by G. BRAUN 548.522

VOL. 12 No. 5 OCTOBER 1957

Philips Research ReportsEDITED BY THE RESEARCH LABORATORY

OF N.V. PHILIPS' GLOEILAMPEN FABRIEKEN, EINDHOVEN, NETHERLANDS

R 328 Philips Res. Rep. 12, 385-414, 1957

MATHEMATICAL TREATMENTOF A TYPICAL ZONE-MELTING PROCESS

SummaryThis paper deals with the mathematical theory of a typical exampleof a zone-melting process as described by Van den Boomgaard in apreceding Report. The process in question aims at the realization of amost homogeneous distribution of a solute (which may be volatile) inan ingot. This is effected by a large number of identical steps andamounts to an extension of the procedure proposed by Pfann andReiss. The intermediate distributions tending to the limiting one areconsidered in detail, as well as the number of steps necessary to arriveat such a distribution. The mathematics determine the asymptoticbehaviour of the solution of a system of differential-difference equa-tions by means of a generating function.

RésuméCet article considère la théorie mathématique d'un exemple caractëris-tique d'un procédé de fusion par zone comme a été dëcrit par M. Van. den Boomgaard dans un article précédent de ces Reports. Le procédéen question concerne la réalisation, autant que possible, d'unedistribution homogène de produits (même volatiles) dissous dans unlingot. La méthode, qui est effectuée par un grand nombre de procédésidentiques, revient à une extension de celle proposée par Pfann etReiss. On s'occupe tout spécialement des distributions intermédiairesqui s'approchent de la distribution finale, ainsi que du nombred'itérations nécessaires pour atteindre une tellé distribution. La më-thode mathëmatique détermine Ie caractère asymptotique de lasolution d'un système d'équations différentielles et de différences, àl'aide d'une fonction génératrice.

ZusammenfassungDieser Artikel behandelt die mathematische Theorie eines typischenZonenschmelzverfahrens, so wie es von Van den Boomgaard in dieserZeitschrift beschrieben wurde. Dieses Verfahren hat den Zweck, einemöglichst homogene Verteilung eines löslichen Stoffes (der auchfiüchtig sein kann) in einem GuBblock herzustellen. Die Methode,die aus einer grollen Zahl identischer Pro zesse besteht, ist eine Aus-dehnung des Verfahrens von Pfann und Reiss. Es werden die Ver-teilungen, die einer Grenzverteilung zustreben, im einzelnen unter-sucht. und es wird die Zahl der Prozesse berechnet, die notwendig ist,urn eine derartige Verteilung zu erzielen. Das mathematische Problembesteht in der Bestimmung des asymptotischen Verhaltens der Lösungeines Systems von Differential-Differenzengleichungen mit Hilfe einererzeugenden Funktion.

386 . G. BRAUN

1. IntroductionIn the last few years several variants of the zone-melting process, origin-

ally described by Pfann 1), have proved very effective to the purposc ofeither purifying a given substance A (elimination of impurities of amaterial B) or producing a prescribed distribution of a substance B inthe substance A 2). Among the distributions which can be arrived at bymeans of such zone-melting processes, the unifor~ distribution of B in Awith a certain concentmtion is of special importance. .

In this paper some mathematical problems connected with a special typeof such a process 3) will be discussed. In spite of simplifying assumptions(which go back to Pfann himself), this model is very representative andshows the essential features' of the process; this justifies a detailed mathe-matical treatment.An investigation of similar models used for the same purpose would

probably give no new aspects.Let us consider a bar of solid material A of length L + 1, in which a

volatile element B is solved with a certain initial distribution. A zone oflength 1 is molten and is moved at constant velocity v through the bar(from the left to the right, say). When the zone has reached the end of thebar, the velocity is reversed and the zone is moved from the right to theleft, and so on. In order to avoid an escaping of the volatile element B out ofthe molten zone, the whole process is performed in a closed vessel in anatmosphere ofvapour, ofthe element B, which is held under constant pres-sure, and therefore, at constant temperature. The passing of the moltenzone through the bar is carried out as often as necessary in order to approachthe final distribution within prescribcd limits, the nth course being carriedout from the right (left) to the left (right), if the (n - l)th course wasperformed from the left (right) to the right (left).'

As explained by Reiss 2), there are three effects, owing to which thecomposition of the melt, and thus of the solid in equilibrium with it, maychange, viz: (1) reaction between liquid and vapour, (2) segregation at the"segregating" end of the molten zone, (3) dissolving of solute at the "dis-solving" end of the molten zone. This situation is explained schematicallyin fig. 1.

--- ....v

solid ~ solid::.~ "

xL l

92461

Fig. 1. Schematic explanation of the zone-melting process.

MATHEMATICAL TREATMENT OF A TYPIÇAL ZONE-MELTING PROCESS 387

In accordance with Pfánn's model the following assumptions are made:(1) the solute B is at any moment homogeneously distributed in the liquid A[instantaneous diffusion in the liquid), (2) the diffusion in the solid isnegligihly slowand there is no reaction hetween solid and gas, (3) theequilibrium between various phases in the gas (for instance, atomic andmole-cular), and between the liquid and the solid is reached practically at once, and(4) the concentration ratio k, fixing the distribution between the liquid andthe solid, is a constant (smaller than 1) and independent of the concentration.

A fulfilment of the assumptions (1), (2) and (3) restricts the velocity v of thezone moving. Hence v is limited by the diffusion coefficients of B in the liquidand the solid, and by the velocity of the reaction at the liquid-solid interfaces.Assumption (4) is fairly well satisfied in most cases. It holds, for instance,rigorously if the solubility curve and the freezing-point curve may be approx-imated by straight lines; this applies, in particular, to small concentrations.

The calculations will refer to a bar of cross-sectional area of 1 cm2• Thecoordinate x of the "segregating" surface of the molten zone is alwaysmeasured in the +v direction. The abbreviations used represent thefollowing quantities:Cn(x) the concentration of B in the solid A after the nth course; k-1Cn(x)the coneentration of B ill the liquid during the nth course; Cequ theequilibrium concentration of B in the liquid as determined by theequilibrium between Bliq and Bgas; the quantity k1 ~ 0 the liquid vapourreaction coefficient; :finally,k-lda~)(x), k-lda~)(x) and k-lda~)(x) the quant-ities of the substances B coming into the molten zone owing to the effects (1)to (3), respectively, during the displacement ofthe zone over' a distance dx.

The levelling effect of the zone melting is due to the coexistence of thetwo flows, -k-lda~)(x) and k-lda~)(x), which are different in the case ofunequal concentrations of B in Asol at x and x + Z. These quantitiesmentioned represent the masses passing from the liquid to the solid throughthe "segregating" surface, on the one hand, and from the solid to the liquidthrough the "dissolving" surface, on the other hand. .

This coexistence produces a transport ofB through the molten zone inthe direction of decreasing concentration, i.e., a levelling effect. The reac-tion with the gas phase determines the equilibrium concentration andinfluences the transport effect.

2. Derivation of the equations

The increment, k-1dan(x), of the quantity of B in the molten zone isequal to the sum of the elementary increments:

1 1 . 1 1k dan (x) =k da~)(x) +k da~)(x) +k da~)(x). (1)

388 G.DRAUN

In view of the definitions of k, kl' Ccqu and v, the following equationshold:

(I)or

(2)

(k-lda~)(:I;) is proportional to kl' to the length of the molten zone 1, tothe reaction time dt, and to the 'difference between the equilibrium con-centration Cequ and the momentary concentratien k-lCn(x) = (ljkl)an(x));

1k da~)(x) = - Cn(x)dx,(11)

or

(3)

(-k-lda~)(x) is equal to the quantity of B in the sedimenting piece oflength dx, i.e., Cn(x)dx);

~ da~)(x) = Cn--l (L-x)dx,(Ill)

orda~)(x) k-d-x- = T an--l (L - x)

(k-lda~)(x) is equal to the quantity ofB in the melting piece oflength dx,i.e., Cn--l(L-x)dx; this expresses the property ofidentical concentrationsin the cross-section X + 1 before the nth course and in the cross-sectionL + 1- (x + 1)= L - x after the (n -l)th course).

(4)

Concenirutions

92462

Fig. 2. Schematic representation of the concenti ation distributions Cn(x), Cn-l(L-x), etc

MATHEMATICAL TREATMENT OF A TYPICAL ZONE.MELTING PROCESS 389

The situation which leads to ,eqs (1) to (4) is shown schematically infig. 2. Substitution of (2) to (4) in (1) leads to

dan(x)~ ~ a an(x) = A + B an-l(L-:-x) (n . 1,2, ... ), (5)

where the following abbreviations have been introduced:

a = k1/v + kil, B = kiL and A = k1lkCequlv, while Co(x) = ao(x)ll

denotes the given initial concentration in Asol, i.e., the concentration beforethe first course.

In order to derive the boundary conditions, we bear in mind thatfor n = 2, 3, 4, ... the concentrations Cn_1(x) = k-1Cn-l(L - 0)(L + 0 ~ x ~ L + l) of Bin Asol after the (n - l)th course is equal to theconcentration k-1Cn(0) of B in Aliq. This yields the equations:

(n = 2, 3,4-, ... ) .

For n = 1 the concentration k-1C1 (0) of B in Aliq is equal to the mean valueL+I

of the initial concentratien (Ill) f Co(x)dx, i.e.,L

L+I

a1(0) = ~ f ao(x)dx.L

For n -+ 00 the system tends to a state of equilibrium. We then getan(x) '"'-' an_l(x) '"'-' ac:oand ac:o= AI(a-B). Putting for brevity ç = ax,L1 = aL, f3 = 1- B]« (0 ~ f3 < 1) and 1'n(x) = ac:o.:_an(x), the systemof eqs (5) to (7) can be written as

d1'n(çla) . Bdç + 1'n(çla) = -; 1'n-l((Ll - ç)la) (n=I,2,3, ... ), (8)

1'n(O) = 1'n-1(L) (n = 2,3,4, ... ),and

L+I

. k f1'1(0) = ac:o(1 - k) +T 1'o(x)dx.L

The quantity l-l1'n(çla) is the deviation of the concentration Cn(~/a)from its equilibrium value, which implies that 1'n(x) -+ 0 for n -+ 00. Theproblem amounts to the determination of the rapidity of this tending tozero, that is, one wants to know how many courses must be carried out inorder to make the relative deviation 1'n(çla)lac:o smaller than a prescribedquantity 8. J?Q~ further çalçulations the equation~ .

(6)

(7)

(9)

(10)

390 G.BRAUN

(n = 2, 3, 4, ... ) (ll)

may be noted too.Before starting with the calculations, it may be useful to get a survey of

the essential parameters of the problem and the possibility of varying theseparameters. The velocity v can be chosen in a certain range but is limitedby our assumptions (1) to (3) (sec. 1), and by the finite value of the heatavailable per second for the melting process. The coefficient ~ = k,./~(~= VIS; V is the volume of the bar, S the surface area of the bar exposedto the liquid-gas reaction) depends on ~, which can be varied; k and kr areconstants of the system A-B. The other parameters of the system, namely,uo(x), 1,L, Cequ, depend on the geometry and the vapour pressure of Bgas;they may be chosen rather arbitrarily.

For small n the system of eqs (8) to (10) can be solved by successiveintegrations. However, this method becomes prohibitively involved forlarger n. It is therefore not suitable to the determination of the asymptoticbehaviour for large n.

The method of generating functions proves to be very adequate for thispurpose.

Let us define a generating function by

(12)

z = x + iy being complex, and Izl chosen so small that the series in the.right-hand side of (12) converges for all ~ throughout [0,L1]. (This x shouldnot be confused with the coordinate used in the beginning of this paper.)Since 7:n(~/a) -+ 0 as n -+ 00, Izl may be chosen different from zero. There-fore, G(~,z) is analytic in a region around z = O. A combination of (8) to(10) leads to .

àG(~,z)à~ + G(~,z) - zG(L1 - ~,z) = (BJa) 7:0«Ll - ~)Ja), (13)

(14)

and

. à2G(~,z) - (1- Z2)G(~,Z) = F(~,z); là~2

where

F(~,z) = (BJa) ~à7:0«L~; ~)Ja) :- 7:0«L1- ~)Ja) - Z7:o(~Ja)~.

(15)

The further calculations are connected with the conformal mappingw = 1"1- z2, which, therefore, is considered in the next section.

Let us put w = u + iv = w(z) = -vI- Z2 • (16)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE-MELTING PROCESS 391

3. Discussion of the conformal mapping w = -v1- Z2 '

To make w one-valued, let it be defined by,w,(O) = 1 and by analyticcontinuation in the plane of z cut along the segments (- 00, -1) and(+1, +oo) (principal value). This definition is equivalent to

w = 11- z21iexp [ti arg (1- Z2)],in which arg(I- Z2) = arg(I- z) + arg( 1 + z) and [arg (1 =+ z)l < n.

By means of this transformation the half-plane Re z > 0; cut along thereal x-axis from z = 1 to z = +00, is mapped on the half-plane Re w > 0,cut along the real u-axis from w = 1 to w = + 00. Figures 3a and 3b showthis conformal mapping. The curve c~is mapped on the curve (c~). Inthis case z = -VI- w" (principal value), and the mapping w = w(z) isinvolutory.

Imz

w..-iz

z-plane

w...-iz

w=+i~ w=-i~+1

w=+i~ Rez

92463

Fig. 3a. Conformal mapping w = (1 - z2)i, z-plane.

Also the half-plane Re z < 0, cut along the real x-axis from z = - = toz = - 1, is mapped on the half-plane Re 10 > 0, and the curve C; on (C;).In this case we have z = --v1- w2 and now the transformation w =w(-z)is involutory. ' ,

The functions -VI± .w are likewise defined by the principal values of thesquare root. Hence -VI- w2 = -VI+ w -VI- 10 for all w in the cut w-plane.Accordingly -vI- w = =Fi -Vu - 1 if 10 = U ± iO (u ~ 1).

FQr fl1rth~r çalçulations it may be noted tga,t a 'transfçrma'tion z -? :«*

392 G.BRAUN

W-pTane W-plane

(C2) (C2)

z= -i UV2=1 z: ilrw2=1+1 z=i~ +1 z=-iVT

92464

Fig. 3b. Confermal mapping w = (1 - Z2)t, w-plane.

(z* = conjugated complex to z) corresponds to a transformation w -~ w*,-VI·1= w --+ -VI 1= w* = (-VI 1= w)*.

4. Solution of the system of equations (13) to (I5)

Equation (15) has the solutione

G(~,z) = ~gl(Z) + 2~ f F(1],Z)e-WT}\d1]~ ewl;+o

+ 19,(,) + 2~ /F(n,,),"'dn l,~.,.e

The functions gl(z) and g2(Z) can be determined with the aid of (13) and(14). It is easy to prove that every solution of (15) satisfying (13) for ~= 0solves the latter also for any ~. We may therefore confine ourselves to (13)for ~ = 0 and (14). It is convenient to introduce the abbreviations

(17)

(18)

These quantities are connected by the relations

hl(Z) -vl- w 1= h2(Z) -VI +w ewL, =

= B .0(0) ~-VI- w ewL, 1= -VI+w~ ± B .0(L) ~-Vl +w ewL, 1= -VI - w~. (19)a a

1IIATHEMATICAL TREATlIrENT OF A TYPICAL ZONE-MELTING PROCESS 393

Here and in the sequel, the upper and lower signs correspond to Re z > 0and Re z < 0, respectively. Substitution of (17) in (13) (for ~..:....0) and (14)yields the system of equations

=F-VI+ w gl + -vI':_ w e-wL,g2=

= B 2_ ~-00(0) 1'1- w =Foo(L) -VI+ wt,.Ó: a,2w

(1- (J - z ewL,)gl + (1- (J - z e-wL')g2=(20)

which has the following solution:

gl(Z) =1 ~ B ,/- B ,/_w-{J

= ( --rI-W1:l(O)=F-rI+w--T.o(O)+Nl,2 tv) a a 2w

B ,/- w + {J ,/- w - (J L ~+ -yI-w-- oo(L) ± yI+w--ew 1 h2,a 2w 2w (21)

g2(Z) =1 ~ B ,/-- L B ,/-- w + {J L=N ( ) =F- rI+wew'ol(O)--rI-w--eW,oo(O) ±

1,2 w a a 2w

B w-{J w-{J ~±- -VI+w-- ewL, oo(L) ± -vI+w __ ewL,hl .a 2w 2w

Here Nl,2(W) is an abbreviation for

N1,2(w) = (w + (J) -vl- w =F(w - (J) -VI+ w ewL, (22)

where the subscripts 1 and 2 correspond to the - and the + sign, respectiv-ely. By subsitution of (21) in (17), we get

~ L,

G(~,z) = 2~ ~fF(1'J,~) ew(hld'17+ f F(TJ,z) eW(q-~ldTJ~+o e

+ N \ ) ~±w - {J-VI+ w ewL, [h2ewg+ hle-we]-1,2 w ( 2w

_ B 01(0) [-vl- w ewg± -VI+ w ew(Ldl] =Fa

=pB oo(O)~ [(w-{J) -vI+weWg± (w+ (J) -vI-wew(Ld] +a 2w

+ B oo(L) 2_ [(w + (J) -vI- w ewe± (w - (J) -VI+ w eW(L,-el]~. (23). a 2w. ~

394 G. BRAUN

In view of the definition of w, the integrals in (23), ~(z) and h2(z), areanalytic in the cut z-plane (see figs 3a, 3b). All occurring square-roots areanalytic likewise there. Hence G(~,z), too, is analytic throughout the cutz-plane with the exception ofthe zeros of NI(w) and N2(w) and of the pointw = 0 (z = ±I).

As already mentioned in sec. 2, there exists a region arou~d z = 0 in .which G(~,z) is analytic (N1.2(W) =1= 0 at z = O).'Cauchy's formula can there-fore be applied and we get

( ) _ (B )11-1 1 [G(~,z)dz.'On ~/a - - -. a 2ni . zn

Ct

(n = 1,2, ...), (24)

where Cl denotes a small circle around z = 0, lying in this region.Since the characteristic features of the problem d~ not essentially depend

on the particular choice of the initial concentration, we specialize the latterto be independent of x, i.e., ao(x)= ao'Equations (7) and (10) then trans-form into

al(O) = kao,A

To(~/a)= ---ao = A2'a-BA

'Î'1(0) = -- - al(O) =Al'a-B

(25)

After some reduction we arrive at the following result:

G(~,z) = (B) _1_ Xa I-z

X ~A2 N 1 [AI(I-z)-A2(I-,B-z)] [yI-w ewe±yI+w eW(Ld)]~.1.2(W) (26)

All symbols introduced so far will henceforth refer to this special case..Substituting (26) in (24) and carrying out the integrations for the first term,we are led to

(n=I,2, ... ), (27)

where Fn(~,z) is an abbreviation for

AB2 AB [AB A ( A)]i2(~/a) = ( ) 2 +- eÇ-Ll+ - +- e-L1 1 - - e-e> O.a - B a 2a2 a2 a 2a

(30)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE-MELTING PROCESS 395

The same transformations can be applied in the case of an arbitrary initial distribution.The term containing the integrals, which corresponds more or less to (Bja) A2j(l - z), canhe calculated by straightforward integration. The other terms in (23), which depend onN1•2(W), can he evaluated in the same manner as established here with the term containingFn(~,z).

Before continuing the general theory, we shall discuss some simpleproperties of 7:n(~/a) in the case aa = 0 and derive a theorem of the theoryof functions which will he used in the subsequent sections.

5. Discussion of an important integral relation

Integration of (8) an application of (9) results in the relation. e

7:n(~/a) = (~)f e(1j"f;l't'n-d(Ll-'11)/a]d'11+'t'n-l(L)e-f; (n=2,3, .... ). (29)o .

In the special case aa = 0, Al = A2, we write Tn(~/a) instead of 7:n(~/a).The first two functions in(~/a) are of the form

AB Ail(~/a) =_- +- e-e > 0,

a-B a

The following interesting properties can he derived next:(a) All in(7:/a) (n = 1,2, ... ) are non-negative. This can he proved by meansof (29) and (30).

. (h) The relation (31)

holds for n ~ 2. For n = 2 equality is possible: Cl has the value Cl =(1- {J)+ {Je-Lt.Proof: Determine Cl > 0 by the condition

Min [ï'2(~/a) - Cl il(~/a)] = 0 (0 ~ ~ :::;;Ll). It then proves to have theindicated value. For n ~ 2 eqs (29) 'are also valid for

in(~/a) - Cl 7:n-l(~/a)instead of 't'n(~/a), where the former quantities are non-negative too. Thesame reasoning as in the case of remark (a) completes.the proof.(c) Likewise we can prove the inequality

'in(~/a) < C2 ï'n-l(~/a), (32)which holds for n ~ 2. For n = 2 equality is possible; C2 denotes thequantity C2 = Cl + t(1 - {J) (1- e-L1)2/[(1 - {J)/ {J+ e-Lt] > Cl'

Proof: Determine C2 > 0 by the conditionMax [i'2(~/a) - C2 il(~/a)] = 0 (0 ~ ~ :::;;Ll), andproceed as in case (h).For Ll ~ 1, C2 ~ Cl and i'n(~/a)/in-l(~/a) is known. ,

396 G.BRAUN

6. A theorem of the theory of functions

Theorem: Let f (z) = u + iv be analytic in a region G, C a simple closedcontour traversed in the positive direction while f(z) =1= 0 on C. Denote byal' a2, ••• , an = al the ordered sequence of zeros of Re f = u on C, if any.Denote by Llcargfthe variation of argfalong C, by Lli argfthe variationof arg f along C between ai and ai+h and by N the number of zeros offwithin C. Then the following relations apply:

1 f f' (z) 1. ,1 nN = - -- dz = - LIc argf = - ~ Lli argf,

2:n;i f(z) 2:n; 2:n;i=1c

(i) (33)

(ü) (34)

Proof: The first part of (i) is the theorem of the logarithmic residue 4), thesecond part follows from the definition of Lli argf; (ü) is proved by the state-ment that Lli arg f :;:;;;:n;holds. 5)

7. Further discussion of the relation (27)

We now consider the quantity Fn(~,z) as defined by eq. (28). Theoccurring of the expressions w = (1 - Z2)t and (1 T w)! suggests a many-valuedness of this function in the z-plane: however, a closer inspection ofFn(~,z), taking into account the definitions of the terms wand (1 T w)!given in sec. 3, shows that this function is single-valued in the whole z-plane.In order to prove this property, we investigate the connection relations ofFn(~,z). Once going around the branch points z = ± 1 means a transforma-tion

z --+ z, w --+ -w, NI,2(W) --+ ± e-L,W NI,2(W) and[(1- w)! ewe ± (1+ w)t e'·(Ldl] --+ ± e-L,W [(1- w)t ewE± (1+w)t ew(L,-el],

so that Fn(~,z) remains unaltered. Therefore, Fn(~,z) is meromorphic andsingle-valued in the whole z-plane.

Let us discuss the quantity

~ f Fn(~,z)dz-~ f Fn(~,z)dz, (35)2m .2m. c. . C,

C2 being the closed curve, composed of the parts C~, ... C~, as shown infig. 4. As Cl lies within C2, the quantity (35) represents the sum of the resi-dues of Fn(~,z) at the poles of Fn between Cl and C2• For lzl ~ 1, Fn(~,z)becomes very small; to be more specific, we find, after somereductions,

(Im.z~ 0), (36)

so that the integral (lJ2:n;i) f. Fn(~,z)dz vanishes as C~recedes to infinity. c,

(37)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE-lIIELTING PROCESS 397

(This shifting to infinity of the whole contour C2 may be applied, sinceFn(~,z) is single-valued in the z-plane.) The calculation of the residucsbetween Cl and C2 requircs the knowledge of the poles of Fn(~,z) in thisregion; since the point z = 0 does not lie in this rcgion, the poles of Fn(e,z)that remain to be considered are the zeros of N1•2 (w) and the points z = ± 1.The main problem of the further investigation concerns the ·discussion ofthese very zeros.

z-plane

Imz

CT ~'Rez

92465

Fig. 4. Cut z-plane with contour C2 = Cl + C22 + ... + C26.

According to sec. 5, all in(~/a), (n = 1,2, ... ), are nonnegative, A weU-known theorem of the theory of functions 6) then states that the singularityofthe generating function G(~,z) (for ao = 0) closest to the origin is situatedon the positive real axis of N1,2(W) (Nl(w) in this case). Moreover Nl(w)and N2(w) do not depend on the initial concentration but only on (3 andLl; the same result therefore holds for any initial distribution and, a fortiori,

for the initial distribution ao(x) = const =1= o. A calculation shows the real-valuedness of N1(w(z» for 0 :::;;z ::::;;1, whereas Nl(w) < 0 at z = 0 andNl(w) > 0 at z = 1. As a consequence the zero of Nl(w(z» in question, Zo

say, is situated between 0 and 1. The definition of the radius of convergenceof a Taylor series implies

which already gives some information about the asymptotic behaviour of7:n(~/a) as n -? 00. However, much more information is contained in the

398 G.BRAUN

formula (86) to be derived later on. As a matter of fact the latter includesthe present result. Our further analysis will be based on the value of thequantity (35). In fact, the knowledge of (35) enables us to replace the inte-gral in (27) by the difference of an integral along the contour C2 and the sumofthe residues at the poles of Fn(g,z) between Cl and C2• As shown above,the contour C2 does not contribute at all (provided it is shifted to infinity)so that only the residues need to be calculated. The discussion of the zeros ofNl•2(w) will be accomplished in secs 8 to 10. The final evalutation in secs11 and 12 is based on the results summarized at thc end of sec. 10.An under-standing of these latter results, without their detailed derivation, will sufficefor a reader only interested in the practical consequences of the generaltheory.

8. Discussion of the zeros of Nl,2(W)

The following remarks are useful for the general argument.(a) From the definition (22) of Nl•2(W) we get the relations

(Rez~O). (38)

The transformation z --* z* involves w _,.. w*, (1 =f w)t _,.. (1 =f w*)t =[(1 =f w)t]*, Nl•2(W) --* Nl•2(W*) = (N1,2(w»*, and Fn(g,z) --* Fn(g,z*) =(Fn(g,z»*. According to this remark a zero w of N1•2(W) proves to be con-nected with the other zeros, -w, w*, -w· of Nl•2(W), all having the samemultiplicity.(b) The behaviour of N1,2(W) can be easily investigated on the real andimaginary axes. Both functions ,Nl(w) and N2(w), have zeros for 0 ~ tv < 1and no zeros for w > 1. In addition N2(w) has a zero for w = O. In the samemanner Re Nl•2(W) are shown to have no zeros for w > 1. However, thereexist zeros of Nl•2(W) on the imaginary axis. All these statements followatonce from the definition (22).(c) The functions Nl,2(W) get the following simple forms:

(Re z~O), (39)

if Iwl ~ 1. Here wt = Iwl leiiargw, -1(; < arg w ~ 1(;.Formula (39) can bederived straightforwardly using. the relations (-w)t = 11vlieiiarg(-w>,-1(; < arg (_·w) ~ 1(;.

As a consequence the functions Nl•2(w) have no zeros for sufficientlylarge Iwl. The remarks (b) and (c) imply that-the contours C~ and C:, ifvaried arbitrarily, while keeping fixed the point w = iv, do not cross asingularity of Fn(g,z), provided Iwl is sufficiently large on C~ and C:.(d) We choose two positive numbers c and M,M being an integer, and bothnumbers ~ 1. Let.

MATHEMATICAL TREATMENT OF A TYPICAL ZONE-MELTING PROCESS 399

nu = number of zeros of Nl(w) in 0 < u <: c, 0 < v < 2MnJLl'nl2 = number ofzeros of Nl(w) in u = 0, 0< v < 2MnJLl'nl3 = number of zeros of Nl(w) in 0 < u < 1, v _O.

Further, let

n2l = number of zeros of N2(w) in 0 < u < c, 0 < v < 2MnJL1'n22 = number of zeros of N2(w) in u = 0, 2nJL1 < v :::;;;2MnJL1'n23 = number of zeros of N2(w) in u = 0, 0< v :::;;;2nJLl'n24 = number ofzeros ofN2(w) in 0 < u < 1, v = 0,2 n25 + 1 = number of zeros ofN2(w) for u = 0, v = 0 (This zero is of oddmultiplicity in view of (38)).Since the functions N1•2(W(Z)) have no zeros for Re z = 0, the parts ofthe

contours C~,C; (see fig. 3a) parallel to the imaginary axis may be shiftedso as to coincide with the imaginary axis (el ~ 0), so that C2= C~+ C;.(e) Let us denote the closed curve as drawn in fig. 5 by (Ca)' In view of the

92466

-1 +1

W-plane

Fig. 5. Cut se-plane with contour (Cs),

remarks (b) to (d), we may conclude that the number of zeros of N1(w)in the interior of (Ca) is obtained by adding the number of zeros of N1(w)in the followingsets:(i) the interior of (C~), containing 2nll + nla zeros;(ü) the interior of the image of (C~) with respect to the imaginary axis,containing 2 ~l + nl3 zeros;(üi) the interval -2MnJLl < v < 2MnJLl' u = 0, containing 2 nl2 zeros.

400 G. BRAUN

The function N1(w) has therefore

4nll + 2n12+ 2n1a (40)

zeros in the interior of (Ca). In the same manner, the number of zeros ofN2 (w) in the interior of (Ca) equals the sum of the number of zeros of thisfunction in the following sets: .(i) the interior of (C;), that is, 2 n21 + n24 zeros;(ii) the interior of the image of (C;) with respect to the imaginary axis,that is, 2 n21 + n24 zeros;(iii) the interval-2MnjLl < v < 2MnjL1, u = 0, that is, 2 n22 + 2 n2a +2 n25 + 1 zeros.

The function N2(w) has thus

(41)

zeros in the interior of (Ca). From (40) and (41) we conclude that thefunction

has

N = 4(nll + n21) + 2(~12 + n1a + n22 + n2a + n24 + n2S) + 1 (43)

zeros in the interior of (Ca).

Im(ciJw

2 w-plane<,

<, 1 \ 2M;(/ (c~) /

/

........',@M+l) 'ït\L, / 1 // /

(ciJ f u >;/ /..... \

/ ~ / I '):Tt (Cll, \ ~5\Çt'Ç12M-- -<,<, \ / ~ , L,.... \ / 0°........\ 1/C '"~, c R,,// \'

/ 1 \ <,-> I

\ "(cfJ

/ / \ ,/ I \ . ......<, (C!J// / \ .....

// / \ <, <,// / \ ' <,

,/'/ / - \(cIJ

<,

(CiJ 1

924

ew

67

Fig. 6. w-plane with contour (C,).

Since N(w) has no zeros for lul ~ 1, v = 0, the curve (Ca) may bereplacedby (C4), as shown in fig. 6, without altering the number of zeros within(compare remark (c». In order to apply the theorem of sec. 6, the zeros ofRe N(w) on C are yet to be determined.

Re N(w) = Iwl3 ~-cos 311'(1 + e2L,u cos 2L1v) + sin 311'e2L,u sin2Llv~, (45)

Im N(w) = Iwl3 ~-sin3p(1 + e2L

,u cos2L1v) - cos 311'e2L,u sin2L1vt,in which 11'= arg w. Let us denote the various parts of the curve (C4) by(Cl), ... , (Cn as shown in fig. 6, and let an be the zeros of Re N(w) on(C4) so that the an are the special roots of the equation

-cos 311' (1 + e2L,u cos 2L1v) + sin 311' e2L,u sin 2L1v = 0 (46)

which lie on (C4). ,

(a) We fix the integer M::p 1 and let c tend to infinity. On (C~), u = ctends to infinity and therefore 11'to zero. Hence (45) can be simplified to

Re N = -lwl3 e2L,u cos (311' + 2L1v),Im N = -lwl3 e2L,u sin (31P + 2L1v).

(47)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE.MELTING PROCESS 401.

9. Determination of .the zeros of Re N(w) on (C,,)

On (C4) the relation (42) reads

N(w) = w3~-1-e2,L,w~, (44)

or written in an alternative form,

The points am situated on (C4), next are to be determined by means ofthe equations

an ~ c + i(n + t)n/2Ll' (48)

so that there are 4M zeros an on this part of (C4)·

(b) On (c:) sin 2L1v is negative and cos 2L1v ,positive for u > O. Zeros ofN(w) therefore are only possible if cot3p is negative, which amounts ton/6 < 11' :::;;n/3. For these values of 11'and for large M eq. (47) holds, sinceL1u::p 1 applies here. For 11'= n/6 we have Re N < 0, while Re N> 0for 11'= n/3. Moreover, Re N is monotonic in th~ intermediate interval.It has therefore exactly one zero, a4M' in the range n/6 ~ lP ~ n/2 on(C:).On the imaginary axis (u = 0) Re N has one zero

(49)

on (C4), provided M is large. -(c) On (C:) we have sin 2L1v > 0 and cos 2L1v > 0, provided u is negative.Therefore, zeros of Re N(w) are only possible here if cot 311' is positive,that is, if 2n/3 < 11'< 5n/6. Since -L1u:;:p 1 holds in this interval, (45)assumes the simpler form

Re N = Iwl3 ~- cos 311'+ sin 31P e2L,u sin 2L1Vt,-Im N = Iwl3 ~- sin 311' - cos 31P e2L,u sin 2L1v ~•

(50)

402 G.BRAUN

Hence Re N is negative for rp = 21"&/3and Re N positive for rp = 51"&/6oh(C~). Since it is monotonic in the intermediate interval, we have exactlyone zero of Re N, a4M+2' on (C:), provided u is negative.(d) On (C:) the coordinate u =- c tends to - 00, the argument rp thereforeto 1"&,Re N remaining positive. There is therefore no zero of Re N in thisrange •.(e) ·Combination: of these results leads to a total number of 4M + 3 for thezeros of Re N(w) on (C4)for Im w ~ O. By reason ofsymmetry we thus getSM + 6 zeros on the closed contour (C4). According to the theorem of sec. 6we obtain

(51)

fot n = 0,1, ... , SM +. 5. The number of zeros of N(w) in the interior of (C4)

then proves to satisfy the inequality . .

N~4M+3. (52)

A final application of (43) gives the important result

2(nll + n2I) + nI2 + nI3 + n22 + n23 + n24 + n25 ~ 2M + 1. (53)

The discussion in the following section amounts to an estimation of theindividual terms of (53) from which their exact values can be derived there-áfter,

10. Determination of nw ... , n25

In the following we frequently have to use RoUe's theorem.

(a) Determination of nI2

For w = iv, v > 0 the relations

w ± {J = (v2 + (J2)! ei erg (iv±/l)

(0 ~ arg (iv + (J) ~ ; , arg (iv _:_ (J)= 1"&- arg(iv + (J»), (54)

and

(1 ± w)! = (1+ v2)i eiierg(l±iv)

(0 ~ arg (1 + iv) ~ ; , arg (1 - iv) = .; arg (1 + iV»), . (55)

are verified at once. A substitution yields

NI (iv) = (1+ v2)i (v2 + (J2)t eli[org (l;1=iv) - 2 erg (iv+/l»)'

~ei[2Ilrg (iv+/l) - erg (I+iv») _ ei[Llv+.n)~ • (56)

MATHElIlATICAL TREATMENT OF A TYPICAL ZONE-MELTING PROCESS 403

Thezeros of N1(iv) are the roots ofthe equation. (iv+ ,8)2

2 arg (iv + ,8)- arg (1+ iv) = arg ( . ) = L1v - (2n -I)n. (57). I+w

For short we put· \v[,8(2 - ,8)+ v2]

f(v) , ,82+ (2,8_ I)v2 (58)

and arrive at the relation

arctan f(v) = LIv - (2n + m -I)n,

in which m and n are integers and the arctangent has its principal value(larctanf(v)1 < n/2). For m we derive the values:(i) m = 0 for 2,8-1 > 0;(ii) Sm = 0 for 2,8-1 < 0 and v < Vo= ,8/(1- 2,8)t

? m = 1 for 2,8- 1 < 0 and v > Vo'Case (i), 2,8- 1 ~ O. The properties f(v) > 0 for v> 0, f'(v) > 0 forv ~ O,f(O)= 0 andf(oo) = 00 apply here. All zeros ofNI(iv) are situatedin the intervals (2n -I)n < LIv ::::;;(2n -l)n (n = 1,2, .;., M). In eachof these intervals f(v) - tan LIv is continuous and changes sign at leastonce, so that NI (iv) has at least one solution there. Hence ~2 ~ M.Case (ii), 2,8-1 < O.The correspondingproperties aref(O) = O,f(v) > 0and f'(v) > 0 for 0::::;;v < Vo, f(Vo - 0) = 00, f(Vo + 0) = -00.

Further wehave 'f( v) < 0for Vo< v < 00 and f( 00 )=- 00. Let us definev by the inequality (2v - 2)n ::::;;LIvO< 2vn, assuming M> v. An in-vestigation hased on Rolle's theorem proves the followingresult:For n = 1, 2, ... , v-I there exists at least one solution of (59) in each

interval (2n -I)n < LIv ::::;;(2n -l)n. For n = v + 1, ... , M we obtainat least one solution in each interval (2n -l)n ::::;;LIv ::::;;2nn. For n = vthere also exists at least one solution of (59), either lying in the interval(2v-l)n < LIv < (2v-l)n or in the interval (2v- !)n ::::;;LIv < 2v;7t.It may therefore he concludedthat nI2 ~ M holds here onceagain, so as tohave in either case

The inequality here accounts for multiple roots or more than one rootin one interval, if any.

(h) Determination of ~3

The function NI(w) is real for 0 ::::;;w ::::;;1.SinceNI(O)> 0 and NI(I) < 0one zero, at least, doesexist~ the intervalO < w < 1; a fortiori, all possiblesolutions are to he found in the interval ,8< u < 1. In view of thesearguments we get

(59)

(60)

(61)

404 G.BRAUN

(c) Determination of. n22In analogy to (a) we arrive at the following expression for N2(iv) if

v> 0:

N2(iv) == (1+ v2)i (v2+ fJ2)! e!i[arg(l+iv)-2arg(iV-fJ») ~i[2arg(iv+fJ)-arg(1+iv») _ eiL1V~. (62)

The zeros of N2(iv) thus satisfy the equation

(iv + fJ)22 arg (iv + fJ) - arg (1+ iv) = arg . = L1v - 2(n-:-l)n. (63)

(l+w)

Finally we obtain the relation

arctanf(v) = L1v - (2n + m-2)n, (64)

in whichf(v), arctan, m and n have the same meaning as in (59).Case (i), 2fJ -1 ~ O.Each ofthe intervals (2n-2)n < L1v ~ (2n-3/2)n(n = 2, ... M) proves to contain at least one solution of (63).Case (ii), 2fJ -1 < O. Let the integer 'Vhave the same meaning as before.Analogous arguments lead to the following result: There exists at least onesolution of (63) in each interval (2n-2)n <L1v ~ (2n-3/2)n (for n = 2,... , 'V-1). The samle applies in every interval (2n - 3/2)n ~ L1v <(2n -1)n (for n = 'V+ 1, ... , M). For n = 'Vwe get at least one solutionof (63), either in the interval (2'V - 2)n < L1v ~ (2'V ___.:.3/2)n or in theinterval (2'V - 3/2)n ~ L1v < (2'V -1)n. These considerations show that

(65)does hold in either case.

(d) Determination of n23 and n25

In order to calculate n23 and n25, we investigate the function

f(v) - tan (L1v)for small v and obtain

f(v) - tan (L1v) ~ [2/(1 + L1) - fJ] (1 + L1)V/ fJ. (66)

It seems reasonable to discuss the cases (a) fJ> 2/(1 + L1), (b) fJ=2/(1 + L1) and (c) fJ < 2/(1 + L1) separately.Application of Rolle's theorem to the function f(v) - tan (L1v) in the

interval 0 < L1v ~. 2n, and properly accounting for the points of diseonti-nuity, leads to the following result:

case (a):case (b):case (c):

n23 ~ 0,n23 ~ 0,n23 ~ 1,

n25 = 0,n25 ~ 1,n25 = O.

(67)

MATHE~IATICAL TREATMENT OF A TYPICAL ZONE.MELTING PROCESS 405

(e) Determination of n24The function Nlw) is real if 0 ~ W ~ 1. In addition N2(0) = 0, N2(1) > 0,

N2({3) > O. (dN2(w)/dw)w=o is negative, zero, and positive in the cases(a), (b) and (c), respectively: We ge~then the following result:

case (a): n24 ~ 1,case (b): n24 ~ 0, (68)case (c): n24 ~ O.

Moreover, all existing roots prove to lie in the interval (0,{3).From a combination of eqs (53),,(60) and (65) we infer

1 ~ 2(nu + n2l) + (n23 + n24 + n25) , (69)which implies

nn= 0, (70)

Application of (61), (67) and (68) shows that the relations

case (a):case (b):case (c):

n23 = 0,n23 = 0, .n23 = 1,

n24 = 1,n24 = 0,n24 = 0,

n25= 0,n25= 1,n25= 0

(71)

hold, respectively. As a consequence the inequalities cannot occur, so thatwe have n12=M, n13=1, n22=M-l. (72)

We now summarize the results of this section:(i) Both Nl(w) and N2(w) have only zeros which are either realor purelyimaginary.(ü) There are the following real positive zeros ofNl(w) and N2(w):(1) one simple zero of Nl(u), Wo = Uo say, in the interval ((3,1);(2) one zero of N2(w), w = 0, ofmultiplicity one, three and one in the cases(a), (b) and (c), respectively;(3) one simple zero of N2(u), Wo =Uo say, in the interval (0,{3)in case (a).(ill) All purely imaginary zeros of Nl(w) and N2(w) are simple. The follow-ing appear to exist, if v > 0:(1) one zero of Nl(iv), Wn = ivno in each of the intervals

(2n-l):n: < Llvn< 2n:n:(n=1,2, ... );(2) one zero of N2(iv), wn = ivno in each of the intervals

(2n - 2):n: < L1vn < (2n - 1):n;(n=2,3, ... );(3) one zero of N2(iv), wl = ivl, in the intervalO < Lli\ <:n; in case (c).(iv) The further roots of Nl(w) and N2(w) can be calculated using eq. (38)and the remark thereafter.Remembering these results and the formula % = :f: (1 - w2)i shQWSthe

406 G. BRAUN

real-valuedness of all zeros of N1•2(w(Z» in the z-plane. The zeros Zo =(1- u~)t, Zn = (1+ v~)t (n = 1,2, ... ), - 1 and zn = - (1 + v~)i (n _:_2, 3, ... ) exist throughout, whereas the additional zeros Zo = - (1 - u~)t'and Z1 = -(1+ V12)i only occur in the cases (a) and (c) respectively.

11. Further evaluation ofthe relation (27) representing the quantity 't'n(s/a)

The evaluation is achieved in the following 'way. Apply the results ofsecs 8 to 10 to eq. (27). With the aid of (35) and (36) and the remarks at theend of sec. 7, we transform the contour Cl into C2and shift C2 to infinity.A substitution then leads 'to the equation

B n . co

"n(~/a)=(~) ~A2+Resz=zoFn(~,z)+Resz=lFn(~,z)+ ~lResz=zmFn(~,z) ++ U((J - 2/(1 + L1» Resz=zoFn(~,z) + ResZ=_l Fn(~,z) +

+ U(2/(1 + L1) - (J) ResZ=ZlFn(~,z) + m~2Resa=zm Fn(~,z)~. (73)

Here U(x) denotes 1for x> 0 and 0 for x ~ O.To arrive at the final resultwe only have to calculate these residues and order the various termsaccording to their magnitude, which is determined hy the magnitudesof the Zm and zm (m = 0, 1, 2, ... ). First let us have a look at the termsResZ=l Fn(~,z), ResZ=_lFn(~,z). Calculation of these terms with the aidof eq. (28) yields the formulas

(74)

and 0 (J =1= 2/(1 + L1)

Re,,__ i Fn(l',» . )(-1)" 3:!P ; [A,p+2(A,-A,)] (J= 2/(1+L1)·(75)

Before we evaluate the other terms, we hear in mind that thederivation of formula (73) assures the convergence of the two seriesco co~ Resz=zm Fn(~,z) and ~ Resz=zmFn(~,z) for all values of ~(0~~~L1)'m=1 m=2The result of these calculations is of the following form:

1 ' 2(y'I-w~+I-w~)Res F (t: z) - - -:----"----.,,.:'.,----,.:-,,---;;-0------'----.,,--,----,----;:----:-:-

Z=Zm n \,i, - z:; (w~ _ (J2)+ 2(J(1 - w~) + L1(1 - w~) (w~ - (J2)

~(AI - A2) (1- y'1- w~) + A2(J~ X ~cosh wm~ - : sinh Wm~~ , (76)m

1 2(-y'I-w~+I-w~)Resz=zm Fn(~,z) = ~ (w~ _ (J2)+ 2(J(1- w~) + L1(1- w~) (w~ - f32)

~(Al - A2) (1+ y'1- ~~) + A2(J~ X ~cosh wm~ - :m sinh Wm~~ . (77)

, (B)n7:n(g/a) ~ ~ Resz=zoFn(g,z). (80)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE·MELTING PROCESS 407

A rearrangement of the various terms of (73) according to their order ofmagnitude and application of (74) leads to the following:

B n .7:n(g/a) = (~)~Resz=zoFn(~,z) + U(fJ-2/(I +L1» Resz=zoFn(g,z) +

+ ResZ~-l Fn(g,z) + U(2/(I + L1) - fJ) ResZ=ZlFn(~,z) +ResZ=ZlFn(g,z) + ...t. (78)

For brevity's sake we confine ourselves to the first term, and neglect thesmall corrections due to the followingones. We then obtain the"asymptoticresult

O((~Jn)7:n(~/a) = (~)nResz=zoFn(g,z) + 0(( ~ )n)

o((~Jn)fJ = 2/(1 + L1), (79)

Here we have used the éonvergence of the series in eq. (73), whichproperty makes it possible to have all remaining termes included inO( (B/a zo)n), O( (B/at) or O( (iJ/az1t). In the case of Resz=zoFn(g,z) =1= 0these O-terms too' can he neglected and we get

12. Derivation of the final expression for the concentration distribution

To calculate Resz=zo Fn(g,z) we need some information about Wo = Uo'We put for brevity U = [fJ(2 - fJ)]! and M1(u) = (1 + u)!N1(u). SinceN1(fJ) is positive and N1(U) negative, this root Uo lies between fJ and U

fJ< Uo < U. (81)

In order to find an approximation for Uo we investigate the function

M1(U) . 'VI u2 (u + fJ) - (1 + u) (u - fJ)é1". (82)

It can be proved that M~'(u) < 0, Mi"(u) < 0 for fJ < u < U.' As aconsequence the inequalities

M1(U) + M{(U)(u - U) + t Mi'(U)(u - U)2 << M1(U)'< M1(U) + 'M{(U)(u- U) (83)

hold so that we get U2 < Uo< Ui' U1 and U2 denoting the zerosof the first'and the last term in (83), respectively. It is easy to derive that the function

408 G.BRAUN

Uo = uO((3,L1) satisfies the relations auO/aL1 < 0 and a!lo/a(3 > O. Goodapproximations for uO((3,L1) are given by the formulas

Uo = [(3(2 - (3)]t ~1-(~-=~2 L1 + t ~~:=:~:(4(32-9(3 + 3)L~~ (L1~:l),

u'o f::::j (3 (L1~ 1), (84)

if L1 ~ 1 an:d L1~ 1, respectively. The function uo = uO((3,L1) has heencalculated and plotted for the values of the parameters, (3= 0, 0,1, ... , 1,

T{)

o·

~ ~ ~I,~ V/V V

~ .:V Juo(IJ.L,)

Cl/ / VJIr/ 1 j

L,,,,,O.~yW VI IfI- ~L,,,,,O

/ 1/ JI- L,=O.()~JL1=0.,-111 J XI

AYlj 1~,-0.s IJI 1/ J '1-,,,,,1 /VlI!l r7 f71y/ L,=S; V,1(/ 1/iLl-la

/ ~~oo

fJ / 'IJlij V/!/

1 I/V1/

IJ

oe

0·7

0·6

0·5

0·3

o-

o 0·1 oe 0-3 004 Q.5 0·6 0·7 0-8 O·g 1__'f3 i924M,

Fig. 7. Zero Uo = uO({J,L1) of the function N1(u) •.

B 1-{JAl({J,Ll) = - = , ::::;;1

azo 1'1- 1I~

is required for further calculations. It satisfies the relations OAl/àL1 < 0and àAl/O{J < O.The numerical values ofthis quantity have been calculatedand are plotted in fig. 8 (for the same values of the parameters). A speciali-

(85)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE·MELTING PROCESS 409

and Ll = 0, 0,01, 0·05, ... , 10 and 00 in fig. 7. The quantity

o

I~ -t:::::::: 'L,=CJ.07 L(=O1'-.. t--t--

~"~ r-, L7-0·' t---

~ ~ -."~ "'"

<,8

.~

~<,

r-, L,=0.5"t(13,Lt)

~<,

""~VT r-, I7 , r-,~~t=5 [\.

Lt=~\ r-.,

~ \Lt=~

" 1\5 1\ \

, ,~ \\ \

~3 \

t\·2 \·T

\,0 0" O~ 0·3 0"' 0·5 0·6 0·7 0·8 O·g 1{)

----f> f3 92469

Fig. 8. Quantity Al = Al({J,Ll)·

T·O

O·g

o

o-

0·6

O·..

o

O·

o

410 G.BRAUN

zation of eq. (76) to m = 0 and substitution of the values for Al and A2results in the formula -

(86)

which will be discussed subsequently. As cosh uo~ - (f3luo) sinh uo~ ismonotonically decreasing in the intervalO::::;;; ~ ::::;;L1, we may confineourselves to Tn(O); the property mentioned amounts to the inequalitiesITn+1(O)l= ITn(L)I ~ ITn(~/a)l::::;;; ITn(O)I· Formula 86 then transforms into

Tn(O) aco - ~n(O) n . 2f3[-Vl-u~ +1- u~]--'-= I::::i Al Xaco aco (u~- f32) + 2f3(I-u~) + L1(1-u~) (u~- f32)

~ao 1--V1-u~ ( ao)~X -(1-k) + 1-'-- .aco f3 aco

(87)

The influence of the various factors can easily be discussed with the aid ofthis equation. The most important factor is the first one, because it depends onn. The second factor, which is plotted in fig. 9, is of the order of 1 for f3< 1.To be more specific, it assumes values between 1(f3 = 0, L1 ;;:::0) and 3·295(f3 = 0·9, L1 = 00). This factor thetefore can be treated as a constant ofthe order of 1 in most cases. The last factor is a monotonically decreasingfunction of ao, in view of (1- k) [1- (1- lI~)t]< 1- (1- u~)! < f3,which inequality results in a negative coefficient of aolaco in this factor.If we disregard the practically unimportant case ao/aco ~ 2/(2 - k),we can prove that the initial concentration ao = 0 constitutes the mostunfavourable situation for approximating the final distribution within pre-scribed limits, the number of courses n being given. On the other hand thespecial value ao/alX)' which makes this last factor equal to zero, that is,Resz=zo(~'z) = 0, is especially favourable for that purpose. In this caseTn(~/a) tends much more rapidly to zero and we have to use a further term,of (78).for our formula. Whenever this situation does not occur it is sufficientfor the discussions to replace the right-hand side of (87) by A~. This proce-dure transforms (87) into

, (88)

If we want to arrive at 7:n(O)/alX) < e, the necessary number, Unec, ofcourses is

1lnec= ln(1/e)fln(1/A1). (89)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE-MELTING PROCESS 411

3

ft( 2)(vR+I~U:) I I1M-p2) +2p(I-ug)+ L,rt-ugJ(ULp2)

~-: ·D

IfjVI I

Ll-oo Vf/ /---~/ V. / V _./_ // / 1.ljS /

/

/ / Lly V VV

V V / c:::Vvi~o.s/

71~V ~ e= I-- L,= ~5 L!"if-O.fl....:::::=:

1.1. [~{OL1=0·01

2

oo 0·2 0·6 . 0.0 f.O

--- (J 92470 .

. 2.8[(1-uo2)s + 1- U02]

FIg. 9. Factor (U02-,82) + 2.8(1 -uo2) + LI(1 -u02)(U02_,82)

A look at fig. 8 shows that llnec becomes small for (J -+ 1 and L1 -+ oo ,In: this case ;;n(O) fao:> assumes the smallest values for a certain n.

In addition to llnec the time Tnec = Tnnec = L1 {Jllnecfk1 (T = durationof one course), needed for the whole process, and the time TnecfL, neededfor the latter per unit length of the bar, are of importance. In view of (89)we get

and

Tnec In (l/e) 1--=L v ln(lfÄ1)

(90)

--- ~_._----

412 G.BRAUN

(92)

This new quantity A2({J,L1) has the properties A2({J,0) = 1, A2({J,=) , 00and A2({J,L1) > 1 for arbitrary L1> 0. It is plotted in fig. 10. As a con-sequence of these properties the inequality Tnec;;;::(In (l/e»/k1 holds.The equations (87) to (92) defining the "simplified" problem only containthe independent parameters k, I, v, ao/a, (J and L1' since the quantityk1 = vk{J/l(1 - (J) can be eliminated.

6

9 -,-,

8 1.2=À2(IJ,LI)\\

\

7 1\

\\

<,r-, LI =IO~

~<,<, \

<,r-.1'<=5I'\.

'\1\

\1\\----___

r---'""-._L L, =1_ L,=0'1 _r---..L L,-0·5 r---Ll=O L,=().01L,=0·05 / T -=t--t:-- <,1 t--

0-1 Q'2 0-3 0-4 0·5 0·6. 0·7 0·8 0·9 Hl~{J 92471

Fig. 10. Quantity À2= Àa(fJ,Ll)'

5

3

2

(95)

MATHEMATICAL TREATMENT OF A TYPICAL ZONE-MELTING PROCESS 413

The demand ofa minimization of nnee, TneejL and Tnee, respectively,imposes different conditions on the parameters. In order to get Unee orTneejL1 as small as possible, L1~ 1 and fJ......,..Imust hold, which amounts.to fJj(I - fJ) ~ 1. The relation kj1v = kfJjl(I- fJ) then yields the resultk1jv ~ kjl, which corresponds to a strong gas phase.

On the other hand, the quantity Tneebecomes small if L1 ~ 1 and fJ ......,..1holds. Once more these conditions would imply k1jv ~ kjl. As can be seenfrom the definition of a, these requirements are not consistent because thesmallness of L1 = aL > aLmin would require small a > kjl + ~jvmax,which can only be achieved for small k1jv, kjl being fixed. Therefore, acompromise has to be found to optimize kl. This determination of theparameters probably would result in a weak gas phase.

All our calculations were based on the assumption fJ=1= 0, that is, k1 > 0.But it seems interesting also to investigate the situation ~ = 0, fJ= 0,Ceq = 0, which corresponds to the absence of a gas phase. A repetitionof the considerations, which will not be given here in detail, leads to theformula

7:n(çja) = A2 + ResZ=l Fn(ç,z) + ResZ=-l Fn(~,z) +

which will replace (73). A calculation of the terms ResZ=l Fn(ç,z) andResz=-l Fn(ç,z) yields the result

Resz=-l Fn(ç,z) = 0, (94)

whereas the remaining terms prove to be O(Ijz~). Finally we obtain theformula

which yields an"""'" k(L + l)jl(I + kLjl). Since there is no gas phaseat all in this case, the total quantity of material B in the bar,ao(I + Ljl), must- remain unaltered. In fact, this quantity amounts toan(Ljl + Ijk) = an (1+ kLjl)jk = ao(l + Ljl) and this checks the result.

In order to get an impression of such calculations, some numericalexamples are worked out below and are listed in a diagram. Here propervalues for the quantities l, L, k, k" 6, v, ao and B are chosen, whereas theother quantities are calculated.

414 G.BRAUN

I H HI IV

l(cm) 2 2 2 2L(cm) 20 40 40 40k 10-3 10-1 ~0-1 10-1kr (cm/sec) 10-6 10-4 10-4 10-4(cm) 1 1 10-1 1v(cm/sec) 0·002 0·002 0·002 0·02ao(g) 0 0 0 0e 0·001 0·001 0·001 0·001kl(l/sec) 0·000001 0·0001 0·001 0·0001a(l/sec) 0·001 0·1 0·55 0·055B(I/sec) 0·0005 0·05 0·05 0·05P 0·5 . 0·5 0·909 0·091L1 0·02 4 22 2·2Uo 0·863 0·560 0·909 0·238Ä.1 0·990 0·604 0·218 0·936Ä.2 1·006 3·960 15-40 3·02Unec 695 14 6 105Tnec (days) 80·4 3·2 1·23 2-42Tnec/L 4·2 0·08 0·03 0·06(days/cm)

The author is greatly indebted to Dr H. Bremmer for help in the calcula-tions of sec. 2 and for many stimulating discussions. He would like to expresshis gratitude to !VIr J. van den Boomgaard for putting forward thisproblem.

Eindhoven, December 1956

REFERENCES

1) W. G. Pfann, J. Metals, N.Y. 4, 747-753, 1952.2) H. Reiss, J. Metals, N.Y. 6, 1053-1059, 1954.8) J. van den Boomgaard, Philips Res. Rep. 10, 319-336, 1955.') E. C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, 2nd ed., p. 116.6) Idem, p. 123.G) Idem, p. 214. .;/