Ind. Eng. Chem. Fundam. 1084, 23, 143-147
143
2 = composition in W phase, mole fraction
Literature Cited
Subscripts and Superscripts I, 11, I11 = phase boundary arb = arbitrary E = equilibrium in = inlet out = outlet
Cassamatta, G.; Bouchey, D.; Angeline, H. Chem. Eng. Sci. 1978, 33, 145. Klng, C. J. “Separation Processes”, 2nd ed.;McGraw-Hill: New York, 1980. Li, N. N. U.S. Patent 3650091, 1972. Misic, D. M.; Smith, J. M. Ind. Eng. Chem. Fundam. 1971, 70, 380. Wankat, P. C. Ind. Eng. Chem. Fundam. 1980, 19, 358. Ward, W. J., I11 AIChE J . 1970, 16, 405. Way, J. D.:Noble, R. D.;Fiynn, T. A.: Sloan, E. D.J. Membr. Sci. 1982, 12, 239.
Greek Letters density, mol/m3
Received for review November 29, 1982 Accepted August 15, 1983
p =
Kinetics of Phase Transitions in the System Sodium Sulfate-Water David Rosenblatt,+’ Stephen B. Marks,$ and Robert L. Pigford’’ Institute of Energy Conversion and Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The rates of growth and sdution of single crystals of sodium sulfate decahydrate and anhydrous sodium sulfate were measured under conditions of low driving force and low solution velocity past the crystal. The rates of growth and solution of sodium sulfate decahydrate were found to be controlled by the rates of heat and mass transport between the crystal surface and the bulk solution. The rate of growth of anhydrous sodium sulfate was controlled by the rate of incorporation of ions into the crystal lattice at the crystal surface. This information can be used to predict the rates of phase transitions in a system containing sodium sulfate and water, where the crystal populations, nucleation rates, and the fluid mechanics at the crystal surfaces are known.
Introduction Sodium sulfate decahydrate (Glauber’s salt) is an attractive material for use as a phase change thermal energy storage material. Its melting point (32.3 “C)and enthalpy of fusion (60 cal/g) make it potentially capable of storing solar thermal energy for residential heating purposes particularly in so-called “passive” installations. Thermal energy is stored by using solar energy to melt the phasechange material; later, when the solar energy is no longer available, the melted material crystallizes and releases the stored energy. In order to be of use in a practical system, the phase-change material must possess certain properties: suitable phase-change temperature, large heat of fusion, and sufficiently rapid rates of melting and crystallization. The first two criteria are somewhat obvious; however, the importance of the third point has not always been appreciated. Many solar thermal energy storage systems must be capable of storing and releasing energy in a 24-h period; therefore, the phase change material must be able to melt and crystallize within this time. Generally, the rate of melting is a function of the heat and mass transfer characteristics of the material, however, for many materials the rate of crystallization is often controlled by a chemical reaction a t the crystal surface. Thus, the rate of crystallization may be limited by factors other than mass and heat transfer. In such a case, the enthalpy of crystallization, i.e., the stored energy, cannot be released faster than the material can crystallize and so this process becomes rate determining. It is critical to determine the factors governing the crystallization and melting kinetics of any candidate phase change material such as Glauber’s salt. Therefore, a systematic study of the system’s phase-change t Institute of Energy Conversion. t Department of Chemical Engineering.
#E.I. du Pont de Nemours and Co., Inc., Richmond, VA. 0196-4313/84/1023-0143$01.50/0
kinetics has been undertaken in order to determine which factors control the rates of solution and crystallization of Glauber’s salt. T h e Sodium Sulfate-Water System Figure 1 shows a partial phase diagram of the system over the range of temperatures and concentrations of interest. A detailed description of the Na2S04-H20system can be found in the work of Wetmore and LeRoy (1951). Curve AB represents the solubility of Na2S04.10H20as a function of temperature, BC is the solubility of Na2S04 vs. temperature, and DE is the composition of pure Na2S04-10H20.The intersection of AB and BC represents the “melting point” of the decahydrate; however, since B is to the left of D, the decahydrate is said to “melt” incongruently; i.e., a crystal of decahydrate decomposes to give solution and excess Na2S04in the ratio 10 mol of H 2 0 to 1 mol of Na2S04. A kinetic study of the system is complicated by incongruent melting of the decahydrate a t compositions to the left of E. In order to simplify the project, it was decided initially to study the rates of crystallization and solution of individual crystals of Na2S04 and Na2S04.10H20under varying conditions of temperature, supersaturation, and solution flow rate past the crystal surface. Experimental Section An apparatus was constructed in which a sodium sulfate solution, whose concentration and temperature were precisely adjustable, flowed at a known velocity over a single crystal of either Na2S04or Na2S04.10H20.The solution was preconditioned in a large holding tank so that the desired degree of supersaturation was obtained. The conditioned solution was pumped at a known rate over the crystal which was suspended from a glass rod in a glass tube. A thermocouple was suspended in the tube near the surface of the crystal. The solution was then pumped into another holding tank in which the temperature was de0 1984 American Chemical Society
144
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
o m
'
IO
1
I
20
30
1 1 40
I
I
50
60
WEIGHT P E R C E N T N o 2 S 0 4
Figure 1. Partial phase diagram of the sodium sulfate-water system.
liberately set to undersaturate the solution so that incipient nuclei or crystallites were prevented from being circulated through the system. The solution then was filtered and returned to the first holding tank for reconditioning and recirculation. At periodic intervals the suspended crystal was removed from the flow tube and weighed. In this manner data concerning the rates of growth and solution of Na2S04and Na2SO4.10Hz0were generated. It is estimated that, because of the large volume of solution (ca. 35 L), temperature fluctuations were less than 0.05 "C at flow rates of up to 1.0 L/min. The solution temperature was measured within 0.1 O C by an ironconstantan thermocouple located near the suspended crystal. Solution concentrations were determined by pycnometry within f0.005 M. All solutions were prepared from Baker's analyzed reagent grade anhydrous Na2S04 and distilled water. The velocity of the solution was determined by a rotameter placed in series with the crystal cell. Results and Discussion A. Sodium Sulfate Decahydrate. If a crystal retains its shape as it grows, its mass (M) and surface area (S) are related to its characteristic linear dimension, E , as
m = ppE3
(1)
S = BL2 (2) The decahydrate crystal was approximated as a rectangular parallelopiped with A = 1.0 and B = 6.0. The change in crystal mass, m,was related to a change in the average linear dimension of the crystal, assuming the rate of change of the crystal's mass is proportional to the exposed surface area This equation, developed from eq 1, was first used by Hixson (1951) in analyzing the growth of crystals. The results of the experiments are presented in Figure 2, which shows the dependence of the rate of change of the crystal dimension, L , on the concentration driving force defined in eq 4. The data were measured in the solution temperature range 25.7 to 27.4 "C for growth rates and 23.8 to 28.6 "C for solution rates. Supersatyration ratios, (C, - CE)/CE, from -0.116 to +0.166 were used. A comparison of the rates of growth and solution shows that: (a) the rate of growth per unit driving force was slightly greater than the rate of solution; (b) both the rates of growth and solution were strong functions of the fluid velocity. It appears that the rate of incorporation of molecules was not a rate-limiting factor in the velocity range studied.
-030
0.20 -0.10
0.10
0
( cb-c.
UNDERSATURATION ( g molc/litcr )
0.20
0.30
(cb-ca ) SUPERSATURATION ( g molc/liter )
Figure 2. Variation of crystal growth and solution rate with concentration driving force.
The rate-limiting step in the process was the rate of heat and mass transport between the crystal and the bulk solution. The flux of sodium sulfate to the interface is given by
I::::[
N , = kC,F In
-
(4)
In this expression, derived from diffusion film theory, F = N , / ( N ,+ N,) is the ratio of the flux of salt to the total flux of material and Xi and Xb are the interface and bulk mole fractions of Na2S04. This mass transport expression takes into account the fact that the fluxes of water and salt are coupled. At steady state the water and salt must arrive a t the interface in the proportions required to form the decahydrate. The salt flux carries with it water superimposed upon the diffusive salt flux (Sherwood et al., 1975). The rate of salt transport is therefore greater than predicted when the coupling of the fluxes is neglected. In the limit of a small difference between Xi and xb,N , = k [ F / ( F - X,)](C, - Ci) > k(CB - Ci). The effect of heat transport can be estimated by calculating the temperature change a t the interface from a combined heat and mass balance
h(Ti-
Tb)
I;::[
= k(-hHf)C,F ln
-
(5)
AHf is the heat of fusion of the decahydrate. The ratio of the mass to heat transport coefficients, k l h , requires information about the coefficients independently. The ratio will be assumed to be that estimated from the "creeping flow" estimate to Brian and Hales (1969)
Pr and Sc represent the Prandtl number and the Schmidt number, respectively. Re is the Reynolds number based on solution velocity and average particle diameter. Since there is apparently no resistance due to surface kinetics the interface composition, Xi, is set equal to the equilib-
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984 15000
I
1
I
I
I
0
Sh: [40+1 21 ( R ~ S C ) ~ ' ~ ] ' ' ' Sh = K i / D
12000
9000
0
-
1
I
i -
145
q-
A
c
/ A
0 Sh2
l x l o - 6 ~
Figure 3. Dependence of crystal growth on fluid velocity and comparison with diffusion theory.
rium concentration C, a t the interface temperature. The rate of change of the crystal dimension is related to the flux by a mass balance a t the crystal surface (7) pm is the crystal molar density. The mass transfer coef-
ficients extracted from the data in Figure 2 using eq 4 are presented in Figure 3, for two cases: (1)the interface temperature was assumed equal to the bulk temperature of the solution; (2) the interface temperature was predicted by the combined heat and mass balance, eq 5 and 6. The resulting mass transfer coefficients, e_xpressedin dimensionless form, as Sherwood numbers, kL/D, were compared with values estimated from the creeping flow model of Brian and Hales (1969) in which the diffusivity was estimated to be 5 X lo4 cm2/s and the crystal density pm was 1.464 g/cm3. The following conclusions can be drawn: (a) The mass transfer coefficients were somewhat larger than predicted by the creeping flow model. This is reasonable because the crystals are not spherical. (b) The mass transfer coefficients computed from the rate of growth data were larger than those from the rate of solution experiments, possibly due to the effects of natural convection. (c) Inclusion in the computations of the effect of heat transport led to a two to 12 9% increase in the mass transfer coefficients. The density of the solution is a strong function of concentration. Natural convection currents occur due to the concentration and temperature differences between the interface and the bulk solution. In the range of fluid velocities studied the effect of natural convection may be significant. During growth the fluid a t the crystal surface is less dense than the bulk solution. The fluid rises past the surface, reinforcing the upward forced convection flow. During dissolution the fluid at the interface is denser than the bulk solution. Then the resulting natural convection patterns act opposite to the forced convection, reducing the net upward flow. Hayakawa et al. (1973) report data on the rate of growth and solution of single crystals falling freely through a column of solution. From the rate of growth data, surface concentrations and temperatures were calculated from mass and heat transfer correlations developed from the rate of solution data. A surface rate equation was extracted from the data based on the calculated interface properties. Their equation is
This expression can be used to estimate the concentration
Ix 1 0 ' ~ 0.01
0.10
I.oo
( Cb'CI)
CONCENTRATION DRIVING FORCE ( p mole/litcr )
Figure 4. Dependence of rate of growth and solution of anhydrous NaaSOI on concentration driving force.
excess at the crystal surface, Ci - C,, required to produce the rate of growth of decahydrate observed in our experiments. At a growth rate on the order of cm/s at 25.0 "C the concentration driving force at the surface, Ci - C,, would be on the order of g-mol/L, compared to an observed overall driving force, C b - C,, of lo-' g-mol/L. The resistance introduced by this surface rate process is negligible compared to the mass transport resistance. Data reported by McCabe and Stevens (1951) for copper sulfate pentahydrate and by Liu et al. (1971) for magnesium sulfate heptahydrate show appreciable surface resistance at low solution velocities. The lower rates might be explained by dislocation theories of growth like that of Burton, Cabrera, and Frank (1951). (1)Sodium sulfate is more heavily hydrated than either the copper or the magnesium hydrates. (2) The additional water molecules surrounding the ions weaken interionic forces in the crystal. (3) The crystal lattice is more easily deformed and dislocations are more easily created by stresses in the decahydrate. (4) Dislocations a t the crystal surface act as sites for growth. The greater the dislocation density the more likely a molecule a t the surface will find a favorable site for attachment and the faster the crystal will grow. One therefore expects the crystals of decahydrate to grow a t a faster rate than the other hydrates. B. Anhydrous Sodium Sulfate. The change in crystal mass was converted to a rate of change of the characteristic linear dimension, 2, using eq 3. L is the distance from the crystal origin to the (111)face along the face-normal. The shape factors were assumed to be A = 9.32 and B = 34.19. The results of a series of rate-of-growth and solution experiments are shown in Figure 4. Driving forces were created by heating or subcooling a 3.04 M solution. The results show that: (a) the effect of solution velocity on the rate of growth is small compared to its effect on the rate of dissolving; (b) the rate of growth is an order of magnitude smaller than the rate of solution. These results indicate a significant resistance to growth at the surface, compared to the mass-transport resistance in the solution phase. A further set of experiments on the rate of growth at constant temperatures yielded the results shown in Figure 5. The data show a significant effect of temperature. The experimental results were analyzed as follows to determine the surface kinetics in the rate of growth pro-
146
I d . Eng. Chem. Fundam., Vol. 23. No. 2. 1984
0 V . 8.2c:m/.ec x V. 16.3
OV.8.2 oV.16.3
E
1.10-5
y/, ,f ,,,,I , , ,
EO. 10
IXlO-‘
0.01
0.10
1.00
Figure 5. Determination of activation energy for rate of growth of anhydrous N,S04.
!
I Figure 6. Electron micrograph of anhydrous crystal surface
Table I
soln vel, cm/s
from expt
8.2 35.0
39 117
creeping flow eq of Brian and
Hales (1969) 64 100
cess: (a) The heat of solution of sodium sulfate is 0.28 kcal/g-mol. The effect of the heat generated on the temperature at the interface is insignificant, so the effect of heat transport can he ignored. (b) The rate of solution is mass-transport controlled. Mass transfer coefficients were extracted from the data with eq 4 and 7. The values are listed in Table I. The effect of the mass transfer resistance on the rate of growth can be estimated by using the mass transfer coefficients extracted from the rate of solution data to calculate the interface concentration at different growth rates. Rearranging eq 4 and 7
For rates of growth on the order of 10%cm/s at a fluid velocity of 8.2 cm/s the mass transport resistance leads to a 1% difference in concentration between the interface and the bulk solution. The interface concentration was calculated for the data in Figure 5 by using eq 9. The rate of growth data were fit to the following empirical equation in terms of the interface composition and the temperature
-a-- 1.484 X 105 exp(-13.7
X 103/RT)(Ci - Ce)1.6 (10) dt where Ci - C. is the concentration driving force a t the interface in g-mol/L. The activation energy for the p m w is 13.7 kcal/g-mol compared to an activation energy of 3-4 kcal/g-mol for a diffusion-controlled process. There were no other experimental data available for comparison with these results. Electron micrographs of a (111)face of an anhydrous crystal are shown in Figures 6 and 7. The crystal used for these pictures was taken from the growth cell during a typical growth experiment, dipped briefly in distilled water and placed in a container fded with petroleum ether. The dried crystal was thoroughly rinsed in the ether and
Figure 7. Electron micrograph of anhydrous crystal surface.
placed in a desiccator. The dried crystal was gold coated and viewed under an electron microscope at 640X magnification. Both micrographs show a step-like array of surface defects. Figure 7 contains a small crystallite of sodium sulfate growing on the surface among these steps. It is possible that the visible steps are actually “bunches” of much smaller steps of smaller height, where the actual process of addition of ions into the crystal lattice occurs. A potential cause of the formation of “step-bunches”is the presence of impurities on the crystal surface which impede the motion of individual steps across the surface, causing the following steps to “bunch-up”behind the impeded step. The advance of these “bunches”of steps across the crystal surface during the growth process has been described by Frank (1958) by use of kinematic wave theory. It is impossible to say whether the visible features are the product of the growth process alone or whether the steps were heavily etched during the cleaning process. In any event, it is probable that the rate of growth of the (111) crystal face is limited by the advance of such steps or “step-bunches” across the crystal surface.
Conclusions The rates of growth and solution of single crystals of sodium sulfate decahydrate and anhydrous sodium sulfate were studied under conditions of low fluid velocity and low driving forces such as those likely to be encountered in a thermal energy storage process application. The following
147
Ind. Eng. Chem. Fundam. 1984. 23, 147-153
conclusions can be reached: (a) The rates of growth and solution of sodium sulfate decahydrate are controlled by the rates of heat and mass transfer near the crystal surface. (b) The rate of solution of anhydrous sodium sulfate is also controlled by the rate of mass transport from the crystal surface. (c) The rate of growth of the (111) face of the anhydrous crystals is limited by the rate of incorporation of the ions into the crystal lattice a t the surface.
Acknowledgment We gratefully acknowledge the support of the Department of Energy in the form of Research Grant No. DEFG02-79ET-00088. The work reported here has been taken in part from the Master of Chemical Engineering Thesis (June, 1981) of David Rosenblatt. Nomenclature A = volume shape factor B = surface area shape factor C = concentration of sodium sulfate in solution, g-mol/cm3 C, = total concentration in solution, g-mol/cm3 C, = heat capacity of the solution, cal/(g "C) D = diffusivity of sodium sulfate in solution, cm2/s F = ratio of the salt flux to the total material flux g = gravitational constant, cm/sz h = heat transport coefficient, cal/(cm2 s "C) AHf = heat of fusion of sodium sulfate decahydrate, cal/g-mol AH8 = heat of solution of anhydrous sodium sulfate, cal/g-mol k = mass transfer coefficient, cm/s 5 = crystallization rate coefficient, (cm/s)/ (g-mol L)1.5 L = characteristic crystal dimension, cm m = crystal mass, g Ns = flux of salt to the interface, g-mol/(cm2 s) N , = flux of water to the interface, g-mol/(cm2 s) r = equivalent crystal radius, cm R = gas constant, cal/(g-mol K) T = temperature of the solution, K t = time, s V = bulk solution velocity past the crystal, cm/s X = mole fraction of sodium sulfate in solution
Gr = p:ge3/3c(AC)/p2, Grashoff number Nu = hL/K, Nusselt number Pe = (Re)(sc),Peclet number Per = (2r/D)(dr/dt), Peclet number due to radial velocity Re = eVp,/p, Reynolds number Sc = p/p,D, Schmidt number Sh = kL/D, Sherwood number Greek Letters cy = fractional solution specific volume change pT = temperature coefficient of fractional volume change, O C - ' /3c = concentration coefficient of fractional volume change, (g-mol/ L)-l K = solution thermal conductivity, cal/(cm s "C) pc = crystal phase density, g/cm3 pm = crystal phase molar density, g-mol/cm3 ps = solution density, g/cm3 p = solution viscosity, g/(cm s), P Subscripts i = refers to interface properties b = refers to bulk solution properties e = refers to equilibrium properties Registry No. Sodium sulfte decahydrate,7727-73-3;sodium sulfate, 7757-82-6.
Literature Cited Brian, P. L. T.; Hales, H. B. AIChE J . 1969, 75, 419-425. Bruton, W.; Cabrera, H.; Frank, F. C. Phil. Trans R . SOC. London 1951, AZ43, 299-358. Frank, F. C. "On the Kinematic Theory of Crystal Gyowth and Dissolution Process" in "Growth and Perfection of Crystals", International Crystal Conference", Doremus, R.; Turnbuii, J., Ed.; New York, Wiley: 1958; pp 4 11-41 8. Hayakawa, T.; Matsuoka, M. Heat Transfer Jap. Res. 1973, 2 , 104-115. Hixson, A. W.; Knox, K. L. Ind. Eng. Chem. 1951, 4 3 , 2146. Liu, V. Y.; Tsuei, H. S.; Youngqulst, 0. R. Chem. Eng. frog. Symp. S e r . 1971, 170. McCabe, W. L.; Stevens, R. P. Chem. Eng. f r o g . 1951, 4 7 , 168-174. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. "Mass Transfer"; McGraw-Hill: New York, 1975. Wetmore, F. E. W.; LeRoy, D. J. "Principles of Phase Equilibria"; McGraw-Hill: New York, 1951.
Received for review November 9, 1981 Accepted October 10, 1983
Interstitial Flow Intensification within Packed Granular Bed Filters: Experiments and Theory Robert W. L. Snaddon' and Peter W. Dletz General Electric Company, Corporate Research and Development, Schenectady, New York 1230 1
The mechanical collection efficiency of a granular bed filter is determined by the flow within the bed. I n spite of the demonstrable sensitivity of existing models to their proposed flow structures, most do not attempt accurate characterization of the flow. I n particular, interstitial flow intensification and flow separation are usually neglected. In the present paper an analytic model and collection measurements are combined to examine these effects. A potential flow model is developed in which the magnitude of the flow intensification is characterized by a single "intensification parameter". The experimental data are then employed to develop a unique correlation of this parameter with Reynolds number.
Introduction In recent years considerable interest has been expressed in the use of packed granular bed filters for separating fine particles and droplets from gas streams. This interest stems from the inherent simplicity, reliability, and low cost 0196-431318411023-0147$01.50/0
of these devices as well as their ability to remove fine submicron particulates, especially when augmenting agencies such as electrostatics are employed as shown by Kallio et al. (1979), Kallio and Dietz (1981), Dietz (1981), Zahedi and Melcher (1976), and Zahedi and Melcher 0 1984 American Chemical Society
