Ind. Eng. Chem. Res. 1987,26, 23-27
23
Growth of Radially Symmetric Ice in a Supercooled Sucrose Solution John L. Bomben,* John Newman,?and C. Judson King? Western Regional Research Center, Berkeley, California 94710
Simultaneous heat and mass transport during radially symmetric growth is described. It is shown that even in the case of variable properties a similarity transformation could be used. The equations are applied t o a sucrose-water solution, which has a diffusion coefficient strongly dependent on concentration and is a model for the fluid in plant tissue. The coupled, nonlinear equations are solved by the method described by Newman.
As a f i s t approximation to the problem of solidification, the growth of a radially symmetric particle in a solution may be used. Such an analysis ignores constitutional supercooling and dendritic growth (Fleming, 1974; Mullins and Sekerka, 1963; Sekerka, 1965). The problem of the growth of a bubble in a binary solution has been thoroughly described by Scriven (1959). His analysis is applicable to the case of solidification, as well, when the properties of the solution are not affected by temperature or composition (Frank, 1950). However, there are cases such as the phase transitions in metals and the freezing of foods in which the properties, especially the diffusion coefficient, are marked functions of composition and temperature (Atkinson, 1968; Atkinson et al., 1973). Mass transport problems in solidification can be investigated if the velocity of solidification is known a priori and supercooling is assumed negligible (Fleming, 1974). This cannot be done in the case of the freezing of a supercooled solution, such as occurs in the intracellular freezing of a cellular food material, since the local velocity of the freezing interface is unknown. The freezing of a sucrose solution may be considered an approximation of the intracellular freezing of fruits and vegetables. Sucrose solutions do not freeze as eutectics; rather, they form a composite of ice and concentrated, solidified sucrose-water solutions. Such a solidified solution occurs because concentration polarization prevents further formation of ice (Bellows and King, 1973). The following analysis examines the general problem of phase change in a supercooled solution having variable properties, and it applies the results to ice formation in a sucrose solution, specifically examining the effects of concentration polarization as will occur in the intracellular formation of ice in a cell. Formulation of the Problem
dp
at
a + -r21 -(pr2u,) ar
=0
When there are no chemical reactions, identical body forces on all species, no viscous dissipation, no pressure effects, and only ordinary diffusion, the energy equation for a binary solution can be written as (Bird et al., 1960)
The material balance for water and Fick's law lead to
Aside from pressure effects, Dufour energy flux, and thermal diffusion, eq 1, 2, and 3 describe the mass and energy transport in a binary, spherically symmetric system. In order to make the condition specific to the phase transformation of water to ice, boundary conditions must be specified. The solution is at a uniform temperature, To,that is lower than the equilibrium freezing point of the solution. Thus, we are examining the initial relief of supercooling immediately after nucleation, such as would occur in the sudden freezing of a solution inside a cell. As initial conditions, the solution will be at uniform temperature and concentration at t = 0
T = To
(4)
w, = wwo
(5)
-
Furthermore, we will consider the case of an unbounded solution, and thus, as r m
T w,
-
To
(6)
wwo
(7)
It is assumed that nucleation occurs at the origin of a spherical coordinate system at time t = 0 and that radial symmetry is preserved throughout the subsequent growth process. Furthermore, it is assumed that the radius is sufficiently large so that the effect of surface curvature on the solubility can be ignored. The physical properties of the solution depend on temperature but not on pressure. Any velocity is due only to growth. Also, thermal diffusion, the DuFour energy, and all pressure effects will be neglected (Bird et al., 1960). In spherical coordinates, the continuity equation for the solution may be written as (Bird et al., 1960)
In the actual case of a supercooled solution, many nuclei would form simultaneously. Equations 6 and 7 assume that the nuclei are sufficiently far apart that heat and mass transport around one are not affected by the presence of the others. The other boundary of the system is the moving spherical surface of the ice. The surface will be considered in equilibrium with the solution immediately adjacent to it, and therefore, a thermodynamic relationship between the composition and temperature will apply
*Present address: SRI International, Menlo Park, CA 94025.
T = f(ww) (8) For a dilute solution, eq 8 would be linear; however, it does not appreciably complicate the analysis to leave it more general.
Address while work was being done: University of California,
Berkeley, CA 94720.
0888-588518712626-0023$01.50/0 0 1987 American Chemical Society
24 Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987
As another boundary condition, we can make an energy balance at the moving spherical surface, in which the rate of enthalpy release by freezing equals the rate a t which heat is conducted away from the solidifying surface
The new variables in the transformation are (24)
(9)
A mass balance at the solidifying surface for the two species gives two more equations. To be completely general at this point, the equilibrium distribution of each species between the two phases would have to be taken into account, but for aqueous solutions and ice, that equilibrium coefficient is very small and will be assumed equal to zero. Thus, with no solute entering the solid phase at r = ro,
and
(11)
The fluxes in eq 10 and 11may be eliminated by using Fick’s law and the definition of the mass-average velocity. We can then say that
By use of eq 10 and 11 again,
Equations 1-3 with the initial conditions (eq 4 and 5) and with the six boundary conditions (eq 6-9,12, and 13) specify the problem completely. This set of simultaneous nonlinear partial differential equations can be transformed to a set of ordinary differential equations by a similarity transformation.
(25) ~ ( 7 )=
rour
(26)
The quantity 0 will be referred to as the growth constant and is one of the unknowns in the formulation of the problem. Scriven (1959) defined the growth constant as ro/2[k/(pCpt)]1/z,but this definition is not convenient when considering a solution with variable properties. Note especially that since p is a constant, ro is proportional to t1I2, even when the variation of physical properties is taken into account. Aside from neglecting Dufour energy flux, thermal diffusion, and pressure effects, eq 14-23 are generally applicable to the problem of radially symmetric solidification in a supercooled, infinite medium. Note especially that the transformation has not required any assumptions about the properties of the solution except that they depend only on q. Although the equations have been written for a binary solution, the equations for a multicomponent solution, using the appropriate definitions of mass and energy fluxes, could be transformed in the same manner. A solution for these equations with the limitation of constant solution properties and a linearized boundary condition for eq 20 was obtained by Scriven (1959). The case of variable-solution properties requires a finite-difference calculation. Although the effects of a variablesolution density, thermal conductivity, and the heat flux due to mass transport can be included in the calculations, these are small for the sucrose-water solution used as an example. The diffusion coefficient for this solution, on the other hand, varies considerably with concentration and temperature; therefore, it will be the only property assumed to vary in the calculations.
Finite-Difference Solution
PDWSWW” +
+ pD,,’ + D,,p‘ + Ppq - PU + k’ + /3pCpq - pCPv P
-D,,w’(R,’ M
pqp’
- Rs’)= 0 (16)
W
+ pu’ + u p / = 0
8’= 0 The transformed boundary conditions are at T = To
(16)
-
(17) (18)
w, = w,o
(19)
T = f(w,)
(20)
and at 7 = 1
The method described by Newman (1968, 1973) for linearizing and solving coupled, nonlinear, ordinary differential equations was applied to eq 14-23 for the case of a sucrose-water solution. Bomben (1981) describes the details of this finite-difference calculation. So that accuracy could be enhanced with the use of less computer memory, a logarithmic independent variable, y = In q, was used, and eq 14-23 were further transformed. In the numerical computation, one must determine a value of qmaxsuch that any further increase will not alter the value of the growth constant a t the desired accuracy. It was found that qmar > 100 was sufficient to ensure no change in p to six significant figures. The values reported for p are ones obtained by linearly extrapolating to h2 = 0 from values calculated at h = 0.001 and 0.002 (White et al., 1975). Data for the freezing point depression for sucrose-water solutions (International Critical Tables, 1928) were fit with a least-squares regression to give
T* = 5.421( w,’
PIP
= -(1
PDW,
- a,)
- 1)
+ 1.8151(
2
- 1)
+ 273.15
0.4221 Iw, I1.0 Data for the diffusion coefficient of sucrose-water so-
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 25 Table I. Properties and Growth Constants for 15% Sucrose Solution const diff. coeff 1o-5mfs, T* - To,K k , W/(m K) J/kg 101oDw: mz/s 1Ol0/3,m2/s T1- To,K 0.01 0.523 3.316 1.687 0.015 733 6 0.0009 0.5 0.523 3.308 1.656 1.098 91 0.0615 1.0 0.522 3.299 1.625 2.27001 0.1249 5.0 0.517 3.231 1.392 8.073 2 0.4200 7.0 0.515 3.197 1.287 10.0 0.511 3.145 1.140 9.853 3 0.4996
T* = 272.1368 K p =
1063 kg/m3
wW1
0.8489 0.8017 0.7604 0.5521 0.4275
variable diff. coeff m2/s T1- To,K 0.015 692 0 0.00091 1.045 18 0.058 5 2.065 89 0.1140 5.608 0.296 1 5.830 0.304 5 5.587 0.292 5
wW1
0.8489 0.8014 0.7594 0.5476 0.4874 0.4239
= 917 kg/m3 C, = 3807 J/(kg K)
pI
Diffusion coefficient a t bulk concentration and temperature.
Figure 1. Temperature profiles ahead of ice front in a freezing sucrose solution. Curve, wwo, a,,, AT, (K), TI- To(K), 1Ol0/3 (m2/s); A, 0.85,0.4239, 10,0.2925,5.587; B, 0.50,0.3413, 10,0.0098,0.13859; C, 0.85, 0.8014, 0.5, 0.0585, 1.04518; D, 0.50, 0.4867, 0.5, 3.0014, 0.192 631.
Figure 2. Concentration profiles ahead of ice front in a freezing sucrose solution. Conditions the same as in Figure 1.
CONSTANT DIFFUSION COEF.
lutions (Gostingand Morris, 1949; English and Dole, 1950) were fit with the relationship
-
D , = D W S O ex.[
(+ A) ] -
where D,: = 2.4254 X - 4.2940 X lo4 (1 - w,) + 1.5298 X lo-* (1 - w,)2(m2/s) and -AED/R = 2493.8 552.78 In (w, - 0.25) (in kelvin) where 0.25
