Chemical Reactions in Continuous=
Flow Systems HETEROGENEOUS REACTIONS H U G H M. HULBURT Hunter College, New York, N. Y. General kinetic equations governing reacting, flowing mixtures are applied to reactions catalyzed at the walls of a cylinder. Dimensionless ratios for correlating yields, diffusivity, and process variables are presented. Simple formulas for estimating the influence of diffusivity on yield are given and extended to turbulent flow in tubes for which the ratio, LID, of length to diameter exceeds 60. Extension to pelleted catalysts and to catalysts poisoned by products is discussed.
Substituting Equations 3 and 4 into Equation 2, expanding the divergence operator, and using Equation 1, m
2
=
mrb
+ V .V DkmCk
(6)
There is thus one equation for each component of the mixture. I n type A reactions the velocity of any Component which accumulates in the reactor is zero. For such components Equation 2 replaces Equation 5 and reads:
H
ETEROGENEOUS reactions in continuous-flow reactors fall into two classes. Type A reactions, in which one or more products accumulate in the reactor, are exemplified by “coke” production in catalytic cracking of petroleum fractions, chromatographic analysis, and zeolitic ion exchange reactions. Type B reactions are those in which all reactants and products are continuously removed. Most catalytic processes fall in this class, since the products are usually readily and completely desorbed from t h e catalyst into the stream of unchanged reactants and products. In countercurrent extraction there is a transfer of material from one phase t o another, each of which is removed at independent rates. Both classes of reactions can be analyzed kinetically as special cases of the general differential equations for flow systems presented in a previous paper (6). These may be recapitulated briefiy for isothermal systems:
I n this case the density, m, refers to the mean density of matter in the reaction space, whether actually flowing or not. Thus catalytic packing may be considered as a fluid component of zero velocity. I n countercurrent absorption towers, the velocities, vk, of absorbent fluid are not zero but are set as independent design variables at predetermined values. In this way all possible cases are formally included. Mathematically type A reactions result in problems in which the concentrations are functions of time, increasing continually for accumulating components. In type B reactions, however, concentrations reach a steady state in which time is not an explicit variable. This paper considers in detail the application of these equations to heterogeneous catalyzed reactions. SURFACE REACTIONS IN A CYLINDER
(1) (2) whereck = concentration of kth component of fluid mixture, (moles gm.-l) m = mean denmty of flowing mixture, grams cm.-* vb vector velocity of lcth component, cm. sec.-l V = mean vector velocity of mixture, cm. set.-'t = time, sec. rr = rate of production of lcth component in fluid stream, moles gram-1 sec.-l u o ijj = v.v a v = i G j b + k . as 5
-
I
+ +
In homogeneous phases, Equation defining a diffusion velocity, ub
=
can be simplified further by
vj - v
(3)
and using Fick’s law of diffusion, -mckUb
V DbmCh
where Dk = diffusivity of kth component, om.’ sec.-l
(4)
One of the simplest heterogeneous reactors consists of a cylinder into which reactants are passed‘; the cylinder wall has catalytic action. Although it presents too little surface t o serve as a Dractical reactor, the unpicked cylinder deserves consideration an approximate model for the tortuous passages t o be found in actual pelleted catalytic packing. A careful analysis of this case should indicate a t least the qualitative kinetic behavior of more complex packings and the significant dimensionless variables, in terms of which more complex cases can be analyzed empirically. Consider first a reaction in which all products and reactants concerned in the kinetic rate expression are continuously removed from the reactor. A steady state will be reached in the reactor for which dCk/dt = 0 everywhere in the interior of the cvlinder. Let it be further assumed that the flow is adequately represented by a mean axial velocity, V, uniform in any cross section of the reactor. At the surface. however. the velocitv is zero. in accordance with known facts. This introduces the approximation of an infinite velocity gradient at the surface but alters no essential features of the chemical behavior of the system. Concentrations now depend upon two space variables: 2, the axial distance from entrance to the reactor, and r, the radial distance from the axis. Equation 5 becomes for this two-dimensional case:
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Vol. 37, No. 11
INDUSTRIAL AND ENGINEERING CHEMISTRY
1064
where u = radial component of mean velocity V w = axial component of mean velocity V However, since the mean velocity is purely axial, u = 0 and reactants reach the walls by diffusion. Moreover, a previous paper (6)showed that almost all practicable conditions result in negligible axial diffusion in comparison wit,h the mean axial velocity. Hence
Since there is no reaction in the bulk phase ( m r k = 0), Equation 0 becomes:
the presence of inert components is accounted for. The previous paper (6) gave a more complete discussion of Equation 17. The assumption that the mass velocity is constant in any cross section is partially justified by proper choice of G and Dk. If the constant velocity is taken as the mean velocity of the cross section, mass velocity G gives the mean flow per unit crosssectional area in the tube and is easily measured experimentally. If the flow is laminar, Dk is independent of the mean velocity, but this case is seldom of practical importance. I n turbulent flow, Dk is given by the usual mass transfer coefficient as a function of Reynolds number and mean velocity. Any of the usual approximations (7) may be used to estimate its value. Since these may be defined to reproduce the radial mass transfer from a turbulent stream in the form
(7) From Equation 1 under these conditions,
At the surface T = R, the velocity w = 0, and
(9) As an initial condition, let us choose
mcb
=
mock (z = 0 )
(10)
Le., uniform concentration over the entire entry cross section. Equations 7 to 10 form a determinate system of partial differential equations, boundary and initial conditions. Before integrating them, however, it will be instructive t o transform to dimensionless variables : z/L;
tl
T/R;f = ck/ckO
Noting that Equation 8 can be integrated at once t o give mw where G = mass velocity (grams cm.2 sec.-l), a constant,
(r = 0 )
mf = mo
(11)
-
G, The various possibilities for mrh have been extensively discussed (4). Let us suppose, by way of example, that the surface is only sparsely covered with adsorbate and that the reaction on the surface is much slower than the rate of adsorption. The surface concentration is then given effectively by the adsorption isotherm in the absence of reaction. For a first-order reaction on the surface, under these conditions,
(15)
If the net reaction which occurs is we define v =
aa
+
f5
=
mrk =
where (I refera to any chosen reference reactant &. The density m can be expressed in terms off, since m = mo/P P(1 where ma = density of feed mixture
m r , = -ICk;Wr (18) moles adsorption sites per cm.a of surface = fraction of sites occupied by t h e kth sort of molecules k i = fraction of molecules adsorbed which react per second
where 0
When reaction occurs between adsorbed molecules and impinging molecules in the gaseous phase,
- Zaj
Zb,
it is not necessary to consider the radial velocity gradients further. Hence, if mass transfer coefficients are used rn defined, the approximation of constant velocity in any cross section would be expected to cause negligible error in the kinetic equations. Equations 12 to 15 are mathematically tractable only for the case in which the mean fluid density is unchanging throughout the reactor. For gas reactions v i t h no increase in the total number of moles ( V = O ) , for most liquid reactions, for reactions in dilute streams (Macao < < l), and in the initial stages of any reaction (f = I), this condition will he satisfied. Taking m = m, and defining the space velocity as S = G / w L (in set.-'), Equations 12 to 15 become:
- f)l
(17)
= vMocm
- k z Qtbrncj
(19)
When reaction occurs bimolecularly between adjacent molecules absorbed on the surface,
mrk
=
-k'z
Q)B#k
(201
For a sparsely covered surface,
Mo = mean molecular weight of feed mixture The factor Maceois the mole fraction of the ath component in the feed mixture. Since the mean molecular weight of mixture M is given by N
where V = molar volume of mixture
where ,.CI k d = specific rates per cm.l for adsorption and desorption, respectively. For the first-order case, which also includes a bimolecular reac: tion with one component in large excess, Equations 18 and 21 give : mrk = IClrnockoQj where IC1 = k & / l c d
-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Novombr, 1945
1085;
and Equation 13A becomes
tds, 1)
=
- Rkihl xf
(22)
The set of Equations 12A, 22, 14A, and 15A is identical with that considered by Paneth and Herzfeld in the kinetic analysis of the removal of metallic mirrors by free radicals (6). It is readily solved by standard methods. Writing K
Dr/SR';
(Y
= Rklhl/DI; f = Z(r).R(s)
Equation l3A reduces to two ordinary differential equations:
where X is a separation constant with values to be determined. Equation 23 has the solution
z
= e-Ant
(25)
while Equation 24 has the general solution
R
=A J e ( 6 )
+ BYo(l/i;~)
(26)
where JOand YOare the Bessel functions of zero order. Equation 14A becomes:
R,(O) = 0 = AJA(0)
+ BYd(0)
Figure 1. Chart for Determining Xo, the First Root of Jib)/J&) a/%
But Y i is infinite at the origin whereas Ji is zero at the origin; hence B = 0. Equation 22 indicates that X must satisfy the relationship,
(YJdfl
=
-miXfi = + f l J l ( f l
(27)
p
Thus, from Equation 29,
Suppose Equation 27 to be satisfied for the series of values X = Xi, deferring their evaluation for the moment. Equation 24 then has the solution, Rj = AjJo(dGr)
(28)
Aj
=
2 Ji(dG1 V % [ J ? ( f i A JE(dG)I
+
(35)
and from Equation 31,
and Equation 15A becomes 03
1 =
At J o ( V %
?I)
(29)
J-0
Substituting for J1
(.;/xi)
from Equation 27
which may be solved for Aj as will be seen. (37)
The complete solution is, then, m
The m5an conversion over the entire cross section of the reactor is 1 f, where
-
There remains only the evaluation of A, from Equation 27. This may be done grephically in any case, as shown in Figure 1, where 1 / 6 is plotted against J l ( & ) / J a ( d i ) . Writing z = Equation 27 becomes:
d
