On the examination of zone melting

ON THE EXAMINATION

ooc?-2509,&1

s3.M) + .a0

G 1984 Pergamon

Press Ltd.

OF ZONE MELTING

KATALIN VASARHELYI Hungarian Oil and Gas Research Institute, Veszprim, J6zsef A. u. 34, Hungary (Receitxd

4 February 1983; accepted in revisedform 14 Jzdy 1983)

Abstract-Mathematical model of zone melting of systems with variable distribution coefficient was solved using numerical methods under the condition of complete liquid mixing. Theoretical multipass distributions were predicted for one- and multi-zone processes, and parameter-sensitivity studies were made to find conditions under which the effectiveness of separation can be increased.

INTRODUmION

technique, especially in the case of variable distribution coefficient, however, it has been devoted little attentionrll]. To complete the above mentioned works, we have developed a computer program for calculation the multipass distribution of component of interest along the ingot during one-zone and multi-zone zone melting processes under the condition of complete liquid mixing for the case when the distribution coefficient is either constant or changes with the concentration of component as well as the length of the molten zone may change after each zone pass. This program was used for studying the effect of some important parameters of zone melting operation on the shape of the component distribution and on the yield of zone melting.

Melt crystallization is a separation method based on the phase equilibrium, in which the two distinct phases (solid and liquid) are formed entirely from the components to be separated, so no chemical species that require subsequent removal are added. There are several melt crystallization techniques among which zone melting is probably the bestknown one used mainly for purification. In zone melting a small volume of solid mixture to be separated is melted and this molten zone is caused to move slowly along the lengthy ingot of solid from one end to the other, either by means of moving the solid or by moving the heater which generates the molten zone. The separation is reached by the fact that the components of mixture to be separated accumulate either in the direction of the zone movement or in the opposite direction, depending on their distribution coefficients. The equilibrium distribution coefficient of the component of interest is the ratio of the concentration of the component in question in solid to that in liquid according to the phase diagram of the mixture to be separated. The separation effect produced by zone melting is usually characterized by distribution of components along the length of ingot. Each subsequent zone pass following the first, changes the distribution, i.e. contributes to the separation of components until the ultimate distribution is attained. The ultimate distribution, belonging theoretically to infinite number of zone passes, is the distribution which does not change any more upon passage of an additional zone pass. To improve the efficiency of separation, several zones may pass through the solid ingot simultaneously in the so-called multi-zone zone melting techniques.

1.1 Mathematical model of zone melting To predict the component distribution along the ingot, a mathematical model of multi-zone systems must be developed. The multi-zone operation can be carried out in two basic mode[2]. In the one mode all heaters--each producing a molten zone-travel through the ingot from one end to another. The difficulty with such a procedure is that the ingot is completely within the heating array only part of the time-in the beginning only the front of the ingot has zones and at the end only the tail end does. More practical and effective is the use of reciprocating heaters in the so-called intermittent mode. The heating array, comprising heaters of number i, is slowly moved a length equal to the distance between the heaters (h) and then rapidly moved back the same distance before the zones can freeze. In this way the melt in the first zone is transferred into the second zone, meantime the melt initially formed in the second zone is transferred into the third one and so forth. Further on we are going to study the intermittent multi-zone systems. From mathematical modelling point of view these systems can be regarded as separate one-zone subsystems of number (i + 1) each with the length of h except for the (i + 1)th subsystem which is I in the length, as shown in Fig. 1. During derivation of equations of mathematical model for multi-zone systems, we used the wellknown differential mass balance equations of one-

1. PREDICTION OF COMPONENT DISTRIBUTION There have been several papers on the prediction of single and multipass distributions resulting from one-zone zone melting processes with the asdumptions of complete liquid mixing and either of a variable distribution constant [ l-31 of a or cbefficient[4-71. There have been solutions for single pass distributions under the condition of no liquid mixing[%lO]. As for the determination of multipass distributions produced by a multi-zone zone melting 3

K. VAS~HELYI

Fig. 1. Schematic drawing

of a multi-zone

zone zone melting included the following assumptions: (I) the molten zone is completely mixed; (2) no diffusion in solid; and (3) the densities are independent upon temperature and composition [l, 21. Being zone melting primarily a purification technique, the model equations should be derived for the minor component, which is called impurity. The differential equation for multi-zone system is the following: dC,N.m -=f.E.[D_CC”._m(x)], dx

(1)

m = 1,2...,

i

(I.11

and if

D=Cp’“+‘(x+Z-h), m=12.

system divided into subsystems.

relationship can be found from the solid-liquid diagram in the following from: C, = k

C, where k =f(C,).

phase

(5)

The k-the equilibrium distribution coefficient of impurity--usually depends on the liquid concentration of impurity according to f function which is not necessarily a continuous and smooth function of C,. If the distribution coefficient is determined empirically (this is the effective distribution coefficient) the model is valid also in case of non-equilibrium state. For determination of impurity distribution after arbitrary Nth pass of zone, we must know the previous distribution formed after the (N - 1)th pass of zone. Since, the initial impurity concentration of mixture to be separated, C,, is known and uniform along the sample ingot, for the first zone pass the initial condition is given by

and

h--ltx$h

3 t .->i-1

(1.2)

C$m(x)

= CO for m = 1,2,.. ., i + 1 and 0 sx

6 h. (6)

and D=C~~‘~“+‘(x+I-h),ifh-1~x~handm=i (1.3) respectively. After the quick returning of heating array to its initial position, the melt in the last, (i + 1)th subsystem-for lack of any temperature gradient along the length-freezes with composition equal to that of molten zone, that is C~+‘(x)=C~i{h} The boundary

conditions

for OS$X&L

(2)

Impurity distributions for subsequent passes can be determined by substituting the distributions resulting from previous zone passes for C:-‘. Finally we derive the overall impurity balance of multi-zone system: i+, h C,.L= c C:“(x) dx, (7) m=, s 0 where L = i. h + I is the total length of ingot. Moreover, the yield of product of a prescribed purity is needed to know for characterizing the separation effect. The yield of product corresponding to impurity-level of p value was computed by

of eqn (1) at x = 0 arc:

Yp=& and Cy-{O)=C~-‘.m-‘{hj

for M =2,3,...,

i.

(4)

To solve eqn (1) and obtain solid concentrations from liquid ones, it is necessary to specify the relationship between C, and C,. Assuming that the freezing rate (which is aproximately equal to the movement rate of heater) is quite small to allow attaining the equilibrium at the freezing interface, the

100.

(8)

in this equation 1, is the part of ingot where the impurity concentration does not reach the value defined by the impurity-level of p. The impurity-levels are expressed in the form of relative impurity concentration related to the initial impurity concentration. Since the cross-section of the sample ingot is uniform and the solid density does not depend on concentration, the value of Y, gives the weight fraction of product of p impurity-level in per cent of the weight of the total sample.

On the examination

of zone melting

5

The mathematical model for one-zone systems can be easily obtained from the above derived multi-zone model if we take the i = 1 relationship into account in the appropriate equations.

heaters, that is when there were molten zones at the front of each subsystems except the first one. The Appendix contains the computational algorithm of the program in the form of block diagram.

1.2 Development of computer program The previously derived equations of mathematical model were solved by numerical methods. The solution of eqn (1) was approximated by a second-order Rung+Kutta formula and the integrals in eqns (3) and (7) were computed by a trapeze formula. In the case when the distribution coefficient was not constant the approximate solution of eqn (1) needed an inner iterative procedure because of the dependence of k on C,. The accuracy of calculation was checked by the application of the overall impurity balance. The algorithm was programmed in basic code on a Hewlet Packard 35 computer. The program permits to calculate the impurity concentrations at discrete values of length and the yields of product for arbitrary values of initial impurity concentration, zonepass number, length of zones, number of simultaneous zones, distance of zones, impurity-levels and for arbitrary function of k and C,. The computational results were printed out in the moment of return of

By means of the computer program, the effect of some parameters on impurity distribution and yield of zone melting can be studied. These parametersensitivity studies can help us with choosing the suitable conditions of zone melting operation in order to increase the separation efficiency. On some concrete purification task, the impurity concentration of the sample to be purified and the distribution coefficient of impurity in the sample system are given, as well as the purity of product to be achieved is usually prescribed. Therefore we confined our attention to studying the effects of so-called equipment parameters, namely ingot length, zone length, zone-pass number and number of simultaneous zones on the separation efficiency of zone melting.

2. PARAMETERSENSITIVITY

STUDIES

2.1 In one-zone systems The common effect of ingot length and zone length on the impurity distributions should be taken into

-

10’ (0’)

t

::-I/l0

12

Fig. 2. The Impurity

(b)

3

4

0

12

3

4

;i;lbii;l

distributions (C,/C, vs x) in one-zone process for various L/I ratio, coefficient, after I, 2, 5, 10 and 20 passes of zone.

dlstrlhution

‘I

x tcml at k =0.5

5

K.

V.kiR”ELYI

purity at a prescribed purification time, which is directly proportional to zone-pass number. It follows from Figs. 2 and 3 that the usage of low L/1 ratio restricts the available purity. At the same time, high purity can be obtained with application of high L/I ratio, but for high N only, which means long separation time, however. Consequently it is worth gradually decreasing the zone length during zone melting operation instead of applying the same length of zone all over the time of separation. This conclusion is supported by figures in Table I. N = 20

0

5

10

15

20

2.2 ln multi-zone systems Tables 2 and 3 show that the yields of product increase with increasing number of simultaneous molten zones at the same purification times. The excessive increasing of simultaneous zone number on a given length of ingot is restricted, in practice by the requirement that there must be exist a well-defined solid layer between two neighbouring molten zones in order to achieve separation. It can be stated from Table 4 that longer zones are more efficient than shorter ones for small N-values, later on, with increasing number of zone passes the use of shorter molten zones become more advantageous. Taking all the results into account, it can be concluded that the best separation effect can be reached if as many and short molten zones as possible are used from the beginning to the end of separation time. If only a few simultaneous molten zones may be used for the above mentioned or other reasons, the effectiveness of separation can be improved by gradually decreasing the length of molten zones during the separation.

L/L

25

Fig. 3. The yields of product as a function of L/I ratio (Y,,

vs L/1) in one-zone process, at vsrious p wdues and k = 0.5 ifter 10 and 20 passes of zone, in per cent of the sample.

account in the form of L/i ratio. To eliminate the effect of initial impurity concentration (C,), in the following we use relative impurity concentration (CJC,) instead of the absolute one (C,). It can be seen from Fig. 2 that the L/Z ratio unambigously determine the highest purity that can be obtained. With increasing of zone-pass number, the purity of product increases, although the magnitude of change in concentration is getting less and becomes negligible near the ultimate distribution. It can also be stated that at the beginning of separation, for first few passes of zone, the use of lower L/E ratio results in higher change in concentration, than higher L/l ratio makes. The curves in Fig. 3 can be excellently used for choosing the suitable zone length in the case when we want to maximalize the yield of product of a given

3. EXPERIMENTAL

RESULTS

Zone melting experiments were made to control our theoretical results. Naphthalene-/I-naphtol organic system was chosen for this aim, which system forms complete solid solution all over the range of concentration, so the function between k and C, could be found in the form of a continuous, smooth

Table I. The yields of one-zone process after 20 passes of zone l’he

yields

in

Impurity-

cent

of

the

sample

iv - 20

-levels

CEJ'Co

per

k = 0,4 + 0,5.C,

k = 0.5 L/l

=

10

L/l

=

20

L/l

-(a)*

L/l

= 10

L/l=(a)=

10-2

27

14

27

44

53

5.10-1

50

34

55

56

68

10-l

58

46

67

61

72

5.10-l

72

80

83

71

80

(a)“:

L/l

= 2.5 for

N = 1,

L/l

= 13 from

iI = 6 to

L/l

= 5 from I = 2 to B = 5, 10 and L/l

= 20 from N = 11 to 20.

On the examination of zone melting

7

Table 2. The yields of one-zone and three-zone processes at (L/i

The yields

cent of

in per L/i.1

k

=

k = 0.4 + 0,s

_

_

_

-

-

-

. C, L

i-3

i-3

i=l

I I

the eample

10

0.5

=

I) = LO

t3

tl

t2

t3

t1

tz

t3

_

_

_

_

-

16

-

-

4

-

7

28

;I

t2

_

_

-

3

18

-

7

25

-

-

13

-

11

32

53

2

24

55

8

28

55

7

6

27

Table 3. The yields of three-, five- and eight-zone processes The yields L = 15 cm.

Impurity -levels

C,/%

1 = 1

i-3 *1

t3

t2

of the cm,

sample k

1=5

_---

5.10'2

_

_

5

10-l

-

3

13

31

=

0.5

158

t2

tl

t4

10-2

5.10-1

in per cent

t3

t4

t1

t2

t3

t4 35

-

-

6

19

-

6

21

14

-

8

19

38

3

20

35

47

13

25

-

18

34

51

9

23

46

58

42

63

19

46

66

-75

23

47

69

77

Table 4. The yields of three-zone process at various zone lengths

Impurity-levels

The yields 1 - 3.

per cent

in

1=1cm

CB’Ccl

t1

t2

t3

of

I - 6 cm.

t4

*5

t1

the

aample

k - 0.5 1 - 0.5

cm

t2

f4

t3

t5

10-2

-

-

10

19

20

-

-

6

22

40

5.10'2

-

20

29

43

43

-

9

29

46

62

10-l

1

29

44

46

47

-

20

31

58

69

57

57

71

71

31

62

72

75

75

5.10-l

53

function. Solid mixtures from this system were filled into glass tubes of 1Omm in diameter. Through this sample ingot one or more heaters were passed from the top of the ingot to the bottom in vertical equipment in order to increase mixing of molten zones by means of free heat-convection stirring. Very slow movement rate of heaters (about 0.5 cm/h), which means the freezing rate of melt, was used to give enough time to attain the equilibrium between solid and liquid at the freezing interface. The separation obtained was to some extent poorer than predicted from the results of this paper. This must be attributed to the fact that the assumptions made at

modeling of zone melting were not completely fulfilled under the experimental conditions. NOTATION

C 1 x i h D k S

impurity concentration, in weight fraction length of molten zone, cm distance along the ingot, cm number of simultaneous molten zones distance between simultaneous molten zones, cm defined in eqns (l.l), (1.2) and (1.3) equilibrium distribution coefficient function between k and C,, defined in eqn (5)

total length of sample ingot impurity-Level in relative impurity concentration yield of product having impurity-level ofp, in per cent of the sample weight length of product having impurity-level of p. cm

time in tables, set maximum zone-pass

Greek symbols p

density,

g/cm3

number

N m

I

s 0

denotes denotes denotes

liquid phase solid phase initial values

zone-pass number index of subsystem REFERENCES

[I] Pfann W. G., Zone Melting. Wiley, New York 1958. 121 Zief M. and Wilcox W. R., Fractional Solid&at&z, Vol. I. Dekker, New York 1967. (31 Heringlon

E. F. G., Zone Melting of Organic pounds. Wiley, New York 1963. [4] Nelson E. T., Brooks M. S. and Armington A. F., Chew. 1964 36 931. [5] Gag&n C. G., Sobotevskij V. V. and Rode V. V., Khim. Tekhn. 1975 9(31 446. [6] Matz G., Ghan.-lng.-‘?echnik 1964 36 381. [7j Gouw T. H. and Jentoft R. E., Anal. Chim. Acta.

[8] Subscripts

denotes denotes

[9] [lo] [II]

Com-

Anal. Tear.

1967 39 383. Wilcox W. R. and Wilke C. R., A.2.Ch.E. J. 1964 lO(2) 160. Wilcox W. R., Ind. Engng Chum. Fundls 1964 3(3) 235. Singh D. C. and Mathur S. C., Sep. Sci. 1972 7(3) 243. Aleksandrov Ju. I., Martinson 1. G. and Ncvikov G. A., Tear. Khim. Tekhn. 1981 15(l) 121.

On

the examination

of zone

9

melting

APPENDIX

Black

diagram

of the computational

Determ. by Eq.( I

*or

deter-m.

ot D 1.2)

I

SUBRUTIN

i

algorithm

of

c_ Nm

-values

at

1

1.

discrete

x

by

Eq.

I11