TECHN!SC;:E HOGGSCHOOL DELFT VLISGTUIGEOL'VVKUNCE
BIEüüTHEEK
2 2 JUU ^963
CoA. R e p o r t No. 159
THE COLLEGE OF AERONAUTICS
C R A N F I E L D
THE F R E E - M O L E C U L E FLOW O F A POLYATOMIC GAS
by
T. M a r s h
REPORT NO. 159 May. 1963.
THE C O L L E G E OF AERONAUTICS
CRANFIELD
The Free-Molecule Flow of a Polyatomic Gas b y
-T. Marsh, B . S c , D . A . E .
SUMMARY
A study has been made of the free-naolecule flow of a polyatomic gas past a body, with special reference to the evaluation of accommodation coefficients.
A model of the gas-surface interaction is devised, based on the phenomenon of physical adsorption. Using this m.odel, expressions for the thermal
accommodation coefficients of the various energy modes of a polyatomic gas a r e developed by postulating that the energy exchange at the surface is governed by equations similar in form to the gas-phase relaxation equations. The expressions so obtained a r e dependent on the relevant integral heat of adsorption and the various relaxation times involved. Some suggestions are put forward as to how these relaxation times could be evaluated since, once their behaviour is known, the results of this report will provide a simple method for estimating the
accommodation coefficients in a wide range of flow conditions.
This report is based on work conducted by the author at The College of Aeronautics in partial fulfilment of the requirements for the Diploma in Advanced Engineering.
CONTENTS P a g e S u m m a r y L i s t of Symbols 1. Introduction 1 2. Incident p h a s e 2 2 . 1 . B a s i c equations 2 3. G a s - s u r f a c e i n t e r a c t i o n 4 3 . 1 . M o n a t o m i c g a s - s u r f a c e i n t e r a c t i o n 7 3 . 2 . E x p e r i m e n t a l e v i d e n c e . Monatomic g a s e s 11 4 . P o l y a t o m i c g a s - s u r f a c e i n t e r a c t i o n 13 4 . 1 . E x p e r i m e n t a l r e s u l t s . P o l y a t o m i c g a s e s . 17
5. Flow o v e r a flat plate 17
6. C o n c l u s i o n s 19 7. A c k n o w l e d g e m e n t s 20
8. R e f e r e n c e s 21 F i g u r e s
T h e s y m b o l s defined below a r e t h o s e used throughout; o t h e r s which a p p e a r b r i e f l y a r e defined in the text a s they o c c u r .
E . E n e r g y flux c a r r i e d to a solid s u r f a c e by incident m o l e c u l e s
E E n e r g y flux c a r r i e d away from a solid s u r f a c e by r e f l e c t e d m o l e c u l e s E D e n o t e s E when T = T ( s e e below)
w r r w
K Function of speed r a t i o and incidence o c c u r r i n g in E.
N. N u m b e r of m o l e c u l e s s t r i k i n g unit a r e a of s u r f a c e p e r unit t i m e Q Heat of a d s o r p t i o n p e r m o l e of a d s o r b e d gas R G a s constant r e f e r r e d to unit m a s s R M o l a r g a s constant T . T e m p e r a t u r e of incident s t r e a m T T e m p e r a t u r e of r e f l e c t e d s t r e a m T T e m p e r a t u r e of solid s u r f a c e S R a t i o of s t r e a m speed to m o s t p r o b a b l e m o l e c u l a r speed j N u m b e r of d e g r e e s of f r e e d o m in an e n e r g y mode p . N o r m a l m o m e n t u m flux to s u r f a c e ( incident m o l e c u l e s )
p N o r m a l m o m e n t u m flux away from s u r f a c e ( r e f l e c t e d m o l e c u l e s ) r p D e n o t e s p when T = T '^ w r r w a T h e r m a l a c c o m m o d a t i o n coefficient rj N o r m a l m o m e n t u m a c c o m m o d a t i o n coefficient 0 Angle of incidence
T Relaxation t i m e a s s o c i a t e d with t r a n s l a t i o n a l and active m o d e s a T. Relaxation t i m e a s s o c i a t e d with i n e r t e n e r g y m o d e s T" A v e r a g e t i m e of a d s o r p t i o n T P e r i o d of v i b r a t i o n of a d s o r b e d m o l e c u l e s n o r m a l to s u r f a c e o Affix I r e f e r s q u a n t i t i e s to l o w e r s u r f a c e of flat p l a t e ( s e e F i g . 1), u r e f e r s to upper s u r f a c e
List of Symbols (Continued)
For quantities associated with the separate energy modes of a molecule : No prime refers to translation mode
One prime refers to active mode Two primes refer to inert mode
1. Introduction
If a gas flowing over a body satisfies the two conditions, that the mean free path of the molecules is large compared with the dimensions of the body, and also that an element of volume of the gas contains a sufficiently high number of molecules to determine the macroscopic properties of the flow, then the flow system is designated a free-molecule flow. That such flow conditions can exist in practice is shown *''^' by the fact that when the mean free path in the upper atmosphere is 10 feet, the number of molecules in a cubic inch is about 10 Experimental evidence i n d i c a t e s ' ^ ' that when the ratio of mean free path to a characteristic linear dimension of the body is greater than 10, then free-molecule theory is applicable. The above ratio is known as the Knudsen Number, after Martin Knudsen who pioneered work in this field.
A considerable amount of work on the free-molecule flow of monatomic gases has been published, and complete accounts of the basic theory can be found in (1), (2) and (3), to name but a few of the available sources. There i s , as yet, no completely self-sufficient theory of free-molecule flow. All the investigators in this field have introduced certain average flow parameters, which will be
defined later, where these parameters are determined experimentally. The same course has been followed in this work, where the aim is to predict the trends followed by the aerodynamic characteristics and the heat transfer under real gas flow
conditions and not their absolute magnitudes.
Since the term polyatomic covers a large number of different species of molecules, (all having different numbers of degrees of freedona) for the purposes of this investigation a simplified model has been adopted. In our model, the molecules have an arbitrary number of internal modes of energy, but all modes
except one maintain thermal equilibrium with the translational energy during any change of state. A physical example of this type of gas would be a diatomic gas, with an
active rotational mode of energy and an inert vibrational mode. The t e r m s active and inert are used here as in (4) to describe modes having negligible and significant relaxation times, respectively. Furthermore, it is assumed that in the gas upstream of the body all the internal modes of energy are fully excited, i . e . in thermal
equilibrium with the translational energy. We assume also that chemical reactions do not occur anywhere in the flow system. These assumptions, whilst admittedly restrictive, do leave a range of applicability of the results, which is wide enough to be of interest.
One of the consequences of the basic definition of free-molecule flow is that the molecules incident on a surface do not interact with the molecules reflected or re-emitted from the surface. Thus the flow can conveniently be broken down into the incident phase, the gas-surface interaction phase, and the reflected phase, and each phase may be considered separately. However, since the reflected phase is really the end product of the gas-surface interaction, we need only consider the incident and gas-surface interaction phases.
2
-2. Incident phase
The interaction of the gas with the surface gives rise to physically
measurable quantities; the rate of heat transfer, normal pressure and the skin friction. These quantities are related to the amounts of energy, normal
momentum and tangential momentum which are carried by the molecules to unit area of the surface in unit time. The method of obtaining expressions for these quantitites can be found in several of the references given in this report, but for convenience we will give a brief derivation here.
Consider a flat plate moving with velocity £ = (u, v, w) at angle of incidence 6, through a uniform polyatomic gas. The temperature of the undisturbed gas is T^ and the number density is n-. Let us define co-ordinate axes fixed relative to the
plate as shown in Fig. 1, then, viewed from these axes, the molecular velocity components are u = t r + U , v = V + V , w = w + W , where U, V and W are the thermal velocity components of a molecule. We will assume that the molecular motion in the undisturbed gas ahead of the flat plate is Maxwellian.
2 . 1 . Basic equations
The number of molecules which strike unit area of the plate in unit time and have velocity components in the range u, u + du; v, v + dv; w, w + dw; is then n^ vf du dv dw. f is Maxwell's velocity distribution function given by
\ 2 , r C' )
- 3CV2C'
e
a
where C = (U, V, W) and C is the mean square of C.
The total number of molecules striking unit area of the plate in unit time is now found by integrating the above expression over all possible values of u, v and w. Thus on the lower surface of the plate the limits of integration on V are 0 to » , while on U, W they are from - «> to + " . To calculate the corresponding quantity for the upper surface, we integrate -n^ vf du dv dw since v must be negative, and the limits of integration on V are - » to 0. If we let N^ and N. represent these number fluxes for the lower and upper surfaces respectively, then
Nj = n t T5 rp Q 2 4 ^ [ e ' ^ + S^ V^ ( l + e r f S ^ ) ] (2.1) I R T. - s ' N " = n. J ^ — i f e "" - S V5^ (1 - e r f S ) 1 (2.2) 1 I N 27r L ^ ^ J
where S = —— is the ratio of the mass velocity component normal to the surface ^ 2 pv -x' to the most probable speed of the incident molecules, erf S = -^ j e dx,
o
and R is the gas constant referred to unit m a s s . Note that if we define S = q/Cj^, then Sy = S sin 6.
To obtain the translational energy flux to the lower surface we integrate ^ n^ mc^ V du dv dw where m is the m a s s of a molecule and c = (u, v, w). The limits of integration are as before. Denoting the translational energy flux to the lower and upper surfaces by E^ and E.u respectively, we obtain ,e „ e , r S ,2 4 e x p ( - S ) + 5 S V w ^ d + e r f S ) E : = 2 m N : R T . j f + — \ (2.3) ^ ^ ' ^ 4rexp(-S'') + S V F ( l + e r f S ) l L V v V J
i
«-^it
r „2 E " = 2 m N " R T. 1 1 1 4exp(-S'') - 5 S Vn^ (1 - erf S ) I H-- . ^-
1(2.4)
4rexp(-S ) - S VF(1 - e r f S ) ! L V V V J 1 - erf S )1 -^ Let us write these as0 9 0
E. = (2 m N. R) (K T.) (2.5)
E " = (2 m N " R) ( K ^ T . ) (2.6)
1 1 1
where K and K are plotted for various angles of incidence in Figs. 2 and 3. The internal energy which is carried to the surface by the incident molecules can be allowed for by assuming that all internal modes are fully excited and that classical equipartition of energy applies*. Let us distinguish here between the active and inert modes of energy for use in later sections of this report. We will suppose that there are j ' degrees of freedom in the active energy modes and j * in the inert energy modes. Then according to the assumptions above, the amounts of active and inert energy carried to unit area of the lower surface in unit time a r e , respectively,
( E * ) ' = i m N . j ' R T. (2.7)
(E.*)" = i m N * i"R T. (2.8)
1 1 •* 1
and similarly for the upper surface,
( E " ) ' = è m N " j'R T^ (2.9)
( E " ) " = i m N " j"R T. (2.10) In a similar manner, by calculating the amounts of normal and tangential
momentum carried to unit area of the surface per unit time, we can arrive at the following expressions for normal p r e s s u r e p^, and skin friction T . , due to the incident molecules.
* This assumption is not strictly necessary and a completely general value for the energy in the internal modes could be taken. The classical value is used here for convenience only.
4 -I Pi u Pi = m N = m N.
e
r « ^ i
wB. T 2S V •fir 2S L VF+
1 + e r f S exp(-S^) + S v ^ l + e r f S ^ ) 1 - e r f S e x p ( - S ^ - S VF(1 - e r f S ) V V V ( 2 . 1 1 ) ( 2 . 1 2 ) e T. 1 m N. q cos 0 1 ( 2 . 1 3 ) u u T. = m N. q cos 0 1 1 ( 2 . 1 4 ) 3. G a s - s u r f a c e i n t e r a c t i o nIt h a s been found e x p e r i m e n t a l l y that when a gas c o m e s into contact with a solid at a different t e m p e r a t u r e , the e n e r g y and m o m e n t u m e x c h a n g e s between gas and solid a r e not, in g e n e r a l , c o m p l e t e . We s a y that the solid d o e s not c o m p l e t e l y a c c o m m o d a t e the g a s . To define t h i s d i s c r e p a n c y in t e r m s of e n e r g y and m o m e n t u m fluxes, c o n s i d e r a m o n a t o m i c gas flowing o v e r a s u r f a c e whose t e m p e r a t u r e T i s different from T . . Suppose that dynamic e q u i l i b r i u m e x i s t s , so that the n u m b e r of m o l e c u l e s N leaving unit a r e a p e r unit t i m e i s equal to N. ( a s defined in s e c t i o n 2 . 1 ) . Let E^, denote the a c t u a l e n e r g y flux away from the s u r f a c e , and E ^ the flux which would e x i s t if c o m p l e t e t h e r m a l a c c o m m o d a t i o n o c c u r r e d . Then l E . - E [ < | E . - E | in g e n e r a l , and we define the t h e r m a l a c c o m m o d a t i o n coefficient a s a E . - E 1 r E . - E 1 w ( 3 . 1 )
T , T the n o r m a l and tangential m o m e n t u m f l u x e s , w h e r e r w
If we define p , p '^r '^w
the suffices r and w denote the s a m e p r o p e r t i e s a s in E^. and E ^ , define m o m e n t u m a c c o m m o d a t i o n coefficients. then we can n = and P i P i -T . 1 Pr Pw - T r T . - T 1 W ( 3 . 2 ) ( 3 . 3 )
When d i s c u s s i n g a polyatomic g a s , r] and cr can be defined e x a c t l y a s for a m o n a t o m i c g a s , but the definition of the t h e r m a l a c c o m m o d a t i o n coefficient will obviously have to include the i n t e r n a l e n e r g y flux. We can define e i t h e r one coefficient for the total e n e r g y flux, or s e p a r a t e coefficients for the different m o d e s of e n e r g y . T h u s if E . i s the t o t a l e n e r g y flux E , E .
and s i m i l a r l y for E E." the t r a n s l a t i o n a l ,
1 a c t i v e and i n e r t fluxes r e s p e c t i v e l y , then E. = E. + E ' + E*
and E we can define the total t h e r m a l a c c o m m o d a t i o n coefficient w
o
and we can define E . -1 Ë -1 • E r - Ë w ( 3 . 4 ) and E ' 1 E'. 1 E " 1 E:' 1 - E ' r - E ' w - E " r - E" w (3.5) ( 3 . 6 )
for the active and i n e r t e n e r g y m o d e s r e s p e c t i v e l y . T o g e t h e r with (3.5) and (3.6) we h a v e , of c o u r s e , a a s defined by (3.1) for the t r a n s l a t i o n a l e n e r g y flux. It i s e a s y to s e e that
a(E. - E ) + a'ÏE: - E ' l + a " r E " - E " ]
^ = —J: ^ ±^ JiLl L_i ^-' (3.7)
E . - E 1 w
When d e a l i n g with a m o n a t o m i c g a s we s e e that all the q u a n t i t i e s needed to e v a l u a t e the o v e r a l l flow p a r a m e t e r s can be obtained. E . , p. and T . can be evaluated in t e r m s of known q u a n t i t i e s a s in s e c t i o n 2 . 1 . F u r t h e r m o r e , by m a k i n g the a s s u m p t i o n that the v e l o c i t y d i s t r i b u t i o n is Maxwellian E , p and T can be e v a l u a t e d , a s will
w w w
be shown l a t e r . T h i s l e a v e s Ej,, pj, and TJ.. NOW the t o t a l e n e r g y t r a n s f e r p e r unit t i m e i s equal to E^ - E j . , the total p r e s s u r e is equal to p^ + pj. and the total skin f r i c t i o n i s equal t o T^ " T^., and t h e s e t h r e e q u a n t i t i e s a r e a l l capable of being
m e a s u r e d e x p e r i m e n t a l l y . Thus with the aid of e x p e r i m e n t we can find the v a r i a t i o n of a, n and cr with changing flow p r o p e r t i e s .
In the c a s e of a p o l y a t o m i c g a s , h o w e v e r , it i s not p o s s i b l e to m e a s u r e d i r e c t l y the e n e r g y t r a n s f e r s a s s o c i a t e d with the v a r i o u s m o d e s of e n e r g y , but only t h e i r
combined t o t a l . T h u s a t h e o r e t i c a l r e l a t i o n s h i p between a, a' and a" , o r a p r e d i c t i o n of t h e i r r e s p e c t i v e v a r i a t i o n s with flow p r o p e r t i e s , would be of g r e a t u s e . As a f i r s t s t e p in t h i s d i r e c t i o n we m u s t p r o c e e d to e x a m i n e the m e c h a n i s m of the g a s - s u r f a c e i n t e r a c t i o n in m o r e d e t a i l .
All the p a r a m e t e r s defined above will be dependent on the m e c h a n i s m of the e n e r g y i n t e r c h a n g e between the gas and the s o l i d . When Maxwell f i r s t studied t h i s type of p r o b l e m , he postulated that a fraction f, of the incident m o l e c u l e s , would be t e m p o r a r i l y t r a p p e d by the s u r f a c e whilst the r e m a i n d e r would be
r e f l e c t e d s p e c u l a r l y , i . e . with an angle of r e f l e c t i o n equal to the angle of i n c i d e n c e , and a r e l a t i v e s p e e d equal to the speed of the incident m o l e c u l e . T h o s e m o l e c u l e s t r a p p e d by the s u r f a c e would be c o n s i d e r e d a s coming from a g a s inside the s u r f a c e , at the s a m e t e m p e r a t u r e a s the s u r f a c e . T h i s l a t t e r type of r e f l e c t i o n Maxwell t e r m e d diffuse r e f l e c t i o n . It can be s e e n that when the velocity d i s t r i b u t i o n of the diffusely r e f l e c t e d m o l e c u l e s i s a s s u m e d Maxwellian then, for a m o n a t o m i c g a s ,
a, r) and a can be e x p r e s s e d in t e r m s of f. However, t h i s m o d e l of the r e f l e c t i o n
p r o c e s s i s nowadays c o n s i d e r e d to be too s i m p l e , and will not be adopted h e r e . In fact, t h e r e i s e x p e r i m e n t a l evidence that for m o d e r a t e v a l u e s of the s p e e d r a t i o S,
a s defined in s e c t i o n 2 . 1 . s u r f a c e i n t e r a c t i o n s ' " ' .
the value of f i s unity in the m a j o r i t y of g a s
-The m o r e r e a l i s t i c m o d e l of the i n t e r a c t i o n , which i s adopted h e r e , i s a s follows. When a gas m o l e c u l e collides with a solid s u r f a c e , one of two a p p a r e n t l y different f o r m s of c o l l i s i o n m a y o c c u r . Thus the m o l e c u l e m^ay rebound i m m e d i a t e l y at s o m e angle u n r e l a t e d to its angle of incidence and at the s a m e t i m e undergo in g e n e r a l a change in e n e r g y . A l t e r n a t i v e l y , the m o l e c u l e m a y be a d s o r b e d on the s u r f a c e for s o m e t i m e and then be r e - e m i t t e d with different e n e r g y , and again the angle at which it l e a v e s the s u r f a c e will be-u n r e l a t e d to its incident a n g l e . In the above we a r e r e f e r r i n g to p h y s i c a l a d s o r p t i o n , in which the m o l e c u l e s a r e held to the s u r f a c e by Van d e r W a a l s f o r c e s (6) We a r e not c o n s i d e r i n g c h e m i s o r p t i o n in which t h e r e i s an i n t e r -change of e l e c t r o n s between the s u r f a c e and the g a s , d i s s o c i a t i o n , or any such p r o c e s s r e q u i r i n g high activation e n e r g y . The phenomenon of p h y s i c a l a d s o r p t i o n a r i s e s a s a d i r e c t consequence of the f o r m of the m u t u a l potential e n e r g y existing b e t w e e n the m o l e c u l e and the solid body. T h i s potential e n e r g y can be
r e p r e s e n t e d by the curve shown in the s k e t c h below ;
\ ^« . Z ' , , L—.J I • ^ .
V i s the m u t u a l potential e n e r g y and z is d i s t a n c e m e a s u r e d along the outward n o r m a l from the s u r f a c e . A m o l e c u l e approaching the s u r f a c e will have i t s k i n e t i c e n e r g y of t r a n s l a t i o n i n c r e a s e d by an amount D a s it
p r o c e e d s from z =« z^ w h e r e V " 0 to z = z^. On p a s s i n g through the point z = ZQ, it will m e e t the r e p u l s i v e f o r c e , s i n c e for z < ZQ V i n c r e a s e s with d e c r e a s i n g z, and t h i s stage of its j o u r n e y r e p r e s e n t s its collision with the s u r f a c e . If we now c o n s i d e r the s u r f a c e smooth on the m o l e c u l a r s c a l e and a l s o r i g i d , the m o l e c u l e will rebound with the m a j o r portion of i t s kinetic e n e r g y i n t a c t , and will thus be able to e s c a p e completely from the s u r f a c e . On the o t h e r hand, if the s u r f a c e i s rough on the m o l e c u l a r s c a l e and a l s o n o n -r i g i d , the m o l e c u l e will give up the m a j o -r p o -r t i o n of i t s e n e -r g y without n e c e s s a -r i l y a c q u i r i n g a sufficiently l a r g e velocity component n o r m a l to the s u r f a c e , and will t h e r e f o r e not be able to e s c a p e from the potential well. An o s c i l l a t i n g m o t i o n , between points A and B in the s k e t c h , will be set up. Since s u r f a c e s a r e composed of a t o m s v i b r a t i n g r e l a t i v e to each o t h e r , with c r a c k s or i n t e r s t i c e s b e t w e e n , the second s u r f a c e m o d e l i s m o r e r e a l i s t i c .
From the preceding paragraph it is clear that the quantity D is closely
related to the so-called heat of adsorption which has been measured experinnentally for a large number of gases and surfaces(S). it has also been demonstrated(8) that D varies over the surface of the adsorbent (the solid taking up the gas). Therefore, if a gas molecule, on first colliding with a surface atom, hits that atom, head on, on a part of the surface where D is smaller say, there is a possibility that the molecule will rebound with enough energy to escape the potential well. Furthermore, if we consider gas-surface combinations with smaller and smaller D, the probability of molecules escaping after one collision i n c r e a s e s . On this basis there is no necessity to differentiate between single inelastic collisions and physical adsorption, since we may take the one as being a limiting case of the other.
Before we can complete the outline of our model of gas-surface interaction, we must decide on the process by which those molecules which are trapped in the potential well, eventually escape from the surface. There is no experimental evidence to guide us on this point, and we shall assume with Zwanzig'*'', that this energy of desorption is communicated to the molecule by the chance coming together of sound waves in the surface lattice; these sound waves transfer a sufficient
amount of energy to the molecule for it to escape the potential well.
This, then, completes the outline of our gas-surface interaction model. A point to be noted is, that if we assume that the internal energy of polyatomic molecules is not affected by the mutual potential energy existing between the gas molecule and the surface, then the above model will suffice for both monatomic and polyatomic molecules. To fill in the detail of our model we have to consider what happens to the molecules whilst they are trapped on the surface, with particular regard to their energy changes. We shall do this first of all in the simpler case of a monatomic gas, since it is easier in this case to ensure that any assumptions we make do not violate available experimental data.
3 . 1 . Monatonaic gas-surface interaction
Consider a monatomic gas at a certain temperature T^ in contact with a solid surface at constant temperature T ^ ; free-molecule conditions being applicable, of course, as throughout this report. Then if a state of dynamic equilibrium exists, the average energy of the molecules in the adsorbed layer will be constant with respect to time. In the particular case of T^ > T^, this constant energy in
the adsorbed layer is maintained by the following process. Molecules are continually being adsorbed and desorbed. The average energy of all thernolecules which became
adsorbed will increase during their time of contact with the surface. Thus dynamic thermal equilibrium is set up by energy being transferred at a constant rate from the solid to the adsorbed layer, this process being balanced by the continuous desorption of the higher energy molecules and the continuous adsorption of the lower energy molecules. Note that the heat of adsorption plays no part in this flow of energy since the body has to give it all up again as heat of desorption.
A common assumption made, and one having some experimental justification, is that the adsorbed molecules do not interact one with the other'"'. We will follow this assumption here, so that all the energy interchange is between the solid and the adsorbed layer of molecules and not between the adsorbed molecules themselves. The molecules in the layer receive their energy changes by way of impacts with the
8
-vibrating atoms of the surface. The vibration of the atoms will not all be normal to the surface, so that the adsorbed molecules will probably move over the surface in a hopping motion, rather than undergo oscillations on one spot. The degree of mobility of the molecules is discussed in (6).
The amount of energy a trapped molecule can receive in any one collision is limited by the frequency spectrum of the vibrating atoms of the surface. Furthermore, molecules with higher energy, i . e . the faster moving molecules within the adsorbed layer, will be more likely to deform the surface lattice which they come into collision with, and these will be more likely to undergo an energy change. This is very similar to the energy exchange mechanism of the internal modes of energy of a polyatomic molecule in the gas phase. For such gas phase systems, the well known' ' relaxation equation can be shown to describe the rate of energy transfci' between translational energy and internal energy. Therefore we will apply this type of equation here, and examine the results of such an analysis in the light of experimental evidence.
Since we are considering a state of dynamic equilibrium, the following method of considering the energy exchange is permissible. We can consider the N^ molecules which impinge on unit area of the surface per unit time, as all entering into energy exchange with the surface at the same time, staying an average time T (the average time of adsorption) and all leaving the surface after a time interval T . The energy they c a r r y away can then be equated to E^.. Thus if E is the energy of the Nj molecules at a time t after adsorption, we postulate that a good approximation to the rate of change of E with time is given by
| E . . 1 (E - E ) (3.8) 8t T w
a
where E is as defined in Section 3, and Tg^ is the relaxation time of the energy exchange. To draw the analogy in detail, E is equivalent to the inert internal energy naode of a gas, which can only receive energy from the translational energy mode via collisions. It cannot be stressed too strongly, however, that T is an entirely different quantity from the gas phase relaxation time.
The relaxation time in the gas phase is determined by the energy transition rates of the particular inert internal mode of energy being considered. One does not need to introduce the transition rates of the translational energy since the translational energy of a molecule is changed at every collision; translational energy changes occur without involving the internal energy. Thus the relaxation time is dependent only on the slower transition r a t e s . These slower transition rates are in turn dependent on the translational motion and the energy involved in each internal mode energy change'^'.
The relaxation time we are concerned with will also be determined by the slower transition r a t e s , but several differences are apparent. Since we are not allowing direct energy exchanges between adsorbed molecules, we may safely assume that they will have slower transition rates than the surface atoms. These latter will be continuously exchanging energy between themselves. However, the frequency of collision, and the probability of an energy exchange resulting from it, will now be determined largely by the motion of the adsorbed molecules, whilst the total energy available will be dependent on the temperature of the solid ( i . e . the vibrational motion of its atoms). Thus it is not unreasonable to expect that T^
will show a m a r k e d dependence on T^ and T ^ .
T h e solution of equation 3 . 8 can be w r i t t e n down i m m e d i a t e l y s i n c e T ^ i s constant (with T ^ ' c o n s t . ) , o r
- t / T a E = A e + E
w
F u r t h e r m o r e , when t = 0, E = E^ and a l s o when t = f", E = E^.
- r h
E = (E. - E ) e + E (3.9) r 1 w w whence _ - T / T a = 1 - e •* (3.10) Note that a s T - <» , a - I and a s T^^ •• <» , a - 0, so that the l i m i t i n gb e h a v i o u r i s a s we would e x p e c t . F r e n k e l * ^ " ' shows that — M w
T = T e o
w h e r e TQ i s the a v e r a g e period of v i b r a t i o n n o r m a l to the s u r f a c e of the a d s o r b e d m o l e c u l e s , Q i s the a v e r a g e heat of a d s o r p t i o n p e r mole of a d s o r b e d g a s , and R]y[ i s the m o l a r gas c o n s t a n t . The v a r i a t i o n of T" with t e m p e r a t u r e i s thus known, provided that the v a r i a t i o n of Q with t e m p e r a t u r e i s available from e x p e r i m e n t .
The e x p e r i m e n t a l evidence on t h i s point i s inconclusive and we s h a l l follow B r u n a u e r ' s ^ ' advice in r e g a r d i n g Q a s independent of t e m p e r a t u r e for m o d e r a t e t e m p e r a t u r e
v a r i a t i o n s .
Before leaving the d i s c u s s i o n on T" one l a s t point m u s t be clarified, i t s v a r i a t i o n with S, the speed r a t i o of the incident m o l e c u l e s . Nocilla*^^' g i v e s an e x p r e s s i o n for t h i s v a r i a t i o n , b a s e d on the a s s u m p t i o n that the n u m b e r of m o l e c u l e s in the a d s o r b e d l a y e r i s independent of S. He d e r i v e s t h i s e x p r e s s i o n in o r d e r to explain the v a r i a t i o n of the t h e r m a l a c c o m m o d a t i o n coefficient with S, a s r e p o r t e d by Devienne*^'*'. F r e n k e l , in h i s a n a l y s i s , a s s u m e s a s t a t i c g a s but d o e s not specify s u r f a c e satui'ation ( i . e . that all available a d s o r p t i o n s i t e s a r e o c c u p i e d ) . To a s s u m e the n u m b e r of a d s o r b e d m o l e c u l e s constant without knowing the conditions p r e v a i l i n g in the e x p e r i m e n t d o e s not a p p e a r to have any s p e c i a l m e r i t o v e r a s s u m i n g -f c o n s t a n t . T h e r e f o r e in the r e s t of t h i s work we will take -Tas constant for m o d e r a t e r a n g e s of S. F u r t h e r m o r e t h i s v a r i a t i o n of a ( a s d e t e r m i n e d e x p e r i m e n t a l l y ) with speed r a t i o can be a t t r i b u t e d to a n o t h e r c a u s e ( s e e s e c t i o n 3 . 2 ) .
On the b a s i s of the above we can now p r e d i c t the changes in -Fdue to changes in t e m p e r a t u r e and speed r a t i o . The c o r r e s p o n d i n g v a r i a t i o n of T^ h a s not been obtained e x p l i c i t l y , but s o m e guidance a s to i t s probable b e h a v i o u r can be found in e x p e r i m e n t a l r e s u l t s and will be d i s c u s s e d l a t e r .
A r i g o r o u s proof that equation 3 . 8 i s applicable to our p r o b l e m h a s , up to now, not been f o r t h c o m i n g . H o w e v e r , we m a y s a y that such e q u a t i o n s have been shown to hold for d e v i a t i o n s from e n e r g y e q u i l i b r i u m , w h e r e the r e t u r n to e q u i l i b r i u m i s l i m i t e d to one quantum e n e r g y change at each effective c o l l i s i o n . A c c o r d i n g to S t r a c h a n , cited in*^^) such a l i m i t a t i o n a p p l i e s h e r e , and we s h a l l now outline
1 0
the s t e p s which led us to p o s t u l a t e the applicability of equation 3 . 8 .
A s s u m e a solid m a d e up of h a r m o n i c o s c i l l a t i o n s all of the s a m e f r e q u e n c e y
V . T h i s i s the m o d e l E i n s t e i n used in his specific heat t h e o r y . F u r t h e r m o r e ,
following S t r a c h a n , the e n e r g y exchanges between solid and a d s o r b e d l a y e r a r e l i m i t e d to one quantum, hv p e r e x c h a n g e . L e t K - , , -(e, e + hf) be the r a t e at which an a d s o r b e d m o l e c u l e with e n e r g y in the r a n g e e, e + d e , t a k e s up e n e r g y
hv from a s u r f a c e atom with e n e r g y (i + l ) h i ' . S i m i l a r l y , Kj ^^.^(e + hv, e) is
the r a t e at which an a d s o r b e d m o l e c u l e with e n e r g y in the r a n g e e + hv , e + hv-t- de gives up hi/ to a s u r f a c e atom with e n e r g y i h i * . ( i = 0 , 1 , 2 e t c . ) . Let Ng de be the n u m b e r of a d s o r b e d m o l e c u l e s with e n e r g y e, e + de and n^ the n u m b e r of s u r f a c e a t o m s with e n e r g y i h v .
Now it i s well known'^' that when c o n s i d e r i n g t r a n s i t i o n r a t e of h a r m o n i c o s c i l l a t o r s , we can take out a f a c t o r t o allow for r a d i a t i v e effects and w r i t e
K. , . ( e , e + hi^) = (i + 1) K ( e , e + hi/)
1+1 , 1 10
and K. (e + hv, e) = (i + 1) K^,, (e + hv, e)
T h u s we can w r i t e 8E
8t hv Z (i + 1) / n . , , N K (e, e + hi/) - n.N ^. K (e+hv.e) de . „ J I 1+1 e 10 ' 1 e+hv 01 J i = 0
If we invoke the P r i n c i p l e of D e t a i l e d Balancing we can w r i t e , when E = E
n. N ^ (i + 1) K (e + hv e) = n. , N (i + 1)K ( e , e + hv)
1 e+hv 0 1 1+1 e 10
w h e r e d e n o t e s e q u i l i b r i u m v a l u e s . Now s i n c e we a r e c o n s i d e r i n g the c a s e w h e r e T i s m a i n t a i n e d at a constant v a l u e , we will a s s u m e that " i ^ " i
w ~ —iS— . "i+1 "i+1 T h i s will be even b e t t e r if we l i m i t o u r s e l v e s to s m a l l v a l u e s of I T. - T 1. ' 1 w ' We m a y t h e r e f o r e w r i t e " r N K (e, e + hv) ^ flE " e 10 * *
- f = h. , - U ^ l ) / [ -
^ — • N^,,^K,(e + h . e )
-1=0 J '^ N K ( e , e + hv) N ^ K (e + hi/, e) de e+hy 01 J 00 . -.= hi/ £ n. (i + 1) [ N* ,_ K* (e + hi/, e) - N . K (e + hi/, e) de ._ 1 j (_ e+hi/ 0 1 e+hv oi J s i n c e i=0 o N K ( e , e + hi/) e 10 is of o r d e r unity for T. of o r d e r T . * * "^ 1 w N K (e, e + h i ) e 10
on the energy levels involved in the transition. Thus we may postulate that the following type of relationship will hold
t^ / . u \ e+hv A K„, (e + hv. e) k T T
where k is Boltzmann's constant, A is a non-dimensional constant and T has units of time. Making a similar assumption for K {e + hv, e), we can write
8E C C *
—— ^ —»- E - — E where C and C are non-dinaensional
a t T w T , .
a a constants.
Then on the assumption that E., E and E do not differ greatly from each other C / T * = C / T SO that
a a
1^ = - -^ (E - E ).
9 t T w
a
As we stated at the outset, the foregoing analysis is not intended as a rigorous proof of the applicability of this equation to our problem, but rather as the basis on which we postulate its applicability.
In the next section we shall draw some conclusions as to the behaviour of T with changing T. and T , which contradict the views of Jackson and Howard^^^' and Devonshire \ 12). Their theories were based on the assumption that each gas molecule suffered only one inelastic collision with the surface ( i . e . adsorption was disallowed) and this may explain the difference between their results and that given here.
To summarise, we have an expression for a the accommodation coefficient of a monatomic gas, whose behaviour we can predict in a general way for variations of T., T Q, the integral heat of adsorption, and S, the speed ratio of the incident molecules. In this next section available experimental results will be used to test the validity of our expression.
3.2. Experimental evidence. Monatomic gases.
There are a great number of experimental results available, giving the
accommodation coefficients of monatomic gases on various surfaces. In the main these results all show the same trends, although in a number of cases vastly different values have been obtained for the same gas-surface combination. Most w r i t e r s in the field put this down to different surface conditions, some experimienters having taken more care than others in ensuring a clean surface and a pure gas.
However this need not worry us unduly since the main effect of a surface having an adsorbed layer of impurity,will be to change Q and hence T. This does not affect the applicability of our expression for a.
The results we have chosen to demonstrate the variation of a when T - T,
w 1
12
-These results were obtained for helium, neon and argon on a tungsten surface, varying T. from 100°K to 300°K and keeping T - T. constant at 18°K. They are shown in Fig. 4, along with our theoretical curves. The values used in plotting the theoretical curves are also shown in the figure. It can be seen that for the lower values of the accommodation coefficient, suitably chosen constant values of T . / T Q are sufficient to give quite a good approximation to the experimental results. For the argon curve, a suitable variation of T^/TQ with temperature would have to be introduced in order to obtain a better fit. However, even in this case, the assumption that T^^/TQ is constant gives the main trend of the experimental r e s u l t s . It should be mentioned that De B o e r ' " ' states that argon on clean tungsten will have a value of Q = 3,000 cals. per mole approx-imately. He also inaplies that over the temperature range concerned, he would expect a for argon to have a value of unity, so that we may feel justified in choosing a lower value of Q.
For the results showing the variation of a with T^ - T^, keeping T^ constant, we turn to (2) where results of Oliver are cited. These are for helium and argon on tungsten. The re s u lts , as given in (2) are shown in Fig. 5. These results are replotted as a against T / T assuming Q = 2000 cals. per mole, from which is derived Fig. 6, showing TQIT^^ against T^ - T^. As can be seen T Q / T ^ tends
to zero as T^ - T^ tends to zero. This is, at first sight, a surprising result since it means that as T; -• T ^ , T - «>. However, it should be borne in mind
1 w' a
that Ta may behave quite differently from the gas-phase relaxation time. Here again the value of Q for helium on tungsten is high according to De Boer, but we may safely assume that the surface was contaminated in some way, otherwise the low values of a found by Thomas and Schofield would have been obtained.
The results of Oliver were obtained with a static gas and, as we have seen, can be explained by assunaing our prediction of the behaviour of T^ to be correct. Consider now a flat plate placed in a free-molecule stream, and let T^ the temperature of the plate, be greater than Tj. Let us assume that a is measured on the lower surface (referring now to Fig. 1) for increasing values of the speed ratio. We see from Fig. 2 that at constant incidence, as S increases, K increases, and eventually, if T ^ is maintained at a constant temperature, we would have
K T- = T,.,. Then, since we could replace T- in the static case by K^ TJ in the
1 w '^ 1 • ' 1
non-static case, we would expect the measured values of a when plotted against S to decrease to zero at the value of S, where K^T^ = T^ if our prediction of T^^'S behaviour is correct. The previously mentioned results of Devienne were obtained with an insulated radiating flat place normal to the flow. In working out his r es u l t s , Devienne assumes that the accommodation coefficients of front and r e a r surfaces will be equal. However, we see from Figs. 1 and 2 that in the case of an insulated flat plate normal to the flow, the naolecules striking the front surface of the plate would be giving it energy (K > 1), whilst those striking the r e a r surface would be taking energy away from the plate (K'^ < 1). Under these circumstances it is difficult to see how the results can be used to throw light on the variation of
thermal accommodation coefficient with speed ratio. For this reason, these results will not be used as a comparison with the results of this paper.
This is as far as we will take the comparison with experiment at this stage. The experimental results quoted follow trends which have been observed in more than one experiment, and we see that our theoretical results follow these trends
in quite a satisfactory manner. We will, therefore, proceed in the next section to apply similar methods to a polyatomic gas. However it would not be fair to leave this section without noting that some results, notably those of R o b e r t s ' ^ ' j can only be explained by our model if we assume that Q increases with temperature or, alternatively, that T^/TQ descreases with temperature.
4. Polyatomic gas-surface
interaction-Before we can proceed to apply the methods of section 3 . 1 . to a polyatomic gas, we have to decide what the values of the translational, active and inert energies of our molecules a r e , immediately after adsorption. In section 2 . 1 . we obtained an expression for Ej which was made up, partly of the thermal motion of the molecule and partly of the mean m a s s motion of the incident stream. When the gas strikes the surface, the distinction diappears, and we may write, as in
equations (2.5) and (2.6), E^ = 2m N^ R KT^. Now at a later time the translational energy of the molecules will be E. Let us define a quantity T,by T= __E__
amN^R Then as E varies from Ej to E^, T will vary from K Tj to T^.. Now consider the active energy which the Nj molecules carry to the surface. From equations (2.7) etc. , this is given by E'^ = ^ m Nj j'R T^. Let E ' denote the energy of these molecules at a later time, t, and define T' = E /^ m N. j ' R , then T ' will change from T to T ' as E ' changes from E ' to E ' .
1 r " 1 r
Here we are considering an active mode and, by definition, an active mode adjusts itself immediately to any change in the translational mode energy. In the gas phase this adjustment takes place via molecular collisions. Since in this case there are no inter-molecular collisions, we will take "active" as meaning the active mode will adjust itself to the surface temperature at the same rate as does the translational mode. In the same way we define T" for the inert mode and note that it v a i i e s from T. to T .
1 r
When, in the previous section, we were only concerned with the translational energy, it was reasonable to assume that E^ was the state to which E would tend. However, now the internal energy only comes into contact with the surface,insofar as it is brought into a collision by the translational motion. Furthermore, we know that energy transition rates are closely linked with the binding forces applicable to the mode of energy concerned. For example, in the gas phase the transition r a t e s of, say, a vibrational energy mode are much slower than the transition rates of the translational energy, and the forces binding the atoms together into molecules a r e nauch stronger than the mutual forces between the molecules. If, in the case of adsorption, the forces which bind the molecules to the surface become such that the transition r a t e s for the translational and active energy modes are less than the transition rates of the inert energy, then we have chemi-sorption, and probably dissociation. Thus for physical adsorption, we postulate the following equations as governing the energy exchanges :
1^ . - 1 (E - E ) - -V (E -
E;)- -i- (E - E")
a t T W T ' T . ^ a 1
If' . - i . ( E ' - E ; ,
aI f " " - T - <^"-^:^
114
-w h e r e T , T ' and T . a r e the r e l a x a t i o n t i m e s a s s o c i a t e d -with the t r a n s l a t i o n a l ,
a 1
a c t i v e and i n e r t m o d e s r e s p e c t i v e l y , E, i s the e n e r g y the t r a n s l a t i o n mode would have if it w e r e in e q u i l i b r i u m with the active m o d e ; E'^ is the e n e r g y the t r a n s -l a t i o n a -l mode wou-ld have if it w e r e in e q u i -l i b r i u m with the i n e r t m o d e ; E j and E^' a r e the e n e r g i e s the active and i n e r t naodes r e s p e c t i v e l y would have if they w e r e in e q u i l i b r i u m with the t r a n s l a t i o n a l m o d e . We now want to s a y that T ' i s negligible when c o m p a r e d w i t h T J . H o w e v e r , for a moving (K ^ 1) g a s , t h i s i m p l i e s an i n s t a n t a n e o u s change in E and E ' at the m o m e n t of a d s o r p t i o n , s i n c e when the gas b e c o m e s a d s o r b e d a l l its t r a n s l a t i o n a l e n e r g y i s naade up of t h e r m a l m o t i o n . However, if the gas i s s t a t i c t h i s problena does not a r i s e , so we will p r o c e e d with the s t a t i c c a s e and d e a l with K 'i' 1 l a t e r .
Let us now neglect T' with the r e s u l t that our equations of e n e r g y change b e c o m e | E , . _ L (E . E ) . i _ (E _ E« ) 8 t T W T. ' a 1 ' 8 E 1 . // „ « . _ , . _ (E - E ; 1
w h e r e E now d e n o t e s t r a n s l a t i o n a l plus active e n e r g y . If E = (2m N. R + i m N. j ' R) T
E" = (2m N. R + I m N. j ' R) T* ( 4 . 1 ) 2 1 1
and E^' = i m N. j " R T
then t h e s e equations a r e quite e a s y to s o l v e . The assunaptions n e c e s s a r y to justify t h i s s t e p a r e that the m o l e c u l e s a r e r e - e m i t t e d from the s u r f a c e with a Maxwellian velocity d i s t r i b u t i o n c o r r e s p o n d i n g to a t e m p e r a t u r e Tj-, and that T ^ - T j i s s m a l l . A c c e p t i n g t h e s e a s s u m p t i o n s the equations b e c o m e : ^ = - 1 _ (T - T ) - — (T - T " ) 8t T W T . a 1 9 T " _ _1_ at ' ' T . ( T " - T) 1 T h e solutions a r e T = A e " ' ' * + B e - ' ^ * + T w and
T"= A T . ( ^ + ^-^e-'^'+Br. (^ + -ï- - . V ' +T
w h e r e A and B a r e a r b i t r a r y c o n s t a n t s andNow when t ^ O , T » T " = T . whence f 1 / T -V _ n - ih E = ( E . - E ) e"^"" + r 1 Vi <- ^^ - V n - V a -l/T - e + E 1 / T -V — n - 1/T — -^ E ' = ( E ' - E ' ) i —^ e-"^ + ^ e-''-' + E ' r I W L A ' - I ' IU ~ V J w E " f T. ( 1 / T - I / ) ( 1 / T + 1 / T . - / U ) - T.(M - 1 / T ) ( 1 / T + 1 / T . - I ^ = ( E * - E " ) - i 2 ^^ ^ e-^-^ + - i S ^ L - e - " ^ (+E 1 VJ (- H - V w
F r o m t h e s e e q u a t i o n s it i s e a s y to obtain e x p r e s s i o n s for a, a' and a" : M- 1 / T a a' = I 1/T -V a -uT e M - y M - I' a -i/T e a n d ( 4 . 2 ) T. ( 1 / T - I / ) ( 1 / T + 1 / T . - M ) - T.(M - 1 / T ) ( 1 / T + 1 / T . - y ) „ , 1 a a 1 -UT 1 a a i -vr a - I - e - e M - 1/ ( 4 . 3 ) Note t h a t as T •• oo a s T" •• 0 a s T •• •» a a s T .. 0 a s T. -1 a s T. a, a' and a" - \ a a and a" - 0 a, a ' a n d a" •• 0 J ' 1 " 1 " T / T ; a and a - * l , o - » l - e ^ a and o' •• I - e ^ , a" •• 0 / , « , -T/2Ta a, a and a - I - e *
T h u s our e x p r e s s i o n s for the t h r e e a c c o m m o d a t i o n coefficients behave a s we would expect in the v a r i o u s l i m i t i n g c a s e s . It i s i n t e r e s t i n g t o note t h a t , a s we would e x p e c t , in the two c a s e s T . = •" , T . = 0 we obtain e x p r e s s i o n s i d e n t i c a l in f o r m t o the e x p r e s s i o n obtained e a r l i e r for a m o n a t o m i c g a s .
Before d i s c u s s i n g the m e a n i n g of t h e s e r e s u l t s , we m u s t c o n s i d e r the c a s e of a g a s with a m e a n naotion (K 4 D- As s t a t e d e a r l i e r , in t h i s c a s e it i s not p o s s i b l e to think of the a c t i v e mode a s being in e q u i l i b r i u m with the t r a n s l a t i o n a l motion, and the s u b s t i t u t i o n s of equation 4 . 1 a r e c e r t a i n l y no longer valid. The only way round the difficulty, and one which can be b a s e d on p h y s i c a l a r g u m e n t s , i s to a s s u m e that the active e n e r g y , whilst having the s a m e t r a n s i t i o n r a t e s a s the t r a n s l a t i o n a l e n e r g y , l a g s behind the l e v e l of the t r a n s l a t i o n a l mode by a constant a m o u n t . T h i s
16
-constant amount is represented by i m N. j'R (KT. - T.). Here we are invoking the same argument as before, that the active mode transition rates cannot be greater than the translational mode transition rates in the type of adsorption we are considering. Similar renaarks apply to the inert mode energy, and on this basis it is obvious that we would expect the same expressions to hold for a, a' and a" in the case K =^ 1 as for K = 1.
The dependence of T and T as functions of temperature and speed ratio would be expected to be the same as for a monatomic gas, and in the next section limited experimental evidence will be shown to support this view. Unfortunately, due to lack of experimental evidence and lack of success in theoretical investigations, it has not been possible to throw any light on the behaviour of T.. It should be reasonable however,to assume that T. will show little dependence on T - T. and will be largely
1 ^ w 1 ^ ^ dependent on the gas temperature and the particular gas being considered. In this respect we expect that T. will show some similarity to the gas phase relaxation time. Fig. 7 shows a - a' plotted against a" for the full range of values of T / T .
H 1 and -T/T , whilst Fig. 8 shows a, a' and a" plotted against temperature. The value of Q = 31)00 cals/mole is representative of the heat of adsorption of a number of the heavier gases. In addition, we have assumed that T and T. will be of the same
^ a 1
order of magnitude, i.e. T / T . = 1, and the value of T /T =100 seems to be of the ^ a 1 a o
right order from our study of monatomic gases. This curve then should be
representative of the variation of a, a' and a" with temperature, assuming T - T. constant, for a gas such as CO^ which has its bending mode excited in the temperature range considered. In the same figure we have shown the variation of a for a naon-atomic gas, using the values of Q and T /T given above.
(18)
An expression for a" has been obtained by Herman and Rubin . They consider the inert mode's energy exchange as completely divorced from the translational and active modes, and also consider that the re-emitted molecules
carry with them an average amount of inert energy which is equal to the average amount of inert energy of the adsorbed layer viewed as a whole. We can best compare our expression with theirs by taking the value of a" when T = 0, i. e.
"T IT' / / I
a"= 1 - e ^. Their expression written in our notation is then a = -—; T-_ .
'^ 1 + T . / T
1 These are compared in Fig. 9. We see that our expression gives larger
values than that of Herman and Rubin as we would expect, since different assumptions are made. The available experimental data are not sufficient to say which is more realistic with regard to estimating acconamodation coefficients, but to assume that molecules just arriving at the surface have as great a probability of leaving as those that have been there some time, as Herman and Rubin do, does not qualitatively fit the observed facts of physical adsorption.
4 . 1 . Experimental r e s u l t s . Polyatomic gases
The most famous experiment performed to find the accommodation coefficient of a polyatomic gas, is the one performed by Knudsen^^O) ^ith hydrogen on platinum. Knudsen used a flat strip of platinum which was bright on one side and blackened on the other, thus giving different accommodation coefficients on each surface. By relating the p r e s s u r e acting on the surface to the translational accommodation coefficient of the surface, in a manner which will be shown in the next section, he was able to demonstrate that the translational and internal accommodation coefficients were equal for each surface. This result is often quoted (c.f. Ref. 19) as demonstrating that the accommodation coefficients of all the modes of energy of a molecule will be equal. This i s , however, a very doubtful assumption since Knudsen, judging from the apparatus he used was alnaost certainly working at about room temperature, although in (20) he does not state the working temperature. Thus the vibrational mode of the hydrogen would not enter into his experiment at all. The only safe conclusion to draw from his result is that he provided some evidence in favour of assunaing that the t r a n s -lational and active accommodation coefficients of a gas will be equal, thus supporting the line we have taken in this report.
At the present time there are no experimental results directly applicable to the results we established in the previous section. There are experimental values for the accommodation coefficients of diatomic gases, Hg and Nj , but the experiments were conducted at temperatures so low that the vibrational energy mode would almost certainly not enter into the energy exchange p r o c e s s . These results* , as we expect from our theory, follow in the main the same trends as the results for mon-atomic gases quoted e a r l i e r . An exception to this are the the results of Blodgett and Langmuir for H^ on W, T - T. being maintained approximately constant. The values they obtain for the accommodation coefficient decrease with tentiperature from 200 K to 500 K and then increase over the range 500 K to 1000 K. It is not easy to explain these results on the basis of our theory without assuming that Q increases with temperature. It should be noted that at room temperature, hydrogen forms a stable chemi-sorbed layer on tungsten which only beconaes unstable at around 2000 K, so that the results of Blodgett and Langmuir are for H on an adsorbed layer of H on tungsten.
The results of Devienne, mentioned e a r l i e r , also include values for polyatonaic gases, but the sanae objection with regard to the interpretation of his data applies as before. Since, as we have stated, all available data appertain to the translational and active modes only, we will proceed at once to the next section where we will consider I n detail the flow over a flat plate in order to see what can be done experimentally or otherwise to illuminate this problem a little m o r e . 5. Flow over a flat plate
Knudsen was the first to point out that, when dealing with a polyatomic gas, the parts played by the t ranslational naode and the internal modes in the heat transfer are not separable experimentally. In order to obtain values for their separate
contributions, one has to relate them in some way to other measurable quantities, such as the skin friction or the normal p r e s s u r e . Now if, as
we have assumed, there is no specular reflection, the skin friction is expressible in known quantities. E a r l i e r we defined cr = ^[i__]_^ . Now, when all the
incident molecules are temporarily trapped by the surface, on re-emission they will show no directional preference, so that T = T = 0 , and T . conaprises the total skin friction. (Note that here T. refers to the skin friction force and should not be
18
-confused with the relaxation time. Since the skin friction only enters briefly into this report, it was thought best to use the conventional symbol). Thus we have to rely on the normal p r e s s u r e to provide us with the necessary extra relationship. Fortunately, in the case of a flat plate, this proves possible.
We have defined P i - P r '' ' Pi - Pw
where p. has been given in section 2 . 1 . Now p is easily expressible in t e r m s of known quantities by assuming that the molecules are re-emitted at temperature T with a Maxwellian velocity distribution. In this case p = i m N v 2 JT R T
^ r i r Note that this implies T. - T small. If we now consider the thermal
accommodation coefficient of the translational mode, and make the same assumption as to velocity distributions, we can write
K T - T 1 r K T. - T 1 w whence T = K T. - a ( K T. - T ) r 1 1 w
Now the total p r e s s u r e acting on one face of the flat plate is p = p. + p = p. + i m N. V 2jrH T
Substituting for T
p = p. + è m N. v' 2jrRrK T. - a ( K T. - T )] and from section 2 . 1 .
p. = i m N . ^ 2jrR T. é
1 1 1 *^
where 1^ is given by equation 2.11 or 2.12, (we do not need to specify which surface we are considering yet).
. ' . p = i m N. ^ 2 7rR i/)/r. + / [ K T . - a ( K T . - T )I1 (5.1.)
1 »- 1 1 i w J
This is the type of analysis Knudsen used, as mentioned e a r l i e r . Knudsen, however, who was dealing with a static gas, then went on to linearise this equation by
assuming T - T. very small, and he was then able to relate the p r e s s u r e difference a c r o s s the plate to the difference of the two translational accommodation coefficients. We will proceed along different lines, as follows.
If p can be naeasured now on one surface of the plate, equation 5.1 will yield a. The difficulty is the measuring of p and a for one surface of the plate only, since in free-molecule flow experiments, the plate must be kept small. However Devienne^^)
4 . 1 . Experimental r e s u l t s . Polyatomic gases
The most famous experiment performed to find the accommodation coefficient of a polyatomic gas, is the one performed by Knudsen^20) ^vith hydrogen on platinum. Knudsen used a flat strip of platinum which was bright on one side and blackened on the other, thus giving different accommodation coefficients on each surface. By relating the p r e s s u r e acting on the surface to the translational accommodation coefficient of the surface, in a manner which will be shown in the next section, he was able to demonstrate that the translational and internal accommodation coefficients were equal for each surface. This result is often quoted (c.f. Ref. 19) as demonstrating that the accommodation coefficients of all the modes of energy of a molecule will be equal. This i s , however, a very doubtful assumption since Knudsen, judging from the apparatus he used was almost certainly working at about room temperature, although in (20) he does not state the working temperature. Thus the vibrational mode of the hydrogen would not enter into his experiment at all. The only safe conclusion to draw from his result is that he provided some evidence in favour of assunaing that the t r a n s -lational and active accommodation coefficients of a gas will be equal, thus supporting the line we have taken in this report.
At the present time there are no experinaental results directly applicable to the results we established in the previous section. There are experinaental values for the accommodation coefficients of diatomic gases, H^ and Ng . but the experiments were conducted at temperatures so low that the vibrational energy mode would almost certainly not enter into the energy exchange p r o c e s s . These results* , as we expect from our theory, follow in the main the same trends as the results for mon-atomic gases quoted e a r l i e r . An exception to this are the the results of Blodgett and Langmuir for H^ on W, T - T. being maintained approximately constant. The values they obtain for the accommodation coefficient decrease with temperature from 200 K to 500 K and then increase over the range 500 K to 1000 K. It is not easy to explain these results on the basis of our theory without assuming that Q increases with temperature. It should be noted that at roona temperature, hydrogen forms a stable chemi-sorbed layer on tungsten which only becomes unstable at around 2000 K, so that the results of Blodgett and Langmuir are for H on an adsorbed layer of H on tungsten.
The results of Devienne, mentioned e a r l i e r , also include values for polyatomic gases, but the same objection with regard to the interpretation of his data applies as before. Since, as we have stated, all available data appertain to the translational and active modes only, we will proceed at once to the next section where we will consider i n detail the flow over a flat plate in order to see what can be done experimentally or otherwise to illuminate this problem a little m o r e . 5. Flow over a flat plate
Knudsen was the first to point out that, when dealing with a polyatomic gas, the parts played by the t ranslational mode and the internal modes in the heat transfer are not separable experimentally. In order to obtain values for their separate
contributions, one has to relate them in some way to other measurable quantities, such as the skin friction or the normal p r e s s u r e . Now if, as
we have assumed, there is no specular reflection, the skin friction is expressible in known quantities. E a r l i e r we defined a = ""^i " ''^r . Now, when all the
incident naolecules are temporarily trapped by the surface, on re-emission they will show no directional preference, so that T = T = 0 , and T , comprises the total skin
r w 1
18
-confused with t h e r e l a x a t i o n t i m e . Since the skin friction only e n t e r s b r i e f l y into t h i s r e p o r t , it w a s thought b e s t to u s e the conventional s y m b o l ) . T h u s we have to r e l y on the n o r m a l p r e s s u r e to p r o v i d e us with the n e c e s s a r y e x t r a r e l a t i o n s h i p . F o r t u n a t e l y , in the c a s e of a flat p l a t e , t h i s p r o v e s p o s s i b l e .
We have defined P i - P r " ' Pi - Pw
w h e r e p . h a s been given in s e c t i o n 2 . 1 . Now p i s e a s i l y e x p r e s s i b l e in t e r m s of known q u a n t i t i e s by a s s u m i n g that the m o l e c u l e s a r e r e - e m i t t e d at t e m p e r a t u r e T with a M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n . In t h i s c a s e p = i m N v 2 ir R T
r '^r "^ i r Note t h a t t h i s i m p l i e s T - T s m a l l . If we now c o n s i d e r the t h e r m a l
a c c o m m o d a t i o n coefficient of the t r a n s l a t i o n a l m o d e , and m a k e the s a m e a s s u m p t i o n a s to v e l o c i t y d i s t r i b u t i o n s , we can w r i t e a K T - T 1 r K T. - T 1 w w h e n c e T = K T . - a ( K T. - T ) r 1 1 w
Now the t o t a l p r e s s u r e a c t i n g on one face of the flat p l a t e i s p = p. + p = p. + i m N. V 2jrR T S u b s t i t u t i n g for T p = p. + i m N. iT 2 7rRrK T. - a ( K T, - T )l 1 1 L i i w J and f r o m s e c t i o n 2 . 1 . p. = I m N. ^^ 2 JTR T . é 1 1 1 *^
w h e r e ip i s given by equation 2.11 o r 2 . 1 2 , (we do not need to specify which s u r f a c e we a r e c o n s i d e r i n g y e t ) .
P = I m N. V 2ffR i/)/T. + / [ K T. - a ( K T. - T )] ( 5 . 1 . )
1 L i 1 i w j
T h i s i s the type of a n a l y s i s Knudsen u s e d , a s mentioned e a r l i e r . Knudsen, h o w e v e r , who w a s d e a l i n g with a s t a t i c g a s , then went on to l i n e a r i s e t h i s equation by
a s s u m i n g T - T . v e r y s m a l l , and he w a s t h e n able to r e l a t e the p r e s s u r e d i f f e r e n c e a c r o s s the p l a t e t o the difference of the two t r a n s l a t i o n a l a c c o m m o d a t i o n c o e f f i c i e n t s . We will p r o c e e d along different l i n e s , a s follows.
If p can be m e a s u r e d now on one s u r f a c e of the p l a t e , equation 5 . 1 will yield a. T h e difficulty i s the m e a s u r i n g of p and a for one s u r f a c e of the plate only, s i n c e in f r e e - m o l e c u l e flow e x p e r i m e n t s , the p l a t e m u s t be kept s m a l l . However Devienne^l^)
has already given the outline of an experiment in which the speed ratio was varied. This being possible, one could choose values of T , T , S and 0
t w 1
such that K T. is equal to T , see Fig. 1, and the heat transfer to the lower 1 w ^
surface of the plate becomes zero. In addition, the p r e s s u r e on the lower surface becomes calculable in ternas of known quantities. One need then m e a s u r e only the total heat transfer to the plate and the total normal p r e s s u r e force. To obtain the inert accommodation coefficient is then a simple matter. We defined in equation 3.4 the combined accommodation coefficient. Hence, having measured a and knowing Q(= a') from the p r e s s u r e measurement, one could evaluate a". This would then, with the aid of Fig. 7, give values for T / T . and Til , and if the relevant value of Q were known, values of T /T a 1 a a o and T . / T could be calculated. The above r e m a r k s obviously apply to making
K " T . (see Fig. 2) equal to T . With K T. = T , the plate would have to be heated in 1 o T ^ 1 w
o r d e r t o maintain constant T ^ , whilst when K " T . = T it would have to be cooled. "' 1 w
The above procedure would allow us to examine the behaviour of the separate accommodation coefficients and hence T and T . , on both sides of the plate, for
a 1
various values of K T. and T . It ;M3 uld not allow us to examine their behaviour 1 w
with varying S or 0 whilst keeping T constant, and this behaviour, in the light of e a r l i e r r e m a r k s , would be of great importance. It does not seem possible to utilise the results of theory to suggest an experimental procedure for this case, and to obtain such results one would have to rely either on the use of sufficiently small p r e s s u r e pickups and heat transfer gauges, or one could possibly shield one surface of the plate from the flow. It is felt to be essential, in experimental work in this field, to use'a flat plate or surface if the results are to be of any use in giving the behaviour of a, a' and a" for varying speed ratio and angle of incidence.
6. Conclusions
At the present time there are neither experimental nor theoretical results available from which to form estimates of the accommodation coefficients of polyatomic gases. Most investigations into the problems of free-molecule flow make use of one of the two following assumptions. They either assume a"= 0 or
a = a' = a" . This latter assumption seenas to be founded on a mis-interpretation
of Knudsen's result (c. f. section 4.1 and Ref. 19). In this report, expressions for the various accommodation coefficients have been obtained which, with the aid of experinaental data, would enable one to estimate their values and variations with temperature, etc. A limited comparison with results obtained with monatomic gases indicates that these expressions have the correct behaviour as regards varying temperature and gas-surface temperature difference. The expressions show the dependence of the coefficients on three factors. These are the heat of adsorption, the relaxation time of the energy exchange between the surface and the active energy modes, and the relaxation time of the inert mode relative to the active modes. (We use the t e r m active here as including translational). The first of
these quantities is known from experiment for a large number of c a s e s . The relaxation times are at the moment almost completely unknown quantities; but some r e m a r k s can be
made about their probable behaviour. It is reasonable to suppose that T , besides its temperature dependence, will be dependent on the m a s s e s of the gas naolecule and
20
-the surface atoms. Thus, for -the same surface, we would not expect T to vary much between gases whose molecules have similar m a s s e s . A more
complete survey of the available monatomic gas results than was possible
within the scope of this investigation should yield information of use in connection with polyatomic gases. Also, it is not unreasonable to suppose that r . will be of the same order of magnitude as the gas phase relaxation time, and this, together with the above, could be used as a basis for estimating the values of the
accommodation coefficients. These two a r e , however, secondary to a direct experimental investigation of the problem, and also, if possible, a rigorous wave-mechanical investigation.
As a by-product of our approach to the problena, it appears that the speed ratio and flow incidence only affect the acconamodation coefficients in so far as they affect the energy difference E - E..
We would therefore expect that the variation of both S and 0 in the flow case would have the same effect as the variation of T. in the static case, if T is
1 w
naaintained constant.
Finally it naust be stressed that experinaental work with polyatomic gases, along the lines indicated in section 5, is of prime importance in order to test the validity of the gas-surface interaction model proposed herein and the theory built up from it.
7. Acknowledgements
The author is indebted to Professor G.M.Lilley, who not only suggested this topic for investigation, but also provided great assistance throughout the ensuing work.
