ORNL-4148.txt

    
 
   
     
  
   
   
 
 
   
  
 
 

   
 

"/RECENE.D BY DTIE SEP111951

 

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T ORNL 4148
UC-80 - Reucfor Technology

  

 

GAS TRANSPGRT linRE MODERATOR GRAPHITE._;,_;‘ i

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B REVI Ew OF THEORY AND S
COUNTERDIFFUSION EXPERIMENTS s

A P Mallnauskas g
T SduL. Rutherford =~ - =
. _ RBEvcnslil L

 

 

 

 

 

 

OAK RIDGE NA'I'IONAI. I.ABORATORY
opercted by T P
UNION CARBIDE CORPORAHON
- forthe .
u s ATOMIC ENERGY commssuon

 

. BISTRIBEUTION OF THIS DOCUMENT, IS UNLIMITE |

 

 
 

 

 

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This rapor! was prepcrod as an account” of Govefnment sponsored work. Naitlrer the \United Stuf@s,_

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ORNL-4148

Contract No. W-7405-eng-26

CE3II ERICES

00} ux 55/

KG §

REACTOR CHEMISTRY DIVISION

GAS TRANSPORT IN MSRE MODERATOR GRAPHITE.

REVIEW OF THEORY AND COUNTERDIFFUSION EXPERIMENTS

A. P. Malinauskas
J. L. Rutherford

- R. B. Evans Il

 

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" Odk Ridge, Tennessee
operated by
UNION CARBIDE CORPORAT_IO_N

'SEPTEMBER 1967
~ OAK RIDGE NATIONAL LABORATORY

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_ for the
U.S. ATOMIC ENERGY COMMISSION

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BISTRIBUTION OF THIS, DOCUMENT {5 UNUMUAED
 

 

 

 

 
 

 

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CONTENTS
CADSEIACE ..o s e s 1
I INEIOQUCHION ..oeoieeiinieiicce ettt st sae st s s con bbb sb b et s e s ana s s ebne besasnnbasns 1
‘,? . NOMENCIALUTE .....c.ooceeeeeereeneieenir e sseassee ete et ies e oo b st sbae cesbas e et e bt et smebsas bt sbbassbens eebertsabessaneans 2
\f;_ L v - .
r‘;’ III. Theory of Gas Flow in Porous Media .........oiiimincininirnenirirscccsnsssesinssessnsessssancsessecsannes 5
o~
* Velocity and FIux Definitions ..........cccmimmmmimerrcimnimiiamimesinsssnsseoissesesssssnestsosssssssmarens 5
Permeability Concepts ......... euserersesnsatesas esneL e sbet ke bns et s aR e AR AR RS SR e R R KRR ORS00 8
_ Binéry Gas Mixture TranSport ... oierrerercrnnenrssesesssnstensnmmirssnesssssssses ttteritreneesnssesssessiaresanas 12
Summary ....................................... 23
TV. EXPEIiMEntal .......eoeovroicereiiieiciiiieicnieiia s snnisssssasssenesissstosss serssssessssnensnressnesssansssans sobtns sesrnrmneranensn 23
Description of MSRE Graphite and the Experimental SPecimen ........ocveveenrecrerrrcserssnes 23
Gas Transport Characterization of the Diffusion Septum ... 26
Summary eeveeetaseeeeeseese s eeeateen s ea et e e es St e e reRe S SRR S SRR RS SRR A e AR e SR re RS 37
Vo ADPPENMIE - oovvoovoveeecesarseeies asssesessssssssssiesssssssssssesssss s s e sstssassssssssa s ssssses s sesssasesssbnasnssasesssen 37

 
 

 

 

 

 

 
 

 

 

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GAS TRANSPORT IN HSRE MODERATOR GRAPHITE.
I. REVIEW OF THEORY AND COUNTERDIFFUSION EXPERIMENTS
. . \ . - . '

A. P. Malinauskas' J. L. Rutherford R. B. Evans IIl

ABSTRACT

The authors develop equations describing gas transport in porous media. Since the
report is directed chiefly to those with little familiarity with gas transport, many simplify-
ing assumptions are made in denvmg the formulas. Development of the theory proceeds
logically from gas transport of a pure gas in a single capillary, to transport of a binary gas
mixture in a single capillary, to gas flow through a bundle of capillaries, and, fmally,_ to
gas flow through a porous medium. Egquations are gi{ren for each type of transport; Practi-
cal applicetions of the' theoretical concepts are also shown for a moderator graphite of the
type used in the Molten-Salt Reactor Experiment (MSRE).

The experimental findings are limited but significant. Under MSRE conditions it appears
quite justifiable'to'ignore normal diffo_sion e_t‘fects in gas transport computations. This
means that gll the gsseous diffusion information necessary to correlate fission product
migration data may be gained through simpie permeability measurements; the more complex
_interdiffusion experiments are not requ-ired. Thus a complete flow-property survey of all
MSRE moderator materials can be performed with a minimum expenditt_rre of time and effort.

 

1. "INTRODUCTION

~

Much has been written on the subject of gas transport in porous media; hence one is somewhat

| apprehensrve in writing another report on the subject, lest he add to the extant confusion rather

._ than clarify some of the concepts which have become confused Nonetheless we have encountered

sufflcrent misinterpretations or misapplications of derived expressions to warrant an additional work

~ as desirable, partrcularly for those with little or no famlharrty w1th gas transpo‘rt

Furthermore, the compllcatrons introduced by the presence of a porous medium have spawned
numerous models, most of which do little more than add computatlonal complexlty or can easily

mlslead the uninitiated into malung totally mcorrect correlatlons among geometric parameters. This

- report_has therefore been written with two primary putposes in mind: first, we seek to convey to

the reader an appreciation of the concepts associated with gas transport in general, and second, we
attempt to demonstrate how the geometric aspects of the problem which are introduced in dealing

with porous septa may be handled efficiently.

~
.

 
 

 

Smce our pr1nc1pal audience is intended to be those w1sh1ng to become famlhar with the sub-

| ject, rather than co-workers in the fleld we have striven to keep the theoretxcal treatment as smple |

_as possible. Thus, for example, only 1sothermal transport is consxdered Similarly, in some in-

stances mathematical rigor has been compromlsed for clarity in presentation, although the rigorously

derived expressions are likewise given and noted_ accordingly. Blbhographlcal references have

also been omitted; all too frequently these prove to be bothersome interruptions. For those wishing

& more detailed treatment, we strongly recommend the treatises listed in the Appendix.
We shéll begin our discus_sion by introducing the various dgfinitions of velocity and flux which |
wili be encountered throughout this work, and then tum our attention to the _actuai task at hand,
namely, the presgéntation of the concepts associated with gas transport. This will be done by con- | .

sidering several types of gas transpbrt The simplest of thesé hence the first to be treated, in-

volves pure. gas flow in a smgle capxllary as the result of an apphed,pressure drop Next, trans- x
" port in a binary gas mixture will be considered; here pressure- and concentratzon—mduced transport - = ¢
will be treated, but we shall still limit the discussion to only a single caplllary This limitation : -

will then be removed by first allowing the gas to be transported through a bundle of identical capz1
laries, in order to gain some famlhanty with the geometrical aspects of the problem, and then we
shall proceed to the case involving a porous medium.
The theoretical portion will be essentially completed thh the latter problem, but to conclude
here would probably be an injustice to those seekmg practical applications of the theory. Accord- ' -~

mgly, we have included a second section; this part is experimental in scope. In order to demon-

- strate the application of the theoretical concepts and to present a reasonably detailed description

of the experimental aspects, the gas transport characteristicé of a particular graphite specimen are
determined by way of example. Although any porous medium would have sufficed, the experimental

data which are presented have been determined for a graphite of the type employed in the Molten-

Salt Reactor Experiment (MSRE). The data thus serve an additional purpose; they may be used at \,
least as an estimate of the extent of gaseous fission product migration in the MSRE graphite. <
Il. NOMENCLATURE B 8-
a, = Scattenng factor for ith gas component.
A= Superf1c1a1 area normal to flow in porous medla, cm?.
B = Viscous flow parameter for a porous medium, cm?. -
c= Subscnpt or superscnpt md1catmg a capillary or caplllary model. ,
= Mean thermal speed of an ith gas particle, ~cm/sec. '
- °6 = Modified transport coefficient with }0 contributions factored out.
.C = Transport coefficient referred to L. ‘ | \
C, = Modified transport coefficient referred to 1. ’ o
- d = Subscript or superscupt indicating dust or dust model - L g
d; = Inner diameter of diffusion septum, cm. ‘ '
. 3
 

 

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-

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d j= Diameter of collision lo; i-j hard spheres, cm.

dj = A combination of driving forces, cm™!.

d, = Outer diameter of diffusion septum, cm. \ -
D = Subscript indicating diffusive flow component.
dv, Volnme element in velocity space, cm®/sec3.
D, = Gas-dust diffusion coefficient, cm?/sec. ;
D, = Combined Knudsen-normal coeff1c1ent for zth-component diffusion, cm?/sec."
N, ; = Normal diffusion coefficient for an i-j binary mixture in free space, cm?/sec.
Dy
(DK> ‘= Knudsen diffusion coefficient for a uniform gas mixture, cm?/sec."

= Knudsen diffusion coefficient, cm?/sec. .

f = Fraction of diffuse reflections or scattering.

f(vl) = Velocity distribution function, particles sec® ecm—5.

Fy= Foroe exerted on a dust particle, dynes.
h = Height of a cylinder, cm.,

J = Net fluxl of all particles, particles or moles per cm? sec.

2

J, = Flux of particles through any one of identical pores, mole per cm sec.

.2

J; = Diffusive flux! of ith particles, particles or moles per cm* sec.

k = Boltzmann’s constant, p/nT, ergs particle~! (°K)™1.
K = Subscript indicating Knudsen diffusion, _
K = Combined Kmidsen-viscous flow permeability coefficient for porous medium, cm?/sec.
K, = Knudsen flow coefficient.
"~ I'=Tre length of a tortuous cap1llary or connected pore, cm.
L = Superficial length along flow path in a porous_ medtum, cm.
~ m = Subscript denoting a particular pore in a.porous medium, cm,
m, = Particle mass, g/particle.

- M, = Molecular weight, g/mole,
M, j= Rate of momentum transfer from 1th to ]th component gcm sec™2,

M i = Rate of momentum transfer from ith component to wall, g cm sec™ 2,

n = Total particle density of real gases, particles or moles per cm3 .

= Density of dust partlcles, particles or moles per cm®.

- Patticle density of ith component, particles or moles per cm3.

n’ = Total particle den51ty includmg n a’ partxcles or moles per cm3. :

N = Number of caplllanes
: p = Total gas pressure, dynes or atm per cm?,

p = Atmospheric pressure, dynes or atm per cm?; ‘
'p,; = Partial pressure of ith component, dynes or atm per cm?.

p’= Fxct:tlous gas pressure teferred to n’ dynes or atm per cm?,

( p) Anthmetlc mean pressure, dynes or atm per cm?,

Ap = Pressnxe drop across specimen, dynes or atm per cm?,

 
 

q = Effectrve tortuosity factor for porous media. -
q= Tortuosrty factor, for identical caplllary bundle = (I/L)2
j = Tortuosity factor referred to a partrcular transport coefficient. |
Q. Volumetnc flow rate measured at atmospheric pressure, P, cms/ sec,
r = Radial coordinate, in general, cm. '
= Particle radius of ith component,\cm.
’o\ = Capillary radius, cm.
<r>.-—- Mean pore radius, cm.
( 2) = Mean-square pore radius, cm. : | " S -
Ar = Distance defining average plane of last collision, cm, - \
R = Gas constant, atm cm® (°K)~! mole™
T = Absolute temperature, °K.
u = Total number-average velocity,! ]/n, cm/sec.
' u, = Average linear velor:ity,1 same as Y, em/sec‘.
= Slip velocity at_ T, , cm/sec. '
V= Subscnpt mdrcatmg viscous flow component
v, = Total mass-avetage velocity,! cm/sec.
v, = Average Imear velocity! (same as u, ), J/n;, cm/sec.
I_/ = Average d1ffusron velocity® referred to v,, also called ‘“‘peculiar velocity,”’ cm/sec.
= Particle or mole fraction of 1th component.
x;’= Pertrcle or mole fraction of ith component referred to n”.
z = Linear flow coordinate, cm. -
o = Subscript generally indicating capillary or pore radius.
a, = Any quantity which is a function of v,
= Average value of any quantity which is a functmn of v
yj = Normal fraction of total admittance for i i diffusion.

I' = The parameter causing a flux, -
J; = Knudsen fraction of total admittance for i diffusion.
a/ ar = Operator indicating partial derivative, cm 1.
€ = Fraction of bulk volume comprised of open pores. Porosity ‘‘seen’’ by equrhbrmm gas
(no flow). . . |
€’ = Fraction of open porosity engaged in linear steady-state flow.
€/q= Porosrty-tortuosrty ratio for a capillary bundle.

€’/q = Effective porosity-tortuosity ratio, D j/.lg ij + for porous med1a ;o

7= Coefficient of viscosity, poises, dynes cm sec =2

v = Number of components in system.

 

All partmle fluxes and velocrtres may be broken down mto drffuswe and viscous components. For ex-

ample,] ]1D+J v—v +v Ju =y +u v
 

 

h

N

 

 

 

- conditions is .

77 = Transcendental number,_ 3.1416.

p = Total mass density of real gases, g/cm?.

p’ = Total mass density including dust partiéles, ‘g/cm3,
&i ;= Modified diameter for an i-j collision, cm. |
3, = Symbol indicating sum.

flg;'l’* = Collision integral for diffusion.

IIl. THEORY OF GAS FLOW IN POROUS MEDIA

Velocity and Flux Defini-tions_

The molecules which comprise an ordinary gas mixture do not possess a single, common

veloc1ty but exhibit a broad range of values. Thus, in describing the motion of a gas in terms of

_the motmns of the 1nd1v1dua1 molecules one utilizes a statistical approach. It is convenient there- -

| fore to define a velocxty}dxstnbutmn function” f(v) which represents the number of molecules per

unit volume whose velocities lie within the rangé dv about v (where v is a volume vector in velocity
space). In a gas mixture, one such distribution function f(v,) is defined for each component. If

n, is the total number of molecules of type 1 per unit volume, then

- 'ff(y}i) &, - | ®

. where the integration is carried out over a veloeity volume containing all possible values of v,.

The average value &, of any quantity which is a function of v, is given by

fa(v ) £(v,) dv,
f f(vi) dv

 

~@/n) faw)fv)dv,; | @
thus, as an’ example, the average velocity of component i in a gas mixture‘ is
| v, = (I/ni)' 'fvi'.f(.vi) dv,. IR B - - (3)
In a uniform gas mixture at rest, | | | |
Fi =0 (all 1) ;

this should not be confused with the_average speed c,, however, since its value under the s_ame

' VSkT,l/z"‘w ' o : " '
1\ 7m, = | Lo o o
where m, denotes the mass of the i-type molecules, k is Boltzmann’s eonstant' and T is the absolute

temperature. The difference between these two quantities is that ¢, represents the average value

of v, when only the magnitude, but not the direction, is considered.

'
e

 
 

- Alternatively, we could write an expressiori for J which is similar in appearance to Eq. (5), thus:

 

 and (7), we see that the equatrons are consistent provxded

 

We are concerned in most laboratory experiments with the number of i molecules which traverse

a given cross section during a'specified period of time, and for this purpose we introduce the flux
, _ ) ‘

.]i:-l l : S o !

Ji=n7;, - o ®)
which is defined as the rate of transport of the i\-t_ype molecules per unit area, The total flux of
the gas is obtained simply by adding up the fluxes of the individual components, so for a v-com-

ponent mixture,

J=EJ5=Y =% . e

o

. J=nmu, o . ) M
in which n = 2 0, represents the molecular density of the gas as a whole. If we compare Egs. (6)

N

v ' _ ' - ‘
‘u=(1/n) z nv, ;- | ' (8)

i=1

,thus u turns out to be just the number-average velocity of the gas mixture.” Note, however, that a

gas mixture at rest (u = 0) does not necessarily imply that transport within the mixture is absent.
Similarly, when momentum transport is of interest, it is convenient to employ a ‘“‘mass-average
velocity”’ Vé such that one can describe the momentum of the gas per unit volume as if all of the

molecules possess the same velocity. This quantity is defined by the relation

V = (I/P) Z nimi i? (9) | ¢
i=1 ‘ C l ‘ =T
where p = z nm, is the mass densxty of the gas. Finally, it is often advantageous to employ what i o -

is descnbed as the * pecuhar velocrty” Vi’ which is defined by the relation

.V=Vi—V0. ] : - ‘ _ (10)

The pecuhar veIomty thus represents the average velocrty of the i-type molecules measured w1th

respect to the mass-average velocity of the gas as a whole. In other words, we allow our co-

~ordinate system itself to move with the velocxty v,- From Egs. (9) and (10) we therefore obtain

the relation
Y amV - o | S ay
Unfortunately, V is also referred to as the “drffusro,n velocrty” as a consequence, n, V is - u

often misinterpreted as the diffusive flux of component i, and Eq. (11) misapplied to yreld er-

/ " o : 3
 

 

 

v

i_')

roneous results. Later on in this report we shall have occasion to define a diffusive *veiocity,

and we caution the reader that Eq. (10) is not to be equated with this quantity. Accordingly, we

-will differentiate between the velocities by referring to V as the peculiar velocity, and will

introduce another symbol for the diffusion velocxty

Thus far we have accepted the fact that either the gas as a whole or several of the com-
ponents which comprise it are in motion, and we have formulated various definitions to aid us
in describing the motion. In order to introduce additional, equally useful quantities, we now
consider the mechanisms of gas_trausp’ort. 'Un_der isothermal coudit‘ionsr, these modes of trans-

port fall into two distinct categories: (1) forced or viscous flows, which result from gradients

~ of the total pressure, and (2) diffusive flows, in which gradients of partial pressure provide the

driving force. We now associate with each of these types of flow a corresponding flux, so that
J,, is interpreted as the flux of component i due to viscous ttansport and J. = represents the
flux resultmg from dlffuswe transport. Each of these fluxes is associated with a cortespondmg

velocity. Thus the viscous velocity of component i may be defined as
Uiy = ]iv/ni , o - ' i (12)
and the diffusive velocity by the expression
. %p =Jip/my - | (13)

Now consider th_e_'flow of a binary gas'inixtulre', of components 1 and 2, in a capillary. If the

flux of component 1 is J 1 , and that for component 2 is J,, we can immediately write

h=ly+lp, o (14a)
Li=lay+lw, | | 48)

‘ The total flux J, on the other hand, tuay bel.v’vritten either in‘ Athe form ' ' _
A (s
1%11:"7-’2,' o | ,J ' : - , .(‘1’55) |

The problem now is to ensure that there is no external coupling between the J,/ /-and the J D
in other ‘words, we must defme the fluxes (or the velocities) in such a manner that viscous
terms do not appear in the expression for J jp Dot that diffusive terms appear in the formula for
Jiv It turns out that this can be done very easily provxded we account for surface effects in -
terms of a diffusive mechanism. To be sure, the equations are still coupled but this coupling
is indirect; it occurs through the boundary condltlons and the comp051t10n dependence of the
transport coefficients associated with the two modes of transport. ‘As a result, it is usually
necessary to solve the viscous flow equatlon and the diffusive flow equatmn sxmultaneously,

and this can become qulte complicated.

 
 

 

The v_iscous part, when defined as outlined above, is nonSeparative; this permits us to ap-

portion the total viscous flux to the individual components in accordance with their mole frac-

tions, thus.:

]lv .Xl v ? | h , | . . , (168)

A - asy

-

" In terms of velocrty, this nnphes that the vrscous velocity assocrated with ] is common to all

 of the components in the mixture. That is, in the case in questron,

uiv = u2v *

’ Unfortunately, a sumlar apportionment for the dxffuswe part is not possrbie The reason for

this stems pnmanly from the two different viewpoints which are used to describe the mechamsms, ' ’ S
in treating viscous transport we can look upon the gas as a continuum, but in dealmg with dif-
/fusive flow (mcludmg surface effects) one must drfferentlate among the types of encounters . : »
which the mdzvrdual gas molecules undergo. (* o

The solution to a given problem can therefore be reduced to obtalnmg expressions for the

relations
Jy=Tp txd,,
Ty=To+ 3,
J=Jp+],,

-

\

in terms of the driving forces and the characteristics of the gas and porous septqu Although )

the most general case would involve a multicomponent mixture with an unspecified number of

| components, the most complicated case considered to date has been that for which only two

components are involved. This presents no difficulty in applying the equations to multicom- - ' .
ponent systems in which all but one of the componeots are present in trace quantities, however, {
because under this condition all other trace compoqenrs can be safely ignored when considering . _ -

the transport of any one, S L T

'

It is now instructive to take up the problem of the flow of a pure gas through a single straight

capillary, since this provrdes the srmplest illustration of the concepts and definitions which have

- been presented above. In this case the problem degenerates to writing a solution only for the

- equation

_r_‘J‘=Jv+JD- o . .

T . o Permeobiliti Concepts

" Viscous Flow in Capillaries. — In this section we consider the 1sotherma1 steady-state - g U

transport of a pure gas ‘through a long, stralght caplllary under the mfluence of a pressure

<
 

 

 

 

N

#

»

gradient. If we do not allow turbulence and confine the treatment to the hydrodynamc region,

then the equat:on of motion of the gas is given by

(dp/dz) = (1/rxa/ar) [nr(av /oy, ~an

. in which (dp/dz) represents the pressure gradient, r is the radial distance parameter, and 7

‘denotes the viscosity coefficient of the gas. Integtatxon of this equation over the limits r=0

andr=r y1elds, after some manipulation,

712 dp = 27rr dz) [T](aV / ar)] - : | (18)

which is simply the force ‘Balance expression for a cylinder of fluid of cross-sectional area i

and length dz. The left-hand side denotes the applied force on the fluid, whereas the right-hand -

side represents the shear force (tangential stress). If the fluid is not accelerated, then these

forces are, of course, equal.
An expression for the mass-average velocity v, can now be obtained by integrating Eq. (18)

over the limits r=rand r = ros where r, is the radius of the capillary. Thus

vo(® = (2 — A/dn) (= dp/dz) + 4y , - a9

in which u,=v (r ). We therefore see that under conditions of laminar flow, the mass-average

velocity proflle is parabolic.

* So far we have found it convenient to describe the gas transport in terms of the mass-average

'veloc1ty, but in the laboratory we are concemed instead with the number-average velocity. At

this point it is therefore advantageous to seek out a relation between these two average quan-

tities. In the case of a pure gas no difficulty is encountered; as can be readily seen from Egs.

{8) and (9), the two veloci't‘ies turn out to be identical, and we can immediately write

u(r)=v(r)=[(r2—rz)/zgrfl(-&p/dz)w . | O (20)

All that remains to be done now is to average u(r) over the (assumed umform) cross section of

‘the caplllary The result is glven by

u= (t2/817)(-- dp/dz) +uy . s - o (2D

" The flux of molecules whlch pass thtough any given cross section of the tube is then ob-

tamed from the relatxon

J‘&_nu. .- - o "_.

Thus, by“substituting for n _the well-known'_formula :

n= p/kT

we derive an expressmn whlch relates the measured flux to the viscosity of the gas, the geom-

etry of the cap1llary,« and the pressure gradxent which causes the gas to flow; this is given by

J= (r:/ 87 Xp/ kl‘)(- dp/ dz) + nu, . | ' (22)

 
 

thus the individual dxffuswe fluxes, J, , are given by

10

Noth_ihg has been said about the extra term, nu,, which appears in Eq. (Zi). We shall main-

tain this silence for a little while longer, except to point out that it appears as the result of a

boundary condltlon

If we retrace the denvatxon ‘of Vo th;s time for a gas mxxture, we agam find that the mass-

average velocity avetaged over the cross section of the capillary is glven by

vV, = (r§/8n)(— dp/dZ) +uy,

~where n_‘_now refers to the viscosity of the mixture. One can therefore always write

0, = vy — 1 = G3/BrX—dp/d), €

.

JV = nuv=(r§/871XP/ kTX—dp/dz) . L G . '. , : 7(‘24_)

This is the definition of the viscous flux which we had mentioried earlier. In order to obtain an ™

expressmn for the dlffuswe flux, we manipulate Eq. (8) into the form

u= (1/n)r. n(¥, +uf—-v)+u

~

Jip =0ty =0y + vy — v) =V, +uy). o ' : (25)

The oorresponding diffusive velocities therefore represent the average velocities of the molecules
measured with respect to a hypothetical mass-average velocity which is derived from the équation .
of motion under the assumption v (r ) = 0 [see Eq. (19)] ' _

By means of these definitions we have’ solved the viscous or forced-flow part for all of the

cases in which it arises; the answer is

J, = (c3/8qXp/kTX~dp/dz) ; | | (26) e
. _{{,
we shall now turn our attention to the diffusive part of the problem. o

Sllp Flow in Capillaries. — Equation (26) tumns out to be a rather good apprommatxon at hlgh

_ pressurés for flow through large tubes, but at low pressures and for small-diameter tubes, the

“shp flow’’ contubutxon nu_, can become quite significant. We must therefore express nu  in

’
terms of those quantltxes wh:ch are amenable to measurement in the laboratory. To 'do this, we
shall take advantage of the separability of the viscous and diffusive parts of lhe problem. Con- |
ceptually, then, in the case of a pure gas, we are considering the transport of n molecules per
unit volume which have a drift v_elocity u, and are under the influence of a pressure gradient.
Now consider a volume element —'rrr2 dz within the capillary. ‘The molecules will receive a net

forward momentum per unit time equal to -7Tr2(dp/ dz) dz. If the gas is not to be accelerated

_ this momentum per unit time must be lost to the cap1llary walls. ' , o ] o
 

 

 

]

e

y

¥

Th1s expressmn is usually presented in the form

11
Of the n'ntg dz molecules, let the fractionl'(l Qf) eollide with the wall in a specular mahner; |
in this type of collision the angle of incidence equals the angle of rebound, and there is no
change in the z component of the velocity (in this case u,, on the average). For these colli-

sions there is no net transport of momentum; thus they can be ignored in the rate-of-momentum-

transfer balance. On the other hand, let the remaining fraction f be collisions in which the mole-

'cufee-rebbuhd from the wall in a completely random manner (diffuse scattering). For these

collisions, on the average, the z component of the momentum which is transferred in the direc-
tion from the wall to the gas is zero, so that the nef rate at which the momentum is lost to the
wall is simply the rate at which it is transferred in the direction from the gas to the wall. The

rate at which the molecules strike the surface is (1/4)nc(27rr, dz), and of these collisions, per

unit time, _(f/--’l)nE(Z'nrr0 dz) actually transfer momentum to the wall. In each case, on the average,

the momentum mu, is transferfed, so the momentum balance is given by
(muo)(f/-‘-'l)(nc':)(Z'nrro dz) = —'n‘rg (dp/dz) dz ;
thuS _ - _
nu, = (rn /mcX2/f(—dp/dz) . '_ (27)
Although the derivation just presented is by no meyans rigorous, it is correct in spirit and
is consistent conceptually with a similar type of derivation which will be given later in connec-

tion W1th bmary gaseous diffusion,. Another derivation, which likewise 1s lacking in mathematical

rigor, yields (2 f)/f in place of the factor 2/f. Since f appears to be very nearly equal to

~ unity, the two expressions differ by about a factor of 2. Equation (27) does in fact overestimate

the effect of slip flow, pnmanly because of simplifications in the denvatlon, so we shall-adopt

the commonly quoted result,
- nug = (r,/mc) [2 - 6/ f1(—dp/ fIZ) . . - (28)
The diffusive flux J, is therefore given by | |
o J, =nu, - (r,/mc) [(2 - £)/1] (— dp/dz) , - | o 29
and the total flux 1s obtamed by addmg Eqgs. (26) and (29) to yxeld | ’

J=J,+ Iy {(rz/Sfl)(P/kT) + (11/8) (e c/kT)[(2 - n/f]; (— dp/dz) - (30)

'f=F—p+-cK]<H>< S

 where the vxscous-flow coeff1c1ent” B and the “Knudsen-flow coefflcxent" K are defined by

By, o - G

K, = 3/16Xe,/2)[(2 - n/Al. - - - (32b)

 
 

 

12 | S
. I ] . - i &
_ Since the slip term was regarded as a diffusive flux, we could have immediately written - -

Iy = =D (dn/ds), S e

l and then attempted to express the “Knudsen diffusion coefficient,” D, in terms of the charac-
teristics of the caprllary and the gas. By comparing the slrp term in Eq. (31) with Eq. (33), we"
see that the result should be equrvalent to '
DK ='-§-c K : L | - _ 349)
' and we shall accept this result without further justifrcatron. , ,
Thrs completes the drscussron of pure gas transport; we now turn our + attention to the trans-

port charactenstrcs of binary gas mlxtures. Since the viscous part of the problem has already

been worked out, we need only consider the dlffusrve aspects. We shall therefore begin by _ _ a

ignoring viscous flow completely.

- K Binary Gas Mixture Transport

Counterdiffu-sion in Copilluries. — A typical experimental setup for investigating diffusion _
processes in capillaries or porous media is sketched in Fig. 1. The septum (either a single
capillary, an array of parallel capillaries, or a porous medrum) is sealed into a container so that
its ends may beswept with two initially pure component gases. The extent of the counterdrffu-
sion through the barrrer is then determmed from measurements of the degree of contammatron of
the two sweep streams. . C

To simplify the sign convention, we shall choose the positive z drrectron as the direction of
transport of the lighter component; this component will always be desrgnated as component 1.
We now seek to de.scribe the transport, in particular the diffusive transport, in terms of the
| characteristics of the two components and the geometry of the septum (in this case a single
capillary). To accomplish this end, we again consider the rate of trensport of momentum under x
steady-state conditions. ' |

Within the volume element Trr ‘dz in the caprllary, the molecules of component 1 will receive

W
.

1

a net forward momentum per unit time which is e_qual to — ‘To"t:(dpl /dz) dz. This is the same ex-
pression written down earlier, except that we now employ a gradient of partial pressure. How-
- ever, it is now possible for the component 1 molecules to lose this momentum in two ways: (1)
to the capillary walls, as-in the previous case, and (2) to component 2 molecules. B
~ Note that there can be no transfer of momentum due to collisions among molecules of the
same component. Tnis is readily demonstrate’d by considering a head-on collision (partly for
simplicity) between two molecules which are identical in every respect except rrelocity Let .
molecule A, with velocrty Vas collide head-on with molecule B, whose veloc:ty is vg- As a
 result of the cors ervation of momentum, the velocity of molecule A will be v after the colh-
_s_.ron, whereas that of molecule B will be v, . But since the molecules are identical, we can | U

: intercfiange even our designations A and B after the collision. In other words, to an obsetver

\
 

t

.

N)

.

13

GAS { . |
AINLET X

 

 

 

 

 

- ORNL-LR-DWG 57904R

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘, |
« %m&z -
PRESSURE. GAGES —
| | | |
| Cseerom Y
W
- "//////////////// 1
7 j'_i '
~ ANALYSIS | | iANALYS!S‘_
aorns _ %Tmfl

 

 

 

 

 

 

!

" Fig. 1. Diagram of o Typicol Counferdif'fusion Ex-

‘ perlmenf

who is watchmg the event it would appear that the two molecules never really did collide but o

passed through one another instead!

-If we denote by M, . the rate of transfer of momentum from the pomponent 1 molecules to the

wall and by M, , the rate of transfer to component 2 molecules the momentum balance equation -

~ for the component 1 molecules becomes ,

Mlx +M,,= —Wrg(dpl'/dz) c{z ,

@5

and a similar equation canlikewise be written for component 2.

 
 

 

‘we are only employing the diffusive velocities, u

9, =0

14

i

As a tesult of the derivation of ¥, ;t which was presented in connection with the diffusive

transport of a pure gas, 'we can immediately write

#
-

M, = (o) [2(2 0] (n c. )(2771' dz) , R ' | | (36)

‘where u;'D represents the average diffusive velocity of the component 1 molecules, and the

_' factor (f/4) has been adjusted to comply with _Eq. (29). This result, it is recalled, .was obtained

by considering the average number of collisions which the molecules make with the walls in

unit time, and then mupliplying by the average momentum which is transferred in a single colli- -

~sion. We shall employ the same procedure to evaluate M, ,, but once more emphas1ze that al-

though the method is correct in principle, it is lackmg in mathematxcal rigor. Note also that
u,. In other words, our reference frame is mov-
ing along the tube with the viscous velocxty u,. Subsequent addition of the diffusive and viscous
velocltles of more ptopetly the diffusive and v1scous fluxes, in effect fixes the reference frame |
to correSpond to the laboratory coordinate system _

'The average number of COlllSlOl’lS which occur between unlike molecules in the volume ele-
ment 1Tr dz in unit time is given by n n,7d, 2 C" 2 )1/ 2 ('n‘r dz), where 7Td tepresents the
cross section of the sphere of influence for unhke-molecule collisions. In each of these col-
lisions, the average amount of momentum whlch is transferred in- the z direction from molecule
1 to molecule 2 is [m m,/(m, + m )](u ,p)» SO the momentum lost by the component 1

molecules per unit time as a result of coll:smn with component 2 molecules is given by

12,

If we insert this expressxon along with Eq (36), into Eq. (35) and sunphfy, we obtain

]ln{ [(2 0]( )} N A R {(Skr)lfz[fm +m)],/2nm1p }‘_1

 

 

-- (dp‘)
o

2-p] /nec\)? . 7kT\'/?|(m, + m,)|*/? 1 |-!
Jw{rol:_——[ ] (—8—1)} +(x,J,p _XIJZD){ _8_) [n:1m2 -2] mrdi }
)

' The first term in braces is just the Knudsen diffusion coefficient for component 1 D, ., whereas

the second term in braces is ‘the binary diffusion coefficient 19 , of the ‘system 1-2. (Note that

215 this can be shown by mterchangmg subscnpts in the flux equations. ) Equatlon (38)

can therefore be wntten in the form

M., =nnmd}(c? 44'32)1/2[m /(m +m )](u - )('m-z‘dz) - (37

R

#

.
 

 

¢

N

"

Jyp +x2]“'>'— x,Jap ____ffi

 

 

. 39)
Dlx D 12 dz .
Had we accounted for the transport of momentum via intermolecular collision in a mathe-
matlcally r1gorous manner, the actual expression for the binary diffusion coefhcxent would be
given by
-3 /mkT\!/? m+m 172 1. o
N12=Z<T ! fl (1, 1%’ “0)
| | m, J amo - o

in Which ‘7TO-'2 0(12'1)* represents the colhslon cross section for diffusion. Unlike the simple

expression 77d , this quantity is temperature-dependent and is evaluated from a detailed con-

12*
sideration of thg dynamics of the collision process. The simple derivation once again gives

an overestimate of the momentum 'transport_ed, being approximately four-thirds times the rigorously
derived result. |

In order to account for viscous effects, we need _only insert the expressions
Ji=Jip+tx oy )
Jy=Tp + %0
J=J +J,

mto Eq. (39), temembetmg that ] is given by Eq (26).. The fmal result is conveniently ex-
pressed by the relation

 

  

 

, dn, .
v Jy=-D, /7 x,8,] — x, nkT (41)
where
101 1 I - -
—— e, . T @2
Dl D D ' R : ' ) (2a)
b, - bl'x
= —-—_—-—- , (42b)
T, B0,
: D, D‘ | L - o

Equation (41) is symmetri_c with respecl; to an interchange of species subscripts; hence the cor- '

__responding equation for J, is R B : c

o o fdn B.p (dp - ) ) _
]2 = —Dz(—'d—:) +Vx J 2y2 flI:T (d ) ' ‘? o 7 (43)

 
 

LM

16

A third equation is obtained by addmg Eq. (39) and the corresponding expressron for com-

poneni 2,
Jip +~’20 ___dnf
Dix Dy 92
The result can likewise be cast into a form which involves the total fluxes J, and ]2 Gie., the /

fluxes actually measured in the laboratory) Thus

._'fi.+..']_2.=— 1+(.:L )..( ) . (44
Dig Dy | Du; - R

Although we have wntten three equatrons to describe binary gas transport through caprllarzes,
namely, Eqs (41), (43), and (44), only two of these are mdependent any one can be derived by

[t

suitably combining the other two. The limiting forms of these relatlons in the free-molecule
| region (p»0) and the hydrodynamic region (p -» oo) can be readrly obtained by inserting the

values preseénted in Table 1 for the various diffusion parameters.

- .

‘Table 1. High and Low Pressure Limits of
Diffusion Parameters®

 

o Limitas n &, D, 8 v ny, -

 

p->0 0 o0 1K 0 1 0

D
p-» oo 00 0 fl .1 0 flt@

12 12/D1K

 

2Note that n]g , and D, are pressuré-independent
- quantities. : . .

A very important result is obtained from Eq. (44) for diffusion under conditions of uniform _ ' oz

pressure. For this case the right-hand side of the equation vanishes, and one therefore obtains

‘Since the quantity (2 N/t is reasonably independent of the gas, the ratio of the Knudsen dlf-

fusion ooefflcrents can easily be reduced to yield

72 ~ N .
: m : , |
—"!J' = (——2— ) . ) - : (45)
J, \my J ) ' |
_This result is expected for tranSport under free-molecule conditions and is generally accepted .
as Graham’s law of effusxon Equation (45) has been derived under no special conditions rela- | ‘
tive to the pressure of the system; it is therefore apphcable at all pressures, not only in the B Q j

‘_ free-molecule region. Although this relation was also stated quite explicitly by Graharn, in fact

 

"
 

 

 

my

17

several years prior to his effusion studies, it apparently was either forgotten or misinterpreted.

Nonetheless, this is Graham’s law of diffusion, and has been experimentally verified by many

" mvestxgators Note also that it is xmposszble to obtain zero net flow (] 0) under uniform

_pressure conditions unless D, =D,

Thus far we have restncted the treatment to transport through a smgle capillary; in the next
section we extend this treatment first to a bundle of parallel capillaries and then to porous media.
Pore Geometry and Overall Coefficients. — Consider a cylindrical solid with a bulk volume

given by 7re?L which contains N identical -capillaries, each of volume ‘rrrgl. If the axes of the

individual capillaries do not coincide with the exis of the cylinder, then the length I will be

-

greater than L; therefore let g% = I/L. The porosity or relative void volume € of the cylinder

is given by

N(me2D)

7L

 

If we insert the definition of the tortflosity g into the above expression and reartange, we find

.that the number of identical cépillaries in the cylinder can be described by the relation’

oS (rz) | | N ,
61(2 rg * : . : |

The total flow of ‘mdlecules me asured relative to the geometry of the solid is J(77r?); this
must be numerically equivalent to N J O(Wt:), where J denotes the flux of molecules through

any one of the N identical pores. Hence

- However, the flux J is expressed as the product of two quantities, the gradlent of some parameter

which is causmg the flux and a proportlonahty constant. In general, then,

) VA o
J,=-C, ) -

and

' By suitably rearrangmg these two expressxons and by inserting for ]/] the relatlonshxp given

prevmusly, we see that the ratio of the coefficients is given by |

(46)

 

 
 

 

 

18

 Two of the coefficients of interest incorporate T, to some power; thus it is advantageous to

~account for this fact by writing C, = c{)rz"2 , so that Eq. (46) may be rewritten in the form .
Cley, =(/Dri=?  (1=234).

In the case of hormal‘diffusion, co = 0 4o and (1 — 2) =0, but for Knudsen diffusion,
m_
-8‘ [(2 - )/ f]

and the exponent (j — 2) is umty, whereas for viscous transport co = p/87 and (1 2) 2.

The situation mvolvmg a bundle of uniform rdentrcal caprllarres is highly idealized; althou

47

gh

we can consider many other geometrical models which are more complex but still tractable math-

ematrcally, their exposrtron will provide little insight conceming the geometrical characteristics

‘of a typicai poroua medium. In the lariguage of the pore or eapillary concept, such septa must

be regarded as consisting of amyriad of nonuniform interconnected capillaries of widely varying

lengths. One is therefore faced not only with averagmg the pore radrus r,, over the number of
caprlIanes (from m = .0 to m = N), but also over the individual lengths I of the capxllanes

Moreover, all of the void volume € need not contnbut_e to flow (blind pores, for example), and w

.

shall denote this fact by using the symbol €” to signify that part of € which is actually involved

in gas transp ort. -

It is therefore quite obvious that the specxflcatron of the geometry of a porous med1urn re- . . |

quires such fine detail that a complete solution of the problem will almost certainly never be
obtained. Nonetheless, we can set up the gas transport equatrons in a formal manner and there
reduce the problem to obtammg only a few parameters experunentally '

We start by defining the effective porosity € in the following way:

'anrI

m m m

2 €’ <e,.
‘L

Now, however, the equivalent expressions for the flux yield
2y _ 2
J(me®) = Z T (),

or, in terms of the transport-'ooefficients_,'

C(me?) (=9T/OL) = mcy Z ri(~dT/d1,)  (i=23,4).
.This immediately gives

D
S, r? ‘ |

where r again represents the radius of the medium which contains the capillaries. If we insert

Eq. (48) into the above expression, we obtain

by

- (48)

(49)

Wy

in

o

A

"

r
 

 

 

 

»

22

19

C €2 LD
'“'=—'_'""T'

c, 2

mrmq

" Finally, we can obtain a form similar to Eq. (47) provided we define the tortuosity '('1'1 and the

average value of r/—2 by

(tj--2> _'2 (rj/‘auz)-

; (50
-q-j z r2—1/2 | ( )

 

thus
LS o 61
€, qj ‘ ' ; _

- Ve should like to impress upon the reader that '(j}, unlike €’, is a function of the paramefer ,
j, and we have added the subscript to indicate this fact. Furthermore, except for the case j = 2,

( i- 2> and T, g; are defmed as a group; for the one exceptlon we have

2=1/2
T, rags

2T RS

. The permeability eduation, that is, the gas transport equation for a single gas under the

influence of a pressure gradient, is commonly written in the form

Pl -

in which the flux ] and the pressure gradient are measured relative to the geometry of the porous

medium. Equation (51) is identical to Eq. (31), the expression for a single capillary, only in
- /

appearance, for B;, and Ko have been rede_fined;_

B-_(q)(, A ‘. R B D

where €’ and the grouping (rf 2) /g, are given by Egs. (48) and (49) Note also that the

verages ( i~ 2) actually represent a seoond averaging process. In other words, we have °

 

" tacitly assumed that such factors as cxoss-lmkmg, nonun1form1ty down the length of a gwen pore,

~and shape have already been taken into account.

- Fortunately for everyone ’s sanity, permeab:hty measurements are not too difficult to perform,
so we let nature do_ the averaging processes for us. Except for the fac_tor fin K, which is
independent of the gas to a good approximation anyway, both B, and K, depend only upon'the‘

geometry of the medium. Under steady-state conditions, then, we can integrate Eq. (52) to yield

 
 

 

20

kKTL B, 4_ - S

okt <p> +55K, o 6
where Ap = p(0) — p(L), (p) [p(0) +p(L)], in Whlch p(0) and p(L) are the pressures at the
two faces of the porous medium, and JkT is the flow, in pressure-volume units per unit time, per
unit cross section of the medium. Hence B and K can be obtained from the slope and inter-
cept, respectively, of a plot of the left-hand side of Eq. (54) vs ( p). Once determine'd these

parameters are invariant to the ch01ce of gas and appear to remain teasonably constant with

| respect to temperature and tune.

It can likewise be shown that, for transport in porous media, the diffusion equatlons given
earlier for diffusion in capillaries remain unchanged in form prov1ded we replace the diffusion -

coefficient .@ , by an effective diffusion coefficient, D”, where .

D, =(e7a"W, o = 69
3 r2—1/2 ‘ ! ‘ '
q° = ‘-1.2 EW (55a)

The quant1ty (e’/q")is hkewxse dependent only upon the geometry of the septum and is most

convemently determined through counterdiffusion experiments which are performed at uniform
pressure. k | |

In concluding this sectlon we should like at least to partially dispel the impression which

the reader may have received from our discussion with regard to the utility of porosity and pore-

size distribution data. It is quite true that if one is interested in small differences in the trans-_
" port characteristics of two septa, for example, any inferences which are drawn from the pore-size

" distributions of the two samples and later verified by experiment are unquestmnably fortultous.

As a rule of thumb, however one ¢an state that the permeability generally increases with in-
creasing porosity and that of two specimens having approximately the same poros1ty , the one
with the larger potes will give the higher permeability values. Moreover, one can make inferences
from pore;size distribution data if large differences are involved, but even these should be
verified by experiment. ' | | | -

Gas Tronspott in @ Static Dust Environment. — In our derivations of the diffusive part of the
gas transport problem, we nonchalantly made a number of assumptions in order to keep the presen-

tation and the mathematxcs as simple as possible. For example, we assumed that the average

| rate of momentum transfer was equal to the average number of collisions times the average

momentum transfetred per collision. The average of a prj.oduct, in general, only approximates the
product of the averages. In fact, such approximations led to overestimates of the rate of mo- -

mentum transfer in the simplified treatments, but it was a relatively simple matter to ‘‘properly”

“adjust the co."i~iente because we knew what the answer was beforehand. The cormrect expres-

sions were no. untai.: - - an application of the capillary flow concept, but rather from a

theoretical investigation of gas ’rar_nsport in a static dust environment, so per'ha-ps we should at

“least outline how the rigorous derivations were obtained.

.

1L ]
¥

 

"

 

of gases 1 and 2 and the dust d.

21 -

The physical description is as follows: Sfippose we have an agglomerate of giant gas mole-

‘ cules (dust) which are uniformly distributed and fixed in épacie. For simplicity, let all of these

molecules be of exactly the same size. If two gases are allowed to interdiffuse through the ag-

glomerate the process can be described as diffusion of a ternary gas mixture, that is, a mixture

4

The diffusive flux relationships for such systems under isothermal condltlons are obtamed

from the Stefan-Maxwell diffusion equatlons.

E {n I’J”]"1 (] - n]jD) n’d,,

l#i
where dj repfesents a combination of driving forces. The primed quantities indicate that the
dust is to be included in the counting process; thus n” =n, + n, + ny = n +n,, where the un-
primed quantities refer only to the gas. This poses no problems, since n’D' =nD, antl dn’/dz =
dn/dz, the latter by virtue of the postulated uniform density of the dust (dn d/dz 0)

~ For the ternary mixture considered here, we have three equations of the form given above -

one each for j =1, 2, and d — but only two of these are independent. Also note that although the
gases are not acted upon by an external force, there is an external force F; which acts upon the
dust, namely, that clamping force which keeps the dust particles statlonary The clamping force
is

F, - _1_ dp o ‘

. dz | _ | .

o

where p refers to the true gas pressure. Fot j = 1, the Stefan-Maxwell equation becomes

_nm dlnp n’nm '
L4+ n’{x] L 2n.F, .
PP

If we insert the :expression above for F g4 into the eqt_lation., the relationship simplifies to yield

xJ2p = X3J1p _ P4)1p :

D,, 7 ,"D1d

 

 

| nd]1D+x2]1D - X,J5p =_'d”1 |

nD, Dy, dz.

 

which is identi_cal to Eq (39) with

1d _
—— =D, .
aoa . ’

 

The gas—dust diffusion coeff1c1ent Di d is given by

""T) (_..>’” e (1+_)]“1

where r, denotes the radius of the dust particles and g, repres_ents a .scattering factor which is

related to f£.

 
 

 

- , | - 22

The viscous flux, on the other hand, is obtamed from Stokes law. The force F on the

. particles due to viscous drag is given by

Fy4 =—677rd17 (f) . | : o - )

If we equate the right side of this equatron to the right side of the clampmg force expressron

and rearrange, we obtain

A

We have made no assumptlons which would prevent us from orienting the dust in such a

~ manner as to form a capillary. However, this ‘model is couched 1n a language which fortunately
" excludes a duect connection between Ty the radius of the dust, and ro, the radius of the capil-

lary. Only the simplest sort of “geometncal factor” is required, namely, €’/q’, as in Eq. (55),

and th1_s will apply in the same form to all parameters. The important point is that the model

. separates the geometrical aspects of the problem from the characteristics of the gas, and ‘more-

‘over does so in a mathematically rigorous manner.

The extension to a porous medium is performed in much the same way as that done pte-
viously, except that one now takes some suit’ably averaged value of the dust radii. The trans-
port coefficient ‘expressions for gas flow in porous media which are obtained from the capillary

model and the dusty-gas model are presented for comparison in Table 2.

\

Table 2. Mathematical Expressions of the Gas Transport Coefficients for Flow in Porous Media

 

) Model® - ,
Transport Coefficient ' — / b
Capillary ' : Dusty-Gas

 

e’ .
2 ]
D,,, cm”/sec (E—)Lou : ( )‘012

(morma! diffusion)

 

 

 

 

K

,&:;S@ ctiffusion) ‘13)( |:(2— ](’3) < )[ ( >(s +a)]
i RO <—:—:>[e‘md\<re>r

(viscous flow)

®The expression for Jg g in terms of molecular properties is given by Eq. (40) The quantities ql and
q” are defined by Egs. (50) and (55a) respectively,
bNote that we have retained the caprllary concept in defining (€ 7q”) for the dusty-gas model.

'
¢

“

wi

 

23
Summary
In the preceding sections of this report we have attempted to present, in as simple a manner
as possible, the various flow equations which are encountered in dealing with isothermal trans-
port in porous media. We can best summarize this portion by pointing out that any isothermal

gas transport problem involving a porous septum is completely specified by Egs. (41) and (44),

provided the coefficients D

iK'’ D”, and Bo are modified to take into account the nuances of

pote geometry
Unfortunately, an a priori method for evaluatmg the smtably modified coeff:cxents is unlikely

to be had; recourse must therefore be made to experiment. However, the only measurements

_tequired are a few permeability determinations with a single gas and a few counterdiffusion ex-

periments with a single gas pair. This is rrelatively easy to accomplish Once this is done,

the septum is completely chatactenzed that is, the transport behavior of any gas under a given

set of cond1t10ns may be predxcted with confidence.

In the experimental portion of this work we shall demonstrate' (1) how the geometry of the

~ septum is characterized through permeablhty‘ and counterdiffusion experiments, and (2) how the

results may be applied to gases and conditions other than those employed in the experiments.

.Appropriafitely, we have chosen to use a graphite specimen of the MSRE type.

IV. EXPERIMENTAL
Description of MSRE Graphite and the Expelrimentul :Specimen

Little will be gained at this time if we consider details of the manufacture of the MSRE
graphite. To be sure, the fabrication procedures significantly affect the transport characteristics
of the finished material and become quite important if property variations within the specimen
or an intercomparison of various types of graphite are of interest, However, in the ptesent
case we concern ourselves only with a single type. of graphite and tnoreover concenttate on the -

transport characteristics of the material as a whole. A detailed consideration of its manufacture

“thus becomes academic, so only those aspects whlch are pertinent to this limited ob]ectlve are

presented. |
- In the original design concepts of the molten-salt reactor, intrusion of the salt into the

graphlte was regarded as an intolerable contmgency As a result a material of low permeabxhty

. | was demanded Such low-permeability graphlte is usually obtamed by applying addmonal

Spec1al treatments, begmmng with a mod1f1ed porous nuclear-grade graphite. These treatments

involve injecting a suitable 1mpregnant into the base stock wl’uch upon undergomg heat treat-

ment, deposxts a char within the pores of the graphite,
As is illustrated by the photomlcrograph comparison of NC-CGB-BS (base stock) and NC--

CGB (1mpregnated stock) in Fig. 2, impregnation treatments con51derably lessen the pore space

within the graphlte. ‘This difference in pore size likewise accounts for the observed d1fference

in the penetrability of the two graphites by molten salt, wh1ch is also shown in Fig. 2.

. ‘ '

 
 

"

24

. ’ . - ’ »
} . - ‘ jx
PHOTO 86976¢ - .

 

 

 

 

 

 

 

 

 

 

100X

 

 

 

BASE STOCK AFTER TREATMENT
(NC-CGB-BS) (NC-CGB)

 

Fig. 2. Photographs Showing the Effect of Multiple
impregnation Treatments on the Microstructure and
Molten-Salt Penetration of NC-CGB Graphite. - Light
areas in the upper photomicrographs indicate void

spaces (pores). Light creas in the lower radiographs
indicate the presence of a **nonwetting®’ salt (BULT,
14.0-50) which invaded the samples during @ 100-hr

exposure to molten salt at 704°C and 11 atm pressure. w

One might logically expect that, as a result of impregnation treatments, the end product
would exhibit property variétions along directions norm;tl to the impregnation surfacés,_ particularly
near the surfaces of the graphite, where impfegnation should be especially effective. Insofar as
MSRE graphite is concerned, the nonhomogeneity is probably mitig’ated somewhat by subsequent
macrhining‘operations which are required to produce the final dimensions of the material, and we. -
shall explore this facet in a later -report.r For the present, however,_v\vJe.c}'xoose to concentrate
on the material as a whole, ‘
The MSRE utilizes the graphite in the form of 6-ft-long bars which have a cross section of
3.08 in.2. All four sides of each bar are slotted along the entire 6-ft length; these slots provide u

the flow channels for the molten salt.” The flow specimen was machined from one of these bars.
 

w

-}

25

!

Our choice of sample geometry and location in the MSRE graphite bar was govemed by the
followmg ob;ectwe, namely, to obtain information regarding the relatlve contributions of Knudsen
and hydrodynamic transport to the overall flow pattern. This tequlres both permeability and
counterdiffusion .experiments and these, in turn, require samples which have a large surface-
area-to-thickness ratio, as well as a reasonable degree of umfonmty Since the bar was expected
to exhibit considerable nonuniformity and a lugh Knudsen contnbutmn in the regions near its
surfaces, we decided to obtain the flow specimen from its center, .This position is defined in
Fig. 3; in this location uniformity, normal diffusion effects, and porosity may be considered

maximal.

. ORNL-DWG 66-12740
DIMENSIONS ARE IN INCHES |

PERMEABILITY-DIFFUSION =
- SEPTUM DIMENSIONS (6in. LONG) -

  
 

SALT FLOW
CHANNELS

PORCSITY PLUG

1

Fig. 3. Position ond Dimensriens of the Diffusion Sepfumiund Perokliy. Scmple. )

The speeimen, "hereafter designated as the diffusioiz septum, was in the ferm of a thin-walled- '

cylmder whose axis coincided with the extrusmn axis of the bar, For this geometry, the area-

to—length ratio A/L is obtained from the radial steady-state flow relationsl'up for a umform

mater:al

| A/L=2nh(In (d,/d)]!

 
 

 

 

 

2%

- where h denotes the height of the cylinder (not to be confused with the length of the fio_w path
L) and d, and d, represent the outer and inner diameters tespectlvely The septum is thus

characterized by the following geometrical parameters:

h=6in.,
d, = 0.800 in.,
d, =0.600 in.,
A/L =203.6 cm,
A=77.57 cm?.

Gas Transport Characterizafion of the Diffusion Sepfum

Apporctus and Procedure. — Mutual diffusion coefficient determmatlons mvolvmg binary gas
mixtures are generally made under transient conditions in an apparatus whose geometry is well
defined and known. Moreover, the system is closed throughout the course of the experiment,
thereby forcing the diffusion rates of the two components to be equal. In the present work,
hov&ever, we employed a steady-state method, and this required that the system be open.

The approach used bjr us was originally deve10ped by Wicke for his investigations of ad- '
sotbed CO, surface diffusion in porous media; a C‘Oz-N2 mixture was swept across one face of
a porous septum, whereas the opppsite‘ face was swept with a stream of pfire N, in such a manner
that no pressure gradient was imposed across the septum. Althoughthe CO, diffusion rate was
determined in his studies, Wicke unfortunately ignored the N, diffusion rate. Somewhgt later, |
 Hoogschagen adopted the Wicke procedure and added one impdrtant modification; he monitored |
the degree of contamination of both_swee’i) streams. This led to the rediscovery of Graham’s
law of diffusion. (Ironically, Hoogschagen’s rediscovery of Graham’s law and Soret’s earlier
use of this law to verify the formula O, for ozone were also confused by workers in the field!)
Figure 4 is a photograph of the diffusion cell assembly which was used in this work; the
components, from left to right in the figure, are: Ar sweep-gas outlet tube and thermocouple;

septum container; diffusion septum, container cap, and fittings; and the He sweep-gas flow. guide

ahd septum end cap. The end caps were attached to the graphite cylinder with epoxy resin to ef- -

fect a gas-tight seal and to define the surface of the septum available to gaseous diffusion.
The counterdiffusion experiments were performed by sweeping the inner surface of the diffusion
' septum with He and the outer surface with Ar énd analyzing the effluent streams for the corre- |
'sponding" contaminant. Pure helium was introduced into the upper T-joint fifhich is shown in Fig,
4 and made to flow down the annulus formed by the %- and ? %4-in. tubing‘ The gas then entered
the inner section of the septum in the region of the upper end cap and was thhdrawn at the .

| base of the flow gu:de through the Y %4-in. tubing. In a similar manner, pure argon was admitted
- to the outer surface of the spec1men through the T-joint adjacent to the contamer cap and with-

* drawn at the base of the container through the ¥ 7,-in. outle_t tube.
A drawing of the entire flow system is shown in Fig. 5. Uniform pressure conditions were

obtained by adjusting the control valves'R , R,, R,, and R, until the pressure drop Ap across

/

¢

 
 

 

 

wt

PHOTO 36002

 

 
 

 

 

 

¢ : ORNL-DOWG 64-2439

- "GAS SUPPLY { .~ - GAS SUPPLY 2

Ry

Ry S . - -
o | +T——Ap : C
-MANOMETER

 

 

 

 

 

 

 

 

 

ATM

 

Fig. 5. Line Drawing of the Diffusion and Perme-

ability Measurement Apparatus.

- the sample (as determined with a mercury differential manometer) was zero. . After sweeping its

respective side of the porous medium, each effluent stream was passed through one of a pair of
thermostated thermal conductivity cells (T) forgas épmposition analysis. Continuous compari-

son with streams of the corresponding pure gases under identical flow conditions was accom-

' ‘phshed by adjusting the control valves R, and R until the rotameter paxrs (F) indicated equal

flow rates. Back-diffusion of air into these reference streams was minimized by venting the
gases through 12 ft of coiled ¥ 4 -in. copper tubing, whereas the sweep streams were passed
hrough d1buty1 phthalate bubblers (B) before bemg admitted into the calibrated wet-test meters
(S). In about half the expenments the gas compos1t10n analyses were obtained with'a mass

spectrometer. For these muns the samples were w1thdrawn from samplmg ports located at T.

,Prevmusly cahbrated Bourdon gages (G) provided measurements’ of the pressures at which the

experiments were performed. _
The permeat _lity data were likewise obtained with thlS apparatus ThlS was accomplished
by closing one of the inlet valves _and the outlet valve of the opposite flow stream (e.g., R, a_nd

R). o

i

 
 

 

 

'

3

-t

 

29

All of the gases employed in this work were found to be at least 99.9% pure. Analyses of
the he_liixm and argon s'upplies' indicated a Eree oxygen content in the range between 1 and 4 ppm
and water contents from 10 to 15 ppm. Thus, no furfher attempts at purification were undertaken.

Permeability Results. — Since the techniques Vusuallry'employed to obtain permeability data
appear in abundance in the open i'itefature, a detailed discussion on our part is unwarranted.

We therefore merely outline the calculational procedure in this section.
The integrated steady-state equation that applies to the diffusion septum permeability meas-

urements is given by

2,0, = K(A/L) Ap,

' where the permeabilitycoeffic_ient of component i is

K, = (Bo/q,.)(_p) + Dy - (56)

In our experimenfs, the -effluent volumetric flow rate ‘Q,, is determined at the barcmetric
pressure p_ by means of the calibrated wet-test meters; the pressure drop Ap.= p(0) — p(L) _
across the septum is measured with the mefcury differential pressure manometer whiph is shown
in Fig. 5; and, finally, the arithmetic mean pressure <p> = %[p(0) + p(L)] is determined from "
readings of the barometuc pressure and the calibrated Bourdon gages. ,

' Diffusion septum permeablhty coefficients were determined at 22.5°C for three gases‘ hydro-
gen, helium, and argon. The resultant experimental data are presented in Table 3 and are
graphically displayed in Fig; 6 as a function of the mean pressure {p) . In accord with the
linear relation, Eq. (56), these data have beeri smoothed using a linear least-squares procedure;
the solid lines which appear in Fig. 6 thus represent the smoothed data and form the basis for
the determination of the permeability parameters which are tabulated in Table 4.

_ Although the quantity \/flTi,D ;i and the viscous-flow coefficienf B, should depend only upon

the graphite structure and therefore be independent of the gas, some variation in these values

'has been noted. 'The"se discrepancies are probably 'indicative of the experimental errors in-

volved; hence an average of the values .‘forb'\/flTi D k and B, which were obtained from the helium

and the argon data has been taken to be representative of these quantities when we consider the

He-Ar counterdiffusion data. | o . _ - | '
Helium-Argon Counterdiffu sion'Results. All of the counterdiffusicn experiments were con-

ducted under conditions of uniform pressure hence the data were correlated in accordance with

the constant-pressure form of Eq (41),

 

Juo =Dy \—= )+ J. - -~ 6D
, For this case Dfle and '8*'“. are ccnstant over the length z=0toz = L, and Eq. (57) can be
integrated and rearranged to yield an expression for the effective diffusion coefficient Dya,in

 
 

30 .

Table 3. E_;cperimenral Values of the Pe}meabilify Coeffiéiehf of the NC~CGB Graphite Diffusion
: Septum at 22.5°C as Determined with Hydrogen, Helium, end Argon

 

 

 

 

 

 Hydrogen - . ~ Helium o . Argem
(;) K ( p) K : (p) K
(atm) = (cm?/sec) (atm) {cm?/sec) (atm) (em?/sec)
x10° x1074 x10° x 1074 x10° x107*
1.366  7.801 1.275 5.366 1318 . 2.016
1.406 .~ 8.000 . ©1.450 5.453 1.506 2.112
1.525 8.09 1.674 - 5.550 1.782 . 2.264
1.652 8.250 1.925 5689 |  2.083  2.352
1.756 8.411 2.176 5.813 2.265 2.449 -
1,848 8.452 2.451 ~ 5.980 2.482 © 2.564
1.935 . 8.582 2732 .6,105 2.827 2.705
2.032 8.763 2.970 6.248 - 3,073 2816
2.195 8.841 - 3.197 6.350 3.258 2.853
2.286 - 8.983 3.712 . 6.638 . | 3.526 2.998
© 2,490 9.174 . 4.205 _ 6.874 . 3.794 " 3.073
2.704 - 9420 4685 7.091 ' 4.049 - 3.214
2.964 9.759 5.252 7.391 4.331 3.348
3,257 10.05 5.729 7.650 . 4,983 3.614
3.537 ' 10.37 | 6.216 7.900 . 5.201 3.763
3759 10.63 | . 6.740 8.200 5.725 . 3.929
3.986 10.91 7.185 8.418 © 6.030 4.074
4.241 11.16 7.630 8.667 b e2mm 4.200
4.485 11.44 ST 6.590 4.333
4.985 - 11.95 _ 6.970 . 4449
5.243 12.34 o 7.269 4,582
5.567 12,71 - ' o 7.529 " 4,688
5.993 13.14 S - ‘ '
6.530 . 13,73 |
7.0001  14.20
7.514 14.87

 

 

 

Ty

Table 4. Summary of the Permeability Parameters of the NC~CGB Graphite Septum at 22.5°C

 

 

 

n j \/E: - D B, - ' VM, Diox O
Gas - _ (poise) (g/mole)l/ 2 (cmzlsec) , (cm?) _ (g” 2 cm? sec™ mole"l/ 2)
%104 x10°  x107* %1078 x10~*
Hydrogen 0.8863 1.420 . 6.40 9.82 . 9,09
Helium 1.971 2,001 4.70  10.07 940
Argon 2.235 6.320 1.48 9.47 9,35

He-Ar average ’ ' ‘ 9,77 - 9.38

 

 

L &)
 

 

 

 

)

31

¢

“ORNL—-DWG 66—12744

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(xib-4) ‘
) 14 _ /}/
e g2 :
‘ . ,
ol "
E 10 - O —
w
S / .
' W . ) *—
, % 8 . Aaod:’j . ._;‘_',.of
O A _ _ _ ‘_./0 [~
t / ' KH"e ._—'.
- o
- - -(‘./
m 6 i .—“.
a .o-®
2 //.. '
= .
x KAr ' D,D/EM
2 o]
/
o . ,
o 1 2 3 4 5 6 7 8
¢p>, MEAN PRESSURE (atm) '
Fig. 6. Pressure Dependence of the f’emeability Coefficients of the NC.CGB
_ Graphite Diffusion Septum at 22.5°C.
terms of the experimentally determined'variables. Thus
- JA{A [ ~ 8, U/1.) %, (L)]
< “Hear = T
. : n. _!L 1~ 8 U/Jr-re) x,,(0)
where |
SHe =D [D +DHeAr] * ’

HeK

In view of the theoretical relatro‘n for umform-pressure diffusion, namely,

S, \V?
=< Ar)  =3.16,
VMHe o

a measurement of either Jue nr ']

obtains two mdependent values of J if both 1nd1v1dual fluxes are determrned and these may be

- JHe
Ja

 

 

r

averaged in order to enhance the accuracy of the results. This procedure was in fact employed

in analyzmg the present data.

uy

. is in effect a measurement of the net flux T One therefore '

(58)

59

 
 

 

result of this aulelary equation the expressmn for D,

‘culates an “‘apparent’’ value of nD

- gas mixture are present only in trace quantities, one can safely ignore all other trace components

The importance of obtaining valge's‘df the Knudsen coefficient D, . before performing dif- ' g
fusion experiments is readiiy realized by noting its appeerant:e in Eq. (59). Furthermore, as a
Heas EQ- (58), is a transcendental rela-
tion. Accordmgly, it must be solyed by an iterative techmqfie.' The number of iterations re-
quired of course depends upon the value that is chosen as a first approximétidn to Dn A, @nd
if one makes a poor choice the convergence can be pamfully slow. It is therefore desirable to
obtain as good a first approxunatxon as possible. If several diffusion expenments are performed ,
at different pressures, the most convement method is as follows: For each expenment one cal-

He A, from Eq. (58) by takmg Oy, =1. If the tec1procals of

the values so obtained are then plotted against 1/p, the intercept corresponds to the “‘true’’

value of nD, ;\r since, as we have rema_fked eerl_ier, &> 1as 1/p » 0. Moreover, the plots fre-
quently appear to be almost linear, so that the required extrapolation is generally straightfor-

ward. - - ) . ' ' '_ . 7 -

‘The He-Ar 'counferd'ijffusion data which were obtained with the diffusion septum are presented _
in Table 5. Two series of experiments were petformed: in the first, the extent of contamination
of the two sweep streams was adjusted to be about 1 mole % and the analyses were performed

with a mass spectrometer; in the second series, the degree of contamination was held at about

0.2 mole %, and thermal conductivity cells were employed in the sweep-stream analyses. The

diffusion coefficients which are tabulated have been eomputed in accordance with Eq. (58),
where the value of D, has been taken to be 4.69 x 10~* cm?/sec, as discussed previously.

The first approximation to nD ., , as determined by the intercept method described above, was

- 2.85 x 10~ 8 mole cm™! sec™!; the rapid convergence which was obtained by this method is ? _ o

readily seen by comparing this value with the “final’’ results in the table. B

Except for the geometric factor (¢°/q”), the diffusion coefflment DH A, is given by Eq. (40);
thus the values presented in Table 5 should vary linearly thh reciprocal pressure. This de- .
pendence is illustrated in Fig. 7. | | A _

Parameters for Fission Product Diffusion in MSRE Graphites. — We are now in a position to : B
apply the mformatlon which has been obtained thus.far to cases of mterest to the Molten-Salt ' '
Reactor Expenment. Typxcal of these is the migration of xenon and krypton agamst a helmm
atmosphere in the graphite. . However, we shall not work out the problem in detail, nor shall we
even write down the flux expressions; instead, we confme ourselves only to a dxscussmn of the
flow parameters.

We begin this section by once more emphasmmg that if all but one of the components of a

- in describing the diffusion characteristics of any one component. For example, if we wish to

charactenze the transport of trace amounts of xenon and krypton in a helium atmosphere it is

unnecessary to consider the effect of xenon on the transport of krypton and vice versa. Our

_object here, therefore, is simply to obtain values for the quantities D, and D, ., where i N o ‘ - |

_ represents either krypton or xenon, which may be apphed to the MSRE condltlons. These two

parameters are sufficient to completely describe the migration of the two fxssmx; products.

 
 

 

o

 

 

33

Table 5. Helium-Argon Interdiffusion Data Obtoineci:at 24°C with the NC-CGB Graphite Diffusion Septum

 

Diffusion

 

 

 

_ ' ' o , " Normal Diffusion
Pre(ssure, P Diffueion Rate (mole /sec) - Rate Ratio, Coetz'fjcient Constant,
atm) Y -J.. /] ' (cm”/sec)
- . UHe' A)exp UArA)exp' UA)nv ’ .He Ar —_— ' nDHeAr
D, Dyoas (mole cm—1! gsec—1)
x10° x107® " x10”6 x10~8 . x10° = x107* x10~* x10"3
1.36 3.30 —-1.07 228 - 3.08 245 516 | 2.88
1.57 . 3.57 ~1.14 . 2.45 3.13 2,30 4,51 2.86
1.78 3.96 -1.19 2.63  3.33 212 3.86 2.82
2.11 4.29 =137 2.94- 313 1.94 3.31 . 2.87
271 478 —-1.63 3.39 293 = 1.68 2.62 2.91

1 3.73 5.43 —1.84 3.85 2,95 1.35 1.90 2.90
4.91 6,11 - —1.93 4.17 3.7 1.07 1.39 2.81
7.70 6.95 —~2.33 4.89 2.98 . 0.76 0.91 2.88

' | Av 3.09 10.10 . Av 2.87 £0.03
‘ Xyl =X,,(0~0.19 Mo!e_%
1.2 3.07 —0.97 210 - 3.16 2.55 5.61 2.81
1.36 ~3.29 —-1.06 227 - 3.10 2.44 5.10 2.84 _
1.53 3.52 ~1.13 2.42 312 . 2.29 4.48 2.81
173 3.78 -1.12 *  2.61 3.09 2.14 3.96 2.81
2.00 4.15 - ~1.32 2.85 - 3.14 1,98 3.43 2.82
2.54 474 —1.56 0330 3.04 1.74 2.76 . 2.88
3.09 5.37 -1.74 " 3.72 3.09 1.56 2.34 2.9
4.32 6.02 = —1.94 4,16 3.10 1.21 1.63. 2.89
5.00 6.19 -1.94 422 3.19 1,05 1.36 2.78
5.68 " 6.66 ~217 4.62 3.07 0.99  1.25 2.92
6.42 7.10 <229  4.90 3.10 - 0.91 1.13 _ 2.99
7.48 7.40 —2.43 5.15 © 3.04 0.81 0.98 3.01
| | . Av3.10+003 - Av 2.88 1 0.06

Ve have selected Kr and Xe for present considerations becafise the propettiesrof these gases
are, to a close apprommatlon, representatxve of the average values for the volatlle Spec1es in
the so-called light and heavy fractxons of the total products formed by fission.

First, however, it is necessary for us to ‘make a few assumptlons " The most obvious of these

 is that all of the MSRE graph1te bars do not sxgruhcantly dev1ate from the diffusion septum with
~ respect to internal geometry. In other words, (e’/ q’)is about the same throughout Further,

we shall assume that (¢”/q") is reasonably mdependent of temperature, so that the only tempera-
ture dependence which is exhxblted by the gas transport is due to the gases themselves. Finally,
we shall choose T 936°K and p= 2 36 atm (20 psig) as the conditions characteristic of gas
transport in the MSRE

 

 
 

 

 

 

 

 

 

 

 

 

 

 

34 .
ORNL-DWG 66-12742
(x10°%) /
. o

55 // }
5.0 — : [

o EXPERIMENTS WITH: Xy (L) =Xy (0)~O.A9mole % ' ‘

o EXPERIMENTS WITH: X, (L)=X, {0} ~091mole %o
45 o

 

&
o

-o\

 

o
t

 

< |

 

.\,\
o

/

 

n
o

/

(Do ot NORMAL DIFFUSION COEFFICIENTS {cm/sec)

 

i ®
,/
y

 

10 ,/
-

/
05 - /

 

 

 

 

 

 

 

 

 

 

 

 

o Ot 0.2 0.3 04 0.5

<p>"' RECIPROCAL MEAN PRESSURE

06

.(cnm)'1

0.8 09

Fig. 7. Pressure Dependence of the Normal Diffusion Coefficient c;f the NC-CGB

Graphite Diffusion Septum for the System Helium-Argon at 24°C.

If we insert the expression for the average speed c, into Eq. (34), we obtain

D 4 /8 RT 1/?K
iK—--s- ‘”Mi. o’

(60)

) where K is given by the relation presented in Table 2 and depends 'élmost completely upon the -

geometry of the medium. Simply by rearranging Eq. (60) we see that the grouping (M 4 /2D iK

. is likewise dependent only on the geometry of the graphite. Thus, if the characteristic value of

&,

 
 

 

 

wi

o e

.:,‘Q,

but 1s still retamed inD

i the value adopted from the data in Table 5 is

" The problem now reduces to solvmg Eq (62) for (e’/ q )wrth the value of pD

 

35

- VM, D, is taken to be equal to 9.38 g!/2 cm? sec—! mole~ Y2 at 22.5°C, as discussed pre-

viously, we obtain |
 (M/T)'/2 D, =5.45.105 g!/? cm? sec! mole~1/2 deg~!/2 . (61)

With this result it is then possible to obtain the Knudsen diffusion coefficients of the three

gases concemed. These results are listed in Table 6.

Table 6.- Parameters for He-Xe and He-Kr Diffusion in MSRE Graphites at 9360!(
and 2. 36 otm Pressure (MSRE Operating Conditions)

 

 

G VM, 1/2' ' L;ix : " ]gl-ret - Dyes ;)i
as (g/mole )" (cm*/sec) (cm? /sec) (¢m2 /sec) (cm” /sec)
. i 3
L o x1074 | x1073 - x107*
He 2.001 8.33 o |
Kr 9.154 1.82 1.81 ‘ 1.69 1.64

" Xe 11.46 1.45 1.63 1.52 132

 

We now tum to an evaluatlon of the normal drffusron coefficient. As 1s shown in Table 2,

‘this coefficient is the product of two factors:

Dyy=(e/aWyys (62)

in which the “free-space” coefficient DHe ; is simply the nomal diffusion coefficient as de-

~ termined for a known pore geometry. The difference between the script and the printed coef-

fxcrent is that the intemal structure, so to speak has been removed from the former coefficient,

Heli’
Thus far, all we have is a value- for L whrch is characterrstrc of the drffusron septum

i -

aD,_, =(2.87:I:0.,05)x10“‘8'mole.cnr""sec"‘f, S (63)

-

which refers toa temperature of 24°C. However, 1t tums out for our purposes to be more cof-
venrent to work with the group pDy, ..» where the pressure pis eXpressed in atmOSpheres The

correspondmg value for the drffusxon septum is then given by

pD'

Hear = 6" 99>< 10~ teltm'cm3 sec‘"__‘_ . - o o _v | (64)

Hear BiVEN above

and a value of pfl This latter quadtity can be determined from Eq (40) provxded the colli-

HAr

' sion cross section for diffusion, 'nrcri g 0‘112 1* is known. Alternatively, one can employ experi-
' rnentally determined values of the diffusion coefficients if these are available. The results are

often expressed in the form

 
 

et st e o s

 

 

10810 (Plgu) = A(logu‘) T) +B, LT S (65)

where T is the temperature in °K and A and B are constants. For the systems of interest in this

- work, the following equations have been proposed in the literature:

He-Ar ! _ :
log,, (pNHeA ) = 1.684 (1og10 T) — 4.2902 , o (66a)
He-Kr: _ | - o
log,, (p&HeKr) 1. 688 (log,, T) — 4. 3844 - (665)
H_e-Xe:. 7 ‘ - _ ,
 log,, (9, 4,) =1.720 (log,, T) — 4.5251 . B | (66c)

i

‘Although these equations reproduce the experimental data only over the temperature range O to

120°C, the error introduced in employmg the equations at higher temperatures is normally quite
small. From Eq. (663) we obtam at 24°C, '

plgmu;r = 0.748 atm cm? sec™! ; | - ' (67)
thus |
(€°/q") = 9.34 x 10~* . o | (68)

It is now possible to obtain normal diffusion coefflment values for any gas pair for this

particular graphite simply by employing the relation

D,,=(934x1079 0 . | o | | . (69)

As an example, we have also presented in Table 6 the characteristic values for He-Xe and He-Kr

diffusion for approximate MSRE operating conditions.

Similarly, one can predict the overall coefficient for diffusion D, from the relation

1
(70)

 

1 1
) Di_ b <DHe1t ‘ -

1

These coefficients for Kr and Xe diffusion are also listed in Table 6. The important thing to

" note is that D, and the corresponding D, dif_fer by only about 10%; in other words, normal dif-

fusion associated with the coefficient D, does not contribute significantly to overall trans-

| port in the dlffusxon septum. Moreover, th1s septum was sectioned from the MSRE graphite in

that region where normal diffusion can reasonably be assumed to be maximal. It therefore ap-

pears quite ]ustxflable,to ignore normal diffusion effects in oonsxdermg gas transport in MSRE

~ graphite,

 
 

 

+

R

 

37

Summary

7

In the experimental section of this report our primary concern was the determination of the

_ parameters sz' B,, and D , (or €’/q") for a g,ivén septum.” We have demonstrated that the

former two coefficients can be evaluated on the basis of ithye pressure dependence of the per-

meability coefficient, whereas D,, is conveniently determined from diffusion experiments which
are performed under conditions of constant total pressure.

The importance of these parameters lies in the fact that they specify completely the char-
acteristics of the medium geometty; once evaluated, these quantities may then be manipulated .
so as to describe gas transport under a variety of conditions. In other words, additional experi-
ments need not be performed if gases other than those employed in the ““calibration’’ of the
medium are of interest, or if the pressuxe and/or temperature conditions are varied.

Pethaps the most significant feature with regard to MSRE application is that the Knudsen
mechanism predominates in describing gaseous diffusion through this type of graphite. This
fisdihg provides a most welcome simplificetion to analyses of fission product migration.

Although the data presented here for the MSRE graphite may be employed to obtain rough
estimates of its gas transport characteristics, we wish to caution the reader of the possibility -
that the sample used in this work may or may not be typical of all of the MSRE material. Our
specimen was machined from Bar # 23 [ORNL lot # 1, NCC lot # 12 (of 14)]. Provided the fab-
rication of this bar has been reasonably duphcated in the manufacture of the remainder of the
material, the flow parameters may be adopted with a fair degree of confidence. Otherwise, how-
evet, the results presented here can be least typical of MSRE graphites.

In conclusmn we also wish to emphasme that impregnation treatments frequently 1mpart
nonhomogeneity in the structure of the fmlshed graphite material. This can lead to rather dif-

ferent results regarding fission product transport and retention, pariicula'rly since radioactive

" decay must be taken into account. We shall consider this problem for the case of MSRE graphites

in a later report.

V. APPENDIX

More detafled presentatlons of the kinetic theory of gases and gas transport through porous

| medla may be had by consultmg the followmg selected references.

P. C. Carmen, Flow of Gases Through Porous Medla, Academxc New York, 1956

R. D. Present, Kinetic Tbeory of Gases, McGraw-Hill, New York, 1958.

J. 0. erschfelder, C. H. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids,
John Wiley & Sons, New York, 1954, : '

R. D. Present and A. J. deBethune, Phys Rev. 75, 1050 (1949).

E. A Mason, A. P. Malinauskas, and R. B. Evans 111, J. Chem. Phys, 46, 3199 (1967)

 
 

 

 

 

 

 
 

 

“Kwivemt

- 1. Biology Library 84.
- 2—4. Central Research Library 85.
5-6. ORNL - Y-12 Technical Library 86.
: Document Reference Section 87.
7-40. Laboratory Records Department 88.
41. Laboratory Records, ORNL R.C. 89.
42. R. E. Adams 90.
43, C. F. Baes : 91.
44, C. D. Baumann - ’ . 92—10].
45. R. L. Bennett 102,
46. E. S. Bettis 103.
47. H. Beutler 104.
48. F. F. Blankenship: 105.
49. C. M. Blood 106.
- 50. E. G. Bohlmann 107.
51. G. E. Boyd 108.
52. R. B. Briggs 109.
53. H. R. Bronstein : | . 110.
- 54, 'T. J. Burnett - ' 111=119.
55. M. M. Chiles ' 120.
56. W. H. Cook 121,
37. S. J. Cromer 122.
- 58. F.L.Culler 123.
59. W. Davis, Jr. 124.
60. S. J. Ditto 125.
61. W. P. Eatherly 126.
62-71. R. B. Evans Il 127.
72. J. |. Federer 128.
73. D. E. Ferguson 129.
74. S. H. Freid . 130,
75. M. Fontana 131,
76. W. R. Grimes - 132,
-77. A. G. Grindell - 133
78. R. P. Hammond 134,
79. W. 0. Harms , 135,
80. P.N. Haubenreich : 136,
81. G. H. Jenks - 137.
- 82. P.R. Kasten ! . 138,
83. R

 

39

INTERNAL DISTRIBUTION

J. Kedl

139. J. A. Swartout, Union Carbide Corporation, New York _
140-417. Given distribution as shown in TID-4500 under Reactor Technology cotegory (25

coples—CFSTl)

EXTERNAL DISTRIBUTION

ORNL-4148

uc-80 —Reactor Technology

. Keilholtz
. Kennedy

. Lackey

. Larson i
. Lindemer
. Lorenz

. MacPherson
. MacPherson
. Malinauskas
. McCoy

. Moore

. Moore

. Nicholson

. Odkes

. Osbome

. Parker

. Perry

. Rosenthal

. Rutherford
. Scott

E. Sessions -
. Sisman .

J. Skinner

. Tallackson
. Thoma

. Trauger

. Wantland

. Watson

R
E
B
L
M
M. Weinberg
R
C
E
C
P.
rm

Fr=ZTO0TMOrFrfr9oMotmMO>»OmMMA =

ORLEONDEEPREMACIPAINANEND

. Weir

. Weir

. Whatley

. White

Wichner

an Hackerman (consultant)
. L. Margrave (consultant)
. Reiss (consultant)
. C. Yogel (consultant)

o

”Ihzflfigfik?P%

——