The sublimating film gas bearing

Wear, 30 (1974) 149-178 1; Elsevier Sequoia S.A., Lausanne ~ Printed.in The Netherlands

THE SUBLIMATING

149

FILM GAS BEARING

C. DAYSON National Research Council of Canada, 3904 West 4th Avenue, Vancouver, B.C. (Canada)

(Received May 3, 1974)

SUMMARY

The mechanism by which a block of dry ice can be supported upon a flat surface on a film of gas formed by its own sublimation, is put forward as the basis for the design of a new family of gas bearings. A general criterion for the design of such bearings is established and a number of arrangements for both journal and thrust bearings are suggested. A half-journal bearing of this type is modelled theoretically and its operating characteristics obtained. These indicate that such a bearing could operate in a stable condition in the manner envisaged. Calculations at specific operating conditions with carbon dioxide as the working medium indicate that bearings of this type could be usefully applied in cryogenic systems especially where their negligible friction, low pressure gas supply requirements would be advantageous.

NOMENCLATURE

A, A’

Points on characteristics denoting initial operating conditions Suffix denoting conditions at shaft-solid film interface Symbol denoting new operating condition. : b Thickness of shaft wall; suffix denoting operating conditions at bearing surface or relating parameters to conditions there C, Cr, C,Constants of integration Radial bearing clearance, suftix denoting solidification : Suftix denoting condition at internal shaft radius Shaft eccentricity Suffix denoting average condition in gas film or relating parameters to 7 conditions in film Suffix denoting conditions in gas supply zone or relating parameters to 9 conditions of gas supply Heat transfer coefficient at the exposed gas-solid film boundary Hs h Thickness of gas film I? Non-dimensional thickness of gas film Suffix denoting conditions in solidified film ‘K Thermal conductivity L Heat liberated or consumed during change of state from gas to solid or

150

Qd ril m P

I, Pd

q R

sh

T T

A E 9 0

“f “i

;d \’

P

Y 0

C. DAYSON

vice versa (for unit mass) Axial length of bearing Non-dimensional rate of solidi~~~tion or subIimation at exposed surface or in bearing arc Modified non-dimensional rate of solidification or sublimation at exposed surface or in bearing arc Rate of solidification or sublimation at exposed surface or in bearing arc Suffix denoting maximum or minimum Gas film pressure (gauge); suffix denoting effect of pressure flow Non-dimensional gas film pressure Modified non-dimensional gas’ film pressure Heat flux per unit area per unit time External shaft radius Suffix denoting effect of rotation Suffix denoting sublimation condition or condition of change of state Suffix denoting shaft properties Temperature or absolute temperature Ratio of absolute temperatures Thickness of solid film Gas velocity measured in direction x Bearing load Non-dimensional bearing load Modified non-dimensional bearing load Circumferential distance measured from centre of bearing arc Non-dimensional circumferential distance from centre of bearing arc Radial position in gas film measured from bearing surface Non-dimensional temperature parameter Ratio of shaft wall thickness to radial bearing clearance Ratio of absolute temperatures of sublimation and solidification divided by corresponding ratio of latent heats Symbol denoting a change in magnitude of a variable Ratio of shaft eccentricity to radial bearing clearance Absolute viscosity Angular position measured from the mid-point of the exposed half of the shaft Product of radial clearance, heat transfer coefficient at exposed surface of shaft divided by conductivity of dry ice Ratio of mean thermal conductivity of gas film and thermal conductivity of solid him Ratio of thermal conductivities of solid film and shaft material Non-dimensional variation in thickness of solid film Modified form of A Kinematic viscosity Density Ratio of solid film thickness to radial clearance Product of /I and K~ Angular velocity

THE SUBLIMATING

GJ”

FILM

151

GAS BEARING

Grashof number Nusselt number based on shaft diameter Prandtl number Reynolds number defined on basis of shaft diameter and rotational velocity Reynolds number defined on basis of film thickness and surface speed

NUD Pr

ReD Re,

INTRODUCTION

A lump of dry ice when placed to rest on the horizontal surface of another body can, if the mating surface of the dry ice is sufficiently flat, be supported entirely by the gas film generated as a result of the flow of heat from the surface and the consequent sublimation of the solid carbon dioxide; Fig. 1. One has in effect a gas film bearing that functions at the expense of a reduction in the size of one of the bearing components, i.e. the dry ice itself.

GAS FLOW

Y

I

I

HIGHER

TEMPERATURE SOLID I

Fig. 1. The support

of a block of dry ice on a gas film formed

by its own sublimation.

Useful applications of this bearing effect must be very restricted where this sacrificial form of lubrication takes place. Nevertheless the effect has certain characteristics that might be advantageously employed, were it possible to design a bearing which operates on this principle but did not suffer from this disadvantage. The presence of the gas film between the surfaces means that such a bearing would operate with negligible friction like a conventional hydrostatic gas bearing. What is more important however, and what could in certain circumstances give a bearing of this type an advantage over the conventional hydrostatic type, is the fact that a high pressure supply of gas would not be needed for it to operate effectively even at low speeds. SYSTEM

ENVISAGED

Any process by which the dry ice is periodically replaced or in which the dry ice member is continuously manufactured and fed mechanically to the load bearing zone, must be inconvenient or complex respectively. Therefore if the effect is to be considered as the basis for the design of a practical bearing, some way must be found of automatically replacing the ice as fast as it is consumed. The thesis of

152

C. DAYSON

the present work is that such replacement can be achieved as long as the block of dry ice is replaced by a thin film of it on a surface which is sliding over a bearing surface. Where, as shown in Fig. 2, the sliding surface is held at a temperature T, below that at which carbon dioxide sublimes, this gas (supplied at atmospheric pressure to the cavities between the pads) will solidify on it, and then subsequently sublime when it passes into the bearing zones just as it does from the block of dry ice in the manner described above. More generally for such a bearing to function, the temperatures of the two mating surfaces must be one above and the other below the sublimation temperature of the lubricating medium, and the hot surface must have gaps in it of significant size and be sliding sufficiently rapidly relative to the cold one. The system shown in Fig. 2 might for instance be considered as a thrust bearing with the cold surface as the slider and the hot ones as the pads.

I

SLIDER

-

T < T,

(

SUBLIMATIO

GAS SUPPLY

Fig. 2. Basic system and mode of action

of a continuously

acting

sublimating

gas bearing

A more detailed description of the process involved would be given thus. As the slider passes between the pads with the gas solidifying on it, the film is formed which is of increasing thickness towards the next pad. As the film passes through the load bearing zone between the pad and the slider, its thickness is reduced by sublimation and thus it provides the gas which in order to escape rapidly despite its viscous nature causes pressures to be generated in the film which can be utilised to sustain the bearing load. This process of solidification and sublimation can obviously go on indefinitely as the slider passes each cavity and pad, so the system would be therefore capable of continuous operation. The speed must be great enough so that the thickness of the film that builds up on the slider is not greater than the distance between it and the pad. Although the thickness of the solid film varies along the pad it cannot fall to zero, as long as the speed of the bearing is great enough to provide sufficient solid for generating a gas film that is thicker than the unevenness of the surfaces. While a gas film exists it cannot be in direct contact with a surface the temperature of which is below that at which it solidifies. It may be further argued that under typical running conditions the variation in the thickness of the solid film will be small in comparison with its average thickness. This will be true because the densities of substances in their gaseous form are many orders of magnitude lower than they are as solids, so that relatively large volumes of gas would be liberated by a small decrease in the thickness of the film. Only at very low speeds when a point on the slider spends an excessive time passing the pad should there be a

THE SUBLIMATING

FILM

153

GAS BEARING

BEARING

BEARINGS

THRUST

BEARINGS

Fig. 3. Sketches of possible gas film principle.

arrangements

for journal

and thrust

bearings

operating

on the sublimating

significant reduction in the film thickness. In the analysis given below it is assumed that the variation in the thickness of the solid film along the bearing is negligible. Figure 3 illustrates the various basic thrust and journal arrangements for such bearings. It is apparent that the common factor in all the examples is the discontinuity of the bearing surface of the higher temperature member, so that the colder surface is regularly exposed to the gas without the hotter surface being in close proximity. THEORETICAL

ANALYSIS

The arrangement analysed here is a 180 degree journal bearing with a symmetrically loaded hollow shaft; Fig. 4. The inside surface of the shaft is assumed

COLD

I

‘SHAFT!

HIGHER

TEMPERATURE

BEARING I

Fig. 4. Diagrammatic

representation

of semi-cylindrical

sublimating

film gas bearing.

C. DAYSON

154

to be at a temperature below that at which the lubricant sublimates, and the bearing at a higher temperature than this. A model with a hollow rather than a solid shaft was chosen in order to simplify this initial analysis of the problem; an analysis with a solid shaft would require that the three dimensional conduction of heat from it be represented. As indicated in Fig. 4 the thickness of the solid film on the shaft is assumed constant around its circumference for the reason given above; it will be shown below that this was a reasonable assumption. It is also assumed that the shaft stays in the symmetrical position indicated in Fig. 4; i.e. with its centre always somewhere on the vertical line through the bearing centre. In making this assumption the off-setting effect that accompanies any hydrodynamic action in the gas film (generated by the shaft’s rotation) is being ignored, and the possibility of effectively allowing for hydrodynamic action in the film is forfeited. However since appreciable hydrodynamic action in hydrostatic gas film bearings can only be achieved at very high rotational speeds, such an assumption will not effect the validity of the results obtained for the slow or medium speed ranges at which it is envisaged these bearings would normally operate. The method of analysis involves the examination of the heat flows into the shaft for both parts of the bearing system, the viscous flow of sublimated gas from the load bearing film and the manipulation of the relationships obtained to derive the governing equations. In this way expressions can be obtained for the film pressures, load capacity, rates of sublimation and solidi~cation and the thickness of the solid lilm at given shaft eccentricities and shaft, gas supply and bearing temperatures. Heat traasfer

iu gas solidifying

zone

The space above the shaft is supplied with the lubricant in its gaseous form, so the temperature therein must be greater than the sublimation temperature at the supply pressure. At the interface between the solid film and the gas, where it is envisaged that solidification is taking place, the temperature must be that of sublimation, i.e. lower than the gas temperature. It follows that heat must be transferred from the gas to the solid film. This flow of heat may be described by the equation YY= H&T,- T,)

(1)

As it solidifies on the shaft the gas gives up its latent heat which along with the heat received from the gas is transferred through the film to the colder shaft. This heat balance can be expressed thus q*y= Y,+&%

(2)

Also the transfer of this heat through the solidified film and the wall of the shaft is governed by the following equations respectively

THE SUBLIMATING

FILM GAS BEARING

155

By describing this heat flow with these simple linear equations it is being assumed in this instance that the thickness of the solidified film and the wall of the shaft are small in comparison with the shaft radius. Eliminating T,, and substituting for the heat flow parameters in eqn. (2)

Td) = H,(T,- T,,)+L,riz,

Ki(Kg-

f (I+ bKi/tK,)

(5)

which relates the rate of solidification of the gas to the gas supply and internal sfraft temperatures. Introducing the following non-dimensional parameters

(6) (7) and a non-dimensional $g

=

rate of solidification parameter defined as

L~cc

(8)

c T,gKi

eqn. (5) becomes

(9) Heat transfer in load bearing zone

It can be shown (Appendix I), that at typical conditions under which a bearing of this type would operate, the flow of gas between the bearing and the shaft will be laminar. Where this is the case it can also be demonstrated (Appendix II) that most of the heat emitted from the surface of the bearing is removed by conduction across the gas film to the surface of the solid film rather than by convection along the gas film to the outlet. The relationship governing the heat transfer at any point along the bearing arc therefore is qb

aT,-T,f)

=

h

(10)

At the surface of the solid film there is a heat balance between the heat flows to and from it, which is similar to the one derived above, except that where sublimation is occurring, heat is being consumed rather than produced. Thus qb

(11)

Lsrizs + qdf

=

The transfer of heat through the solid layer and the shaft is again described by two equations qdf

=

Ki(

T$-

T,f) t

(12)

156

C. DAYSON

qdf

bl(T,f- T,)

=

(13)

h

which are almost identical with eqns. (3) and (4). The different subscripts for the heat flows and temperatures allow for the higher sublimation temperature and the consequently greater heat flow through these materials that result from the fact that the pressures in the film will be greater than that of the supply gas. The four eqns. (10) to (13) may be used to derive an expression for the rate of sublimation from the solid layer on the shaft at any point on the bearing arc, in terms of the shaft and bearing temperatures, i.e.

Kf(Tb-T,f) =

L

k

s

h

Ki(T,-T,)

+

s

1f hKi/tK,,)

tf

(14)

Where

and defining a non-dimensional

rate of sublimation

eqn. (14) may be written as n;3 _ “J(Tb-1) -s K

l--TdS Z+prCi

(17)

Gas flow iH jilm

The flow of the sublimated gas from the film is generated by the pressure that supports the applied load and is resisted by the viscous nature of the gas. Where the flow of the film is laminar and it is assumed to occur in a circumferential direction only, i.e. the bearing is considered to be long in comparison with its radius so that side leakage can be neglected, the balance between the viscous and pressure forces may be expressed thus dp_ d2a dx - Yf dq’2

(18)

This is a very simplified form of the Navier-Stokes equation for a viscous fluid where fluid inertia and the variation of the fluid viscosity in the direction perpendicular to the film are neglected. Integrating twice with respect to y this becomes u

=

L . .dp* -+ 9 Y,

dx

2

c,y+cz

Any hydrodynamic effect on the pressures developed in the gas film are now neglected in comparison with the hydrostatic one already described. This assumption is specified mathematically by considering that as far as the gas flow is concerned, the shaft’s rotation may be neglected and the velocity of the fluid u is taken to be zero at the surface of the bearing and the solid film. That is u=O

at y=O

andy=h

THE SUBLIMATING

FILM GAS BEARING

157

and

u=-dp

L’ Zrr,(h-Y)

’

,_Jx

(‘9)

for any point on the bearing arc. The corresponding flow at that position is -h 4x

=

udy

J0

Substituting for u 1 qX=-29/.dx

““ih

h3

=-

0

(yh-y’)dy

dp

(20)

12~3 dx Generally the flow in the film should increase with increasing distance from the centre of the bearing as a result of the sublimation that is occurring. Where the gas is assumed to be incompressible this flow variation is represented by the following continuity condition.

dqx _=s

ti

dx

P/

(2’)

The gas has been assumed to be incompressible although for instance over the range of pressures that would occur in a properly designed bearing using carbon dioxide as the lubricant, i.e. 15 p.s.i.a. to 70 p.s.i.a. (the latter being the pressure at triple point), the effect of compressibility would be appreciable. Nevertheless in a preliminary treatment such as this where it is important that as many simplifying assumptions as possible are made, neglecting compressibility should be tolerable and not have a major effect on the results’, particularly where values for the gas properties at a typical film pressure and temperature are assumed (see Appendix III). Pressure gradient

Substituting parameters

in gas film

for rit, and qx in eqn. (21) and defining the non-dimensional

PC4L P/ ’ = 12R2r,y7$Ki’

x=x

R

(22)

we have (23)

158

C. DAYSON

Now the film thickness /z c;Ln to a close approximation

be represented by

/I = c + L’cos 0 - t

(24)

I;= 1-CCOS.U-5

(25)

1:= c:c .

(26)

or where Thence substituting for I? in eqn. (23) and integrating’

-T/I+c 1_(@--4(1 T+/?Ki

As the hydrodynamic effect has been neglected and because it has been assumed that the shaft is supported in a symmetrical manner, the pressure distribution in the film should also be symmetrical about the centre of the film. So the pressure gradient in the film at this central point must zero; that is at X=0,

dj$dx = 0

Thus

and the pressure gradient is described by E-(1-r)cosX (l_r)+cO~

1-T,,)(2,) 1 1 -UC +;

+ -r+/jlii

With this equation and eqn. (25) the non-dimensional pressure gradient can be calculated at points along the film for arbitrary non-dimensional values of the bearing and shaft temperatures. the eccentricity and the thickness of the solid film. Thence starting at %=x/2, the symmetrical pressure distribution may be found. Eccentricity-solid

film thickness

relationship

An expression defining the relationship between the eccentricity and the thickness of the solid film has yet to be obtained. This essential relationship is obtained by satisfying the requirement that for the bearing to run continuously the net rate of sublimation must be equal to the net rate of solidification. Since the net amount of material sublimated must all flow from the bearing as gas this balance may be expressed as

Substituting for qXand riz, and non-dimensionalising 1 - i& r(7;,-1) 5 + /3h.i

2I. = - 7

gives (29)

(30)

THE SUBLIMATING

159

FILM GAS BEARING

Thence by eqn. (27) we have 1 - 7;,,

- I(;i;,-1)

= r

T+fiKi

2Q(iiib- 1)

n((l-+&2)*

[sin-I(+&)

which is the relationship governing the interdependence

(31)

+ 91 -s} I

of T and E.

Bearing load

Where the load acts vertically through the centre of the bearing it is given by -3~12 w=

-1 J

xl2

pR cos 6dO

Defining a non-dimensional w’=

(32)

load parameter as

WC4L, p/ 12R31rj,TfKi

(33)

and taking into account the symmetry of the pressure distribution about the centre of the film this becomes

Thus where the pressure at any point in the film is dependent on the eccentricity (and the related thickness of the solid film) the above equation represents the relationship between the eccentricity and the load acting. It is therefore the final equation needed to completely describe the action of the bearing. Fluctuations

in the thickness

of the solid film

Despite the fact that it has been assumed that the variation in the thickness of the solid film around the shaft is negligible in comparison with the thickness itself, it is still possible to estimate the magnitude of the variation as long as it is small. Where At is the difference between the thickness of the solid film at the ends of the bearing arc and n/o is the half period of shaft rotation then At

=

m,? Pi0

Expressing At as a ratio of t, and substituting for rit, and t At -= t

T, KinlCi, L,C'piOT

Where I, a modified form of this ratio is introduced such that )c = fttLcc2Pio then

tT,KiK

(35)

160

C. DAYSON

SIMPLIFICATION

OF EQUATIONS

To obtain the data that will describe the characteristics of the bearing, eqn. (31) must be solved for given properties and characteristics of the lubricant as described by the parameters I, ICYand r and at arbitrary values of all but one of the other parameters. By making one simplification and rearranging the equations, it is possible to considerably reduce the number of independent variables. With most of the substances suitable as a working medium there would not be a great difference between the temperature at which the substance changes its state in the load bearing zone and the temperature at which it does so outside. For the case of carbon dioxide this difference could not possibly be more than 40”F, i.e. the difference between the sublimation temperature at atmospheric pressure and the temperature at the triple point of C02. Under typical conditions the difference with carbon dioxide would generally be less than 20°F. Where temperatures are expressed absolutely the error involved in assuming that the sublimation and solidification temperatures are identical and equal to T, the average of the two is small. Thus it will be taken that

The possible differences between the latent heats of solidification and sublimation are also small when expressed as a percentage of a typical value, so it may also be assumed that L,=

L,=

L

(40)

It then follows that q=

qg=

qs

(41)

and r is unity. Making the above simplifications eqn. (31) becomes l-Z,(t+Y)

=

2Zb(Z + Y) 7r((1-+&2)+

[sin-i(&)

+ t]

1

(42)

where (43)

(Z-1) z”=“f(l_j=)

(44).

YJ=plCi

(45)

Equation (27) which describes the pressure gradient in the film can also be modified so that its parameters are consistent with those of eqn. (42). h3

4%

dx

_

2 z+Y

((l-$6”):

THE SUBLIMATING

161

FILM GAS BEARING

where Pd =

l-1-z

(47)

and 7i is given by eqn. (25). Also defining a new load parameter

W, as

w

-

(48)

wd= 1-‘iid

n/2

IVd= 2

i

jd cos X&

(49)

0

Correspondingly modified versions of the parameters introduced as follows. Where

&Is, tic and L can also be

then eqn. (17) becomes

Where hj,, =

&

(52)

l-T,

Equation (9) becomes

(53) and where

then eqn. (38) becomes

(55) SOLUTION

OF EQUATIONS

AND PRESENTATION

OF DATA

Computed values of Zb were obtained by the solution of eqn. (42) for specific values of the independent variables Z, and Y and ranges of values of z and a. The chosen values of Z, and Y include that combination of values calculated in Appendix III for the 2 in. diam. bearing using carbon dioxide as the working medium, i.e. 0.3 and 1.0 respectively. The computed values of Z,, for these values of Z, and Y, are plotted as functions of z and E in Fig. 5. These values of Z, and Y relak to the situation when the shaft is made of stainless steel of wall thickness 0.3 in. and the temperature ratios Tg and Td are 2.5 and 0.8 respectively. Similar characteristics were plotted for other combinations of values of 2, and Y.

.0

NON-DIMENSIONAL LOAD PARAMETER ij, -

I

N 0

0

0 co

F 0

PARAMETER Zb k

_

MAXIMUM NON-DIMENSIONAL PRESSURE

0TEMPEYTURE 0 Jtl b in

Fddm

-.

THE SUBLIMATING

FILM

163

GAS BEARING

Distributions of the non-dimensional pressure & in the load bearing film could be obtained from eqn. (46) where the values of 2, as calculated above were known. Then corresponding values for the non-dimensional load parameter Wdwere obtained by numerical integration on the basis of eqn. (49). Thus relationships for the maximum value of jd and for IQ corresponding to these for Z, could be obtained. These characteristics for Z,=O.3 and Y = 1.0, are described by the constant z lines in Figs. 6 and 7. Similar relationships for the other values of Z, and Y were obtained and plotted. A relationship exists between the non-dimensional minimum gas film thickness & and z and E, which does not depend on either Z, or Y. It follows from eqn. (25) that this is given by i;,=

l--Z-&

(56)

which is represented by the constant r lines in Fig. 8. 9 3 Y

1.0

::

E 3 0.5 iz % 5”

0.2

z ii

0.1

;: zi v, 0.05 ci 5 1002 e

0

0.2 0.4 ECCENTRICITY

0.6 RATIO

Fig. 8. Theoretical non-dimensional Z, and Y. Effect of Z,.

0.6 E

1.0

minimum

gas film thickness

characteristics

at typical

values

of

As E varies along the constant z characteristics of Figs. 6, 7, and 8, the temperature ratio parameter Zb is also varying according to Fig. 5. It would be more meaningful to have characteristics described by the controllable bearing temperature rather than by the dependent solid film thickness ratio. Such relationships have been obtained by transferring the eccentricity values at each curve of z for constant values of Z, (e.g. in Fig. 5) to the corresponding curves for j&,,, Wdand i;,; (Figs. 6, 7 and 8). It should be noted that one of the values of Z, chosen for presenting the data in this form is 0.3, i.e. the value of this parameter calculated in Appendix III for the typical operating condition of the example bearing. From other curves of Zb, &,,,, Wdand fi,,,alluded to earlier for other combinations of Z, and Y, characteristics to illustrate the effect of these latter parameters were obtained in a similar manner. Thus for example the effect of variations in Z, on the Wdand fi,,,characteristics is shown in Figs. 9 and 10.

164

C. DAYSON

,,E

z

1.0

i!

5

0.5

F 3 G z

0.2

c1 2 T -I

O’ 0.05

ii H ? ECCENTRICITY

Fig. 9. Theoretical of z,.

RATIO

E

non-dimensional

load capacity

Fig. 10. Theoretical non-dimensional Z, and Y. Effect of Z,.

ECCENTRICITY

RATIO

0.02

B

minimum

characteristics

gas film thickness

0

0.2

0.4

0.6

ECCENTRICITY

at typical

values

characteristics

RATIO

0.6

I

E

of Z, and

Y. Effect

at typical

values

of

E

Fig. 11. Theoretical non-dimensional internal shaft temperature.

load

capacity

characteristics

at typical

values

of Y. Effect of

By considering data for combinations of Zb and Z, where the values of both these variables increases in the same ratio each time, it can be seen from their definitions, that one is simulating the effect of a change in 1 - Tdor G the internal shaft temperature. Relationships describing the effect of such variations in these param-

THE SUBLIMATING

165

FILM GAS BEARING

eters were obtained in the above mentioned manner from data computed for various equal values of Z, and ZB with Y = 1.0. The effect of such variations on the load parameter k!$ is shown in Fig. 11. Where constant values of Z,, Z, and !P are considered a unique relationship exists between the parameters P& E and 2. Thus for example from Fig. 7, (2,=0.3, Y = 1.0) values of z and E may be obtained that correspond to arbitrary values of wd for Z,=O.3. For each combination of a and E the distribution of the pressure Is, around the bearing arc can be obtained in the manner already indicated. Also from eqn. (51) the corresponding distributions of the non-dimensional rate of sublimation from the dry ice film can be calculated. Such distributions for one half of the bearing arc are plotted in Figs. 12 and 13 for a range of values of PQ. At a given bearing load similar relationships between o!and E may also be obtained for varying values of any one of the variables Z,, ZB, Zb = Z,, and Y. Figure 14 illustrates for example the effect that varying the bearing temperature has on the pressure distribution. Also the corresponding effect on the variation of the gas film thickness around the bearing arc can be obtained from eqn. (25) and plotted as in Fig. 15. According to eqns. (53) and (55), $I&,,and & are functions of Z,, Y and z only. The dependence of these variables on Z, and r for Y= 1.0 is shown in Figs. 16 and 17. IF

;-

3-

I

1

I

20 ANGLE

X

-

DEGREES

1

I

40 ANGLE

X

-

I

/

60

80

DEGREES

Fig. 12. Distributions of non-dimensional other controhing parameters.

film pressure for various load capacities at typical values of

Fig. 13. Distributions of non-dimensional values of other controlling paranieters.

rate of sublimation for various load capacities at typical

C. DAYSON

i2 ANGLE

X

-

01

20

0

40

ANGLE

DEGREES

Fig. 14. Distributions of non-dimensional of other controlling parameters.

film pressure

Fig. 15. Distributions of non-dimensional other controlling parameters.

for various

gas film thickness

x

values

_

60

80

DEGREES

of Z, and

typical

at two values of Zh and typical

values

values of

.f

5,.,I 0

0.2

SOLID

0.4 FILM

0.6

THICKNESS

0.6 RATIO

Fig. 16. Non-dimensional of Y.

rate of solidification

Fig. 17. Non-dimensional

solid film thickness

TYPICAL

DIMENSIONAL

O.lo

1.0 T

CONDITION

0.2 SOLID

characteristics

variation

for various

for various

AND PARAMETRIC

0.4 FILM

0.6 THICKNESS

0.0 RATIO

values of Z, at typical

values of Z, at typical

T

value

value of 1y.

VARIATIONS

Computed results at a particular condition in the example bearing

At the particular conditions used in Appendix III to calculate the required

THE SUBLIMATING

FILM GAS BEARING

167

properties of the carbon dioxide in the example bearing, i.e. with Z,= Zb= 0.3, Y = 1.0 and z =0.3, values for the load parameter L& and the maximum pressure parameter pa,,, can be obtained from Figs. 6 and 7 along with a value for the eccentricity E. Figure 8 gives the corresponding value for the minimum film thickness variable i;,. These values are rd = 68.0,

&,,, = 85

E = 0.58,

t;,= 0.12

From the definition of J$, and its above value, the load capacity W of the example bearing may be calculated at the specified condition. A value for the maximum pressure pm may be similarly obtained. These values are W = 63.7 lbs.,

pm= 40 p.s.i.g.

The maxiqum pressure is less than the triple point pressure of carbon dioxide, i.e. 70 p.s.i.a. Should the pressures in a bearing of this type rise significantly above this value, the increasing effect of the carbon dioxide changing from a solid to a liquid and then to a gas rather than directly to a gas would at first render the present analysis inapplicable and at higher loads (and pressures) cause the bearing to be inoperable. Where the COZ is supplied at atmospheric pressure as in the example, the critical maximum gauge pressure is therefore approximately 55 p.s.i. The corresponding value of jj,, in this case is 117. This limitation can be represented on Fig. 6 by a straight horizontal line, and by noting the values of Eat which it crosses lines of constant CIthis limit can be transferred to the Vdw E relationship, Fig. 7. Thus it can be clearly seen what load can be applied without causing any liquid to be formed at any bearing operating condition specified by the values of the other controlling parameters. The corresponding value of the parameter I; which gives the variation in the thickness of the solid film around the circumference of the shaft, was obtained from Fig. 17, i.e. Ad= 1.56

From the definition of &, and for a shaft whose speed of rotation is 1,000 r.p.m., At/t = 0.059 that is a six percent variation in thickhess. These value: of W, p,,, and At/t obtained from the computed non-dimensional data, serve to indicate that the particular bearing being considered would operate effectively with an appreciable load capacity and in the manner envisaged, i.e. without any large variation in the thickness of solid film on the shaft. Effect of controlling parameters on bearing operating conditions 1. Load

From the load parameter relationships it is seen that the eccentricity of the shaft always increases with increasing values of this variable. This means that a bearing of the type being considered as well as being capable of supporting a loaded shaft, will also do so in a stable manner for an indefinite period of time. It is instructive to examine the characteristics in somewhat more detail. If

168

C. DAYSON

T’ABLE I EFFECT OF VARIATION

IN CONTROLLING

VARIABLES

ON BEARING

OPERATING

CONDITIONS

Case

Z,

Z,

Y

Wd

5

I

0.3 0.3 0.3 0.3 0.3 0.5 0.3 0.4 0.3 0.3

0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.4 0.3 0.3

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.7 1.0

26.5 68.0

0.5 0.3

0.29 0.58

0.21 0.12

22.3 85.0

0.365 0.47

f0.29

-0.2

- 0.09

68.0 68.0

0.6 0.3

0.26 0.58

0.14 0.12

58.0 85.0

0.325 0.47

+0.32

-0.3

- 0.02

68.0 68.0

0.3 0.15

0.58 0.745

0.12 0.105

85.0 94.5

0.47 0.37

+0.165

-0.15

-0.015

30.0 40.0

0.473 0.140

0.34 0.72

0.19 0.143

28.6 54.8

0.38 0.48

+ 0.38

- 0.333

- 0.047

68.0 68.0

0.555 0.3

0.29 0.58

0.155 0.12

60.5 85.0

0.5 0.47

+0.29

-0.255

- 0.035

2 3 1 5

one takes for instance the particular values of Z,, Z, and Y used in the calculations for the example bearing and considers the operating condition at two different values of the load parameter mdb,a further insight into the manner in which the bearing functions can be obtained. The values of all the relevant pa#;imeters at two such conditions have been taken from Figs. 6, 7, 8 and 16 and are given in Table I, Case 1. The two conditions considered are marked A and B on the characteristics. A is the initial condition and B the final one. It should be emphasised here that where the shaft temperature (as specified by Td) does not vary, the film pressure bearing load and rate of solidification are directly proportional to ijd, rd and ni,, respectively. The nearly three fold increase in load causes a marked increase in the shaft’s eccentricity, about two thirds of which resulted from a decrease in the thickness of the dry-ice film and one third from a reduction in the minimum thickness of the gas film. The process by which such a change occurs after an increase in the applied load, can presumably be described in approximately the following manner. The immediate effect of the increased load is a reduction in the minimum thickness of the gas film. Thus there is a reduction in the resistance to the flow of heat from the bearing to the solid film, which causes an increased rate of sublimation from its surface. Where the rate of solidification at the free surface of the shaft is initially unchanged the effect of this must be a reduction in the thickness of the solid layer. However with a solid film of smaller thickness the rate of heat transfer through it and the shaft must increase, the effect of which is to reduce the rate of sublimation and increase that of solidification. Both these effects tend to reduce the difference between the rate of solidification and sublimation that resulted from the initial increase in load, and it can be appreciated that as the solid film continues to get thinner, the rates will gradually approach each other until a new lower equilibrium thickness of the ice film (r) is established and the two rates are again identical but greater in magnitude. The increase in &lc, is apparent from the table, and the manner in which the overall increase in Ii& occurs can be appreciated from Fig. 13. It is also significant that the maximum pressure has increased in a greater proportion than the load. This results from the increased eccentricity which tends to increase the film pressures near the centre of the film while having the

THE SUBLIMATING

FILM GAS BEARING

169

reverse effect towards the edges; see Fig. 12. The load-eccentricity characteristics also indicate that the bearing should be capable of supporting a definite load when the shaft is in a concentric position or even when the eccentricity is negative, i.e. with the shaft’s centre above that of the bearing. This may seem surprising to those familiar with the manner in which journal bearings usually operate. In that case where the lilm is generated hydrodynamically the eccentricity can only approach zero. and for this to be so the applied load must be very small. The reason for the difference is that the sublimating film bearing does not operate on a hydrodynamic principle, in fact it should be capable of operating effectively for short periods of time without any rotation of the shaft, i.e. for the length of time it takes the solid film in the load bearing region to be removed by sublimation. This is to re-emphasize therefore that despite the lack of a high pressure gas supply the bearing can operate in a similar manner to hydrostatic gas bearings. The ability of the bearing to operate effectively for short periods without the shaft turning could make this type of device particularly useful for reciprocating or rapid start-stop situations. The form of the rate of sublimation distributions in the bearing arc are worthy of further discussion. At the lightest load condition represented in Fig. 13, i.e. for Wd=20.4, the rate of sublimation is positive over the whole of the bearing arc. At higher loads however, the rate becomes negative towards the outlet of the film; gas that has been formed by sublimation in the central portion of the film is resolidified before it can escape at the outlet. This effect will occur because with the thinner ice film the gas film thickness towards the exit becomes so great that heat conduction across the film becomes less than the heat flow through the ice in this zone. Thence (see eqn. (11)) the rate of sublimation becomes negative; i.e. solidification occurs. 2. System temperatures The effect of a change in bearing temperature can be obtained by examining the conditions described by a line of constant load A’B on the k&-c relationship of Fig. 7. The data thus obtained for two values of Z, are presented in the table as Case 2. The fifty per cent increase in the absolute temperature of the bearing causes a large increase in the eccentricity of the shaft. It is clear that this increase is almost completely due to the reduction in the thickness of the solid layer and is only slightly affected by the small decrease in the gas film thickness. An appreciation of the reasons for these changes could be obtained by considering in the above manner the effect of a sudden increase in the bearing temperature. The decrease in ti,,, is somewhat unexpected; with the same load it would be envisaged that a greater film thickness would be required to allow for the increased rate of the gas flow (higher ;\;I,,). However, it must be remembered that the tabulated value is the minimum one and as such does not provide a measure of the film thickness towards the ends of the film. Thus even where there is a small decrease in the minimum film thickness as indicated here, because of the greatly reduced thickness of the ice film the gas film thickness at points outside the central region of the film will be increased; see Fig. 15. So despite the reduction in ti, it is evident how the increased gas flow is accommodated, and also why the maximum film pressure should be higher although the load supported is the same. The effect ofchanging the bearing temperature on the

170

C. DAYSON

pressure distribution is indicated in Fig. 14. The increase in the maximum pressure is clearly shown along with the corresponding reduction in pressures near the ends of the fdm. The effect of an increase in the temperature at which the low pressure gas is supplied to the space above the exposed surface of the shaft can be obtained by considering points A and B in Figs. 9 and 10. The data obtained from the graphs for these two conditions is presented as Case 3 in Table I. The increase in the absolute temperature of the gas by a factor of two thirds results in the considerable increase in the shaft eccentricity, most of which is again due to the decrease in the thickness of the solid film rather than any appreciable change in the minimum thickness of the gas film. Although this behaviour is similar to that occurring with a change in the bearing temperature, the increase in eccentricity, is in this case considerably smaller. Points A and B in Fig. 11 describe some of the conditions at the two shaft temperature conditions represented in Case 4 of Table I, where corresponding values of the other parameters are also given. It will be noted that although the two points represent conditions of equal load capacity ( W), they have different k& values. This is because II$ is dependent on 5 as well as II? From the tabulated values it appears that with increasing shaft temperature there is a marked increase in the eccentricity at which the shaft operates. Again this increase is due in the main to a large decrease in the thickness of the solid film and to a smaller extent to a decrease in the minimum thickness of the gas film. The Case 5 data show the effect of a change in clearance or shaft wall thickness. A general examination of the changes in eccentricity, solid and gas film thicknesses in Table I shows that although the controlling shaft, gas supply and bearing temperatures have considerable effect on the ice film thickness and as a result of this on the eccentricity, they generally have a much smaller effect on the minimum thickness of the gas film. The load on the other hand as well as having an important effect on the solid film thickness and eccentricity, also has a marked effect on the gas film thickness. DISCUSSION

OF RESULTS

On the basis of the theoretical model set-up to describe the action of the suggested sublimating film bearing, it appears that such bearings may function in an effective manner. The results indicate that with one geometrical arrangement (a half journal) and one working medium (carbon dioxide) a bearing designed to function on this basis could be capable of supporting substantial loads under certain conditions. Furthermore the characteristics of such a bearing indicate that it would operate in a stable manner under varying conditions of operation. Several major assumptionswere made in deriving the model in order to achieve certain mathematical simplifications. However it is felt that the introduction of these assumptions has not had a significant effect on the findings of the study. A major assumption made in the analysis, namely that the variation of the thickness of the solid film as it is carried around by the shaft is small in comparison with the average thickness, has also been substantiated by the results obtained. This is a gratifying result in that the analysis was greatly simplified on this basis. Another

THE SUBLIMATING

FILM GAS BEARING

171

assumption i.e. that side leakage of gas from the lilm be neglected, is not considered to have had any major effect on the essential findings. That is not to say however that in a practical situation where the bearing is of a typical axial length and not infinitely long, the gas or the solid film thickness will not be appreciably smaller than would be predicted here, for this would be the case. Although this is true such bearings (like more conventional gas bearings) would still be capable of providing a useful means of supporting rotating components under specific conditions. The other major assumption made was that the gas in the film behaves in an incompressible manner. This assumption has the effect of reducing the predicted load capacity in the bearing because allowance for the compressibility of a gas causes higher pressures to be predicted at points towards the outlet of the film. Thus it is to be expected that a more rigorous analysis that accounts for compressibility would predict greater load capacities than have been obtained here. GENERAL

CONCLUSIONS

The fact that the bringing together of a surface of a solid which sublimates at ordinary pressures with the matching surface of a warmer body is resisted by the pressures required to remove the sublimated gas from the gap between them, has been recognised as a principle upon which it should be possible to design a new type of hydrostatic gas bearings. A criterion for the design of such bearings has been established. It is, that the temperatures of the two bodies being lubricated by the sublimating working medium, be of temperatures that are one above and one below the temperature at which the medium sublimates and solidifies, and that the surface of the cooler one be periodically exposed to the gaseous phase of the medium. A bearing designed to satisfy this criterion should function as a result of the solid film of the working medium that would be deposited from the gaseous phase on the colder surface, and which would be subsequently removed by sublimation while that surface is in close proximity to the higher temperature surface. The restricted path for the egress of the gas gives rise to pressures between the two surfaces that are sustained by the applied load. Several arrangements for both thrust and journal bearings that would function on this basis have been suggested. It is felt however that these do not by any means represent all the types and geometries of bearings that could be designed to function in this way. A theoretical model of one particular form such a bearing might take has been developed. This bearing was a semi-cylindrical journal operating at the higher temperature with a cold rotating shaft. It is considered that a similar analysis could be made of bearings of other configurations; e.g. thrust bearings. The relationships governing the characteristics of the bearing were obtained using a preliminary theoretical model. Despite the simplifying assumptions made, it is felt they indicate that the operation of such a bearing would be feasible. Calculations based on these results for typical conditions in a bearing ,using carbon dioxide as the working medium suggest that such a bearing could operate effectively in a practical system. Although bearings of this type must necessarily have a restricted range of applications, e.g. (cryogenic temperatures with carbon dioxide as the lubricating

172

C. DAYSON

medium), under such conditions they could have several advantages over more conventional types of bearings. Like any gas bearing they operate with negligible frictional resistance and could therefore be suitable in high speed applications. Also since the gas is supplied to the film via the solidified layer. a high pressure gas supply is not required as is the case with conventional hydrostatic bearings. Also the sublimating bearing should be capable of operating effectively with a self generated film of gas at quite low speeds. It is possible for the two components of such a bearing to be relatively at rest for short periods without the gas film being destroyed. Such bearings could therefore be employed to advantage where reciprocating or rapid stop-start motions are involved. APPENDIX LAMINAR

I FLOW

IN THE GAS FILM

It has been shown3 that where a Reynolds number defined as Re, = 2hu,/v,

where u/ is a film velocity which is given in terms of a mean pressure generated velocity up and a rotational velocity u, by u/ = UP+ uJ2 that the flow of the gas film between two concentric cylinders is laminar as long as Re, < 1500

In the bearing the maximum value of u/ occurs at the end of the gas film on that side where the shaft surface is leaving. Here up is a maximum, up,,,, and we can consider the positive value of uJ2. Then u,/2

14fm= Upmf

The film thickness is also a maximum here so the check for laminarity at this point must be the most severe one that can be made. The mass flow that corresponds to the maximum value of up is given by half the rate at which the gas is solidifying on the free surface of the shaft, i.e. up,,,hp f = 71Rri2,/2 where tit, is given by tic= TvKi( 1 -T~)Ll~~/L,C From the typical bearing conditions represented by the following values of 2, and Y and 7 (see Appendix III) for a unit with COz as the working medium z, = 0.3,

Y = 1.o,

7=

0.3

it can be seen from Fig. 16 that k,

= 0.469

Thence where values for the other parameters and the working medium properties

THE SLJBL.IMA-I-IN<; FILM

GAS HEARING

173

are also taken from Appendix III, the above equation gives I&.= 1.8 x 10~’ lb (s in”) The film thickness at exit from the film is given by /I _ (.( 1--. 7) so at the chosen value of the radial clearance and the above mentioned

WILICcjf r.

this gives It = 3.85 x: IO .’ in. Then taking the calculated value of the mean gas film density frc)m Appcndik III the above oq~l;ttion containing ul>,,,was used to obtain u~,~= 486 in.+ Considering a fairly high shaft speed of lO.MO r.p.m. as a possihlc uppa limit at which such LLbearing might be cxpectcd to perform u, = ZnR N/ 60 = 1046 in.,‘s So the above equation for usIltgives lfj-,,,= 1009 in.is and where ;I value for the gas viscosity is obtained from Appendix III Rc,, I.: 1._7’0

This va11reis s~lrnewh~ltless than the limiting value of the Reynotd’s ntimhcr for laminar llo\v. so the validity of using this assumption in the :tnalysis ix substantinted.

The rate of heat transferred per unit area across the filtn by c.ondu&in 1‘; given by eqn. (10). It follc,)ws that where II, is a typical film thickness the tt,ttal rate of heat transferred i11this way is Qh = nRfk’,( 7h- Q//I, For the exemplary bearing of Appendix 111, for the typical opcr;ttins <.t)!kci:ir
174

(‘. DAYSON

The rate at which heat is removed from the bearing by convection in the film depends on the total mass flow of gas from the film and its increase in temperature after sublimating. Assuming that its mean exit temperature is ( Th+ 7Ji2 the total heat flow by convection is Qc where Qc= Rlriz,C,( r,-

T,).i2

Again for the conditions given in Appendix III n’~, may be calculated (8) and (52) for the value of k, given in Appendix I; i.e.

from eqns.

ii/l,, = 0.469 SO +I,=

1.84 x lO-4 lb/(s in2)

At a mean temperature

in the gas film defined

as

i.e. 170°F under the above conditions. and a typical film pressure of three atmospheres the specific heat of the gas C, is 0.216 B.T.U.!(lb. ‘,F). Thence Q, = 0.0675 B.T.U.;s Comparing these values for Qb and Qc it is seen that at this typical operating condition the rate of heat removal by convection was less than IO’<,,of that transferred across the film by conduction. Therefore it appears that the assumption made in the analysis. i.e. that the former effect be neglected. was a reasonable one in a preliminary analysis such as this. APPENDIX

111

CALCIJLATIOE; DIOXIDE

OF

WORKING

SUBLIMATING

MEDIUM

FILM.GAS

PROPERTIES

IN

AI\; EXEMPLARY

CARBON

BEARING

Bearing and shaft dimensions and properties A 180 degree journal bearing of the form described in the main part of the paper is considered. Its dimensions and those of the shaft arc specified by in. R=l I = 2 in. (’ = 0.0055 in. in. h = 0.3 The shaft is made of stainless K,=

steel with a thermal

conductivity

K,, given by

14.5 B.T.U.!‘(h -F/ft ft”)

Conditions of‘ sublimation and solidification Where the working medium is CO2 and where for effective operation the maximum pressure in the load bearing film is limited to that of the triple point. i.e. 70 p.s.i.a.. a typical average film pressure of 25 p.s.i.g. was taken. Where the ambient pressure is considered to be 15 p.s.i.a. then the typical absolute film and gas supply pressures are respectively

THE SUBLIMATING

FILM GAS BEARING

175

PJ = 40 p.s.i.a. ps = 15 p.s.i.a. Thus mean sublimation and a solidification temperature can be obtained4. They are l$ = -88°F T,,= - 1lO”F Neglecting the difference between these two values and replacing them by their average T, T,= -99 “m-100°F

The corresponding value of the heat of sublimation obtained from the same source. Thus

(and solidification)

can be

L = L, = L, = 243 B.T.U./lb. Shaft and dry ice temperatures and conditions

Taking 0.8 as the value of the ratio Td the internal shaft temperature &= T,*T,=0.8x360=288”R=

-172°F

Since the rate of heat flow through the dry ice and the shaft must be equal KiAi? _ &AT, b

t

but AT+AT,=

T,-T,

Thus

K-T, = 1 +BKi/Z Now

j3 = b/c = 0.3/0.0055 = 54.5 and for an assumed mean ice film temperature Ki= 0.315 B.T.U./(h ft2 “F/ft) according’ to Fig. A.l. Thus Ici is given by Ici= Ki/K, = 0.315/14.5 = 0.0217 With an arbitrary value of 0.3 for z AT = 14.6”F Thus the mean ice temperature

T is given by

17i’=- lOO+ (- 14.6/2) = - 107.3”F

of - 105°F

is

176

C. DAYSON

TEMPERATUffE

FigA

I, Thermal conductivity

-

OF

of dry ice

This is close to the jnitiaI~y assumed value of 7 so that where great accuracy is not required the initially assumed value of ki is adequate and will be used in the following calculations. The density of the dry ice is taken4 to be its value at - 1lO’F pi = 97.656 lb/R3 = 0.0564 lb/in3

Loudheuriily

pus fitrn

wnditioi7s

Assummg that the temperature

ratio T is 2.5 the bearing temperature

is

& = 7;bT, = 2.5 x 360 = 900”R = 440°F Thus the mean gas film temperature

T, is

~~(~+~)~2=(3~+9~)/2=63~

R= 170°F

At this temperature and at atmospheric pressure the conductivity of carbon dioxide6 is 0.0118 B.T.U.,‘(h ft2 “F/ft) Where the small effect of pressure on conductivity taken as K,. Thence

is neglected this value may be

Q ='K,/K, = 0.01 M/O.315 = 0.0375 The density of the gas at the mean film condition is Q~= 1.515 x 10e4 lb/in3 and the viscosity which depends only upon temperature is yls = 9.64 x lo-’

lb/(s in.)

THE SUBLIMATING

Conditions

I-‘ILM GAS BEARING

in gas solidifjing

177

zone

Where the temperature

ratio T, is taken to be equal to 7;b

T,=T:,17;=2.5~360=900’R=440-F and the gas supply pressure is 1 atm. we have’ that the density. conductivity the viscosity of the carbon dioxide at this condition are respectively

and

py = 3.875 x IO-’ lb/in3 K, = 1.94 x lo-’

B.T.U,:(h ft2 “F/ft)

q9 = 1.3 x IO-’ Ib/(s in.) A relationship exists’ governing the turbulent heat transfer to or from horizontal rotating shafts. This empirical relationship will be used here to describe the magnitude of the heat transfer coefIicient at the semi-cylindrical solidifying CO2 surface, although the equation is based on data obtained with a complete shaft. Where a Reynold’s number defined on the basis of shaft diameter and angular velocity as Re,, = wnD2:v,

is greater than 8,000 the heat transfer coefficient is described by the following relationship for a Nusselt number defined in terms of the shaft diameter Nir,,= H,D/‘K,=O.lt

l(O.5 Rc~i+Gr)Pr)“.~~

In this case Re,) is greater than 8,000 when 0 > 204 r.p.m. so the above relationship speed of 3,000 r.p.m. Re,,=

should be generally applicable. At a typical rotational

1.18 x 10”

The Grashof number is given by GY= (p;g( T/T,-

l)D3)/q;

Neglecting the negative sign that arises because YrslTgis less than unity in this case G’i.= 1.65x IO6 The Prandtl number6 for 900 R and 1 atm is given by Pr = 0.702

In calculating the Nusselt number using the above equation the Grashof number is’ so small in comparison with the Reynold’s number term that it can be neglected. This is to say that the heat transferred as a result of natural convection is much less than th>t transferred as a result of the shaft’s rotation. Thus NM,>= 272 Thence H,= NuDK,JD = 31.7 B.T.U./h ft2 “F)

178

C. DAYSON

and 1= H&/xi

= 0.046

Values of the independent variables

The corresponding values of the independent variable Z,, Zg and Y which arise in the governing equations are thus given by ,.=

Icf( z - l)/( 1 - Q = 0.281 z 0.3

z,=

2(~-1)/(1-~)~0.345~0.3

Y =plCi= 1.18~ 1.0 For convenience in computing and for ease of presenting the theoretical data the values have been rounded off in the manner indicated. In view of the various assumptions that have been made in ob~ining these property values it seems that such an approximation can be tolerated. REFERENCES 1 J. W. Powell, The Design of Aerostatic Bearings, Machinery, London, 1970, p. 52. 2 R. D. Carmichael and E. R. Smith, ~ut~~ticul Tables and For~uias, Dover, New York, 1931, p. 260. 3 C. Gazley, Heat transfer of the rotational and axial flow between concentric cylinders, ASME Annual Meeting, New York, Nov. 25-30, (1956). 4 Physical properties of carbon dioxide, Tech. Bull. No. 48, Liquid Carbonic Canadian Corporation Ltd., Montreal 9, (1969). 5 Personai communication, Liquid Carbonic Canadian Corporation Ltd., Montreal 9, August (1969). 6 Tables of thermal properties of gases, Circuiar 564, U.S. Dept. of Comm. Nat. Bureau of Stds., Nov. 1 (1955) 138-198. 7 F. Rreith, Principles of Heat Transfer, Int. Textbook Co., Scranton, 1964, p. 324.