theoretical study of interphase heat and mass transfer in saturated porous media

Pergamon PII: S0020.7225(96)00067-5

Int. J. Engng Sci. Vol. 35, No. 2, pp. 171-185, 1997 Copyright © 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0020-7225/97 $17.00 + 0.00

T H E O R E T I C A L STUDY OF INTERPHASE HEAT AND MASS TRANSFER IN SATURATED POROUS MEDIA

K. MURALIDHAR and JYOTI SWARUP Department of Mechanical Engineering, Indian Institute of Technology, Kanpur-208 016, India

Abstract--Transient convective heat and mass transfer between the solid and fluid phases of a saturated porous medium has been considered in the present study. A geometrically simple model that is capable of predicting retardation of the species has been developed. The equations governing species transport in the liquid and solid phases of the porous medium are coupled due to heat and mass transfer at their interface. The species concentration in the liquid phase is determined by analytically solving the system of differential equations along with experimentally motivated values of the interphase parameter. The following problems have been separately considered: (1) transient heat transfer between a heated fluid and an initially cold matrix; (2) mass transfer of a radioactive nuclide from a contaminated matrix into a clean fluid; (3) transport of a radioactive nuclide present in a fluid flowing through an initially clean matrix; (4) effect of a non-zero dispersion length. Results show that interphase transport between the solid and fluid is an important factor in retarding species transport in the porous region as a whole. Equilibrium in species concentration between the two phases is seen to be attained only under special conditions. Copyright © 1997 Elsevier Science Ltd

INTRODUCTION

Unsteady convective heat and mass transfer in a saturated porous medium is considered in the present work. The study is motivated by the need for analysis of spreading of chemical and radioactive wastes by groundwater [1-3]. When the transport process reaches steady state it is commonly assumed that an equilibrium in terms of the concentration of a chemical species or temperature exists between the fluid phase and the solid matrix. Hence a local volume- averaged concentration and temperature can be defined for each point in the porous medium. In transient problems transport rates in the solid and the fluid phases will be distinct and equilibrium will be observed only under restrictive conditions.

One of the most important consequences of non-equilibrium is the occurrence of retardation of the transported species. Retardation refers to a reduction in the distance covered by the species dissolved in the fluid and hence a drop in its concentration at a given station below the value expected from equilibrium calculations. Retardation has been observed in both laboratory and field experiments when the solute reacts with the solid phase [4-6].

Deviation from equilibrium during a transient transport process in a saturated porous medium will be significant when the solid fraction is high, i.e. the porosity is low and when the conductivity ratio between the solid and the fluid is small. Even when the ratio of the molecular conductivities of the matrix and the fluid is close to unity, dispersion in the fluid phase will increase the fluid conductivity and hence lower the effective conductivity ratio. Under these conditions the transport processes in each of the phases along with an interface equation relating them must be separately considered.

Methods of handling non-equilibrium in concentration between the solid and a reacting solute in the fluid phase are described by Bear [7] and Bear and Bachmat [8]. For sorption reaction of a species from the fluid phase into a solid matrix, a linear reversible instantaneous chemical reaction of the form

S =K~C (1)

is assumed. Here S is the amount of material transferred from the fluid to the solid matrix and C is the instantaneous concentration of the species being studied in the fluid phase. Kr is a reaction constant that must be experimentally determined. This approach results in a

171

172 K. MURALIDHAR and J. SWARUP

mathematical problem where time is divided by a constant factor/3 that is greater than unity. Hence the process of sorption from the fluid to the solid phase in effect retards the transport process in the bulk of the porous medium. Values of 13 reported in the literature vary from 1 to 1000 [1] and for some species it can be as high as 10 000 [9].

Heat transfer in gas flows through a porous matrix finds application in fluidized beds and regenerative heat exchangers. Two-equation models that treat the solid and fluid phases separately have been considered by Vortmeyer and Schaefer [10]. Analytical models developed also include axial dispersion and intra-particle conduction. These models are a convenient starting point for determining flow parameters such as solid-to-fluid Nusselt number and dispersion coefficients.

The present study is concerned with transport of stable and decaying species in a liquid-saturated porous medium in the absence of any chemical reactions between the species and the solid phase. A survey of the literature shows that two-phase modeling of non-reacting flows through porous media is restrictive and adopts one of the following approaches: (1) multiplication of the time derivative by an adjustable factor (>1) [11]; (2) accounting for thermal capacity ratio but assuming the interphase resistance as zero [12, 13]; (3) determination of the interphase resistance from Nusselt number correlations that are valid for high Reynolds number [14]. While approaches (1) and (2) are clearly approximate, approach (3) is also inadequate because heat and mass transfer resistance at the solid-fluid interface primarily determines the extent of retardation of the transported front and the absence of equilibrium between the two phases. In the present study, separate transport equations have been written for the solid and fluid phases with inflow, outflow, farfield and interphase conditions. The equations have been analytically solved. The interphase parameter has been suitably chosen from low Reynolds number data and cross-checked against a laboratory experiment. The model developed has been used to address the following problems: 1. Flow of a heated fluid into an initially cold porous medium. 2. Flow of a clean fluid into an initially contaminated porous medium. 3. Flow of a contaminated fluid into an initially clean porous medium. 4. Effect of dispersion in the fluid phase.

THEORETICAL MODEL

The model developed here represents the following physical configuration. A saturated porous region is suddenly exposed to contaminated water at its inflow plane. The solid matrix may itself be initially contaminated. The transport mechanisms in the liquid phase consist of diffusion and advection. Besides, the species can have a finite half-life and may further be transported to or from the solid matrix. We consider a class of applications such as nuclear waste disposal where (a) the species in the liquid phase forms a dilute solution and (b) the porosity e can be taken as small. Hence the temperature changes owing to decay are neglected. Since the porosity is small, the solid fraction is large and has a large surface area. Further, the solid phase conductivity with respect to the diffusing species is usually smaller than that for water by an order-of-magnitude. Under these conditions the diffusion process in the solid phase is close to one-dimensional in a direction normal to the surface of the solid particle.

The transport process in the solid phase is sketched in Fig. l(a). Here 8 is the depth of penetration of the solid particle of size D. The assumptions of small e and conductivity guarantee that 8]D is also small, i.e. 3/D << 1 and gradients normal to the particle surface predominate over those parallel to it. Since the ratio of radii of curvature is close to unity, curvature effects are neglected and the interface between the liquid and solid phase is treated as planar. This is shown in Fig. l(b). Hence species transport is reduced to a two-dimensional flow problem in the small parallel gap of opening 2d and one-dimensional diffusion in the

Interphase heat and mass transfer

(a) (b)

Fig. 1. (a) Transpor t in a particle of a porous medium, (b) equivalent model in Cartesian coordinates.

173

y-direction in the solid phase of size L. The ratio d/(d + L) (= d/L, when d is small) now represents e, the porosity of the region.

M A T H E M A T I C A L F O R M U L A T I O N

Let C be the concentration of a species in the liquid phase and c S be its concentration in the solid phase. With reference to Fig. l (b) , the equations, initial and boundary conditions governing C and c ~ are as follows [8]. Liquid phase

C t d- u C x q.- U f y = D C x x + OtCyy - AC

C(x, y, O) = O, C(O, y, t) = e -~', C(~, y, t) = 0

C(x, - 2d , t) = cS(-2d, t) and C(x, O, t) = c~(O, t). (2)

Solid phase

s m s m C t -- D s c y y AC s

OC s c~(y, O) = Cs, ay (0, t) = - K ( C - c ~) and

a c s

Oy (L, t) = 0. (3)

In equations (2) and (3), u and v are the x and y components of velocity, A is the decay constant, D and O t a r e the longitudinal and transverse dispersivities of the species under study in water, Ds is the molecular diffusivity of the species in the solid phase and K is an interface heat/mass transfer coefficient. Equation (3) is general enough to handle species transfer from or to the liquid phase since the sign of acS/ay changes according to whether C > c s or C < c s. The boundary condition at y = L reflects the fact that it is a symmetry plane. The limit of a heat transfer problem is reached when A is set equal to zero. When the liquid is clean and the solid matrix is contaminated, equation (2) is modified by the boundary condition x = 0, C = 0. In an initially clean solid cs in equation (3) is zero.

Since d /L is assumed to be small in the present study the liquid phase equations can be simplified by defining an average concentration

1 / ' 0

c(x, t) = ~-~ J-2d C(x, y, t) dy.

Equation (2) is integrated termwise across the depth of the liquid layer. To obtain an equation governing c(x, t) we assume u to be a constant and v = 0. Depar ture from these assumptions


will lead to additional dispersion that can be accounted for through the longitudinal dispersion coefficient. The equation governing c(x, t) can be shown to be

Dt ct + ucx = Dcxx - hc + ~ C>,ly=o.

Here we evaluate the source te rm in the above equation as

- D , c ~ l y = o : -Dsc ; , l ~=o -- -D,~(c - c % , ~ o .

The interface pa ramete r k is now based on the difference between the bulk concentration in the fluid phase and the surface concentration in the solid phase. With this formulation, c(x, t) is de termined f rom the equation

ct + UCx = Dcxx - ( h + D--~ )c + DSk s ~ - c [y ~o (4)

with the initial condition c(x, 0) = 0 and the boundary conditions, c(0, t) = e -~' and c(~ , t) = 0. In the present work, A, D, Ds and d / L are t reated as constants and u and k are prescribed parameters .

E X P E R I M E N T A L S P E C I F I C A T I O N OF k

The transport coefficient k as defined above is related to the heat transfer coefficient defined on the basis of surface tempera ture and the bulk mean tempera ture of the fluid. It is the most important paramete r in the model and must be determined f rom experiments. A commonly

used correlation for heat transfer between the solid and fluid phases is [15]

Nu = 2 + 1.1 Pr °'33 Re °'6. (5)

This formula implies a limiting Nusselt number of 2 as Re approaches zero. On the other hand, Heggs [16] has presented data that shows Nu to approach zero for Re << 1.

To address the question of the value of Nusselt number at low Reynolds numbers, a step response exper iment in a water-glass medium was per formed in the present study. The average fluid velocity was 0.4 m/h , the glass bead size was 0.5 mm, the tube diameter was 36 m m and the tempera ture difference imposed at the inflow plane was 30°C. Precautions taken for the exper iment include complete saturation of the porous medium, permeabil i ty tests to determine packing configuration of the glass beads and accounting for thermal losses to the ambient.

Figure 2 shows a comparison between the experimental tempera ture profiles in the porous

t , , ~ - - - k = OAm- o.6 ',,\ , , ~ c c = o . o s m

T "\,I ""',,, _. \ 0.4 i\ . ~ - . . ~

o2 ~ \ o'-., \ ~ , . , . "L,~,

0'00 0.2 0.4 0,6 0.8 1.0 112

x~rn

Fig. 2. Comparison of experimental temperature profile with the analytical model.

Interphase heat and mass transfer 175

region along the tube axis and the analytical solution developed in the present work. The case

of k - - 0 corresponds to local equilibrium between the phases and k > 0 represents the possibility of loss of thermal equilibrium. G o o d agreement between experiments and the analytical solution is seen when k is taken as 0.1 m -1. At k -- 0, the predicted front moves much faster than the experimental t empera ture profile. For k > 0.1 m -~ the front is practically immobilized near the inflow plane. The experimental Reynolds number based on the particle d iameter is 0.069 and the Nusselt number (= k •dp) is 10 -4. This is much smaller than 2, a value predicted by equat ion (5).

In view of the experimental result given above, the magnitude of k in the present work has been taken f rom heat transfer data for packed bed-type regenerative heat exchangers. For a characteristic velocity of 10 m / y r and particle diameter of 1 mm, the Reynolds number is 4 × 10 -7. The Nusselt number for Reynolds number range of Re = 10 3 to 10 -2 varies f rom

10 -6 to 10 4 (Fig. 5 [16]) and is smaller for smaller Reynolds number. The relation connecting k

and Nu is k = (D/Ds)(NU/dp), where dp is the particle diameter. Hence the corresponding values of k are in the range of 0.0001 m -~ to 0.1 m -1.

M E T H O D OF S O L U T I O N

Equat ions (3) and (4) are coupled through the source term in equation (4) and the boundary condition in equat ion (3). The derivation of the analytical solution for the depth-averaged liquid phase concentrat ion c(x, t) has been presented by the authors elsewhere [17]. The final result is given in the Appendix.

D A T A FOR S I M U L A T I O N

The following data appropr ia te for the radioactive waste disposal application has been used in the present study: d = 0.005 m, L = 1 m, Ds = 2.52 m2/yr, D(molecular) -- 25.2 m2/yr, u = 1 and 10 m/yr , interphase transfer coefficient k = 0, 10 -4 and 10 -3 m -~ and half-life = 100 and oo

yr. The saturating fluid is taken as water and fluid velocities correspond to normal groundwater velocities near the surface of the Earth. The longitudinal dispersion coefficient is calculated as

D -- D(molecular ) + au, where a is the dispersion length. A wide range of dispersion length scales are possible [18]. For a natural formation a will scale with its longitudinal dimension and so values of a = 0 and 10 m have been used in the present work.

R E S U L T S

Specific results with u, k and half-life as parameters are given below. When the half-life is o¢ the species is stable. This represents t ransport of passive scalars such as thermal energy. In the examples considered here, the dispersion length a is taken as zero except in the section where a - - 1 0 m. The porosity is 0.5% in the discussion below. However , calculations show that the trends are found to remain unaltered for a porosity up to 2.5 %.

Heat transfer

Function c(x, t) is now interpreted as the mean tempera ture of the fluid averaged over the fluid gap height. Figure 3(a) and (b) shows a plot of c(x, t) as a function of x when u -- 1 m/y r at t = 10 and 100 years respectively. Two values of k, namely 0 and 0.001 m -1, have been considered. For k = 0 , the time scale (defined in the Appendix) is T - - 6 0 y r and for k = 0.001 m -1, T = 2yr . Hence k - -0 .001 m -1 is a large interface coefficient. As seen in Fig.

176 K. M U R A L I D H A R and J. S W A R U P

1.0

0.8

0.6

c

0.4

0.2

t ~ i u = i~m/yr t

~ . t = I0 yr

\ I0 20 50 4 0 50

x , m 60

1.0

0.8

0.6 c

0.4

0.2

0

t =100

50 60 90 120 150 180 210 240

x~m

Fig. 3. C o n c e n t r a t i o n d i s t r ibu t ion of a s tab le spec ies as a funct ion of x (u = I m/y r ) . (a) t = 10 yr, (b) t = 100 yr.

3(a) and (b) a tempera ture front steadily moves through fluid region for k = 0. However when k > 0 the front becomes immobile for t > 10 yr. Under these conditions a steady state exists in

the porous medium with the concentration flux at x = 0 being completely passed into the solid medium. A transient may however be reinitiated in high porosity systems when the solid phase attains a spatially uniform temperature. Figure 4(a) and (b) shows the variation of c with time at x = 1 and 10 m respectively. When k > 0, heat transfer to the solid phase can be seen to lower the steady state tempera ture attained by the fluid. This reduction is prominent at large distances. As seen in Fig. 4(b) interphase transfer also sharply reduces the time scale and steady state is attained in 100 yr when k = 0 and 4 yr when k = 0.001 m -1.

Figure 5(a) and (b) shows a plot of c versus x at t = 10 and 100 yr when u = 10 m/yr. The

characteristic t ime scale for this problem is T = 1 yr when k = 0 and 0.67 yr when k = 0.001 m -1. The sharp reduction in T is related to the front-like movement of the tempera ture field when u is raised. Steady state can be expected to be reached at a characteristic distance X = ~ - 5 m for both k = 0 and k > 0 in 1 yr. This can be seen in Fig. 6(a) and (b). When k = 0 steady state represents the at ta inment of unit temperature. However for k > 0, c can be less than unity and will depend on the distance f rom the inflow plane. The affected region over which c varies f rom unity to nearly zero increases with u and t when k = 0. For k > 0 this increase is only marginal and beyond a certain period the size of the affected region becomes a constant.

Contaminated solid

(Half-life = 100yr, k =0 .001m-~ . ) In the problem considered here, the solid is initially contaminated ( c , > 0 ) and the incoming fluid is clean (c(0, t ) = 0 ) . Figure 7(a) shows concentrat ion profiles in the fluid phase at two different instants and for two different velocities


C

I ' 0 / ' , , , ~ j

0.8 x = l m

0 .6 (O) U = I m / y r

I I I I

0 I 2

t , yr

,o o zp 4p 6o 8p ,oo I~m-'

o a k ~ x=lOm

oL/ 0.1

I

0 I 2 :5 4 5

t , y r

Fig. 4. I n c r e a s e in c o n c e n t r a t i o n o f a s t ab le spec ies wi th t ime (u = 1 m / y r ) . (a) x = i m, (b) x = 10 m.

1.0

O.8

0.6 C 0.4

0.2

' ~ ' u'= i0 m/yr t = I0 yr \ \ore

\ ~o ~o ~o ~o ,oo ,~o , , o ,~o

x , m

x , m

800 880 960 1040 1[20 1200 I.O ~ , ' u =lOm/yr

0.8 ' ~ X t= lOOyr

o o \ ~ °~-' c

0.4i

0.2

0 I I I ~',,,.J ~""--- 20 40 60 80 I00

x~m

Fig. 5. C o n c e n t r a t i o n d i s t r i b u t i o n o f a s t ab le spec i e s as a f u n c t i o n o f x (u = 10 m / y r ) . (a) t = 10 yr , (b) t = 100 yr.


e l y i I/ \ k . o =

. 0.6 r ( o ) i i 0 0.2 0.4 0.6 0,8 1.0

t , y r

1.0 0.5 u =lOm/yr ' ' f x = I00 m /

0.8 / k = O m-I 9,4

0.6 0,3

C IOC 0 4 0.2

/ ~ I0- 3m-I 02 L ~ ~ ' " - !0.I y - (b)

J d r - i I I 1(3 0 4 8 12 16 2 ~

t , yr

Fig. 6. Increase in concentrat ion of a stable species with time (u = 10m/yr) . (a) x =1 m, (b) x = 100m.

I . O , i

I :i0 -3 m -I l O''S i 0,6

3

0.4 4

u i~yr t 02 io

Ioo 0 I00

o 2'0 40 ;o 8~ ,oo x~m

0.5 , , , 1.0 k k=lO_3m_ I

04 . . ~ u,m/yr x,m Q8 I I I0

~ 2 I0 I0 0.3 ~ \ , ~ , , ~.6

c \ \ \ ~ ,o , ,oc

0 I00 200 300 400 t, yr

Fig. 7. Contamina ted solid: (a) concentrat ion distribution as a function of x, (b) decay in concentrat ion with time.

I n t e r p h a s e h e a t a n d m a s s t r a n s f e r 179

(u = 1 and 10 m/yr) . At each time c increases from a zero value at the inflow boundary to attain an equilibrium value that matches the concentration level in the solid phase. The increase is rapid at higher velocities owing to the predominance of advection over diffusion. Figure 7(b) shows a plot of c versus time at two different x locations and two different velocities. At each location c increases rapidly from zero to attain a maximum value and this is subsequently followed by decay. Once the fluid concentration attains a value equal to the solid phase value the net transport (advection + diffusion) is reduced to zero and c(x, t) decays as exp (-At) . The solid and fluid phases will then be in equilibrium with each other. At smaller values of x the solid phase concentration is always larger than c(x, t) and the latter decays to zero under the combined effects of advection, diffusion, desorption and decay. Hence the onset of decay can be momentari ly delayed as seen in the curve marked 2 in Fig. 7(b).

Contaminated liquid at inflow plane

(Half-life = 100yr.) In the problem considered here, the liquid approaching the porous medium is contaminated (c(0, t) > 0) and the solid phase is initially clean (cs = 0). The effect of

fluid velocity and dispersion are separately discussed below. (i) u = lm/yr . Figure 8(a) and (b) shows concentration profiles in the fluid phase at times

10 and 100 yr respectively. Three different values of k equal to 0, 10 -4 and 10 -3 m -] have been considered. The retardation effects on the concentration profile become increasingly significant

I . O I I I I I

~ , . t = I0 yr 0.8 ~ u=lm/yr

o. , /o,

0 I0 20 30 40 50 60 x,m

0.5 ~ ~ I I t=llOOyr I

0.4 ~ ~ U = I m/yr

0.2

o., \

I t I 0 30 60 90 120 150 180 210 240

xtm

Fig. 8. Contaminated liquid at inflow plane: concentration distribution as a function o fx (u = 1 m/yr). (a) t = 10 yr , (b) t = 100 yr.


L O I I I

~. ---x=Im L_11~% x = I0 m

0.8 F-~'~ u = I mlyr

C Om-I

0.4 -- ~ ' ~ k =lO-3m-I

,o-.o-, -

0 200 400 600 t , y r

Fig. 9. Contaiminated liquid at inflow plane: decay in concentration with time (u = 1 m/yr).

at large t imes even when k is as smal l as 10 -4 m -~. F igure 9 shows the change in the m a g n i t u d e

of c wi th t ime at x = 1 and 10 m respec t ive ly . A t x = 1 m the so lu t ions for k = 0, 10 -4 and

1 0 - 3 m 1 a re nea r ly ident ica l . The m a g n i t u d e of c increases sha rp ly in the beg inn ing at bo th

x = 1 and 10 m s ignal l ing the ar r iva l of the c o n c e n t r a t i o n front . This is fo l lowed by a p rocess

tha t is d o m i n a t e d by r ad ioac t i ve decay . T h e rise t ime of c f rom zero is r a p i d and is no t shown in

Fig. 9. The p e a k va lue a t t a ined by c at a g iven loca t ion d r o p s with increas ing va lues of k

because of a d r o p in the af fec ted length ove r which c is non-ze ro .

(ii) u = 10 m/y r . F igu re 10(a) and (b) shows a p lo t of c(x, t) as a func t ion of x at t = 10 and

100yr . Va lues of k = 0 and 1 0 - 3 m 1 are cons ide r e d here . W h e n k = 0 the prof i le d isp lays a

f ron t - l ike b e h a v i o u r t hough the flat p o r t i o n of the curve adjusts i tself to account for r ad ioac t ive

decay . H o w e v e r for k = 10 -3 m ~ a nea r ly i m m o b i l e d i s t r ibu t ion is o b t a i n e d tha t is u n c h a n g e d

b e t w e e n 10 and 100 yr. T h e overa l l concen t r a t i on level r educes owing to r ad ioac t ive decay . A s

in the hea t t r ans fe r p r o b l e m the af fec ted reg ion with mass t rans fe r is a b o u t twice the size as

when u = 1 m / y r . F igu re 11 shows a p lo t of c aga ins t t ime at x -- 1 and 100m. A t x = l m the

so lu t ion for k = 0 and 10 -3 m -~ a re nea r ly ident ica l and mass t ransfe r effects a re no t significant.

The f ron t - l ike prof i le for c(x, t) at smal l x forces a r ap id a t t a i n m e n t of equ i l i b r i um b e t w e e n the

fluid and sol id phases and fu r the r p r even t s the poss ib i l i ty of r eve r se spec ies t ransfe r f rom the

sol id to the fluid phase . A t x = 100 m and k = 0.001 m ] the t rans ien ts a re long-l ived. H o w e v e r ,

1.0

0.8

0.6

C

0.4

800

0.2

I I I

~ ~ k = t =lOyr u = iO mlyr

Om-I

50 I00 150 ZOO

x,m

x , m 900 I000 I100 200

0.5 ~--'~ ~ i

0 " 4 t ~ ~ tu ==I?O0~;Y r"

0 20 40 60 0 8 0

x,m

Fig. 10. Contaminated liquid at inflow-plane: concentration distribution as a function of x (u = 10 m/yr). (a) t = 10 yr, (b) t = 100 yr.


1.0 i i i 1.0

~ t ~ u = I0 m/yr

0.8 0.8 ' % ~ - - - x = l m . . , o o m

0.6 '~"~ ~ ,~-3_-i 0.6 ~

c

0.4 0.4

0.2 0.2

1 [ I 0 I00 200 300 400

t , y r

Fig. 11. Contaminated liquid at inflow plane: decay in concentration with time (u = 10 m/yr).

this is not significant since the concentration levels have already fallen to a low value at this

location. (iii) a = 10 m. Data on dispersion length is scarce and exhibits scatter. There is some

evidence that o~ will scale with the overall size of the porous region rather than the average pore diameter. For this reason a = 10 m has been used in the present calculations. The velocity

u is 10 m/yr and the species half-life in the present discussion is 100 yr. Figure 12(a) and (b) shows a plot of c(x, t) versus x at t = 10 and 100 yr for k = 0 and 10 _3 m -1. For both values of k

700 0.5

0.4

0.3

C 0.2

O.i

1.0 i i i

~ . t =lOyr u = I0 m/yr

0.8 ~ ~t=lO m

0.6

C k=O

0 50 I00 150 200 x~m

x,m 800 900 I000 I100 1200 1300

' ' ', • Ioo y,' u-lO m/yr ~.= I0 m

k=O

20 40 60 80 I00 120 x~m

Fig. 12. Effect of dispersion: concentration as a function of distance. (a) t = 10 yr, (b) t = 100 yr.


1.0 i i = I i m 5 - - - - - - X

,\ ~ x : 1 0 0 m

~X J k = 0 u=lOm/yr 0.8 ~ 1 0 - 3 m-i ~Z__lO m 4

0.6 rk=O "'~..~ 3~ C _ _ o_

0.4

0.2

I I I 0 I00 2 0 0 3 0 0 4 0 0

t , y r

Fig. ]3. E f fec t o f d ispersion: concen t ra t ion as a func t ion o f t ime (u = 10 m / y r ) .

the effect of non-zero a is to stretch the affected length over which c is non-zero parallel to the flow direction. The corresponding results for a = 0 are shown in Fig. 10(a) and (b). The increase in the affected length is more pronounced when k = 0 and reduces when k is increased. For k = 0, the effect of dispersion is seen only in that part of the concentration profile where c changes with x. Over the portion of the curve where c is fiat, advection predominates and the result is independent of a. Hence the possible increase in the spreading of the concentration front due to dispersion in high velocity flows is offset by spreading by advection, especially at large times. When k > 0, increased spreading due to a non-zero a increases the exposed surface area of the solid phase and increased mass transfer. Hence the front becomes immobile once

again. Figure 13 shows a plot of c(x, t) versus time for x = 1 m and 100 m. The corresponding graph

for a = 0 is shown in Fig. 11. At x = 1 m advection is dominant and dispersion effects are not visible. Hence the profiles for k = 0 and 10-3m -~ nearly coincide. However at x = 100m dispersion leads to a slower build-up of concentration and a lower peak value. The total duration of the transients is identical to the problem where a is zero and is barely affected by

additional dispersion.

D I S C U S S I O N

Solutions of equations (3) and (4) display the following general trends. In the absence of mass transfer between the fluid and solid phases, k = 0 and the solid phase concentration if it is initially non-zero will uniformly decay with time. The fluid concentration will develop a distribution in the x-direction that changes with time. At increasing velocities the distribution approaches the shape of a front which translates with the same speed as the fluid. When the decay constant is non-zero the shape of the fluid concentration distribution continues to be determined by the relative importance of advection and diffusion, but the average concentration level at a point decreases with time. Hence the build-up of fluid concentration at a point and its consequent decay will depend on the distance of this point from the inflow plane, the magnitudes of the mean velocity, the longitudinal dispersion coefficient and the nuclide

half-life. In the presence of mass transfer into a clean solid (k > 0) a fraction of the species goes into

the solid thus reducing the extent of the affected region in the fluid phase. This in effect is a retardation mechanism for species transport in the liquid phase. Mass transfer effects are increasingly felt at large distances from the inflow plane. In heat transfer problems an equilibrium is possible between the fluid and the solid. This requires that the porosity be large


so that the solid can become saturated with the species being transported. The fluid and solid phases will then attain a uniform value that depends on the location in the porous medium. If the porosity is small, as in the problem studied here, a diffusion front moves into the solid with a speed proportional to X/-Ds/t. Since Ds is small the front becomes nearly immobile beyond a certain time. When steady state is reached the solid phase concentration Cs depends on the y-coordinate and the values of c S and C(x , y, t) match only at the solid-fluid interface. When steady state is reached in the fluid phase the heat/mass transfer at the inflow plane is balanced by that into the solid matrix. The latter decides the length of the region over which c is non-zero, being large when k is small and small when k is large. In transport of decaying species in an adsorbing porous medium the absorbed material and that present in the fluid simultaneously undergo decay with time. Hence solid-fluid equilibrium is not attained even after a long time.

In mass transfer from a contaminated solid into a clean flowing fluid equilibrium can be attained in the spatial direction. The solute concentration in the fluid which is zero at x = 0 increases with distance till it becomes equal to the solid concentration. At a given location the mass transfer into the fluid phase will eventually balance the transport due to advection and diffusion. The concentration in solid and fluid phases will decrease with time because of the finiteness of the species half-life, though at any instant, the concentration field will be spatially uniform.

C O N C L U S I O N S

1. Interphase heat/mass transfer of non-reacting solutes in a saturated porous medium leads to retardation of the temperature/concentrat ion front. For stable species, the extent of retardation is pronounced over a range of fluid velocities. It is practically unaffected by dispersion. It is also unaffected by a finite half-life, except that the respective values of the transported quantity diminish with time. 2. The approximation of equilibrium in concentration between the solid and the fluid phases is realized when the solid phase is contaminated and the incoming fluid is clean. It is not seen in the transport of a decaying species in a contaminated fluid flowing into a clean matrix. It is likely to be observed for heat transfer in high porosity systems. 3. In problems involving contaminated inflow with a decaying species the possibility of the solid phase concentration exceeding that of the fluid phase exists in principle. However, for porosities upto 2.5% studied here the solid phase concentration does not increase sufficiently for this effect to be visible. 4. Over the range of parameters studied, retardation dominates dispersion. Dispersion is expected to become important when the species is unretarded, for example when the porosity is high.

R E F E R E N C E S

[1] Broyd, T. W., in Nuclear Containment, ed. D. G. WALTON. Cambridge University Press, U.K., 1988. [2] Huyakorn, P. S., I_ester, B. H. and Mercer, J., Water Resources Res., 1983, 19, 841. [3] Kladius, N. and Prasad, V., Trans. ASME, J. Heat Transfer, 1990, 112, 675. [4] Neretnieks, I., J. Geophys. Res., 1980, 85, 4379. [5] Neretnieks, I., Eriksen, T. and Tahtinen, P., Water Resources Res., 1982. 18, 849. [6] Rubin, J., Water Resources Res., 1983, 19, 1231. [7] Bear, J., Dynamics of Fluids in Porous Media. Elsevier, New York, 1979. [8] Bear, J. and Bachmat, Y., Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic

Publishers, London, 1990. [9] Krishnaswamy, S., Graustein, W. C., Turekian, K. K. and Dowd, J. F., Water Resources Res. 1982, 18, 849.


[10] Vortmeyer , D. and Schaefer, R. J., Chem. Engng Sci., 1974, 29, 485. [11] Kim, S. Y., Kang, B. H. and Hyum, J. Y., lntJ. Heat Mass Transfer, 1994, 37, 2025. [12] Beji, H. and Gobin, D., Numer. Heat Transfer, Part A, 1992, 22, 487. [13] Kuznetsov, A. V., lnt J. Heat Mass Transfer, 1994, 37, 3030. [14] Tung, V. X. and Dhir, V. K., ASME J. Heat Transfer, 1993, 115, 503. [15] Genett i , W. E., in Handbook of Heat and Mass Transfer. Vol. I, ed. N. P. Cheremisinoff. Gulf Publishing Co.,

1986, p. 559. [16] Heggs, P. J., in Low Reynolds Number Flow Heat Exchangers, ed. S. Kakac, R. K. Shah and A. E. Bergles.

Hemisphere Publishing Corporation, New York, 1983, p. 341. [17] Muralidhar, K. and Swarup, J., ASME J. AppL Mech., 1994, 61, 740. [18] Dagan, G., Flow and Transport in Porous Formations. Springer, Berlin, 1989. [19] Krylov, V. I., and Kruglikova, L. G., Handbook of Numerical Harmonic Analysis. Israel Program for Scientific

Translat ions Ltd, 1969. [20] Muralidhar, K., Verghese, M. and Pillai, K. M., Numerical Heat Transfer, 1993, 23, 99.

(Received 5 October 1995; accepted 3 June 1996)

A P P E N D I X

The analytical solution of equations (3) and (4) given in the text for the depth-averaged concentrat ion in the liquid phase is given below. The following notat ion is introduced first.

The time scale for the present problem is denoted as T and the length scales as X and Y. It can be shown that

T = I A + - - + 4D

X : X / D T and Y = D~/~.,T.

In the following formulas, x, y and t refer to the dimensionless coordinates x/X, y / Y and t/T. Let w - DskT/d and b = uX/2D. In the following expressions, A refers to the product AT and k to kY.

Let /3 , , (m = 1, 2 . . . . ) be the roots of the equat ion/3 tan/3L = k. The norm associated with the eigenvalue problem in the solid phase is denoted as N(/3,,) and is given as

1 2(/32, + k e)

N(/3m ) L ( / 3 ] , + k 2 ) + k

and the eigenfunction Y is Y(flm, Y)= c o s f l m ( L - y ) . For the special case of k =0 , / 3 , , , - 0 is a root and the corresponding values of Y and 1IN are 1 and 1/L respectively. Let A m = k/[N(/3m)/3,,'~-~,~2,~2,n +k2]; again k = 0 is a special case and for/3, , = 0, Am = 1. Let

A,,/3,, 2 2 Cm

For the special case of k = 0, c,, = 1. Let U(x, t) stand for the composite function

U(x, t) ~wc~e bx-~l-a-h2)' d r e erfc /3 r = - e hx erfc . 3o t L 2 V t - r - m Cme "

+ e,,[ee' erfc t j+ Let ~ = 1 - ~m c,,;

p):[exp(-y-d l_exp( . . . L \ 4 ( t - r ) ] \ 4 ~ - ~ ) ] expt°~x p) ( l - A ) ( t - r ) )

and for i , j = 0, 1, . , ( M - I),

[ ,+I l ] F,j- i!(1-+xy + i! '~J,,t '~-T-- , ~ tzU(p, r ) r ' + ~,,,/3mC,,,r &'U(p, ¢hr)e ~''('b ')~d& .

Let r = (1 - x)/(1 + x) and s = (1 - t ) /( l + t). The coefficients a 0 are solutions of the system of coupled equations given by

f l l f l l (U(r,s) "K~ ~ ~ Tt(r)Tk(s)drds - r,jao ) l~i~_ r21X/~_ s2 = u ,

for each l, k = 0, 1 , . . . , (M - 1). Here Tt and Tk are lth and kth order Chebyshev polynomials in r and s respectively. Define the function Z(x, t) as

=.S, ~ t ' Z(x,t) ~ i ! ( l + x y "

The analytical solution of c(x, t) can now be written as

c(x, t) = U(x, t)Z(x, t )exp(-At) .

The summat ion involved in the solution obtained above has been carried out using 20 terms. Multiple integrals

Interphase heat and mass transfer

Table A1. Comparison of numerical and analytical solutions of c(x, t): u = 1 m/yr, k = 0.001 m -1, Half-life = 100yr

185

x(m) 0 2.5 5 10 15 20 30 40

t = 10 yr Analytical 0.932 0.69 0.51 0.278 0 . 1 5 1 0.082 0.032 0.0067 Numerical 0.932 0.67 0.49 0 . 2 6 5 0 . 1 4 5 0.079 0.023 0.0063

t(yr) 0 5 10 20 40 60 100 190 310 400

x = l m Analytical 0 0.854 0.827 0.77 0.678 0 . 6 3 5 0.474 0.23 0.097 0.049 Numerical 0 0.83 0.80 0.75 0.65 0.59 0.44 0.23 0.12 0.06

appearing in the definitions of U(x, t) and aij are evaluated using a ten-point quadrature formula [19]. The number of approximating functions for Z(x, t) is taken as 5, i.e. 25 different values of aij are computed. Size convergence of the solution with respect to these values has been independently tested and found to be good.

The analytical solution for c(x, t) has been compared against the finite difference solution of equations (3) and (4) using operator-splitting [20]. The numerical comparison between the two sets of results is shown in Table A1 for the property data given in the text and grid sizes of Ax = Ay = 0.I m and At = 0.1 yr. The comparison is seen to be satisfactory.

N O M E N C L A T U R E

c Depth-averaged concentration in liquid phase (dimensionless)

C Local concentration in the liquid phase co Initial concentration in the liquid phase c s Initial concentration in the solid phase c s Concentration in solid phase dp Particle diameter (m) d Equivalent opening of fluid gap in Cartesian

model (m) D Longitudinal dispersion coefficient in liquid

phase (m2/yr) D s Diffusion coefficient in solid phase (m2/yr) D t Transverse dispersion coefficient in liquid

phase (m2/yr) k Inter-phase heat/mass transfer coefficient

based on average liquid phase concentration c(x, t) (m -1)

K Inter-phase heat/mass transfer coefficient

based on local liquid phase concentration C(x, O, t) (m 1)

K r Reaction constant L Equivalent size of solid medium in Cartesian

model (m) u, v x and y components of liquid velocity (m/yr) x, y Cartesian coordinates

X, Y Length scales (m) t Time (yr)

T Time scale (yr) T 1 Chebyshev polynomial of order 1 a Dispersion length /3 Retardation factor

/3., Eigenvalue h Decay parameter = 0.693/half-life (yr -1)

Subscript x, t O/Ox, O/at

theoretical study of interphase heat and mass transfer in saturated porous media

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