PARTICLE GROWTH PROCESSES D 0 N A L D W . B E H N K E N , dmetican Cyanamid Co., Stamford, J A C 0 B H 0 R 0 W I T Z , dmerican Cyanamid Go., Penracola, Flu. Y. S T A N L E Y K A T Z , American Cyanamid Co.. .\'ew York,
Conn..
A mathematical formulation is given for determining particle size distributions in mixing vessels, where each particle, while in a vessel, grows or shrinks according to a prescribed differential law. The analysis is deliberately made in rather abstract terms to include such areas of application as crystallization processes, certain solid reactions, and catalyst-aging studies. It allows accordingly for arbitrary dependencies of instantaneous particle growth and shrinkage rates on particle size. The application studied is a catalytic fluidized reaction system, where the catalyst particles are recycled between two zones-one in which they are loaded with reagent, and one in which they are stripped of product. Steady-state size distributions are developed, together with expressions for the average loading and stripping rates. Procedures are given for following transients.
A
V E S S E L ? whose contents are vigorous1)- mixed: is continuously fed and depleted by a stream of particles. Each particle, during the time it is in the vessel: gro\vs (or shrinks) according to a prescribed differential law. TVe ask for the distribution of particle sizes in the vessel. To formulate this question mathematically, let
t = ZL' = li(iL8)
=
S
=
Q = (ze,t) = g(c:t) =
time, seconds size of some particle a t some time, dimensions unspecified rate a t which a particle of size ZL grows (or shrinks) while in vessel, (dimensions of E ) (sec.) inventory of particles in vessel, particles vessel feed (take-off) rate, (particles) (sec.) distribution of particle sizes in vessel (and take-off) a t time t , (dimensions of Z L ) - ~ distribution of particle sizes in feed a t time t , (dimensions of E ) -l
-'
Several remarks on this system of nomenclature are in order. Distributionsf and g are formally probability densities in icthat is,
[
-V
j(ze:r)d;.]
Q
=
Q
g ( w , t ) dic
-
fJ t ~ ' . t )
d u +H(x,t) - H ( J . ~ ) ( 4 )
\-e1 bally. [the rate at ivhich the number of particles in the vessel in the size range ( X J ) is increasing] = jtlie rate at which they are fed] [the rate a t which they are taken off] [the rate a t which particles gro\\ into the size range at x ] [the rate a t \vhich they grow out a t ]
+
Recalling Equation 3. Ice may \\rite the gro\cth (elms in Equation 4 as
ff(zu,t)dzc! is the fraction of the particles in the
\-esse1 (and take-off) that a t time t have sizes in the range (xJ);
Then, taking a n arbitrary size interval ( x ! J). we can \\rite a n integral balance on particles in this size range in the form
and
f
and then rearrange Equation 4 into the form
g(zu,t)dw is the fraction of the particles in the
feed that a t time t have sizes in the range (x:y). That J represents the particle size distribution equally in the vessel and in the take-off simply says that the contents of the vessel are so vigorously mixed that the take-off is at any time a representative fraction of the vessel contents a t that time. Finally: the dimensions of particle size are deliberately left unspecified, since different choices are appropriate in different engineering contexts. We can now write a differential equation in the vessel distribution function, f. T h e feed and take-off arrangements are as shown in Figure 1. Taking R to be a growth rate: so that the increase of ic for a particle in the vessel folloivs the differential equation
\\here 9 = S/Q = mean holdup time of particles in vessel, seconds
Finally we argue that since the integral in Equation 5 vanishes for a n arbitrary size interval ( x g ) . the integrand must itself vanish identically. and this leads us to 17)
Equation 7 is the desired partial differential equation i n the vessel distribution function, f. i2'ith a specification off(zc.0). it determinesf(2f.t) in terms of the (presumed known) functions
I
\ \ e note that. for any size ic. Hl2e.t) = .Vj[Zf!,t) x R(U) =
212
the rate a t Lvhich particles in the vessel are growing past size IC a t time t : (particles)(sec.)-1 l&EC FUNDAMENTALS
-4 Figure 1 .
VESSEL N,f,R . .
P-
Feed and take-off arrangements
-
R(ib) and g(ir3.t) and the parameter. 6. If the particles in tlie vessel shrink rather than grow at rate R,so that the decrease of IC for a particle in the vessel follows the differential equation
Q, f i
the differential equation forJ becomes
Equatioris like 7 and 9 arise in various chemical processing contexts. I n a continuous crystallization study, for example. f \could be the distributi'on of crystal sizes, LC'? here normalized not to unity: but rather to the [changing) number density of crystals in the system; a. term Ivould be added to Equation to represent the rate of formation of new crystals. and this term as \vel1 as the cr),stal grojcth rate. R, \could have a n explicit dependence on the concentration of solute in the liquor. I n a study of a gas-solid reaction carried out by fluidizing the reagent solids ivith the reagent gas,J \could be the distribution of solid reagent particle sizes zc; here Equation 9 jvould apply: and the pai~icleshrinkage rate? R:\vould have an explicit dependence on the concentration of the gaseous r e q e n t in the fluidizing gas. I n the present study. Tve are concerned ivith neither of these tivo situations, but rather rvith a catalytic fluidized reaction system. where the catal) st particles are recycled between t\vo zones-one in tvhich they are loaded rvith reagent, and one in tvhich they are stripped of product. Here the "size" of a particle is measured by the amount of reactive material on i t . so that ZL has in effect the dimensions of mass: rather than of length? as is commonly i.aken in the applications noted above. T h e differential equations for this problem and their stead>-stat? solution are developed belo\c; it is also sholvn Iioiv certain transient analyses can be carried out.
-
The Two-Vessel Problern
1I-e consider here a system in \chic11 particles are recycled benveen tivo vessels. groiving in one and shrinking in tlie other according to prescribed differential laws. \Ve have in mind the catalytic reaction sl'stem noted above, where the catalyst particles are recycled between t\vo fluid beds (or two zones of a single bed)-one in \vh:ich they are loaded Ivith reagent. and one in ivhich they are stripped of product-the "size" of a particle at any time being measured by the amount of reactive material on it. T h e analysis, hoicever? \vi11 be couched in the general terms of Equations 1 to 9. Consider accordingl! the system of Figure 2: \vith the follo\ving nomenclature .adapted from Equations 1 and 6 : time, seconds size of some particle a t some time, dimensions unspecified Q = particle recycle rate. (particles) (sec.) R l ( w ) = rate a t which a particle of size w shrinks \vhile in vessel 1, (dimensions of w ) (sec.) R * ( w ) = rate a t which a particle of size zc' groxvs lvhile in vessel 2, (dimensions of w ) (sec.) -I t =
i ~ s=
,2', = inventory of particles in vessel i , particles 8, = *YJ'Q = me,an holdup time of particles in i vessel i? seconds ji(a',t) = distribution of particle sizes in vessel i at -l time t , (dimensions of ZL')
VESSEL 1 STRIPPER
VESSEL 2 LOADER N2 , f P ,R2
Q,f2 Figure 2.
and hence Equation 9 applies. \vith fll for 6, R1 for R . f l for /, and f 2 for g. I n vessel 2. Equation 2 for the increase of io becomes diu - = R?(LL') dt
-
and Equation applies. \vith 62 for 6.RZ for R.f?forJ. and f , for. j1.e have accordingly
c.
Equations 10 characterize the distribution functions jl and \Vith a specification of fl(zc.O), f2(w.01. they determine , f j ( u . t ) . J > ( i ( . t ) in terms of the (presumed kno\vn) functions R1(zt) and Rzjit). and the parameters B1 and 8 2 .
f?
The Steady State
\Ye develop here steady-state solutions for the two-vessel problem. I t turns out that solutions d o not always exist for arbitrary choices of the rate functions, Ri and R?. However, tic0 cases of engineering interest for \vhich solutions d o exist are presented, and explicit formulas are given for the steadystate distributions. In the steady state? distributionsjl andJ? no longer depend on tlie time, and the partial differential Equations 10 become
,ll)
a pair oi' ordinary differential equations in the distribution functions j l ( z c ) > f 2 ( w ) . Together with the differential Equations l l ? \ce need certain side conditions to ensure the unique determination of distribution functions J 1 and 1 2 . \Ye take these side conditions to be. first. a regularity requirement on i': and.f?
-
lim
:1
=
1.2
\Ve can readily adapt the earlier development to rhc present situation. I n vessel 1: Equation 8 for the decrease of i ~ 'bc. comes
System for two-vessel problem
Rlile)Jl(zc) =
m
ic-
lim R ? ( z ~ ) j ? / z=c )0
(12)
m
d!>d second, the requirement that J 1 and f 2 be normalized as p:'oper probability density functions
Lrn
fl(2L')dZU =
Lrn
f?(&)dL(! = 1
(13)
Condition 12 appears to be a reasonable regularity requirement! and indeed a considerably stronger version of it will be needed in making &hetransient analyses. Condition 13 is of VOL. 2
NO. 3
AUGUST 1963
213
course necessary for a physical interpretation of the solutions. Except for certain singular cases ( 3 ) ,which cannot easily be incorporated into our general treatment, Equations 11 and 13 can be solved in a straightforward way. d
; i ; (elRl(w)jl(w) - e , ~ ~ ( ~ ~ ) i ~ (=w o) i
which. with Equation 12 gives elRl(w)fl(a) = e 2 R Z ( m ) f 2 ( t L )
(141
Solving Equation 14 forfz and inserting it into the first line of 11 give a differential equation in O,R,fl
'
