natural convection from point source embedded in darcian porous medium

Fluid Dynamics Research 6 (1990) 175-184 175 North-Holland

Natural convection from point source embedded in Darcian porous medium

Noor Afzal 1

Department of Mechanical Engineering, Aligarh Muslim University, Aligarh 202001, India

and

M. Yahya Salam Department of Mechanical Engineering Kuwait University, P.O. Box 5969, Kuwait

Received 1 December 1989

Abstract. The buoyancy induced convection arising from a point heat source in a fluid saturated Darcy porous medium bounded by an adiabatic conical surface, of apex angle 28,~, has been studied. The point heat source is located at the apex and the axis of the conical surface coincides with the direction of buoyant force vector. The self-similar equations are obtained for 0 < 0,~ < ~r and Rayleigh number Ra from zero to infinity. The numerical solutions of self-similar equations have been obtained for 9,~ = ~r (free plume), 3~r/4, ,n/2 (plume bounded by horizontal surface), and ~r/4. It is shown that for a given Ra, the centre line velocity and temperature increases as 0,o (the bounding space) decreases. A perturbation solution for Ra --, 0 has also been studied to present closed form solution.

1. Introduction

The mechanics of b u o y a n c y induced convec t ion a r o u n d an i so la ted hea t source in a f luid sa tura ted porous med ia is of interest in mode l l ing of geo the rmal processes.

I t was shown by W o o d i n g (1963) tha t the so lu t ion for s teady s ta te convec t ion f rom a line or po in t thermal source in a Darc i an po rous med ium, at large Rayle igh numbers , can be ob ta ined f rom direct ana log with Schl icht ing p l a n a r or ax i symmet r i c l amina r j e t (Schlicht ing, 1968). The work of W o o d i n g employed the b o u n d a r y layer a p p r o x i m a t i o n at large Ray le igh number s Ra and is valid in the region far away f rom the source.

The po in t heat source at low Rayle igh number s has been s tudied by Bejan (1978) in terms of a regular pe r t u rba t i on expans ion in the powers of Ray le igh number . F o r s t e a d y s tate convec- t ion results to o rde r Ra 3 and for t rans ient convec t ion to o rder R a were ob ta ined . The work of Bejan was based on a local hea t t ransfer cond i t i on in the i m m e d i a t e n e i g h b o u r h o o d of the source ra ther than g lobal heat f lux condi t ion . As a consequence his resul ts are val id to o rder R a

as R a ~ 0. Hickox and W a t t s [4] for the po in t hea t source s tud ied the numer ica l so lu t ions for Rayle igh

numbers f rom 0.1 to 100 in two s i tuat ions of u n b o u n d e d m e d i u m and semi- inf in i te m e d i u m b o u n d e d by an insu la ted hor izonta l plane. Thei r s tudy cons idered d i f fe ren t fo rmula t ions of self-similar equat ions for the two cases. The self-s imilar var iables e m p l o y e d for u n b o u n d e d

i Current address: AI-Fateh University, Engineering Academy P.O. Box 13406, Tripoli, Libya.

0169-5983/90/$3.50 © 1990 - The Japan Society of Fluid Mechanics

176 N. Afzal, M. }~ Salam / Convection in Darcian porous medium

medium were due to Squire (1951) and for semi-infinite medium were due to Yih (1965). It is desirable to have a unified formulation for the two cases.

The present work deals with the buoyancy induced convection arising from a point heat source in fluid saturated porous medium bounded by an adiabatic conical surface of arbitrary apex angle (26~). The point heat source is located at the apex and the axis of the conical surface coincides with the direction of buoyant force vector. In terms of similarity variables the self-similar equations are obtained for arbitrary values of Oo, and Ra. In the two limiting cases Ra = 0 and 1 / R a = 0 the closed form solutions have been obtained. For Ra ~ 0, a perturbation solution in the powers of Ra is obtained. The numerical solutions to the general self-similar equations have been obtained for entire range of Ra from zero to infinity for the various values of semi-apex angle 0~, in the range 0 < #~, < ~r. The special cases of 0,0 = ~r correspond to the plume in an unbounded medium and 6,o = ¢r/2 to the plume in a semi-infinite medium bounded by a horizontal surface.

The buoyancy induced flow, in an unbounded medium arising from a point source generating simultaneously heat and mass transfer was studied by Poulikakos (1985) in a power series in Ra. This work like the work of Bejan (1978) was also based on local heat and mass transfer conditions in the immediate neighbourhood of the source. In the present work an analysis of heat and mass transfer based on global conditions is described in the appendix.

The problem of a two dimensional line source in a Darcian porous medium is not amenable to self-similar treatment. By employing the method of matched asymptotic expansions Afzal (1985) has considered the third-order boundary layer effects.

2. Governing equations

The equations governing the steady state buoyancy induced motion in a Darcian porous medium are

div U = O, (1)

I~U = - k [ - v p + ~,13(T- To)], (2)

U" grad T = et V 2T. (3)

Here U is the velocity vector, T is the temperature, p is the pressure difference between actual static pressure and local hydrostatic pressure, T O is the temperature of reference state and ~ is the gravitational acceleration vector. Further, t~, ~t, 13 and k s t a n d for thermal diffusivity, molecular viscosity of fluid, volumetric thermal expansion coefficient and permeability of porous media respectively.

The spherical polar coordinate system (r, 0, ~) with axis /9 = 0 pointing vertically upwards shown inflg, l a is considered and the point heat source is situated at the origin. The porous medium is bounded by an adiabatic cone (of apex angle 28,~) with its apex at the origin of coordinates and its axis coinciding with the vertical upward direction 0 = 0.

The special case of 8,0 = ~r corresponds to a plane emerging in an infinite expanse of porous media, and 60, ~- ~r/2 to the plume bounded by a horizontal adiabatic surface. As the problem is symmetric in the g-direction around the vertical axis, the motion is independent of q~-coordinate, and the velocity vector U has only two components: u, the radial component in r-direction and v, the angular component in 6-direction. The stream function if' that satisfies the continuity equation (1) in spherical polar coordinates is defined as

1 i~,/, 1 3 ~ u = r 2 sin 0 30 ' v = r E sin 0 Or ' (4)

N. Afzal, M. Y. Salam / Convection in Darcian porous medium 177

{b} ) Fig. 1. Axisymmetric plume bounded by a conical adiabatic surface which coincides with one of the coordinate surfaces 0 = 0,o in a spherical coordinate system and axis coincides with the direction of buoyant force vector. The point heat source is at the apex of coincal surface. (a) Impermeable bounding surface: heat transfer. (b) Bounding surface permeable at apex: heat and mass transfer.

Further, the pressure terms in equation (3) can be eliminated through cross differentiation. The momentum and energy equations in terms of the stream function • now become

r 2 a0 sin 0 00 + sin-----0 0 r 2 1, 0 " ~ + r sin --~-], (5)

1 r 2s in0 00 Or Or -~ ~-~ k 0 r ] + r 2sin----O 00 s i n 0 - ~ . (6)

The boundary conditions on the vertical axis are those of symmetry with regard to velocity and temperature distributions,

Ou OT 0 = 0 , - ~ = v = - ~ = 0. (7)

As the plume is confined within an adiabatic cone of apex angle 20,~, the boundary conditions there are

0T 0--___0~, v = 0 , u, T-- finite, - ~ = 0 . (8)

The integration of the energy equation (6) for a large control volume enclosing the point heat source leads to

foOO' o ~ - oOrl Q [ ~ - ( T - T o ) - a r 2 sin Or ] dO= 2~p----~p' (9)

where Q is the heat released from the point heat source.

3. Analysis

It has been pointed out by Wooding [1] that the problem of a point heat source in a porous medium is analogous to the problem of an appropriate similarity variables are

~t" = a f t ( t ) ,

Q h T - ro = 2~C,; (~)'

= ½(1 - cos 0), ~ = ½(1 - cos 0,~).

axisymmetric jet in a Newtonian fluid. The

O0a)

(10b)

OOc)

178 N. Afzal, M. Y. Salam / Convection in Darcian porous medium

Substituting these variables in eqs. (5) and (6) we get

f " = [2(1 - 2~)h] ' Ra, ~ (11)

[ ~ ( f - a ) h ' ] ' = ½ ( f h ) ' . (12)

The boundary conditions at the axis of the plume (7) are

~ O, f ~ O, ~ l / 2 f , , ~ O, ~ l / 2 h " -'+ 0 (13)

and at the confined adiabatic cone are

~ , f ~ 0 , f ' = f i n i t e , h ' ~ 0 . (14)

The heat flux relation (9) becomes

fo¢~(2 + f ' ) h df = 1. (15)

Here the characteristic Rayleigh number Ra is

Ra Qgflk pCp 2~a2v " (16)

Based on the heat released from the source and the properties of the porous medium and the fluid, for a free plume in an unbounded porous medium 0~ - ,~ ( ~ = 1) and for the plume in a semi-infinite medium bounded by an insulated horizontal surface 0~ = ,~/2 ( ~ = ½).

An analytical solution in the limit of small Rayleigh number (Ra -+ 0) can be obtained by the perturbation analysis. Expanding the variables f and h in the powers of Ra and solving the first- and second-order equations we get (~ 4= 0)

{3~,(1 - ~ ) ln(1 - ~ ) - (1 - ~) ln(1 - ~) / = L) Ra+

+ (~" - ~'~,) [~'o,(2~" - 4~',~ + 3) + 3(1 - ~'~)2 In(1 - ~',~)] } Ra2 + O(Ra3), (17a)

1 1 h = 2 ~ 493 {(1 - g~)[g= In((1 - ~')/(1 - ~,)) + ln(1 - g~,)]

+~ , (~ - ~-~,~ + 1 ) ) Ra + O(Ra2). (17b)

For the special case of ~'~, = 1, for the plume in an infinite expansion of fluid, the higher-order perturbations have been estimated to get

f = ~(1 - ~)[Ra + ~-(1 - 2~) Ra + ~ ( 1 + 15~- 15~ "2) Ra 3

+ 5-~(184~ "3 - 270~ "2 + 174~ + 234) Ra 4 + O(RaS)], (18a)

h = ½ + ~-(1 - 2~) R a + 2~--g(-1 - 30~+ 30f 2) Ra 2

- 7 ~ ( 9 2 f 3 - 138~ 2 + 16f + 15) Ra 3 + O(Ra4). (18b)

In the present results (18a) and (18b) the first two terms in f and h agree with the work of Bejan (1978) whereas the higher-order terms differ (see eqs. (26) and (27) of Bejan). We shall return to this point later in section 4.

For large values Ra the plume becomes very thin, and so the eqs. (11)-(15) are not appropriate as their numerical integrations would require a very fine step size. The independent variables in eqs. (11)-(15) may be eliminated in favour of X as

X = ~ Ra. (19)


The resulting equations, even though appropriate at high Ra, would be difficult to integrate at low values of Ra. In order to obtain the solutions for the entire range of Ra two sets of equations are needed: the set (11)-(15) for low or moderate Ra and the corresponding set in terms of variable (19) for moderate and large values of Ra with a change over at certain value of R. It is therefore desirable to have a single set of equations that remains valid for all values of Ra [9]. This in turn needs a single independent variable that reduces to ~ as Ra ~ 0 and to X as Ra ~ or. The new independent variable 77 may be defined as

* / = { ' { - l + R a " (20)

Based on the variable ~, eqs. (11) and (12) after integrating them once with respect to 77, yield

f ' = 2(1 - { ) ( 1 - 2{7/)h + d, (21)

277({77 - 1 )h ' = f h , (22)

where d is the constant of integration. The boundary conditions (13) and (14) become

f (0 ) = 0, f(~/,~) = 0, f '(~/,~) = finite (23)

subject to the heat flux relation

fon~(f'+ 2{)h d~/= 1. (24)

Here the parameter *1~ is given by

~,~ 0.5(1 - cos 0~) (25)

For Ra = 0, { = 1 and as Ra ~ ~ , { ~ 0. Therefore, the entire natural convection regime from Ra = 0 to ~ is described by parameter { in a finite domain 0 ~< c ~< 1.

At the two asymptotes { = 0 and 1, the closed form solution of eqs. (21)-(25) can be obtained. The closed form solution for { = 1 (Ra = 0) is

f = 0 , h = ( 2 L ) - I . (26)

This corresponds to the state of pure conduction around the point source in the confined space, and there is no fluid motion. The closed form solution at the other asymptote 1 / R a = 0, { = 0 is given by

4a*/ 2a = 3. (27) f = 1 + a*/ ' h (1 + a n ) 2 ' a

At the axis of the plume, the velocity and temperature are given by

f ' ( 0 ) = 1.5, h(0) =0 .75 . (28)

It may be noted that the solution (27) for c = 0 is independent of ~,~, the conical space bounding the plume. This is due to the fact that as Ra ~ {~, the plume becomes very thin and to this order it does not feel the presence of the bounding space.

The solution (27) for c = 0 can as well be expressed in terms of more familiar form. As Ra ---, oo, the plume is thin and the variable 77 = (1 - cos 0 ) /2{ can be approximated for 0 ~ 0 as

T/=-] Ra 0 2.

If the coordinate x is along the plume and y is the radial coordinate, then for small angles 0 = y / x the variable becomes

y 2 -- Ra( ) , (29)

the variable of Wooding (1963).


4. Results and discussion

In the present work, a general formulation for the natural convection in a fluid saturated porous medium due to a point heat source situated at the apex of the insulated conical surface, (of apex angle 20o,), is presented in terms of self-similar equations (21)-(25) for all values of Rayleigh numbers. Two special cases of 0,0 = ~r and ¢r/2 correspond respectively to the plume in an unbounded medium and in a semi-i/afinite medium bounded by a horizontal surface. In contrast, the work of Hickox and Watt (1980) dealt with these two special cases by employing two different formulations. The case of simultaneous heat and mass transfer is described in the appendix.

For the two asymptotic cases of zero and infinite Rayleigh numbers the closed form solutions have been presented. For weekly buoyant flow, a regular perturbation solution in the powers of Ra has been obtained for arbitrary values of 0~,. The higher-order perturbations of series solutions (18) differ from Bejan (1978) and Poulikakos (1985). This difference is attributed to the manner in which the condition for heat flux from the source is imposed.

On global heat flux relation (9) if a condition that temperature difference T - T o is independent of angular position 0 is imposed, the relation (9) reduces to

2a7" (30) - r "~- = 4 Cp'

the local heat flux relation employed by Bejan (1978) and Poulikakos (1985). It may be noted that the condition of 0 being independent of T - To implies that the heat transport is purely by conduction. Clearly, eq. (30) considered by Bejan (1978) and Poulikakos (1985) ignores the convective transport of heat. An order of magnitude analysis shows that the ratio of the conduction to convection terms is of order Ra -1. For Ra ~ 0 the conduction is dominant whereas for Ra---, oo convection is dominant. In conclusions, the condition (30) is valid only for Ra = 0 and therefore to order Ra the solutions of Bejan (1978) and Poulikakos (1985) should be in error.

However, to this order the constant of integration in the first-order equation for the temperature incidenfly vanishes and the second terms in his series (26) and (27) are fortuitously correct. Beyond the second term their results (Bejan, 1978; Poulikakos, 1985) are in error.

Eqs. (21)-(25) were integrated numerically by the Runge-Kutta-Gill method on a VAX- 11/780 for various values of Ra and in the ranges 0 < 0o, < "~ and 0 < ( ~< 1. The two missing boundary values a and h(0) have been guessed by the Newton-Ralphson method such that the boundary conditions (23) and (24) are satisfied. The difficulty,, due to singular nature of the equations at the boundary points, was circumvented by estimating the derivatives there as

h'(0) = - ½f'(0)h(0), (31)

h ' ( , l~ ) = 0, c ~ , ~ 1,

= ½ f ' ( ~ , ~ ) h ( ~ , o ) , ,~/,~ = 1, (32)

and starting the solution directly from boundary *1 = 0. The domain (0, ,/) was divided into 400 step sizes for ½ ~< c < 1 and 700 for 0 < c ~< ½. The solutions were obtained for various values of 0,~ = ~r (free plume), 3~r/4, ~r/2 (plume bounded by horizontal surface), and ir/4.

The numerical solutions for f '(0), the vertical velocity at the centreline of the plume is displayed in fig. 2 against c for various values of 0,~. For a fixed value of O,0, f ' (0 ) increases as c decreases and for c = 0 and 1, and it agrees with the closed form solutions (24) and (26). For O~--~r and 3~r/4, the function f ' (0) is monotonic, while for O~, = ~r/2 and ~r/4 there is a maximum. For a fixed Ra, as O,~ decreases from ~r to ~r/4, the centreline velocity increases. The numerical solutions for centreline temperature h (0) are displayed in fig. 3 against ~ for various


((o) Ow

i i I I , ~ , ] , ~

O~ 02 03 Oc 0.5 06 07 {)8 09 10

Fig. 2. Centreline velocity in an axisymmetric plume bounded by conical adiabatic surface in Darcian porous medium

t f (0) = (df /dn)n=0, ¢ = 1 / ( 1 + Ra).

values of 0,0. For a fixed 0o,, the centreline temperature h (0) increases as ( decreases to become maximum at a certain value of (, and then as c ---, 0, it decreases rapidly to the asymptotic value of 1.5 for unconfined plume. The value of ( at which h(0) becomes maximum decreases as 0,~ decreases from ~r to ~r/4. For a fixed ( ~ 0 the temperature h(0) increases as 0~ decreases, the rate of increase approaches to zero as c ~ 0.

h co)

2

Ow

13"

0.I 02 03 0 ~ 0.5 06 07 O~ 09

Fig. 3. Distribution of centreline temperature in an axisymmetric plume bounded by conical adiabatic surface in Darcian porous medium.

182 N. Afzal, M. Y. Salam / Conuection in Darcian porous medium

12

1 8

16

f(o) R o 14

I

I

T7" /2

I L I J I

t 2

h(o) 1.o

0.8

0.6

0 01

- - 0 w

1 7 / 2

I L I , I , I 10 100 ~000

Ra

Fig. 4. Comparison of centreline velocity and temperature for a plume in an unbounded medium (0,~ = ~) and in a semi- infinite medium bounded by horizontal adiabatic plane (0,~ = -~/2) represented by the asymptotes. (a) Velocity at the axis f ' ( 0 ) = ( d f / d ~ ) , l _ 0. (b) Temperature at the axis h(0).

A comparison of the results for a plume in an unbounded medium (8,~ = ~) and a plume bounded by a semi-infinite adiabatic horizontal plate (0,~ = ~r/2) is displayed in figs. 4a and 4b. For a given O,~, the centreline velocity and temperature attains a maximum. The value of Ra at which maxima occurs decreases as 0~ decreases. Fig. 4 shows that the centreline velocity and temperature increases in the confined plume when compared to free plume. This is due to the fact that in an unconfined plume the entrainment is from the entire space whereas in a bounded plume the entrainment is restricted due to the boundary confining the plume.

Appendix

Buoyancy induced convective heat and mass transfer from a point source in an unbounded Darcian porous medium studied by Poulikakos (1985) had been based on local heat and mass transfer conditions (see eqs. (20) and (21) of Poulikakos (1985)). In this section an analysis based on global heat and mass transfer consitions is presented. The boundary surface is permeable at the apex (see fig. lb) to permit the mass transfer th in addition to heat flux Q from the source. Based on the variables

"It = arf(~), ~ = ½(1 - cos e ) ,

Q h rh T - T O - 2~rp-~Cpr ( ~ ) ' C - C o = 2~roacprg(~ ). (A.1)

Eqs. (11)-(13) of Poulikakos (1985) for momentum, energy and species conservation reduce to

f " = [2(1 - 2 ~ ' ) ( h - Ng)]" Ra, [ ~ ( ~ - 1 ) h ' ] ' = ½(fh)' ,

[~'(~" - 1 )g ' ] ' = ½(fg)' . (A.2)

N. Afzal, M.Y. Salam/ Convection in Darcian porous medium 183

The boundary conditions are

~ 0, f , ~l/2f,,, ~l /2h t ' ~1/2g, _ . O,

~ , o , f o O ,

h', g' --* 0 for ~ ~ 1 and (1 - ~,o)1/2h ' or g' ~ 0 for f~ = 1. (A.3)

The global heat and mass flux conditions determined from integration of energy and species conservation equations for a large control volume, under the similarity transformation reduce to

fo~(2 + f ' ) h d r = 1, fof~(2 + Le f ' ) g d~'= 1. (A.4)

Here parameter N and Lewis number Le are defined as

/~c ~ (A.5) N=--~- , L e - D '

where fie is the coefficient of concentration expansion, fl is the coefficient of thermal expansion, D is the mass diffusivity and a is the thermal diffusivity.

When Lewis number Le = 1, the energy and species conservation equations require h = g and the momentum equation reduces to

f " = (1 - N ) [ 2 ( 1 - 2 f ) h ] ' Ra.

This equation is the same as (11) provided Ra(1 - N) is replaced by Ra. Therefore the solution for the mass transfer case can be obtained from the corresponding zero mass transfer case provided Ra(1 - N) is replaced by Ra.

For Le 4: 1, the equations have to be integrated numerically for various values of N, Le and Ra. However, at low Ra a series solution can be obtained by expanding the variables in power series as

f= fo + Ra ./1 + Ra2f2 + . . . . h = h o + Ra ha + Ra2h2 + . . . .

g = go + Ra gl + RaZgz + ... (A.6)

The equations for the successive approximations can be obtained by substituting these expansions in the governing similar equations. The solution to these equations obtained for a free plume (f~ = 1) is given below:

First order:

' (A.7) fo = 0 , h o = ½ , go = 7-

Second order:

fa = (1 - N ) f ( 1 - ~'), h I = (1 - N) (1 - 2 f ) / 8 ,

gl = ( 1 - - N ) Le(1 - 2 f ) / 8 . (a .8 )

Third order:

f2 = (1 - N) (1 - U Le)~'(1 - ~')(1 - 2~')/6,

h2= - ( 1 - N) { [ 5 - N(3 + Z Le)](~ - ~ + ~ 2 ) - l + N } /48,

g2 = - Le(1 - N) { [5 - U(3 + 2 Le)] ( I - ~" + ~.2) _ 1 + N } /48 . (A.9)

Fourth order:

f3 = (1 - U) (1 - U Le)~'(1 - ~')

× [ 3 ( 1 - N ) ( - 1 - 3 ~ ' + 3 ~ "2) + 2 ( 1 - U L e ) ( 1 - 3 ~ ' + 3 ~ ' 2 ) ] / 1 4 4 . (A.10)


The centreline velocity, temperature and concent ra t ions are given by

f ' ( 0 ) = (1 - N ) Ra + I (1 - N ) ( 1 - N Le) Ra 2 + 1-~(1 - N ) ( 1 - N Le)

× [ - 3 ( 1 - N ) + 2(1 - U Le)] Ra 3 + . . . ,

h(0) = ½ + ~ - ( 1 - U ) R a + ~8(1 - N ) [ - 3 ( 1 - N ) + 2 ( 1 - U Le)] R a 3 + . . . ,

g(0) = ½ + ~-(1 - U ) Le Ra

+ ~ g ( 1 - N ) [ - 3 ( 1 - N)~q - 2(1 - N Le)] Le Ra 3 + . . . (A.11)

In these relations the first two terms are the same as ob ta ined by Poulikakos (1985) bu t the third term is different. This is due to the fact that the local heat and mass flux condi t ions employed in by Poulikakos (1985) are valid in the immedia te ne ighbourhood of the source and ignore the convective t ranspor t of order Ra.

References

Afzal, N. (1985) Two dimensional buoyant plume in a porous medium: higher order effects, Intern. J. Heat Muss Transf 28, 2029-2041.

Afzal, N. (1986) Mixed convection in buoyant plumes, in: Handbook of Heat and Mass Transfer, Vol. 1. Heat Transfer, Operations, ed. N.P. Cheremissin (Gulf Publ. Co., Texas) pp. 389-440.

Bejan, A. (1978) Natural convection in an infinite porous medium with a concentrated heat source, J. Fluid Mech. 89, 97-107.

Hickox, C.E. and H.A. Watts (1980) Steady thermal convection from a concentrated source in a porous medium, Z Heat Transf. Trans. ASME 102, 248-252.

Poulikakos, D. (1985) On buoyancy induced heat and mass transfer from a concetrated source in an infinite porous medium, Intern. J. Heat Muss Transfer 28, 621-629.

Schlichting, H. (1968) Boundary Layer Theory, 6th Ed. (McGraw-Hill, New York). Squire, H.B. (1951) The round laminar jet, Quart. J. Mech. Appl. Math. 4, 321-329. Wooding, R.A. (1963) Convection in a saturated porous medium at large Rayleigh or Peclet number, Z Fluid Mech. 15,

527-546. Yih, C.S. (1965) Dynamics of Non-Homogeneous Fluids (Mac-Millan, New York).

natural convection from point source embedded in darcian porous medium

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