Comparison of two-dimensional models for fixed bed catalytic reactors

Chewsid Dylinecring Printed in the U.S.A.

Science Vol. 39, No. 6, pp. 1017-1024,

19114

030%2w9/84 Rrg8mc.n

COMPARISON OF TWO-DIMENSIONAL MODELS FIXED BED CATALYTIC REACTORS

$3.00 + .oo Rns Ltd.

FOR

S. I. PEREIRA DUARTE Departamento de Ingenieria Quimica, Facaltad de Ingenieria, Calle 1 y 47, 1900 La Plats, Argentina Centro de Investigaci6n y Desarrollo

G. F. BARRETO en Procesos Cataliticos (CMDECA), Plata, Argentina

N. 0.

PINMATE,

Calle 47 No. 257 1900 La

LEMCOFF*

Departamento de Industrias, Fact&ad de Ciencias Exaetas y Natnrales, Ciudad Universitaria, 1428 Buenos Aires, Argentina

(Received 19 ApriI 1982; accepted 27

September

1983)

Abstract-The performance of different two-dimensional models of fixed bed catalytic reactors is analyzed for different situations. The deviation of the response of the simplified models with respect to the heterogeneous one is explained in terms of the operative conditions and the value of dimensionless groups. A very simple expression to predict the errDr in the conversion for the models considered is derived. A reasonable apreem ent with the errors calculated from the numerical solution of the two-dimensional

models is foun;i

1. JNTROlMJCI-ION modelling of non adiabatic tabular fixed bed reactors has received considerable attention and various attempts have been made to develop twodimensional models. A review of the different models The

was recently presented by Froment and Bischoff[l]. The models can be classified as pseudo-homogeneous or heterogeneous, depending whether the distinction between conditions in the solid and in the fluid phase! is taken or not into account. When interfacial gradients are significant, an heterogeneous model should be used, and separate balance equations for the fluid phase and the catalyst are written. The correct way to set up the energy balance equation is to consider separately the radial heat transfer in the solid and fluid phases by means of effective heat transfer parameters for each phase, as it was done by De Wasch and Froment[2]. However, most authors[3-7l have only considered the heat transfer in the fluid phase, and used lumped effective thermal parameters. This approach is conceptually wrong[l]. Therefore, three different models can be distinguished: pseudo homogeneous (model I), heterogeneous, written hi the wrong way (model II) and heterogeneous, written in the correct way (model III). The first two models have been frequently used in the literature whereas model III started to be used in the last decade and very few authors have used it[2, 81. The objective of this work is to analyze the results

obtained with the three types of two-dimensional models in the simulation of a non-adiabatic tubular tied bed reactor. Heterogeneous model III is considered the most adequate to simulate fixed hed catalytic reactors since the essential transfer mechanisms of tixed bed are taken into account. Therefore a method to quantitatively estimate the errors of models I and II with respect to model III has been developed, ‘which in general does not require the numerical integration of the balance equations. This method also facilitates the qualitative analysis of the influence of the system parameters on the differences between the models. Conditions at which the greatest deviations with respect to model III occur can also be predicted. 2. MATHEMATICAL. DESCRIPl’tON

steady state equations for the different models analyzed and a single reaction are presented in Table 1. Constant properties and plug flow are supposed. It is also assumed that the influence of the temperature on the fluid velocity is negligible. In addition, for the reactors considered, the contribution of the axial dispersion is in general not important (1) and therefore only the radial heat and mass dispersion is taken into account. A 6rst order irreversible reaction was considered The

r,=kC

(1)

where *Author

dressed.

to whom

all correspondence

should &

ad_ 1017

k = &e-EIRT

(2)

1018

S. I. PBRBnu Dt_mRm et al.

Table 1. Two-dimensional models of fixed bed catalvtic reactors

_-

-

conditions for

boundary

C = Co T --0

To

models

at

z = 0 a11

T

at

r

72

I

and

II

1’

ac ar

= 0 all

__?2--0 I

ar us

-

ac

1

a % ta7+-

=cDer

kg a”

(C -

$3

r

az c;,

=

-

kg +

(C -

c:>

n TA

ac -=d ar

at

- A,,

2.5

= law (P

-

“3 pg

Cp

=

x

er

a+

i

( a--;r + r

aT

= Rt

all

2

TW)

ar aT a

r

I

3 h* a” (TZ - T)

y)

although the analysis that follows can be applied to any type of reaction. In general the temperature variation inside the catalyst pellet is negligible. The maximum temperature increase for the conditions of this work has heen calculated from Prater’s equation (9).

Therefore we can consider that T; = T, and the isothermal expression for the effectiveness factor can be used,

is a great uncertainty in some of them. There is a lack of experimental data for a= and in addition very different values are predicted from correlations for h. and a,,!, especially at low values of Re and of the ratio d,/d,_ Actually, in these conditions the heterogeneous model is necessary for a correct description of the heat transfer phenomena in a fixed bed. For estimating the particle-fluid heat transfer coefficient a correlation proposed by Bhattacharyya and Peri[lO] and Balakrishnan and Pei[l l] was used .J*= 0.018 (Ar/Re&JO,” 4:”

(6)

and for the mass transfer coefficient the analogy ‘I’=i

tanh’(34)

-’

34 >

for spheres

(4)

with

3. CORRELATIONS F-OR THE ESTIMATION TRANSFER PARAMETEXS

JD=J,,

(7)

was used. The wall fluid heat transfer coefficient was calculated from the Yagi and Wakao correlation proposed for mass transfer [ 121 OF THE

It will be assumed that all the parameters arising in the equations in Table 1 are known, although there

U$P_ As

0.6 Pr’l’ Relfl 0.2 Pr”’ Re”.8

Re < 40 Re > 40’

(8)

Comparison of two-dimensional models for iixcd bed catalytic reactors

Olbrich[ 131developed a correlation for the wall-solid heat transfer coefficient

in predicting the behaviour of a fixed bed reactor, an error was defined From the radial mean conversions

_

% G2.12+

(9)

P

The effective conductivity for the solid phase may be estimated by the equation proposed by Kunii and Smith[l4] where the radiative contribution has been neglected, because in the cases analyzed the tomperature did not exceed 4OO”C[15]

(10) where y is the effective length of a solid particle for heat transfer in a bed of unconsolidated particles divided by pellet diameter; 4 is the effective thickness of the fluid film adjacent to the surface of two solid particles divided by pellet diameter; /l is the effective length between centers of two neighboring solid particles in direction of heat flow divided by pellet diameter. For radial mass transfer the equation Pe,

= 9( 1 + 19.qdP/4)z)

(11)

derived from the correlations of Dorrweiler and Fahien [ 161 and Fahien and Smith[ 171 was used. The effective conductivity for the fluid phase may be estimated From the equation used by Dixon and Cresswell[ 181. +n&+;&

AL + A:,

(13)

a, = a,‘+ a,“.

(14)

We shall see later that the method of evaluation of the lumped effective thermal parameters only modifies slightly the differences between models. The Crank-Nicholson finite differences method was used to solve the differential equations of Table 1. Ten radial intervals were set and it was verified that an increase in this number did not improve the results. A sulbciently small axial integration step was used so that no change in the results was found by decreasing it. 4. ERRORS

OF MODELS

ei =

_

xi - Xl11

-

~I11

i =

I, II

(15)

where

The radial mean conversions for the different models are evaluated at the same length and are obtained from the numerical solution of the differential equations (Table 1). The magnitude of the error will depend on the position at which it is evaluated. It is obvious that at infinite length it will tend to zero and, therefore, a characteristic length must be chosen. For exothermic reactions, the axial position corresponding to the maximum temperature of the solid phase according to model III can be elected. 4.2 Evaluation of an estimated error In order to avoid the numerical integration of the two-dimensional equations of Table 1, we will develop a simple method to estimate the error in the radial mean conversion (eqn 15) of models I and II. This method is also important since it facilitates the analysis of the factors that influence the error of models I and II. The radially averaged conversion at a certain length in the reactor is proportional to the volume averaged rate of reaction

(12)

where r is a tortuosity factor and Ai, = l p,Cd-),,. The lumped effective thermal parameters &, and a, of models I and II can be evaluated in different ways[l g-201. In our calculations we have chosen the simplest one, that is to add the corresponding effective parameters of both phases: L=

1019

I AND II

4.1 Definition of the error In order to analyze the performance of the models

(16)

and, therefore, the estimated error can be evaluated from (17) . I

-

v,~ represents the estimated volume averaged rate of reaction between the reactor inlet and the characteristic length. However, the average rate will be equal to the rate of reaction evaluated at certain intermediate conditions (radial mean fluid composition and temperature) between those two points. Therefore, the problem of evaluating an estimated error lies in determining an appropriate set (I?, 0. The way this set will be chosen will be discussed later. The resulting equations For the volume averaged rate of reaction are: For model I,

where WA

(tlrJr,u = (tlr&

F

(18)

Since for model II the rate of reaction is a Function of the conditions on the solid surface, it is necessary

S. I. PeReuu DUAR~ ef aL

1020

Table 2. Numerical values of the parameters cP = Pf

Oy

pP

to include

kcal/ky

‘K

PT = 0.7

h

= 0.05

kg/n

= 0.66

kg/m3

e

= 0.4

To = 573°K

= 2097

kg/m’

T = 1.5

Tw = 510-K

0.0408

k.xll,nPK

= 0.490

kcal,m°K

L -

SC = 1.21

h

h

De

I 0.0036

m2/h

P = 1 atm

the solid phase equations

(W”)“, Ilc = &Q&C - C9 = (Vl)C?/, Q hp,(lp;’ - Q = ( - dH)(tlrr)+, Similarly

dt

= 0.119

1y -

is

0.24

for model

i;o

m

0.45

m

G -

dP= kcal/kmol

AH=-2500@

co

-

8.10

-4

h

O-Olrn

E = 38000

kmol/m3

kg/m2

kcal/kmol

A0 = 9.381.1019

II-l

where it has been assumed that the radial temperature profiles of the solid phase are parabolic. (19a) Wb)

5. ANALYSIS

OF

THE

ERRORS

OF

MODELS

I AND

where in eqn (20b) the heat transfer from the solid phase to the refrigerant is considered to be proportional to the difference between the radial mean temperature of the solid phase and the wall temperature, similarly to the one-dimensional analysis. a/ is the one-dimensional heat transfer coefficient for the solid phase oh the bed side. This parameter can be calculated from the two-dimensional models parameters according to [2 l]

II

values of C and f for the evaluation of the average rate of reaction and, therefore, of the estimated error, must now be determined. In general, these could be estimated from the solution of a simple one-dimensional model, e.g., the pseuclo-homogeneous model. This involves much less computing time than the original problem of solving numerically the two-dimensional differential equations. However, in the conditions of this work (exothermic reactions and To 9 T,), since the increase in temperature is not very large, the problem can be simplified further. It is reasonable to accept that the volume averaged rate of reaction will be proportional to the rate of reaction evaluated at feed values, (qr,),, Torand that the proportionality constant will not differ very much between models. Therefore, we can substitute in eqn (I 7) the ratio of volume averaged rates of reaction by the ratio of rates of reaction at feed conditions. A comparison between the errors calculated from eqn (1.5), by numerical integration of the twodimensional model equations, and the estimated valThe

III

--_----

cstlmated calculated

error error

Fig. 1.Errors calculated (eqn 15) and estimated (eqns 17-20) at critical zone vs 2 except G and A, which is mod&d

5950

to keep (c - I’) constant

Icq (cq

17))

11511

Re. (Conditionwas in Table at the reactor inlet for model II).

Comparison

of two-dimensional

models for fixed bed catalytic reactors

ues evaluated from eqn (17) Was carried out for different values of the parameters of the system. The values of the parameters shown in Table 2 are those corresponding to the basic case and are fixed for all the situations considered, except for the parameters in the last column that are varied in the figures and their values are indicated in them. The results are presented in Figs. l-3 and a very close agreement is observed. This allows us to establish that the principal sources of errors of the models I and II can be inferred from the analysis of eqns (17H20). If the mass and heat transfer resistances in the particle-fluid interface ‘are negligible (CS‘ z C, T;’ E T) the three models will give the same result.

%c

------

estimated calculated

error error

lcq.[17)) 1ca.11511

1021

The error of model I is principally due to the fact that the rate of reaction is calculated in terms of the fluid phase conditions. The error will depend on the temperature and composition difference between the pellet surface and the bulk fluid and on the sensitivity of the reaction rate to variations of composition and temperature. The error of model II originates from the erroneous consideration of the heat balances in the fluid and solid phases. It will only lead to an agreement with model III when the solid phase radial heat transfer conduction is negligible with respect to the other terms in the energy balance equation. 5.1 Inzuence of the parameters on the errors of model I and II For a single tist order reaction the systems of algebraic equations (19) and (20) are reduced to single equations. By defining the dimensionless variables

T-T,

(21)

I=-

TO

Y-----l

from eqns (19a) and (19b) it follows for model II that

and from eqns (20a) and (2Ob), it follows III that I

0

I

I

5

10

dp [mm1

Fig. 2. Errors calculated (eqn Is) and estimated (eqns 17-20) at critical zone vs 4. (Conditions as in Table 2 except d, and A, which is mod&led to keep Ada constant).

____-_

-10

est imried calculated

error Irq.ll7)) error[rQ.llSIJ

‘\

t-

Fig. 3. Errors calculated (eqn IS) and estimated (eqns 17-20) at critical zone vs E. (Conditions as in Table 2 except E and A, which is modified to keep A,, exp (- vO)constant).

” - ’ = 1 + c&h@)

for model

(1 - 2) - P(T, - 3.

From eqn (17) it follows

for model

(23)

I

where ?z is obtained from eqn (23) In eqns (22) - (24), h(t) = q*(i) exp(y,t/l + t) with y,,= E/RT,. It can easily be seen that the error of model II is generally positive for an exothermic reaction since the calculated interfacial temperature drop will be higher than the actual one and a higher rate of reaction will be considered. The magnitude of the error will depend on the second term in the r.h.s. of eqn (23). When S’+O, models II and III will give the same response. The error depends also on the difference (< - f,) and, through it, on the dimensionless wall temperature and the dimensionless reaction parameters & and C,. It depends also on the sensitivity of the reaction rate with temperature (yO). From eqn (24) it can be seen that for model I, whereby the mass transfer resistance is neglected, the

1022

S. I. Pmmm Dvmm et al.

rate of reaction tends to increase (factor between brackets). The effect of temperature for model I will depend on the system conditions since two opposite effects exist. According to eqn (23), the solid surface temperature may be higher or lower than the bulk fluid temperature even for an exothermic reaction. An analysis of the influence of the parameters on the error of each model follows. From the correlations used in this work: (24)

(25) and (26)

with

In order to analyze the effect of the variation of some parameters of the system (Re, d, and E) on the error of model I and II it was necessary to modify adequately A,, to compare reactors in similar conditions. 5.1.~ Model ZZ. From eqn (26) it follows that the error of model II must decrease as the Reynolds number, the ratio d,/d, and the tube diameter increase. The influence of Reynolds number is represented in Fig. 1, where the interfacial temperature difference at the reactor inlet in model II (eqn 22) was kept constant.

The influence of the pellet diameter is represented in Fig. 2. In this case Adlo was kept constant. The increase in the error with dp is due to the increase of 6” and of the interfacial temperature gradients caused by the reaction. In order to analyze the influence of the activation energy, the product A,exp (yO) was kept constant. Since the sensitivity of the reaction rate with temperature increases with E, it is obvious that higher errors will be obtained at higher activation energies (Fig. 3). As E 40 the same conversion is obtained for model II and III even though the temperature profiles do not necessarily coincide. 5.1.b Model I. The influence of Reynolds number is represented, in Fig 1. As Re increases, b’_decreases and 5 increases from values lower than f (positive errors) to values higher than r(negative errors). There is a value such that the error is zero due to the compensation of the different factors. As it can be seen in Fig. 2 when the pellet diameter

decreases the interfacial resistances decrease and so does the error. An increase in the pellet diameter increases the interfacial gradients due to the reaction and, for these conditions, it is only partially cornpen sated by an increase in S”. Therefore, the errors of model I become more negative. At low values of the activation energy, the rate of reaction is slightly sensitive to the temperature and the positive errors due to the conversion difference predominate (Fig. 3). Since for the conditions in the figure the solid temperature is higher than the fluid one, as the activation energy increases, the errors become negative. 5.2. Injluence of the lumped effective heat trwfeer parameters on the errors of models Z and ZZ The errors calculated from eqn (15) for models I and II corresponding to the extreme values of the parameters analyzed are shown in Table 3. They are denoted by e: and the simple sum of parameters was

Table 3. Comparison of errors estimated (eqns 17-20) and calculated (eqn 15) for the axtreme values of the parameters used in Figs. l-3. e;, lumped effective heat transfer parameters evaluated from eqns (13) and (14); el. lumped effective heat transfer parameters evaluatedfrom equations proposed by Dixon and Cresswell[ 181

Re = 10

Re = 3000

I

d

= O.lrmn

95.89

96.88

a.54

5.06

5.35

-1.77

0.84

1.39

1.51

-15.84

-15.55

13.62

15.55

16.09

12.78

12.05

12.05

-16.62

-17.08

-16.34

43.82

47.01

47.72

-17.96

-16.54

-16.32

-0.93

-1.88

-17.50

91.35

.

P

dP= I4 lmm E a loo0

E = 42000

rcal/kmol

kcal/kmol

.1921

13.46

.1853

16.00

.1853

17.08

1023

Comparison of two-dimeasionalmodels for fued bed catalytic reactors T(r-01

['Cl

Rt- 600

___

I

Fl -

model model mtdcl

I ---II .---.ilI -

Fig. 4. Axis temperature and radial mean conversion profiles predicted by models in Table 1 vs dimensionless axial distance. (Conditions as in Table 2 except A,,= 1.932 x 10”h-‘, G = 1190).

Fig. 5. Axis temperature and radial mean conversion profiles predicted by models in Table 1 vs dimensionless axial distance (Conditions as in Table 2 except A, = 1.337 x 1oMh-l, d, = 0.012 m).

used to evaluate the lumped effective heat transfer parameters. In the same table the errors obtained when Dixon and Cresswell equations are used to estimate the lumped parameters, e;, are shown. It can be seen that there is a small difference between both errors. The trend in the error arising from the approximate expressions, eqns (17x20), is very close to that found for the numerically calculated errors. Since the mentioned expressions arc independent of the lumped effective parameters, it seems that the magnitude of the error does not depend significatively on the way these parameters are evaluated.

models

5.3. O@erence between the temperature profiles Although the analysis carried out was based on the conversion, it is relatively easy to relate the error in conversion (eqn 15) with the differences that would be found for the temperature profiles. Axial temperature profiles in the axis and radial mean conversions have been plotted in Figs. 4 and 5. It can be seen that the trend observed for the conversion profiles is the same as that corresponding to the temperature, since the reaction rate and the heat generation rate are proportional and the heat transfer capacity to the surroundings is similar for the three models.

6. CONCLUSIONS The

performance

of different two-dimensional

in the conversions by models I II with respect to those predicted by model A very simple expression a reasonable accuracy in the conversion has been derived. to determine of error when models I and II are used simulate of fixed It can be seen that for high Re the performance of model II tends to that of model III since the importance of the heat transfer across the solid phase is reduced. However, model II should not be used because, in addition to being conceptually wrong, it does not present any computational advantage over model III. Model I is consistent and, in addition, requires a lower programming and computing time than model III. It can be used in those cases when the interfacial gradients are small. The temperature difference depends on the balance between the heat generation rate and the radial heat transfer through the solid phase. It has been shown that at low particle diameter the error introduced by this model is low. Acknowledgements-The authors are grateful to the Consejo National de Investigaciones Cientificas y T&&as (Argentina) for their financial assistance which made this work possible.

s. I.

1024

PERaluA DUARTBer ai.

NOTATION

4

AD frequency factor Ar

a,

c, c; CP

D D,

Archimedes number, dp3gpg(pp - p&i external particle surface area per unit reactor volume, m-l reactant molar concentration in the bulk fluid, at solid surface, kmol/mF specific heat of fluid, kcal/kg”K molecular diffusivity, m2/h effective diffusivity within the particle,

m’/h effective radial diffusivity, m2/h pellet diameter, m internal tube diameter, m activation energy, kcal/kmol superficial mass flow velocity, kg/h m* acceleration of gravity, m/h2 heat of reaction, kcal/kmol heat transfer coefficient between fluid and pellet, kcal/hm*“K JD j-factor for mass transfer, (k,p,lG) SC*/’ J* j-factor for heat transfer (h,/C,G) Pr*” k reaction rate coefficient, h-’ ks mass transfer coefficient between fluid and pellet, m/h L bed length, m pew Peclet number for radial effective diffusion, Q&D, Pr Prandtl number, cplL/I& R gas constant Re Reynolds number, dnG/p, I&/(1 - E) ReFm m&d&d Reynolds &&er, R, tube radius m r radial coordinate, m r.4 rate of reaction per unit bed volume, kmol/m3h SC Schmidt number, p!/pp T, T,, T; temperature in the bulk fluid; inside solid; at solid surface, “K or “C t dimensionless temperature, T - TO/TO u, superficial fluid velocity, m/h V reactor volume, m3 x fractional conversion 2 axial coordinate. m Greek symboIs wall heat transfer coefficient: lumped; for fluid phase; for solid phase, kcal/m2 h”C dimensionless adiabatic temperature

fi=, (-dWCo/~$pTo dimensionless activation energy, dimensionless group, k;/d+,, void fraction of packing

EIRT

effectiveness factor for solid particle rll% effective thermal conductivity with respect to radial direction, lumped, for the fluid phase, for the solid phase, kcal/hmX thermal conductivity of fluid (kcal/hm”K effective thermal conductivity in a solid particle, kcal/hm”K fluid phase viscosity, kg/mh dimensionless group, (- dH)(qr,M T&&l dimensionless group, (qr,)&,k~ fluid density, kg/m; pellet density, kg]m,” Thiele modulus shape factor, & = 1 for spheres effective value or estimated inlet condition at the wall

value

REFERENCES [l] Froment G. F. and Bischoff K. B., Chemical Reactor Anclrysis and Design. Wiley, New York (1979). [2] De Wasch A. P. and Froment G. F., Chem. Engng Sci. 1971 26, 629. [3] Carberry J. J. and White D., Ind. Engng Chem. 1%9 61 [4] zth T. G. and Carberry J. J., Advan. Chem. 1972 133, 362. [5] Young L. C. and Finlayson B. A., 2nd. Engng Chem. FundIs 1973 ---- 12 -1 412. [6] Finlayson B. PL., Cur. I&u-Sci. Engng 1974 10(l) 69. [7] Jordan R., Ks rsbenbaum L. and L&z Isunza F., 7 Simposio Iberoamerica no de Cakiiisir, July 1980. [a] Holton R. D. and Trimm D. L., Chem. Engng Sci. 1976 -1 11-?7. rrn J1, [91 Prater C. D., C&m. Engng Sri. 1958 8, 284. [IO] Bhattacharyya D. and Pe.i D. C. T., Chem. &gng Sci. 1975 30 293. [l l] Balakrishnan A. R. and Pel D. C. T., Ind, Engng Che&. Proc. Des. Den 1974 13 441. [12] Yagi S. and Wakaa N.. A.1.Ch.E. Journal, 1959 5 79. [13] Olbrich W. E., hoc. Chemeca 70 Conj: Buttenvorths, London 1970. 1141Kunii D. and Smith J. M., A.1.Ch.E. J. 1960 6 71. il5j Yagi S. and Kunii D., A.I.Ch.E.J. 1957 3 373. [16] Domveiler V. P. and Fabien R. W., A.I.Ch.E. J. 1959 = .*n ‘? [17J JFallen R. W. and Smith J. M., A.1.Ch.E. J. 1955 125. 114 _ _ Dixon A. G. and Cresswell D. L., A.1.Ch.E. I. 1979 25 [I91 ?I& W. B. and Smith J. M., Chem. Engng Prog. 1953 49 443. [20] J3eek J., ABun. CXem. Engng 1962 3 303. [21] Pereira Duarte S. and Martinez O., private eommunieation.