Distillation under Moderately High Vacuum, illustrated by the Vacuum Distillation of Zinc f r o m Lead--Theoretical T. R. A. DAVEY Development Department, The National Smelting Co. Ltd., Avonmouth Mathematical treatment is given to the problem o f distillation under vacuum, in a pressure region above that at which " molecular distillation " can occur. The treatment is related to the vacuum separation of zinc from lead, which is established as a lead refining process, and will shortly be used for producing zinc. In the practical applications, the velocity o f zinc vapour distilling is a sizeable fraction q f the projected mean molecular velocity, and under these conditions the normal concept o f pressure must be abandoned, and corrections applied to pressure terms. The corrected distillation rate cannot be determined analytically by this method, but can be obtained by successive approximations. A comparison o f theoretical and practical distillation rates shows a satisfactory agreement.
I. Introduction
stress laid on the practical application of the results to the specific case of vacuum dezincing in lead refining. Thus there was no detailed discussion of the limitations of the equations derived. In addition, no reference was made to an obvious discrepancy, in that the rate of distillation calculated for very low pressures would be only half the rate for molecular distillation.
The object of this paper is to determine the distillation rate to be expected when a solution containing one volatile element is exposed to a vacuum not complete enough to permit "molecular distillation " to occur : when the residual pressure within the distillation space has been reduced almost to zero, but the " mean free path " of the distilling atoms is still substantially less than the distance traversed between evaporating and condensing surfaces.
The distillation rates calculated according to referencet are adequate for design purposes for vacuum dezincing applied to lead refining, as no very great accuracy is required : an error of ±50 per cent in the rate calculated would result in a range in the zinc content of dezinced lead, of 0.03-0.09 per cent Zn, if 0.05 be aimed for, which is not of very great consequence in practice, the residual zinc being easily removed by stirring the lead with caustic soda. Recently, however, a new application for vacuum dezincing has been found 6, in which it may well be desirable to obtain greater accuracy. This new application (for the production of high-grade zinc from lead circulating through the condensers of the Imperial Smelting Process furnaces) may well be widely used throughout the world, and therefore it seems appropriate to publish a fuller account of vacuum distillation in this industrial high-vacuum range.
A rather brief account of a process operating in this pressure range was presehted some years ago 1. In connexion with the recovery of zinc from desilverized lead by vacuum distillation, it was desirable to be able to calculate theoretically the dependance of the distillation rate on such factors as : concentration of zinc in lead, temperatures of distilland and distillate, operating vacuum, and distillation p a t h - i.e. distance between evaporating and condensing surfaces. In 1947, when the practical development of continuous vacuum dezincing of lead was undertaken at the Broken Hill Associated Smelters Ltd., Port Pirie, Australia2, no reference could be located in which mathematical consideration had been given to the problem of distillation at pressures intermediate between atmospheric and the very low range where " molecular distillation " occurs. A batch vacuum dezincing process had been developed at Herculaneum, U.S.A. 3, the design of plant having presumably been worked out empirically, on the basis of smaller-scale experiments. The original proposal came from Kroll4, who conducted experimental work, but also did not attempt any fundamental evaluations of the distillation rate to be expected.
An account of the practical side of vacuum dezincing, and a comparison of practical with theoretical distillation rates, will appear in this journal subsequently.
2. Outline of the problem This account, while illustrated for the system lead-zinc, is of general application to any system with only one volatile component, where the vapour pressure of this component, and its activity in the distilland, are known. Reference to lead and zinc is made for the purpose of defining the pressure ranges and general conditions being considered. Under
In 1948 Carman5 presented an account of vacuum distillation in the pressure region under consideration, and his treatment is considered invalid for reasons already givenl. The writer's previous accountZ was incomplete in several respects, the fundamental side being highly condensed and 83
84
T. R. A. DAVEY
I TO COLD T~AP AND VACUUMPUMP
___
ZINCY LE.~) SUPPLY
6
~ ~
6
,~ ~
~
~
~
6
IO
DE.ZINCE.D LEA.D LE~,VE$
ZINC PRODUCE.D ~
W~,TER COOLING
~
MOLTENLE~'.O
~
MOLTENZiNC
~
Zl~. COI~E
m
INSUL~,TIQN
FIG. 1. This figure, slightly modified from reference8, shows schematically a vacuum vessel comprising a barometric leg (1) through which zinc-containing lead enters, an annular reservoir of lead (2) between the cylindrical outer shell (3) and the conical spreading surface (4), which is attached to the outer shell by brackets (not shown) near its top, and joined to the outer shell at the bottom through a flexible joint (5). Lead passes from the reservoir to an annular distributing channel (6), divided into a number of segments, each fed by one of the pipes (7). These pipes, by offering a resistance to flow, cause a difference in head between (2) and (6). Lead flows from the distributing channel over a rounded weir (8), down the spreading surface (4), is collected in the channel (9), and leaves the vessel through the barometric leg (10). A slight departure from levelness of the weir (8) can be tolerated because the difference in head between (2) and (6) ensures that some lead enters each segment of distributing channel, ensuring fairly even flow of lead over a great length of weir (about 20 ft at Port Pirie). Zinc evaporates from the thin, turbulently flowing film of lead on the spreading surface, and condenses on the surface of a crust of solid zinc (11). This is cooled by water pipes (12), so designed that the thermal gradient through the zinc crust (11), due to inflow of h e a t - by radiation from the spreading surface, and by latent heat of condensation of zinc v a p o u r - gives a surface temperature of 420 °C at a reasonable thickness of crust. This is the " l i q u i d zinc condenser " referred to in this paper. (A solid zinc condenser is made by providing increased water-cooled surface area, as detailed in references 2 and 7.) Liquid zinc dripping from the condenser surface is collected in a funnel (13), discharging through a barometric leg to atmosphere. Some uncondensed zinc vapour leaves the vessel through the vacuum line (14), which is kept free either by heating, to prevent solid zinc from condensing before the cold trap, or by provision of mechanical means of scraping solid zinc crystals off the walls. The condenser is integral with the vessel top, which is sealed to the vessel walls by means of a large rubber O-ring between water-c0oled machined flanges (15). The top may easily be removed for inspection, cleaning or repairs--and in the case of solid zinc condensation, for zinc removal. In the latter case the vacuum line connexion is made through the vessel bottom, not the top2, 7.
Distillation under Moderately High Vacuum, illustrated by the Vacuum Distillation of Zinc from Lead--Theoretical certain conditions some of the simplifying assumptions made here may not be permissible, and conceivably other simplifications could be made in some cases. Desilverized lead contains 0.55~.60 per cent zinc, of which about 90-95 per cent may be economically recoverable by vacuum distillation, leaving about 0.05 per cent zinc in the dezinced lead. The permissible range of working temperatures is limited to those in which steel vessels may be used. Provided that reasonable care is exercised in the design, a mild steel vessel may be used up to about 620°C, although it is preferable that temperatures remain below about 600°C, lest rapid distortion and collapse occur. The working pressure, or indicated vacuum, is produced in practice by a Kinney single-stage mechanical pump, which is capable of giving a vacuum of 5~ (0.005 torr) against a blank, or 20-100/~ in a large industrial vacuum chamber, depending upon the care taken to minimize leakage through joints, valves, etc. For various practical reasons it is desirable that the distillation path (as defined in the second opening paragraph) be of the order of 30 cm, whereas molecular distillation could not be expected to occur unless this distance were a fraction of 1 cm. It has proved convenient to condense zinc to the solid state 7, which means that the temperature of the condensing surface is kept well below 420 °C, the melting point of pure zinc. In contrast, the Imperial Smelting Process application6 involves lead with a zinc content of the order of 1 per cent, at a temperature not above 560°C, and a condenser surface kept exactly at the melting point of zinc, not by any imposed controls, but inherently in the designS. (Reference to Fig. 1 and the accompanying text will show the type of apparatus which has been developed to carry out the process to which the following calculations should apply). We are therefore concerned with a distillation process with a residual gas pressure of 20-100~z, the partial pressure of the distilling element in the distilland being of the order of 2000/z, decreasing to about 200/~, in the case of lead refining, and around 1700# in the case of zinc production. The partial pressure of distillate will be negligible in the former case, and 154/~ in the latter, this being the vapour pressure of liquid or solid zinc at 420 °C, the melting point. The areas of evaporating surface and condensing surface will be assumed equal in the first instance, and corrections applied subsequently to allow for the inequality in practice. The evaporating and condensing surfaces are also assumed to be parallel, with no obstructions to vapour flow between them, and any resistance due to end effects at the walls of the containing vessel is also assumed to be negligible--these conditions can be met in practice. Finally, the residual gas pressure is assumed (i) to be equal at all points equidistant from the condensing surface, and (ii) to be a maximum immediately above the condensing surface--i.e, it is assumed that the still has been designed to be " self-pumping ". Assumption (i) is of doubtful validity, although not of great significance in the lead-zinc distillation case, whereas
85
assumption (ii) can of course always be met by appropriate design. Fig. 2 represents graphically the case being considered. For purposes of mathematical treatment, the distillation process is regarded as comprising three distinct parts: evaporation, condensation, and migration or movement of zinc vapour between evaporating and condensing surfaces. Expressions are derived for each of these three parts, and these are equated in order to obtain a solution for the overall distillation rate under steady state conditions. A difficulty arises in considering exactly what temperature means, as applied to atoms distilling through a vacuum space, and so it would be difficult to define rigorously what is meant by adiabatic or isothermal behaviour of these atoms. However, both conditions have been considered formally in the treatment of the stage involving transfer of vapour between evaporating and condensing surfaces, and fortunately the difference between isothermal and adiabatic behaviour, so considered, is small enough to be negligible over a large range of conditions. The isothermal expression was subsequently used for evaluations, since it is much simpler to manipulate, although it is realized that in the literal sense the behaviour of atoms distilling must be adiabatic. Similar difficulties, more important in their significance, arise in the consideration of what pressure means, in a gas whose molecules are moving predominantly in one direction with a net mean velocity approaching the molecular velocity. In the following treatment this velocity effect is first neglected, and later an approximate correction is developed for modification of the solution so obtained, to allow for the condition that the net mean velocity of distilling molecules is a sizeable fraction of the free molecular velocity. This correction brings the new expression into line with the well-known one for molecular distillation, under the appropriate conditions. The method of solution thus follows this plan : (i) development of expressions for rate of evaporation, and rate of condensation, (ii) development of expressions for both isothermal and adiabatic movement of vapour between evaporating a n d condensing surfaces, (iii) development of overall distillation rates for both isothermal and adiabatic conditions, (iv) numerical evaluation of both isothermal and adiabatic expressions to show that they yield approximately the same results in ranges of practical interest, (v) application of distillation rates to process results by integration, and development of simpler approximations for particular conditions, (vi) extension of previous treatment, which assumed equal areas of evaporation and condensation, to the general case, (vii) development of an approximate correction to allow for high velocities of diffusion, which had been neglected thus far.
86
T. R. A. DAVEY
a
NOMENCLATURE ~ constant in equation (7). See Table I. a = 0.472/~/(T) for zinc.
A
~
surface area (cm2).
AC
=
condensing surface area (cm2).
AD
~
area through which gas diffuses (cm2).
AE
=
evaporating surface area (cruZ).
~x
~
a c c o m m o d a t i o n e~efficient (~ = 1 for most metals).
~
~
proportionality constant for interdiffusion of two components, A and B.
b
=
constant in equations (4) and (7). b = (0.796DP/TL) for zinc.
~
mass of lead (g).
C
~
constant in equation (7), see Table I.
E
~
pc~/(T1/Tc).
mean projected molecular velocity (cm/sec). g = (2/~/3)~/(2RT/~M).
C
~
Sutherland's constant.
C
=
rate of condensation (g/cm2.sec).
~ I", ~ . ~ I",
Condensin9 Surfac¢ P~ ~T=)
Pr¢ssur¢t Evaporatin9 C5urfoc~
See Table I.
B
c-
P~; ~ ~1~,)
///~VacuumfLin¢
~
O
~
Distance
L
F~6. 2. Schematic representation o f the distillation o f zinc vapour through nitrogen. Zinc vapour has a partial pressure o f p' in the evaporating surface, and Pc in the condensing surface. Zinc partial pressure in the distillation space, p~, falls from pz (at a temperature TI) above the evaporating surface, to p~ (at a temperature T2) above the condensing surface. The total pressure, P, remains constant throughout the distillation space, comprising PA ~ P~. The nitrogen partial pressure, p~, rises to V at the condensing surface, this being the measured V f l C U U ~ .
D
--
Maxwell's diffusivity (cm2/sec).
3. Evaporation rate
e
--
exponential function.
E
--
rate of evaporation (g/cm2.sec).
I
=
integral in equation (6).
D u e principally to the high thermal conductivity of metals, under vacuum boiling does not occur, and evaporation is primarily a surface p h e n o m e n o n 4. In view of the fact that the relative fractionation obtained by vacuum distillation is frequently less than the theoretical, especially at high distillation rates 1, 9 it may well be that conditions below the evaporating surface should also be considered. Here, however, the surface alone will be considered, and it will later be assumed that conditions in the liquid distilland are turbulent enough for the surface concentration of the volatile element to be kept at the same value as the bulk concentration. In the pressure range being considered, the effect of residual gas.molecules cannot be ignored, and thus there will be some hold-up of distillate atoms above the evaporating surface, and these by virtue of their pressure exert a tendency to recondensation on the evaporating surface. The net evaporation rate, from Langmuir's expression, will therefore be :
.pl p0.2 d p I =
-
2 P
P
I, m
~
constants in equations (2b), (4a) and (7a).
L
=
distance between surfaces (cm).
M
=
rate of m o v e m e n t (g/cm2.sec).
M
~
N
evaporating of
zinc
and
condensing
vapour
distilling
molecular weight (g). velocity of diffusion (cm/sec).
p
~ moles of gas diffusing (g-mole/cm2.sec). ~ pressure of zinc vapour (torr).
pt
m
P
= total pressure (torr).
y
--
R
= gas constant.
t
~ time (see).
T //
-- temperature (°K). ~ E / b in equations (14), (17) and (18).
V
~ vacuum, or residual gas pressure (torr).
V
-- atomic volume (cm3).
partial pressure of zinc in lead (torr).
~
ratio ( v / ~ ) .
E=
~
~(p'--pl)/(~ST1 ~
R = 82.07 cm3 atm/deg.mole.
partial density (g/cm3).
velocity of zinc vapour distillation (cm/sec).
w
~ weight per cent zinc in lead.
x
~
distance (cm).
x, y ~ constants in equations (12), (13) and (14)--see Table I.
)
g/cm2.sec
(1)
~
where the symbols are as defined under Nomenclature, and their significance can be seen at a glance in Fig. 2. The a c c o m m o d a t i o n coefficient e was shown at an early date to be unity for metals, and it is of interest that Hickman and Trevoyg(b) found it to be unity for a number of organic liquids, if the evaporating surface were very clean, in cases where it had formerly been found to be very much lower. The case where distillation rates are so rapid that the distilling atoms are moving with high velocities away from the evaporating surface, and thus affecting Pl, the backward partial pressure of these atoms, will be considered subsequently. F o r zinc, P ' --Pl E = 0,472 ~/(T1)
g/cm2'sec
(la)
87
Distillation under Moderately High Vacuum, illustrated by the Vacuum Distillation of Zinc from Lead--Theoretical
4. Condensation rate I n similar m a n n e r the net c o n d e n s a t i o n rate for zinc c a n be arrived a t : C ~ 0.472 ~P2 / T 2 --0.472 ~/P~c g/cm2"sec'
T h e rate of m o v e m e n t o f zinc between e v a p o r a t i n g a n d c o n d e n s i n g surfaces m a y b e d e t e r m i n e d b y considering the diffusion o f o n e gas (A) t h r o u g h a n o t h e r s t a g n a n t gas (B), following the p r o c e d u r e of Lewis a n d Chang10 : A t the steady state, using symbols as defined in N o m e n clature :
\ dx /
: a
S u b s t i t u t i n g NA =
OA OB MA Me
~h
QA~A OB M-A × M ,
~ a
C=
r2%/(yA~ V B ! ) 7 3 L VA ~ + V a ~ I
(10)
(b) Adiabatic diffusion T o simplify the evaluations, the value o f T in the t e r m (1 + C / T ) of e q u a t i o n (5) was set at T = 873 °K. V a r i a t i o n in this T does n o t greatly affect the value o f DP. Substituting D P ~- 3.60 × 10-ST 1.5 in e q u a t i o n (3), T0.5 dp M . d x = 2.86 × 10-5 - - - P--p where the suffix A h a s been d r o p p e d f r o m the t e r m p, representing partial pressure of zinc. Before integration, the s u b s t i t u t i o n m u s t be m a d e T = k p 0.4 (where k = [T1/plo.4]), since zinc v a p o u r is m o n a t o m i c , then :
PB ~ RT
~B
(CACa)
As p o i n t e d o u t by Gambillt2, A r n o l d ' s p u b l i s h e d expression for C is in error, the index ] a p p e a r i n g in e q u a t i o n (10) being o m i t t e d in all four places. Values o f D P are t a b u l a t e d in T a b l e I, a n d thus the c o n s t a n t b c a n be evaluated w h e n values are assigned to T a n d L.
.
#A ~A -- = rate o f diffusion (mol./cm2.sec) MA -- M a dpA NA = - × • a~B dx Substituting ~ B
(5)
where S u t h e r l a n d ' s c o n s t a n t ,
5. Migration rate
[~//'d°A\
0.00837 T3,/2 ~/(1/rMA + 1/MB) (VA ~ + VB~)2 (1 + c / T )
DP =
(2a)
H e r e the t e m p e r a t u r e s of a t o m s c o n d e n s i n g a n d those r e - e v a p o r a t i n g are n o t necessarily e q u a l ; hence two t e r m s a p p e a r o n the right h a n d side o f e q u a t i o n (2a). A m o d i f i c a t i o n o f the first term, w h e n h i g h velocities o f diffusion are concerned, will be considered subsequently.
--
F o r e v a l u a t i o n o f DP, A r n o l d 11 r e c o m m e n d s the use of Jeans' formula :
M . d x = 2.86 × 10-5~/(k) (p0.2 dp)/(P--p). I n t e g r a t i n g f r o m x = 0, p = P b to x = L, p = P2,
--RT
NA =
apB RZT 2
S u b s t i t u t i n g Maxwell's D =
NA ~
aP
×
dpA dx
•
M L = 2.86 × 10-5~/k
= diffusivity (cm2/sec),
Placing
I =
-- D P dpA R T ~ × dx (m°l'/cmZ'sec)"
P u t t i n g M(g/cmZ.sec) = 65.4NA(mol./cmZ.sec) R = 82.07(cm3atm) ; a n d p~ ~ P--PA,
M T 0.796DP
ax =
--dpA P - - P- -A .
for
,
(6)
zinc ; TABLE I
Values o f constants
(3)
(a) Isothermal diffusion I n this case T i s c o n s t a n t , so t h a t o n i n t e g r a t i o n e q u a t i o n (3) becomes : MTL P--P2 = In . 0.796DP P--pa Reference to Fig. 2 shows t h a t P m a y be replaced by P2 -~- V, so t h a t : V M = b In , (4) V + P2--Pl 0.796DP . TL
.f~/~1 pO.2 dp T~ 2 / ~ - - ~ ' a n d k = plo. 4 2.86 × 10-5 1 Ta 0.5 M . . . . . L p~0.2
As the t e r m D P is i n d e p e n d e n t o f b o t h x a n d PA (see below), e q u a t i o n (3) c a n be i n t e g r a t e d f r o m x = 0, PA = P l , to x = L, PA = P2, p r o v i d e d t h a t a decision is m a d e as to w h e t h e r the gas behaves isothermally or adiabatically. B o t h cases will b e considered.
where b = c o n s t a n t =
ID t p0.2 dp 2 ~to_p- °
Temp. (°c) 480 500 520 540 560 580 600 620
10DP 102a 105b(]') 10c
7.11 7.48 7.83 8.15 8.58 8.97 9.28 9.69
1.72 1.70 1.68 1.66 1.64 1.62 1.60 1.58
2.51 2.57 2.62 2.67 2.73 2.78 2.83 2.89
1,61 1,63 1.65 1.67 1.69 1.71 1.73 1.75
0-0.5 0.5-1.0~Zn 1.0-2.0 ~ Zn
0.42 0.62 0.91 1.31 1.82 2.54 3.46 4.62
x
y
x
y
0.36 0.53 0.79 1.14 1.60 2.22 3.04 4.06
0.03 0.04 0.06 0.08 0.11 0.16 0.22 0.30
0.28 0.42 0.63 0.92
0.11 0.15 0.22 0.30
1.30
0.41
1.82 2.51 3.39
0.56 0.75 0.97
*y=0. ~" Assuming L = 30 cm. The partial pressure of zinc as a function of weight per cent zinc in lead was calculated from Kelley'sl3 values of the vapour pressure of pure zinc, and Lumsden's14 expression for the free energy of mixing of lead and zinc. The values calculated for 0, 0.5, 1.0 and 2.0 weight per cent zinc in lead were used to yield linear relations for the values between these points, expressed in the form : P ' g n ~= xW + y.
88
T. R. A. DAVEY
6. Overall distillation rate By e q u a t i n g net e v a p o r a t i o n , migration, a n d c o n d e n s a t i o n rates, the overall distillation rate is obtained. B o t h isot h e r m a l a n d a d i a b a t i c c o n d i t i o n s will be considered. (a) Isothermal diffusion Placing E = M-----C in e q u a t i o n s
(la),
(2a) a n d
(4), (7)
p ' ~- V + c -? (2E/a) -- Ve E/b,
where a, b, c, are c o n s t a n t s : a = 0.472/~/T1, b = 0.796DP/TL, c = pc~/(T1/Tc). Values of a, b, c are s h o w n in T a b l e I for a r a n g e o f temperatures. W h e n E is greater t h a n 8b, the value o f the e x p o n e n t i a l t e r m in e q u a t i o n (7) is negligible--less t h a n 1 per cent. T h u s : a
E = 2 ( p ' -- V --c),
(8)
provided that : p'>
Isothermal diffusion: T ~ 600°C, L = 3 0 c m , O. 1O, c -- 0 (cold condenser) Evaluation o f equation (7)
16b +V+c. a
W i t h a cold condenser, a distillation p a t h of 30 cm, a v a c u u m o f 100/~, a n d a n e v a p o r a t i n g t e m p e r a t u r e of 600°C, e q u a t i o n (8) is valid d o w n to a zinc c o n t e n t of 0.04 per cent in l e a d ; f o r 580°C e v a p o r a t i n g t e m p e r a t u r e t h e corresp o n d i n g limit is 0.05 p e r cent Zn. W i t h higher v a c u u m (i.e. lower values o f V) the a p p r o x i m a t i o n is valid to even lower limits. T h u s the simplified e q u a t i o n (8) m a y be used over practically the whole r a n g e of zinc c o n t e n t s in practice ( 0 . 5 6 ~ . 0 5 per cent Zn). W i t h a liquid zinc c o n d e n s e r (at 420°C, the melting p o i n t o f zinc) a v a c u u m o f 100/~, a n d a distillation p a t h of 30 cm, the a p p r o x i m a t i o n h o l d s d o w n to 0.22 per cent zinc in lead with a n e v a p o r a t i n g t e m p e r a t u r e o f 540°C, or 0.32 per cent Z n at 520°C. T h u s the a p p r o x i m a t i o n m a y easily be m a d e in practice where zinc c o n t e n t s in the Imperial Smelting Process use of v a c u u m dezincing a p p r o x i m a t e 1 p e r cent. (b) Adiabatic diffusion Placing E = M = C in e q u a t i o n s (la), (2a) a n d (6) does n o t p e r m i t of a n analytical solution, because u n f o r t u n a t e l y the value of the integral I c a n n o t be expressed in t e r m s of simple functions. A n u m b e r o f graphical evaluations was m a d e for t h e values T1 = 873°K, L = 3 0 c m , V = 0.1 or 0.05 torr, a n d in each case it was f o u n d t h a t in the r a n g e of practical interest in lead refining (i.e. w = 0.05-0.56 p e r cent) the value of Pl lies extremely close to P. T h e physical significance o f this is t h a t the n i t r o g e n p a r t i a l pressure just a b o v e the e v a p o r a t i n g surface is negligibly small, a n d since P:P2+
V,
(9)
Pz : P l -- V.
[,
]0.,
Solution o f e q u a t i o n s (la), (2a), a n d (9) yields : -- c = a
1 --
V = 0.1 or 0.05 (i.e. v a c u u m of 100 or 50/0 e v a p o r a t i n g surface at 600°C, fixing the value o f a at 0.016, c o n d e n s i n g surface cold (¢ = 0) or at 4 2 0 ° C ( c - 0.173), with a distillation p a t h of 30 cm. Tables II a n d III illustrate the m o s t c o n v e n i e n t m e t h o d of evaluating e q u a t i o n s (7) a n d (10), a n d the results are s h o w n graphically in Fig. 3. A t very low zinc c o n c e n t r a t i o n s the difference between c o r r e s p o n d i n g expressions is nearly 20 p e r cent, b u t is mostly less t h a n 10 p e r cent, a n d at higher zinc c o n c e n t r a t i o n s (above 0.2 p e r cent Z n ) is a b o u t 1 p e r cent or less. It is quite evident that, a l t h o u g h a d i a b a t i c c o n d i t i o n s a l m o s t certainly prevail, .the m u c h simpler expression for isothermal c o n d i t i o n s c a n be applied w i t h o u t i n t r o d u c i n g very great inaccuracies in the case o f lead refining (0.05-0.56 p e r cent Z n in lead) a n d with negligible inaccuracy in the case of zinc p r o d u c t i o n (approx. 1 per cent Z n in lead). TABLE II(a)
1 _
-- V
(10)
a
7. Comparison of adiabatic and isothermal distillation E v a l u a t i o n s were m a d e o f e q u a t i o n s (7) a n d (10), a p p l y i n g values of the c o n s t a n t s w h i c h could be of practical interest :
I
E/b
E × 105
50~
:
p'
0.0625 0.125 0.25 0.5 1 2 4 8 16 32 64 128 256 512
0.176 0.352 0.705 1.41 2.82 5.64 11.28 22.6 45.1 90.2 180 361 722 1444
,J w
0.0033 0.0063 0.0120 0.0215 0.0351 0.0503 0.0632 0.0782 0.1064 0.1628 0.2756 0.5012 0.9524 1.8548
V = 0.05 or
100~ p'
0.00091 0.00177 0.00335 0.0060 0.0098 0.0140 0.0176 0.0218 0.0297 0.0454 0.0769 0.140 0.266 0.517
w
0.0063 0.0122 0.0231 0.0412 0.0667 0.0936 0.1123 0.1282 0.1564 0.2128 0.326 0.551 1.002 1.905
0.00176 0.00340 0.00645 0.0115 0.0186 0.0261 0.0314 0.0358 0.0437 0.0594 0.0909 0.154 0.279 0.532
By setting E at multiples of b, it is simple to calculate p'. Values of p' are converted to values of w by reference to data for the lead-zinc system--see Table I. TABLE II(b) Isothermal diffusion: T ~ 600°C, L = 30 cm, V - ~ 0.05 or 0.10, c = 0.173 (liquid zinc condenser) Evaluation o f equation (7)
50~ E/b
E × 105
0.0625 0.125 0.25 0.5 1 2 4 8 16 32 64 128 256 512
0.176 0.352 0.705 1.41 2.82 5.64 11.28 22.6 45.1 90.2 180 361 722 1444
100~
p:
W
0.176 0.179 0.185 0.195 0.208 0.223 0.236 0.251 0.279 0.336 0.449 0.674 1.125 2.028
0.0491 0.0500 0.0516 0.0542 0.0580 0.0622 0.0658 0.0700 0.0779 0.0936 0.1251 0.188 0.314 0.565
See note to Table lI(a).
..I
P"
W
0.179 0.185 0.196 0.214 0.240 0.267 0.285 0.301 0.329 0.386 0.499 0.724 1.175 2.078
0.0500 0.0516 0.0546 0.0597 0.0668 0.0743 0.0796 0.0840 0.0919 0.1076 0.139 0.202 0.327 0.580
Distillation u n d e r Moderately H i g h V a c u u m , illustrated by the V a c u u m Distillation of Zinc f r o m L e a d - - T h e o r e t i c a l
L,ooo
500 ~,
•
too
.
~
~
~"~
~~~ ~~ ~, ~/,// ,;~.
¸
SO
~'_~,,
./,/ ~
%
I0
I'0 0-00
/ 0.05
O.IO
Wt. °/o 'Zinc in
O'15
Leatl ~
FIG. 3. Comparison of adiabatic and isothermal distillation rates. Temperature 600°C.
L = 30cm.
V = as shown. Adiabatic diffusion. Isothermal diffusion.
, 0.20
89
90
T. R. A. DAVEY TABLE I l l ( a )
Adiabatic diffusion: T -- 6 0 0 ° C , L = 30 c m , V = 0 . 1 0 0 Evaluation o f equation (10) Cold condenser (c = O) Pl
P2
105E
0.11 0.12 0.13 0.14 0.15 0.20 0.27 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20
0.01 0.02 0.03 0.04 0~05 0.10 0.17 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10
25.8 45.7 64.2 82.1 99.4 183.4 297 347 508 669 829 991 1150 1308 1471 1630 1792
p' 0.1261 0.1486 0.1702 0.1913 0.2121 0.3147 0.456 0.518 0.718 0.918 1.118 1.32 1.52 1.72 1.92 2.12 2.32
Liquid zinc condenser (c -- 0.173) w
105E
p'
w
0.0352 0.0415 0.0475 0.0534 0.0592 0.0878 0.127 0.145 0.200 0.256 0.312 0.368 0.423 0.479 0.535 0.591 0.647
--
--
--
fo
or
dt = 1 0 0 ~
wf
~
sec.
(11)
N o w a s s u m i n g t h a t it is p e r m i s s i b l e to a p p r o x i m a t e v a p o u r p r e s s u r e (p') as a l i n e a r f u n c t i o n o f w e i g h t p e r c e n t (w), n a m e l y as : p' -- x w + y, (12) t h e n e q u a t i o n (7) b e c o m e s :
w ~=
1 ( 2E -E/b) V + c -- y + -- Ve x a
.
(13)
S u b s t i t u t i n g (13) in (11) a n d i n t e g r a t i n g : -21 71 232 393 553 715 874 1052 1195 1354 1516
-0.283 0.344 0.545 0.745 0.945 1.145 1.345 1.545 1.745 1.945 2.145
-0.0790 0.0961 0.152 0.209 0.264 0.320 0.376 0.431 0.487 0.543 0.601
See note to Table IIl(b).
100At
ax
I/ ~lli e-U / du c m 2 . s e c / g , E f + b x d tff u
Ei
2 =
B
In
04)
E w h e r e u --- b ' suffix i = initial, f = final. Values of the exponential integral, taken from Smithsonian T a b l e s , a r e s h o w n in T a b l e IV. TABLE IV
Values o f exponential integral,
fU
e -u du
-~
U
TABLE I I I ( b )
Adiabatic diffusion: T 600 °C, L = 30 c m , V = 0.050 evaluation o f Equation (10) Cold condenser (c 0) PI
P2
105E
P'
0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.22 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
0.01 0.02 0.03 0.04 0.05 0.10 0.15 0.17 0.20 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
22.9 40.9 58.4 75.2 91.7 174 254 286 336 414 574 735 894 1054 1212 1370 1540
0.0743 0.0956 0.1165 0.1370 0.1573 0.2586 0.3585 0.399 0.460 0.559 0.759 0.959 1.16 1.36 1.56 1.76 1.96
Liquid zinc condenser (c = 0.173) w
105E
p'
w
0.0207 0.0267 0.0325 0.0382 0.0439 0.0722 0.1001 0.111 0.128 0.156 0.212 0.268 0.323 0.379 0.434 0.490 0.547
--
--
--
-10 60 138 298 459 618 778 936 1094 1254
-0.226 0.287 0.386 0.586 0.766 0.986 1.186 1.39 1.59 1.79
-0.063 0.080 0.1077 0.1635 0.219 0.275 0.331 0.387 0.442 0.498
It is most convenient to evaluate by taking pairs of values o f p l and P2 which satisfy equation (9). Then E = C is evaluated from equation (2a). Next p' is evaluated from equation (la). w is obtained from p" by reference to data for the lead-zinc system--see Table I.
8. Application to process To interrelate the rate of distillation with the time of the process, area available, zinc concentration, and amount of l e a d to b e t r e a t e d , c o n s i d e r first a b a t c h p r o c e s s . F r o m a n a r e a A c m 2 o f B g o f lead, t h e r a t e o f e v a p o r a t i o n is E g / c m 2 . s e c a t t i m e t sec a f t e r t h e start, w h e n t h e c o n c e n t r a t i o n o f z i n c is w (wt p e r cent), a n d t h e c o n c e n t r a t i o n c h a n g e s b y - - d w p e r c e n t in t i m e dt sec, t h e n :
-- B d w EAdt =
100
g
u
Integral
u
Integral
0.0625 0.125 0.25 0.5
2.2592 1.6242 1.0443 0.5598
1 2 4 8
0.2194 0.04890 0.003779 0.000037
E q u a t i o n (14) m a y b e a p p l i e d to e i t h e r a b a t c h o r c o n t i n u o u s p r o c e s s , s i n c e in t h e f o r m e r case, f o r a g i v e n v a l u e o f t h e r i g h t h a n d side o f (14) (i.e. f o r a g i v e n r e d u c t i o n o f zinc content of lead under stated conditions of vacuum, t e m p e r a t u r e , etc.), t i m e r e q u i r e d , a r e a r e q u i r e d a n d m a s s o f l e a d a r e r e l a t e d so t h a t b y fixing t w o t h e t h i r d m a y b e calculated. F o r a c o n t i n u o u s p r o c e s s B i t m a y be r e p l a c e d b y F, a flow r a t e o f l e a d t h r o u g h t h e still, a n d t h i s is r e l a t e d to A, t h e d i s t i l l a t i o n a r e a . S o m e e v a l u a t i o n s o f e q u a t i o n (14) a p p e a r in Fig. 4. T h e s e e v a l u a t i o n s w e r e m a d e b y s e t t i n g E a t m u l t i p l e s o f b, w h e r e u p o n w is f o u n d f r o m e q u a t i o n (13), a n d A t / B f r o m e q u a t i o n (14), a n d t h e t w o m a y b e p l o t t e d a g a i n s t e a c h o t h e r . T h i s p r o c e d u r e c a n b e f o l l o w e d in T a b l e V. A s in t h e d i s c u s s i o n o f e q u a t i o n w h e n u > 8 , i.e. w h e n w > l / x (16b/a s e c o n d t e r m in e q u a t i o n (14) a n d e q u a t i o n (13) b e c o m e negligible, becomes : 100At
B
....
2
ax
In
(8), it will b e s e e n t h a t + V + c --y), then the t h e e x p o n e n t i a l t e r m in s o t h a t e q u a t i o n (14)
x w i + y -- V - - c xw[q-yV-c
(15)
In t h e p r a c t i c a l a p p l i c a t i o n o f v a c u u m d e z i n c i n g in l e a d refining, e q u a t i o n (15) is v a l i d f o r all e x c e p t p e r h a p s t h e e n d o f t h e d i s t i l l a t i o n p e r i o d , w h e n t h e z i n c is l o w e s t - - s e e t h e d i s c u s s i o n o f e q u a t i o n (8). I n t h e a p p l i c a t i o n to z i n c p r o d u c t i o n , t h e a p p r o x i m a t i o n (15) will b e v a l i d f o r t h e w h o l e r a n g e e n c o u n t e r e d in p r a c t i c e . T h i s n a t u r a l l y g r e a t l y simplifies practical calculations.
Distillation u n d e r M o d e r a t e l y H i g h V a c u u m , illustrated by t h e V a c u u m Distillation o f Zinc f r o m L e a d - - T h e r o r e t i c a l
91
I'0
0"I0 8
6
~
._____------. ~ o ~ ,
~..~or ~._~~,~_¢_._-
~
_ ~
,..
0"01 0
4 -
At 8 -
6 cm~.sec/g
~
FIG. 4. Distillation process times. Temperature 560°C.
L = 30 cm.
Cold condenser.
I~
I0
92
T. R. A. DAVEY TABLE V Evaluation o f equations (13) and (14).
A t 560°C, a = 1.63 × 10 -2, b = 2.73 × 10-5, c = 0
V = 0.10 L = 30 u
w
q
256 128 64 32 16 8 4 2 1 ½ ¼
Term 1
1.0
0
0.525 0.281 0.164 0.107 0.0790 0.0650 0.0573 0.0476 0.0340 0.0210 0.0115
0.43 0.44 0.44 0.44 0.44 0.44 0.44 0.43 0.43 0.43 0.43
Term 2 0
V = 0.05 L = 30 Cumulative total
Total 0
0 0 0 0 0 0 0.07 0.84 3.19 6.36 9.03
0
0.43 0.44 0.44 0.44 0.44 0.44 0.51 1.27 3.62 6.79 9.46
0.43 0.87 1.31 1.75 2.19 2.63 3.14 4.41 8.03 14.82 24.28
w
Term 1
Term 2
Total
Cumulative total
1.0 0.496 0.253 0.137 0.0810 0.0531 0.0393 0.0320 0.0254 0.0179 0.0109 0.0058
0 0.45 0.44 0.44 0.44 0.44 0.43 0.43 0.43 0.43 0.43 0.43
0 0 0 0 0 0 0 0.04 0.42 1.59 3.18 4.51
0 0.45 0.44 0.44 0.44 0.44 0.43 0.47 0.85 2.02 3.61 4.94
0 0.45 0.89 1.33 1.77 2.21 2.64 3.11 3.96 5.98 9.59 14.53
Term 1 and Term 2 are the two terms of the right hand side of equation (14) ; their sum gives the " Total ". The evaluation was made from each value of w to the next. The column " Cumulative total " represents total units from w = 1.0 % Zn. The first " Term 1 " is obtained using the approximation of equation (15). I n a n o t h e r range, t h a t o f very dilute solutions o f zinc, a different a p p r o x i m a t i o n m a y be made, by neglecting the first t e r m o f the right h a n d side o f e q u a t i o n (14)--this is less t h a n 1 per cent o f the second term w h e n E is less t h a n a b o u t 0.01 b u n d e r the conditions o f vacuum a n d t e m p e r a t u r e o f the m a g n i t u d e s being considered elsewhere. Thus, w h e n : 1 (
W~
X
b
V + c - - y + 50a
_Ve_O.Ol )
,
then, neglecting the term 2 E / a in e q u a t i o n (13) : V E = b ln lOOAt - B
and
=
(16)
V + c - - x w -- y '
V
~ui
e -u
-b x
J uf
u
du
(17)
'
V
where
u = In
Now
co n
9. Correction for inequality of evaporating, condensing and distillation areas Previously, in deriving e q u a t i o n (7) f r o m equations (la), (2a) arid (4), the areas available for evaporation, c o n d e n s a t i o n a n d distillation were a s s u m e d to be equal. In general the areas need n o t be equal, a n d this is the case in practical applications to date. I f A E ~ evaporating area, A C = c o n d e n s i n g area, AD = distilling area (for v a p o u r flow, neglecting for the m o m e n t that if A E =# AC, then A D is n o t a constant), 1 = AC/AE, and m ~ AD/AE, then equations (la), (2a) and (4) become, for unit evaporating area : E
= a (p' - - p l ) ,
(lb)
C
= la (P2 - - c),
(2b)
V+c--xw--y e- u -u du=
?l2
--0.5772 - - I n n + n - -
M
2l ~ - k -
V V+p2--Pt
= mbln
"
(4a)
The solution yields : tt3 + 3[_3 - - " . . . . .
(18)
a n d w h e n u f i s very small, all terms in (18), after the first two, vanish, so that (17) becomes : 100At - B
V
= bx
V =
bx
(--0.5772--1n ttf)
( _ 0 . 5 7 7 2 _ 1 n l n V + c - - x w -V -y
) cmZ.sec/g (19)
E q u a t i o n (19) is o f academic interest only, in the case o f v a c u u m distillation o f zinc f r o m lead, as it refers to such low limits o f zinc in lead as would be impractical to recover, a n d such zinc c o n t e n t s would, if necessary, be r e m o v e d f r o m lead by other means.
(l+1) l
E a
= p ' -- V -
c + Ve-E/mb
(7a)
in c o m p a r i s o n with the f o r m e r : 2E = p" -- V a
c + Ve-E~,'b.
(7)
In cases where the exponential term can be neglected in (7) it will also be negligible in (7a), p r o v i d e d that m does n o t differ very far f r o m unity. It is therefore n o t necessary to allow for the fact that A D is n o t constant. As the right h a n d sides o f equations (7) and (7a) are nearly equal, the corrected distillation rate will be increased, f r o m (7) to (7a), by the factor 21 .
~+ l
2 AC ~
A c + A~"
Distillation under Moderately High Vacuum, illustrated by the Vacuum Distillation of Zinc from Lead--Theoretical This was determined for unit area of evaporating surface, and the actual area to .be taken in any real case is 2 A E A C / ( A c + AE) which is the h a r m o n i c m e a n of A E and A c. This is the reciprocal of the m e a n of the reciprocals, and the physical significance is that the evaporating and condensing areas m a y be regarded as conductances, or reciprocal resistances, the m e a n of which is to be taken. The approximation used above, viz. neglect of the exponential term in (7) or (7a), is valid for b o t h the applications of v a c u u m dezincing, in lead refining and in zinc production, so the harmonic m e a n of the areas of evaporation and condensation may be taken as the effective distillation area.
93
It is necessary to determine r and this can be done by successive approximations : a
E1 ~_~ ~ (p' -- V -- c) g/cm2.sec.
(8)
Zinc vapour occupies : 22400
T
65.4
273
760 p~
.'. Velocity of zinc vapour :
=
T 954 -- c m 3 / g . Pl
954a T(p' -- V -- c) cm/seco 2 Pl
F r o m equations (la) and (8), a
E = a(p' --Pl) = 2 (p' -- V - - c ) ,
10. C o r r e c t i o n f o r h i g h v e l o c i t i e s o f d i f f u s i o n The expression derived for isothermal diffusion (equation (8)) reduces, when the residual pressure is so small as to be negligible, and the condensing surface is so cold as to prevent any re-evaporation, to : ap' E = -2 (20)
whence 2pt = p ' + V + c, 954aT(p'-.'. v =
V--c) cm/sec.
p'+V+c
2 I(2R_T~ N o w ~ = ~/3 ~ / \ ~ M J
cm/sec.
when V = 0, c = 0, Ve-Ejb = O. N o w under these conditions the formula for " m o l e c u l a r distillation ", E = ap' (21) should be valid. It is obvious therefore that expression (20) understates the expected rate of distillation by half at very low residual gas pressures, and presumably is also not valid at moderately low pressures. The source of this anomaly is to be found in considering the expressions derived for evaporation and condensation (la) and (2a), where the pressure of the distilling atoms, Pl or P2, was in effect considered to be equal in all directions. H o w e v e r it is evident that if the atoms are moving with a large net average velocity f r o m the evaporating to the condensing surface, the effective back pressure above the evaporating surface will be less than Pl, and the effective forward pressure above the condensing surface will be greater than p2, where Pl and P2 refer to the pressures exerted at right angles to the direction of rapid motion. D u e to the m e t h o d used in deriving M, in (4), the expression for isothermal diffusion of vapour is unaffected by these considerations, and it is still permissible to write P = pz 4- V, so that equation (4) is still valid. If the average net velocity of zinc vapour across the distillation space be v cm/sec, and the average projected velocity of zinc atoms be ~ cm/sec, then since p~ct~ 2, the expression pj must be replaced by (1-r)2p~ in equation (la) and P2 by (1 4- r)2p2 in equation (2a) where r = v/& It is to be noted that the symbol g is here used for average projected velocity, and not the average velocity, as is usual. It is thus 1/~/3 times the usual value. W h e n r < < 1, then the solution of equations (la), (2a) and (4) remains as before : equation (8). Otherwise the solution becomes : a
E = 2 + 2r 2 [p' (1 q- r) 2 -- V(I --r2) 2 - - c ( l --r)2].
(22)
Thus a first approximation to rl = v/~ m a y be obtained, inserted in equation (22), and a second approximation to/!7 obtained, say E2. N o w a second approximation to r may be found as E2/E1 = r2, and so on. This correction has been evaluated for one set of conditions, and plotted in Fig. 4. Over most of this range of compositions, the corrected distillation rate is just over double that calculated from equation (8), and in fact is calculated to be over 90 per cent of the " molecular distillation " rate, except at low zinc concentrations (around 0.1 per cent Zn) where the velocity of zinc vapour falls to only a small fraction of the molecular velocity. The correction has also been evaluated for conditions which might be of practical interest in zinc production, and the results are summarized in Table VI. It can be seen that, at around 1 per cent zinc in lead, the corrected distillation rate is about 2} times that calculated f r o m equation (8). Referring back to equation (22) it can be seen that this still does not reduce to the correct expression for molecular distillation : E = a (p' -- Pc),
(23)
in the case where V becomes negligible but c does not. This arises because, in the correction terms for equation (22), the condensing surface was considered, not as a source of atoms which could distill across to the evaporating surface, but simply as a source of atoms which, by contributing to the pressure above the condensing surface, provide a barrier retarding the distillation. Thus a further term should be added to equation (22) : minus q~ ac on the right hand side, where q~ is the probability of a zinc a t o m evaporated f r o m the condensing surface reaching the evaporating surface. This case will not be considered further here, since ~0 is approximately zero in the examples being considered.
94
T. R. A. DAVEY TABLE V I
Corrected distillation rate o f zinc in lead under conditions: Distillation p a t h 30 cm approximately Evaporating surface temperature 520 °C Condensing surface temperature 420 °C Vacuum lOO/z (Projected mean molecular velocity o f zinc vapour at 520°C is 29,200 cm/sec)
~ Zn in lead
0.5 0.75 1.0 1.5
Mean E1 × 103 Ratio Velocity (calcd. f r o m C o r r e c t e d C o r r e c t e d E of Zn eqn. (8)) E × 103 vapour × (g/cm2.sec) (g/cm2.sec) Et 10-3 (cm/sec) 1.55 3.07 4.83 7.55
5.36 9.02 12.6 18.6
3.46 2.94 2.61 2.46
Corrected E as ~ o f " molecular distillation rate"
11.4 15.2 17.2 19.7
71 85 89 95
II. Comparison with practice At the time of writing no plant for zinc production has yet operated, but the continuous vacuum dezincing plant at the Broken Hill Associated Smelters, Port Pirie, South Australia, has been operating for lead refining for nearly a decade. Exact data are difficult to obtain, but the following data apply approximately : Av. lead flow through vacuum vessel
27 tons/hr
Av. zinc in input lead
0.58 per cent
Av. zinc in output lead
0.04 per cent
Mean evaporating surface temperature
560 °C
Operating vacuum
50-100/z
Evaporating surface area (visually estimated at 1/3 coverage of total surface available)
26 ft 2
Condensing surface area
21 ft 2
Referring to Fig. 4, we find the calculated area required to reduce zinc from 0.58 per cent to 0.04 per cent is 3.02 -- 0.18 = 2 . 8 4 c m z for a lead flow of 1 g/sec. This represents 23.3 ft 2 for 27 tons/hr of lead flow. The harmonic mean of 26 and 21 is also 23.3, but since the evaporating surface area cannot be estimated accurately, the agreement may not be so good as it appears. However, it can certainly be stated that, to a first approximation, the method of calculation here developed gives results in accordance with practice.
sarily in some particulars, but is not stringent enough in others. It has been assumed : (i) that the evaporating surface is clean (i.e. free of mechanical hindrances) and sufficiently agitated that the surface temperature and concentration do not differ from those in the bulk of the distilland : (ii) that the space between evaporating and condensing surface is clear of mechanical obstructions to vapour flow, and that the resistance of the sides of the containing vessel is also negligible, so that the total static pressure is equal at all points ; (iii) that the difference between adiabatic and isothermal passage of vapour can be neglected ; (iv) that the condensing surface is parallel to the evaporating surface ; (v) that the area of the distillation path remains constant throughout its length; (vi) that the vapour flow from evaporating surface to condensing surface so preponderates that flow in the reverse direction can be neglected ; (vii) that any vapour condensing while distilling reaches the condenser, and does not fall back into the evaporating surface ; (viii) that only one volatile species is present in the distilland ; (xi) that the residual gas is chemically inert to the vapour ; (x) that the residual gas pressure is a maximum at the condensing surface, i.e. the still is " self-pumping ". If any practical experiment yields results which indicate a distillation rate greatly less than the calculated, then the above assumptions should be examined. Frequently experiments are designed which do not conform to conditions (i), (ii) or (vii) above, although attention to these items might increase many-fold the practically attainable distillation rate. The analysis may be applied to the vacuum distillation of a pure substance, or a solution of one volatile substance in an involatile solvent. The distillation rate E is related to the partial pressure of distilling substance in the distilland, p', the residual gas pressure V, and the length of distillation path L by the equation : p'
V + c 4-
2E a
- -- Ve E./b
where a, b, c are simply determined for the particular substance distilling, and the inert residual gas, at given values of T for evaporating and condensing surfaces, and of L. This can frequently be approximated to : a
E=
2(P'--V--c)
when
12. Conclusion It is important to consider the conditions of applicability of the formulae and techniques of this paper, the more so since a recent Russian publication15 examining the author's previous account 1 would restrict the applicability unneces-
(7)
p'>
16b -
a
+ V+c
,
or even to : a
E = 2 (P'-
v),
(8)
Distillation under Moderately High Vacuum, illustrated by the Vacuum Distillation of Zinc from Lead--Theoretical when the partial pressure of the condensate is negligible. If, in any binary system, the partial pressure of the volatile species p ' can be related linearly to its weight per cent w, in the distilland, as : p'=xwq-y,
then the relation between weight of distilland B, area of distilling surfaces A, and time of distillation process, t, for the case of equation (8), is given by : lOOAt
2
B
ax
In
x w i + y -- V - xwf+y--
V--e
c
(15)
The solution of the full equation (7), involving the use of standard tables for an integral, has also been illustrated. W h e n the solution, obtained as above, indicates a velocity of distilling atoms or molecules which is a sizeable fraction of the free molecular velocity, then a correction must be applied to allow for the fact that pressure can no longer be regarded as being equal in all directions. It m a y then be found that distillation rates approaching the " molecular distillation rate " are capable of achievement, although the residual gas pressure is too high, by formal definition, for this to occur. When the resistance to vapour flow is small, then the harmonic mean of evaporating and condensing surface areas (where these are different) is taken for the distillation area. When the resistance to vapour flow is not negligible, then any change in area along the distillation path should be allowed for when integrating equation (3). W h e n assumption (ii) above is not valid, then the resistance offered to vapour flow by the vacuum vessel walls, as distinct from the inert residual gas, must also be allowed for, and
95
equation (4) must be replaced by the appropriate expression. This case has not been considered in the present contribution.
Acknowledgements The author is deeply indebted to Mr. J. Lumsden of A v o n m o u t h for discussions and criticism, and to Mr. R . Cornelius of Sulphide Corporation, Cockle Creek, N e w South Wales, Mr. D. R. Blaskett of B.H.A.S., Port Pirie, South Australia, and Dr. I. G. Matthew of A v o n m o u t h , for constructive criticism of the draft, and is grateful to the M a n a g e m e n t of The Broken Hill Associated Smelters Pty. Ltd., Port Pirie, Australia, and to Imperial Smelting C o r poration Ltd., A v o n m o u t h , for permission to publish this paper.
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