_4E-01

I!.. , ,'l.'!E-O1

~.141F,:O'

7.90'i£-~1

7.676£-0.1

7.496£-01

9.643£-01

9.~27£-01

9.412J'!-01

9.2-

.

-",-,.,

.

1.361P.·OO

.".

-

--

I. 36 1 F.-OO

-~---

,.

1.

3~

2.1nE-Ol .. ... 7.4nr.-04

1.773F.-04

l. q ~f; F.- O~

9. OM p.-O~

2.572P.-06

'l.11~P.-04

2.601E-04

7.42QF.-OS

2.121£-000

'.146£+011._ .. 1.363£+01) ---

1.091E-02

-----

~--.-----.-

2.456P'-04 6.'l04P.-O~ _.". ,--.-"'--,-..

3.0968-01 -"--

....

•

-

1

1.0Q3P-02 .----

-,_._-

--~---

",""

4. 62A£-0 1

-

~-------

1.401P.-Ol •

04 JP.-O 1

~.

-

2.'10qP,-OI

LTi,2f.+ 00-· -. CfioE+ 0·0 -Cf'lilfi-ob-·· 1.-:11, f. > 0-0"

=

7.

3.619E-03 ... ... 'l. 14n- 0;>

f: 1 0-6F >DO--·-l.111~'O(i

Cq (.I)

~----

R. 4.18 E-03

2.'146f-Ol ..

'

_ __ •

1.9598,,02

'"

",

_

4.515£-02

2. M5P.-O 1

--"

1. 5l31!-~4

~

1.3Aor+00

04Ar-1) 1

'"

6. A09£-04

.. __ ."_" __

1.379£_00

ij.

,

Q.'i52E:-0:l

J . .11 ?J:,,-,,+",,0 0

~.615E-Ol

,

1.376E-Ol.

_. __ . __ ,_ _ _ _ . _____ " __ , __ . _ ... _ .. _ _ _ _ ._

I.J72E_OO -

1.4Q6f-Ol

-

l.q7AE-Ol

.... _.

-~------~---,.

1.30.1

Inlet concentration of unsaturated compounds. kmol/ms •

Reduced molar concentration of hvdrocrackable com• • p":JlE1ds. dimellsionless Reduced molar concentration of nitrogen-containing compounds. dimensionless

..

•

PROCESS DESIGN OF TRICKLE-BED REACTORS 237 •

c,.ox

Reduced molar concentration of oxygen-containing com- ' pounds, dimensionless

dp

Catalyst particle diameter, m

d,

Equivalent spherical diameter of catalyst panicle, m

D

Diffusi\'ity of reacting molecules in liquid phase, nlls

DH

2

Molecular diffusion coefficient for hydrogen, cm /s 2

D lleff

Effective diffusion coefficient for hydrogen, cm /s

Dr

Inside diameter of reactor, m

E

Activation energy, kJ/kmol

HIM

H ydrogen/oil mole ratio

RI

Reaction rate constant for type I sulfur

k~

Reaction rate constant for type 2 sulfur

kH,S

Adsorption equilibrium constant for hydrogen sulfide

Ka

Axial thermal diffusiyity in packed beds, m~/s

K;

fherage adsorption equilibrium [in Eg. (7-50)] constant for aromatic h vdrocarbons •

LHSV

Liguid hourly space velocity

P

Total pressure, bars

p.;

Partial pressure of aromatic hydrocarbons, bars Peclet number for mass and axial dispersion in liquid phase, dimensionless

.,

\ .

b~rs

PH

Partial pressure of hydrogen,

Po

Vapor pressure of sulfur compound, bars

~R

Global reaction rate, kmol/(s . m:l) l

5

Sulfur concentration in the product, kmollm: or [in Eq. (750)] amount of a single sulfur compound

5'!

Initial content of type I (aliphatic) suI fur, kmolim:l

s~

Initial content of type 2 (thiophellic) sulfur. kmol/m:

T

Absolute temperature, K

l

Absolute inlet temperature of feed •

to

reactor, K

Tr

Reduced temperature, dimensionless

u

Superficial velocity of the gas phase, m!s

UL

Superficial velocity of the liquid phase, m!s

•

238 CATALYTIC REACTOR DESIGN

,

Mole fraction of liquid vaporized, dimensionless

Vi

,

v

Total volume of catalyst beds of the reactor, m

3

,

Gas volume in catalyst beds of the reactor,

01

3

Liquid volume in catalyst beds of the reactor, m

3

Volume of catalyst particles in packed beds of the reactor, 3 m

",'HSV

Weight hourly space velocity

x

Mole fraction of compound S in liquid, dimensionless Mole fraction of compound S in feed liquid, dimensionless

z

Axial coordinate in the reactor, m

z

Dimensionless axial coordinate in the reactor Total reactor length, m Reaction order, dimensionless

0:

Heat of reaction, kJlkmol

'

Effectiveness factors of various reactions

TJ

Void fraction, dimensionles'i

""