VARIABLE DENSITY EFFECTS AND AXIAL DISPERSION IN CHEMICAL REACTORS . M. K. B.
J
D0 UG
L A S , The Atlantic Rejning
B ISC H0 FF
,
Co., Glenolden, Pa.
Department of Chemical Engineering, The C’nivetsity o j Texas, Austin, Tex.
The effects of variable density on conversion in chemical reactors with axial dispersion have been considered for laminar and turbulent flow models. For the turbulent flow case the density effects tend to cancel if ratios with respect to piston flow are used. This is not true of the results for the laminar flow model.
N RECENT YEARS
much attention has been devoted to predict-
I ing the effects of mixing on the design of chemical reactors. For empty or parked tubular reactors the mixing phenomenon is usually described by an effective diffusion or dispersion model. In general this model is complex, but in many situations (where transverse gradients are unimportant) a simplification can be made, leading to what is called the axial dispersed plug flow model (5). Most previous work concerning dispersion in chemical reactors has been limited to constant density cases. This paper describes the importance of expansion effects for a few simple reaction mechanisms. Axial Dispersion
Experimental data a t high gas rates (6, 72, 79) indicate that the axial Peclet number in packed beds is approximately equal to 2.0 and is independent of Reynolds number. This is in good agreement with theoretical predictions ( 7 , 79). At very low ReynoIds numbers, R e < < 1 , the dispersion coefficient is equal to the molecular diffusivity, and thus the Peclet group must be proportional to velocity (8). In addition to these gas phase experiments, there have been several investigations of liquid systems. A compilation of all this information may be found elsewhere (5). All of the gas phase experimental work has been concerned with nonreacting systems. However, the results indicate that the axial dispersed plug flow model adequately describes the mixing in tubular and packed bed reactors, and in several theoretical studies the effects of dispersion on isothermal reactor design have been based on this model. These include discussions of analytical solutions ( 7 7 , 76), boundary conditions (4, 73, 22), the effect of reaction order (74, 78),and selectivity
(27). There have also been a few investigations of nonisothermal reactor performance. These include both theoretical studies (9, 70) and a n experimental investigation of a second-order, adiabatic reaction in a liquid systcm (20). The experimental results agreed well with the theoretical model, and thus lend support to its widespread use. Variable Density Reactions
Many gas phase reactions cause a change in the total number of moles present, and therefore the density of the system varies as the reaction proceeds. Expansion effects are encountered 130
I & E C PROCESS D E S I G N A N D DEVELOPMENT
for cracking, dehydrogenation, and cyclization reactions, while contraction effects may predominate in polymerization and alkylation reactions. An approach for determining the importance of this phenomenon was outlined by Hulbert (76) in an early paper, but has received little attention since that time. The present development is slightly different from Hulburt’s, as a different definition of Fick’s law has been used. When the generalized diffusion equation ( 3 ) is simplified to describe a steady-state, one-dimensional system, the result is :
p udf-
1 d -L dZ
=
dZ
( );: p D --
+
RL
The total continuity equation gives:
(2)
pLi =
To simplify Equation 1 , it is necessary to distinguish between the cases of laminar and turbulent flow, since experimental data in packed beds indicate that for laminar flow D is a constant, while for turbulent flow D is proportional to the velocity (5). Thus for the turbulent flow case:
);(
D pD = p o l l O - = pOUO
IJ
=
POD0
(3)
and Equation 1 may be written as:
(4) For the laminar flow case it is necessary to have a relationship between density and conversion. This is normally written as (77): P
-
=
Po
1 -___
(5)
1+E(l-f)
which when substituted into Equation 1 gives:
1 1 4-E (1
-Y+
-f) dZ2
[(l
+
+ Rd
(6)
T h e rate groups depend on the form of the kinetic expression of interest. I n this study two forms were considered: a
homogeneous rate expression and a simple adsorption rate equation, Using the procedures given by Hougen and Watson (75) these can be written as:
Table 1.
s = l
E
Homogeneous
Effect of Expansion Factor
f
V/V, = R/R,
- 'I2 0.0386 2.37
0 0.0991
1 0.194
2.16
2.03
4
0.352 1.97
s = 2
-"
(1
+ E(l
)'
- f)
(7)
Heterogeneous
f
+ E(1 - f )
._
Lk P
1
-
1
+ E(1 -R'ff) + Rzf
-
(8)
Three of the possible combinations of rate expression and dispersion model were chosen for this study : Turbulent, Homogeneous
Discussion of Results
Turbulent, Heterogeneous df Y .-- 6: dZ dZ2 1
Rlf
-~
+ E(1 - f) 4- Rzf
Laminar, Homogeneous, First-Order
df- dZ
d:
1
+ E(l
d:tf -
- f) d Z 2 +
For the constant density case, E = 0, Equations 9 and 11 are identical. However, in most situations the equations are nonlinear and analytical solutions are not available. Therefore, numerical solutions were performed on a n IBM 704 digital computer. A fourth-order Runge-Kutta method was used for the integration, and 32 increments taken along the reactor length provided sufficient accuracy. 'The procedure was essentially the same as described by L.evenspie1 and Bischoff ( 78).
[l
+ Rof
1
+ E(1 - f)
with the boundary conditions (4.22):
Numerical Solutions
When s = 1 and E == 0 in Equation 9, the equation reduces to a linear form and the analytical solution is well known (22). Similarly, when R 2 = E in Equation 10 the result is linear and has the same analytical solution, where R1/(1 E ) = Ro.
T h e results of the computations are given in Figures 1 to 10. T h e graphs show the ratio of the reactor volume to the plug flow reactor volume giving the same conversion plotted against the fraction of unconverted feed, with parameters of the dispersion and rate groups. Both expansion and contraction effects are illustrated for the various reaction mechanisms. As a reference value, the equation describing mixing on a molecular scale in a continuous-stirred-tank reactor also has been plotted. This value does not necessarily correspond to the minimum conversion possible, and therefore is not necessarily the correct value as D approaches infinity. In fact, it is likely that the axial dispersed plug flow model would predict a value between those calculated on the basis of complete segregation and maximum mixedness ( 6 , 2 3 ) . The results of the numerical solutions of Equations 9 and 12 are given in Figures 1 to 4. Reactor performance (conversion per unit volume) decreases as the value of the dispersion group increases when s, R,, and E are held ronstant. For given values of s, R,, and b: the conversion decreases as E increases; the conversion for expansion, E > 0, is less than the constant density case, E = 0, and the conversion corresponding to contraction, E < 0, is greater. The importance of these effects increases markedly as the reaction order increases.
+
10.0
8.0 8 0
-
BACKMIX ACKMIX REACTOR
- E = 4 0E = 4.0
F-
6.0
6 0 4.0
40
'.
,R,=
20
4;
a!, oc"
'I>"
4 >" 2.0 , 2 0
I .O 0 05
0.1
02
04
06
0.8 I O
I .o 0.05
0.I
0.2
f
Figure 1.
Turbulent model, first-order reaction
0.4
06
0.8 1.0
f
Figure 2.
Turbulent model, first-order reaction VOL. 3
NO. 2
APRIL 1 9 6 4
131
All these results are in agreement with the directional trends obtained from an examination of plug flow and perfectly mixed reactor behavior. While the conversions vary widely with E a t a constant value of Ro and C, the values of V / V , = R / R , remain relatively constant. For instance, in Table I, the values of fraction unconverted and V/V, = R / R , are given as a function of Efor a case where d: = 1.O and Ro = 5. ’Thus the expansion effects in an axial dispersed plug flow model are approximately the same as in the plug flow model, and apparently “cancel” to a certain extent when the ratio is considered. This implies that the constant density cases studied by others (74, 78, 22) can be used to predict the effect of dispersion on variable density reactions. The actual reactor volume needed should, of course, be based on the plug flow volume determined by standard methods (75, 77) using the correct value ofE; V = ( V / V p ) B = o ( V p ) B .
- E = l O
=I en
0 05
02
01
04
06
08
10
f
Figure 3.
Turbulent model, second-order reaction
Adsorption Rate Equation with Turbulent Flow 40
Figures 5 through 8 show the results of the computer solutions for Equations 10 and 12. A qualitative understanding of the behavior can be obtained by comparing the heterogeneous and homogeiieous rate equations, Equations 7 and 8. This comparison shows that when R2 = E in Equation 8, the reaction rate will be similar to a constant density first-order E ) . Hence, the curves in reaction, where Ro = R 1 / ( l Figure 5 corresponding to R2 = 1.0 and E = 1.0 are identical to those in Figure 1 for the case where E = 0. A further analysis of Equation 8 reveals that when R P < E, the rate acts approximately like a first-order homogeneous reaction with a smaller value of E, and that when Rz > E the behavior is similar to contraction in the homogeneous case. Thus the curves have the same general characteristics as those for homogeneous reactions, but are somewhat more complex because of the interaction between RP and E. Reactor performance decreases as backmixing increases, decreases as E increases for given values of R2 and C ,and increases with increasing values of R2 when E and C are fixed. For given values of R1,L, and R,, V / V , = R / R , is essentially independent
-E=l
---- E.0
0
=I.”2 0 >I$
+
I O
0 05
01
04
02
06
08
I O
f
Figure 4.
Turbulent model, one-half-order reaction
6.0
-
BACKMIX R E A C T O R
E = 1.0
of E. Homogeneous Reactions and Laminar Flow
Graphs comparing the laminar and turbulent flow dispersion models for both expansion and contraction effects are given in Figures 9 and 10. The two models are identical when E = 0 (Figure 1). Examination of the graphs shows that the laminar flow model corresponds to better performance when the expansion factor is positive and poorer performance when it is negative. When compared to the constant density case, positive values of E in the laminar model act like contraction in the turbulent model, and vice versa. The curves for the contraction case (Figure l o ) , also show that the performance is sometimes poorer than that corresponding to mixing o n a molecular scale in a backmix reactor. Another difference in the behavior of the laminar model is that the ratio of V / V , = RJR, varies widely with E.
0 05
01
04
02
06
08
IO
f
Figure 5.
Turbulent model, adsorption reaction R2
= 1.0
-
6’o b B A C K M I X REACTOR
E.I.0
Conclusions
As has been pointed out by Carberry (7) and Beek ( Z ) , the effects of axial dispersion can often be neglected, since turbulent flow is normally encountered and the values of the dispersion group are usually very low. However, in very short packed reactors sometimes used for laboratory kinetic studies, and 132
l&EC PROCESS D E S I G N A N D DEVELOPMENT
01
0 05
02
0 4
06
08
f
Figure
6.
Turbulent model, adsorption reaction Rq 2.0
1.0
r C
4.0
“!,e“
2.0
,E.-L
~
K
I O 0 05
2
I REACTOR X
M
0 2
01
04
06
08
IO
f
Figure 7.
Turbulent model, adsorption reaction
possibly for some industrial reactors which have a low length to diameter ratio, such as SO2 converters, dispersion effects might be important. When this occurs, it is a simple matter to predict expansion effects for the turbulent flow case from a knowledge of plug flow behavior and the effects of dispersion for the constant density case. I n situations where laminar flow prevails, axial dispersion effects can be extremely important. For these cases the simple technique of predicting variable density effects does not apply, and the directional rffects of the expansion factor appear to be opposite to what would be expected from analysis of plug flow and perfectly mixed reactor behnvior.
R> = 2.0
Nomenclature
D
effective dispersion coefficient particle diameter E expansion factor = fraction of reactant unconverted f k = rate constant K = adsorption constant L = reactor length d) = D,,,/l.CT0= dispersion group = moles of fred per unit mass n, nAo = initial moles of rcactant per unit mass P = total presslire Pe = d D 1 7 / D= Peclet number R = reaction rate Rd = RI,/Cr,,po = dimensionless reaction rate = L k ( P/no) ’(n.,,) --I / p o U,, Ro R1 = LkP/pC,Uo?i,, R2 = KPn.4,,/n0 S = reaction order U = velocity V , VI, = reactor volume: and plug flow reactor volume Z = dimensionless reactor length = = =
d,
0.I
0 05
04
0.2
0 6
08
1.0
f
Figure 8.
Turbulent model, adsorption reaction R2
6.0
= 0.5
-
l i
LAMINAR
----TURBULENT
GREEK = density
p
4.0
literature Cited
“la“ >.I>” 20
\
1.0
005
0.I
04
02
08 I O
06
f
Figure
9. Laminar model, first-order reaction E = 1
6o
(1) Aris, R., Amundson, N. I
