The Performance of a Continuous Well Stirred Ice Crystallizer Geoffrey Margolis,' Thomas K. Sherwood,2 P. 1. Thibaut Brian, and Adel F. Sarofim Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 02f 39
As part of a continuing research program in water desalination by freezing, studies have been made on the performance of a well stirred continuous crystallizer producing ice by direct contact refrigeration. Photographs of the effluent revealed that the ice particles were disk shaped and that the particle size distributions in all cases passed through a maximum. Growth rates correlated well with values predicted from heat and mass transfer rates in the size range 0.6-2 mm but increased sharply at the smaller sizes. For average residence times ranging from 6.5 to 13 min, measured subcoolings varied from about 0.02 to 0.03"C and formed a very small fraction of the overall temperature driving force, while nucleation rates varied from 9 to 30 nuclei/(cm3 of slurry mirt) and could be correlated with the solution subcooling and the moments of the distribution but were independent of refrigerant temperature. Permeabilities of ice beds formed from the crystallizer product were found to be a strong function of both the method of bed formation and the size and shape of the crystals. An important practical result of the study was the indication that much larger ice production rates per unit crystallizer volume should be possible without sacrifice of crystal size.
o n e of the more promising techniques for water desalination in medium sized plants is the secondary refrigerant freezing process (Barduhn, 1965, 1967, 1968; Brian, 1968). Ice is produced from a saline solution by direct contact refrigeration, separated from the slurry, washed, and then melted by direct contact condensation of the compressed refrigerant vapor. The ice, produced as a slurry in the crystallizer, is separated from the brine and washed as a consolidated moving bed in a counter-current wash column. Wash column capacity depends on the permeability of the ice bed, which is strongly dependent upon the size and size distribution of the particles produced in the crystallizer. Thus the design and operation of the crystallizer have a major effect upon the wash column cost. Furthermore, the overall undercooling in the crystallizer adds to the compression poxer requirement, and so this too is affected by the design and operation of the crystallizer. However, the effects of residence time, undercooling, and agitation upon the size of the crystals produced and the subsequent effect that these crystals have on the permeability of ice beds formed in the washer are not well known. Consequently, the design and optimization of the process cannot be made on a sound basis. The objective of this study was therefore twofold: first, to obtain a quantitative understanding of the processes involved in the growth and nucleation of ice in a continuously operated well mixed crystallizer, and secondly, to relate crystallizer performance to the permeability of ice beds formed from the product ice. Theory
Population Balance. The mathematics required to describe systems in which a spectrum of particle sizes exists, such as in crystallization, was recognized by Bransom, et al. (1949), and is finding increasing application To whom correspondence should be sent. Present address, Department of Chemical Engineering, University of California, Berkeley, Calif.
to crystallizers. The work of Randolph (1964), Randolph and Larson (1962), and Hulburt and Katz (1964) has been notable. The approach followed has been to perform a balance on particles within a small size range in an arbitrary volume. If the sizes of the particles in a continuous, well mixed crystallizer can be specified by a single length dimension, r , the population balance yields a partial differential equation with the particle size distribution function, f, as the independent variable
The terms in eq 1 represent, for a given size range, accumulation, net change in number due to growth, net nucleation, and withdrawal in the exit stream. I t is assumed that the feed stream contains no particles and that the crystallizer is well mixed, so that the exit slurry is identical with t'hat throughout the crystallizer. The linear growth rat,e G and the net nucleation rate distribution H in eq 1 will in general be functions of the temperature and salinity in the crystallizer as well as particle size. The population balance, together with material and energy balances, completely models the well mixed crystallizer. Clearly then, the particle size distribution function is an important means of describing the crystallizer operation and is uniquely determined by t,he growth and nucleation functions, G and H . Nucleation of I c e Crystals. The function H is a net nucleat.ion rate distribution and consequent,ly must include effects such as at'trition and agglomeration in addition to t h e so-called normal processes of nucleation. T h e literature 011 such phenomena is sparse. Very pure mater in small droplets can be subcooled about 35OC before nucleation (Masoii, 1958) (presumably homogeneous nucleat'ion) occurs. Laboratory distilled water in a 2-1. flask, without special precautions regarding cleanliness, was found (Sadek, 1965; Sherwood, et al., 1966) to nucleate Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
439
when subcooled about 4 or 5OC; this is presumed to occur by heterogeneous nucleation on dust particles or perhaps on the vessel wall. However, when growing ice crystals are present in subcooled water, nucleation occurs (Sadek, 1965; Sherwood, et al., 1966) a t undercoolings of less than 0.5”C. This phenomenon is called secondary nucleation because it is presumably caused by the presence of the growing ice crystals. I n a continuous back-mixed crystallizer, secondary nucleation is generally believed to be the only nucleation mechanism of importance. There is little or no theory describing secondary nucleation, although it is generally thought to occur by the shedding of dendrites grown on the surfaces of the parent crystals. The rate of nucleation has been observed to increase with agitation and with the degree of subcooling. Cayey and Estrin (1967) have observed a strong relationship between supersaturation and secondary nucleation rates of magnesium sulfate. Melia and Moffit (1964) observed similar trends with potassium chloride in addition to noting increased nucleation rates a t higher agitation rates. Interestingly, they did find that nucleation would not occur until dendrites began to form on the seed crystals. Even less has been reported on the secondary nucleation of ice. Sadek (Sadek, 1965; Sherwood, et al., 1966) performed batch freezing experiments with water and salt solutions cooled by direct contact refrigeration using isobutylene. It was found that the nucleation rate could be correlated either by the product of the refrigerant flow rate and the total number of crystals present or by the product of the refrigerant flow rate and the total crystal area and consequently it was postulated that the nucleation process was initiated by the contacting of the cold refrigerant with the ice crystals. The Growth Rate of Ice Crystals. The growth of suspended ice crystals in a subcooled salt solution depends on diffusional processes in addition to the intrinsic kinetics of crystallization, which is a measure of the rate of attachment and rearrangement of water molecules on the solid surface in accordance with the crystal habit. The diffusion of heat and the diffusion of salt away from the ice-water interface are both important and may limit the overall crystallization rate if the intrinsic kinetics are fast relative t o these transport processes. Furthermore, other processes such as agglomeration and attrition can greatly modify the apparent crystal growth rates, and little is known about. the rates of either of these processes. The intrinsic growth rates along the two major growth axes of ice have been measured, and it is observed that the intrinsic growth rate along the a axis direction is much faster than that along the e axis direction. This results in a disk-like morphology, observed for example by RiIason, et al. (1963), and arakawa (1954), which rapidly becomes dendritic a t any sizable growth rate. Sperry (1965) has summarized most of the reported a axis growth rate data and has pointed out the possibility that the reported data may in fact include heat transfer limitations due to the rapidity of the intrinsic kinetics. The fact was further emphasized by the studies of Fernandez and Barduhn (1967), who have successfully correlated a axis growth rates, of the same magnitude as the previously reported intrinsic data, by a purely heat and mass transfer theory. Intrinsic growth rates in the c axis direction have been reported by Hillig (1958) and Michaels, et al. (1966). Both observed enhanced growth rates when the face of the crystal became damaged. I n order to analyze the heat and salt diffusion effects on ice crystal growth, transport coefficients must be known. A correlation is available (Brian, et al., 1969) for heat and mass 440 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
transport to spherical particles suspended in an agitated liquid, but no such correlation exists for disk-shaped particles. Methods for coupling the mass and heat transfer processes occurring when ice grows from saline solutions have been reported by Sherwood and Brian (1964) and Harriott (1967). Both used a simplified film model to account for transpiration of water through the boundary layer. Recently, however, a more general method has been presented (Brian and Hales, 1969) that includes the effects of transpiration and changing diameter on the heat and mass transfer to spheres. Permeability of the Ice Beds. The permeability Bo of a bed of particles is defined by D’arcy’s law
AP L
- puo
gcB0
(2)
For uniformly packed beds of spheres the well established (Carman, 1956) Kozeny-Carman equation (eq 3) further relates the permeability to the porosity E of the bed, the surface to volume ratio of the particles in the bed SO,and a Kozeny-Carman constant k. The constant k is presumed equal to I C ~ ( L ~where / L ) ~ ko is a “shape factor” and Le/L is the tortuosity. For beds of spheres k is 5.0 and So = 6/de so t h a t the permeability can be also expressed in terms of a n equivalent spherical diameter, de, and porosity.
Ice particles produced in crystallizers, however, are somewhat irregular disks varying in size over a wide range. Permeability data on flow through beds of small disks are scarce. Wyllie and Gregory (1955) measured both So and the permeability of beds of plastic disks 3.18 mm in diameter and 1.58 mm thick. Porosity was varied by employing different methods of forming the beds. The results show So decreasing with decreases in porosity, presumably because of greater particle alignment. Tortuosities, on the other hand, obtained by measurements of the electrical conductivity of liquids filling the voids, were greater a t the lower porosities. Thus, with decreasing e, So decreased but k increased (assuming ko unchanged), so that the permeability Bo varied less with E than suggested by eq 3. Coulson (1949) tested steel plates of 6.4 X 6.4 mm and either 3.2 or 1.6 mm thickness. Ritter (1969) reports permeabilities of ice beds formed from slurries containing about 10 w t % ice in 4 and 7 w t % sodium chloride. In both cases trends were similar to those observed by Wyllie and Gregory (1955). I t appears, therefore, that particle shape and size distribution can have important, but as yet not well characterized, effects on permeability of beds. Equation 3 must therefore be considered an empirical equation when applied to the present system. Scope of the Present Study. It is evident t h a t general methods for the prediction of nucleation rates are lacking. Whereas some procedures for determining growth rates are known, uncertainties exist regarding the intrinsic kinetics and also in the application of available heat and mass transfer correlations to ice disks. Furthermore, no techniques exist for relating the permeabilities of ice beds to the intrinsic properties of the packed bed and the particles t h a t make up the bed. The purpose of this study was, therefore, twofold: first, to measure the effects of residence time, agitation rate, and undercooling on the particle size distribution of the ice produced in a well mixed crystallizer and t o use these results to obtain ice growth and nucleation rates; and second, to relate crystal-
I so. VENT
4-
ISO. FEED
ORIFICE METER
EXIT ROTAME ER
EXIT CONTROL VALVES
Figure 1 . Flow diagram of apparatus
lizer operating conditions to the permeability of ice beds formed from the product ice. Experimental Section
The equipment employed was designed to permit studies of the continuous steady-state crystallhation of ice from sodium chloride brine. Refrigeration was provided by direct contact evaporation of isobutylene sparged in a t the bottom of the crystallizer. A diagram of the equipment is shown in Figure 1, and full details of the design of the apparatus are reported elsewhere (Margolis, 1969). The cylindrical crystallizer 93/s in. i.d. by 20 in. tall was fed a t a steady rate with precooled brine, and the effluent was examined photographically to determine the concentration and the characteristics of the ice particles formed. The crystallizer contents were well agitated by two downward pumping marine impellers on a central shaft driven by a variable-speed motor. Ice-brine slurry flowed out of the crystallizer through an exit tube that faced downward into the upward flowing slurry. This arrangement was designed to obtain near isokinetic sampling of the slurry withdrawn and thus to avoid particle classification a t the exit port. Two gas disengaging tubes removed the major portion of the isobutylene vapor entrained in the exit slurry stream. The degassed ice slurry was photographed as it passed through a narrow rectangular horizontal channel formed between flat Plexiglas plates. The slurry entered this channel through tapered end sections that spread the crystals evenly across the channel and left through a similarly tapered section. Three different channel heights were employed for different ice concentrations and flow rates. The cell dimensions were
varied so as to cause an even spread of crystals not more than about 2 or 3 crystal layers thick across the viewing area. Photographs were taken with a Honeywell Spotmatic SLR camera fitted with extension tubes and a 145mm f/4.5 lens. A parallel light ray source provided the illumination. Motion pictures of the moving particles were also taken. After leaving the photographic cell, the slurry passed through a discharge pump, a melter, and a rotameter. The mass fraction ice in the exit slurry was computed from the salinity of the exit stream after the ice melter and the salinity of a brine sample from the slurry a t the crystallizer exit port, this sample being withdrawn through a hypodermic needle that excluded the ice crystals. The slurry level in the crystallizer and the crystallizer pressure, which controlled the refrigerant vaporization temperature, were automatically controlled. The temperature of the slurry in the crystallizer was recorded continuously by means of a platinum resistance thermometer and bridge circuit capable of detecting temperature variations of *O.O0loC. The salinity of the exit brine before and after the ice melter was determined every half hour. An experimental run lasted about 3.5 hr. Steady-state operation, as indicated by slurry temperature and weight fraction of ice produced. was reached within about 1.5 hr. Photographs of the exit slurry were taken after 3 hr of operation. The undercooling of the brine was determined a t the end of a run by noting the rapid temperature equilibration of the solution that occurred after the aqueous and refrigerant flow rates were stopped simultaneously. These measured temperature changes had to be corrected for the effect of refrigerant holdup in the crystallizer, as discussed later. Crystal counts were made on photographs enlarged to eight Ind. Eng. Chem. Pundom., Vol. 10, No. 3, 1971
441
TO V A L V E
SECTION THRLJ' EXIT PORT
AND R O T A M E T E R
$r
Table I. Experimental Conditions of Runs and Solution Undercoolings Av no.
SECTION T H R U ' TOP B A L L OF VALVE 'A" PLEXIGLAS8 TUBING VALVES A R E 1" N O M i k A L B A L L VALVES. ( b ) P I P I N G 1" ID
Mixer rPm
Wt%
Cora solution undcrcooling (20
ice
Tal, "C
-
38 13 -5.43 375 7.93 0.024 51 13 -5.43 375 7.82 0.025 65 13 -5.43 375 8.95 0.021 39 13 -5.43 375 3.15 0.024 48 13 -5.43 375 3.51 0.021 53 13 -5.43 375 2.94 0.025 54 13 -5.43 375 2.95 0.023 62 13 -5.43 375 3.20 0.021 40 13 -4.69 375 3.67 0.026 42 13 -4.69 375 3.73 0.022 50 13 -4.69 375 3.48 0.025 64 13 -4.69 375 3.48 0.024 45 6.5 -5.43 375 3.81 0.033 46 6.5 375 -5.43 4.28 0.031 44 6.5 -5.43 687 3.77 0.017 47 6.5 -5.43 687 4.82 0,022 a Corrected undercooling = measured undercooling 0.012OC. All runs performed with a steady-state brine concent,ration of about 6 wt % (fp -3.71"C). c No P.S.D.'s were obtained for runs 46,44, and 47 owing to difficulty in counting particles.
+
NOTE W B A L L
SLURRY
residence time, min
Refrig evap temp, OC
73
INLET
1
SCREEN SOLDERED TO B O T T O M OF BRASS T U B E
Figure 2. Permeability test device
Small corrections for brine held in spaces outside the ice bed were necessary. times the actual size. Particles were taken to be single crystals if they approximated circular or hexagonal disks irrespective of whether they appeared singly or in clumps because movies showed that the separate particles in a clump would sometimes separate and later recombine. Rectangular particles, as shown below, were counted as single disk-shaped ice crystals on edge of diameter D and height h.
D Particle size distributions were obtained from the resulting particle count histograms. Upon termination of a run, permeability measurements were immediately made on the slurry remaining in the crystallizer. By nitrogen pressurization of the crystallizer, the slurry ice was forced to flow a t about 900 ml/min to the permeability test device. The permeability test equipment, shown in Figure 2 , consisted of a 2.54-cm i.d. PVC tube held between two 2.54-cm ball valves 15 cm apart. A fine metal screen set in the top valve served to retain ice carried by upward-flowing slurry fed a t the bottom. Pressure taps drilled into the tubes were connected to a mercury manometer; these could be flushed briefly with flowing brine. The pressure drop and the effluent brine flowmeter reading were noted when the accumulating ice reached the Plexiglas tubing below ball valve B, a t which time both valves were closed, isolating the test bed. The entire procedure required about 6 min. The pressure taps were then closed and the bed was allowed to melt. Analyses of the brine in the feed slurry and of the melted ice-brine bed permitted calculation of the bed porosity. 442
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
Results and Discussion
The effects of four independent variables upon the operation of the crystallizer were studied. The average residence time was varied from 6.5 to 13 min; the weight fraction of ice, or slurry density, was varied from 4 to 8 wt %; the refrigerant evaporation temperature was varied from - 5.43 to -4.69OC; and the agitation rate was varied from 375 to 687 rpm. All runs were performed so as to produce a slurry containing a brine of approximately 6 wt Yo NaCl (equilibrium freezing point -3.706"C). The experimental conditions of each run, together with the solution undercooling, are tabulated in Table I. Table I1 shows values of bed permeability, Bo, and bed porosity, e, calculated directly from the data. Values of residence time and agitator speed refer to operation of the crystallizer during the production of the slurry tested. The column "melting" is the per cent of the ice leaving the crystallizer which had melted during passage through the lines and the permeability test device, as a result of heat leaks. (The permeability tester was precooled before each run to minimize the heat capacity effects.) The amount of melting was calculated from titrations of brine effluents from the crystallizer and from the ice test bed. Figures 3 and 4 are representative photographs of the ice crystals produced in the crystallizer, and the particle size distributions obtained for the different runs are shown in Figures 5 through 9. Particle clasqification a t the slurry outlet port, due to nonisokinetic sampling, could cause the particle size distributions of the crystallizer contents and the exit stream to be different. This effect is considered to be negligible here because the weight fraction ice in the effluent showed good agreement with that in the crystallizer a t the end of a run. Furthermore, the cross-sectional area of the exit port was 787, greater for run 54 than it was for run 62
Table II. Ice Bed Permeability Data Run" no.
Av residence time, mi"
Agitation rpm
Bed por01ity.
Permeabilityb
Ba X l o g f t l
d , mm
Melting
%
so, mm-l
Key=
40 375 0.559 6.95 0.359 9.53 0 375 0.561 6.76 0.352 9.7 8.84 0 375 0.625 11.8 0.337 3.9 8.99 0 51 375 0.517 3.92 0.332 8.18 X 31 9.37 X 375 0.612 8.15 0.299 32 375 0.548 5.27 0.330 ... X 2.2 9.12 X 375 0.570 5.22 0,299 ... 0 45 375 0.573 5.39 0.296 30 9.95 0 375 0.588 6.28 0.295 8.47 0 28 375 0.488 4.32 0.404 50 8.95 0 5.3 375 0.603 7.46 0.299 64 12.2 7.58 A 375 0.594 6.01 0.278 45 6.5 ... A 375 0.568 5.14 0.296 fi.5 46 0.264 18.2 ... 0 2.57 0.521 6.5 687 44 0.275 33.1 ... 0 2.76 0 520 47 f.~~ i5 687 ~. In runs 38, 51, and 65 the slurry was approximately 8 wt % ice; in the remaining runs it was roughly 4 wt %. For definition of Bo see Nomenclature. The brine flaw was laminar in the forming ice bed; Reynolds number varied between the extremes of 4.7 and 1.9. The "key" identifies the points on Figures 19 and 20.
38 51 65 39 48 61 62 40 42
13 13 13 13 13 13 13 13 13 13 13 6.5
0
and Figure 7 reveals no appreciable effect of this differeiice in sampling velocity upon the measured particle size distribution. Similarly, high-speed movies showed t h a t all ice particles traveled through the photographic cell a t approximately the same velocity, indicating t h a t particle classification was not taking place within the cell. Figure 10 indicates the measured crystal heights as a func-
tion of diameter. Within the precision of the data, a constant height-to diameter ratio of 0.34 is indicated. Growth Rates of the Ice Crystals. For a continuous well mixed crystallizer operating under steady-state conditions, eq 1 reduces t o
d(Gf)= H - ; f dr Clearly there are inany combinations of II(r) and G(r) t h a t
Figure 3. Typical ice crystals produced in the crystallizer. Run 38: wt fraction ice = 0.0793; 7 = 1 3 min; agitator = 375 rpm; ATom 0.024OC; refrig evap temp = -5.43"C. The dark rectangular regions are the light disks on edge. The circular images are isobutylene bubbles. The two dark lines are 1 mm apart
Figure 4. Typical ice crystals produced in the crystallizer. Run 64: wt fraction ice = 0.0348; T = 1 3 min; agitator = 375 rpm; ATom = 0.024"C; refrig evap temp = -4.69OC Ind. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
443
1000 0 RUN 65 0 RUN51
A RUN 38
z
0
10
I 0
I 02
I
I
I
I 0.6
0.4
RADIUS
I
I
0.8
I
14
?
1 .o
mm
Figure 5. Particle size distribution: av res time = 13 min; wt % ice 'v 8%; agitator = 375 rpm; refrig evap temp = - 5.43 "C
can yield the same distribution f ( r ) . If nucleation, agglomeration, and particle frachring do not act to add or remove particles within some size range, T I to r2, then H = 0 throughout this range, and eq 4 can be integrated to give Gljl = G2j2
+
(5)
l:fdr
It is often assumed that all nucleation occurs a t very small size, and the present photographs showed little evidence of fracturing of crystals. If agglomeration and fracturing are
2
I
I
I
I
I
I
I
I
I
I
I
I
assumed to be absent and if all nuclei are assumed to be born a t near zero size, eq 4 becomes -(Gj) d dr
=
-f (for r > 0) 7
(6)
with the boundary condition
fG = N
(at r = 0+)
(7)
Integration gives
For the special case of a growth rate which is independent of crystal size, integration of eq 6 gives
f = (;)exp(-
B4 0
0.2
0.6
0.4 RADIUS
0.8
1.o
mm
Figure 6. Particle size distribution: av res time = 13 min; wt % ice 'v 4%; agitator = 3 7 5 rpm; refrig evap temp = -5.43"C 444
Ind. Eng. Chem. Fundam., Vol. 10,
No. 3, 1971
&)
(9)
which yields a straight line on a semilog plot of j us. r. Figures 5 through 9 reveal that all the measured particle size distributions passed through a maximum, which would appear to indicate a size-dependent growth rate. Assuming H = 0, the variation of G with r can be obtained from the experimental curves of f ZIS.T by use of eq 5, but this requires, in addition, knowledge of G a t some value of r. It is tempting to use eq 8 with the experimentally determined nucleation rate, but incomplete particle size distributions a t small sizes preclude this. A possible alternative is to note that for large crystals the curves in Figures 5 through 9 appear to be fairly straight. It was shown by eq 9 that if G is independent of r , the plot would be a straight line with a negative slope l / G r . Clearly if the same were true for the situation where G is independent of r in a limited size range then this would form a convenient point of departure.
500
"I
h:.34 D ***@. c
F 05
2a
z
0
40,50,42,64 45
A RUNS 01 -
0
+RUN l
l
l
l
I
1
I
l
I
1
l
1
U
w u
i
0
0
0.2
0.4
0.6
0.8
1.0
1.2
RADIUS m m Figure 8. Particle size distribution: av res time = 13 min; wt % ice 'v 4%; agitator = 375 rpm; refrig evap temp =
z
L-
1
0 c
-4.69"C z 0 i
3 -
m E
0
02
0.4
0.6
0.8
1.0
mm Figure 1 1 , Particle size distribution of run 65 RADIUS
RADIUS
mm
Figure 9. Particle size distribution: av res time = 6.5 min; wt % ice 'v 4%; agitator = 375 rpm; refrig evap temp = -543°C
If, then, over some range the semilog plot off with a slope of - s f = ae-8'
(for rl
5 r 5 rz)
US.
r is linear (10)
and if H = 0 over this range, eq 4 yields 1
-
-
~-
ST
GI -
1 -
Gz
-
es(T2-Tl)
=
L j?
(11)
s7
Therefore it is seen that G need not be constant and equal to S ST when eq 10 holds over a limited size range, but if eq 10 holds over a considerable range, such that f varies by a large
factor over this range, it is clear from eq 11 that GI must be very nearly equal t,o 1/s7; ot,herwise G would have to change enormously with variat,ionsin r over that range. Equat'ion 11 also reveals a general characteristic of eq 5. If different starting values of GPare chosen a t r = r 2 and eq 5 is used to calculate G os. r back t,o smaller values of r , using a n experiment,al curve of f us. r , the coniput'ed curve of G US. r will rapidly become insensitive to t,he starting value, GP, as r decreases from rz. Therefore for r values substantially smaller than rP,t,he curve of G us. r d l be determined largely by the curve of j us. r and not by the value chosen for GP.Equation 11 shows t,his quantitatively for the case of a linear semilog plot, of j u s . r , but this behavior is true in general when f increases rapidly as r decreases because the integral in eq 5 becomes large relative t o the term G2fz when r1 is substantially less than rp. These considerations suggest a niet,hod of determining t'he function G ( r ) from the experimental curves of fvs. r which will be illustrated for run no. 65. Figure 11 shows the experimental f curve. For values of T between 0.5 and 1.0, t,he dot'ted straight line is a fair representation of the data, alt'hough the curve is thought to be somewhat' better. The slope of t.he dot,ted straight line corresponds to G = 0.0114 mm/min with T = Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
445
t
1
l
RUN 6 5
t
rk
\
0.01-'
RUN 4 8 j
!~~~~~~~~~~
0.006 0.2
0
0.8
0.6
0.4
1.0
1.2
RADIUS m m Figure 12. Illustration of the sensitivity of the measured growth rates to G,,,$
13 min. Assuming this value of G down to r = 0.5 and then integrating the curve back to smaller values of r , employing eq 5, yields the dotted curve of G us. r shown in Figure 12. At r = 1.01 mm, the solid curve in Figure 11 has a higher slope than the dotted line, and this slope corresponds to G = 0.0085 mm/min. Using this value and integrating the solid curve back from r = 1.01 mm yields, according to eq 5, the solid curve of G us. r in Figure 12. To show the sensitivity of the computed curve to the value chosen for G a t r = 1.01 mm, this calculation \vas repeated with starting values of
I
1.0
0.1
I
1
I
I
1
I
I
1
1
I
0
1.4
0.2
F\ 1
1.0
1.2
RADIUS mm Figure 14. Measured and predicted growth rates; ATo, 0.023"C used in predicting growth rates
=
twice 0.0085 and half of 0.0085, and these results are shown once again in Figure 12. For values of r less than 0.7 mm, the computed curve is quite insensitive to the starting value used for G. The calculational method which produced the solid curve in Figure 12 appears to be satisfactory, although there is, of course, some uncertainty in G for r values between 0.7 and 1.0 mm. The method illustrated for calculating the solid curve (using the local slope a t rz to determine G? and then using eq 5 ) was employed for all of the runs, and the results are shown as the measured growth rates in Figures 13 through 16. They all show a sharp increase in G as r decreases below 0.4 mm,
I 4
MEASURED
MEASURED GROWTH
0.8
0.6
0.4
I
RATES
GROWTH RATES
PREDICTED GROWTH -EFFECTS
OF HEAT
RATES MASS
a
TRANSFER
PREDICTED GROWTH RATES E F F E C T S O F HEAT 6 TRANCFFR
f ; A C C
50
40
-t
t-I 0
0.2
0.4
0.6
08
1.0
1.2
mrn Figure 13. Measured and predicted growth rates; ATo, = 0.023"C used in predicting growth rates RADIUS
446
Ind. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
0
I
I 0.2
I
I 0.4
I
1
I
0.6
RADIUS
I 0.8
11-1
RUN 4 2 ' I
1.o
1.2
mm
Figure 15. Measured and predicted growth rates; AT,, 0.023"C used in predicting growth rates
=
MEASURED GROWTH RATES
0.1
PREDICTED GROWTH RATES -EFFECTS
I .oil 0
OF HEAT
a
MASS
I
TRANSFER
I
I 0.2
I
I 0.4
I
I
I
I
0.8
0.6
RADIUS
I
I
I
1
1.2
1.0
mm
Figure 16. Measured and predicted growth rates:
AT,,
=
0.033"Cused in predicting growth rates corresponding to the sharp maxima in t'he particle size distribution functions. Ice crystal growth rates were initially predicted based on t,he assumpt,ionthat the intrinsic kinetics were infinitely rapid and therefore that growth was limited by the rate of heat transfer and the rate of mass transfer of salt. Since salt is excluded from the growing crystal surface, the salinity of the solution adjacent to bhe surface exceeds the salinity of the bulk solution, causing salt to diffuse away from t'he advancing surface. This increased salinity a t the surfa,ce lowers the freezing point and decreases t,he t.emperature driving force for heat t,ransfer from the surface to the bulk solut'ion. Techniques for analyzing t.he combined effects of heat and salt diffusion have been present,ed by Sherwood and Brian (1964) and Harriott (1967) based upon a film theory model to account for transpiration through t'he boundary layer. Brian and Hales (1969) present more complete analysis which accounts for transient effects due to part'icle growth as well as for transpiration effects. However a t t'he slow growth rates considered here the effects of transpiration and changing size are negligible, and t8hese various methods become identical. They simply solve for a temperature and salinity a t the cryst'al surface which lie on the freezing point curve and which balance the heat transfer rat,e with the salt diffusion rate. I n order to make t'hese calculations, heat and mass transfer coefficients t,o the suspended particles must be known. A correla't'ion is available for spheres (Brian, et al., 1969) but there is none for disks. I n these calculations it was assumed t'hat, for a disk of diameter D and height h , the t'ransport coefficients to the edge and to the face of the disk could be approximated by the transport coefficients t o spheres of diameter h and D , respectively. This assumpt,ion cannot be defended on any grounds other than the fact that no correlation exists for disks and some guess had to be made, but it would be expected t,o give answers of the right order of magnitude. In applying the correlation (Brian, et al., 1969) for heat and mass transport, the agitat,ion power is required. The value employed was 3.8 cal/(min) (1.) a t 375 rpm, which was the value measured in the present syst.em in the absence of ice crystals and isobutylene sparging. The measured subcoolings (T, - T,) had to be corrected for the possible effects of evaporation of residual refrigerant after the termination of a run, before they could be used to predict gron-th rates. The ext'ent of the correction was determined by sampling the solution in the crystallizer and not,ing the sampling times on the recorder output, from the platinum resistance ther-
mometer. These samples were titrated for salt content and their equilibrium freezing points were determined from the data of Scatchard and Prentiss (1933). It was assumed t h a t equilibrium was established a short while after run termination so that the recorder temperature trace could be fitted to the freezing points obtained from the titrations. This procedure established both the instrument calibration and the actual temperature of the solution (mean of the fluctuations) before the run ended. Salt concentrations of the brine in the crystallizer during steady-state operation permitted the calculation of the equilibrium freezing point. The difference of these two temperatures was the correct overall subcooling. This procedure was followed for only runs 53 and 55 (experimental conditions of run 55 similar to those of runs 3865) and the subcoolings exceeded by 0.013 and 0.011 the measured temperature rise when the aqueous and refrigerant flows were stopped simultaneously. The average of these two corrections, O.O12OC, was therefore added t o this temperature rise for each run to produce the "corrected" solution undercooling values given in Table I. Calculations of the growth rates were then made (Margolis, 1969) for solution undercoolings of 0.023 and 0.033OC a t a n agitation rate of 375 rpm. The growth rates were computed for particles of different diameters assuming h = 0.340. The edge growth rates, drldt, are plotted as the predicted curves in Figures 13-16. These calculations were also modified to include the intrinsic crystallization kinetics. For the growth of the edge the results of several investigations for a axis growth as summarized by Sperry (1965) can be represented by Gaaxis(mm/min) = 96AT,,l
611
(12)
The work of Fernandez and Barduhn (1967) supports Sperry's belief that eq 12 does not represent a true kinetic rate but that the intrinsic rate must be faster than that indicated by eq 12. Kevertheless, eq 12 was used in the calculations and the results indicated t h a t including the intrinsic kinetics lowers the edge growth rate only 10 to 20% for particles of 0.15 to 1 mm radius; if the true kinetics are faster than those given by eq 12, the reduction would be smaller than this. Inspection of Figures 13-16 shows remarkably good agreement between measured and predicted growth rates in the size range 0.3 to 1 mm, considering the approximations made in predicting transfer coefficients and the inherent difficulties encountered in measuring the small solution undercoolings. The measured growth rates are seen to increase sharply as the disk size is reduced and to exceed by far thp predicted values. This is the result of the rapid decrease in f for values of r less than about 0.3 mni. The curve of f us. r which corresponds to the predicted curve of (f us. r can be calculated by integrating eq 4 to give
assuming H = 0 in the range rl to r2.Employing this equation with the predicted diffusion-limited growth curve in Figure 13 yields the predicted particle size distribution curve shown in Figure 17. Since eq 13 yields only the ratio of f values, the result may be multiplied by an arbitrary constant, which corresponds to sliding the predicted f curve up or down in the semilog plot of Figure 17. The curve shown was so adjusted to show agreement with the measured f curve for run 65 a t large crystal sizes. Figures 13 and 17 are two different ways of comparing the predicted growth rates with the measured rates for run 65. Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
447
Table 111. Wt
PREDICTED PARTICLE SIZE DISTRIBUTION
MEASURED PARTICLE S I Z E DISTRIBUTION -
c -
-1
a-
E E >
Z J
,“ In W
Y
J
U
locok0
0.2
I
R U N 65
0.4
0.8
0.6
RADIUS
1.0
mm
Figure 17. Predicted and measured particle size distribution
The absence of counted particles less than 0.3 mm in radius in Figure 17 corresponds t o the very high measured growth rates in Figure 13. One possible explanation for the discrepancy is that the small ice particles may agglomerate rapidly, in which case H is not zero. This would give rise t o an apparent “growth” mechanism which is more rapid than the diffusionlimited growth. Another possibility is that very small particles are caught or nucleated in the cold regions adjacent t o a drop or vapor bubble of refrigerant where the undercooling might be as high as 1-2OC, and this could result in high growth rates. However, calculations indicated that the residence times of the refrigerant bubbles are probably too short to explain appreciable particle growth even at such high rates. A third possible explanation for the discrepancy is that melting of the ice particles in the slurry line before the photographs were taken could possibly have affected the observed particle size distributions and caused these effects. Only 5y0 of the total ice must be melted in order to change the predicted f curve in Figure 17 to the observed f curve for T < 0.3 mm. The heat leak into the slurry line is estimated to be high enough to melt 3y0 of the ice, but shifting from the predicted to the measured f curve in Figure 16 implies preferential melting of some crystals and not others. If all crystals of a given size melted to the same extent, even though different size crystals melted to different extents, then melting would displace each point on an f curve to the left, although different points might be displaced by different amounts. Such a process could never shift the maximum point to larger values of T and could surely not change the predicted f curve to the measured f curve in Figure 16; this would require melting particles of the same size by different amounts. Therefore it seems unlikely that crystal melting caused the discrepancy observed in Figures 13 and 17. The cause remains undetermined a t present, but of the three possibilities the agglomeration postulate appears t o be the most likely explanation. 448
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
Run no.
38 51 65 39 48 53 54 62 40 42 50
64 45
% Ice from
Particle Counts
W t % ice from chem analysis
Wt ice from part. counts
%
Counted/ chemical wt % ice
Measd nucleation rate, nuclei/ (cm30f slurry min)
7.93 7.82 8.95 3.15 3.51 2.94 2.95 3.20 3.67 3.73 3.48 3.48 3.81
4.06 4.32 5.89 2.19 2.66 2.30 2.27 2.73 2.17 2.72 2.43 2.54 2.43
0.51 0.55 0.66 0.69 0.76 0.78 0.77 0.85 0.59 0.73 0.70 0.73 0.64
16.0 13.7 20.0 5.2 9.5 6.96 6.8 10.1 7.4 7.4 7.8 8.4 9.4
Finally, it should be noted that the ice particle shape is a unique function of the a and c axis growth rates, which are dependent on the intrinsic growth kinetics as well as heat and mass transfer characteristics and possibly other effects such as agglomeration. Reported values of the c axis intrinsic growth rates measured in capillaries (Hillig, 1958; Michaels, et al., 1966) are exceedingly small at the undercoolings encountered in this study so that very thin disks would be expected. This is different from the crystal shape observed in this study, suggesting either that the kinetics measured in capillaries are not representative of a free crystal or that agglomeration of thin crystals has produced thick ones. However, both Hillig (1958) and Sperry (1965) have observed enhanced c axis growth rates when the crystal face becomes damaged, and this possibly could have accounted for the increased face growth rates. Nucleation Rates. The particle counts obtained from the photographs give directly t h e number of ice crystals counted in a known slurry volume in the photographic cell. Dividing t h e number of crystals by this slurry volume and multiplying by the volumetric flow rate of slurry yields directly t h e net rate of production of crystals. This rate divided by t h e volume of slurry i n t h e crystallizer is t h e net nucleation rate, as indicated by the particle count, a n d is tabulated for each run in Table 111. As a check on the completeness of particle counting, the mass of ice counted was computed for each run. The volume of each crystal counted was calculated as 0 . 6 8 on ~ ~the~ assumption that it was a disk of thickness equal to 0.6%. hlultiplying this volume by the density of ice yielded the mass of the ice crystal. Using this procedure the total mass of ice crystals was determined in the slurry volume in the photographic cell in the area counted; thus the weight per cent ice in the slurry, as indicated by the particle count, was determined for each run. These values are compared in Table 111 with the values determined by chloride analysis, and it is seen that the slurry density indicated by the counting procedure is low by 15-50%, or about 3oYOon the average. This discrepancy could have been caused by many factors such as difficulty in counting particles in the darker regions and covering of particles by gas bubbles, but the major error is considered to have been the result of the fact that the areas counted were chosen on the basis of clarity and this frequently led to lower particle concentrations. (The relative distribu-
Table IV. Corrected Slurry Densities and Nucleation Rates and Experimental Conditions of the Different Runs
Run no.
Chem. wt ice
%
7.93 7.82 8.95 3.15 3.51 2.94 2.95 3.20 3.67 3.73 3.48 3.48 3.81
38 51 65 39 48 53 54
62 40 42 50 64
45
AV res. time, min
Refrig evap temp, " C
Mixer rpm
Cor solution undercooling (To OC
13 13 13 13 13 13 13 13 13 13 13 13 6.5
-5.43 -5.43 -5.43 -5.43 -5.43 -5.43 -5.43 -5.43 -4.69 -4.69 -4.69 -4.69 -5.43
375 375 375 375 375 375 37 5 375 37 5 375 375 375 37 5
0,024 0.025 0.021 0.024 0.021 0.025 0.023 0.021 0.026 0.022 0.025 0.024 0.033
rml,
tions were hopefully not affected since the particles traveled in clumps.) I n order to account for this discrepancy the computed particle counts were multiplied by the ratio of the ice mass determined by chemical analysis to that computed from the counting analysis. The corrected nucleation rates together with the relevant operating conditions for each run are shown in Table IV. Also included in Table IV are specific crystal areas and perimeters computed by assuming the crystal to be a disk with h = 0.6%. The data suggest that the refrigerant evaporation temperature appeared to have litt,le effect on nucleation rate. Whereas comparison of runs 38-65 with runs 39-62 and 40-64 would indicate refrigerant flow rate to affect nucleation rates, run 45, made at high refrigerant flow rate but half t,he average residence time, shows nearly the same nucleat.ion rate as found in the similar tests a t half the refrigerant flow rate. Evidently the nucleation rate appears to correlate better with slurry density, which is proportional to the refrigerant flow rate mult,iplied by the average residence time. However, it should be iioted t,hat the specific cryst>alarea and perimeter were found t,o be approximately constant for all runs. This fact, coupled wit'h the observation that the ratio of two successive moment's of the dist.ribut'ion was very approximately equal t,o GT indicates further that the nucleation should also cor-
N e t refrig flow rate, g/min
Slurry density, g of ice/ cm3 of slurry
Specific crystal area, cma/g of ice
98.5 98.1 99.0 46.3 45.8 45.4 45.4 46.3 47.2 46.8 47.2 47.2 98.5
0.082 0.081 0.093 0.0326 0.0364 0.0307 0.0305 0.0330 0.0380 0.0386 0.0362 0.0361 0.0394
104.6 97.2 98.6 89.9 103.0 93.8 95.4 100.3 108.9 109.3 93.1 98.3 83.2
Cor nucleation Specific rate, crystal nuclei/ perimeter, (cm3 of slurry cmfg of ice min)
1324 1157 1185 960 1261 1078 1098 1239 1237 1031 1108 1166 830
31.3 24.8 30.4 7.44 12.6 8.9 8.9 11.8 12.6 10.1 11.1 11.5 14.8
relate about equally well with the product of either the crystal area or perimeter and the undercooling raised to a positive exponent. Since little theoretical background on the phenomena involved in secondary nucleation is available, models for correlating the data were based either on the observed trends in the data or on previous studies with ice. Five such models were proposed. Four of these attempted to model data trends while the fifth based on the studies by Sadek (1965) was a proportionality between nucleation rate and the product of crystal area and isobutylene flow rate. The data were statistically fitted to the models (Margolis, 1969) utilizing the techniques of Lindley (1947) for the situation where both independent and dependent variables are subject to error, and the results are shown in Table V. The variance of the estimate in Table V is a measure of how well the proposed model fitted the measured data. It is evident that those models that mere based on the data trends (models 1-4) correlated the data much better than the model that was based on Sadek's results and Figure 18 shows graphically the fit of the measured data to one of these correlations (model 1). However, it is clear that since all four models correlated the data about equally well, it is not possible to choose between them without further inputs regarding the nucleation process.
Table V. Statistical Analysis of the Proposed Nucleation Rate Models Model
(1) AT = (2) N = (3) N (4) S = (5) N =
1
Regressionvalues of constants and exponents
kid1 k*JlATo,a ksAAT0,b kdPATo,C
N N N
kdFIw
N
= = = 11' = =
328 851 979J1ATom0306 92 5AAT0,O 57 5PAT0,' 4 3 0 047AF1so
Variance of estimate
4 5 5 4 33
0 16 22 58 6
n,
Variance of estimate
=
( Y %- YtY n, - f
where Y , = predicted nucleation rate from regression model, yz = corrected measured nucleation rate, n, = number of measurements, and f = degrees of freedom = 1 for models 1 and 5 and 2 for models 2 , 3 , and 4.
0
0.02
M-SLURRY
0.04
0.06
DENSITY - 9
0.08
0.10
0.12
ICE/crn3 SLURRY
Figure 18. Nucleation rate as a function of ice slurry density Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
449
I
x -l
Figure 19. Ice bed permeability as a function of porosity (for key see Table II)
Only the results of Sadek's (1965) batch tests were available for comparison with the present study. It is difficult to analyze data from batch tests, since conditions change throughout the run, and in Sadek's case particle counts were made only a t the end of a run. I n most cases the mean particle diameter was only 0.5-1.0 mm, so secondary nucleation was taking place with smaller particles than in the present tests. Sadek reported limited data on subcooling, which apparently varied considerably during the course of a run. I n view of these factors it IS probably not surprising that the model proposed by Sadek did not correlate the present data. However, as a test of the sensitivity of Sadek's data to different correlating models, an attempt (llargolis, 1969) was made to correlate Sadek's data by a nucleation model in which the rate was proportional to slurry density as was found in this study. A fairly good fit mas obtained, and this suggests therefore that Sadek's nucleation data could probably have equally well been correlated by the model proposed in this study. Clearly the determination of definitive models of the nucleation process from overall data without prior knowledge of the microprocesses occurring still remains a difficult task. Kevertheless, Sadek observed an increase in the number of crystals produced when operating a t large refrigerant uncoolings [T,- T, = 4-5OC], although there was considerable scatter. Since the refrigerant undercooling in this study varied only between 1.0 and 1.74OC the lack of an observed effect of refrigerant undercooling on nucleation rate is understandable. The effect of varying the degree of agitation on nucleation rate was not determined quantitatively because particle counts could not be obtained a t the high agitation conditions. However, the permeabilities of beds formed from the ice crystals produced in the crystallizer were measured under conditions of low and high agitation. The permeability is an indirect measure of the size of crystals forming the bed and is also a function of the bed porosity. An equivalent spherical diameter ( d e ) determined by the use of the Kozeny-Carman equation, however, is independent of porosity for spheres but IS a weak function of porosity for nonspherical particles such 450 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
as ice. Both the permeability and equivalent spherical diameter should be lower for smaller particles. Table I1 summarizes the results of the permeability tests which will be discussed later. It is clear that runs 44 and 47, performed at high agitation rates, produced smaller crystals than the low agitation runs. This, as expected, is an indication of larger numbers of smaller particles being produced and must be the result of either increased nucleation rates or decreased rates of agglomeration since the higher agitation rates would have been expected to increase the growth rates. Permeability of Ice Beds. Figure 19 shows t h e permeability as a function of bed porosity. The considerable variation in measured porosity within each set of tests can be explained only as t h e result of variations in the manner in which t h e bed was formed as ice accumulated from the flowing slurry. The experimental technique was t h e same in each case, and t h e reasons for the variation in bed structure are not known. However, the fact t h a t there can be such variations is a n observation of considerable practical importance. The data are seen to fall in three groups in Figure 19. The two lowest points are the two tests at high agitator speeds, in which much smaller ice particles were produced. The three middle curves, represented approximately by the line A-B, show the results with slurries containing 4 wt yoice. The top line in Figure 19 represents data obtained with the higher slurry density of 8 wt yo ice. Since the slurry flow rate was the same, these beds were formed in half the time, with about half the total brine flow. Particle size distributions, a s indicated earlier, were similar to those with 4 wt % ice, and it is not evident why the permeabilities were greater. Visual observations indicated somewhat greater clumping of particles in the thicker slurry, and this may have increased the effective particle size in the test bed. It is also possible that the more rapid bed formation may have caused the disk-shaped particles to be so oriented as to provide more uniform passages for brine flow resulting in higher permeability, even a t the same porosity. The dashed line in Figure 19 represents the KozenyCarman equation with kSo2taken as 0.182 X 109, independent of porosity. The data show a smaller variation of B,, with E than called for by the equation: Le., kSo2 is larger a t the higher porosities. Figure 20 shows the same data plotted as de 11s.E . This is
c
0.45
E
W I-
0.40
z
W 1
6
L
3
E
1
I
I
1
I
t
0'35L 0.30
W
'0
0.40
0.45
0.50
0.55
0.60
0.65
€ -POROSITY
Figure 20. Equivalent spherical diameters calculated using the Kozeny-Carman equation, as a function of porosity (for key see Table II)
I'
I
I
I
I
Table VI. Comparison of Disk Data from Three Sources (E 0.52) Key (Figure 21)
A, Coulson (plates) C
B, Wyllie and Gregory (disks) D, Margolis (ice) F
W J
>"
-
-
0.8-
0.6-
0
**
3
s,
UPI
-
-
2.8 1.7 2.1 0.38 0.33 0.36 0.35
mm-'
k(So/so)'
1.89 3.15 2.52 9.12 8.8 9.3 7.58
6.44 6.3
SO,
6.4
15 21.4 15.2 20.7
k-
0.4
-
0
0
2
0
0.2 I
de, mm
I
I
I
I
I
included because it is common pract'ice to report values of de as indices of ice bed permeabilit,ies. The several curves are labeled t,o indicate variables in crystallizer operation. As noted earlier, the particle size distributions were essentially the same for all but the two tests a t high rpm. This graph shows the same trends as Figure 19: the equivalent diameter of uniform spheres varies considerably with bed porosity, and d e is smaller a t high porosities. Deviations from the Kozeny-Carman equation appear as substantial departures of both kSo2and de from const'ancy. As noted above, brine titrations indicated that some melting occurred between bhe crystallizer and the outlet of the permeability test equipment. This appeared to have no effect on the results. No trend with "per cent melting" can be discerned in Figures 19 and 20; tests with widely different per cent melting but, the same porosity showed similar permeabilities (e.g., runs 38 and 51). Data from the four sources are compared in Figure 21 as d e us. e. The key is given in Table VI. The points are from Rit'ter, the solid circles being for 4 wt % NaCl and the open circles for brine containing 7 wt. NaC1. Curve D represents the approximabe posit'ion of the three middle lines of Figure 20. Each of the sets of data shows de to decrease with increase in porosity; the trend is more pronounced with ice, which has a wide range of sizes of somewhat irregular shape, than for the plast'ic and metal disks of uniform size. As noted earlier, k for randomly packed spheres is 5 , but it is usually greater for particles of irregular shapes. The constant k cannot be obtained from the present d a t a , since So,the surface-to-volume ratio of the solids in contact with the flowing stream in t'he formed beds, could not be measured. However, it is possible to debermine so, the surface-t'o-volume ratio for t'he same particles separated and out of contact with each other, from the particle size distributions and shape. Thus the dimensionless group k(So/so)2can be used to compare the data on ice with the disk data of Coulson and of Ryllie and Gregory. The quantity k(So/so)2should equal k for spheres but be less than k for disks and other irregular shapes that provide opportunit'ies for contacts of flat surfaces in a packed bed. The comparison is shown in Table VI for the same porosity of 0.52 in each case.
The large values of k(So/so)*indicate that k must be much larger than the value 5 for spheres, since So/so is less than unity. This increase in k is evidently due primarily to increases in t0rtuosit.y since the shape factor ko is but little greater for disks than for spheres. Extrapolation of the ice data to large values of E , for which So/so is near unity, suggests values of k of perhaps 30-40, corresponding to tortuosities of 3.4-3.9 if ko is about 2.6 (Carman, 1956). The low permeability of the ice beds is evidenced by the fact that the values of so of 7-10 mm-l are equivalent to spheres 0.6-0.86 mm in diameterabout twice the measured equivalent diameters for flow. This low permeability, also evidenced by the large value of k , must be due primarily to the wide range of particle sizes, with small particles blocking the passages between the larger ones and greatly increasing the tortuosity. This conclusion is supported by the results of Wyllie and Gregory (1955)) who report the relatively high tortuosity of 3.26 for mixed spheres of three sizes (3.0, 0.711, and 0.028 mm) as compared with 2.01-2.09 for spheres of a single size. Practical Implications of the Study. ;In important result of this study is t h a t hardly any change was detected in t'he product particle size distribution when both slurry density and average residence time were varied twofold, at constant agitation rate. This would indicate t'hat larger ice production rates per unit crystallizer volume should be possible without sacrifice of crystal size. Indeed a nucleation rate which is proportional to slurry density and independent of solution subcooling implies t h a t the average particle size is inversely proportional to t h e one-third power of t h e average residence time, suggesting t h a t increased throughput will be accompanied by a n increase in crystal size. This was found t o be the case in r u n 45, but t h e range of residence time st'udied was only twofold, and t'his conclusion should be corroborated over a larger range. I n t,his connection it should be noted t h a t the volumetric ice production rates in this study were similar t o those reported by Struthers (1967) for a continuously operated pilot scale ice crystallizer, but the part,icle sizes produced were generally 50-100% larger than those found by Struthers (1967) or in a similar study conducted by Kawasaki a n d Umano (1963). I n both studies t h e agitation rates were lower than those of t h e present study, b u t t h e residence times were greater. Increased agitation rates were found to produce smaller crystals, so that it would appear that agitation rates lower than those employed here would be preferable. However, it is expected that this would result in a lower freezing capacity if turbulence alone is relied upon for augmenting heat transfer to the refrigerant. This points out the need for better methods of refrigerant dispersion other than t,urbulence produced by agitators. This need is further emphasized by the fact that' the temperature difference required t'o transfer heat from brine to Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
451
refrigerant presently forms the major thermodynamic loss in the freezer, as has been pointed out by Harriott (1967). A reduction in this temperature difference would have an important economic consequence. Finally, it was observed that at the highest attainable porosities in the permeability tester, the disk-like shape and the wide range of particle sizes formed in the crystallizer produced permeabilities that were only one-quarter to onethird the corresponding values based on single-sized spheres having the same measured surface-to-volume ratio of the ice. In addition, it was found that for similar ice particles the method of bed formation resulted in large variations of bed porosity and hence permeability. These are observations of considerable industrial importance: clearly, the method of bed formation, in addition to the size and shape of the crystals produced in the crystallizer, can have a profound influence on wash column capacity. Acknowledgment
The authors gratefully acknowledge the financial support provided by the Office of Saline Water of the Department of the Interior. Nomenclature a
A
= defined byeq 10 = total crystal area, cm2of ice/cm3 of slurry = bed permeability = puoL/APg,,ft2 = solute concentration, g/cc
Bo C D = diameter of disk-shaped ice particle, mm f ( r , t ) = particle size distribution, no./(cma of slurry mm) d e = equivalent particle diameter = [Bo(l - e ) W O / e3]1/~
$
h
H
= = = =
k ko
= = =
Le JI N
= =
P
=
AP
=
Q r
= =
So
=
so
=
L
=
s
=
t
To
= = = =
T,
=
T,
=
T, T,
ATei =
452
gravitational constant, ft lb of mass/sec2 lb of force linear growth rate = dr/dt, mm/min height of disk-shaped ice particle, mm net nucleation rate distribution function, no./(cm3 of slurry mm min) Kozeny-Carman “constant,” dimensionless shape factor, k / ( L , / L ) * ,dimensionless length of porous bed over which pressure drop is measured, ft length of liquid flow through bed, ft slurry density, g of ice/cm3 of slurry corrected net nucleation rate, nuclei/(cma of slurry min) crystal perimeter, cm/cma of slurry hydraulic pressure drop across the porous bed, lb of force/ft2 volumetric throughput, cm3/min radius of disk-shaped ice particle = D / 2 , mm ratio of internal bed surface in contact with flowing stream to volume of solids in bed, ft-’ ratio of surface to volume of particles forming bed, separated and not in contact, ft-l or mm-I defined by eq 10 time,min interface temperature, “ C temperature of the bulk solution, “ C equilibrium freezing temperature of the bulk solution, “ C equilibrium freezing temperature of the solution a t the ice-brine interface. “ C refrigerant evaporation teiperature, “C
(T,- T,)> “c
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
ATrnr= (T,- T,),O C ATo, (To - T,),O C uo
I‘
= superficial liquid velocity in the porous bed, ft/sec = crystallizer volume, cma
GREEKLETTERS t = bedporosity T = average resident time = V/Q, min = brine viscosity, lb m/ft sec p
SUBSCRIPTS 1 2
= v a l u e a t r = r1 = v a l u e a t r = rz
Literature Cited
Arakawa, K., J . Fac. Sci., Hokkaido Univ., Ser. 2 4, 311 (1954). Barduhn, A. J., Chem. Eng. Progr. 63,98 (1967). Barduhn, A. J., Desalination 5, 173 (1968). Barduhn, A. J., International Symposium on Water Desalination, Paper No. SWD/88, Washington, D. C., Oct 3-9, 1965. Bransom, S. H., Dunning, W. J., Millard, B., Discuss. Faraday SOC. 5, 83 (1949). Brian, P. L. T., Mech. Eng. 19 (2), 42 (1968). Brian, P. L. T., Hales, H. B., A.I.Ch.E. J . 15,419 (1969). Brian, P. L. T., Hales, H., Sherwood, T. K., A.I.Ch.E. J . 15, 727 (1969). Carman, P. C., “Flow of Gases through - Porous Media.” Butterworths, London, 1956. Cayey, N. W., Estrin, J., IND.ENQ.CHEM.,FUNDAM. 6, 14 (1967). Coulson J. M., Trans. Inst. Chem. Eng. 27, 237 (1949). Fernandez, R., Barduhn, A. J., Desalination 3, 330 (1967). Harriott, P., A.1.Ch.E. J . 13, 755 (1967). Hillig, W. B., in “Growth and Perfection of Crystals,” R. H. Doremus, B. W. Roberts, and D. Turnbull, Ed., p 127, Wiley, New York, K.Y., 1958. Hulburt, H. M., Katz, S., Chem. Eng. Sci. 19, 555 (1964). Kawasaki, S., Umano, S., Kagaku Kogaku (Abr. Ed. Engl.) 1. 69. il963). ~ ~ . Lindley, D. V., Roy. Stat. S O ~ Ser. . , B 9, 218 (1947). Mar olis, G., Sc.D. Thesis, Massachusetts Institute of Technoyogy, Cambridge, Mass. 1969 Mason, B. J., Advan. Phys. 7, 221 i1958). Mason, B. J., Bryant, G. W., Van den Hewell, A. P., Phil. Mag. 8,505 (1963). Melia, T. P., Moffit, W. P., IND.ENG.CHEM.,FUNDAM. 3, 313 (1964). \-__-,-
Michaels, A. S., Brian, P. L. T., Sperry, P. R., J . A p p l . Phys. 37,4649 (1966). Randolph, A. D., Can. J . Chem. Eng. 42,280 (1964). Randolph, A. D., Larson, 31.A., A.I.Ch.E. J . 8, 639 (1962). Ritter. D.. Sc.D. Thesis. Massachusetes Institute of Technolonv. -“, Cambridge, Mass., 1969. Sadek, S., Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1965. Scatchard, G., Prentiss, S. S., J . Amer. Chem. SOC. 55, 4355 (1933). Sherwood, T. K., Brian, P. L. T., “Research on Saline Water Conversion by Freezing,’’ Massachusetts Institute of Technology Desalination Research Laboratory, Report 295-6, Oct 31, 1964; also Office of Saline Water Research and Development Progress Report No. 179, June 1966. Sherwood, T. K., Brian, P. L. T., Sarofim, A. F., Smith, K. A., “Research on Saline Water Conversion by Freezing,” hlassachusetts Institute of Technology, Desalination Research Laboratory, Report 295-9, Dec 19, 1966; also Ofice of Saline Water Research and Development Progress Report No. 242, Mar 1967. Sperry, P. R., Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1965. Struthers Scientific and International Corp., Interim Report to the Office of Saline Water, U. S. Department of the Interior, Apr 30, 1967. Wyllie, 11.1. R. J., Gregory, A. R., Ind. Eng. Chem. 47, 1379 (1935). RECEIVED for review July 20, 1970 ACCEPTEDMAY4, 1971
