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Evaporation of water film in an enclosure filled with a porous mediumWu‐Shung Fu a & Wen‐Wang Ke a

a Department of Mechanical Engineering , National Chiao Tung University , 1001 Ta Hsueh Road, Hsinchu, 30050, Taiwan,Republic of ChinaPublished online: 03 Mar 2011.

To cite this article: Wu‐Shung Fu & Wen‐Wang Ke (1999) Evaporation of water film in an enclosure filled with a porous medium, Journal of the ChineseInstitute of Engineers, 22:3, 325-339, DOI: 10.1080/02533839.1999.9670470

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EJ061199900325Journal of the Chinese Institute of Engineers, Vol. 22, No. 3, pp. 325-339 (1999)

EVAPORATION OF WATER FILM IN AN ENCLOSURE FILLED

WITH A POROUS MEDIUM

Wu-Shung Fu* and Wen-Wang KeDepartment of Mechanical Engineering

National Chiao Tung University1001 Ta Hsueh Road, Hsinchu, 30050, Taiwan

Republic of China

Key Words: evaporation, porous medium, latent heat, sensible heat.

ABSTRACT

A double diffusive natural convection in an enclosure filled witha porous medium is investigated numerically. To enhance the heattransfer rate of a heat wall, the heat wall is wetted with a thin waterfilm, then sensible and latent heat transfer occur simultaneously. TheDarcy-Brinkman-Forchheimer model is used and the factors of heatflux, porosity and diameter of bead are taken into consideration. TheSIMPLEC method with iterative processes is adopted to solve the gov-erning equations. The results show that the effect of evaporation onthe heat transfer rate is apparent, and the latent heat transfer plays animportant role in the high Darcy-Rayleigh number situations. The av-eraged Nusselt number Nu is approximately 2.5 when Ra*<4. In therange of 4<Ra*<4469, the relationship of the averaged Nusselt numberwith the Darcy-Rayleigh number is linear in logarithmic scale.

I. INTRODUCTION

Double diffusive natural convection in a porousmedium has many important applications in energy-related engineering problems, such as moisture mi-gration in fibrous insulation, diffusion of pollutantsin saturated soil, and flow in the wick of a heat pipe.Although numerous investigations have studied theproblem of double diffusive natural convection in aporous medium numerically and experimentally, dueto the complexity of temperature and flow fields inthe porous medium, the subject has continued to be afocus of research in the last decade.

Trevisan and Bejan (1985) investigated the ef-fects of heat and mass transfer on natural convectionin a porous medium by numerical simulations andscale analysis. The results of the scale analysis werein agreement with those produced by the discrete

numerical experiments, and sorted out many effectswhich influence overall heat and mass transfer resultsof numerical experiments. Lai and Kulacki (1991)studied a vertical plate embedded in a saturated po-rous medium and utilized a similar method to solvegoverning equations derived from Darcy's law andBoussinesq approximation. The results showed thatthe effect of the Lewis number on the concentrationconvection field was more pronounced than that ofthe temperature and flow fields. Hsiao et al. (1992)conducted a numerical investigation of natural con-vection of a heated horizontal cylinder in an enclo-sure filled with a porous medium. The results pointedout that the effect of thermal dispersion on the natu-ral convection was small from ranging from low tomoderate Rayleigh numbers, and the numerical solu-tions were in better agreement with the experimentaldata. Alavyoon (1993) also used a numerical method

*Correspondence addressee

•5/3

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326 Journal of the Chinese Institute of Engineers, Vol. 22, No. 3 (1999)

to study unsteady and steady convection in a fluid-saturated vertical and homogenous porous medium.The results indicated that as the Lewis number waslarger than 1, there existed a minimum aspect ratiobelow which the concentration field in the core re-gion was rather uniform and above which it was lin-early stratified in the vertical direction. Alavyoon etal. (1994) subsequently studied the above subject withopposite and horizontal gradients of heat and soluteconditions. The results revealed that under a certaindomain the convection oscillated and outside thisdomain the convection approached a steady state.Nithiarasu et al. (1997) studied a generalized non-Darian porous medium model with constant and vari-able porosities for natural convective flow. The re-sults indicated the wall Nusselt number was signifi-cantly affected by the combination of Rayleighnumber, Darcy number and porosity in the non-Darcyflow regime, and the model was able to predict thechanneling effect. Murray and Chen (1989) carriedout an experimental study to examine double-diffu-sive convection in a porous medium, and the resultsshowed that as the temperature difference was reducedfrom supercritical to subcritical values, the heatflux curve established a hysteresis loop. Lauriat andPrasad (1989) investigated non-Darcian effects onnatural convection in a vertical porous enclosure,and the results showed that the effect of Prandtlnumber on Nussult number was not straight forwardand depended on the flow regime and otherparameters.

Because of the difficulties associated withthe theoretical determination of effective thermalconductivity for the corresponding porous mediumsituation, a number of experimental studies investi-gating the effective thermal conductivity wereconducted. Singh et al. (1973) conducted experimen-tal studies of the effective thermal conductivity of liq-uid saturated sintered fiber metal wicks, and a newcorrelation for predicting the effective thermal con-ductivity was proposed. Koh and Fortini (1973)performed experimental studies for prediction ofthermal conductivity and electrical resistivity of po-rous materials including 304L stainless steel powdersand OFHC sintered spherical powders. The resultsshowed that the thermal conductivity and electricalresistivity could be related to the solid materialproperties and porosity of the porous matrix, re-gardless of the matrix structure. Hadley (1986)presented both experimental measurements and atheoretical model for the thermal conductivity of aconsolidated mixture of a two-metal powder. Thepredictions showed good agreement with publishedtwo-phase data from a wide variety of sources.Shonnard and Whitaker (1989) conducted an experi-mental study to determine the effective thermal

conductivity for a point-contact porous medium, andthe results verified the functional dependence of(kjkf) predicted by Batchelor and O'Brien. Besides,the experimental data and the theory of Batchelor andO'Brien were in good agreement with the experimen-tal results of Swift. Prasad et al. (1989) conducted anexperimental study to evaluate a correlation for stag-nant thermal conductivity of liquid-saturated porousbeds of spheres. The results showed that a mixingrule based on the volume fraction could not be usedto predict stagnant thermal conductivity unless thethermal conductivity ratio was almost unity.

Differing with the above literature, Renken andAboye (1993) analyzed film conductivity condensa-tion within inclined thin porous-layer coated surfaces.The dependence of the average heat transfer coeffi-cient on the surface subcooling, the gravity field andthe thin porous-layer coating characteristics weredocumented.

Based on the above research, the temperatureand concentration boundary conditions on the surfacesare almost always designated as respective constants.However, for some situations, such as the enhance-ment of a heat surface wetted with liquid, and evapo-ration of sweat on skin, etc, the temperature andconcentration boundary conditions induced by evapo-ration of liquid on the heat surface can no longer beregarded as constants in advance, and mutual conju-gation often results. The heat transfer mechanism ofthe above situations becomes complicated and is hardto investigate.

The purpose of this paper is to numerically in-vestigate double diffusive natural convection phe-nomena of an enclosure filled with a porous medium.The channeling effect, constant porosity and Darcy-Brinkman-Forchheimer model of the porous mediumare taken into consideration. To reduce the tempera-ture on the heat wall, the heat wall is wetted with athin water film. Due to the occurrence of latent andsensible heat transfer on the left wall, the boundaryconditions of temperature and concentration of theheat wall are no longer designated as the known val-ues in advance, and the above boundary conditionsare then conjugated. The SIMPLEC numericalmethod is adopted to solve the governing equationsand iterative computing processes are indispensablein obtaining temperatures, concentrations and veloci-ties on the heat surface and in the enclosure. Theresults indicate that the effect of the evaporation onthe reduction of temperature of the heat wall is re-markable and the latent heat transfer plays an impor-tant role in the heat transfer mechanism under highDarcy number situations. The relationship of aver-aged Nusselt number with Darcy-Rayleigh numbersis almost linear in logarithmic scale for high Darcy-Rayleigh numbers and the averaged Nusselt number

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W.S. Fu and W.W. Ke: Evaporation of Water Film in An Enclosure Filled with A Porous Medium 327

is approximately 2.5 for low Darcy-Rayleighnumbers.

II. PHYSICAL MODEL

A physical model of a two-dimensionalrectengular enclosure filled with a porous medium isshown in Fig. 1. The width and length of the enclo-sure are labeled H and L, respectively. The gravityis downward and two horizontal walls of the enclo-sure are adiabatic. The left wall is subject to a uni-form heat flux qw and the relatively low temperature,concentration, and relative humidity conditions ofthe right wall are designated as TR, CR and <pR,respectively, and are constant. In order to enhancethe heat transfer rate of the left wall, the left wallsurface is wetted with a water film. The thickness ofthe a water film is assumed to be extremely thin andstationary such that water does not penetrate into theregion of the porous medium. The water film isevaporated by the heat flux imposed on the left walland the condition of relative humidity of the left wallsurface is saturated. Consequently, the sensible andlatent heat transfers of natural convection occur inthe enclosure simultaneously, and the temperature,TL(y) and concentration, CL(y) on the left wall are nolonger regarded as constant in advance and conjugatemutually. The working fluid in the enclosure becomesan air-moist fluid.

To facilitate the analysis, the following assump-tions are made .(i) The porous medium is made of non-deformable

pure copper spherical beads (&s=386 Wm~loC~')which are not chemically reactive with the fluid,

(ii) The flow in the enclosure is laminar, steady andtwo-dimensional. Corresponding to the thermaland concentration boundary conditions selected,condensation phenomenon does not occur in theenclosure.

(iii)The effects of Soret and Dufour induced by masstransfer are neglected, and Boussinesq approxi-mation is held.

(iv)Except for the left wall region, the properties ofthe porous medium are constant and based on thetemperature of the right wall. The properties onthe left wall are based on the temperature of theleft wall under a saturated condition (Kato andMihara, 1977).

(v) The form of the effective thermal conductivity keffdefined in Eq. (1) has been proposed in (Shonnandand Whitaker, 1989). The calculations of the air-moist fluid properties are shown in the the Ap-pendix (Kato and Mihara, 1977). The porosity eis constant and the permeability K and inertia fac-tor F (Uafai, 1984) are defined in Eqs. (2) and(3), respectively.

Fig. 1. Physical model.

keff=kf[4\n(-^)-[\)

K =150(1 -ef

_ 1.75

(1)

(2)

(3)

Based on the above assumptions and with thefollowing characteristic scales of H, TR, qw, vR, andkeff, the governing equations adopted in (Hsiao et cd.,1992), geometry dimensions and boundary conditionsare normalized as follows. The Darcy-Brinkman-Forchheimer model is used in the momentumequations.Continuity equation

(4)

X-momentum equation

l a F aFJ DaY-momentum equation

U

'DaeU (5)

F U

'DaGrt0e+GrcW£ (6)

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328 Journal of the Chinese Institute of Engineers, Vol. 22, No. 3 (1999)

Energy equation

/ ]v±L 4. I /—

UdX + VdY~Pi

Concentration equation, ydW _ 1 (d

2W

(7)

(8)

where

yy — x v — u — —vn

P —

q H '

gf3c{C-CR)Hl

(9)

aR DP pCp

U =

Boundary conditionsOn OP (top wall)

U=0, V=0, i j | = 0 , (10)

On PQ (right wall)

UR=0, VR=0, 6R=0, WR=0

On QR (bottom wall)

(11)

M), V=0, | | = 0, = 0 (12)

On RO ( left wall )

vL=o, eL=eL(Y), wL=wL(Y)

The other important values of evaporation massflow rate mL(y), evaporation velocity uL(y), sensibleheat flux qs(y) and latent heat flux g^(jjare calculatedby the following equations, respectively

PLDL dC

x=0

(14)—Ar2 '

PL (\-CL(y))dx , = 0

DL (CE(Xl,y)-CL(y))(l-CL(y)) K ,

= -k(TE(Xl,y)-TL(y))

ejf

(15)

(16)

(17)

.x = 0

(CE(xvy)-CL(y))

d-CL(y)) —AY

2 '

(18)

III. NUMERICAL METHOD

The SIMPLEC algorithm (Van Doormaal andRaithby, 1984) with TDMA solver (Patankas, 1980).is used to solve the governing Eqs. (3)-(8) of the flow,temperature and concentration fields. Eqs. (3)-(8) arefirst discretized into algebraic equations by using thecontrol volume method (Patankar, 1980) with thepower-law scheme. The under-relaxation factors forthe velocity, temperature, and concentration fieldsrange from 0.1-0.5 for all the fields. The conserva-tion residues (Van Doormaal and Raithby, 1984) ofthe equations of momentum, energy, concentrationand continuity and the relative errors of all variablesare used to examine the convergence criteria whichare defined as follows:

nRe sidue of O equation

c.v.\ o=u, v, e, w

max

max< 1 0 , <3>=U, V, P, d, W

(19)

(20)

To reduce computation time, staggered mesh isused, and the finer meshes are set near the solid wallregions. The meshes are then expanded outward fromthe boundary wall with a scale of 1.03. The accuracyof a similar numerical method is validated in Fu etal. (1996, 1997). A comparison of the results, whichindicate double diffusive natural convection in a po-rous cavity with aspect ratio A=l and Darcy numberD«=10"7 using the Darcy-Brinkman formulation ofGoyeau et al. (1996), with the present method inshown in Table 1. The deviations between these two

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1/1/. S. Fu and W.W. Ke: Evaporation of Water Film in An Enclosure Filled with A Porous Medium 329

Table 1. A comparison of the results of Goyeau et al. [21] with the present study.

Ra

50

100

200

Le

110

1001

102050

1001

102050

100

Nu(Goyeau et al.)

1.981.981.983.113.113.113.113.114.964.964.964.964.96

Nu(present study)

1.99841.99841.99843.15583.15583.15583.15583.15585.09115.09115.09115.09115.0911

Error(%)

0.920.920.921.451.451.451.451.452.582.582.582.582.58

Sh(Goyeau et al.)

1.988.79

27.973.11

13.2518.8929.7241.53

4.9619.8628.1744.0061.09

Sh(present study)

1.99848.9514

28.7543.1558

13.66019.55830.61942.358

5.091120.61529.08344.65860.327

Error(%)

0.921.82.731.453.003.422.941.952.583.663.121.471.26

methods are less than 4%.The following steps provide a brief outline of

the computing processes and numerical results basedon the given data of TR=293K, PR=\.0\3 bar and

Wl + 1 111 + I

(i) Under given heat flux qw and boundary condi-tions on the right wall, the pseudo values of T",C", u" and v" in the enclosure, including the leftwall, are initially selected (n=0). Subsequently,calculate the corresponding properties of theair-moist fluid on the left wall including v'[,PlCplDlandh^.

(ii) Based on the above conditions, utilize theSIMPLEC method to solve Eqs. (5) and (6) andobtain the values of u"E

+l(x,y) and v"E+l(x,y) in

the enclosure, without including the values onthe left wall.

(iii) Based on the values of u"E+{(x,y) and v£+l

(x,y), the SIMPLEC method is successively uti-lized to solve Eqs. (7) and (8) to gain the newvalues of T"E

+](xhy) and C"E+\x\,y) in the en-

closure except the values on the left wall.(iv) Substitute C'[(y) and C"E

+\xhy) into Eq. (14)to calculate the evaporation mass flow ratem'[(y) on the left wall, and use Eq. (18) to ob-tain q'\ sequentially. If q"f is smaller than qw,continue the computing procedures, otherwisetry a new value of TL(y)and return to Step (ii).

(v) Subtract q'\ from qw in Eq.(16) to obtain q".Use qn

s in Eq. (17) to calculate T"L+](y) on the

left wall.(vi) Using the saturated condition on the left wall

and the calculated value of T"L+\y), the new

values of C"L+\y), v'[+\y), p'l + ](y), Cn

p+](y),

\, s ...., >» + \,y) a r e c a i c u i a t e d . ThDn + \L

n + Ifi"L (y) and hn

f*x{y) are calculated. Then

(vii)

C"L+l(y) and C£+l(x,,)0 in Eq. (14), and the

evaporation velocity u'[+[(y), induced by theevaporation mass flow rate mn

L+i(y), is calcu-

lated from Eq. (15).Calculate the values oiqn

( + [{y) from Eq. (18),

then subtract q"( + \y) from qw in Eq. (16) and

obtain q" + l(y).(viii) Compare the values of qn + x(y) and q"{y) ob-

tained from Steps (7) and (5), respectively. Ifthe difference between q" + l(y) and q"(y) andthe convergence criteria of TE(x,y), CE(x,y),uE(x,y) and vE(x,y) do not satisfy the criteria ofEqs. (21) and (19)~(20), respectively, then usethe corrected values of T"L

+ ' (y), C"L+i (y), u'[ +

(y) and the corresponding properties T"E+i

C n +£ W of air-moist fluid to iterate

the computing procedure from Step (2), untilthe above conditions are satisfied.

•\-3(21)

IV. RESULTS AND DISCUSSIONS

m'[+ (y) can be calculated by the utilization of

The dimensionless parameters of four Rayleighnumbers, three porosities and three diameters ofspherical beads tabulated in Table 2 are adopted inthis paper. H, L, T/?=293K, fo=30%, Pr=0.02583and 5c=0.5425 are fixed in the following situations.The ranges of Ra' and Da are from 0.02725 to4469.041 and from 2.963xlO"8 to 1.215xlO"3,respectively.

The grid test for computation is based on thesituation of #a=9.197xlO5, D/?=5xl0"3 and £=0.4.The results of the test are listed in Table 3, and

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330 Journal of the Chinese Institute of Engineers, Vol. 22, No. 3 (1999)

Table 2. Values of Ra, e and Dp adopted in thestudy.

Ra Dp

9.197xlO5

1.839X106

2.758xlO6

3.678X106

0.40.70.9

0.0050.0250.05

Streamlines Isothermal lines Concentration lines

( a ) Ra = 9.197 x 105, e = 0.7, Dp = 0.025 and Da = 1.588 x 10"5

Table 3. Grid tests for 7?a=9.197xlO5, £=0.4, andDp=0.005

A.vi,A.Vl'AylXi

-1) Grids

(NxxNv)Nu

0.0250.0250.0150.0150.0150.010.010.00750.00750.00750.00750.0050.00250.0025

1.031.011.0051.031.011.031.0011.051.031.011.0051.031.051.03

32x3236x3638x3846x4658x5862x6298x9860x6074x74

102x102116x11694x9498x98

132x132

0.995620.995780.995700.996500.99510.99590.99550.99560.99670.99550.99550.99610.99610.9969

the grids of 74x74 are selected for the followingcalculation processes with Ax^A)^ =0.0075 and theratio of AJC,-+I/AX/=1.03, which are symmetric to cen-tral lines of the computing domain.

The distributions of streamlines, isothermal linesand constant concentration lines are illustrated in Fig.2. In Figs. 2(a) and (b), as Ra becomes larger, theconvection is more dominant, the isothermal lines be-come more skewed and the distributions of concen-tration lines are dense near the two vertical walls ofFig. 2(b).

In Figs. 2(c) and (d), the larger the porosity is,the stronger the convection becomes, therefore, theisothermal and concentration lines of Fig. 2(d) aremore curved than these shown in Fig. 2(c). For thesame reason, the larger the diameter of the sphericalbead, the more dominant the convection, and thedistributions of the isothermal and concentrationlines of Fig. 2(f) are more skewed than these ofFig. 2(e).

The distributions of modified dimension tem-perature 0* on the left wall under different physicalsituations are illustrated in Fig. 3. The situationsshown in Figs. 3(a) and 3(b) are the two limiting cases

l6 o -( b) Ra = 3.678 x 106, e = 0.7, Dp = 0.025 and Da = 1.588 x 10-5

( c ) Ra = 1.839 x 106, e = 0.4, Dp = 0.05 and Da = 2.963 x 10'

( d ) Ra = 1.839 X10 6 ,E = 0.9, Dp = 0.05 and Da = 1.215x10 -3

( e ) Ra = 1.839 x 106, e = 0.7, Dp = 0.005and Da = 6.353 x 10-7

( f ) Ra = 1.839xl06,e = 0.7,]

Fig. 2. The distributions of streamlines, isothermal lines and con-stant concentration lines under different Rayleigh num-bers Ra, porosities £ and diameters of beads Dp.

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IV. S. Fu and W.W. Ke: Evaporation of Water Film in An Enclosure Filled with A Porous Medium 331

0 2

0.0 0.2 0.4 0.6 0.8 1.0

conductivity ratio (~) in upper region of the leftkf

wall, is more remarkable than that of the NPEsituation. The highest temperature of the PE situa-tion in upper region of the left wall is lower than thatof the NPE case, and the distribution of temperatureis more even than that of the NPE case.

The effects of Rayleigh number Ra, porosity eand diameter of bead Dp on the local Nusselt Nu(Y)and Sherwood numbers Sh(Y) on the left wall are il-lustrated in Figs. 4(a), 4(b) and 4(c), respectively. Thedefinitions of Nu(Y) and Sh(Y) are expressed in thefollowing equations.

Nu(Y) = ^ = Nu S(Y) + Nu e(Y)keff

(a) Ra=9.197xlO-\ e=0A Dp=0.005 and Da=2.963xl(H (22)

00.0

1

1 1

1 ' 1

NPWE

PWE

PE

NPE

1 i 1

1

/

ym *

m m

^ ^ • * ™

.*• ^ * ^ ^

i

0.2 0.4 0.6Y

0.8 1.0

(b) Ra=3.678xlO6. e=0.9, Dp=0.05 and Da= 1.215x10-'

Fig. 3. The distributions of modified dimensional temperature 6°on the left wall under four different physical situations.

where ht(y) =(TL(y)-TR)

dx (23)x=o

where hc(y) =mL(\-CL(y))

pL(CL(y)-CR)

In general, as Ra, e and Dp become larger,evaporation, mainly induced by convection, becomesmore dominant. Therefore, the local Nusselt andSherwood numbers on the left wall are larger underthe larger Ra, £ and Dp situations.

In Fig. 5, the values of Re and qs indicate theratios of latent heat flux qp and sensible heat flux qs

to the total heat flux qw, respectively, and these val-ues are defined in the following equations.

(24)

of convection capability under the lowest and high-est Ra", respectively. Generally, the effect of theevaporation on heat transfer is remarkable, andthen the temperatures distributed on the left wall de-scend efficiently. The flow resistance in the porousmedium situation (PE) is greater than that in the non-porous medium situation (NPE), therefore convectionin the NPE situation is more active than that in thePE situation. As a result, for most regions of the leftwall, the temperatures of the NPE situation are lowerthan those of the PE situation. However, the convec-tion effect is weak in the upper region of the left wall(the upper left corner). The occurrence of conduc-tion in the PE situation, with a larger thermal

The value of the ratio R means the contributionof above individual heat flux to the total heat flux.As the values of Ra, e and Dp become smaller, theresistance to fluid flow becomes serious, which si-multaneously results in weak convection and littleevaporation. The value of Re is thus smaller underthe smaller Ra, e and Dp cases as shown in Figs. 5(al), 5(a2) and 5(a3). Oppositely, as the values ofRa, e and Dp become larger, based on the reason men-tioned above, evaporation becomes dominant andthe values of R? are larger as shown in Figs. 5(b2),5(c2) and 5(c3). In Fig. 5(c2), the value of Re iseven larger than 1 at F=0.03 which means theevaporation of film absorbs heat not only from thetotal heat flux qw on the left wall but also from the

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332 Journal of the Chinese Institute of Engineers, Vol. 22, No. 3 (1999)

Nu(Y)

5.0

4.0

3.0

2.0

1.0

\ ' I ' I

I , I0.0

I • I

Sh(Y)

10.0

8.0

6.0

4.0

2.0

1

\V

\ \" \ \

_*—

-

, 1

1

Ra

Ra

Ra

1

1 • I *

= 3.678xlO6 -

= 2.758xlO6"

= 1.839xlO6_

= 9.197xlO5 •

_X " .

0.0 0.2 0.4 0.6 0.8 1.0Y

0.00.0 0.2 0.4 0.6 0.8 1.0

Y

( a ) E = 0.4,Dp = 0.05and Da = 2.963x10'

Nu(Y)

10.0

8.0

6.0

4.0

2.0

1 ' 1

\~* \

\*\

" s,

* N

-

'

• I . I

1

" in.

* %. _

i 1 •

1

-

-

- — _ -

1

Sh(Y)

25.0

20.0

15.0

10.0

5.0

i ' I • i

T \ — • — • — 8

; \ s

- \ 8

\\ ^.

•— N >»

N N»V " ^

1 •

1 , 1 i 1 r

I

= 0.9 '

= 0.7 "

= 0.4 -

-

* xN

0.00.0 0.2 0.4

Y0.6 0.8 1.0

0.00.0 0.2 0.4

Y0.6 0.8 1.0

( b ) Ra = 1.839 xlO6 and Dp = 0.025

10.0

Nu(Y)

0.0 0.2 0.4 0.6 0.8 1.0

30.0

20.0

Sh(Y)

10.0

i • r

Dp = 0.05

Dp = 0.025

Dp = 0.005 -

0.0 0.2 0.4 0.6 0.8 1.0

( c ) Ra = 3.678xl06ande = 0.7

Fig. 4. The distributions of local Nusselt and Sherwood numbers on the left wall.

fluid in the enclosure. As Dp or e increases, Daalso increases, which leads to greater permeability ofthe porous medium and an increase in latent heattransfer. Based on the above results, the latent heatflux plays an important role in the heat transfermechanism under a high Darcy number. Besides, the

distribution of the ratio Rf in the low Rayleigh num-ber region is larger than that in the high Rayleighnumber region under Da>10~5.

The distributions of velocity V along the y-axisat X=0.0075 are very close to the left wall are shownin Fig. 6. The evaporation of the water film on

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W.S. Fu and W.W. Ke: Evaporation of Water Film in An Enclosure Filled with A Porous Medium 333

1.0

0.8

0.6

0.4

0.2

- Si.

I . I . I

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8 1Y

<J°

1.0

0.8

0.6

0.4

0.2

°%

"*. \\

--

0

1

\

1 ,

0.2

Ki

0.4

^

\

i

0.6

1

\ \

\ \

i

0.8

-

-

-

V \ ~

1

0.0

0.2

- 0.4

0.6

0.8

( a l ) e = 0.4,Dp = 0.005,Da = 2.963xl(T8 (a2) e=0.4, Dp= 0.025, Da=7.40&<l(r7 (a3) e = 0.4,Dp= 0.05,Da= 2.963x1(1*

R,

0.0 0.2 0.4 0.6 0.8 1.000.0 0.2 0.4 0.6 0.8 1.0

( b l ) e = 0.7, Dp = 0.005, Da = 6.353 ( b 2 ) 6=0.7,Dp=0.025,Da=1.58&<l(r5 ( b3 ) e = 0.7, Dp= 0.05, Da= 6.3 53xlOTs

0.2 0.4 0.6 0.8 1.0

( c l ) e = 0.9,Dp = 0.005,Da = 1.215xlO"5 ( c 2 ) 8=0.9,Dp=0.025,Da=3.30&ia4 ( c 3 ) e=0.9,Dp= 0.05,Da= 1.215xl(T3

Fig. 5. The distributions of R( and /?, on the left wall under different situations. (—-Ra=3.678x106, — -fla=2.758x106, fla=1.839xlO6,—Ra=9.197x10s)

the left wall is taken into consideration, and thedirection of the evaporation velocity uL(y) is to theright. Under the strong convection cases shown inFigs. 6(d), 6(e) and 6(f) mentioned earlier, the mainvelocity of the fluid flow in the enclosure is clock-wise and overcomes the evaporation velocity, and thedirection of the resultant velocity is to the left in thelower region of the Y axis. Oppositely, when £=0.4and Dp=5xl0~\ the convection effect is minute,and the evaporation velocity is relatively more re-markable than that of the fluid flow in the enclosure.Consequently, in this situation, the direction ofthe velocity U is to the right in the lower region ofthe K-axis. In the upper region of the K-axis, thedirection of the main velocity of the fluid flowand the evaporation velocity are the same and to

the right. Naturally, the direction of the velocityU is to the right in the upper region of the F-axis,and the channeling effect is observed in somesituations.

The distributions of velocity V along the X-axisat different positions at both vertical walls areillustrated in Fig. 7. The channeling effect is ob-served in some situations of Figs. 7(b), (d), (e) and(f) where the convection is strong. Due to the occur-rence of evaporation on the left wall, the channelingeffect on the right wall is more apparent than that onthe left wall. For the same reason, as £=0.4 and 0.7,and D/?=5xl0~3, the convection, and there is little evi-dence of the channeling effect on the both walls.

Figure 8 shows the relationship of Darcy-Rayleigh number Ra* with the averaged Nusselt

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334 Journal of the Chinese Institute of Engineers, Vol. 22, No. 3 (1999)

1.0

0.8

0.6

0.4

0.2

•• / i ." i i ' i '

i • !

i ; :

0.00.0 0.4 0.8 1.2 1.6

U( a ) e = 0.4 and Dp = 0.005

-1.0 0.0 1.0 2.0

( c ) e = 0.7 and Dp = 0.005

1.0

0.8

0.6

0.4

0.2

-

-

-

-

J

/ / . ' / •

I'-ll.1;/ " •

r\

-I . I ,

0.0-20.0 -10.0 0.0

U10.0 20.0

1.0

0.8

0.6

0.4

0.2

1 • / j . . - , ; / - • • •

/ ' / / •

1

;.';

I!.1/' • ' • '

; ;

'•i

?/

/, ..L.,«r!/ \ , i

i •

-

-

-

—

i i i

0.08-8.0 -4.0 0.0 4.0 8.0 12.0

u( b ) e = 0.4 and Dp = 0.05

o.o-40.0 -20.0

0.0 20.0 40.0

U( d ) e = 0.7 and Dp = 0.05

o.o-40.0 -20.0

0.0 20.0 40.0

U

( e ) E = 0.9 and Dp = 0.005 ( f ) 6 = 0.9 and Dp = 0.05

Fig. 6. The distributions of velocity U along the /-axis at ^=0.0075 under different situations. (-.-/?a=3.678xlO6, — .Ra=2.758x106,/?a=1.839xl06,—Ra=9.197x105)

number Nu, which is defined in Eq. (25) andexpressed in Eqs. (26) and (27). Under the smallerRa* regions, the averaged Nusselt number Nu is al-most independent of the Darcy-Rayleigh number and

its value is approximately 2.5. Beyond the aboveregion, the relationship of Nu with Ra* is almost lin-ear based on the logarithmic coordinates. Theequations of these two parts are shown in Eqs. (26)

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W.S. Fu and W.W. Ke: Evaporation of Water Film in An Enclosure Filled with A Porous Medium 335

2.00

1.00

V 0.00

-1.00

V

Y

Y

Y

Y

' = 0.00375= 0.1453

= 0.4873

= 0.8547

= 0.99625

I I I . I I

-2.000.0 0.2 0.4 0.6 0.8

X

( a ) E = 0.4 and Dp = 0.005

1.0

-1000.0 0.2 0.4 0.6 0.8 1.0

X

( c ) e = 0.7 and Dp = 0.005

V

400

200

0

-200

\^\

\v

i

i

i ' i

I . I ,

-

-

-

•

\ .\

V

1-400

0.0 0.2 0.4 0.6

X

0.8 1.0

200

100

V 0

-100

-200

\

"O • , • ^ ^

-

i i 1 I

1

i ,

-

-

^ - i

rj

0.0 0.2 0.4 0.6 0.8 1.0

X

( b ) e = 0.4 and Dp = 0.05

800

400

V 0

-400

-800

• \

-

•

> \

1

1

1 1

1

1 1

-

-

\'\ 1"yjV.

J

0.0 0.2 0.4 0.6 0.8 1.0

X

( d ) e = 0.7 and Dp = 0.05

V

1400

1000

600

200

-200

-600

-1000

• I '

%

- w \

-•

-

1

1 •

- . ^ - , w ^

1 , 1 .

1

-

-

\ v • 1• \ \ 'x . v '

\ V'!\ \ ~*\ . i

I , •<•

-14000.0 0.2 0.4 0.6

X0.8 1.0

( e ) e = 0.9 and Dp = 0.005 ( f ) z = 0.9 and Dp = 0.05

Fig. 7. The distributions of velocity V along the X-axis at different heights of K-axis under situation of /?a=3.678x!06 situation.

and (27)

JoNu = Nu(Y)dY

Nu=2.5±0.5 for Ra<4

(25)

(26)

Nu = 1.9463fla*°202 for 4</?a*<4469

V. CONCLUSIONS

(27)

A double diffusive natural convection in anenclosure filled with a porous medium under an

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336 Journal of the Chinese Institute of Engineers, Vol. 22, No. 3 (1999)

1210

8

6

4

Nu

i i mill

-.

-

••

MM Nil

—i i mini—rTTTirm—i i mmr i

>• Present results

~ - Eq.(26) /r?nf97\ /•E,q\ LI ) s

i i i n u l l i 1 1 I I I I I I i i i m i l l i

I I I I I I I 1 1 I I I I I I

/

-

-

i t m i l i i i i n n1

0 0.1 1 10 100 1000 10000

Ra*Fig. 8. The relationship of averaged Nusselt number Nu and

Darcy-Rayleigh number Ra°.

evaporation situation was investigated numerically.The effects of Rayleigh number Ra, porosity e anddiameter of beads Dp on heat transfer were examinedwith the D-B-F model. The results are summarizedas follows:(i) The effect of evaporation on the reduction of tem-

perature of the heated wall is remarkable, espe-cially in the upper corner.

(ii) The latent heat flux plays an important role inthe heat transfer mechanism for high Darcynumbers. The distribution of the ratio R( in thelower Ra region is larger than that in the high Raregion as £>a>10~5.

(iii) The relationship of averaged Nusselt number Nuwith Darcy-Rayleigh number Ra" is almost lin-ear in logarithmic scale when 4<Ra "<4469-, andNu is approximately 2.5 when Ra*<4.

NOMENCLATURE

C Mass or concentration fraction

Cp Specific heat of fluid (kJkg"1 "C"1)

dp Porous bead diameter (m)D Binary diffusion coefficient ( m V )Da Darcy numberDp Dimensionless porous bead diameter,

(dp/H)F Inertial factor; Forchheimer factorg Gravitational acceleration (ms~2)Grc Grashof number for mass diffusion, [gfic

Gr,

Khfsht

HKff

KLrhNuNu

PPPr

QwRa

ScShTuUvVw

x,yX, Y

PE

PWE

NPE

NPWE

Grashof number for thermal diffusion,[(gprqwH4)/(keffV

2R)]

Mass transfer coefficient along vertical wallVaporization enthalpy (Jkg"1)Thermal heat transfer coefficient along ver-tical wall (WnT2 °C-')Dimensional width of porous enclosure (m)Effective thermal conductivity of porousmedium (War1 "C"1)Thermal conductivity of fluid (Wm"1 °C~')Thermal conductivity of solid phase in po-rous medium (Wm"1 °C~')Permeability (m2)Dimensional length of porous enclosure (m)Evaporation mass flow rate (Kgm"2s~')Local Nusselt number along vertical wallMean Nusselt numberDimensional pressure (Nm~2)Dimensionless pressurePrandtl numberHeat flux (WnT2)Rayleigh number, [Gr, Pr=(gP,qwH4)/

Darcy-Rayleigh number, [RaDa =(gPiqwH2K)/(keffvRaR)]Ratio of latent heat flux in left wall to totalheat flux in the left wallRatio of sensible heat flux in left wall tototal heat flux in the left wallSchmidt number, (~)Local Sherwood number along vertical wallTemperature (°C)Dimensional velocity in x direction (ms"1)Dimensionless velocity in X directionDimensional velocity in y direction (ms"1)Dimensionless velocity in Y directionDimensionless concentractionDimensional Cartesian coordinate (m)Dimensionless Cartesian coordinate, (=x/H,=ylH)Porous medium with evaporation situationin enclosurePorous medium without evaporation situa-tion in enclosureNon-porous medium with evaporation situ-ation in enclosureNon-porous medium without evaporationsituation in enclosure

Greek symbols

a

Pc

Thermal diffusivity (m2s ')Coefficient of volumetric expansion withtemperature (°C~')Coefficient of volumetric expansion withmass fraction

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W.S. Fu and W.W. Ke: Evaporation of Water Film in An Enclosure Filled with A Porous Medium 337

V

9e*

Porosity (m3m 3)Viscosity (kgrrr's"1)Kinematic viscosity ( m V )Dimensionless temperatureModified dimension temperature (6/keff)

Fluid density (kgm 3)Relative humidity

0¥

Computation variableDimensionless stream

Superscripts

n——>

The nth iteration indexMean valueVelocity vector

Subscripts

C.V.effEfiIL

PRsSW

Control volumeEffective valueEnclosureFluid phaseIndexLatent heatLeft wallPorous mediumRight wallSolid phaseSensible heatSolid wall

OTHER

I I Magnitude of velocity vector

ACKNOWLEDGMENT

The support of this work by the National Sci-ence Council, Taiwan, R.O.C. under contract NSC86-2212-E-009-042 is gratefully acknowledged. The au-thors also appreciate the instructive suggestions fromDr. Huang.

REFERENCES

1. Alavyoon, F., 1993, "On Natural Convection inVertical Porous Enclosures Due to PrescribedFluxes of Heat and Mass at the VerticalBoundaries," International Journal of Heat andMass Transfer, Vol. 36, pp. 2479-2498.

2. Alavyoon, F., Masuda, Y. and Kimura, S., 1994,"On Natural Convection in Vertical PorousEnclosures Due to Opposing Fluxes of Heat andMass Prescribed at the Vertical Walls," Interna-tional Journal of Heat and Mass Transfer, Vol.

37, pp. 195-206.3. Fu, W.S., Huang, H.C. and Liou, W.Y., 1996,

"Thermal Enhancement in Laminar Channel Flowwith a Porous Block," International Journal ofHeat and Mass Transfer, Vol. 39, pp. 2165-2175.

4. Fu, W.S. and Huang, H.C., 1997, "Thermal Per-formances of Different Shapes of Porous Blocksunder an Impinging Jet," International Journalof Heat and Mass Transfer, Vol. 40, pp. 2261-2272.

5. Fujii, T., Kato, Y. and Mihara, K, 1977, "Expres-sion of Transport and Thermodynamic Propertiesof Air, Steam and Water," Sei San Ka GakuKenkyu Jo, Report no. 66, Kyu Shu Dai Gaku,Kyu Shu, Japan.

6. Goyeau, B., Songbe, J.P. and Gobin, D., 1996,"Numerical Study of Double-diffusive NaturalConvection in a Porous Cavity Using the Darcy-Brinkman Formulation," International Journalof Heat and Mass Transfer, Vol. 39, pp. 1363-1378.

7. Hadley, G.R., 1986, "Thermal Conductivity ofPacked Metal Powders," International Journalof Heat and Mass Transfer, Vol. 29, pp. 909-920.

8. Hsiao, S.W., Cheng, P. and Chen, C.K., 1992,"Non-uniform Porosity and Thermal DispersionEffects on Natural Convection About a HeatedHorizontal Cylinder in an Enclosed PorousMedium," International Journal of Heat andMass Transfer, Vol. 35, pp. 3407-3418.

9. Kon, J.C.Y. and Fortini, A., 1973, "Prediction ofThermal Conductivity and Electrical Resistivityof Porous Metallic Materials," International Jour-nal of Heat and Mass Transfer, Vol. 16, pp. 2013-2022.

10. Lai, F.C. and Kulacki, F.A., 1991, "Coupled Heatand Mass Transfer by Natural Convection fromVertical Surfaces in Porous Media," InternationalJournal of Heat and Mass Transfer, Vol. 34, pp.1189-1194.

11. Lauriat, G. and Prasad, V., 1989, "Non-DarcianEffects on Natural Convection in a VerticalPorous Enclosure," International Journal ofHeat and Mass Transfer, Vol. 32, pp. 2135-2148.

12. Murray, B.T. and Chen, C.F., 1989, "Double-dif-fusive Convection in a Porous Medium," Journalof Fluid Mechanics, Vol. 201, pp. 147-166.

13. Nithiarasu, P., Seetharamu, K.N. andSundararajan, T., 1997, "Natural Convective HeatTransfer in a Fluid Saturated Variable PorosityMedium," International Journal of Heat andMass Transfer, Vol. 40, pp. 3955-3967.

14. Patankar, S.V., 1980, "Numerical Heat Transferand Fluid Flows," Hemisphere, Washington

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338 Journal of the Chinese Institute of Engineers, Vol. 22, No. 3 (1999)

D.C.15. Prasad, V., Kladias, N., Bandyopadhaya, A. and

Tian, Q., 1989, "Evaluation of Correlations forStagnant Thermal Conductivity of Liquid-satu-rated Porous Beds of Spheres," InternationalJournal of Heat and Mass Transfer, Vol. 32, pp.1793-1789.

16. Renken, K.J. and Aboye, M., 1993, "Analysis ofFilm Condensation Within Inclined Thin Porous-layer Coated Surfaces," International Journal ofHeat and Fluid Flow, Vol. 14, pp. 48-53.

17. Shonnard, D.R. and Whitaker, S., 1989, "TheEffective Thermal Conductivity for Point Con-tact Porous Medium: an Experimental Study," In-ternational Journal of Heat and Mass Transfer,Vol. 23, pp. 503-512.

18. Singh, B.S., Dybbs, A. and Lyman, F.A., 1973,"Experimental Study of the Effective ThermalConductivity of Liquid Saturated Sintered FiberMetal Wicks," International Journal of Heat andMass Transfer, Vol. 16, pp. 145-155.

19. Trevisan, O.V. and Bejan, A., 1985, "NaturalConvection with Combined Heat and Mass Trans-fer Buoyancy Effects in a Porous Medium," In-ternational Journal of Heat and Mass Transfer,Vol. 28, pp. 1597-1611.

20. Vafai, K., 1984, "Convection Flow and HeatTransfer in Variable Porosity Media," Journal ofFluid Mechanics, Vol. 147, pp. 233-259.

21. Van Doormaal, J.P. and Raithby, G.D., 1984,"Enhancements of the SIMPLE Method for Pre-dicting Incompressible Fluid Flow," NumericalHeat Transfer, Vol. 7, pp. 147-163.

APPENDIX

Thermophysical properties of air, steam and

air-moist mixture, in T[K], T[°C] and P[bar].(i) Air

,1.5

.1.6

(kgm-3)

(kgm-'s"1)

(kgnr'K-1)

-io r3 ) x ] O3CPfl=(l+2.5xl0-10r)xl0

(ii) Steam

f)0 8](f) (kgm-3)- 6AW=(8.02+0.04/)xl0

^,m=(1.87+1.65xl0-¥/7+5.7xl0-'¥')xl0-2

(kgm-'K-1)

CPt,m=1.863xl03+1.65xl0-V5+1.2xl0-'¥5

(Jkg-'K"1)stm'

(iii) Air-moist mixture

1 + coPm = [0.455(<w + 0.622)1] x 10- 2

1+0.352(1+0.888stm

Vstm

1+0.1727(1 + 1.1259

(kgm-J)

(kgm-'s-1)

1 +0.4018\ 1+0.837 ft (1

Vstm (1

. 115.:1 T. 559.'' T

" stm

l+0.15556\ 1 + 1.195 r^stm (H

(n

559.7 xT }

115.5xT }

(kgm-'K-1)

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W.S. Fu and W.W. Ke: Evaporation of Water Film in An Enclosure Filled with A Porous Medium 339

stm

A,,=8.O7xlO-6r1833

(iv) Others

(Jkg-'K"1)

(mV)

The evaporation latent heat from the liquid phaseto vapor phase for water saturation

hfg=A\ 86.8x(597.3-0.5590

The saturation pressure of the vapor phase forwater

L o g i n ( ^ ) = -[3.1323 + 3.116xl(T6

Humidity ratio

co = P -Ptot sat

Mass fraction of air-moist mixture

C =0)

1 + CO

Discussions of this paper may appear in the discus-sion section of a future issue. All discussions shouldbe submitted to the Editor-in-Chief.

Manuscript Received: Oct. 17, 1998Revision Received: Jan. 29, 1999

and Accepted: Feb. 04, 1999

71=1 zlc

\$.1iii.BM SIMPLEC S5|5^Darcy-Brinkman-Forchheimer model #)W$1?

B*MM ' &M Darcy-Rayleigh W ' WWkW&®&WI§L#}W&fa& ° ^Darcy-

Rayleigh tfc/K5£ 4 ' ^i^Nusselt WLffiU 2.5 ; E Darcy-Rayleigh WJYifc 4 %}

4469 ZF$ ' ¥ ^ Nusselt iffn Darcy-Rayleigh %L&MWL£M}#MM&Mtifr °

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