Effect of Agitation on Rate of Growth of Single Crystals I
A. W. HIXSON AND K. L. K N O X ' C O L U M B I A UNIVERSITY, NEW YORK 97, N. Y.
A
study was made of the effect of fluid velocity on the rates of growth of single crystals of copper sulfate pentahydrate and magnesium sulfate heptahydrate. The rates of growth coefficients of these crystals were found to depend both upon the mass transfer coefficient, which varied with fluid velocity, and the rate coefficient of the surface reaction, which varied with temperature. Equations were developed which permit the mass transfer coefficients to be compared with mass transfer coefficients or with heat transfer coefficients in other systems. The mass transfer coefficients for copper sulfate were given by:
and the mass transfer coefficients for magnesium sulfate by
With the copper sulfate crystals, the rate coefficient of the surface reaction had a value of 1600 pound moles/ (hour)(square foot) at 53.5' C. and an activation energy of 13,600 gram calories per gram mole. With the magnesium sulfate crystals, the rate of coefficient of the surface reaction had a value of 0.331 pound moles/(hour) (square foot) gat 30.5' C. and an activation energy of 24,300 gram calories per gram mole. These results illustrate a means whereby available data on mass and heat transfer may be applied to crystallization problems.
T
HIS study of the unit operation of crystallization is one of the series of contributions to the subject of agitation which was begun with the work of Hixson and Crowell (6). It was undertaken to determine in a quantitative manner the effect of fluid velocity on the rate of growth of single crystals from solution. This study should be of particular interest to those people in the chemical industries who are growing large uniform crystals and to those research workers who are growing crystals for their piezoelectric and optical properties. I n considering the process of crystallization, emphasis is placed upon the relative effects of the mass transfer, surface reaction, and heat transfer in determining the rate of growth of a crystal. The equations for the mass transfer rate are presented so that the mass transfer coefficients for crystallization can be compared with, or predicted from, the data for other mass transfer systems. The Chilton-Colburn (5) analogy provides a mcans whereby the data for heat transfer systems can be used to calculate the rates of mass transfer in crystallization. I n these experiments, rates of growth measurements were made on a single crystal suspended in a flowing stream of supersaturated solution. In treating the results, it was possible to separate the 1
Present address, E.I. d u Pont de Nemours &. Co., Inc., Buffalo, N. Y .
mass transfer coefficients and the rate coefficients of the surface reaction because the former are dependent upon the flow conditions, while the latter are dependent on temperature and independent of the flow conditions in the fluid (2, 15). The crystals were grown from a solution of uniform concentration under conditions such that the rate of nucleat,ion was not important. PREVIOUS WORK
Noyes and Whitney (12) considered the rate of solution of benzoic acid in water to be a diffusional process and to be dependent upon the degree of agitation. For some time it was considered that crystallization was the reverse of the process of dissolution. Marc (10) and his coworkers made a number of studies of crystallization in a vessel fitted with a paddle agitator. They used the conductivity of the solution as a measure of the colicentration. When crystallizing potassiuni sulfate they found a late of stirring above which there was no further incrensc in the rate of crystal growth with an increase in the rate of stirring. With potassium nitrate and potassiuin chloride, the rates of gron th were nearly equal to the rates of dissolution for a range of stirring speeds. With a number of sulfates, the rate of crystallization was proportional to the square of the degree of supersaturation and the rate of crystallization was less than the rate of dissolution. It was concluded that the mechanism of crystal growth and dissolution were not necessarily identical. Berthoud ( 2 ) and later Valeton (16) developed a theory of crystal growth that took into account the rate of diffusion to the crystal surface and a rate of reaction a t the crystal solution interface. This theory wap used to account for the variation in habit of crystals with the conditions under which they were grown. In the limiting case of a ve'ry large coefficient for the cliwiical rate of growth, their theory became identical with thv NoyesWhitney theory. Van Hook (16) performed some experiments on the riitc o f growth of single crystals of sucrose when rotated on a supp)i ting a r m in a supersaturated solution. While hc found a slight iiicrease in the rate of growth with an increasc in the speed of rotation, he reached a limiting rate of growth and decided that diffusion did not play an important part in the process. This view \\:IS substantiated by a consideration of the activation cncrgy for the crystallization of sucrose. In a recent paper Miller (11) proposed a nicthod of designing :i classifying type of crystallizer. This method was bitset1 in pLrt upon a constant chemical rate of growth. No attempt was made to consider the variation of the rate of growth with the degrcc of agitation in the system. I n a general way the agitation rcquircincnts of crystallization are well known. McCabe (9) has stated that too intense ayitation is liable to cause nucleation in a supers:itur:itcd solution :uld possibly attritioh of existing crystals, while too little agitnt ion will permit the btystals to settle out and thcy will not coinc freely into contact with the solution. In most comnicrcai:tl types o f crystallizers, it is desirable to h:Lve sufficient agit:tti(in t o h t v l ) tlw 2144
I N D U S T R I A L ANI) E N G I N E E R I N G C H E M I S T R Y
September 1951
concentration of the solution uniform and to keep the growing crystals in suspension. Similar views are held by Seavoy and Caldwell (14). DISCUSSION OF THEORY
The growth of a crystal from solution can be considered to take place in three steps (2,9,16).First, the material which is crystallizing will have to diffuse from the bulk of the solution to the surface of the crystal. At the surface of the crystal, the material will form itself into the crystal lattice. This second step may be time consuming and the process will be referred to here ad a surface reaction. At the same time the heat which is liberated on crystallization must diffuse away from the crystal and the rate of removal of heat may be a factor in the rate of growth. The rate of diffusion (1) of the crystallizing material from the solution to the crystal surface will be considered to follow an equation of the form
evaluation of the mass transfer coefficient, in terms of the rate of growth of the crystal and in terms of the mole fraction of the solute in the bulk of the solution and in the solution a t the crystal surface. I n order to have the mass transfer coefficients in a form which can be compared with heat transfer coefficients, Bedingfield ( 1 ) suggests basing the mass transfer coefficients on the plane across which there is a zero net transfer of mass. For this case W:Ma
+
0
W;f&fb
(10)
The quantity z' is then given by
and Equation 3 may be rewritten as
&fb
in regions sufficiently near the crystal surface that the effect of turbulence is unimportant. The diffusivity constant, D,, is for the case in which material is being transferred perpendicular to the plane of area A and in which the net number of moles crossing the plane is zero. A quantity, z, may be defined as the ratio of the rate of transfer of the solute, wa, to the rate of transfer of the solution, w, across the plane of area A such that
2145
-
- Y
&fa
The mass transfer coefficient, Fd', corresponding to the plane in solution across which there is no net mass transfer may be defined by
and
z = - Wa W
When the net molal transfer rate of the solution is not zero, the rate of transfer of the solute is given by
The mass transfer coefficient,
Fd,
In order to relate the mass transfer coefficients which correspond t o the case of no net mass transfer across a plane in solution and to the case of a plane k e d relativp to the crystal surface, a quantity p mag be defined as
is defined by
The mass transfer coefficient may also be expressed in terms of Equation 3 so that
(5) The plane across which transfer is occurring may be defined in different ways and the mass transfer coefficient will be different for each case. For the case in which this plane is parallel to the crystal surface and a t a fixed distance from it
w = w a and z = 1 Equations 3, 4, and 5 may be rewritten as -Dm
As the crystal grows, the crystal surface moves toward the solution at a linear rate which is given by
-c,
=
Wa pg
where ps is the molal density of the crystal. For the mass transfer coefficients to be comparable to other cases of mass transfer, the plane across which diffusion is occurring will be taken as a plane parallel to the crystal surface and fixed relative to the solution. The linear rate a t which the solute crosses such a plane it: given by - e - - -
w:
wo
wo
P1Y
PlY
Pa
(17)
Equation 17 and Equation 8 may be combined so that (18)
and (9) Equation 8 is of particular interest because it permits the
The mass transfer coefficient, Fd", corresponding to the fixed plane in solution may be defined by
INDUSTRIAL A N D ENGINEERING CHEMISTRY
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Vol. 43, No. 9
In growing a crystal from solution, the area of the crystal will increase. If the crystal has its equilibrium shape, it will maintain this shape during further growth under similar conditions. In this case, the cube root law of Hixson and Crowell (6) may be used to take into account the change in area with growth. One may write
and
In order to relate the values for mass transfer coefficients based on a plane fixed with respect to the crystal surface to values for mass transfer coefficients based on a plane fixed with respect to the solution, one may define a quantity y such that
(2% j-;5) ): /
w. =
;1(
--Y
y =
dy=Fq
Fdl
(21)
For a crystal growing in a flowing stream of solution, the value of F d will depend on the diffusivity constant, the density, the viscosity, the velocity of flow of the solution, and a shape factor for the system. The relationship between F d and the other variables is usually expressed by a correlation of the form
The values of the constants, c, d, and e in this equation may be obtained from correlations of mass transfer coefficients or from correlations of heat transfer cofficients by meam of the ChiltonColburn analogy (3). When the crystallizing material has been transferred from the bulk of the solution to the crystal surface, it must then form itself into the crystal lattice for the procesa of growth to take place. The rate of the surface reaction is usually expressed in t e r m of the degree of supersaturation. It may be different for the different faces of the crystal. It is usually faster on faces which are not the natural faces of the crystal. The rate of the surface reaction may be expressed empirically by
-
w4 = F R ( ~ / yc)a
Equations 26 and 1 may be combined and integrated between the limits existing a t the beginning and a t the end of a period of growth so that
(23)
The proper form of rate equation for the growth of a crystal may be different from that of Equation 23. The arguments presented here can be readily modified to fit a preferred form of rate equation. If the rate coefficients for the surface reaction and for mas transfer are of the same order of magnitude, it is necessary to consider both of them in accounting for theobserved values. The observed rate of growth of a crystal can be expressed in terms of an over-all coefficient F, by
It fcdlowe that
During the growth of a crystal, $he heat of crystallization will be liberated at the surface of the cryst@. If the cr.ystal is freely suspended in solution, then the heat will be removed from the arystal by convection and conduction and the temperature of the crystal will be different from that of its surroundings. The difference in temperature is determined by the psychrometric relationships of the system. Normally the ratio of the maw transfer coefficient to the h a t transfer coefficient is small for crystals growing in liquids and the temperature of a crystal growing in solution is essentially the same as the temperature of the solution,
- W,,l’S)
3(WlO)2’S (W2’S Ao e
(27)
This equation will hold for growth by either a diffusional process or a surface reaction providing only that the relative shape of the crystal does not change during growth and that the term w. is held constant. The equations presented here outline a means whereby the available data for mass transfer in other heterogeneous systems and the available data for heat transfer from fluids to solids may be utilized in crystallisation problems. I n classifying types of crystallizers, it is possible to obtain a Reynolds number corresponding to the settling velocity of the crystal and such Reynolds numbers may be used in the calculation of mass transfer coefficients. The information on maea transfer coefficients so obtained is useful in the method of design of continuous crystallizers presented by Miller and Saeman (11). I n agitated crystallizers, the maea transfer data may be correlated in terms of a modified Reynolds number involving the vessel dirtmeter and the speed of stirring. Such correlations of ma= transfer rates in agitated veesels have been presented by Hixson and Baum (6). A control of the rates of crystal growth is desirable when large crystals are grown for their piezo-electric or optical properties. Rates of growth which are too fast lead to imperfect crystals, while slow rates of growth are not economical. The agitation requirements of a given system a t a’constant temperature will depend upon the size of the crystal, and the maw transfer coefficient, Fd, will be proportional to Vd/DeId. While the discussion and experimental work presented here are chiefly concerned with the growth of orystals under conditions of forced flow of solution, it is of interest to compare the case of heat transfer by natural convection with the case of growth of crystals in a still solution. For a crystal to grow in a still solution, a difference in concentration must exist between the surface of the crystal and the bulk of the solution. The density of the salution will vary with the concentration and the variation in the density of the solution will cause convection currents. The flow patterns which will exist around such a crystal depend in part upon the concentration coefficient of expansion and the diiTerence in concentration between the surface of the crystal and the bulk of the solution. McAdams (8) gives correlations of the natural convection heat transfer coefficients for heated horizontal cylinders. Such correlations involve the product of the Grashof and the Prandtl numbers. Analogous correlations of natural convection maas transfer coefficients would involve the product of a modified Grtwhof number and a Schmidt number and would be of the form
The application of the equations presented here to the unit operation of crystallization is limited a t the present time by the lack of information about the permissible degreee of supersatwa-
September 1951
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tion. In general, the maxinium permissible degree of supersaturation Fill be aamciated with the allowable rate a t which nuclei are permitted to form during crystallization. EQUIPMENT AND EXPERIMENTAL PROCEDURE
.s
8
-2 I1 il:
i
-
s
In the equipment (Figure l ) , the solution was passed over a bed of crystals and drawn through a filter to a pump. Part of the solution leaving the pump was returned through the by-pass to the bed of crystals and part was led through a rotameter and through a cooling section in which the solution was superaaturated. The solution was then led into a vertical glass ipe and past the growing crystal. After passing t i e crystal, the solution was heated and returned to the bed of crystals. Such a system could be operated in a steady state. The temperature of the solution leaving the bed of crystals and the temperature difference between this solution and the solution passing the growing crystal were measured by means of a copperconstantan thermocouple and a Leeds & Northrup type K-2 laboratory potentiometer. This system had a limit of error of about 1 microvolt (0.025' C.). I n making a determination of the rate of growth of a crystal, the equipment was o erated until the solution leaving the bed of crystag was saturated. This was tested by placing a crystal in the flowing stream and observing whether or not the crystal dissolved. The rate of removal of heat from the heat exchanger was then adjusted to give the desired degree of supersaturation in the crystallizing section. The crystal was measured and weighed before starting a run. The crystal was suspended on a Nichrome wire seed holder in the crystallizing section and grown for a measured period of time. The degree of supersaturation, the temperature of saturation, and the rate of flow were measured during a run. The crystal was again weighed at the end of the run. Weights were obtained both with the dried crystal and with the crystal suspended in a saturated solution in an effort to reduce the errors involved in determining the increase in weight. The pump used for the runs with magnesium sulfate a t 19.9' and 30.5' 6. was a stainless steel centrifugal pump Model E1 made by the Eastern Engineering Co. For the remainder of the runs a Jabsco Model D pump with a rubber .impeller and brass housing was used. A technical grade of copper sulfate and a U.S.P. grade of magnesium sulfate were used in these experiments. EXPERIMENTAL RESULTS
COPPER SULFATE.The result8 obtained on the rate of growth of copper sulfate crystals are summarized in Table I. The term woj represents the observed rate of crystal growth in pound moles of copper sulfate pentahydrate (CuS04.5H10) per hour per square foot of cryfital surface. The duration of the run in hours is represented by e. The temperature of the saturated solution leaving the bed of crystals is given by teat. The linear velocity, V , of the fluid flowing past the crystal is based upon the measured volumetric rate of flow and the net cross-sectional area in the crystallizing section a t the crystal. The equivalent diameter, De, of the crystal was taken as the. diameter of a sphere having the same area as the crystal. This choice of the equivalent diameter is in keeping with that of McAdams (8) for heat transfer to irregularly shaped particles. The mole fraction, y, of the hydrated copper sulfate in the solution was based upon the tsolubility data in the International Critical Tables (7). The values of p were determined for the range of concentrations encountered here, and these values were found t o be nearly constant and nearly unity. Both the surface reaction and the diffusional step in the growth of copper sulfate cry8tals were found to be of importance. The best interpretation of
INDUSTRIAL AND ENGINEERING CHEMISTRY
2148
the data was obtained by considering that the rate of the surface reaction followed an equation of the form Waf
= FR(Yf
- Vi)'
fate crystal. The rate of the surface reaction was considered to follow an equation of the form wUa/ =
(29)
When the exponent a in the equation for the rate of the surface reaction was equal to unity, the average deviation from the mean values was higher than with a = 2. The data could not be correlated with CY = 0. From the correlations of the heat transfer coefficients for spheres in air (8)
Vol. 43, No. 9
FR(Y/
-
yi)a
(34)
When the exponent a in Equation 34 was 2, the average deviation from the mean was higher than with a = 1. The data could not be correlated with 01 = 0. HEATING SECTION
SUPPORT AN0 THERMOCOUPLE WELL
THERMOCOUPLE
where c" is a const,ant a t a constant temperature and concentration. CRYSTALLIZATION SECTION INSIDE DIAMETER 1.02 in. LENGTH OF STRAIGHT VERTICAL TUBING BEFORE THE CRYSTAL 26 in.
and since the values of way, yu, yf, V , De,and are available from the experimental data, it is possible to combine Equations 29,30, and 31 in order t o determine F d , FR, and yf. The calculated results for the various temperatures are shown in Table I. The coefficients for the rate of the surface reaction for copper sulfate are given approximately by
The density of the saturated solution of copper sulfate was determined using a pycnometer and the viscosity was determined using a 2-ml. viscosity pipet. The results obtained are shown in Table 11. The values for the diffusivity shown in Table I1 have been extrapolated from the data of oholm ( I S ) using an activation energy of 5000 gram cal. per gram mole and a linear extrapolation of the plot of diffusivity versus the normality of the solution.
TABLE 11.
V A L U E S OF I7ISCOSITY, DENSITY, AND
SATURATED COPPER SULFATE SOLUTIONS
Temp.,
Density, Gram/Ml. 1.20c 19.3 2.21b 41.4 1.97 1.27 1.92 1.31 53.5 1.94 1.41 71.2 Extrapolated from the data of Oholm (13). From the International Critical Tables (7). From the data of Flottmann (4).
c.
a b e
DIFFUSIVITY OF
Viscosity,
CP.
The mass transfer coefficient, the dimensionless groups
Fd,
Diffusivitya, Lb. Moles (Hour) (Foot) 4.7 x 10-5 7.4 x 10-5 9 , o x 10-5 12.3 X 10-5
COOLING
i
-
t
Figure 1.
The values of these groups are shown in Table I and FdUs -is
D ...,
plotted versus (D+)o"" (L)o'3 in Figure 2. MmDm The values of the mass transfer coefficientsobserved in growing copper sulfate crystals are given by
(33) The experimental results had an average deviation of 19% from E uation 33. ~ A G N E S I U M SULFATE.Results on the rate of growth of epsomite crystals are summarized in Table 111. The term tu./ represents the observed rate of crystal growth in pound moles of magnesium sulfate heptahydrate (MgS04.7H20) per hour per square foot of crystal surface. The terms e, Laat., V ,De,and fl have the same meanings as in Table I. The mole fraction, 11, of the hydrated magnesium sulfate in the solution was based upon the solubility data in the International Critical Tables (7). The surface reaction and the diffusional processes were important in determining the rate of growth of the magnesium sul-
SECTION
ROTAMETER 4
JJ
Crystallization Equipment
The results for magnesium sulfate were treated in the same manner as the results for copper sulfate and the values of F d and FR are shown in Table 111. The value of FR used for the runs made a t 48" C. was obtained by extrapolation from the results a t lower temperatures. The coefficients of the rate of surface reaction for magnesium sulfate are given approximately by
In FR =
can be correlated in terms of
,B
SATURATOR WITH BED OF CRYSTALS AND FILTER
-24,300 7 + 39.16
(35)
The densities and viscosities were determined in the same manner as for copper sulfate. The values of the diffusivity shown in Table IV were extrapolated from the data of oholm (IS)using an activation energy of 4200 gram cal. per gram mole and a linear extrapolation of the plot of diffusivity versus the normality of the solution. In order to correlate the values of the mass transfer coefficients. the dimensionless groups and )*M P Dm were evaluated and listed in Table 111. versus
(L'?),
(
2 -
(F)0'6(L)0'3
/,-
MmDm
was plotted in Figure 3.
The values of the mass transfer coefficients observed in growing magnesium sulfate crystals are given by
The experimental results had an average deviation of 33% from Equation 36. DISCUSSION OF RESULTS
The results obtained with copper sulfate and with magnesium sulfate indicate that the rate of growth of chalcanthite and epsomite crystals depends in part upon the transfer of material to the surface of the crystal and in part upon a surface reaction.
September 1951
.*
9
??h
?.-
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2149
The mass transfer coefficients for the growth of copper sulfate crystals follow the usual form of correlation within the limits of experimental errors. The mass transfer coefficients, which were observed in growing the copper sulfate crystals, are ahout ZOyo less than those calculated from the heat transfer coefficients for spheres in air streams. The difference between the correlation for mass transfer in the growth of copper sulfate crystals and the correlation for heat transfer to spheres in air streams may be due to the difference in the effects of shape on the flow conditions in the fluid. The mass transfer coefficients for the growth of magnesium sulfate crystals fall within the limits of experimental error a t each of the temperatures a t which runs were made,, but the general correlation of the mass transfer coefficients shows a greater variation than might be expected. This may be due in part to the errors involved in the extrapolation of the diffusivity data. The values of the mass transfer coefficients for the growth of magnesium sulfate are about 30% greater than those calculated from the heat transfer coefficients for spheres in air streams. The results for the coefficient,s of the rate of reaction a t the surfaces of the crystals have been expressed in an empirical form. These empirical equations give results in keeping with the accuracy of the data. With the magnesium sulfate, no value for FR at 48.2' C. was obtained because the coefficient of the surface reaction was large compared to the rate of mass transfer. The deviations of the results from the mean values are of the order expected in view of the precision with which the variables could be measured. The measurements of the temperature difference between the saturated solution and the solution passing the crystal may be in error by 3%, the measurement of the weight increase may be in error by 2%, the measurement of the velocity of the solution 275, and the value for the area of the crystal by about 3%. The results are sufficiently precise to demonstrate clearly the part played by mass transfer and by a surface reaction in growing copper sulfate and magnesium sulfzte crystals. CONCLUSION
,-.
The rates of growth coefficients of single crystals of copper sulfate pentahydrate and magnesium sulfate heptahydrate depend partly upon a mass transfer coefficient and partly upon a coefficient for a rate of surface reaction. The equations presented permit these factors to be combined into an expression for the rate of growth of a crystal. Emphasis has been placed upon having the mass transfer
INDUSTRIAL AND ENGINEERING CHEMISTRY
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TABLE IV. FUSIVITY
1,000
300
B
Drn IO0
b
+at x at o at o of
30
19.3%. 41.4%. 53.5 %. 71.2 %.
I
Figure 2.
Correlation of Mass Transfer Coefficient for Copper Sulfate
coefficients in a form which may be compared with the data for mas8 transfer or heat transfer in other systems. The effect of fluid velocity, or of agitation in a broad sense, on the rate of growth of crystals has been expressed by the correlations cf mass transfer coefficients. For copper sulfate, the mea transfer coefficients had an average deviation of 19% from
VALUESOF THE VISCOSITY,DENSITY, AND DIFSATURATED MAQNESIIJMSULFATESOLUTIONS
OF
Density Gram/Idl. CP. 1.29b 19.9 6.8b 30.5 7.66 1.32 40.5 8.15 1.35 48.2 8.50 1.38 Extrapolabed from the data of 6holm (18). From the International Critical Tables (7).
Tzmp
Fd De
Vol. 43, No. 9
Viscosity,
Diff usivityD, Lb. Moles D m (Hour) (Foot) 3.18 X 10-4 8.56 x 10-4 3.92 x 10-4 4.02 X 1 0 - 4
With magnesium sulfate crystals, the rate of the surface reaction followed an equation of the form WE/
= FR(YI
- yi)
in which FR had a value of approximately 0.331 pound mole per (hour) (square foot) a t 30.5" C . and an activation energy of 24,300gram cal. per gram mole. These results aid in interpreting the role of mass transfer in crystallization and emphasize the need for accurate determinations of diffusivity constants in saturated salt solutions, for the determination of heat transfer coefficients for bodies with typical crystal shapes, and for the determination of the rates of crystal growth from solutions under condition? of natural convection. ACKNOWLEDGMENT
The junior author is indebted to Columbia University for the university fellowships under which this work was carried out. NOMENCLATURE
For the magnesium sulfate, the mass transfer coefficients had an average deviation of 33% from
With copper sulfate crystals, the rate of the surface reaction followed an equation of the form Waf
FB(YI -
in which FR had a value of approximately 1600 pound moles per (hour) (square foot) a t 53.5" C. and an activation energy of 13,600 gram cal. per gram mole.
A
= area parallel t o the crystal surface and across which
A0
= = = =
diffusion is occurring, square feet area of a standard crystal containing W,Omollinear velocity of growth of the crystal surface, feet/hour C, constant a t a constant temperature and concentration c" diameter of a sphere having the same surface area as the D. crystal, feet Dm = diffusivity coustant as defined by Equation 1, pound moles/( hour)(foot) d = prefix, indicative derivative Fa = mass transfer coefficient as defined by Equation 4 for the general case and as defined by Equation 8 for the case in which transfer is occurring across a plane arallel to the crystal surface and a t a fixed distance From it, moles/( hour)(square foot) Fi = mass transfer coefficient corresponding to the plane in solution across which there is a zero net transfer of mass, pound moles/(hour)(square foot) F: = mass transfer coefficient corresponding to a plane parallel to the crystal surface and fixed relative to the solution, pound moles/( hour)( square foot) F, = an over-all mass transfer coefficient, pound moles/(hour) (square foot) FR = rate coefficient of the surface reaction g = acceleration due to gravity, feet/sec.2 Ma = molecular weight of the solute Ma = molecular weight of the solvent M, = mean molecular weight of the solution R =. gas constant, gram cal./(g;ram mole)(' K.) T = absolute temperature in K. t,.t. temperature of the saturated solution, O C. V = linear velocity of flow of the solution past the crystal, feet/second W , = total number of moles in the crystal F,o =f total number of moles in a standard crystal w = molal rate of transfer of solution across a plane parallel to the crystal surface, pound moles/( hour)( square foot) wa = molal rate of transfer of solute across a plane parallel to the crystal surface for the general case and for the case in which the plane is a t a fixed distance from the crystal surface, ound moles/(hour)( square foot) = molal rate of transper of the solute with respect to a plane w! parallel to the crystal surface and across which there is a zero net transfer of mass, pound moles/(hour)(square foot) =i
+
at 19.9'c.
x at 305.G. o at 40.5t: A at 48.2'E.
loo
1,000
.3
Figure 3. Correlation of Mass Transfer Coefficient for Magnesium Sulfate
September 1951 W.“
=
W;
=
X
= =
Y
=
2
=i
=
INDUSTRIAL AND ENGINEERING CHEMISTRY
molal rate of transfer of the solute across a plane parallel to the crystal surface and fixed relative to the solution, pound moles/( hour)(square foot) molal rate of transfer of the solvent with respect to a plane parallel to the crystal surface and across which there is a zero net transfer of mass, pound moles/ (hour)( square foot) distance in the direction of transfer of material, feet mole fraction of solute in solution ratio of the molal rate of transfer of the solute to the total rate of transfer across a given plane as defined by Equation 2 exponent of (yf - y,) in Equation 23 for the rate of the surface reaction ratio of mass transfer coefficients as defined by equation
15 = ratio of mass transfer coefficients as defined by Equation m*
Zl
e
,= time
fi
= viscosity of the solution = coefficient of volumetric expansion of the solution based
?r
on the mole fraction of solute = density of the solution, pounds/cubic foot = molal density of the liquid, pound moles/cubic foot = molal density of the crystal, pound moles/cubic foot
p p1
p,
Subscript
f
= conditions in the solution at the liquid-crystal interface equilibrium conditions at the surface of the crystal conditions in the bulk of the solution
i
=I
u
=i
2151
LITERATURE CITED (1)
Bkdingfield, C. H., Jr., and Drew, T. B., IND. END. CHEM..
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RBCEIVED October 3, 1949. Presented before the Division of Industrial and Engineering Chemistry at the 116th Meeting of the AMERICANCHEMICAL S O C I ~ Atlantio Y, City, N. J .
Countercurrent Solid-Gas Fractionation Process ALLEN F. REID’ COLUMBIA UNIVERSITY, NEW YORK,
To effect a continuous multistage solid-gas fractionation, a process has been worked out with a system utilizing countercurrent operation. The apparatus used to carry out this process was a helical tube of six turns, rotating on a horizontal axis partially submerged in a coolant with vaporizing heat applied to the upper part. For a benzene-cyclohexane mixture at proper operating conditions a separation factor of over 100 was consistently realized. From this and other fractionations effected, separation of vaporizable solids appears feasible in the multistage systems described. In some cases such fractionations may be economically practical.
A
N APPARATUS was constructed for the purpose of separat-
ing mixtures of materials in the solid phase by taking advantage of the differences in the rates of vaporization, condensation, and diffusion of the components. The separations achieved were highly satisfactory and compatible with a Rayleigh’s law type of separation. A description of the apparatus and some typical results are included in the discussion, together with an analysis of the process in regard to some of its possible applications. Present address, Southwestern Medical School, University of Texas, Dallas. Tex. 1
N. Y . PRINCIPLES O F SEPARATION
Standard engineering practice has long included numerous methods for the separation of components of mixtures by taking advantage of the difference of the rates of vaporization and condensation between a liquid and a gas. This has been very practical because efficient multistage systems could be devised and operated in which there was countercurrent motion of the two phases. A comparable type of separation between gas and solid phases can be expected in any single stage. However, such methods of separation have not been widely used where a large number of vaporization-condensation cycles are necessary because no practical methods have been available to provide continuous countercurrent motion of the phases similar to that experienced in fractional distillation. The method described here provides such count,ercurrent motion by means of the unidirectional motion of a series of alternate cold and hot regions along an extended passage which contains the mixture to be fractionated. Figure 1 shows schematically the arrangement of such a system. The process functions somewhat as follows: Material is supplied in the assage and frozen down to a cold spot with alternate hot and cofd zones progressing uniformly and continuously from left to right. As the cold zone moves dight\y
