S T A B I L I T Y OF A D I A B A T I C PACKED-BED R E A C T O R S Efect of Axial Mixing S H E A N - L I N
LIU A N D N E A L R . A M U N D S O N
University of Minnesota. Minneapolis 74, Minn.
A previous study of the stability of an adiabatic packed-bed reactor is extended to the more complicated case in which axial mixing in the fluid is considered. The partial differential equations describing the transient behavior 'of the reactor are solved b y an iterative finite-difference method. The effects of the changes in interstitial velocity and diffusion coefficient upon the temperature and partial pressure profiles are investigated both for cases in which there is a single steady state and for those with multiple steady states. Under the same initial and inlet conditions, the steady-state reaction zone may reside a t various positions along the bed, depending upon the magnitudes of the interstitial velocity and the diffusion coefficient. Calculations are made for Peclet numbers between 1 and 3.
HE problem of stability of a n adiabatic fixed-bed reactor Twas considered in an earlier paper (9) for a simple model in which axial diffusion in the fluid was neglected, and it was shokvn that under certain conditions there would be nonunique steady-state temperature and concentration profiles, depending upon whether the particle could exist in singlet or multiplet states. I n another paper (70) the stability of the nonadiabatic case was analyzed and it was reported that, for some sets of parameters, the reactor effluent was sensitive not only to small changes in the inlet conditions, but also to a small change in the initial particle temperature. The purpose of the work reported here \vas to study the effect of axial mixing on temperature and concentration profiles for the single- and multiple-state cases. Wicke and Vortmeyer (75, 77-79) considered the packedbed reactor and showed that because of axial mixing in the fluid the reaction zone might move toward the bed entrance. McHenry and Wilhelm ( 7 7 ) made experimental studies on packed beds and reported that the longitudinal Peclet number, uD,l/D*, with the fluid flowing a t high Reynolds number was about 2. This relation was also considered theoretically by others ( 7 > 6: 72, 74). In this paper transient equations for the reactor, including axial diffusion, are written and solved numerically by a n iterative finite-difference method. The steps in the computation are described i n some detail, since to the authors' knowledge computations as extensive as these have not been previously reported. Computations are made for Peclet numbers between 1 and 3. The steady-state equations are analyzed, and by integrating the transient equations the effects of axial mixing and interstitial %velocityupon the steady-state temperature and partial pressure profiles are examined for both the singlet and multiplet particle states.
for mass is
a,.k,M ' P a1 = -
where
PfY
The fractional void volume of the bed, y , the mass transfer coefficient, k,, and other symbols are defined in the table of nomenclature. The heat balance may be written
where I;* is the effective axial-thermal conductivity, t is the fluid temperature, t, is the particle temperature, and
If the thermal diffusivity is assumed to be equal to the mass diffusivity-i.e., E ; * / c , p j = D*-the heat balance becomes
The transient equations on a single particle are identical \vith those given previously ( 9 ): (3)
where with
k
=
ko exp ( - l E / R / , , )
Summary of the Equatiions
so that these equations are highly nonlinear. The boundary conditions a t the entrance to the bed are
Consider an adiabatic fixed bed packed ivith small particles. Fluid is introduced into the bed a t s = 0 and a simple chemical reaction A -+ B, which is first-order and irreversible, takes place on the porous surface of the particle. Let the partial pressure of component A in the gas phase be p, the interstitial velocity be u , the effective axial-diffusion coefficient be D*, and the time variable be 0. Then the conservation equation
and a t the bed exit
x =
VOL. 2
o,e>o
NO. 3
(5)
AUGUST
1963
183
x=L,R>O
(6)
T h e above boundary conditions have been extensively discussed in the literature ( 2 , 4, 7, 76). I t is also necessary to state the initial conditions of the solid and interstitial fluid :
P
=
I
= ti ( x )
Pi ( 2 )
t, =
hi ( x )
PP
,&I ( x ) , R
=
which is the same as Equation 8 in ( 9 ) . It follows that the straight lines, Ql, defining the steady states pass through a common point [cf. Figure 1 in ( 9 ) ] . Although the present model has the same characteristics. but because of the discontinuities of p and t at the bed entrance defined by Equation 5 and the backmixing in the fluid, the steady-state temperature and partial pressure profiles will not be the same as those obtained in the nondiffusion case. Numerical Solution of Transient Equations
= 0,
0
