,4t
‘-0 100 200 300 400 300 600 700 800 900 1000
tion must be a t rest a t the start of the test. Some changes in concentration could have occurred during this period for some of the tests and this could explain the increased scatter a t the higher concentrations. The effect of various concentrations of Polyox additive in natural convection heat transfer is best presented by a cross plot of the results, as shown in Figure 4.Addition of polymer causes a decrease in the heat flux. Additions of relatively small quantities of polymer cause significant changes in heat transfer. The addition of 1000 ppm of the polymer reduces the heat flux approximately 40%.
CONCENTRATION ( pprn 1
Figure 4. Effect of Polyox concentration on laminar natural convection heat transfer
mental and analytical results is presented for the distilled water. The comparison of the experimental data to the correlation of McAdams indicates relatively good agreement. The agreement obtained for pure water indicates the ability of the apparatus to give reliable results. The data for the Polyox additive are also presented in Figure 3. The results for 50 and 100 ppm indicate little deviation from a line drawn parallel to the McAdams correlation for pure water. The 300 and 1000 ppm data scatter more. The scatter may result from poor dispersion of the Polyox a t these higher concentrations for some of the test batches. Every effort was made to assure that dispersion was complete, but owing to the nature of natural convection, the bulk solu-
Literature Cited
Gupta, hl. K., Metzner, A. B., Hartnett, J. P., Int. J . Heat Mass Transfer 10, 1211 (1967). Lumley, J. L., Annu. Rev. Fluid Mech. 1, 367 (1969). McAdams, W. H., “Heat Transmission,” p 177, McGraw-Hill, NPWYork. N - .Y - . . 1CL54. ---Meyer, W:A., A.I.Ch.E. J . 12 (3), 522 (1966). Poreh, At., Paz, V., Int. J . Heat Mass Transfer 11, 803 (1968). Wells, C. S.,Jr., A.Z.Ch.E. J . 14 (3), 406 (1968). I
I
DONALD W. LYONS* JAMES W. WHITE JOHX D. HATCHER Departments of Textiles and Mechanical Engineering Clemson University Clemson, S. C. 29631
RECEIVED for review September 7, 1971 ACCEPTED June 5. 1972
Size Distribution for Crystallization with Continuous Growth and Breakage The unsteady-state differential equation i s formulated for uniform crystal growth and breakage. The model assumes that all crystals have the same linear growth rate, that each has an equal chance of breaking, that breakage occurs randomly at any plane in the crystal, and that the fragment volumes sum to the volume of the parent crystal. The moments of the size distribution are given as a set of ordinary linear differential equations. The steady-state crystal size distribution i s found in closed form for a stirred continuous crystallizer and i s compared with the size distribution that would occur if new particles were formed either by microscopic nucleation or from chipping of the edges of parent crystals.
T h e crystal size distribution (CSD) obtained in a crystallizer depends on the dynamic interaction of crystal growth, breakage, nucleation, and the inflow and outflow of crystals. A simple model of crystallizer performance has been given by Saeman (1956) and extended by Randolph and Larson (1962) for the case where crystallization occurs in a mixed suspension mixed product removal (MSMPR) crystallizer. Their treatment assumes that: (1) no breakage occurs; (2) nucleation occurs spontaneously and continuously within the crystallizer; (3) no crystals enter in the feed; (4) the McCabe AL law holds, which requires that all crystal diameters increase by the same increment in a unit of time, regardless of the crystal size. An extension of this model by Randolph (1969) investigated the effect of crystal breakage. For simplicity it was assumed that crystal breakage produced fragments whose diameters summed to the diameter of the parent crystal. The probability 588
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
of a crystal breaking was assumed proportional to some positive power of its diameter. Numerical solutions were given for the case when breakage occurred in the middle of the crystal and these solutions showed a narrowing of the distribution as well as a reduction in the dominant size. This paper presents a simple model of simultaneous crystal growth and breakage and gives solutions of the general integro-differential equation for the model. For the unsteadystate case a set of moment equations is found while for the steady behavior of an XLISMPR crystallizer a very simple form of the CSD is derived. Uniform Breakage Model
We will now formulate the general differential equation for simultaneous growth and breakage in a crystallizer. In this model we make the following assumptions. (1) ,4 crystal has
characteristic length, L. (2) The volume of a crystal is aLa, where cy is a n invariant shape factor. (3) The crystal phase volume fraction, the crystal size distribution, and the temperature and composition of the liquid phase are homogeneous throughout the crystallizer. (4)McCabe’s AL law holds. (5) New crystals are formed only by breakage-no nucleation occurs, and no crystals enter in the feed. (6) Volume is conserved when a crystal breaks, and geometric similarity of the fragments is reestablished instantaneously. While this last point is not true it seems the best approximation we can make that will allow the characterization of crystal size by a single dimension. ( 7 ) All crystals have the same chance of breaking. (8) Breakage can occur a t any plane in the crystal with equal chance. Using these assumptions we may evaluate the birth, growth, and death terms in the general population balance equation
“oor--2
I
0
P(t) =
no nucleotion
0
number of new crystals per unit time number of crystals present
1
2
3
- Pm
(2)
Thus the general number balance becomes bln V
3t+ndt
=
J,
mil, =
J,
dn +G-= dL
cg.
d In V
d t + m l d-t iGml:-l = 3-i
c Q+ + (-) 3+i
(i = 0, 1 , 2 , 3,
, ,
6
7
8
.)
(5)
Lh(L,t)dL
r m
Link(L,t)dL
are the ith moment of the size distribution of crystals in the crystallizer, and the ith moment of the crystal size distribution in the kth inlet-outlet stream, respectively. Equation 4 can be solved directly for steady-state conditions. For a mixed suspension crystallizer the size distribution is n = mo
A complete set of moment equations can be obtained from (4) by formally multiplying each term by (L)‘ and integrating with respect to L on the interval Assuming that the size distribution function, n, is finite a t L = 0 and vanishes for very large L we obtain dm 1:
5
, mm
Figure 2. Weight distribution of crystal sizes for linear growth with new particles formed either by nucleation alone or breakage alone. Solution is for G/P = 1 mm
mi
The first term on the right is the contribution to any size interval from the breakage of larger crystals; the second term is the loss due to breakage out of the interval itself. I n terms of diameter this becomes
bn
4
Diometer
where
Then m(x,t)dz
5
4
c 0.3.Q
m(x,t)dx
be the number of crystals per unit of crystallizer volume with volume less than v , and let
2 3 Diameter, mm
Figure 1 . Steady-state crystal size distributions for linear growth with new particles formed either by nucleation alone or breakage alone. Solution is for G/P = 1 mm
given by Randolph (1964). The growth term becomes G(dn/ dt) since the linear growth rate is assumed independent of L. The birth and death terms both come from breakage. I n order to handle the restriction that the fragment volumes must sum to the parent crystal volume it is convenient to introduce a new size distribution function based on volume. We let
1
no nucleation
0.25
2 P 4La -2PL (5) 6e x p { T }
(7)
Figure 1 compares this distribution with the distribution for linear growth and continuous microscopic nucleation or breakage by chipping a t the edges. It can be seen that when new particles are formed only by nucleation or chipping most of the crystals are quite small. When the same number of new particles are formed by breakage instead of nucleation, very few small crystals are obtained. The distinction is less apparent in Figure 2, which compares the weight fraction (instead of number fraction) distribution functions. These curves are obtained by weighting the CSD’s of Figure 1 by the diameter cubed, and they are the type of distribution curves that would be obtained from a screen analysis. The weight average size is 4(G/P) for no breakage, and (3‘/2)(G/P) for no nucleation. The dominant size is 3(G/P)) for both distributions. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
589
Summary and Discussion
Possible Extensions
X new model for crystal breakage is given which assumes that fragment volumes (rather than diameters) sum to the parent particle volume. A general equation (eq 4) incorporating linear growth, uniform breakage, and product stream loss is developed and transformed into a set of simple moment equations. These moment equations are readily solved on a computer and yield such important CSD characteristics as numberaverage size, weight-average size, and crystal phase volume fraction. They are particularly suitable for unsteady-state simulation since for a n hlSMPR crystallizer, for instance, they involve only three parameters: G, the growth rate; P, the breakage frequency; and Q, the volumetric flow rate of the product stream. A spontaneous nueleation rate for the formation of infinitesimal nuclei could be added as a fourth parameter. It would change the zeroth moment equation to
There are a number of processes other than crystallization which are characterized by simultaneous growth (or shrinkage) and breakage. The dispersion of large drops or bubbles by agitation and the model of turbulent mixing given by Suzuki (1971), in which clumps of different fluids are broken into smaller clumps by eddies and are blurred a t their edges by molecular diffusion, are but two examples. The equations given in this paper should describe these processes as well as (or perhaps better than) they describe crystallization.
(nucleation rate per unit volume)
(8)
and would leave the higher moment equations unchanged. For the case of simultaneous growth and breakage with no nucleation in an MSMPR crystallizer the crystal size distribution is given in closed form. Although the number distributions for breakage with no nucleation and for nucleation with no breakage look quite different, the weight distributions look much alike. I n fact, with the usual scatter of data obtained from screen analyses it would be difficult to determine which curve gives a better fit. A sensitive method of testing such data is available since a semilog plot of the weight distribution divided by the diameter cubed gives a straight line for nucleation alone, whereas a semilog plot of the weight distribution divided by the diameter to the sixth power gives a straight line for breakage alone. I n a real crystallizer it is likely that both nucleation and breakage occur to some extent, and it is reasonable to assume that the actual size distribution would fall somewhere between the two limiting cases of breakage alone and nucleation alone.
Nomenclature
G(t) = McCabe linear growth constant, mm/hr L = characteristic crystal size m(v,?)dv = number of crystals with a (crystal) volume between v and v d ~ ( m mper ) ~ unit volume of crystallizer n(L,t)dL = number of crystals w t h size between L and L dL mm per unit volume. n(L,t) is commonly called the crystal size distribution (CSD) = ith moment of the crystal size distribution mi(t) P(t) = frecluencv of formation of new particles by ,. bieakage, hr-l = volumetric flow rate of product stream, l./hr &(?) t = time, hr 2, = crystal volume, mm3 = crystallizer volume, 1. a = shape factor, v = aL3 = crystal size distribution CSD LISRIPR = mixed suspension mixed product removal
+
+
v
literature Cited
Randolph, A. D., Can. J . Chem. Eng. 42, 280 (1964). Randolph, A. D., IND.ENG.CHEM.,FUNDAM. 8, 58 (1969). Randolph, A. D., Larson, hl. A., A.I.Ch.E. J . 8 , 639 (1962). Saeman, W. C., A.I.Ch.E. J . 2, 107 (1956). Suzuki, M.,University of Tokyo, personal communication, 1971.
THOMAS J. FITZGERXLD*l TXI-CHENG YXYG Illinois Institute of Technology Chicago, Ill. 1 Present address, Department of Chemical Engineering, Oregon State TJniversity, Corvallis, Ore. 9733 1. for review July 6, 1971 RECEIVED ACCEPTEDJune 30, 1972
Inviscid Flow through a Sudden Contraction An analytical solution i s derived for axisymmetric inviscid flow through a sudden contraction. From this analysis it can be inferred that the axial velocity profile at the inlet of the smaller tube for the flow of a Newtonian fluid at high Reynolds numbers possesses symmetrical maxima with a local minimum at the center line.
T h e analysis of the flow of a Newtonian fluid from a reservoir into a circular tube has been the subject of numerous publications which reflect over a century of effort. Not only is this flow field of practical importance but, in addition, it possesses certain general features which are of interest in a wide variety of viscous flows. Most analyses of this problem involve neglecting excess viscous pressure dissipation in the reservoir and assuming that the axial velocity profile is uniform a t the entrance of the smaller tube. Furthermore, the 590
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
elliptic Navier-Stokes equations are replaced with their parabolic, asymptotic forms, the boundary-layer equations. With this set of assumptions, the resultant solutions must be classified as asymptotic approximations, valid, a t best, in the limit of high Reynolds numbers. The validity of the above assumptions a t high Reynolds numbers and the nature of the flow a t lower Reynolds numbers must be ascertained by the development of finite-difference solutions or asymptotic expansions for the full equations
