A continuous method of zone melting

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A continuous method of zone melting (Receined 2 February 1982; accepfed 8 Februnry 1982l

From the moment Pfann[l] first applied the zone melting-it has found a very wide and various application as a process periodically conducted. On the other hand, continuous methods of conducting the melting process have not found-until now-a larger applicationl2]. The most simple in realization are pseudocontinuous methods consisting in a slantwise placing of heaters and containers with the melted material, placed on a moving band. The same effect can be obtained by heaters moving perpendicularly to the direction of the band motion. The same category of solutions comprises the method of a repeatedly applied process of normal freezing[3]. In a small scale the best solution for low-melting substances is the installation composed of an inclined plate with appropriate milled grooves and heaters in the form of cylinders rolled on the surface of the plate[3]. There is finally known the method of continuous zone melting with free spaces put between portions of the melted material[4]. The above mentioned methods of continuous zone melting are characterized by a minor productivity in comparison to the periodical melting process--conducted under the same conditions-with the application of a multisectional system of heaters. In this paper a continuous method of repeated zone melting has been proposed which allows to obtain higher yield compared to the one gained in a analogical periodical installation, a yield characbrized by a very good equalization of the concentration of impurity in the product in case of distribution coe&ent smaller than one. THE PRMCIPLE OF OPERATION AND BASIC BELATIONSAIPS In this paper a continuous method of zone melting has been proposed, consisting in countermovement of continuously formed element of the melted material and heaters, producing melted zones. The principle of operation of the installation has been shown in Fig. 1. In the extreme section of the apparatus the material contained in the melted zone is taken away at the inflow of feedill and completed by the same quantity of feed. Therefore the heater, while in position when the solidification front of the formed melted zone is in the Section I, has no refining effect. The installation is fed in a continuous way, in the same way the product of an oscillating composition is taken away, whereas the feed contained in the melted zone is periodically taken away as the melted zones pass through the Section I of the apparatus, and periodically completed with the feed. The construction of the apparatus can be solved as it is shown in Fig. I, or in the system of two countermoving disks with heaters and melting material.

The number of zone melting cycles of the above described installation will be equal to the number of heaters which pass above the element of melted feed, being at the initial moment in the Section I in the time needed for the element to pass the active length of the apparatus, which leads to the retationship n=-

The length of the product segment, after which the identical concentration distribution of impurity wilt be repeated can be expressed as follows:

a=sb. u2

As for the time needed for the repeating of identical distribution of impurity concentration in the product falls on one melted zone leaving the apparatus, so considering the steady-state operation, it is sufficient to make a balance in the above mentioned period. After taking into account that the material contained in the volume of the melted zone is being completed by the same quantity of feed, one comes to the balancing equality as below: VC,.,

of withdmwal

of melted

Fig. 1. Scheme of the principle of operation installation.

zones

of the continuous

+ V,Cl = v&l+ v=

VrC0

V,.

(3) (41

If the real distribution coefficient k will be defined as the ratio of the concentration in the solid zone to the one in the liquid zone under the assumption of the equalized concentration in the melted zone, the concentration of the melted zone leaving the installation will be

C,=4. From the dependences (3)_(5). one gets the equation Cnt6PIl

=

o 1,+1 1c-cc a

ok

’

where, in accordance with the mean value theorem

The only unknown quantity on the right side of the eqn (6) is the concentration C, at end of the calculational segment a of the first cycle of zone melting. This concentration can be determined after deriving the dependences, describing the distributions of impurity concentration. DLFl’lUBDTIONS

Section

1

k [ I,?

OF IMPWUTV

CONCENTRATION IN TEE MELTED ELEMENT

At the derivation of the dependences describing the distribution of the impurity concentration a constant value of the real distribution coefficient k as welt as the impurity concentration Ce in the feed has been assumed. At the moment, when at passing of the second melted zone the solidIcation front will be in the position x = II - 1, in the melting front the initial concentration of the function of the distribution

Shorter Communications

1344

concentration appears generated after the passing of the first melted zone. With the increase of the number of melting zone cycles-the number of discontinuity zones having tbe length equal to the width of the melted zone also increases. The scheme of discontinuity zones in each melting cycle has been shown in the Pig. 2. If the lower limit of the mth discontinuity zone is designated by xo., and the upper one by x,,,- the following relation will bc satisfied

where (17) (18) Q”,-“-P(X) =

(8)

*=,l-

where C”‘“(x) dewtes the concentration in the nth discontinuity zone of the nth melting cycle in the position z. Two ranges of solution have been analyzed: n ~(clll) and n > (a/I) for the ratio (o/l) being a natural number. In the first case the extreme discontinuity zones of the cycle n - 1 and n are related by C”‘(0) = c::;(o)

(0)

(IO)

where, as it results*from the scheme (Fig. 2) m,,, = f.

(11)

In the range R s (a/l) the following depcndences concentration distribution were obtained: for the discontinuity zone m = II

-m-2

I 2

B”-p-n-r

a=,

I&Ix+(E+m-l)l-ol’+” 0," -B.-p-,I”.“(X) I

(x + ml)’ +

{C/O-

Co} Jn_p(x)e~‘U”*+ Co

J”(X) = $ [X +(n - l)f]” - P;$;Z ;! Ix +(p - W-B,-,

(12)

- & (13)

B! = 0

(14)

- 1)1]” -P;$;z+

(19)

(21)

In the range a >(oll), (a/l) meaning the natural number, (d/l) # 1, the solution range is divided into subranges being the multiplicity of relation (all). If N denotes the number of melting cycles in the given subrange-where 1 G N 4 (a/l), the relation n=c;+N

describing the

where

B, = $(n

.(x + ml)‘+ D&n

O,%) = I,“(x).

C,‘(a) = c:?r

(~e~k)“~P

p - l)l]“-~

(9)

whereas in the second one

C/(x)=:$

[x+(n -

s $%L

+

C”m(%,m) = C:+‘(z,+,)

&

,=?/I-,

[(P - i)flP kP

(22)

is fulfilled-where c means the subrange number. It appears that for each value “I?’ one has to derive separately the relations describing the concentration distributions; moreover the form of resulting equations becomes more and more complex. The general form of the relation for c = 1 has been given in[5]. The results of calculations lead to the conclusion, that the mean concentration of impurity in the product decreases with the ratio (a/O. For example Fii. 3 shows the solution for a number of values of distribution coefficients in case of n = 2. Therefore the most advantageous from this point of view is the process operation for (all) = 1. In this special case following relations, describing the distribution of concentration, hold:

(13 C,(x) = e-‘“‘)’

Ig &

k"-P{Cp(0)- CO}(T)“-’ +Co

(2%

for the remaining discontinuity zones where {C.(O) - GJ = {C.-i(O)k”-“(C,,(O)-

Co1ek -

I-*-I ,z, &

C,,}

Gm C,(O) = Cf.

c,,

o

a

(25)

1

0 n - number

of melting

m- number

of discontinuity

Fig. 2. Scheme of the designation

of discontinuity

cycle zone

zones.

Fig. 3. Dependence of the mean product concentration values of k and (a/l) for n = 2.

on the

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1345

k= l;C,,,,,,=C,

‘Ml3 GENERAL FORM OF SOLUTION, THE CALCULATIONAL SIhU%lFtCATIONS

k+O;C,,,,,,=O

It can be proved [5] that the expression

takes the form

kY,,P I ; C,,,.,,= 1

R

TID

C,(x) dx = {Cr - C,} Y. + C, T

where Y. denotes

the function of process parameters for the given number of zone melting cycles

Yn= [f(k;)]n The eqn (6) can bc transformed

(26)

calculated

127)

The refining process carried out in the way proposed in this paper is characterized by some interesting properties; independently of the number of melting cycles the concentration at the begjnnittg of the computational section “(1” is equal to kCa, in the range k < 1 one gets a product with a very well equalized concentration of impurity along the melted element. The last property is illustrated in the Table 1.

to the form

COMPARATIVE CALCULATIONS If one assumes as a criterion the ratio of the quantity of product and the quantity of feed in the melted zone removed from the installation, the continuous process for (u/l) = I is

C mean= l-~(l-k)-kjrll”ii]CO. [ n The function

I’, has two properties: lim Y. _a I 11-a

(2%

lim Y, = m. Y-al

(30)

After taking into consideration the properties of Y. (eqns 29 and 30), from the eqn (28) the solutions for special cases result as well as calculational simplification as follows: Table 1. Synopsis of extreme

equivalent to the periodical process carried out at (all) = 2. The comparison of mean impurity concentration in the product in both processes is shown in Fig. 4. Because of the identity of the compared processes in case of n = 1, the concentrations in various melting cycles were referred to the mean concentration, obtained for a single cycle of zone melting. The calculations for batch process were performed

by use of the method described in [63. The comparison with the periodical system separating the feed in the same relation as in the continuous Installation, (by

values of impurities’ concentrations

in the product for several values of distribution

coefficient

-

---

Continuous

CES Vol. 38. No. LN

of mean concentrations

of the product

zone mettlng

Botch zone Ich

Fig. 4. Comparison

(34)

-

melting

n

obtained in the continuous

process and in the batch one.

1346

Shorter Communications

assumption of removal of the material contained in the melted zone, after the nth melting cycle in liquid state) and by the use of a block composed of n heaters located at an identical distance as in the continuous process (by observing the same width of the melted zone) leads to the dependence Z=(n-I)(l-d)+d

(35)

where L d=G=l+b

1

(36) a

The speed of the shifting of the refined material is formulated follows u, = ud.

as

(37)

Hence, the conclusion that the maximum yield can be obtained for the highest value of the dimensionless quantity d. CONCLUSIONS (1) The yield of the proposed continuous method in comparison to the one of the periodical process increases with the number of melting cycles. At the same time for n = 2 the yields are identical-independently of the process parameters. (2) The continuous method in the range k< 1 leads to an equalized impurity concentration along the melted element. The uniformity of the concentration distribution of impurity rises with the increase of the number of melting cycles and decrease of the value of the distribution coefficient (Table I). {3) Because of the mean impurity concentration in the product, taking into account the advantages of the process continuity, one can consider the method as being competitive in the range k > 0.5. A small modification of the installation, consisting in extending the active length of the apparatus and installing a larger number of sections, taking away the material contained in the melted zone, assures the usefulness of the method for a far going purification of the feed, without a manifolding of the number of apparatuses. A. BRYCZKOWSKI Institute of Chemical Processing of Coal Zabne. ul. Zamkowal, Poland

Chmicd Enslncrring Scimcr Vol. 38, No. 8. pp. IMC,MS. Printed in Great Britain.

Falsification

NOTATION

calculational length of the segment of product, after which the distribution concentration is repeated b distance between heaters natural number concentration C”“W concentration in the mtb discontinuity zone of the nth cycle in the section x d dimensionless quantity, characterizing the effect of operation parameters on the yield of installation k distribution coefficient I length of melted zone L active length of apparatus m index of discontinuity zone number of cycles of zone melting IG number of melting cycles in the subrange c u velocity of displacement of heaters with respect to the melted material velocity of material UI velocity of heaters u2 x coordinate of the position of solidification front within the calculational length a V volume Z ratio of the yield of the continuous installation and the batch one 0

Subscripts I refers m refers n refers s refers 0 initial

to the to the to the to the value

liquid phase discontinuity zone number of cycles zone melting feed

REFERENCES

[ll Pfann W. G., I. Metals 1952 194 749. [2] Selecki A., Rordzielanie hfiesranin-Metody Niekonwencjonalne. WNT, Warszawa 1972 (in Polish). 131 Parr N., Zone Refining and Allied Techniques, George Newnes, London 1956. [41 Kennedy J. K., Rev. Sci. Instrum. 1962 33(3) 387. [S] Bryczkowski A., Proce Instytutu Iniynierii Chemicznej Politechniki Warsznwskiej 1980 9(4) 323 in Polish). [6] Bryczkowski A., Ini. Chem. i Proc. 1981 Z(2) 259 (in Polish).

arm-2509183 53.03+ .a, 0 1983 PerBamonPress Lid.

~83

of kinetic parameters

by incorrect

treatment

of recirculation

reactor data

(Received 28 June 1982)

In a number of reviews the recirculation reactor is rated very highlyfl-31; and compared to the fixed bed reactor it does, indeed, have several indisputable advantages: Isothermal operation is easily achieved and external resistances to mass and heat transfer are largely avoided. Also, it is often claimed that interpretation of experimental data from a recirculation reactor is easier than is the case for a tubular reactor-the reason being that the rate is measured at one particular conversion which is supposedly found in every point of the reactor. In two recent experimental investigations on widely different gas phase catalytic reactions we have been confronted with situations where an unsophisticated treatment of the raw data from a recirculation reactor would have led not only to inaccurate results for the kinetic parameters, but to qualitatively wrong results which could not have been improved by a cor-

rection procedure, e.g. as suggested by the graphs in Levenspiel ([141, P. 148). Since others might get involved in similar difEculties we have collected our experience from these investigations, and in the present note we give the results in a graphical form which can be used to avoid the di&ulties by proper experimental planning.

tNFJi.UENCE

OF RECYCLE RATIO ON REACTION REACTION CMDEII

RATE

AND

Figure 1 shows a recirculation reactor with catalyst weight W, recycle ratio R, inlet mole fraction yo, fractional conversion al after the preconverter and a after the reactor. The inert dilution is large enough to ensure that the volumetric flow vg to the preconverter is the same as leaves the reactor.