Appl . sci. Res . Vol. A3

THE SUBLIMATION OF ICE IN VACUUM by H. KRAMERS and S. STEMERDING *)

Laboratorium voor Physische Technologie, Technische Hogeschool, Delft

Summary A n a c c o u n t is g iven of e x p e r i m e n t s on t h e r a t e of s u b l i m a t i o n of ice in

v a c u u m a n d in t he p r e sence of a sma l l a m o u n t of air. T h e r e su l t s s h o w t h a t t h e g a s k ine t i c a c c o m o d a t i o n coef f ic ien t of t h e ice b e t w e e n - - 4 0 a n d - - 6 0 ° C lies n e a r u n i t y . Also i t a p p e a r s t h a t t h e m a x i m u m s u b l i m a t i o n r a t e in v a c u u m a n d a t sma l l a i r p r e s s u r e s is in a g r e e m e n t w i t h t he t h e o r e t i c a l a n a l y s i s g i v e n b y C a r m a n.

§ 1. Introduction. As a part of an investigation into the mecha- nisms determining the rate of sublimation of ice in the freeze drying process a number of experiments were performed on the sublimation of pure ice in vacuum. The measurements have been carried out in an apparatus (to be described in § 2) in which the ice is evaporated from a flat surface to a parallel condenser, as shown schematically in fig. 1.

~\.'X'X\\ " k \ \ \ \ \ \ \ \ \ \ \ \ \ \ k \ \ ~ \ ~//k//>'/'//A

/ / o l G l ~ , v . v J ' P | ,el

Fig. 1.

Now, for the sublimation rate w o from a surface with absolute temperature T. and saturated vapour pressure /7. into a hard

*) Now at Royal Dutch/Shell Laboratory, Amsterdam.

- - 7 3 - -

74 H. KRAM-ERS AND S. STEMERDING

vacuum the well-known Knudsen formula applies :

H --1/ M (1) W O ~ e s " - - F 2 ~ R T , , '

where M represents the molecular weight of the evaporating sub- stance and R the gas constant. In this formula the accomodation coefficient a, giving the fraction of the molecules which on colliding with the substance do not rebound, has been assumed to be unity.

In freeze drying practice 1) a similar formula is sometimes used for the actual rate of sublimation w"

= I H , , . . . . M = I o. (2) 2~RT, ,

Here / is a number between 0 and t, which is called the coefficient of evaporation. In it the following effects causing w to be smaller than w 0 may have been collected:

a) the accomodation coefficient is smaaler than unity. b) the pressure in the vapour space is not zero, so that vapour

molecules have a chance of returning to the evaporator surface. This pressure will be at least the vapour pressure Hcs of the condenser surface, eventually increased as a result of resistance to vapour flow from the evaporator to the condenser, either by diffusion through incondensible gas (air) or by wall friction. The latter effect is important in freeze drying since the layer of dry product consider- ably obstructs this flow.

c) the actual sublimation rate w may be limited by the rate at which heat is being supplied to the evaporator and by the rate of withdrawal of the vapour by condensation, absorption or pumping.

I t will be clear t ha t , if the causes b) and especially c) greatly determine the rate of subIimation, a relation between w and w 0 has no physical meaning as the underlying mechanisms are different. As to the accomodation coefficient a, it has been found experimentally by T s c h u d i n 2) that its value for ice between --60 and --85°C is 0.94 --'_ 0.06, i.e. practically unity. A 1 t y a) and P r ii g e r ~), on the other hand, found for water near its freezing point a ~ 0.04. There is no reason, however, that this value would apply to ice near its melting point. As our own measurements make an a of 1 between --40 and --60°C very probable, we will keep to this value in the following.

THE SUBLIMATION OF ICE IN VACUUM 75

If there is no fr ict ional resistance between evapora to r and con- denser and if the to ta l wa te r vapour plus air pressure in the sys tem is P (P ~> 17c,), the ra te of subl imat ion is given b y

w = w o 1 R-TI in \ P _ _ /7 ' . (3)

This fo rmula is due to C a r m a n 5) who gave a ve ry a t t r ac t ive discussion on this mat te r . The idea under lying (3) is tha t there exist two resistances in series to the t ransfer of vapour . The t e rm w0(1 - - U'/17~,) gives the net t ranspor t b y free evapora t ion be tween the evapo ra to r surface and the adjacent vapour , having a par t ia l water va pou r pressure 17', (17,s < 17' < 17,,). The r ight hand expres- sion represents the diffusional flow of Water vapour th rough air f rom 17' near the evapora to r t o 17;, at the condenser surface, over thei r mu tua l dis tance l.

In connect ion wi th our exper iments , two special cases are of interest.

a) I f the air pressure at the condenser is re la t ively low, or P m/7c, t hen a l s o / 7 ' ~ P, which gives

W = W o 1 - - 17~s " (4)

Most of the measurements repor ted in § 3 belong to this case. b) I f w < w 0, e.g. b y an insufficient supply of heat , i t follows f rom

(3) t ha t 17' ~ / 7 , , . In the absence of air this means t h a t / 7 , , ~ 17~,, i.e. the surface t empera tu res of evapo ra to r and condenser are equal. In the presence of air (3) becomes

~I/IDP ( P - - /7~, ) = R T l in P - - / 7 , , . (5)

The exper iments described in § 4 on the influence of air pressure on the ra te of subl imat ion to a large ex ten t apply to this case.

§ 2. Exper imen ta l method. An invest igat ion into the mechanisms previously discussed" involves the following exper imenta l require- ments . In vacuum, ice mus t be evapora ted f rom one surface and condensed on ano ther surface. The ra te of subl imat ion has to be de te rmined during this process as well as the t empera tu res at which evapora t ion and condensat ion of the ice take place. The appara tus

76 It. KRAMERS AND S. STEMERDING

with which these reqllirements v~ere fnlfillectis shown diagramatical- Iy in fig. 2.

Fig. 2.

Its essential part is the balance I, which is enclosed in the vacuum chamber 2 with removable cover 3. To one end of the balance beam a thermally isolated copper t ray 4 (diameter 75 mm) can be clamped, which contains a layer of ice to be evaporated. This tray is placed vertically so that the mass measurements during evaporation will not be disturbed by a force perpendicular to its plane as a co_nse- quence of an eventual pressure difference between the back and the front. A counterweight 5 is attached to the other end of the beam. The mass variation of the ice on the t ray can be compensated by means of the calibrated torsion spring 6, which can be turned from the outside. A small glass window 7 enables the control of the zero position of the balance. With this arrangement continuous weighing in vacuum was possible with an accuracy of 15 mg.

The heat necessary for the evaporation is supplied by radiation from the electric heating element 8 at the back of the tray. Care has been taken to ensure a uniform flow of heat. Opposite the front of the evaporator and parallel to it the condenser 9 is placed at a distance which can be varied. Petroleum ether, which can be cooled down to --70°C, is circulated through the condenser. The temperature of the evaporator and the condenser can be measured with copper-con- stantan thermocouples with an accuracy of about 0. t5°C.

The space between the evaporator and condenser is enclosed in the

THE SUBLIMATION OF ICE IN VACUUM 77

aluminium cylinder 10, to which the evaporator is attached. By this arrangement the interference of leaked-in air with the vapour flow is reduced, cf. E d e 6). Moreover this diminishes to some extent heat exchange by radiation with the surroundings. On the other hand, it offers some resistance to vapour flow between evaporator and con- denser.

The vacuum is obtained by means of a mercury diffusion pump. For measurements at a reiatively high residual air pressure a con- trolled leak is connected to the pumping line. A McLeod gauge is directly attached to the vacuum chamber. As has also been explained elsewhere 7), this type of manometer gives the total pressure of a water vapour-air mixture ufiless the partial pressure of the water vapouris so high that it condenses on compression in the manometer. In the latter case the McLeod indicates the partial air pressure plus the saturated vapour pressure of water at the ambient temperature.

Before each experiment in a separate arrangement a layer of a few millimetres of distilled water was frozen on the tray, now placed horizontally. This was done in vacuum so that a layer of compact ice was obtained in good thermai contact with the t ray material and free from enclosed air. After the t ray had been clamped on to the balance and the vacuum chamber had been closed and evacuated, the condenser was put into operation and finally the heating was turned on. The amount of heat supplied practically determined the rate of sublimation. Thus, an experiment consisted in measuring this constant rate w by weighing and noting the corresponding temper- atures t e and t c as well as the pressflre in the system. Depending upon the rate, an experiment lasted from 4 minutes to one hour.

I t is to be noted that an experiment like this is non-stationary because in the course of evaporation the ice shifts from the evapora- tor to the condenser. Moreover, a flat ice surface on the evaporator has only a small chance of remaining fiat since the evaporation proceeds more rapidly at points where the layer thickness is relati- vely small. This effect, which in spite of a careful preparation of the samples could not be suppressed entirely, and the non-stationary character of the experiment cause the results to be less accurate than would be expected from the actual accuracy of measurement.

For the application of the theory in § 1, the ice surface temperatures t~ s and to, have to be known. They can be derived from the measured ~emperatures te and t~ by correcting for the temperature drop over the

78 H . K R A M E R S A N D S. S T E M E R D I N G

layer of ice. The latter quanti ty can be calculated from the heat flow corresponding to the measured rate of sublimation, the thickness and the thermal conductivity of the ice layers. In order to check this pro- cedure several experiments were carried out for t~ between --20 and --25°C and t~ between --30 and ~37°C at a residual air pressure less than 10 -3 mm Hg, The rate was of the order of 10 -4 g/cm 2 sec, i.e. 2 to 4% of the maximum rate at these temperatures. Consequently, in these runs t~, practically equalled t~. From the measured temper- ature differences t~ - - t~, the corresponding rates of evaporation and ice thickness, the thermal conductivity was found to range between 5.0 and 6.1 × t0 -3 cal/cm°C sec, giving an average value of (5.7 4- 0.4) 10 -3 at a mean temperature of --25°C. The ag-reement with the generally accepted value of 5.7 × 10 -3 at --20°C is satisfactory.

§ 3. The rate o/ sublimation in vacuum. The following kind of experiment clearly demonstrates the existence of a maximum rate of

\ ! \ ' ,

-6o \ \ *tw dlv/cnt 2 a¢~] O . t2 \

• / . 7

- ,~o \ .4so -Go - . o -~o F i g . 3 .

evaporation. Ice is evaporated at a certain constant rate ~dth a condenser temperature of e.g. --30°C and at the lowest possible residual air pressure. Under these circumstances t~, = t,~. If the condenser temperature is lowered, the sublimation rate remaining tile same, a difference between t~ and tc~ begins to show up. I t in- creases in such a way that finally t~, will become constant for very low condenser temperatures. Preliminary measurements, presented in fig. 3, show this behaviour for a number of different sublimation rates w. It is evident that at the corresponding temperatures t~ and

T H E S U B L I M A T I O N O F I C E I N V A C U U M 79

to, these are the m a x i m u m rates, which can be compared with (4). As an example the lowest curve has a point t,s = - - 57°C, t~s = - - 64°C. Wi th these values (4) gives a ra te of 1.3 x 10 -4 g/cm 2 sec, whereas the measured ra te was 1.2 × l0 -4. The agreement is r a the r good, showing moreover tha t the aceomodat ion coefficient is of the order of uni ty .

Table I gives the results of a numb er of final exper iments . T h e y are divided into three groups wi th a different dis tance l be tween evapora to r and condenser. Air pressures were a l i t t le below 10 -3 mm Hg. As during a run the t empera tu re var ied somewhat ( < 2°C), average values have been presented. The limits of w,~p/w~,~ due to this var ia t ion have been given in the last column. In par t icu lar a t high evapora t ion rates the actual er ror m a y still be grea ter because of the uneven evapora t ion men t ioned before.

T A B L E I

Dis tance e v a p o r a t o r condenser

mm

12

20

44

• Surf. t emp . e v a p o r a t o r

te$ °C

- - 4 4 . 0 - - 4 9 . 2 --52.6

- - 5 4 . 8 - - 5 7 . 0 --58.8

- - 6 1 . 7

- - 4 1 . 3 ---46.2 - - 4 6 . 0 - - 5 0 . 4 - - 5 0 . 4 - - 5 2 . 2 - - 5 4 . 3 - - 5 6 . 4 - - 6 1 . 3

- - 4 3 . 3 - - 4 8 . 3 - - 4 9 . 0 - - 5 1 . 2

Surf. t emp . condenser

L, [ °C

- - 4 7 . 9 - - 5 7 . 0 - - 6 1 . 8 - - 6 5 . 5 - - 6 4 . 9 - - 6 4 . 4 --70.0

- - 4 7 . 6 - - 5 2 . 7 - - 5 4 . 6 - - 5 8 . 9 - - 5 8 . 6 - - 6 2 . 2 - - 6 4 . 2 - - 6 6 . 7 --70. I

- - 4 7 . 6 - - 6 0 . 8 - - 5 9 . 4 - - 6 1 . 7

Subl . r a te calc. (4) 10 4 wcalc

g / c m ~ sec

3.6 3.6 2.4 2.0 1.5 0.80 0.75

7.0 4.4 5.1 3.1 3.0 2.6 2.0 1.6 1.0

4.3 4.5 4.0 3.1

Subl. r a t e exp.

104 Wex p

g / t i l l 2 s e e

5.21 3.54 2.09 1,55 1.16 0.59 0.675

6.60 5.00 3.72 3.31 2.78 2.23 1 . 6 0 1.15 0.607

2.70 2.30 2.65 1.86

wex# .w c,,~.

1.4 4- 0.2 1.0 _+_ 0.15 0,9 4- 0.15 0.8 4- 0.2 0.8 4- 0.15 0.7 ± 0.15 0.9 4- 0.05

0.9 -4- 0.05 I . I 4- 0. I 0 . 7 4 - 0 . 1 I . I -4- 0.1 0.9-4- 0. i 0 , 9 4 - 0. i 0.8 4- 0.05 0.7 / : 0.05 0.6 .-4- 0.05

0.6 '__ 0.15 0 . 5 / : 0 . 1 0 . 7 / : 0 . 1 0.6 -u 0. I

Most of the vMues of w,xp/wc~c are seen to be near unity. This, in a_ general way, supports the theory brought forward by C a r m an. This is especially of importance, as in most of these experiments the mean free path of the molecules was less than the distances l, whereas.

80 H. KRAMERS AND S. STEMERDING

(4) is only generally accepted for cases in which the mean free pa th greatly exceeds the dimensions of the vapour space. For this reason, among other things, T s c h u d i n did not extend his measure- ments above --60°C. As (4) has been derived for an accomodation coefficient of unity, these results show that this assumption is ve ry probably correct in the temperature range from dO to --60°C.

Only at low condenser temperatures and at a relatively great distance between the evaporator and condenser rather small values of w~p/wca~c are found. This was probably due to wall friction and to a minor extent to the pressure of residual air. We did not venture to calculate the first effect.

I t was impossible to extend these measurements to evaporator temperatures higher than --40°C because a few millimetres of ice would have been evaporated in too short a time.

§ 4. The rate of sublimation in the presence o/air. In a number of runs with constant evaporation rate and condenser temperature the air pressure was increased in a few steps from 10 -3 up to 0.4 mm Hg. This caused the evaporator temperature to increase correspondingly. From the temperature and pressure measurements P and/7cs were derived, whereas /7 ' could be calculated from the first part of (3). Only for relatively high air pressures the difference between/Tes and /7' was insignificant. Thus, from each run two to six values of (P - - H')/(P --/7~) were obtained. I t will be seen from (6) tha t for a constant sublimation rate and average absolute temperature T this form has to be independent of pressure. So for each run the average value ( P - H ' ) / ( P - II~) was calculated as shown in the last column of table U. Its variation due to the individual values has been indicated as a measure of the reliability of the experiments.

For purposes of comparison with tile second part of (3) use has been made of the approximate proportionali ty between the diffusion coefficient D and T312/P. Thus

DP Do___Po ~/~,, T To 12

in which the subscript refers to the values at 0°C and i atm. After this substitution (3) becomes

P - - H' ( RTo I~ lw ) P - - = e x p DoPol" t " (6)

T I l E S U B L I M A T I O N O F I C E I N V A C U U M 8 1

T A B L E II

l

m m

20

44

104 Wex p

! g / c m a sec

0.48 , 0.48

0.60 0.76 0.84

l 0 .95 , 1 . 0 6 I 1 . 4 2

1 . 0 8

I 0 .57 i 1.05 I 2.05

I

L °C

- - 2 0 - - 3 5 - - 5 0 - - 3 9 - - 3 7 - - 3 6 - - 4 8 - - 3 6 - - 4 4

- - 4 6 - - 4 7 - - 4 4

! 7 I - - I ° K

254 241 229 238 243 240 235 241 236

233 235 238

0,41 0 .42 0 .54 0 .67 0 .74 0 .84 0 .9a 1 . 2 5 1,49

1.01 1 . 9 8 3.60

Rr.'/_Z.- D"Po '1"I" ~'''~ i ca lc . (6)

i 0.66 0.66 0.58 0.51 0 .48 0 .43 0 .39 0.29 0 .23

0.36 0.14 0.03

exp .

0.68 ± 0.03 0.65 ± 0.02 0.56 i 0.02 0.60 ± 0.08 0.49 ± 0.08 0.48 4- 0.08 0 .29 ± 0 .04 0.49 4- 0. l0 0.22 -4- 0.06

0.40 :~ 0.06 0,15 4- 0 .05 0.17 - - 0.05

T

O @

\ O / ' , . , o R x * t . ~ N m '

° \

F i g , 4 .

W i t h D O = 0.22 cm2/sec the first f ac tp r of the exponen t has va lue of 9.33 × 104 consis tent c.g.s, uni ts . In (6) w is the mass f low of w a t e r v a p o u r per uni t surface of the diffusion space. Now in ou r e x p e r i m e n t a l a r r a n g e m e n t the surface avai lable for diffusion was c i r cu l a r wi th a d i ame te r of 75 m m over a dis tance of abou t 6 m m f rom the e v a p o r a t o r surface. Fo r the res t of the dis tance l a d i a m e t e r of 96 m m was ava i lab le (see fig. 2). F r o m this the average flow den- s i ty ~ can be ca lcu la ted to be 0.73 w~,p for l = 20 m m and 0.66 w~,p for l = 44 ram. These values were used in the c o m p u t a t i o n s for table I I .

A p p t . sci. Res. A 3 6*

82 THE SUBLIMATION OF ICE IN VACUUM

Fig. 4 presents a final comparison of the calculated and measured values of (P--H')/(P--IIcs). Most points agree well with the theoretical curve, which proves the validity of (3). For the large deviation of the two points to the right of the curve no explanation can be offered. I t may have been caused by an additional convective transport as a consequence of uneven evaporation.

This investigation was carried out by the T e c h n i c a 1 P h y- s i c s D e p a r t m e n t T.N.O. and T.H. at the request of and in cooperation with the G e n e r a i T e c h n i c a l D e p a r t m e n t T.N.O., the Hague. Our thanks are due for their permission to publish this work.

Received 27th April, 1951.

REFERENCES

1) F l o s d o r f , E. W.,Chem. Eng. Progress 43 (1947) 343. 2) T s c h u d i n, K., Heiv. phys. Acta 19 (1946) 91. 3) A l t y , T., Phil. Mag. 15 (1933) 82. 4) P r f l g e r , W., Z. Phys. 115 (1940) 202. 5) C a r rn a n, P.C., Trans. Faraday Soc. 44 (1948) 529. 6) E d e, A. J., Nature 159 (1947) 610. 7) F l o s d o r f , E. W., "Freeze drying", Reinhold Publ. Corp; New York 1949.

E de , A. J., J. sci. Instrum. 21 (1947) 198.