Rate of Solution of Crystals R. H. WILHELM, L. H. CONKLIN’, AND T. C. SAUER2 Princeton University, Princeton, N. J.
As a background for solution-agitation studies, an integration without limiting restrictions is presented for the diffusion rate equation involved when particles of sodium chloride are freely suspended and dissolved in water. The integral is solved graphically and is presented as a family of curves involving three dimensionless variables. The following individual variables are taken into account: volume of water, amount of salt, size and shape OF salt particles, concentration OF solution, and increase in volume of solution as dissolution of solid salt proceeds. Experimental evidence is presented to show that the derivation is valid over the entire concentration range of sodium chloride solutions. A n effective diffusion film thickness of 0.016 mm. is calculated for specific agitation conditions.
HE solution of a solid in a liquid involves a diffusional T mass transfer in common with other unit operations such as gas absorption, distillation, extraction, humidification, and drying. For example, the rate of solution of a particle of common salt in water may be expressed by the equation:
This equation is an application of Fick’s law to the diffusion of salt through a film of liquid adhering closely to the surface of the dissolving solid. The loss in weight of the particle in a unit of time, or mass transfer rate, -dw/d0, is a direct function of the specific diffusivity of salt in water, D, of the area of the particle, A , and of the difference in concentration of salt on either side of the film, (c. - e). The side of the film in contact with solid salt is assumed to be saturated and has a concentration, c,; the concentration, c, of salt on the other side of the film is that of the main body of solution a t time 0. Equation 1 also expresses the rate of solution as an inverse of the effective film thickness, 1. However, since the film thickness generally cannot be measured directly as an independent variable, it is included in the ratio D/1, which is called the “diffusion rate constant”, k. For any given system this constant varies with temperature (because of the effect o temperature on diffusivity and kinematic viscosity) and with hydraulic conditions around the dissolving particle (because of the effect of agitation on film thickness). The theory of film diffusion in connection with the solution of solids was first set forth in 1897 by Noyes and Whitney (6) who successfully applied a diffusion equation to the solution of rotating cylinders of lead chloride and of benzoic acid in water. Although the importance of area was recognized, this 1 2
variable was not studied or expressed in the following equation of these authors:
ac/ae
= W(C,
- C)
(2)
Brunner and St. Tolloczko (1) in 1900 extended the experiments of Noyes and Whitney t o five difficultly soluble substances which were dissolved in a fixed volume of water. Area was taken into consideration in calculating diffusion rate constants but was not varied experimentally over a significant range. More recently, in 1923, Murphree (4) integrated Equation 1 after expressing the area of crystals undergoing solution in terms of a linear dimension. The resulting complex analytical solution was simplified for special conditions-for example, the presence of crystals in an amount insufficient, sufficient, and in excess of that required to produce a saturated solution. Murphree proposed the use of the diffusion rate constant, k , as a measure of the effectiveness of agitation in comparing two types of equipment. He cited one experiment on the rate of solution of potassium dichromate. The quantitative approach to the study of agitation was greatly advanced in 1931 by Hixson and Crowell (2) who used a diffusion rate constant as an index of the effectiveness of agitation in numerous experiments. These authors integrated Equation 1 after expressing the surface area of the dissolving solid in terms of its weight. The resulting function, in which the cube root of the weight of solid appeared as a term, was called the “cube root law”. Special cases were defined for an initial weight of solid equal t o that necessary for saturation, for systems with a negligible change in concentration, and for conditions of constant surface. A significant study of the effect on the diffusion rate constant of agitation variables, such as equipment size and type, agitation speed, and fluid viscosity, was made by Hixson and Wilkens (3). Tablets of difficultly soluble benzoic acid were the dissolving solid particles used in the series of experiments by these authors. An integration of Equation 1without restricting limitations is presented in this paper. The solution of the integral, which is expressed graphically in terms of dimensionless groups, is valid over the entire concentration range of sodium chloride in water and takes into consideration the initial volume of solvent, the amount of salt, and the size and shape of salt particles. I n addition, the derivation takes into account the increase in volume of solution as dissolution proceeds, a variation not previously included. A series of integral curves similar to those presented for sodium chloride can be prepared for any soluble solid for which saturation and density data are known. Salt was chosen in the present instance bemuse its cheapness and ready availability in a range of particle sizes makes it potentially valuable as an agitation indicator.
Derivation and Integration of Rate Equation The assumption is made that the surfaces of all salt particles are equally exposed to dissolution conditions-for example, when all particles are freely suspended in water by agitation.
Present address, 383 Mt. Prospect Avenue. Newark, N J. Present address, Hercules Powder Company, Parlin, N J
453
INDUSTRIAL AND ENGINEERING CHEMISTRY
454
Equation 1 should be expressed in terms of salt dissolved rather than solid salt present. This may be done by differentiating Equation 3 with respect to time to obtain Equation 4 and substituting the latter in Equation 1: (3)
Salt concentration may be expressed in terms of weight of salt in solution and of volume of solution: c, =
V, WS
Vol. 33, No. 4
Substitute (16) in (13) to obtain the final differential equation, dX =
de
k an1 /)w),2/3 (Y
- X ) 2 / 3 a-(
p2 / V i
1 1 +
- --)X
(17)
1+oJ:
Equation 17 yields the following integral:
This integral was solved graphically with the aid of a planimeter to obtain functions in X and 2 a t different values of Y . Twelve solutions covering the entire practical range of Y are presented in Figure 2. The value of a! used in the integration was 0.13, obtained from Figure 1.
c = -Wd
V
Use of Integral Curves
Substitute Equations 6 and 7 in 5:
It is convenient to change wdt o a dimensionless ratio X, defined as:
x = Wd/Wa
(9)
X varies between the limits 0 -+l. Substitute Equation 9 in 8, rearrange, and obtain: dX X
= kA dB
(k - 7 )
Both the volume of solution, V , and the area of the dissolving solid, A , are functions of X and must be expressed in terms of this variable. The variation of the volume of a sodium chloride solution with the amount of salt dissolved was calculated from density and saturation concentration data of salt a t lo", 25", and 40' C. The calculated points are presented in Figure 1. One straight line adequately represents the variation of V/Vi with X over the entire concentration range and over the temperature range indicated. The equation of this line is as follows: V = (1 aX)Vj (11)
The integral curves in Figures 2 and 3 for the solution of sodium chloride are expressed in terms of three dimensionless variables, X, Y , and 2, for which consistent units must be employed. Y is the ratio of the weight of salt used to that required to saturate the initial volume of water. Values of Y less than 1.0 represent quantities of salt insufficient to form a saturated solution; values greater than 1.0 represent an excess. A series of curves covering a range of Y from 0.1 to 1.5 is presented. X is the ratio of the weight of salt in solution to that required to form a saturated solution. Its maximum value of 1.0 is achieved as a limit when sufficient or excess salt t o form a saturated solution is initially provided. I n any given experiment under conditions of constant agitation, particle number and shape, and initial volume of solution, 2 is a time variable. An X - 2 curve therefore portrays the increase in solution concentration with time and shows the manner in which final concentration is approached asymptotically.
+
v, = (1 + a)V
icleand the density of salt: =
Vol. 33, No. 4
0.40
0.50 0.59
0.80
1.53 1.93 2.10 2.48 2.88 6.26 Average
h. 0.57 0.57 0.60 0.57
0.57
0.57 0.57 0.57 0.61 0.57 0 59
0.54
0 .59 -
0.577,
3.
E F F E C r OK
VARIATIOS IN PARTICLE SIZE CONSTANT Y k
OF
The variation of the diffusion rate constant with particle diameter a t constant values of Y is shown in Figure 3. The plotted data were obtained from sized rock salt from one source, for which a constant shape factor was assumed. The increase in diffusion rate constant with particle diameter may be approximated by the equation: k = f(d0.26)
(20)
By itself this equation appears to be inconsistent with the constancy of k during any one run in which particle diameters decrease rapidly. However, a counterbalancing variable was found in a series of experiments in which Y was varied, particle size remaining constant. The decrease in k with a 0.80 0 70 k 0.60 0.50
0.40
0.15
02
0.3
04 05 06 07080910
15
Y
FIGTRE 4. EFFECTo x k OF VARIAY , OR AMOUXTOF SaLT U S E D AT CONSTANT PARTICLE SIZE
TION I N
variation in Y (or in w, which is proportional to Y)is shown in Figure 4. The variation of k with both d and Y , or w, is expressed as follows:
Equation 21 demonstrates why k can remain constant with time in a dissolution experiment during which both diameter and weight of salt particles decrease. The ratio of diameter and weight functions is invariant when the diameter exponent is the cube of that of the weight exponent, a condition approximated experimentally. An effective diffusion film thickness may be calculated from the relation:
r
INDUSTRIAL AND ENGINEERING CHEMISTRY
April, 1941 k =
D/E
(22)
Using the average diffusivity of salt in water a t 18' C. as 1.3 x 10-6 sq, per second and a value of k of 0.50, the film thickness is about 0.016 111111.
Nomenclature
weight of solid salt at time e, grams or lb. weight of salt dissolved in initial volume of water a t time 8, grams or lb. w s = weight of salt dissolved in initial volume of water a t saturation between 10' and 40° C., grams or lb. w, = initial weight of salt used, grams or lb. k = diffusion rate constant, weight dissolved/(unit time) (unit area) (c, - c ) , in grams/(min.) (sq. cm.) (grams/ cc.) or lb./(min.) (sq. ft.) (lb./cu. ft.) n = number of uniform particles in initial wei ht of salt, wi = a constant relating area and volume of sa% particles = a 6 for a cube, = 4.83for a sphere w
=
wd
=
".
457
= density solid salt, grams/ct.. or lb./cu. f t . Vr = initialvolume of wLter, cc. or cu. ft. V = volume of solution a t time 8, cc. or cu. ft. V, = volume of saturated salt solution formed from initial volume of water, Vi, at temperature of operation, cc. or cu. ft. 0 = time, min. A = total area of solid salt at time e, sq. cm. or sq. ft. c = salt concentration in solution a t time e, = wd/V, grams/ cc. or Ib./cu. ft. cs = salt concentration in solution at saturation a t temperature of operation = wa/Vs, rams/cc. or lb./cu. f t . LY = a constant relating X and V , b i 2 = film thickness, om. or ft. D = diffusivity of salt in water, sq. cm./sec. d = equivalent particle diameter, calculated from sphere of equal weight, mm. p
Literature Cited (1) Brunner and St. Tolloozko, I . physik. Chem., 35, 253 (1900). (2) Hixson and Crowell, IND. ENQ.CHDM.,23, 923,1002, 1160 (1931). (3) Hixson and Wilkens, Ibid., 25, 1196 (1933). (4) Murphree, Ibid., 15, 145 (1923). (5) Noyes and Whitney, I.physik. Chem., 23, 689 (1597).
Prevention of Fog in Cooler-Condensers A. P. COLBURN' AND A. G. EDISON2 E. I. du Pont de Nemours & Company, Inc., Wilmington, Del.
WHEN of
condensation of a vapor takes place in the presence a n inert gas, some fog is generally formed. The fog then passes out of the condenser with the inert gas. This is undesirable if the fog is composed of valuable or noxious materials; as a result a fog removal apparatus is required. Such equipment is usually large and involves considerable pressure drop of the gas, so that a means of condensation without fog formation would be preferable. The purpose of this study was to evaluate the theoretical mechanism of fog formation and then to develop a practical means for its preventi on.
Mechanism of Fog Formation When a mixture of vapor and inert gas passes over a surface which is sufficiently colder than the mixture so that the equilibrium partial pressure of the vapor at the surface is less than the partial pressure in the mixture, there is a simultrtneous flow of heat and mass to the surface. This means a decrease in temperature and vapor content of the mixture. Whether or not fog will be formed depends upon the relative degree of the decreases in temperature and concentration; if these are such as to cause the mixture to become supersaturated, a fog will form. The information needed to study the cooling history of the mixture includes the vapor pressure-temperature relation and heat and mass transfer rates. The rate of change of partial pressure of vapor, pv, with respect to temperature, tv, was shown (2) to be:
where pi,
ti
= vapor pressure and temperature at interface b e
tween gas mixture and condensed liquid
1 Present address, University of Delaware, Newark, Del. 3 Present address, E. I. du P o n t de Nemours & Company, Inc., Seaford. Del.
A theory is developed to explain the occurrence of fog when vapors are condensed in the presence of inert gases. The path of the concentrationtemperature curve is most apt to cross the saturation curve with vapors of high molecular weight, which are usually valuable or toxic and therefore important not to be lost in the waste gases as a fog. Based on the apparent mechanism of fog formation, a theoretical method of eliminating this undesirable condition is to supply a small amount of heat to the vapor-gas mixture during the condensing process. The method was tested experimentally and found to work on mixtures of air with steam, butyl alcohol, and trichloroethylene.
- p ) ,P
= log mean of ( P = total pressure
- pv) and ( P - pi)
C,p,lc,p,D~= heat ca acity a t constant pressure, viscosity,
therma? conductivity, density, and diffusivity, respectively, of vapor-gas mixture
The term (e" - l ) / a , based on a theoretical study of Ackermann ( I ) , corrects for the condition that some of the heat conducted across the interface is supplied by the removal ef sensible heat from the diffusing vapor and thus decreases the temperature decrease in the remaining gas. This term is important when the rate of condensation is high-i. e., when the relative amount of inert gas present is very low.
