Impurity Trapping during Crystallization from Melts Danny D. Ediel and Donald J. Kirwan* Department of Chemical Engineering, CTniversityof" Virginia, Charlottesville, Va. 22901
An apparent interfacial distribution coefficient resulting from liquid trapping at a dendritic (rough) interface i s successfully used to correlate observed nonequilibrium solid compositions of the tolane-bibenzyl, stilbene-bibenzyl, AI-Cu and salol-thymol systems during growth from the melt.
I t has become apparent in recent years t'liat' the impurity concent'ratiori of crystals g r o ~ nfrom the melt or from solution oft'eri cannot be esplaiiied by applying the equilibrium dist'ributioii coefficient, ko, t'o relate the int'erfacial solid and liquid compositions. Some investigators have suggested that the solid composition is not a t its equilibrium value because of the requirement for a kinetic driving force at the interface (Cheng and Pigford, 1971; Kirwan and Pigford, 1969) l l a n y studies have noted liquid occlusions during crystal growth from solution (Belyustin and Fridman, 1968; Brooks, et al., 1968; Denbigh and White, 1966). I n melt gro\Tth, solid impurity concentrations esceeding solid solubility limits or forming a second phase during zone refining or progressive freezing a t high velocities have been noted (Baker and Cahn, 1969; Hellawell, 1965; Sharp and Hellawell, 19TOa; Wilcos, 1970; Wilcos and Zief, 1967a). This suggests that interfacial breakdown and trapping of liquid a t the microscopically rough interface may account for these observations. Thus Cheng, et al. (1967), who st'udied normal freezing of eutectic forming organic mistures, could correlate their results using the concept of a "surface void fraction": the fraction of the interface that is liquid dependent upon the interface struct'ure. The present study deals n i t h a n investigation of this proposed mechanism of impurity capture and n i t h the development of a theory for the prediction of the dependence of the void fraction on variables such as the freezing velocity and temperature gradient a t the interface. I
Interfacial Breakdown and Solute Trapping
The breakdown of a planar interface to a cellular structure seems to be adequat'ely described by the constitutional supercooling criterion (Sharp and Hellawell, 1970a; Tiller et al., 1953; Tiller and Rut'ter, 1956). The criterion states that, for a stable planar interface
G/VCL(O) > m(l
- ko)/D
face roughness or dendritic structure. This results in the trapping of substantial amounts of interfacial liquid (Cheng, et al., 1967; Hellawell, 1965; Wilcox and Zief, 1967~). Further evidence of significant liquid trapping is provided in Figures 1 and 2. These figures show the results of progressive freezing n ith a controlled interfacial temperature gradient in the salol-thymol system (which exhibits no solid solubility). The solid compositions sampled a t 20% of the charge frozen were determined by uv absorption. The figures show that the solid composition approaches the original liquid composition at high velocities indicating severe trapping of interfacial liquid. The constitutional supercooling criterion for these systems would predict planar interface breakdown at velocities of the order of cm/sec. The crystals grown at the higher velocities were quite soft and wet (at room temperature), and microscopic examination revealed numerous occlusions. Severe interface breakdown and dendritic type growth, even in faceted systems, are postulated as the mechanism for substantial amounts of impurity being incorporated during growth from the melt. What is required, therefore, is a relation between the interfacial liquid and solid compositions during growth under such conditions. Derivation of Apparent Interfacial Distribution Coefficient
In many practical solidification problems, mass transfer in the liquid phase is important. Burton, et al. (1953), related an effective distribution coefficient, k , = C,/CL( ) to the interfacial distribution coefficient. They assumed that the solid was in equilibrium with the interfacial liquid and found a quasisteady-state solution to the resulting diffusion equation to obtain
(1)
The above criterion applies to systems having no solid solubility (ko = 0) as well as solid solution formers. For steadystate crystallization of a solid solution former, C,(O) may be replaced by C L ( ~ ) i i 7 c g . For solidification of dilute metal systems the trapping of solute in grooves between cells seems to he a reasonable esplanation for the small amounts of impurity found in the solid phase (Clialmers, 1964; Wilcos arid Zief, 1967b). For progressive freezing or zone refining of concentrated (>1 7 3 melts. iricluding simple eutectic systems, the observed solute tal suggest a much more severe degree of int'er-
where 6 is the boundary layer thickness, V is the groivt'h rate of the solid, D is the diffusion coefficient, and h i s the equilibrium interfacial distribution coefficient. Solids g r o m a t velocities greater than the critical velocity for the breakdown of a planar int,erface tend to 110 longer retain their equilihrium coinpositioii. If an a1)parent interfacial distribution c,oefficient, k,, could he predicted for such a system, eq 2 would still be valid with Ir, replaced by k,, where
IC, Prevent address, Fibers Industries, h e . , Charlotte, N.C 100 Ind.
Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
=
actual composition of interfacial solid phase C, actual composition of interfacial liquid phase C,(O)
I
-0 0
I
I
I
I
0
2c
e
0
o 8 3 % THYMOL G = 30*C/cm +
.I?
c
94.5 % T H Y M O L G = 50°C/cm
.e
0
F
+
.IC
-0 0
E.
.o
u1
V I
C
I
I
I
IO
)
v x lo3
I
2 .o
1
30
,cm/sec
vx
.
Figure 1 Solid composition during progressive freezing of thymol-salol
At velocities greater than the critical velocity, the cells formed by the original breakdown of the planar interface tend t o become dendritic and project outward from their original cell base. If we idealize this broken interface to be composed of dendritic projections of average radius, p , (Figure 3), we can approximate the fraction of the interface t h a t is actually crystal, f, by the ratio of the projected area of the dendrite (crystal) t o t h a t of the entire cell.
f
=
=:
+ (1 - f)C,(O)
fCC(0)
(4)
ivhere C,(O) is the composition of the dendrite (crystal) at the interface and C,(O) is the interfacial liquid composition. If it is assumed that the crystalline (dendritic) portion of the interface is related to the interfacial liquid by the equilibrium distribution coefficient, C,(O) = koCL(0),then ka
=
C,/CL(O)
=
f(ko - 1)
cm/sec
Figure 2. Solid composition during progressive freezing of salol-thymol
Fi
DENDRITES
P
(3)
4?rP/nX2
If we denote the occluded or apparent solid composition as C,, then by a mass balance Cs
IO’,
+1
(5)
Interfacial equilibrium a t dendrite tips has been confirmed in metal systems (Sharp and Hellawell, 1970a, 1971), b u t the assumption may need to be modified in systems with particularly slow interface attachment kinetics. NOMto relate f and h (Le., f) to growth parameters, some results of cellular and dendritic growth in metal systems will be employed. I n steady-state growth experiments on the Al-Cu and A-Ag systems, Sharp and Hellawell (1970a, 1971) found that the average radius of dendrites varied as (CL(m ) V ) - ‘ I 2 a t a given interfacial temperature gradient. Other studies in metal systems confirmed this dependence (Plaqkett and Wmegard, 1956; Ross, 1967)
where V is the growth rate of the dendrite and C,( m ) is the bulk composition of impurity (that component in a binary melt’with ko < 1). Sharp and Rellawell (19iOa) compared metal data taken a t t x o different temperature gradients using the abore eyuatiou. Although K S had no dependency on G (the interfacial temperature gradient), K 1 varied as G-1. In general, therefore, ZCI must he presumed a function of G, m , ko, D , arid @ ( p is the kinetic growth coefficieiit which relates the growth velocity, I-, of the crystal to the undercooling of the crystal IC2 could be a function of vi,ko,D , and p . face). -11~0,
-
/
-
-
-=-e---
Figure 3. Model of a growth front
Sharp and Hellawell (19?0a, 1970b, 1971) observed that the spacing, X, of simple cells or dendrites resulting from the breakdown of a planar interface was insensitive to CLi m ) and V for a given interfacial temperature gradient, G. This spacing is evidently a characteristic of t h e breakdo\Yn of the planar growth front and is determined (at most) by the variables G, m, ko, D , and p . Xullins and Sekerka (1964), by a stability analysis, found t h a t this initial spacing was proportional to G - ’ I Z h
=
K3 G‘/2
(7)
r h e r e K3 may depend on m, ko, D , and p. Employing eq 6 and 7 in eq 3 and the resulting expression for f i n eq 5, me find, after rearrangement
Xote t h a t the left-hand side of eq 8 represents the actual interfacial distribution of components normalized by the distribution of components a t equilibrium and that 2Rl/’K, may be a function of G, vi,ko, and I). Thus we can use the constitutional supercooling criterion (eq 1 to iiormalize [G/CL( m ) V]”‘ and not introduce new parameters that, 21‘1/K3 might depend on. The constitutional supercooling criterion defines a condition a t which the interface is stable and planar and thus may have equilibrium separation of components. Therefore, it is a logical normalizing factor to use. The constit,utional supercooling crit,erion may be n i t t e n as
Ind. Eng. Chern. Fundam., Vol. 12, No. 1 , 1973
101
1.0
,
I
I
7. T O L A N E G = 26.5 'C/crn
o 90
-
--
0.6
-
0
90 K T O L A N E G = 98.3.C/crn 8 0 % TOLANE G. I O * C / c r n 80 X T O L A N E G=IOO'C/crn
-
/
I
E. 0 . 2 u)
v
-
I
/a
3
I -: 0.4 = x 0.2-
2 5 % STILBENE, PROGRESSIVE FROEZE-
ALL DATA
-
[ka - l]"' ko - 1
=
Ai(G)
[
Dko m ( l - ko
~
G=PO'C/cm
1"'[ I"'+ G CL(m)V
lest of Theory
I n order to evaluate the apparent distribution coefficient , data on the liquid and solid compositions a t the growth front are required. Data on interfacial liquid composition for the eutectic system, salol-thymol, and the solid solution-forming system, bibenzyl-stilbene, were taken by Kirwan (1967)' Kirn-an and Pigford (1969), and Cheng and Pigford (1971) using an interferometric technique. Kirwan and Pigford's work employed an isothermal stage while Cheng and Pigford's work used a stage with a controlled temperature gradient. I n the interferometric work the solid composition was calculated from the flus equation a t the interface, V ( C , CL (0)) = DdC/dz),-o. The solid composition calculated fell far from the equilibrium solid composition in studies on both systems. Some bibenzyl-stilbene samples were analyzed by ultraviolet absorption by Cheng and Pigford. These samples tended to confirm that the solid composition given by applying the flux balance was correct. Two additional checks of the flus balance procedure were conducted using progressive freezing of bibenzyl-stilbene. The values of composition and velocity obtained are well within the range observed by Cheng and Pigford on their interferometer (see Figure 4). Similar agreement with the Cheng and Pigford data was observed in progressive freezing experiments with the salol-thymol system. Hence, these progressive freezing esperiments tend to confirm the validity of using the flus balance to compute solid compositions. Chem. Fundam., Vof. 12, No. 1 , ?973
0.2
I '
0.4
I
0.6
1.0
0.8
A ~ G ~ / ~
A2 is a t most a function of only the physical properties of the system, but AI may also depend on G, the temperature gradient a t the interface since Sharp and Hellawell (1970a) found K1 to depend on a. Equation 10 relates the dependence of the apparent interfacial distribution coefficient of an unstable, broken interface on bulk liquid composition and velocity for a given system. Since eq 10 is derived for a dendritic interface, it holds only for growth velocities greater than Verit (the critical velocity for breakdown of a planar interface as defined by the constitutional supercooling criterion). At velocities less than Vcrit the interfacial distribution coefficient is presumed to have its equilibrium value.
Eng.
I
0
Table I. Regression Coefficients for Eq 1 1 System
102 Ind.
-
' 0
0
Introducing eq 9 into eq 8 gives
-
AI, (OC/cm)'/2
Tolane-bibenzyl 9.6 i 0.l e Stilbene-bibenzyl 18.1 f 0 . 7 Thymol-salol 22.1 f 1 . 0 Salol-thymol 24.0 f 0 . 9 Aluminum-copper 15.9 i 1 . 1 Standard deviation of coefficients.
A$, (cm/OC)'/z
*
0.000 0 . 0 0 1 ~ -0.005 f 0.004 -0.004 f 0.008 0.0 0 . 0 4 & 0.00
Tolane-Bibenzyl System. After the progressive freezing tests gave additional verification t h a t solid compositions obtained by a flus balance method were correct, interferometer tests with a controlled temperature gradient stage similar to t h a t of Cheng and Pigford were performed using the tolane-bibenzyl system. The tolane-bibenzyl system exhibits solid-solid solubility and ( a t the 80 and 90 mole % tolane solutions studied) solidifies with the tolane crystal form. Timmermanns (1957) gives the phase diagram for the tolane-bibenzyl system. Procedures and details of these experiments can be found in Edie (1972). The experiments provided interfacial liquid and solid data (70 growth fronts in all) on the tolane-bibenzyl system grown a t various growth rates from two different melt compositions with two different interfacial temperature gradients. We used the data on this system to test eq 10. Linear regression analyses on the data with various assumed dependencies for A1(G) showed A1(G) proportional to G-'" gave a good fit for all the data indicating that eq 10 should be
where A1 and A z are constants for a given system. The regression coefficients for this system and the other systems studied are given in Table I. Figure 5 is a plot of the predicted us. the observed separation, using the determined coefficients. Note that the value of A2 is zero so that the temperature gradient a t the interface apparently has no effect on the separation when separation occurs over an unstable interface. I n these experiments the group koD/m(l - ko) varied by a factor of 1.5, C L ( m ) V about twelvefold, and G fourfold. The data in Figure 5 indicate that the dependence of the distribution coefficient on these variables is well accounted for
10 o 25 % STILBENE GI 2 0 ° C / c m 4 5 % STILBENE
' CHENG
]
I
.-
I
.
8 PIGFORD,
1971
08-O
----
rn 9 0 % T H Y M O L G * 2O0C/cm CHENG e PIGFORD, 1971
z=. 0.6-
9
x0
0
x
\
\ 1
0
-
06-
-
04-
0.4-
-
2
x
02-
,O
6
0;
0'4
06
A,(k,D/m(l-k,)CL(m
0'8
10
I/2
)V)
Figure 6. Actual vs. predicted distribution for the stilbenebibenzyl system
by eq 11 with A? = 0. 111all experiments the growth velocities were well above the critical velocity for breakdown of the planar interface. Stilbene-Bibenzyl System. Cheng and Pigford (1971), Kirwan (1967), and Kirwan and Pigford (1969) all took d a t a 011 t h e stilbeiie-bibeiizyl system using a n interferoniet'ric technique. K i r i a n arid Pigford used a n isothermal stage interferometer giving a n interfacial temperature gradient of zero (G = 0). Cheng and Pigford used a controlled temperature gradient stage (similar t o t h e one used in this study) a n d an interfacial temperature gradient of about 20°C/cm. These investigators covered two compositions of melt, aiid all tests' grou-t,h velocit,ies were above the critical velocities for planar interface brea kdorvii. T h e stilbene-bibenzyl system (like the tolaiiebibenzyl system) has solid-solid solubility. The apparent interfacial distribution coefficient was calculated using the interfacial solid and liquid conipositions obtained by Cheng and Pigford and Kirwan and Pigford. The data were then fit by regression analysis to give values for A I and -42 in eq 11 (see Table I). Figure 6 is a plot of the observed z's. predicted value of [ ( k , - l ) / ( k ousing eq.11 and tlie constants obtained by regression analysis. .%gain, .41(G) = AIG"' and A n is essentially zero, suggesting t h a t the separation of this organic system is very insensitive to the temperature gradient a t the interface when separation occurs over a n unstable interface, The stilbene-bibenzyl data taken a t G = 0 and G = 20"C/ cm agairi verify the usefulness of eq 11. I n this system koD/ m(1 - ko) varied by a fact'or of 2 aiid C,(m)V'varied one hundredfold. Thymol-Salol System. Cherig and Pigford (1971) and Kirnaii (1967) and Kirwan and Pigford (1969) all t,ook d a t a o n thymol crystals growing from a thymol-salol nielt using an iiiterferometric technique. Xs before, Kirwaii and Pigford used a n isotliernial stage and Cherig and Pigford's temperature gradient was 20°C/cm. Kirwan and Pigford's studies covered three compositions of melt, and Cheng arid Pigford studied one melt composition. Since .the thymolsalol system is of the simple eutectic forni with 110 solidsolid solubility, the thymol crystals grown from this melt Jvere always above the critical velocity for breakdown of the planar interface. The apparent interfacial distribution coefficients for each test made by Cheiig and Pigford as well as Kirwan and Pig-
0 06 08 10
00
02
04
A I (D/m
)V
Figure 7. Actual vs. predicted distribution for the thymolsalol system
ford were calculated. We then fit the data by regression analysis to get the values of A1 and A2in eq 11 (Table I). See Figure 7 for a plot of the observed us. predicted value of [ ( k , - l ) / ( k o - l)]''zusing the regression coefficients in eq 11. Since the constitutional supercooliiig criterion as given by eq 9 is iiicorrect for eutectic systems (ICo = O), the normalizing value of ((C,( m)V)/'G),,i, used in eq 11 was
This is not quite the critical velocity by a factor of (C,(m)/ C,(O))'". This ratio is always less than unity for the impurity but would not be expected to be greatly different from 1 a t the breakdown point. Once again the fact that the data fit well with A1(G) = A1G"2 and a value of A 2 t h a t is essentially zero shows t h a t the separation of this organic system is very insensitive t o the temperature gradient imposed a t the interface once breakdownhas occurred. For the concentrations of thymol-salol studied by Kirwan and Pigford and Cheng and Pigford, D and m varied very little, but C,( a)V varied by a factor of about 40. Salol-Thymol System. Kirwan arid Pigford took d a t a on salol crystals growing from a salol-t'hyniol melt using a n isothermal stage interferometer. Four different melt compositions were used. Siiice this is a eutectic eq 12 must be used for (C,(m)V/G),,it. The data of Kirwan and Figford were fit by linear regression to obtain the values for A1 and A2 in eq 11. Since G was zero for all Kirwan and Pigford's data, 8 2 was set equal to zero, but this would riot be unexpected since t'he previous three orgaiiic systems have all yielded a n A2 = 0 even though G was not equal to zero. Figure 8 shows t h a t the predicted us. the observed separation agrees nicely using these values for the regression coefficients. Over the concentrations studied by Kirwari and Pigford, D arid 771 varied very little, and C,( ) V varied about threefold. Cheiig and Pigford (1971) studied this same system and took data on salol crystals grown using a temperature gradient stage interferometer (G = 20"C;cm). I n this study of the salol-thymol system, they calculated the interfacial solid compositions for four experiments. Figure 8 shows that their four data points vary significantly from tlie twelve data points of Kirwan (1967) and Kirvian and Pigford (1969). YO explanation of this discrepaiicy is available a t this time other than experimental error. Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
103
""1
in lo
0 8 5 % SALOL
0 G a
7~'Chm
SHARP
a
HELLAWELL
86.5% SALOL N
0
02
0
e
CHENG
90% SALOL
A
0.4
PIGFORD, 1971
A , (D/rnC,(m,)V)
O L 0
1.0
0.6
0.6 112
A,(k,D/rn(l
Figure 8. Actual vs. predicted distribution for the salolthymol system
*SALOL- THYMOL THYMOL SALOL 0 STILBENE- BIBENZY
.
0 0
I
I
I
I
I
I
I
I
01
02
03
04
OS
06
07
08
1
A: ( k , D / m C J m l V )
Figure 9. Apparent interfacial distribution coefficient for organic systems
Note that the value of A1 using Kirwan and Pigford's data for the salol-thymol system (24.0) is very close to the value of A1 for the thymol-salol system (22.1). Perhaps this is riot surprising since both the salol and the thymol crystals are grown in the same melt (thus the melts have similar diffusivities) and the thymol and salol crystals have relatively slow growth kinetics (as stated before A1 may depend on g, the kinetic growth coefficient of the solid). For all the organic systems studied, A z = 0 and eq 11 becomes
[kQ - 11'"' ko - 1
[k&l(m(l - ko))]'" C,(m)V
Rearranging and solving for k, gives
Figure 9 shows that (given the value of A1 for each system) all these organic systems obey eq 13. As discussed earlier, for eutectic organics
and eq 13 becomes
104 Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
I
0.2
I
I
I
0.4
0.6
0.8
- k,lCL(m)V) IR
I
I.o
A 2 G I/2
Figure 10. Actual vs. predicted distribution for the aluminum-copper system
This form was used to plot the salol-thymol and thymolsalol systems in Figure 9. Equation 13 shows that as V + m , k , + 1, which would be expected. When V = Vcrit, k, should equal k,; therefore, the k , for each system should not fall below ko. This means that for eutectic systems (salol-thymol and thymol-salol, for instance) k, will decrease to zero as velocity decreases to Vcrit and then remain a t zero for all smaller velocities. Similarly, for the stilbene-bibenzyl compositions studied, k, should decrease to a range of 0.64 to 0.42 (depending upon the composition) as velocity decreases and then remain a t this value for smaller velocities. For the tolane-bibenzyl system, k, should decrease to a value of about 0.1 as velocity decreases and remain a t 0.1 for slower velocities. At V N Vc,it, it is possible that the interface attachment kinetics may need to be accounted for. This might slow the approach of k, to ko as V decreased to the immediate vicinity of Vcrit. We found no evidence that such a correction was needed, but all our data were taken above V = Vorit. Aluminum-Copper System. Sharp and Hellawell (1970a) obtained interfacial solid and liquid compositions using the aluminum-copper system which has solid-solid solubility in the composition range t h a t they studied. They employed a progressive freezing apparatus. The interface was quenched when a sample was desired. The quenched sample was cut and the solid-liquid interface a t the time of quenching was analyzed using a n electron microprobe analyzer. All runs were made a t growth rates above the critical growth velocity for planar interface breakdown (photographs also showed that these interfaces were not planar). They analyzed the interfacial liquid concentration as well as the composition of the solid growth front. Their data covered a variety of melt compositions and two interfacial temperature gradients. A linear regression fit of their data to eq 11 gave the values a t A and A2 in Table I. Figure 10 shows a plot of [ ( k , - l)/(ko - I) predicted by eq 11 using the above values of A 1 and A2 us. those obtained experimentally by Sharp and Hellawell. For this metal system, the value of Az is significantly greater than zero (unlike the organic systems); in fact, AzG'/' = 0.51 for the tests with G = 160°C/cm and AZG'"' = 0.35 for G = 75"C/ cm. This indicates that even though interfacial temperature gradient has no appreciable effect on separation when organics are grown above the critical velocity for interfacial breakdown, an increasing temperature gradient increases separation when metals are grown a t velocities above the critical velocity for planar interface breakdown.
Discussion
The different effects of an interfacial temperature gradient on organic and metal systems may be caused by the ability of a metal dendrite to grow with nearly equal velocity in any direction (isotropic kinetics). Thus i t has the ability to grow perpendicular to the direction of advance of a dendritic growth front if encouraged by a n increased temperature gradient normal t o the interface. An organic typically has very slow growth rates on faces perpendicular to a dendrite axis. I t s growth is strongly determined by orientation of growth planes and thus i t has less ability to grow lateral t o the direction of growth even if a high-temperature gradient is imposed. Thus i t is not surprising t h a t the separation of a metal system growing n-ith a nonplanar interface improves with increasing temperature gradient, but the separation of a n organic system does not. The theory we have developed for the prediction of k, is based on the premise t h a t the interface is dendritic and broken. Many times the solid appears smooth when viewed through the microscope of the interferometer. The results of the progressive freezing experiments may explain the answer t o this seeming discrepancy. The progressive freezing experiments indicated that the organic solids grown a t V > Vorlt grow as a series of very thin staggered platelets (on the order of 10+ em thick), and these platelets sandwich the occluded liquid between them. The interface of a sandwich of these dendritic plates would be difficult to distinguish from a planar interface if the sandwich of plates were perpendicular to the microscope viewing surface. It would Le impossible to distinguish this sandwich structure from a planar interface if the plates grew parallel to the microscope viewing surface because the platelet ends would give the appearance of a flat, solid interface. The interface model proposed in Figure 3 mould be conceptually the same if a series of staggered platelets replaced the rods shown. For the metal system, A? # 0 and thus as V approaches a , eq 11 predicts t h a t k , This seems somewhat doubtful since one would expect k, to approach a constant value of 1 at high growth rates. This may indicate t h a t our theory breaks down at very high velocities in metal systems. In organic systems, since A Zseems to always be zero, this limitation does not exist. A1 and A z in eq 11 are not universal constants for all separations. Their values depend on the system being separated. A I and A z can still be dependent on the kinetic growth coefficient, p. Although data on p for tolane are not available, we would expect t h a t it is greater than the growth coefficient for stilbene since tolane is a more symmetric molecule and can orient itself more easily to solidify. Comparing the values of the A1 constants for the organic systems, we see t h a t
AI,,
< Alsa < A I T S
-
AIS,
For these same systems, the magnitude of the kinetic growth coefficients follows the reverse order (Cheng and Pigford, 1971 ; Kirwan and Pigford, 1969). PTB
> PSB > ClTS
-
PST
This suggests that the variation of A1 from system to system may be partially the result of the value of the kinetic grolvth coefficient for the system. Still, eq 11 predicts the separation of a n organic system over a wide range of V > Vorlt with only one reliable data point required to determine A1 (two points are required for metals since A Zmay also be important). As it stands, eq 11
is applicable t o optimization of design parameters for the separation of a given system since i t predicts the variation of the separation of the system with the variables V , CL(m), G, m, D, and ko. Equation 11 represents the first predictive equation derived to give the separation of a system at growth velocities exceeding those where interfacial breakdown occurs. Acknowledgments
D. D. E. appreciates the NDEA Title IV fellowship support during the course of this work. The support of NSF Grant GK-34754 in the latter stages of this work is appreciated.
function in eq 10 constants in eq 11 impurity concentration of bulk liquid, mole fraction dendrite or crystal impurity concentration, mole fraction interfacial liquid impurity concentration, mole fraction apparent or occluded solid impurity concentration, mole fraction diffusivity of impurity in the melt, cm2/sec fraction of growth front t h a t is crystal temperature gradient in the liquid a t the growth front, “C/cm equilibrium interfacial distribution coefficient based on impurity concentration apparent interfacial distribution coefficient effective distribution Coefficient functions liquidus slope based on impurity concentration, OC/mole fraction average radius of dendrite, cm growth velocity, cm/sec critical growth velocity, cm/sec GREEKSYMBOLS 6
x P
= = =
boundary layer thickness, em interdendritic spacing, cm kinetic coefficient of the growth plane of the crystal
SUBSCRIPTS
TI3 SI3 TS
ST
= = =
=
Tolane-bibenzyl system Stilbene-benzyl system Thymol-salol system Salol-thymol system
Literature Cited
Baker, J. C., Cahn, J. W., Acta M e t . 17, 575 (1969). Belyustin, -4. V., Fridman, S.S., Sou. Phys.-Crystallogr. 13, 298 (1968). Brooks, R., Houton, A. T., Torgeson, J. L., J . Cryst. Growth 2, 279 (1968). Burton, J. A., Primm, R. C., Slichter, W. P., J . Chem. Phys. 21, 1991 (1953). Chalmers, B., “Principles of Solidification,” p 173, Wiley, New York, N. Y., 1964. Cheng, C. S., Irvin, D. A,, Kyle, B. G., A.I.Ch.E. J . 13, 739 (1967). 10, Cheng, C. T., Pigford, R. L., IND.ENG.CHEM.,FUNDAY. 211) (1971 ,--.-,.1 Denbigh, K. G., White, E. T., Chem. Eng. Sci. 21, 739 (1966). Edie, D. D., Ph.D. Thesis, University of Virginia, 1972. Hellawell. A,. Trans. Met. SOC.A Z M E 233. 1516 (1965). Kirwan, D. J:, Pigford, R. L., A.Z.Ch.E. J’. 15, 442 (1969). Kirwan, D. J., Ph.11. Thesis, University of Delaware, 1967. Miillins. W. W.. Sekerka. R. F.. J . AmZ. Phvs. 34. 323 11964). Plaskett, T. S., ’Winegard, W. C., Can: J . Pxys. 34, 96 (1956). Ross, K., B.Sc. Thesis, Oxford University, 1967. Sharp, R. LI.,Hellawell, A., J . Cryst. Growth 6 , 253 (1970a). Sharp, R. >I., Hellawell, A,, J . Cryst. Growth 6 , 334 (1970b). Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
105
Sharp, R. BI.,Hellawell, A., J . Cryst. Growth 11, 77 (1971). Tiller. W. A,. Kutter. J. W.. Can. J . Phus. 34. 96 11956’1. Tiller: W. A,: Ruttei, R. W‘., Jackson, K.A.,’Chalmersj B., Acta Met. 1, 428 (1933). Timmermanns, J., “Physico-Chemical Constants of Binary Systems in Concentrated Solutions,” Vol. 1, p 169, Interscience, New York, N. Y., 1957. Wilcox, W. R., “Fractional Solidification Phenomena,]’ Paper Presented at AIChE Meeting, Dee 1970. Wilcox, W. R., Zief, AI., Ed., “Fractional Solidification,” Chapter 3, Marcel Dekker, New York, N. Y., 1967a.
Wilcox, W. R., Zief, M.,Ed., “Fractional Solidification,” pp 119-129, Marcel Dekker, New York, N. Y., 1967b. Wilcox, W. R., Zief, M., Ed., “Fractional Solidification,” p 87, lllarcel Dekker, New York, N. Y., 1967c. RECEIVED for review February 4, 1972 ACCEPTED October 4, 1972 Presented at the 64th Annual Meeting AIChE, San Francisco, Calif., December 1971, and 73rd National Aleeting of AIChE, Minneapolis, Minnesota, August 1972.
An Analysis of the Stagnant Band o n Falling Liquid Films Roger A. Cook and Reginald H. Clark” Department of Chemical Engineering, Queen’s University at Kingston, Ontario, Canada
A stagnant surface may exist at the exit region of a liquid film flowing into a reservoir. Its presence decreases the rate of heat and mass transfer and is of particular concern in wetted-wall and packed columns. The length and profile of the stagnant zone above the reservoir has been investigated in the laminar regime as a function of flow rate and fluid properties. The influence of the surface pressure of the reservoir relative to the surface pressure of tbe film has also been investigated and a linear relationship with the length of the stagnant zone was established. The main forces determining the magnitude of the stagnant zone were found to b e the surface pressures, the shear stress exerted on the stagnant surface b y the flowing film, and the change of momentum of the liquid as the film enters the stagnant zone. Entrance and exit effects in the stagnant zone are shown to invalidate some of the assumptions made in deriving theoretical models, and empirical correlations are shown to b e of more practical value.
A n excellent review of many aspects of film flow has been presented by Fulford (1964). In this review, he mentions stagnant band formation in passing in a section dealing with the effect of surface active agents onwavy films. However, the stagnant band has been observed by many authors. It’only occurs when the reservoir into which the film flows has a “dirty” surface, that is, a reservoir where the liquid has sufficient residence time for surface active material to diffuse to the surface. This mat’erial then climbs the surface of the flowing film to form a n immobile skin, the height being some function of the surface pressure of the reservoir relative to that of the flowing film. This stagnant zone will create a region in which considerable modification of the mass and heat transfer rates will be experienced. Lynn, et a!. (1955), found that the band height (the band height’ or length is effectively the area covered by the stagnant surface) was independent of the total film length and that a satisfactory correlation could be obtained by plotting h1l2us. &’I3 over the range of Reynolds numbers 5-300. For their mass transfer studies, although they were able to measure a finite rate of gas absorption through a stagnant region, they obt’ained a satisfact’ory correlation of their data by assuming that the rate was negligible when compared to the transfer rate through the iioiistagnant surface and simply subtracted t,he area of the stagnant surface from the total surface area of the wetted wall. Xysing and Kraniers (1958) reached a similar conclusion. Kittler (1962) calculated an “effective” stagnant band to account for the discrepancy between observed and 106 Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
predicted gas absorption rates into the flowing film and found that this “effective” band length was greater than the observed value a t high Reynolds numbers. However, the effective band also included the correction for the acceleration region as well as t’he deceleration or exit region of the film. Merson and Quinn (1965) investigated the presence of a stagnant film on flow in a n inclined channel. They placed a barrier on the surface a t the downstream end of their apparatus and measured. the rate of growth of films produced by several materials. They predicted and found that condensed films and gaseous films grow a t different rates and they also observed a system (benzene-air) where the growth rate decreased to zero Iyhich indicated that desorption from the stagnant layer takes place in this system. Xerson and Quinn’s work was conducted a t very low angles of inclination to the horizontal. The work of other investigators on vertical films show that a n equilibrium film length is rapidly achieved. Merson and Quinn suggest that the most feasible explanation for this equilibrium film length is that the shear stress exerted on the film is sufficient to collapse the monolayer into a “multimolecular third phase.” They consider desorpt’ion from the film to be a n unlikely explanation. Stagnant bands have also been observed on jets flowing into reservoirs (Cullen and Davidson, 1957; llat’suyama,1953) and in liquid-liquid systems (Ratcliff and Reid, 1961). .In extensive theoretical and experimental investigation of the stagnant band phenomenon was undertaken by Roberts (1961). His model is the simplest possible, i.e., flow between
