- Email: [email protected]
Free convective heat and mass transfer flow induced by an instantaneous point source in an infinite porous medium
FLUIDDYNAMICS
RESEARCH
ELSEVIER
Fluid Dynamics
Research
14 (1994) 313-329
Free convective heat and mass transfer flow induced by an instantaneous point source in an infinite porous medium R. Ganapathy Department of Mathematics, National College, Tiruchirapalli-620 001, India Received
10 June 1993; revised manuscript
received 4 April 1994
A simple mathematical theory is proposed for the analysis of the buoyancy-driven heat and mass transfer flow induced by an instantaneous point source in an unbounded fluid-saturated porous medium of uniform porosity, assuming the validity of the Brinkman model. The theory consists of retaining only the leading terms of the series expansions of the dependent variables in terms of the thermal Rayleigh number and is valid within the limit of small Rayleigh numbers only. The heat generating rate is assumed to be not excessive, so that the induced flow is slow. The evolution of the flow field is demonstrated by drawing the streamlines at various times, and the results are delineated by comparing them with those of the Darcy flow model. The significance of the impact of species concentration gradients upon the thermally driven flow has been highlighted. Even though heat was specified to be one of the two diffusion mechanisms, the results apply as well to the case where the source simultaneously generates two different chemical components.
1. Introduction Free convective flow, due to the interaction of the force of gravity and density differences caused by the simultaneous diffusion of thermal energy and chemical species induced by concentrated sources in fluid-saturated porous media, is one of the fundamental subjects of study owing to its wide-ranging applications in a variety of fields such as chemical engineering, geophysics and nuclear science. In spite of such wide-ranging applications, the majority of the published works (Wooding, 1963; Bejan, 1978; Hickox and Watts, 1980; Hickox, 1981; Purushothaman et. al., 1990; Ganapathy and Purushothaman, 1990) are primarily concerned with the phenomenon wherein the driving buoyancy mechanism was induced by the temperature gradients alone. However, very often chemical species concentration gradients greatly affect the flow, and as a result they play a decisive role in the development of the temperature field. Although this class of problems has been to some extent established in the literature (see for instance, Poulikakos, 1985; Larsen and Poulikakos, 1986), most of them were primarily concerned with the Darcy flow model only. However, when the particle diameter is small but not vanishingly small (as in the case of Darcy 0169-5983/94/$9.25 0 1994 The Japan SSDI 0169-5983(94)00002-H
Society of Fluid Mechanics.
All rights reserved
314
R. Ganapat~y / Ffuid Dynamics Research 14 (1994) 313-329
model), then the Darcy law in its usual form is not adequate (Pop and Cheng, 1992), for in this case the medium involves larger void spaces, giving rise to viscous shear in addition to the Darcy resistance. To understand the onset of convection in such a medium, one has to take into account the viscous shear, however small it may be, in addition to the Darcy resistance, and this has not been done so far in respect to double diffusion around concentrated sources in saturated porous media. It is towards this end that we study here one such flow problem, the buoyancy-induced heat and mass transfer flow induced by an instantaneous point source in an unbounded fluid-saturated porous medium taking into account the viscous shear, and obtain an analytical solution using a perturbation analysis. Our interest in this topic is motivated by the fact that the analysis of such flows in essential for the solution of many engineering problems such as the spreading of a pollutant created by an exothermic reaction at an underground site, besides others. By considering the slow viscous flow through a swarm of spherical particles, Tam (1969) gave a theoretical justification of the validity of the Brinkman model which was later confirmed more rigorously by Lundgren (1972), among others. In fact, a full discussion on the merits of the Brinkman equation over Darcy equation can be found in Neale et al. (1973), Ooms et al. (19701,and Joseph and Tao (1964). In the present work, we therefore assume the validity of the Brinkman model and proceed with Boussinesq approximation. We further assume that the heat generating rate is not excessive, so that the induced flow is slow. As the Aow field is thermally weak, the thermal Rayleigh number, which is based on the source’s strength, is assumed to be small. Although the problem of double diffusion depends essentially on two Rayleigh numbers, one based on the thermal field and the other on the concentration field, for the sake of simplicity, we have formulated the problem in such a way that the contribution of species concentration in driving the flow is measured by a parameter N, and therefore our findings determine the impact of species concentration gradients upon the thermally driven flow, In essence, we develop the mathematical theory by assuming power series expansions for the dependent variables in terms of the thermal Rayleigh number, and our analysis is valid within the limit of small Rayleigh numbers only. Following Hickox (1981), we retain only the leading terms of the series expansions. As a summary of what is presented below, the problem is formulated in Sec. 2 and the solution is provided in Sec. 3, with a discussion in Sec. 4 of the results obtained. The evolution of the flow field is demonstrated by plotting the streamlines at various times, and the results are compared with the corresponding results for the case of a point heat source in an infinite medium. A comparison with the Darcy flow model is also provided. We conclude with a review of the results obtained.
2. Matbemati~al formuIation Consider a homogeneous and isotropic fluid-saturated porous medium of infinite extent. The medium is of uniform porosity and the fluid saturating the medium Boussinesq incompressible with the relation
P=PoCl -BV-
~o)-~c(c-co)l~
(1)
where p is the fluid density, /3 the thermal expansion coefficient, /& the concentration expansion coefficient, T the temperature, and C the species concentration; the subscript 0 denotes a reference state. It is further assumed that both the medium and the saturating fluid are in thermal
315
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
Fig. 1. Configuration
of interest.
Spherical-polar
coordinate
system (r, 4,O) with the concentrated
source at the origin.
equilibrium. A spherical-polar coordinate system (r, +,19) is chosen with the C$= 0 axis vertically upward (Fig. 1). A point source is located at the origin of the coordinate system, from which a quantity pcQ of heat is liberated instantaneously at time t = 0, together with a substance at the rate m [kgs- ‘1, where c is the specific heat. Then, neglecting the compressibility effects of the fluid, the equations for the conservation of mass, momentum, energy and species in the medium, based on the Brinkman model, are (Rudraiah et al., 1980): v.q=o,
(2)
dq/dt + (q.V)q odT/dt
= - p-w1
+ (q.V)T=
+ PgTir + pcgcir
aV2T+
lacyat +(q.v)c= Dv2c +
+ v*vq
- vr’q,
(3)
Qd(t)d(r),
(4)
m6(t)d(v),
(5)
where q = (u, u, w) is the mean filtration velocity, PI the pressure in excess of the hydrostatic value, CIthe effective thermal diffusivity of the fluid/porous matrix, D the species diffusivity in the medium when filled with fluid, v* the Brinkman viscosity, v the kinematic viscosity of the saturating fluid, g the acceleration due to gravity, K the medium permeability, A a unit vector along the 4 = 0 axis, i the porosity of the medium, and G the heat capacity ratio given by fJ = i + (1 - 4(P4S/(P4~
(6)
where (PC)~ and (pc)r denote the heat capacity of the solid and fluid phases, J(t)6(r) is the Dirac-Delta function satisfying the relation
respectively.
Here
f
s(t)s(r)dVdt
= 1,
(7)
ssv 0 where the volume
I/ includes
the source.
R. GanapathJj / Fluid Dynamics Research 14 (1994) 313-329
316
Since the vertical axis is parallel to the gravity vector, neither 8, nor the @-component of velocity appears in the analysis. Thus, taking advantage of the continuity equation, we now define a stream function \I/ such that U = ~2
algae,
u = (rsin+)-l a~/~~,
(8)
with ,U= cos& and eliminate the pressure terms in Eq. (3) by taking the curl. Introducing non-dimensional variables R = r/&, c,
t, = cLt/oK,
= (C -
~~)~~/~,
the
I& = Il//c&, T, = (T-
~~)~~/Q)
(9
where k is the thermal conductivity, we obtain for the conservation species in the non-dimensional form (after dropping the asterisk):
of momentum,
energy, and
D2 D2 - f & - y2 ) rl/ +
(11)
where ‘J = (v/v*)‘!~ and P = (~/y2)(v/a)
(Modified Prandtl number),
Ra = p g K Q/av k Le = a/D
(thermal Rayleigh number),
(Lewis number),
D2=-!?+(1--~2t$ (non-dimensional
operator).
(13)
While M( = A/g) represents the ratio of the matrix porosity to that of the heat capacity ratio, the parameter N( = ,!Scmk/,8Q D) measures the relative importance of the chemical and thermal diffusion in causing the density gradient which drives the flow. It is worth noting here that N is zero for no species diffusion, infinite for no thermal diffusion, positive for both the effects combining to drive the flow, and negative for the two effects opposed. Consequently, relative to the no-species generation limit (N = 0), positive values of N strengthen the flow, whereas negative values tend to weaken the same, for here the generated substance acts as a break on the thermally induced flow. Eqs. (lo)-(12) are solved subject to the initial conditions U=O,
V=O,
T=O,
C=O
att=O
(Ida)
317
R. Ganapathy 1 Fluid Dynamics Research 14 (1994) 313-329
and the boundary U-+0,
conditions
V-0,
T+O,
C-+0
asR+co,
v=duja+=aTja+=acja4=0 where
U and
at+=O,n,
V are the non-dimensional
(14b)
form of the velocity
components,
given by (U, V) =
(J’%W, 0). Th e origin, being the seat of the point source, is a singular point for the flow field, and hence the singularity in $ at the origin must be a minimum (Ganapathy and Purushothaman, 1990).
3. Solution Consistent with the hypothesis that the heat generating rate is not excessive, we seek a perturbation solution, by assuming power series expansions for I/I, T and C: (ti, T, C) = fRa”($.,
(15)
T,, CA.
0
Substituting Eq. equations for the conduction there then found from
(15) into Eqs.(lO)-(12) and collecting terms of equal powers in Ra, we obtain the solutions of $i, Ti and Ci(i = 0, 1,2, . . .). AS Ra = 0 corresponds to a state of pure will be no fluid motion, and hence we take rjo = 0. The functions To and Co are the solution of the equations
(16b) From
Carslaw
and Jaeger (1959), we have
1 To = -----exp( q7cp
- R2/4t),
(174
A3 co = ---exp( 8(nt)3’2
- A2R2/4t),
U7W
where A = (MLe) lj2 . The first convective solution of the equation D2 [D2 - P-l
d/at,-
in which the variables D2 - f; Substituting
correction
to the velocity
field is now found from the
Y~]I)~ = y2(1 - ,D’) [(R c?JT,/dR) + N(R X,/aR)]
can be separated
- y2 $1 = ~‘(1 - p2)f(R, >
(1W
by setting t).
(18b)
Eq. (18b) into Eq. (18a) and taking the Laplace
7” - 2flR2 = - (1/4~)fiR”~P~‘~[K_~,~(pR)
transform,
we obtain
+ NA7’2K_3,2(ApR)],
(19)
318
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
where p = 4, differentiation f=
K, is the modified Bessel function of the second kind of order n, the primes denote with respect to R, and fis the Laplace transform off given by
e-“‘f(U) du.
(20)
s0 The general solution of Eq. (19) is f= al(s)/R + az(s)R2 - (l/4~)~~-~‘zR1’2[K_~,2(~R) where ai and a2 are the constants of integration.
+ II~A~‘~K_~/~(A~R)],
(21)
Finally, setting
+I = y2(1 - ~~)Rr’~fr(R, 0,
(22)
we have from Eq. (18b), after taking the Laplace transform, R2fi
-t- Rfl,
- (W2R2 + 9/4)x = R312y(R, s),
(23)
where W2 = y2 + s/P and the overbar denoting the Laplace transform of the function under it. The complete solution of Eq. (23) is
+
&K3,2 (w-9_% s) dx, (24) 04 where 1, is the modified Bessel function of the first kind of order M.The boundary conditions ensure 13,2(
WR)
r
a, = u2 =
ff3
=
0,
u4
=
a3.
(251
On evaluating the integrals in Eq. (24) we obtain C(P + l/R) exp( - PR) - (W + l/R) exp( - WR)] NAP + l/R) exp( - ApR) -
(W
.
+ l/R) exp( - WR)] >
(26) With the aid of the table of Laplace transfo~ inverting Eq. (26) sin2 4
$1 = 4n
due to Campbell and Foster (1961), we have, after
y[F(R, t) f NA- I” G(R, t)-J
(27)
where F(R, t) = (l/yR)erfc(R/2$)
+ +(l - l/~R)exp(~R)erfc(R/2fi
- f(1 + l/yR)exp( - yR)erfc(R/2fi t 4 {(i&
- l/yR)exp[-
+ rfi)
- rfi)
(coyt - iR&)y]
[erfc(R/2$
+ iyfi)
319
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
- erfc(R/2* x [erfc(R/24
+ iy&X)] - iy&)
- (i&
+ l/yR)exp[
- erfc(R/2fi
- (wyt + iR&)y]
- iyJmpt)]j,
(27a)
whenP#landco=P/(l-P), F(R, t) = (l/y&)exp(-
R2/4t) + (l/JJR)erfc(R/2$)
+ $( 1 - l/yR) exp(yR)erfc(R/2& - +(l + l/yR)exp(
- (l/&/%)exp[-
(t + R2/4t)]
+ yj;)
- yR)erfc(R/2&
- y$),
W’b) when P = 1, G(R, t) = (l/yR)erfc(AR/2$)
+ *(l - l/yR)exp(R)erfc(R/2fi
- gl + l/yR)exp(
- R)erfc(R/2fi
x exp[ - (co,@ - iAR&)y]
[erfc(AR/Z$
- erfc(R/2fi
+ iA?,/*)]
- (iA&
x exp[ - (~,yt
+ iAR&)y]
[erfc(AR/2$
- erfc(R/2fi when A,/?
- rfi)
+ yfi) + f{(iA&
- l/yR)
+ iyfi) + l/yR) - iy&)
- iAy,/a)]},
(27~)
# 1 and co, = P/(1 - A2P),
G(R, t) = (l/yR)erfc(AR/Z&)
+ (A/y,,&)exp(
x exp[ - (Pt + R2/4Pt)] - &(l + l/yR)exp(
+ i(l
- A2R2/4t)
- (l/ye)
- l/yR) exp(yR) erfc(R/2fi
- yR)erfc(R/Zfi
+ yfi)
- yfi),
when Afi = 1. The complementary error function with complex arguments evaluated numerically with the help of a power series expansion given by Strand
(27( in Eq. (27) is (1965).
4. Discussion The solution for $r reveals parameters have entered the flow characteristics. When N man (1990), where no species solution for $i reduces to $I = (sin2&4z)(l
the fact that due to the existence of species concentration several new problem, each having its own significant influence on the essential = 0 we recover the results obtained by Ganapathy and Purushothadiffusion was considered, and in the special case when A = 1, the
+ N)F(R,
t),
(28)
where F has been previously defined. In order to exemplify the results, in this case, we demonstrate the evolution of the flow field by plotting the streamlines at different times, choosing y = 1. In
320
R. Gunapathy 1 Fluid D_vnamics Research 14 (1994) 313-329
(b)
(a)
-0.001 -0003
R -25
01
-007
5
:::
Fig. 2. Transient
free convection
flow pattern.
Curves represent
4n$l/(f
+ N) = Const., (a) t = 0.1 (b) t = 0.2 (c) t = 1.0.
R. ~an~~a~~~y / Fluid dynamics
Research I4 (1994)
313-329
321
illustration, we have taken P = 0.7 and plotted the streamlines 47~rl/~/‘(l+ N) = const. for three different times (Fig. 2), from which it can be observed that in the initial stages of the flow development, the streamlines in the vicinity of the source are in the form of closed loops, whose geometry considerably change at higher values oft, although the symmetry about the /A= 0 axis is preserved at all times. Near the origin the streamlines come close together, illustrating the fact that the velocity is infinite at the source and as t increases, the flow pattern present near the source spreads outward, filling the entire space. In fact, during the period immediately following the release of the source’s energy, the flow pattern has a dipole structure and the streamlines in the vicinity of the source are in the form of circular vortices which propagate slowly in relation to their rate of growth; the entire process is dominated by viscous, species, and thermal diffusion. The slight bulging of the streamlines near the source indicates the impulsive effect of the source on the fluid particles located in its vicinity and is created by the fluid particles coverging toward the source in order to replace the fluid particles driven upward under the action of buoyancy. There is no accumulation of heat and chemical species into the vortex ring, and the flow in and around the vortices remains laminar, which is analogous to buoyant plumes that rise from steady sources of heat or other sources of positive buoyancy for overall small Rayleigh numbers in pure fluids (see Morton, 1960). These streamlines of the circulating flow field enclose the stagnation points (points of minimum $) which recede to infinity along the p = 0 axis as t increases, the rate of recession being much faster in the case of larger values of P. Should this be taken as an estimate of the rate of evolution of the flow field in response to changes in the temperature and concentration fields, then we are led to the conclusion that the flow field could be set up only instantaneously if P is very large (P % 1). The distribution of heat and chemical species in the flow field is such that the temperature and species concentration decrease outwards from the centre in all directions with a symmetry about the horizontal plane through the source (Eq. (17)). Among the several new parameters that have entered the solution, the parameter A has a striking effect on the transient flow pattern, especially when the two buoyancy mechanisms are opposed (N < 0). In particular, values of A of order less than unity give life for a downward flow far from the source. This phenomenon makes sense if we realize that A represents the ratio of two different length seales, namely the ratio of thermal penetration length, O(G), to that of species penetration, O(e). Consequently, values of A less than unity imply that the chemical species diffuses faster than that of heat at early times. Hence, outside the region in which the thermal effect of the source is felt - that is, outside the region of O(m - the chemical species concentration gradients act alone, creating a downward flow. In order to exemplify this, we have drawn in Fig. 3 the maps of streamlines 47~\C/~ = const. at three different times, choosing ‘/ = 1, P = 0.7, A = 0.1 and N = - 0.5 (Gebhart and Pera, 1971), from which it is deducible that in the initial stages of flow development the streamlines in the vicinity of the source are in the form of closed loops which are surrounded by a downward flow. However, the geometry considerably changes at higher values of t, although the symmetry about the plane /L = 0 is preserved at all times. As in the previous case (A = l), here too we find that in the early stages the streamlines of the circulating flow field come close together near the origin; but as t increases the flow pattern present near the source spreads outwards, driving the envefoping downward flow to move farther away from the source, giving way for viscous and thermal diffusion to dominate in the ultimate stage. Consequently, the flow field for large times resembles that due to the presence of an instantaneous point heat source embedded in
322
R. Ganapathy
/ Fluid Dynamics
Research 14 (1994) 313-329
-001 -003 R - 05
-0.07 -01
05 ~~
Fig. 3. Transient
streamlines.
Curves represent
4ntj,
= const., (a) t = 0.1 (b) t = 0.2 (c) t = 1.0.
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
323
an unbounded medium (see Ganapathy and Purushothaman, 1990), although the enveloping downward flow field persists farther away from the source even after a long time has passed. Though the coefficient of viscosity in the viscous term is in general different from v, the pore fluid value (Cheng, 1978), it has been a common practice to take them to be equal (Hinch, 1977; Howells, 1974). In fact Lundgren (1972), while attempting to predict CL,,the effective viscosity in the porous medium for a dilute swarm of stationary solid spheres has shown that pL*+ ,uo (the fluid viscosity), as the porosity of the medium tends to unity. Brinkman (1947) too favored setting ,u* = p. because this choice provided the closet agreement between his predictions for the permeability of a swarm of solid spheres and experimental data. This has been confirmed by Koplik et al. (1983), who, although putting forward arguments that ,u* # ,u~, observed poorer agreement with experimental data than when setting p * = ,uo. More recently, Masliyah et al. (1987) have shown that their analytical solution for the creeping flow past a composite sphere with ,u* = p. is in excellent agreement with experimental results on the settling rates of a solid sphere with attached threads (adsorbed long chain polymers). Thus the choice y = 1 is in order for a medium in which the porosity is not too small, which is exactly the situation in the present problem. It is worth noting here that the function G(R, t) appearing in Eq. (27) is a direct consequence of the presence of species concentration, whereas the function F(R, t) is due to thermal diffusion only. In order to have a comprehensive estimate of the overall convection effects produced due to the presence of chemical species, we have drawn, in Fig. 4, the streamlines due to double diffusion
Fi
-a
Fig. 4. Transient
free convection
flow pattern.
Curves represent 47~)~ = Const. _ _ _ without species.
at t = 1.0;
with
species,
around the source for a selected value of G(R, t) at a common time, and compared them with the corresponding ones which arise due to pure thermal diffusion only. In illustration we choose P = 0.7 and plot the streamlines 47c$i = const. at time t = 1.0, by assigning NA-“2G(R, t) an arbitrarily chosen values -0.05 and identify the corresponding streamlines with common alphabets, choosing y = 1. It can be observed from the figure that the rate of momentum transfer is faster when the species concentration gradients act along with temperature gradient than in the absence of chemical species. Albeit tacitly, this then implies that the process of momentum transfer is accelerated by the presence of chemical species, a phenomenon that is characteristic of fluid motion with high Prandtl number. Not withstanding the time for the onset of convection, as the flow field in respect of other values of P as well that of y is similar to the one envisaged in Fig. 3, we forgo the analysis concerning these values and omit the respective graphs for the sake of brevity.
5. Darcy flow We have for the Darcy flow model (Cheng, 1978) the equation for the conservation men turn: 02$=Ra(l-$)
,l’:T-Rg+iV
,
dP
of mo-
(29)
where the variables have been previously defined. Proceeding as before and omitting the details for the sake of brevity, we have $1 = (sin2~/4~)[(l/~)e~p(+ NA [(i/fi)exp(
R2/4t) + (l/R)erfc(R/2~)] - A2R2/4t) + (1//1R)erfc(AR/Zt:i;)],
(30)
from which it is evident that not only is the Prandtl number dependence lost but also the emergence of different wave forms which is characteristic of the Brinkman model is completely eliminated. One can observe from Eq. (30) that the stream function 11/idoes not depend on R and t individually but through the single non-dimensional variable R/2,,/% Setting r~= R/2,/-t, the alternative representation for $i becomes $1 = (siu2~i471~){l(l/t/;;)cxp(
4 NA[(l/&)exp(
- Y’) + (V%)erfc(v)l
- A2r2) + (1~2~~)erfc(~~)]~.
(31)
In order to exemplify the explicit difference between the solutions obtained with Darcy’s law and the corresponding Brinkman extension, we have drawn in Fig. 5 the radial velocity graphs based on the Brinkman modification with those based on the Darcy model, for Brinkman viscosities different from the fluid viscosity, choosing R = 1.0, P = 1.0, N = - 0.5 and A = 0.5. From the figure we infer that the flow fields differ substantially at large distances from the origin and that the rate of momentum transfer is larger in the case of the Darcy flow than with the Brinkman extension. Physically, this is expected since the Darcy flow driven by the body force that arises from the buoyancy effects in the fluid does not account for the viscous resistance due to distortion of the velocity field. Further, in the Darcy formulation, the local acceleration term was left out of the
325
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
Fig. 5. Magnitude
of the radial
velocity
at R = 1.0; ____ Darcy flow, (b) y = 0.5 (c) y = 1.0.
Brinkman
extension.
(a) y = 0.25
momentum equations, even though the solution derived from these equations was later used to determine the flow field due to a time-dependent temperature field and species concentration. Albeit tacitly, this then implies that the Prandtl number is high, which is otherwise reflected in the velocity profiles associated with the Darcy flow. Thus, theoretically, it is the contribution of this local acceleration term to that part of the solution obtained for the Brinkman model that is mainly responsible for driving the Brinkman term so as to inflict such strong differences between the two types of flow. These arguments neither dispute the applicability of the Darcy flow model nor claim that the Darcy model gives rise to steady-state solutions only; they only confirm the fact that for a medium in which the particle diameter is not too small, besides the inclusion of viscous shear in the momentum equations, the retention of the local acceleration term is crucial if all the salient features of the flow, especially in the early stages of the flow development, are to be included in the solution for small and moderate values of the Prandtl number. One of the striking features of the flow in respect of the Brinkman model is that the magnitude of the radial velocity tends to increase as y decreases, which is suggestive of the fact that the rate of momentum transfer tends to be faster when the Brinkman viscosity exceeds its pore fluid value
R. Ganapathy
326
/ Fluid Dynamics
Research
1 i
c; bl 30-
*O
9
I I ’ I I I i I
I 1 I 1
14 (1994)
313-329
/
/ I
/
1 I ’ I ’ I ’ I 10-I 1 I:
Fig. 6. Profiles
showing
the effect of the parameter N on the velocity field. (a) N = - 0.5 (b) N = 0 (c) N = 0.5; Darcy flow, ~_ Brinkman extension.
(v* > $1). If this heuristic argument applies, then we are led to the conclusion that the difference between the Brinkman viscosity and the fluid viscosity should be large enough to attain the limit of the Darcy flow model. With a view to showing the effect of the parameter N on the velocity field for both the models, we have drawn in Fig. 6 the velocity profiles at a radial distance R = 1.0 for three different values of N:N = - 0.5, N = 0, N = 0.5, choosing A = 0.5, P = 1, and y = 1. From the figure it is deducible that, in respect of both the models, there is a stronger impact of the source on the velocity field when the two buoyancy mechanisms combine to drive the flow than in the opposite case, relative to the no species generation limit (N = 0). This is otherwise expected, since positive values of N strengthen the flow whereas negative values tend to weaken it. One could observe from Figs. 5 and 6 that in respect of Darcy flow, when the two buoyancy mechanisms are opposed, soon after the release of the source’s energy, at positions farther away from the source, the radial velocity tends to be negative so that back flow sets in for a small duration, striking a balance between momentum and species diffusion. This should be otherwise expected, for, when A is of order less than one, the rate of diffusion of chemical species is faster than that of heat at early times, so that, outside the region of thermal activity, species concentration gradients act alone, creating
R. Gana~a~hy ,! Fluid Dynamics Research I4 (1994) 313-329
327
a downward flow which in turn is responsible for the occurrence of back flow. Although the above descriptions are equally true in respect of the Brinkman model, in the absence of the resistance offered by viscous forces, the phenomenon becomes more pronounced in the Darcy flow than with Brinkman extension. Finally, it is worth noting that the solutions based on Darcy’s law as well as the Brinkman extension exhibit spatial symmetry about the plane # = 7r/2.
6. Concluding remarks We have presented an analytical study of natural convection from a heat and chemical species generating source embedded in an infinite porous medium of uniform porosity, whose particle diameter is small but not vanishingly small. To simplify the analysis, we have assumed the source to be concentrated at one point, and for the same reason we have formulated the problem in such a way that the contribution of species concentration gradients in driving the flow is measured by a parameter N. consequently, our findings determine the impact of species concentration gradients upon the thermally driven fIow. Of special importance is the finding that a downward flow surrounding the circular vortex near the source comes to life when the two buoyancy mechanisms are opposed, especially when the parameter A is of order one or less. This range of values of A is in fact appropriate for a variety of physical and laboratory models, for instance in the case of diffusion of methanol and hydrogen in water-saturated glass beads, A < 0.5. The solution we have obtained in Sec. 3 for the stream function is valid for small Rayleigh numbers only. It is indeed the Green’s function for this class of problems, and its utility derives from the fact that the flow fields due to sources in the presence of solid boundaries can be synthesized from this result, using the method of images or some well-known reflection techniques. Similarly, from this solution, solution to problems where the heat-generating rate is a general but known function of space and time, in bounded and unbounded domains, can be synthesized using known techniques. As the flow field is thermally weak, the solution for $1 is expected to give a reasonably good picture of the free convection flow induced by the compact isolated source; and in the absence of stability effects, the behavior of the flow is unlikely to change radically for moderate values of Ra also. When the modified Prandtl number is very small - that is, when P 6 1, with a different non-dimensionalization scheme - it can be shown that tl/r is the exact solution for the flow field. When the two buoyancy mechanisms are opposed, the presence of chemical species is found to accelerate the process of momentum diffusion. It is worth recording here that, unlike the region of net upward flow wherein the convective corrections to the temperature and concentration fields produce an increase in the temperature and species concentration for points in the upper half space (0 < # d lt/2) accompanied by an equal decrease in the temperature and concentration in the lower half space, if an isothermal region of downward flow similar to the one shown in Fig. 3 exists, then the lower half-space will be richer in chemical species than the upper one. Among the several new parameters that have entered the solution, Lewis number has a stronger impact on the species concentration field than on the temperature field [Eq. (17b)]. Even though heat was specified to be one of the two diffusion mechanisms, our results apply as well to the case of buoyancy-induced flow from a concentrated source generating simultaneously two different chemical components, and the results are valid only in the diffusion - dominated regime, Eq. (18a), which we have derived from the full momentum equation as the first approximation in the limit of small Rayleigh number, is
R. Gunapathy 1 Fluid Dynamics Research 14 (1994) 313-329
328
tantamount to the Stokes flow approximation, which is consistent with our hypothesis that the induced flow is slow. On account of the fact that the temperature distribution is assumed to be unaffected by the fluid motion, we have not presented the results in terms of a Nusselt number, since their utility in the assessment of the overall convection effects is rather limited. Lastly, we emphasize here that the permeability K is related to the porosity 3, of the medium by K =
d2A3/A(1 - iy,
(32)
where d is the particle diameter and A is the Ergun constant. Hence, for a medium in which the particle diameter is not vanishingly small as in the case of sand, a 40-50% porosity is expected (Scheidegger, 1963), so that K may assume values in the range 10v3d2 and 3 x 10p3d2 when A = 180. Although this range of values of K is not commensurate with the range of values appropriate for the Darcy flow model, it is not large enough to warrant incIusion of the dispersion effects, however small they may be, in the analysis (see for instance Pop and Cheng, 1992).
Acknowledgements
The author is thankful to the referees for their many useful suggestions which led to a definite improvement of the paper.
References A. Bejan (1978) Natural convection in an infinite porous medium with a concentrated heat source, J. Fluid. Mech. 89,97. H.C. Brinkman (1947) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A I, 27. G.A. Campbell and R.M. Foster (1961) Fourier integrals for Practical Applications (Van Nostrand, Princeton). H.S. Carslaw and J.E. Jaeger (1959) C~n~~c~~o~ ofWent in Solids (Oxford Univ. Press, Oxford) p. 256. P. Cheng (1978) Advances in Heat Transfer, Vol. 14 (Academic Press, New York). R. Ganapathy and R. Purushothaman (1990) Thermal convection from an instantaneous point heat source in a porous medium, Int. J. Eng. Sci. 28, 907. B. Gebhart and L. Pera (1971) The nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion, Int. J. Heat Mass Transfer, 14, 2025. C.E. Hickox (1981) Thermal convection at low Rayleigh number from concentrated sources in porous media, Trans. ASME J. Heat Transf: 103, 232. C.E. Hickox and H.A. Watts (1980) Steady thermal Convection from a concentrated source in a porous medium, Trans. ASME, J. Heat Trans- 102, 248. E.J. Hinch (1977) An averaged equation approach to particle interaction in a fluid suspension, J. Fluid. Mech. 83, 695. I.D. Howells (1974) Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects, J. Fluid Me&. 64, 449. D.D. Joseph and L.N. Tao (1964) The effect of permeabjlity on the slow motion of a porous sphere in a viscous liquid, ZAMM 44, 361. J. Koplik, H. Levine and A. Zee (1983) Viscosity re-normahsation in the Brinkman equation, Phys. Fluids 26, 2864. S.E. Larsen and D. Pouhkakos (1986) Double diffusion from a horizontal line source in an infinite porous medium, Int. J. Heat Mass Transfer, 29, 492. T.S. Lundgren (1972) Slow flow through stationary random beds and suspensions of spheres, J. Fluid Mech. 551,273.
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
329
J.H. Masliyah, G. Neale, K. Malysa and T.G.M. Van de ven (1987) Creeping flow over a composite sphere: Solid core with porous shell, Chem. Engg. Sci., 42(2), 245. B.R. Morton (1960) Weak thermal vortex rings, .I. Fluid Mech. 9, 107. G. Neale, N. Epstein and W. Nader (1973) Creeping flow relative to permeable spheres, Chem. Eng. Sci. 28,1865; 29 (1974) 1352 (erratum). G. Ooms, P.F. Mijnlieff and H.L. Beckers (1970) Frictional Force exerted by a flowing field on a permeable particle with particular reference to polymer coils, J. Chem. Phys. 53, 4123. I. Pop and P. Cheng (1992) Flow past a circular cylinder embedded in a porous medium based on the Brinkman model, Int. J. Eng. Sci. 30(2), 2.57. D. Poulikakos (1985) On buoyancy induced heat and mass transfer from a concentrated source in an infinite porous medium, Int. J. Heat Mass Transfer, 28, 621. R. Purushothaman, R. Ganapathy and P.S. Hiremath (1990) Free convection in an infinite porous medium due to a pulsating point heat source, ZAMM, 70, 41. N. Rudraiah, B. Veerappa and S. Balachandra Rao (1980) Effects of non-uniform thermal gradient and adiabatic boundaries on convection in porous media, Trans. ASME J. Heat Transfer 102, 254. A.E. Scheidegger (1963) The Physics ofpow through porous Media (Univ. of Toronto Press) p. 13. O.N. Strand (1965) A method of the computation of the error function of a complex variable, Math. Comput. 19, 127. C.K.W. Tam (1969) The drag on a cloud of spherical particles in low Reynolds number flow, J. Fluid Me&. 38, 537. R.A. Wooding (1963) Convection in a saturated porous medium, at large Rayleigh number of Peclet number, J. Fluid Mech. IS, 527.
RESEARCH
ELSEVIER
Fluid Dynamics
Research
14 (1994) 313-329
Free convective heat and mass transfer flow induced by an instantaneous point source in an infinite porous medium R. Ganapathy Department of Mathematics, National College, Tiruchirapalli-620 001, India Received
10 June 1993; revised manuscript
received 4 April 1994
A simple mathematical theory is proposed for the analysis of the buoyancy-driven heat and mass transfer flow induced by an instantaneous point source in an unbounded fluid-saturated porous medium of uniform porosity, assuming the validity of the Brinkman model. The theory consists of retaining only the leading terms of the series expansions of the dependent variables in terms of the thermal Rayleigh number and is valid within the limit of small Rayleigh numbers only. The heat generating rate is assumed to be not excessive, so that the induced flow is slow. The evolution of the flow field is demonstrated by drawing the streamlines at various times, and the results are delineated by comparing them with those of the Darcy flow model. The significance of the impact of species concentration gradients upon the thermally driven flow has been highlighted. Even though heat was specified to be one of the two diffusion mechanisms, the results apply as well to the case where the source simultaneously generates two different chemical components.
1. Introduction Free convective flow, due to the interaction of the force of gravity and density differences caused by the simultaneous diffusion of thermal energy and chemical species induced by concentrated sources in fluid-saturated porous media, is one of the fundamental subjects of study owing to its wide-ranging applications in a variety of fields such as chemical engineering, geophysics and nuclear science. In spite of such wide-ranging applications, the majority of the published works (Wooding, 1963; Bejan, 1978; Hickox and Watts, 1980; Hickox, 1981; Purushothaman et. al., 1990; Ganapathy and Purushothaman, 1990) are primarily concerned with the phenomenon wherein the driving buoyancy mechanism was induced by the temperature gradients alone. However, very often chemical species concentration gradients greatly affect the flow, and as a result they play a decisive role in the development of the temperature field. Although this class of problems has been to some extent established in the literature (see for instance, Poulikakos, 1985; Larsen and Poulikakos, 1986), most of them were primarily concerned with the Darcy flow model only. However, when the particle diameter is small but not vanishingly small (as in the case of Darcy 0169-5983/94/$9.25 0 1994 The Japan SSDI 0169-5983(94)00002-H
Society of Fluid Mechanics.
All rights reserved
314
R. Ganapat~y / Ffuid Dynamics Research 14 (1994) 313-329
model), then the Darcy law in its usual form is not adequate (Pop and Cheng, 1992), for in this case the medium involves larger void spaces, giving rise to viscous shear in addition to the Darcy resistance. To understand the onset of convection in such a medium, one has to take into account the viscous shear, however small it may be, in addition to the Darcy resistance, and this has not been done so far in respect to double diffusion around concentrated sources in saturated porous media. It is towards this end that we study here one such flow problem, the buoyancy-induced heat and mass transfer flow induced by an instantaneous point source in an unbounded fluid-saturated porous medium taking into account the viscous shear, and obtain an analytical solution using a perturbation analysis. Our interest in this topic is motivated by the fact that the analysis of such flows in essential for the solution of many engineering problems such as the spreading of a pollutant created by an exothermic reaction at an underground site, besides others. By considering the slow viscous flow through a swarm of spherical particles, Tam (1969) gave a theoretical justification of the validity of the Brinkman model which was later confirmed more rigorously by Lundgren (1972), among others. In fact, a full discussion on the merits of the Brinkman equation over Darcy equation can be found in Neale et al. (1973), Ooms et al. (19701,and Joseph and Tao (1964). In the present work, we therefore assume the validity of the Brinkman model and proceed with Boussinesq approximation. We further assume that the heat generating rate is not excessive, so that the induced flow is slow. As the Aow field is thermally weak, the thermal Rayleigh number, which is based on the source’s strength, is assumed to be small. Although the problem of double diffusion depends essentially on two Rayleigh numbers, one based on the thermal field and the other on the concentration field, for the sake of simplicity, we have formulated the problem in such a way that the contribution of species concentration in driving the flow is measured by a parameter N, and therefore our findings determine the impact of species concentration gradients upon the thermally driven flow, In essence, we develop the mathematical theory by assuming power series expansions for the dependent variables in terms of the thermal Rayleigh number, and our analysis is valid within the limit of small Rayleigh numbers only. Following Hickox (1981), we retain only the leading terms of the series expansions. As a summary of what is presented below, the problem is formulated in Sec. 2 and the solution is provided in Sec. 3, with a discussion in Sec. 4 of the results obtained. The evolution of the flow field is demonstrated by plotting the streamlines at various times, and the results are compared with the corresponding results for the case of a point heat source in an infinite medium. A comparison with the Darcy flow model is also provided. We conclude with a review of the results obtained.
2. Matbemati~al formuIation Consider a homogeneous and isotropic fluid-saturated porous medium of infinite extent. The medium is of uniform porosity and the fluid saturating the medium Boussinesq incompressible with the relation
P=PoCl -BV-
~o)-~c(c-co)l~
(1)
where p is the fluid density, /3 the thermal expansion coefficient, /& the concentration expansion coefficient, T the temperature, and C the species concentration; the subscript 0 denotes a reference state. It is further assumed that both the medium and the saturating fluid are in thermal
315
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
Fig. 1. Configuration
of interest.
Spherical-polar
coordinate
system (r, 4,O) with the concentrated
source at the origin.
equilibrium. A spherical-polar coordinate system (r, +,19) is chosen with the C$= 0 axis vertically upward (Fig. 1). A point source is located at the origin of the coordinate system, from which a quantity pcQ of heat is liberated instantaneously at time t = 0, together with a substance at the rate m [kgs- ‘1, where c is the specific heat. Then, neglecting the compressibility effects of the fluid, the equations for the conservation of mass, momentum, energy and species in the medium, based on the Brinkman model, are (Rudraiah et al., 1980): v.q=o,
(2)
dq/dt + (q.V)q odT/dt
= - p-w1
+ (q.V)T=
+ PgTir + pcgcir
aV2T+
lacyat +(q.v)c= Dv2c +
+ v*vq
- vr’q,
(3)
Qd(t)d(r),
(4)
m6(t)d(v),
(5)
where q = (u, u, w) is the mean filtration velocity, PI the pressure in excess of the hydrostatic value, CIthe effective thermal diffusivity of the fluid/porous matrix, D the species diffusivity in the medium when filled with fluid, v* the Brinkman viscosity, v the kinematic viscosity of the saturating fluid, g the acceleration due to gravity, K the medium permeability, A a unit vector along the 4 = 0 axis, i the porosity of the medium, and G the heat capacity ratio given by fJ = i + (1 - 4(P4S/(P4~
(6)
where (PC)~ and (pc)r denote the heat capacity of the solid and fluid phases, J(t)6(r) is the Dirac-Delta function satisfying the relation
respectively.
Here
f
s(t)s(r)dVdt
= 1,
(7)
ssv 0 where the volume
I/ includes
the source.
R. GanapathJj / Fluid Dynamics Research 14 (1994) 313-329
316
Since the vertical axis is parallel to the gravity vector, neither 8, nor the @-component of velocity appears in the analysis. Thus, taking advantage of the continuity equation, we now define a stream function \I/ such that U = ~2
algae,
u = (rsin+)-l a~/~~,
(8)
with ,U= cos& and eliminate the pressure terms in Eq. (3) by taking the curl. Introducing non-dimensional variables R = r/&, c,
t, = cLt/oK,
= (C -
~~)~~/~,
the
I& = Il//c&, T, = (T-
~~)~~/Q)
(9
where k is the thermal conductivity, we obtain for the conservation species in the non-dimensional form (after dropping the asterisk):
of momentum,
energy, and
D2 D2 - f & - y2 ) rl/ +
(11)
where ‘J = (v/v*)‘!~ and P = (~/y2)(v/a)
(Modified Prandtl number),
Ra = p g K Q/av k Le = a/D
(thermal Rayleigh number),
(Lewis number),
D2=-!?+(1--~2t$ (non-dimensional
operator).
(13)
While M( = A/g) represents the ratio of the matrix porosity to that of the heat capacity ratio, the parameter N( = ,!Scmk/,8Q D) measures the relative importance of the chemical and thermal diffusion in causing the density gradient which drives the flow. It is worth noting here that N is zero for no species diffusion, infinite for no thermal diffusion, positive for both the effects combining to drive the flow, and negative for the two effects opposed. Consequently, relative to the no-species generation limit (N = 0), positive values of N strengthen the flow, whereas negative values tend to weaken the same, for here the generated substance acts as a break on the thermally induced flow. Eqs. (lo)-(12) are solved subject to the initial conditions U=O,
V=O,
T=O,
C=O
att=O
(Ida)
317
R. Ganapathy 1 Fluid Dynamics Research 14 (1994) 313-329
and the boundary U-+0,
conditions
V-0,
T+O,
C-+0
asR+co,
v=duja+=aTja+=acja4=0 where
U and
at+=O,n,
V are the non-dimensional
(14b)
form of the velocity
components,
given by (U, V) =
(J’%W, 0). Th e origin, being the seat of the point source, is a singular point for the flow field, and hence the singularity in $ at the origin must be a minimum (Ganapathy and Purushothaman, 1990).
3. Solution Consistent with the hypothesis that the heat generating rate is not excessive, we seek a perturbation solution, by assuming power series expansions for I/I, T and C: (ti, T, C) = fRa”($.,
(15)
T,, CA.
0
Substituting Eq. equations for the conduction there then found from
(15) into Eqs.(lO)-(12) and collecting terms of equal powers in Ra, we obtain the solutions of $i, Ti and Ci(i = 0, 1,2, . . .). AS Ra = 0 corresponds to a state of pure will be no fluid motion, and hence we take rjo = 0. The functions To and Co are the solution of the equations
(16b) From
Carslaw
and Jaeger (1959), we have
1 To = -----exp( q7cp
- R2/4t),
(174
A3 co = ---exp( 8(nt)3’2
- A2R2/4t),
U7W
where A = (MLe) lj2 . The first convective solution of the equation D2 [D2 - P-l
d/at,-
in which the variables D2 - f; Substituting
correction
to the velocity
field is now found from the
Y~]I)~ = y2(1 - ,D’) [(R c?JT,/dR) + N(R X,/aR)]
can be separated
- y2 $1 = ~‘(1 - p2)f(R, >
(1W
by setting t).
(18b)
Eq. (18b) into Eq. (18a) and taking the Laplace
7” - 2flR2 = - (1/4~)fiR”~P~‘~[K_~,~(pR)
transform,
we obtain
+ NA7’2K_3,2(ApR)],
(19)
318
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
where p = 4, differentiation f=
K, is the modified Bessel function of the second kind of order n, the primes denote with respect to R, and fis the Laplace transform off given by
e-“‘f(U) du.
(20)
s0 The general solution of Eq. (19) is f= al(s)/R + az(s)R2 - (l/4~)~~-~‘zR1’2[K_~,2(~R) where ai and a2 are the constants of integration.
+ II~A~‘~K_~/~(A~R)],
(21)
Finally, setting
+I = y2(1 - ~~)Rr’~fr(R, 0,
(22)
we have from Eq. (18b), after taking the Laplace transform, R2fi
-t- Rfl,
- (W2R2 + 9/4)x = R312y(R, s),
(23)
where W2 = y2 + s/P and the overbar denoting the Laplace transform of the function under it. The complete solution of Eq. (23) is
+
&K3,2 (w-9_% s) dx, (24) 04 where 1, is the modified Bessel function of the first kind of order M.The boundary conditions ensure 13,2(
WR)
r
a, = u2 =
ff3
=
0,
u4
=
a3.
(251
On evaluating the integrals in Eq. (24) we obtain C(P + l/R) exp( - PR) - (W + l/R) exp( - WR)] NAP + l/R) exp( - ApR) -
(W
.
+ l/R) exp( - WR)] >
(26) With the aid of the table of Laplace transfo~ inverting Eq. (26) sin2 4
$1 = 4n
due to Campbell and Foster (1961), we have, after
y[F(R, t) f NA- I” G(R, t)-J
(27)
where F(R, t) = (l/yR)erfc(R/2$)
+ +(l - l/~R)exp(~R)erfc(R/2fi
- f(1 + l/yR)exp( - yR)erfc(R/2fi t 4 {(i&
- l/yR)exp[-
+ rfi)
- rfi)
(coyt - iR&)y]
[erfc(R/2$
+ iyfi)
319
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
- erfc(R/2* x [erfc(R/24
+ iy&X)] - iy&)
- (i&
+ l/yR)exp[
- erfc(R/2fi
- (wyt + iR&)y]
- iyJmpt)]j,
(27a)
whenP#landco=P/(l-P), F(R, t) = (l/y&)exp(-
R2/4t) + (l/JJR)erfc(R/2$)
+ $( 1 - l/yR) exp(yR)erfc(R/2& - +(l + l/yR)exp(
- (l/&/%)exp[-
(t + R2/4t)]
+ yj;)
- yR)erfc(R/2&
- y$),
W’b) when P = 1, G(R, t) = (l/yR)erfc(AR/2$)
+ *(l - l/yR)exp(R)erfc(R/2fi
- gl + l/yR)exp(
- R)erfc(R/2fi
x exp[ - (co,@ - iAR&)y]
[erfc(AR/Z$
- erfc(R/2fi
+ iA?,/*)]
- (iA&
x exp[ - (~,yt
+ iAR&)y]
[erfc(AR/2$
- erfc(R/2fi when A,/?
- rfi)
+ yfi) + f{(iA&
- l/yR)
+ iyfi) + l/yR) - iy&)
- iAy,/a)]},
(27~)
# 1 and co, = P/(1 - A2P),
G(R, t) = (l/yR)erfc(AR/Z&)
+ (A/y,,&)exp(
x exp[ - (Pt + R2/4Pt)] - &(l + l/yR)exp(
+ i(l
- A2R2/4t)
- (l/ye)
- l/yR) exp(yR) erfc(R/2fi
- yR)erfc(R/Zfi
+ yfi)
- yfi),
when Afi = 1. The complementary error function with complex arguments evaluated numerically with the help of a power series expansion given by Strand
(27( in Eq. (27) is (1965).
4. Discussion The solution for $r reveals parameters have entered the flow characteristics. When N man (1990), where no species solution for $i reduces to $I = (sin2&4z)(l
the fact that due to the existence of species concentration several new problem, each having its own significant influence on the essential = 0 we recover the results obtained by Ganapathy and Purushothadiffusion was considered, and in the special case when A = 1, the
+ N)F(R,
t),
(28)
where F has been previously defined. In order to exemplify the results, in this case, we demonstrate the evolution of the flow field by plotting the streamlines at different times, choosing y = 1. In
320
R. Gunapathy 1 Fluid D_vnamics Research 14 (1994) 313-329
(b)
(a)
-0.001 -0003
R -25
01
-007
5
:::
Fig. 2. Transient
free convection
flow pattern.
Curves represent
4n$l/(f
+ N) = Const., (a) t = 0.1 (b) t = 0.2 (c) t = 1.0.
R. ~an~~a~~~y / Fluid dynamics
Research I4 (1994)
313-329
321
illustration, we have taken P = 0.7 and plotted the streamlines 47~rl/~/‘(l+ N) = const. for three different times (Fig. 2), from which it can be observed that in the initial stages of the flow development, the streamlines in the vicinity of the source are in the form of closed loops, whose geometry considerably change at higher values oft, although the symmetry about the /A= 0 axis is preserved at all times. Near the origin the streamlines come close together, illustrating the fact that the velocity is infinite at the source and as t increases, the flow pattern present near the source spreads outward, filling the entire space. In fact, during the period immediately following the release of the source’s energy, the flow pattern has a dipole structure and the streamlines in the vicinity of the source are in the form of circular vortices which propagate slowly in relation to their rate of growth; the entire process is dominated by viscous, species, and thermal diffusion. The slight bulging of the streamlines near the source indicates the impulsive effect of the source on the fluid particles located in its vicinity and is created by the fluid particles coverging toward the source in order to replace the fluid particles driven upward under the action of buoyancy. There is no accumulation of heat and chemical species into the vortex ring, and the flow in and around the vortices remains laminar, which is analogous to buoyant plumes that rise from steady sources of heat or other sources of positive buoyancy for overall small Rayleigh numbers in pure fluids (see Morton, 1960). These streamlines of the circulating flow field enclose the stagnation points (points of minimum $) which recede to infinity along the p = 0 axis as t increases, the rate of recession being much faster in the case of larger values of P. Should this be taken as an estimate of the rate of evolution of the flow field in response to changes in the temperature and concentration fields, then we are led to the conclusion that the flow field could be set up only instantaneously if P is very large (P % 1). The distribution of heat and chemical species in the flow field is such that the temperature and species concentration decrease outwards from the centre in all directions with a symmetry about the horizontal plane through the source (Eq. (17)). Among the several new parameters that have entered the solution, the parameter A has a striking effect on the transient flow pattern, especially when the two buoyancy mechanisms are opposed (N < 0). In particular, values of A of order less than unity give life for a downward flow far from the source. This phenomenon makes sense if we realize that A represents the ratio of two different length seales, namely the ratio of thermal penetration length, O(G), to that of species penetration, O(e). Consequently, values of A less than unity imply that the chemical species diffuses faster than that of heat at early times. Hence, outside the region in which the thermal effect of the source is felt - that is, outside the region of O(m - the chemical species concentration gradients act alone, creating a downward flow. In order to exemplify this, we have drawn in Fig. 3 the maps of streamlines 47~\C/~ = const. at three different times, choosing ‘/ = 1, P = 0.7, A = 0.1 and N = - 0.5 (Gebhart and Pera, 1971), from which it is deducible that in the initial stages of flow development the streamlines in the vicinity of the source are in the form of closed loops which are surrounded by a downward flow. However, the geometry considerably changes at higher values of t, although the symmetry about the plane /L = 0 is preserved at all times. As in the previous case (A = l), here too we find that in the early stages the streamlines of the circulating flow field come close together near the origin; but as t increases the flow pattern present near the source spreads outwards, driving the envefoping downward flow to move farther away from the source, giving way for viscous and thermal diffusion to dominate in the ultimate stage. Consequently, the flow field for large times resembles that due to the presence of an instantaneous point heat source embedded in
322
R. Ganapathy
/ Fluid Dynamics
Research 14 (1994) 313-329
-001 -003 R - 05
-0.07 -01
05 ~~
Fig. 3. Transient
streamlines.
Curves represent
4ntj,
= const., (a) t = 0.1 (b) t = 0.2 (c) t = 1.0.
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
323
an unbounded medium (see Ganapathy and Purushothaman, 1990), although the enveloping downward flow field persists farther away from the source even after a long time has passed. Though the coefficient of viscosity in the viscous term is in general different from v, the pore fluid value (Cheng, 1978), it has been a common practice to take them to be equal (Hinch, 1977; Howells, 1974). In fact Lundgren (1972), while attempting to predict CL,,the effective viscosity in the porous medium for a dilute swarm of stationary solid spheres has shown that pL*+ ,uo (the fluid viscosity), as the porosity of the medium tends to unity. Brinkman (1947) too favored setting ,u* = p. because this choice provided the closet agreement between his predictions for the permeability of a swarm of solid spheres and experimental data. This has been confirmed by Koplik et al. (1983), who, although putting forward arguments that ,u* # ,u~, observed poorer agreement with experimental data than when setting p * = ,uo. More recently, Masliyah et al. (1987) have shown that their analytical solution for the creeping flow past a composite sphere with ,u* = p. is in excellent agreement with experimental results on the settling rates of a solid sphere with attached threads (adsorbed long chain polymers). Thus the choice y = 1 is in order for a medium in which the porosity is not too small, which is exactly the situation in the present problem. It is worth noting here that the function G(R, t) appearing in Eq. (27) is a direct consequence of the presence of species concentration, whereas the function F(R, t) is due to thermal diffusion only. In order to have a comprehensive estimate of the overall convection effects produced due to the presence of chemical species, we have drawn, in Fig. 4, the streamlines due to double diffusion
Fi
-a
Fig. 4. Transient
free convection
flow pattern.
Curves represent 47~)~ = Const. _ _ _ without species.
at t = 1.0;
with
species,
around the source for a selected value of G(R, t) at a common time, and compared them with the corresponding ones which arise due to pure thermal diffusion only. In illustration we choose P = 0.7 and plot the streamlines 47c$i = const. at time t = 1.0, by assigning NA-“2G(R, t) an arbitrarily chosen values -0.05 and identify the corresponding streamlines with common alphabets, choosing y = 1. It can be observed from the figure that the rate of momentum transfer is faster when the species concentration gradients act along with temperature gradient than in the absence of chemical species. Albeit tacitly, this then implies that the process of momentum transfer is accelerated by the presence of chemical species, a phenomenon that is characteristic of fluid motion with high Prandtl number. Not withstanding the time for the onset of convection, as the flow field in respect of other values of P as well that of y is similar to the one envisaged in Fig. 3, we forgo the analysis concerning these values and omit the respective graphs for the sake of brevity.
5. Darcy flow We have for the Darcy flow model (Cheng, 1978) the equation for the conservation men turn: 02$=Ra(l-$)
,l’:T-Rg+iV
,
dP
of mo-
(29)
where the variables have been previously defined. Proceeding as before and omitting the details for the sake of brevity, we have $1 = (sin2~/4~)[(l/~)e~p(+ NA [(i/fi)exp(
R2/4t) + (l/R)erfc(R/2~)] - A2R2/4t) + (1//1R)erfc(AR/Zt:i;)],
(30)
from which it is evident that not only is the Prandtl number dependence lost but also the emergence of different wave forms which is characteristic of the Brinkman model is completely eliminated. One can observe from Eq. (30) that the stream function 11/idoes not depend on R and t individually but through the single non-dimensional variable R/2,,/% Setting r~= R/2,/-t, the alternative representation for $i becomes $1 = (siu2~i471~){l(l/t/;;)cxp(
4 NA[(l/&)exp(
- Y’) + (V%)erfc(v)l
- A2r2) + (1~2~~)erfc(~~)]~.
(31)
In order to exemplify the explicit difference between the solutions obtained with Darcy’s law and the corresponding Brinkman extension, we have drawn in Fig. 5 the radial velocity graphs based on the Brinkman modification with those based on the Darcy model, for Brinkman viscosities different from the fluid viscosity, choosing R = 1.0, P = 1.0, N = - 0.5 and A = 0.5. From the figure we infer that the flow fields differ substantially at large distances from the origin and that the rate of momentum transfer is larger in the case of the Darcy flow than with the Brinkman extension. Physically, this is expected since the Darcy flow driven by the body force that arises from the buoyancy effects in the fluid does not account for the viscous resistance due to distortion of the velocity field. Further, in the Darcy formulation, the local acceleration term was left out of the
325
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
Fig. 5. Magnitude
of the radial
velocity
at R = 1.0; ____ Darcy flow, (b) y = 0.5 (c) y = 1.0.
Brinkman
extension.
(a) y = 0.25
momentum equations, even though the solution derived from these equations was later used to determine the flow field due to a time-dependent temperature field and species concentration. Albeit tacitly, this then implies that the Prandtl number is high, which is otherwise reflected in the velocity profiles associated with the Darcy flow. Thus, theoretically, it is the contribution of this local acceleration term to that part of the solution obtained for the Brinkman model that is mainly responsible for driving the Brinkman term so as to inflict such strong differences between the two types of flow. These arguments neither dispute the applicability of the Darcy flow model nor claim that the Darcy model gives rise to steady-state solutions only; they only confirm the fact that for a medium in which the particle diameter is not too small, besides the inclusion of viscous shear in the momentum equations, the retention of the local acceleration term is crucial if all the salient features of the flow, especially in the early stages of the flow development, are to be included in the solution for small and moderate values of the Prandtl number. One of the striking features of the flow in respect of the Brinkman model is that the magnitude of the radial velocity tends to increase as y decreases, which is suggestive of the fact that the rate of momentum transfer tends to be faster when the Brinkman viscosity exceeds its pore fluid value
R. Ganapathy
326
/ Fluid Dynamics
Research
1 i
c; bl 30-
*O
9
I I ’ I I I i I
I 1 I 1
14 (1994)
313-329
/
/ I
/
1 I ’ I ’ I ’ I 10-I 1 I:
Fig. 6. Profiles
showing
the effect of the parameter N on the velocity field. (a) N = - 0.5 (b) N = 0 (c) N = 0.5; Darcy flow, ~_ Brinkman extension.
(v* > $1). If this heuristic argument applies, then we are led to the conclusion that the difference between the Brinkman viscosity and the fluid viscosity should be large enough to attain the limit of the Darcy flow model. With a view to showing the effect of the parameter N on the velocity field for both the models, we have drawn in Fig. 6 the velocity profiles at a radial distance R = 1.0 for three different values of N:N = - 0.5, N = 0, N = 0.5, choosing A = 0.5, P = 1, and y = 1. From the figure it is deducible that, in respect of both the models, there is a stronger impact of the source on the velocity field when the two buoyancy mechanisms combine to drive the flow than in the opposite case, relative to the no species generation limit (N = 0). This is otherwise expected, since positive values of N strengthen the flow whereas negative values tend to weaken it. One could observe from Figs. 5 and 6 that in respect of Darcy flow, when the two buoyancy mechanisms are opposed, soon after the release of the source’s energy, at positions farther away from the source, the radial velocity tends to be negative so that back flow sets in for a small duration, striking a balance between momentum and species diffusion. This should be otherwise expected, for, when A is of order less than one, the rate of diffusion of chemical species is faster than that of heat at early times, so that, outside the region of thermal activity, species concentration gradients act alone, creating
R. Gana~a~hy ,! Fluid Dynamics Research I4 (1994) 313-329
327
a downward flow which in turn is responsible for the occurrence of back flow. Although the above descriptions are equally true in respect of the Brinkman model, in the absence of the resistance offered by viscous forces, the phenomenon becomes more pronounced in the Darcy flow than with Brinkman extension. Finally, it is worth noting that the solutions based on Darcy’s law as well as the Brinkman extension exhibit spatial symmetry about the plane # = 7r/2.
6. Concluding remarks We have presented an analytical study of natural convection from a heat and chemical species generating source embedded in an infinite porous medium of uniform porosity, whose particle diameter is small but not vanishingly small. To simplify the analysis, we have assumed the source to be concentrated at one point, and for the same reason we have formulated the problem in such a way that the contribution of species concentration gradients in driving the flow is measured by a parameter N. consequently, our findings determine the impact of species concentration gradients upon the thermally driven fIow. Of special importance is the finding that a downward flow surrounding the circular vortex near the source comes to life when the two buoyancy mechanisms are opposed, especially when the parameter A is of order one or less. This range of values of A is in fact appropriate for a variety of physical and laboratory models, for instance in the case of diffusion of methanol and hydrogen in water-saturated glass beads, A < 0.5. The solution we have obtained in Sec. 3 for the stream function is valid for small Rayleigh numbers only. It is indeed the Green’s function for this class of problems, and its utility derives from the fact that the flow fields due to sources in the presence of solid boundaries can be synthesized from this result, using the method of images or some well-known reflection techniques. Similarly, from this solution, solution to problems where the heat-generating rate is a general but known function of space and time, in bounded and unbounded domains, can be synthesized using known techniques. As the flow field is thermally weak, the solution for $1 is expected to give a reasonably good picture of the free convection flow induced by the compact isolated source; and in the absence of stability effects, the behavior of the flow is unlikely to change radically for moderate values of Ra also. When the modified Prandtl number is very small - that is, when P 6 1, with a different non-dimensionalization scheme - it can be shown that tl/r is the exact solution for the flow field. When the two buoyancy mechanisms are opposed, the presence of chemical species is found to accelerate the process of momentum diffusion. It is worth recording here that, unlike the region of net upward flow wherein the convective corrections to the temperature and concentration fields produce an increase in the temperature and species concentration for points in the upper half space (0 < # d lt/2) accompanied by an equal decrease in the temperature and concentration in the lower half space, if an isothermal region of downward flow similar to the one shown in Fig. 3 exists, then the lower half-space will be richer in chemical species than the upper one. Among the several new parameters that have entered the solution, Lewis number has a stronger impact on the species concentration field than on the temperature field [Eq. (17b)]. Even though heat was specified to be one of the two diffusion mechanisms, our results apply as well to the case of buoyancy-induced flow from a concentrated source generating simultaneously two different chemical components, and the results are valid only in the diffusion - dominated regime, Eq. (18a), which we have derived from the full momentum equation as the first approximation in the limit of small Rayleigh number, is
R. Gunapathy 1 Fluid Dynamics Research 14 (1994) 313-329
328
tantamount to the Stokes flow approximation, which is consistent with our hypothesis that the induced flow is slow. On account of the fact that the temperature distribution is assumed to be unaffected by the fluid motion, we have not presented the results in terms of a Nusselt number, since their utility in the assessment of the overall convection effects is rather limited. Lastly, we emphasize here that the permeability K is related to the porosity 3, of the medium by K =
d2A3/A(1 - iy,
(32)
where d is the particle diameter and A is the Ergun constant. Hence, for a medium in which the particle diameter is not vanishingly small as in the case of sand, a 40-50% porosity is expected (Scheidegger, 1963), so that K may assume values in the range 10v3d2 and 3 x 10p3d2 when A = 180. Although this range of values of K is not commensurate with the range of values appropriate for the Darcy flow model, it is not large enough to warrant incIusion of the dispersion effects, however small they may be, in the analysis (see for instance Pop and Cheng, 1992).
Acknowledgements
The author is thankful to the referees for their many useful suggestions which led to a definite improvement of the paper.
References A. Bejan (1978) Natural convection in an infinite porous medium with a concentrated heat source, J. Fluid. Mech. 89,97. H.C. Brinkman (1947) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A I, 27. G.A. Campbell and R.M. Foster (1961) Fourier integrals for Practical Applications (Van Nostrand, Princeton). H.S. Carslaw and J.E. Jaeger (1959) C~n~~c~~o~ ofWent in Solids (Oxford Univ. Press, Oxford) p. 256. P. Cheng (1978) Advances in Heat Transfer, Vol. 14 (Academic Press, New York). R. Ganapathy and R. Purushothaman (1990) Thermal convection from an instantaneous point heat source in a porous medium, Int. J. Eng. Sci. 28, 907. B. Gebhart and L. Pera (1971) The nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion, Int. J. Heat Mass Transfer, 14, 2025. C.E. Hickox (1981) Thermal convection at low Rayleigh number from concentrated sources in porous media, Trans. ASME J. Heat Transf: 103, 232. C.E. Hickox and H.A. Watts (1980) Steady thermal Convection from a concentrated source in a porous medium, Trans. ASME, J. Heat Trans- 102, 248. E.J. Hinch (1977) An averaged equation approach to particle interaction in a fluid suspension, J. Fluid. Mech. 83, 695. I.D. Howells (1974) Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects, J. Fluid Me&. 64, 449. D.D. Joseph and L.N. Tao (1964) The effect of permeabjlity on the slow motion of a porous sphere in a viscous liquid, ZAMM 44, 361. J. Koplik, H. Levine and A. Zee (1983) Viscosity re-normahsation in the Brinkman equation, Phys. Fluids 26, 2864. S.E. Larsen and D. Pouhkakos (1986) Double diffusion from a horizontal line source in an infinite porous medium, Int. J. Heat Mass Transfer, 29, 492. T.S. Lundgren (1972) Slow flow through stationary random beds and suspensions of spheres, J. Fluid Mech. 551,273.
R. Ganapathy / Fluid Dynamics Research 14 (1994) 313-329
329
J.H. Masliyah, G. Neale, K. Malysa and T.G.M. Van de ven (1987) Creeping flow over a composite sphere: Solid core with porous shell, Chem. Engg. Sci., 42(2), 245. B.R. Morton (1960) Weak thermal vortex rings, .I. Fluid Mech. 9, 107. G. Neale, N. Epstein and W. Nader (1973) Creeping flow relative to permeable spheres, Chem. Eng. Sci. 28,1865; 29 (1974) 1352 (erratum). G. Ooms, P.F. Mijnlieff and H.L. Beckers (1970) Frictional Force exerted by a flowing field on a permeable particle with particular reference to polymer coils, J. Chem. Phys. 53, 4123. I. Pop and P. Cheng (1992) Flow past a circular cylinder embedded in a porous medium based on the Brinkman model, Int. J. Eng. Sci. 30(2), 2.57. D. Poulikakos (1985) On buoyancy induced heat and mass transfer from a concentrated source in an infinite porous medium, Int. J. Heat Mass Transfer, 28, 621. R. Purushothaman, R. Ganapathy and P.S. Hiremath (1990) Free convection in an infinite porous medium due to a pulsating point heat source, ZAMM, 70, 41. N. Rudraiah, B. Veerappa and S. Balachandra Rao (1980) Effects of non-uniform thermal gradient and adiabatic boundaries on convection in porous media, Trans. ASME J. Heat Transfer 102, 254. A.E. Scheidegger (1963) The Physics ofpow through porous Media (Univ. of Toronto Press) p. 13. O.N. Strand (1965) A method of the computation of the error function of a complex variable, Math. Comput. 19, 127. C.K.W. Tam (1969) The drag on a cloud of spherical particles in low Reynolds number flow, J. Fluid Me&. 38, 537. R.A. Wooding (1963) Convection in a saturated porous medium, at large Rayleigh number of Peclet number, J. Fluid Mech. IS, 527.