Chapter 17
Solute Transfer in Zone Refining of EutecticForming Mixtures
Downloaded by KTH ROYAL INST OF TECHNOLOGY on November 16, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch017
George C. Yeh Department of Chemical Engineering, Villanova University, Villanova, PA 19085
A simple physical model called "Filtration Model" has been proposed to describe the separation process of eutectic-forming mixtures by zone refining, in which the solute transfer through the partially solidified zone (mushy region) is considered to be the rate-controlling step of the overall solute transfer process. Two generalized correlations to account for the effects of free convective mixing upon the macrosegregation rate have been developed. The applicabilities of the proposed model and of these correlations have been demonstrated for a wide range of experimental conditions; and the criteria for the maximum and zero separation were also established from the correlations. Following the e a r l i e r work by Pfann (10) i n 1952 on zone r e f i n i n g many publications have appeared i n the l i t e r a t u r e . In order to describe the l o n g i t u i d i n a l solute transfer i n zone r e f i n i n g , various theories assuming complete mixing (5 & 11) or pure molecular d i f f u s i o n mechanism i n the melted zone using the boundary-layer treatment have been proposed (19 through 23) . Most of these early works have assumed the existence of a constant d i s t r i b u t i o n c o e f f i c i e n t and an equilibrium between the s o l i d and melt with a few exceptions, which have devoted to eutectic-forming mixtures which do not have constant d i s t r i b u t i o n c o e f f i c i e n t s (5, 20 through 23). In a l l early works, i t has been assumed that there exists two well-defined planner interfaces. However, i n the actual experiments a p a r t i a l l y s o l i d i f i e d zone ( P . S . Z . ) or 'mushy region' and a p a r t i a l l y melted zone (P.M.Z.) always e x i s t between the completely r e s o l i d i f i e d and fresh s o l i d s . The existence of a P . S . Z . (mushy region) behind the freezing front i s especially important since i t offers the greatest resistance to the forward solute transfer and can trap the solute flowing backward and can affect the nature and the extent of 'macrosegregation' . The existence of a P . S . Z . (mushy region) and i t s importance i n solute transfer during s o l i d i f i c a t i o n of a binary mixture have been recognized by more recent workers (24 through 29). Simple models
O0^7-6156/90/D438-O230$06.25/0 © 1990 American Chemical Society
In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
Downloaded by KTH ROYAL INST OF TECHNOLOGY on November 16, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch017
17. YEH
Solute Transfer in Zone Refuting ofEutectic-Forming Mixtures
r e l a t i n g the temperature and the l i q u i d mass f r a c t i o n for mushy region (30, 31), and more comprehensive models taking into account the i n t e r a c t i o n of momentum, heat and mass transfer processes inside the mushy region i n order to predict macro-segregation patterns were presented recently (32 through 38). A l l these works dealt with s o l i d i f i c a t i o n i n a stationary system and have not been applied to zone r e f i n i n g , which takes place i n a nonstationary system i n which the P . S . Z . (mushy region) and the melt move i n a s p a c i a l r e l a t i o n . This study has been conducted with the objectives: 1) to analyze and determine the effect of the free convective mixing i n the melted zone upon the o v e r - a l l solute transfer process i n zone r e f i n i n g under various conditions and 2) to take into account the solute transfer both i n the melted zone and i n the adjacent P . S . Z . (mushy region) i n p r e d i c t i n g the rate of o v e r - a l l solute transfer. The temperature d i s t r i b u t i o n , composition, zone t r a v e l speed, zone dimension and other important parameters are considered i n this study. THEORETICAL The conditions of freezing and of mixing i n the molten s o l i d play decisive and delicate roles i n determining the separation r e s u l t s . In normal operation, the molten s o l i d i s cooled externally; as a r e s u l t heat transfer i n both r a d i a l and l o n g i t u d i n a l d i r e c t i o n s , and the c r y s t a l s of s o l i d grow i n the directions opposite to that of heat transfer. The r a d i a l mass transfer becomes important when rapid cooling rates and high zone speeds are employed; under these condi tions, the simple assumption of u n i d i r e c t i o n a l mass transfer across the planner freezing interface deviates greatly from the r e a l conditions. Incomplete separation i s resulted not only by the r a d i a l and the anisotropic segregation due to "coring" (1, 2, 3, 4, & 8) or "constit u t i o n a l subcooling" (2, 3, 8, 12, 13 through 18) but also by the incomplete mixing of the molten s o l i d . The melt may be mixed by free convection. In f l u i d mechanical sense, complete mixing of a f l u i d requires the conditions for l o c a l l y i s o t r o p i c turbulence to e x i s t , and consequently some of the solute which have once been removed and c a r r i e d away from the freezing interface may be transported back by the convective currents. Such a r e d i s t r i b u t i o n of solute i n the P . S . Z , (mushy region) i s referred to as micro-segregation. An increase i n the i n t e n s i t y of free convection always results i n the corresponding decrease of the thickness of the l i q u i d layer or the resistence to the forward solute transfer; as a r e s u l t , the degree of back-mixing of solutes and the rate of resultant contamination i n the P . S . Z . (mushy region) are also increased. Since the results of zone r e f i n i n g depend on the i n t e r a c t i o n of momentum, heat and mass transfer i n the system, a l l the basic factors a f f e c t i n g these three processes, both molecular and convective, have to be taken into consideration. These basic factors are: concentra t i o n W, Density f , v i s c o s i t y heat capacity Cp, temperature dens i f i c a t i o n c o e f f i c i e n t fi , thermal conductivity k, molecular d i f f u s i v i t y D, zone diameter d , zone length L , zone t r a v e l speed u, temperature difference i n zone 21 T and acceleration g. The concentra t i o n W may affect p , JUL , Cp, , k, and D as well as the properties of the P . S . Z . (mushy region). Aside from the concentration W, a l l fc
In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
2
CRYSTALLIZATION AS A SEPARATIONS PROCESS
232
other factors may be combined into the following dimensionless D/uL, D / u L (-D/uL ) t
Parameter for molecular d i f f u s i o n model i n a moving zone, equivalent to the r e c i p r o c a l of Peclet number, dispersion number Reynolds number, Re Prandtle number, Pr Schmidt number Sc Grashof number, Gr, where h - (L' + d )/2 for melted zone.
t
cp/vk 3
Downloaded by KTH ROYAL INST OF TECHNOLOGY on November 16, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch017
h ^ATg(P//0
groups
2
t
The product of Pr and Gr numbers may be used as the parameter to account for free convection e f f e c t . The boundary layer treatment requires the knowledge of the e f f e c t i v e thickness of the layer, which cannot be determined experimen tally. Wilcox (19) and Thomas (13) applied to zone melting the p r e d i c t i o n method for v e r t i c a l f l a t p l a t o r i g i n a l l y proposed by Holmes et a l (6); but i t s a p p l i c a b i l i t y has not been proven. Kraussold (7) has developed a generalized c o r r e l a t i o n between the r a t i o of e f f e c t i v e thermal conductivity for free convection to the molecular thermal conductivity k / k and the product of Pr Gr for a f l u i d enclosed between two p l a i n walls. By the analogy between heat and mass transfer, the values of K / k should be i d e n t i c a l to that of D,/D, the r a t i o of e f f e c t i v e d i f f u s i v i t y for free convection to molecular d i f f u s i v i t y i n a given system. By the d e f i n i t i o n , the e f f e c t i v e boundary layer thickness L may be calculated by the r e l a t i o n s h i p L - L ( D / D ) . The values of L so obtained depend on the degree of free convective mixing, which may be represented by the Pr Gr value of the system. It may be suggested that either molecular d i f f u s i v i t y D be replaced by the e f f e c t i v e d i f f u s i v i t y D or the t o t a l zone length L be replaced by the e f f e c t i v e boundary layer thickness L to account for the effect of free convection. To do t h i s , a new parameter uL/D may be used instead of uL/D i n c o r r e l a t i n g the separation results of zone r e f i n i n g . The solute transfer i n zone r e f i n i n g i s an extremely complicated, unsteady state process involving numerous steps and phenomena. In the d i r e c t i o n of zone t r a v e l , the solute transfers through the P . S . Z . (mushy region) mainly by molecular d i f f u s i o n , and p a r t l y by eddy d i f f u s i o n near the freezing front. In the completely melted zone ( C . M . Z . ) , the solute transfers by both molecular and eddy d i f f u s i o n from the freezing front to the plane of complete melting. Near the front of the outer boundaries of both the P . S . Z . and the P . M . Z . , molecular d i f f u s i o n i s predominant, but near t h e i r inner boundaries eddy d i f f u s i o n plays a major r o l e . In the d i r e c t i o n opposite to the zone t r a v e l , there i s the transfer of solute from the C.M.Z. into the P . S . Z . as a r e s u l t of the r e l a t i v e movement of the P . S . Z . into the melt and the back-mixing of the solute by convective currents. The rate of backward solute transfer depends not only on the degree of back mixing and the zone t r a v e l speed, but also on the nature of the P . S . Z . , the freezing rate, and the concentration difference across the P . S . Z . , as may be obvious. Each of the above mentioned steps i s at a highly unsteady state; the i n d i v i d u a l transfer c o e f f i c i e n t s , the d r i v i n g forces and the t
t
t
t
t
t
t
a
In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
t
Downloaded by KTH ROYAL INST OF TECHNOLOGY on November 16, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch017
17. YEH
Solute Transfer in Zone Refining ofEutectic-Forming Mixtures
resistances vary with distance as well as time. The t h e o r e t i c a l p r e d i c t i o n of a l l of these values i s impossible, and t h e i r experimental determination would be i m p r a c t i c a l . Therefore, the s o l u t i o n of any rigorous mathematical expressions for the rates of both forward and backward solute transfer processes i s impossible. However, the rate of the o v e r a l l process (macro-segregation) may be given by using a simple model which represents the o v e r a l l effect of the solute transfer i n zone r e f i n i n g . Consider a physical model with the following simple mechanisms: 1. A molten s o l i d containing a solute of increasing concentra t i o n i s flowing at a constant v e l o c i t y through a porous medium whose porosity decreases with the distance; or one may v i s u a l i z e the porous medium as moving through the molten s o l i d at the same r e l a t i v e velocity. The effect would be the same. In t h i s case, the porous medium i s the P . S . Z . (mushy region). 2. The porous medium, namely the P . S . Z . , possesses a c e r t a i n f i l t r a t i o n capacity to f i l t e r out the solute. This corresponds to the rate of forward mass transfer across the P . S . Z . i n zone r e f i n i n g . 3. The u n f i l t e r e d solute i n the molten s o l i d transfers through the porous medium, v i z . the P . S . Z . , and f i n a l l y reaches the plane of complete freezing, where i t f i n a l l y s o l i d i f i e s completely. This determines the solute concentration i n the treated s o l i d . In Step 1, one v i s u a l i z e s the increasing solute concentration i n the complete melted zone with time, the varying degree of s o l i d i f i c a t i o n along the thickness of the P . S . Z . , or mushy region and the constant r e l a t i v e v e l o c i t y between the two zones (or the heater and the cooler). In Step 2, one v i s u a l i z e s the rate of forward mass transfer through the P . S . Z . , which i s the r a t e - c o n t r o l l i n g step i n the o v e r a l l forward mass transfer process. In Step 3, one v i s u a l i z e s the rate at which the u n f i l t e r e d solute i s trapped inside the treated s o l i d . If i n Step 1, the r e l a t i v e v e l o c i t y of the two zones approaches zero then the rate of Step 2 becomes i n d e f i n i t e l y large, and according to the phase diagram the separation should be complete, or the rate of Step 3 would be zero. The model proposed above i s analogous to a continuous, unsteady state f i l t r a t i o n process, and therefore may be c a l l e d " F i l t r a t i o n Model". In t h i s model, the concentration of the f i l t r a t e , v i z . the concentration of the solute remained i n the treated s o l i d i s one's major concern. This i s given by the rate of Step 3, which may be expressed by an equation s i m i l a r to F i c k ' s Law including a transmission c o e f f i c i e n t D for the porous medium, v i z . the P . S . Z . and the con centration difference AW across the P . S . Z . as the d r i v i n g force, and the thickness of the P . S . Z . as the distance A x . m
In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
2
CRYSTALLIZATION AS A SEPARATIONS PROCESS
234
The following assumptions may be made i n order to solve Equation 1: 1) Constant A and p 2) P . - P i - P , or A p . 0 3) The amount of solute i n the P . S . Z . having the thickness A x i s n e g l i g i b l e compared to that i n the C . M . Z . , and may be neglected i n the material balance. To determine the value of D from the experimental r e s u l t s , the data of W as a function of z are needed. Thus, from the mass balance the amount of solute i n the treated s o l i d between 0 and z i s m
Downloaded by KTH ROYAL INST OF TECHNOLOGY on November 16, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch017
s
Afj*
%(z)4z
the amount of solute i n the untreated s o l i d between 0 and z+L i s
and therefore the amount of solute i n the C.M.Z. at t - t , or z - z i s
Af[cz+L)W,-rw,(z)dz] The t o t a l mass i n the C.M.Z. i s equal^ to A p L ; therefore the average solute concentration i n the C . M . Z . , W i s L
W , - ^ W o - - t - £ w ( z ) d ;\Z s
(2)
On the other hand, dm/dt may be evaluated as follows:
K ) ^ z
z
+
« = A ? f 0
+
4
Z
4 0 . - 1 *
A m -
t
o
Z
W ( z ) d z s
/Z+AZ
