INT. COMM. HEAT MASS TRANSFER Vol. 20, pp. 43-52, 1993 Printed in the USA

0735-1933/93 $6.00 + .00 Copyright°1993 Pergamon Press Ltd.

ANALYSIS OF ADSORPTION AND DESORPTION PROCESSES IN A POROUS MEDIUM

K. Muralidhar and Mathew Verghese Department of Mechanical Engineering

Indian Institute of Technology Kanpur 208016 India

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT

Advective heat and mass transfer in a saturated porous medium is numerically studied here. A geometrically simple model has been used to study adsorption and desorption effects between the fluid and the solid matrix. The common assumptions of spatial and temporal equilibrium between the two phases has been specifically examined in the present study. Results show that these assumptions are valid only under special conditions.

Introduction

Heat and mass transfer in a fluid saturated porous medium is

a problem of considerable engineering importance. It is

encountered in the analysis of spreading of pollutants such as

radioactive and chemical wastes by ground water [1,2]. At

steady state it is commonly assumed that an equilibrium exists in

terms of the concentration of the transported species between the

liquid and the solid phases. Under these conditions a local

volume-averaged concentration can be defined for each point in

the porous region. In transient problems this equilibrium will

not exist in general and the transport processes in each of the

phases must be separately considered. These processes would be

coupled through an interface equation that permits heat and mass

transfer from or to the respective phases.

We model transport in a porous region in which the incoming

flow is clean or contaminated and the solid matrix is initially

43

44 K. Muralidhar and M. Verghese Vol. 20, No. 1

clean or contaminated. The transport species may have a finite

half-life and will further be adsorbed or desorbed into or from

the matrix. In most applications the porosity of the formation

is small and the solid phase conductivity is at least an order of

magnitude smaller than the fluid conductivity. This ratio of

conductivities will decrease further owing to dispersion in flow.

Under these conditions diffusion in the solid phase is close to

one-dimensional in a direction normal to the particle surface.

The affected region in the solid phase is small and hence we may

assume, i. gradients normal to the particle surface are small in

comparison to those parallel to it and 2. curvature effects to

be negligible. The transport problem can now be modelled as flow

in two dimensions in a small parallel gap of opening 2d and

one-dimensional diffusion in the solid phase (Figure i).

D

t

FIG 1 Model for Transport in a Porous Region

Formulation

The liquid and solid phase concentrations cf(x,y,t) and s

c ( y , t , x ) a r e r e s p e c t i v e l y g o v e r n e d by t h e e q u a t i o n s g i v e n b e l o w .

f = + D t - A c} (i) Liquid: {c t + u c x + VCy D Cxx Cyy

s Solid: {c t = DsCyy A c } (2)

Here u and v are velocities in x and y directions respectively,

is the radioactive decay constant, D and D t are longitudinal and

transverse dispersivities, D s is the solid phase diffusivity and

Vol. 20, No. 1 ADSORPTION/DESORPTION IN A POROUS MEDIUM 45

suffix such as in c represents partial differential of c with x

respect to x. Since the porosity is small the fluid gap is also

small and a depth-averaged concentration c(x,t) defined as,

o

c(x,t) - 2dl [ cf(x'y't) dy, can be used to represent c f

-2d Equation 1 for the liquid phase can be integrated through out and

expressed in terms of c(x,t). To complete the integration, we

assume the fluid to be incompressible. Further the interface

condition at y = 0 is chosen as,

_ac~ = k ( c-c s) where k is a prescribed parameter By]y=0

This condition is general enough to model adsorption when c > s s

c as well as desorption when c < c . The equations

governing the depth-averaged fluid concentration c and the solid

phase concentration c s along with the initial and boundary

conditions are as follows.

Dsk Dsk s ct+ UCx= D Cx~ ( ~+ --~--) c + T c ]y=o (3)

with t = 0 c = 0, x = 0 c = exp(-~t) and x ~ m c ~ 0.

s s s c t = D s Cyy A c (4)

with t = 0 c s = 0, y = 0 -c s = k(c - c s) and y = L -c s= 0. Y Y

In Equation 4 the boundary condition at y = L represents a

symmetry plane. In a heat transfer problem A the decay parameter

is chosen as zero. In a pure desorption problem the initial and

boundary conditions are altered to, t=0 cS=l, x = 0 c = 0,

representing a clean incoming flow and a contaminated matrix.

Based on Equations 3 and 4 the time and length scales can be

identified as TS = i/(~ + u2/4D + Dsk/d), X = V~-~.TS and Y =

.TS. Hence at a characteristic distance of X in the fluid s

phase and Y in the solid phase steady state will be reached in a

time t - TS.

Numerical Solution

Equation 3 given above has been solved using an

operator-splitting algorithm [3]. The source term in Equation 3

contains the interface solid phase concentration c s (0,t).

46 K. Muralidhar and M. Verghese Vol. 20, No. I

Since the OS algorithm is time-explicit cS(0,t) is approximated s as c (0, t n) in the solution for (n+l)th time step for

c (x, t n+l) . Typical values of the parameters used in the

calculations are as follows, d = 0.005 m, L = 1 m, D = 2.52 s-3 -I

m2/year, D = 25.2 m2/year, u = i, i0 m/year, k = 0, i0 m ,

half-life TI/2 = i00, years. Based on the scales defined above

it can be shown that k = 10-3m -I represents strong adsorption.

For u = 1 m/year and TI/2 = i00 years, the time scale TS (k = 0)

is 59 years and TS(k = 0.001) is 2 years. For u = i0 m/year TS

(k = 0) is 1 year and TS (k = 0.001) is 0.67 year. The time

step used for integration is 1 percent of TS.

Results

Solutions of Equations 3 and 4 are presented here for the

following cases, i. Fluid contaminated with a radio- nuclide in

an initially clean matrix, 2. Heated fluid flowing through an

initially cold matrix, 3. Clean fluid flowing through an

initially contaminated matrix. Attention is focussed on the

distribution of the concentration in the fluid phase. In an

adsorption problem (k > 0) a part of the species in the

fluid phase goes into the solid thus reducing the extent of the

region over which the fluid concentration is non-zero. Hence

adsorption effects are increasingly felt at large distances from

the inflow plane. The possibility of an equilibrium between the

fluid and solid phases exists only if k > 0. In a heat transfer

problem (Case 2) the heated fluid will lead to the movement of a

thermal front in the solid phase. The speed of this front can be

estimated from Equation 4 as ~ Ds/t . Hence beyond a certain time

the front becomes nearly immobile. Simultaneously the fluid

temperature attains a steady state when advection and diffusion

effects are in balance. Hence a temporal equilibrium is attained

in this problem though the fluid and solid phase temperatures

remain distinct. Similar processes occur in the transport of a

radioactive species (case i) except that temporal equilibrium is

not possible here owing to decay. In a desorption problem (case

3) the vastness of the solid phase arising from the low porosity

of commonly occurring formations guarantees that the solid and

Vol. 20, No. 1 ADSORPTION/DESORPTION IN A POROUS MEDIUM 47

the fluid phases will attain spatial equilibrium over a distance

that depends of the flow velocity and the magnitude of the

dispersion coefficients.

Case I. The transport of a nuclide whose half-life is i00 years

is considered first (Figures 2). The solid matrix is initially

25 50 _0 I . U - i I I ~% u - I mlyr

k =Orn-I

- - - k = 0.001 rn "I

~ i _ ..I/t-loo yr

o ~o ,~ ,~ ~ o ~ o 300 T - -

1.0 k I I i I m / y r u = I\

I ~ / ~ x ' Im - - k . O m-I

l

0 I00 200 300 4 0 0 t, yr

a. FIG 2

Variation of Fluid Concentration With Distance b. Variation of Fluid Concentration With Time

48 K. Muralidhar and M. Verghese Vol. 20, No. I

clean. Figure 2a shows a plot of fluid concentration versus

distance at two time levels. The mean velocity is 1 m/year in

this figure and values of k = 0 and 10 -3 m -I have been

considered. When k = 0 the size of the affected region increases

with time and the concentration level in the fluid decreases

owing to decay. When k > 0 the size of the region over which c

drops from its value at the inflow plane to zero is nearly

unchanged as t increases from I0 to i00 years. This indicates

that transport of the species into the fluid region is balanced

by adsorption at the solid-fluid region and a partial

equilibrium exists between the mechanisms of diffusion and

advection. The solid and fluid phase concentrations are both

space and time dependent even at equilibrium forced by the

adsorption process. Hence the idea of a continuum in

concentration distribution valid for the two phases is not

applicable here. Figure 2b shows a plot of fluid concentration

as a function of time at x = 1 and I0 m and k = 0 and 10 -3 m -I.

In both cases c increases from 0 to a maximum value and is

subsequenly followed by decay. This magnitude of the maximum

value decreases with increasing x and k. When k = 0 the decay -It

process strictly folows the curve e However for k > 0 the

maxima in concentration c are smaller and the decay is faster due

to additional adsorption into the solid phase.

Figures 3a and 3b show the effect of increasing the mean

velocity from 1 to i0 m/year. The concentration profiles in

Figure 3a at k = 0 show the formation of fronts and a large

increase in the size of the affected region in the fluid phase.

The corresponding plot at k = 0 in Figure 3b shows a rapid

increase in c at both x = 1 and i0 m and a near absence of

diffusion effects. Once a maximum is reached concentration

levels drop due to radioactive decay. When k = 10 -3 m -I there is

a substantial reduction in the size of the region over which c is

non-zero though this size is larger than the corresponding value

for u = Im/year (Figure 2a). There is no formation of a front

and the concentration profile is nearly immobile between t = i0

and i00 years. The plots of c versus time for k > 0 show that the

initial increase in concentration is advection- controlled and is

hence rapid but its fall is determined by adsorption and decay

Vol. 20, No. 1 ADSORFTION/DESORPTION IN A POROUS MEDIUM 49

0 0 2 5 5 0 7 5 I00 u = I0 m/yr

\ [ k-Om -I f7 t -Io yr ~! \ __. k =0.001 m. I

C 0.5,, .I " ' , / ~ ( a ) -

J % ,~ %% t=lOOyr ~ - . . . ' . . /

\ .... I 0 300 600 900 1200

x,m

1.0 k i I , I J ~ k u =lOm/yr I ! \ ~ =-'x=lm k=O m - I I

- - ~ ~ . , o ~ ___~.o.oo, _, I

c 0~ -... ~

I I I 0 I00 200 300 400

),yr

FIG 3 a. Variation of Fluid Concentration With Distance

b. Variation of Fluid Concentration With Time

Hence fluid concentration levels are consistently lower when k >

0 as compared to the case of k = 0.

Case 2. The special case of heat transfer is recovered from

Equation 3 by setting the decay parameter X as zero (Figures 4).

50 K. Muralidhar and M. Verghese Vol. 20, No. l

C

C 0.5

1.0 i'--..... ~ , J I

I;% ....

0 6 0 120 180 240 x,m

1.0

~ k "0 rn" ---k "0.

I 0 25 50 75 IO0

f, yr

a. FIG 4

Variation of Fluid Temperature With Distance b. Variation of Fluid Temperature With Time

These show temperature profiles as well as the increase in

temperature with time for a velocity of im/year. The half-gap

height d is 0.05 m in these figures. Both values of k = 0 and

10-3m -I have been considered. The effect of adsorption resulting

in a retarded movement of the front is seen in Figure 4a. In the

absence of adsorption the steady state value of temperature at a

point is equal to the inflow value, namely unity. With

Vol. 20, No. 1 ADSORPTION/DESORPTION IN A POROUS MEDIUM 51

adsorption this value decreases and the extent of this reduction

increases with increasing values of the x-coordinate. The

possibility of a steady state in the fluid phase and a

quasi-steady state in the solid phase due to reduced thermal

front speeds results in a temporal equilibrium in the system.

However temperature is a function of spatial coordinates

representative temperature for the two phases does not exist.

Case 3. Pure desorption is considered next. The parameters used

here are, half life = i00 years, d = 0.005 m, u = i0 m/year and k

= 10 -3 m -I Figure 5a shows the concentration profile in fluid

phase at two different times. The magnitude of c increases from

zero at the inflow plane to a value that brings it in equilibrium

with the solid phase. Subsequently, the concentration in the two

phases decay jointly with time. Hence, except for the region

near the inflow plane a spatially uniform concentration field is

possible, that changes with time owing to radioactive decay. The

size of the region over which c increases from zero to the

1.0

C 0 . 5

0 150

' ' t = IOyr

(o)

t • I 0 0 yr

u= IOm/yr k=O.O01 m- I

I I 5 0 I 0 0

x , m

FIG 5a Variation of Fluid Concentration With Distance

equilibrium value will decrease with increasing velocity. Figure

5b shows a plot of fluid concentration with time at two different

x-locations. The interface solid concentration at y = 0 is also

52 K. Muralidhar and M. Verghese Vol. 20, No. 1

shown here for comparison. At x = i0 m the solid and the fluid

concentrations are quite different at small time. With

increasing time the solid concentration reduces due to decay and

fluid concentration increases due to adsorption and the two

values approach each other. At x = i00 m the two values are

close to each other for nearly all time.

1.0 %~. , Fluid

/ ~ ' t k . • . - - - - - - Solid / " ~ , , ~ x l u u r n u = l O m / y r . -

~ . ~ . k • 0 . 0 0 1 m- =

- ~~. (b)

c 05 /

i I 0 5 0 I 0 0 150

t, yr

FIG 5b Variation of Fluid Concentration With Distance

Conclusions

Results show that the assumption of spatial equilibrium

between the fluid and solid phases is normally violated in

adsorption problems. A steady state is however possible in

transport of stable species such as thermal energy. Spatial

uniformity in the concentration distribution is observed in a

pure desorption problem.

References

i. J Bear, Dynamics of Fluids in Porous Media, Elsevier, New York(1979).

2. T.W. Broyd, in Nuclear Containment, Edited by D.G. Walton, Cambridge University Press, U.K.(1988).

3. K. Muralidhar, Mathew Verghese and K.M. Pillai, Int J of Numerical Heat Transfer, to appear (1992).

Received October 27, 1992