Effects of thermal osmosis near a buried isolated heat sphere

INT. O3MM. H~tTM~SSTRANSFER 0735-1933/86 $3.00 + .00 Vol. 13, pp. 295-304, 1986 ©Pergamon Press Ltd. Printed in the United States

EFFECTS OF THERMALOSMOSIS NEAR A BURIED ISOLATED HEAT SPHERE D.L.R. Oliver University of Toledo Department of Mechanical Engineering Toledo, Ohio 43606 (C~L,~nicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT The effects of thermal osmosis in a homogeneous porous medium are analytically investigated near a heated spherical container. I t was demonstrated that accurate estimations of the effects of thermal osmosis must account for physical constraints such as continuity of mass. This work has application in predicting release from a radioactive waste site.

Introduction Over the past t h i r t y years nuclear power has been developed into a significant source of electricity in several nations.

This increasing

dependence on nuclear power has resulted in a large quantity of radioactive wastes. These wastes remain highly radioactive for many decades. Several nations have been investigating deep geologic waste repositories to isolate the waste from the biosphere until i t has significantly decayed into stable isotopes. One conceptual design for a deep geologic repository is detailed in the Draft Environmental Assessment [1] for a proposed deep geologic waste repository in basalt.

According to this design the waste would be stored in steel

containers individually burried in a horizontal bore hole about 1,000m below the surface (Fig. 1).

At this depth the waste containers will be well below

295

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D.L.R. Oliv~_r

Vol. 13, No. 3

the water table, and will eventually be exposed to the ground water in the surrounding (assumed) porous rock. Eventually the steel containers which surround the waste will corrode, exposing the radionuclides to the ground water in the surrounding media. The radionuclides are then able to migrate with the surrounding ground water. To determine the extent of any detrimental effects the waste repository might have on future generations i t is important to estimate the transport rates from the corroded waste containers.

To adequately estimate the transport

rates near the corroded containers, the relative importance of the transport mechanisms must be understood.

In this work the relative importance of

thermal osmosis with respect to pure Darcy flow is investigated. As the radioisotopes decay heat is generated in the waste containers. The rate at which heat is generated depends on the i n i t i a l composition of the waste.

Also, the rate of heat generation generally decreases with time.

For

example the relative heat generation rates for waste from both a PWR reactor and a CANDUreactor are illustrated in Fig. 2, per Beyerlein and Claiborne [2]. This generation of heat will produce local thermal gradients near the waste containers.

The presence of sharp thermal gradients might significantly

affect both the flow f i e l d and the release rate of radionuclides near the corroded waste containers.

Most analyses of radionuclide transport near a

waste container consider only pure molecular diffusion coupled with convection driven by a local pressure gradient ( i . e . Darcy's law).

However a rather

simple analyses presented by Carnahan [3] showed that thermal osmosis effects might be significant.

In fact, the simple analysis performed by Carnahan

suggested that the flows driven by thermal osmosis might be orders of magnitude larger than the flows driven by a regional pressure gradient.

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297

I f the estimates of the relative effects of thermal osmosis presented in Carnahan are correct, then much of the prior investigations of transport rates from the region near a waste container might seriously underestimate the transport rates of radionuclides.

It is the intent of this investigation to

further estimate the relative effects of thermal osmosis as i t relates to release rates from an isolated waste container.

If these effects can not be

shown by this, and subsequent investigations, to be insignificant then there will be a need for further experimental and theoretical investigations before release rates of radionuclides to the biosphere may be accurately estimated. Anal~sis The equation of motion in a porous medium in the presence of a thermal gradient may be derived from Eq. I of Carnahan, neglecting all driving forces except those due to thermal osmosis and pressure gradients.

J'-v = [- ~ T where

v*T - KhV* HI,

Jv

is

the Darcy velocity,

Lvq

is the phenomenological coefficient for thermal osmosis,

T

is the absolute temperature,

Kh

is the hydraulic conductivity,

H

is the hydraulic head, H = (P - pfgZ)/(pfg),

P

is the pressure,

Pf

is the ground water density,

Z

is the depth.

The superscript "*" indicates that the operation is dimensional. Following Carnahan's example, the velocity components may be estimated given the following parameters obtained (or derived) from Carnahan [2];

(1)

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D.L.R. Oliver

v*H Kh

Vol. 13, No. 3

= 10-3 m/m, = 5 x 10-10 m/s,

v*T

= 2 °K/m,

Lvq

= 8 x 10-8 m2/s,

(from experiments with Kaolinite-water systems) Jv (Darcy) : 5 x 10-13 m/s,

Jv (thermal osmosis) : 4 x 10-10 m/s. The above simple analysis is lacking in that conservation of mass is not imposed as a c o n s t r a i n t .

However, for the case of a thermal gradient

induced by an i s o l a t e object in a homogeneous medium, with no mass sources, the r e s t r a i n t imposed by c o n t i n u i t y of mass can y i e l d a v e l o c i t y p r o f i l e that is independent of the thermal gradient.

Thus, the above s i m p l i f i e d analysis

might seriously overstate the e f f e c t s of thermal osmosis on the transport rates near the waste container. For example, consider an idealized isolated spherical waste container in the presence of a regional hydraulic head gradient of H'.

The coordinate

system is fixed such that the flow d i r e c t i o n in the undisturbed free stream is in the 0 = 0 d i r e c t i o n , (Fig. 3). I f the heat generation rate is constant with time, a steady-state d i f f u s i o n a l analysis y i e l d s a temperature p r o f i l e of approximately;

T(r) . . . . . . . + T®, 4~KT R where

R~ a

Q

is the heat production rate in the waste container,

KT

is the thermal c o n d u c t i v i t y ,

a

is the radius of the waste container,

R

is the dimensional radial coordinate,

T~

is the regional temperature (assumed constant).

(2)

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~ O S M ~ I S N E A R A ~

~

299

The assumption of considering thermal d i f f u s i o n a l transport only is j u s t i f i e d for many situations since the thermal d i f f u s i v i t y of many rock matrices is so large (s T for basalt is about 7 x 10-7 m2/s).

Thus the thermal Peclet numbers

are often much less than one. The above temperature p r o f i l e may be substituted into a dimensionless form of the flow equation, (Eq 1);

3--v

T. = ['NT i - V~ - vh]

Jv

is the dimensionless v e l o c i t y ;

NT

is the thermal osmosis number,

(3)

where

-Lvq

I

~V

=

--

Kh

,

aT i

I

'

is the dimensionless temperature,

T -T. T

........ , where Ts = T(R=a), Ts - T®

is the dimensionless hydraulic head, h

H

=

a

The principal of continuity of mass is impose by noting that: v • 3-~ =

O,

or

T, v • [-N T i - VT - vh]

=

Given constant values for KT, Lvq and Kh, NT w i l l also be constant.

assume that

T(R:a) - T, . . . . . T~ ......

NT V2 T

-

T® << I ( i . e . ~- ~ i ) .

Also

The above assumptions y i e l d :

V2h = 0 .

Combining Eq. 2 with Eq. 5 y i e l d s ;

(4)

C5) v2h = 0 .

(6)

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D.L.R. Oliver

Vol. 13, No. 3

The boundary conditions on Eq. 6 are: Jv • ?

= 0

at

r = i,

(no flux from the container)

(7)

J--v = - h'® [ ( c o s e ) ~ - (sine)-e] as r ÷ ~ , ( f r e e stream conditions)

(8)

where r is the dimensionless radius, r = R/a, and where h'® is the dimensionless gradient of the hydraulic head along the e = 0 axis as r ÷ ~. Equation 3 may be substituted into Equations 7 and 8 to obtain; ah ;r ]

aT

[-N T

[-N T

ar

~

ar

(9)

: o, r =l

~] ar

_

r

NT aT

1

ah]

r

r

ae

E. . . . . . . . .

ae

+

~

r

+

= - h'® cose,

(lOa)

= h'~ sine.

(lOb)

~

Equations 3 through 10 y i e l d a dimensionless hydraulic head p r o f i l e of;

-NT h(r,e) = ( r

I + CO)

+ h'® (r + 2;~) cose,

(11)

where Co is an a r b i t r a r y constant. Substituting the hydraulic head p r o f i l e into Eq. 3 results in a v e l o c i t y p r o f i l e that is independent of NT ;

~v

NT -

r2

r - [

NT

+ h'® (I -

_1):

1

cose] F + [h'® (1 + - - - ) sine]

2r3

(12) I i = - h'® [ ( i - - ; ) cose ~ - ( i + - - - ) sine ~] . r3 2r3 The v e l o c i t y p r o f i l e of Eq. 12 is the same as would result from pure Darcy flow without considering thermal osmosis.

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~OSMDSISNEARA~SPHEI~

301

The fact that the waste container had no mass sources forced the pressure f i e l d to compensate for the thermal osmosis effects.

This results in a

velocity profile that is independent of thermal osmosis. This fortuitous result might not hold in a transient situation where the response time of the pressure f i e l d might be large. There are steady state situations which are relevant to a repository where thermal osmosis might be important.

For example, i f the repository

were placed in a strong regional thermal gradient, such as gradients due to the release of heat from the i n t e r i o r of the earth.

An aquifer deep in

the earth's crust could act as a source for ground water to the system. For this case, a simple analysis such as that presented in Carnahan might be justified. To estimate the possible effects of thermal osmosis on such a system with a regional thermal gradient the basalt system of Ref. [1] is used. Lvq The assumption will also be made that the ratio . . . . 160m obtained for Kh a Kaolinite-water system (Ref. [ 4 ] ) , is also reasonable for deep basalt formations.

Using this ratio, the relative magnitude of the two velocities

may be estimated as; Jv (thermal osmosis) Jv (Darcy)

Lvq v*T

1

T®

Kn v*H

The regional vertical thermal gradient near the repository proposed in Ref. [1] may be estimated from Fig. 6-5 of Ref. [1] to be v*T - .04 °K/m with T.-

330°K. The regional vertical gradient of the hydraulic head is often BH estimated as -- - 10-3 m/m. With the above values, the ratio of the thermal @Z osmosis velocity to the pure Darcy flow velocity is:

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D.L.R. Oliver

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Jv (thermal osmosis) ..................... Jv (Darcy)

20.

Thus thermal osmosis due to a regional thermal gradient ( i . e . due to cooling of the earth's i n t e r i o r ) could be more important than velocities induced by hydraulic head gradients.

However, thermal osmosis due to heat

production inside the repository is probably of less consequence to the steady-state velocity p r o f i l e near the waste containers.

The chief

difference between these two cases is that the potentially large vertical mass fluxes predicted in Eq. 1 resulting from thermal osmosis due to a uniform vertical thermal gradient might be resupplied by a deep aquifer. On the other hand, the f i n i t e waste container can not continuously sustain any net release of f l u i d without a mass source.

Concluding Remarks An analytic solution was proposed which may be used to estimate the effects of thermal osmosis on the release rate of radionuclides from an isolated spherical waste container. I t was demonstrated that thermal osmosis due to the heat generation inside the container w i l l probably not affect the steady state flow f i e l d in a homogeneous medium near the container.

On the other hand, thermal

osmosis effects induced by the cooling of the earth's core might have a significant effect on the flow f i e l d . Thermal osmosis might also be significant in unsteady calculations or for calculations in an inhomogeneous region. scope of this investigation.

These effects are beyond the

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~O~40SISNE~RAHE~TSD

SPHI~E

References 1)

United States Department of Energy, Draft Environmental Assessment- Hanford Site, Washington D.C., (1984).

2)

S. W. Beyerlein, H. C. Claiborn, J. Heat Transfer, 104, 180, 1982.

3)

C . L . Carnahan, Scientific Basis for Nuclear Waste Management VII, M. R. S. 26, 1023, North Holland, New York, 1984.

4)

R.C. Srivastiva and P. K. Avasthi, J. of Hydrology, 24, 111, 1975.

Surface ........

-....

_ _ -~.-~:.!.:::...:..,._:. ~' :,'-'." ".'2 :2"::.':.'

Rock

)ii .i?::...: ,

'...

... •

, ....,

.,

. .

Rock

,, .

Shaft. .i" .'; "."'.!:

'..~

-.:2 : : : •

.

.

. ,

. . .

, ,

~:i., :.: .:...~.,..., :.j:; , .'."'........

F."".::" ..i ":'......~,.-...... --I

\ l:.V..:v:.::-:.::;{.::..:1 / Waste containers ( Not

to

Scale )

FIG. I Schematic for

High

of

o

Level

Deep Nuclear

Geologic Wastes.

Repository

303

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D.L.R. Oliver

Vol. 13, No. 3

c .oo'

n, kkl Z

PWR -!

I0 W

hi

:~

i6 2

1

~J re,

I0

I0

! z

I0

3

I0

TIME OUT OF THE REACTOR (ym) FIG. 2

Relative heat generation rotes of PWR and CANIXJ spent fuel.

o

FIG. 3 Coordinate 8yatam for an Ioolote Spherical Waste Container with Flow Direction.