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Non-dimensional numbers associatedwith the evaporation from a capillaryporous medium / Nombres sansdimensions associés à l'évaporation ausein d'un milieu poreuxNICHOLAS K. GIDAS a & TRYFON CONSTANTINOU ba Etudes Hydrauliques et Ecologiques, Direction des OuvragesHydrauliques, Ministère de l'Environnement du Québec , 1640boulevard de l'Entente, Québec, G1S 4NG, Canadab Defence Research Establishment Valcartier , National DefenceCanada, PO Box 8800, Courcelette, Québec, GOA 1RO, CanadaPublished online: 25 Dec 2009.

To cite this article: NICHOLAS K. GIDAS & TRYFON CONSTANTINOU (1983) Non-dimensionalnumbers associated with the evaporation from a capillary porous medium / Nombres sansdimensions associés à l'évaporation au sein d'un milieu poreux, Hydrological Sciences Journal,28:4, 539-549, DOI: 10.1080/02626668309491994

To link to this article: http://dx.doi.org/10.1080/02626668309491994

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Hydrological Sciences - Journal - des Sciences Hydrologiques, 28, 4, 12/1983

Non-dimensional numbers associated with the evaporation from a capillary porous medium

NICHOLAS K. GIDAS Etudes Hydrauliques et Ecologiques, Direction des Ouvrages Hydrauliques, Ministère de 1'Environnement du Québec, 1640 boulevard de 1'Entente, Québec, Canada GIS 4NG

TRYFON CONSTANT INOU Defence Research Establishment Valcartier, National Defence Canada, PO Box 8800, Courcelette, Québec, Canada GOA 1RO

ABSTRACT The different factors that produce evaporation of moisture from a capillary porous medium are being investigated. Thermodynamic measurements of the porous medium-air-water system and a mathematical model have permitted the establishment of some correlations between the principal non-dimensional numbers associated with heat and mass transfer between the evaporating water in the porous medium and the external gas stream. These correla­tions lead to the prediction of some important parameters such as the depth of the surface of vaporization and the exchange coefficients of the heat and mass transfer during the evaporation from a capillary porous medium.

Nombres sans dimensions associés à 1'evaporation au sein d'un milieu poreux RESUME La présente recherche met en simulation les différents facteurs qui provoquent 1'evaporation de l'eau au sein d'un milieu poreux-capillaire humide. Les mesures thermodynamiques dans le système milieu poreux-eau-air, et les calculs à partir d'un modèle mathématique ont permis d'établir des corrélations reliant les principaux nombres adimensionnels associés au transfert de masse et de chaleur entre le milieu poreux et l'écoulement gazeux superposé. Ces corrélations permett­ent de prédire certaines grandeurs importantes comme la profondeur de la surface de vaporisation et les coeffic­ients d'échange thermique et massique lors de 1'evaporat­ion au sein du milieu poreux capillaire.

INTRODUCTION

The purpose of this study is to introduce research findings on the correlations between certain non-dimensional numbers associated with the process of evaporation from a wetted porous medium.

In the last few years, the evaporation of moisture from capillary porous media has had many applications in the different installations aimed at heat transfer, in cryogenic energy transmission lines and in other such equipment (Dinulescu & Eckert, 1980; Luikov, 1975).

539

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540 Nicholas K. Gidas & Tryfon Constantinou

4\ •f. ' •1< 7-

-:'-fc

f'i I' \ j è ••

tt-: . '* *"• » * i£*(f

i 5 ? : « i <*. f

Fig. 1 Experimental installation: (a) photograph; (b) sketch. 0 = porous medium; 1 = dry zone; 2 = thermocouple connections with "Solartron voltmeter"; 3 = thermocouple cable; 4 = vaporization surface; 5 = electrical outlet; 6 = heat resistance regulator; 7 = water saturated zone; 8 = porous plate; 9 = water layer; 10 = Dewar container with ice (thermocouple reference temperature at 0°C); 11 = distilled water; 12 = air warmer tunnel; 13 = cryostat; 14 = automatic "Solartron voltmeter"; 15 = registering apparatus; 16 = rotary pumps; 17 = diffuser; 18 = communication set valve between vacuum pump and wind tunnel; 19 = air flow pressure regulator valve; 20 = mercury manometer; 21 = electrical autotransformer for infrared lamps; 22 = electronic hygrometer; 23 = wind tunnel wall; 24 = thermometric probe; 25 = thermal boundary layer; 26 = desiccator.

Similarly, in agricultural hydraulics it is desirable to maintain a

certain optimum level of humidity in the soil for irrigation or

drainage purposes taking into account the process of evaporation-

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Evaporation from a capillary porous medium 541

t r a n s p i r a t i o n (Bouchet, 1965; Brutsaer t & S t r i e k e r , 1979; Royer & Vachaud, 1974).

The present paper forms a sequel t o an e a r l i e r study (Gidas, 1979) and has the following o b j e c t i v e s :

(a) t o compile experimental r e s u l t s of a thermodynamic na ture in and above a porous medium ( F i g . l ) fed by an a r t i f i c i a l sheet of water ;

(b) to e s t a b l i s h a mathematical model pe rmi t t i ng the computation of the p r i n c i p a l non-dimensional numbers a s soc i a t ed with mass and heat t r a n s f e r during the evaporat ion of moisture from a porous medium;

(c) t o e s t a b l i s h non-dimensional c o r r e l a t i o n s p e r m i t t i n g the p r e ­d i c t i o n of t he depth of evaporat ion and the heat exchange c o e f f i c i e n t as a function of the thermodynamic c h a r a c t e r i s t i c s of the system as well as of the geometr ical parameters of the porous medium.

EXPERIMENTAL CONDITIONS

Consider a steady laminar flow of dry air inside a wind tunnel under constant pressure (Godin & Gidas, 1973). The experimental installat­ion is shown by a photograph in Fig.1(a) and by a corresponding diagram in Fig.1(b). The different parts of the apparatus are defined in the legend of Fig.l.

A porous element (no. 0) is placed in the lower wall of the tunnel. The bottom of the porous substance (no. 8) receives water hydrostatically from a supply tank (no. 11) maintained at a constant temperature. Variation of the thermal state of the porous-air system is achieved by means of a thermal regulator as well as two infrared radiation emitters (no. 21). The latter are placed above the air flow in such a way that the thermal flux is spread out uniformly on the surface of the porous medium. This medium is made up of glass grains of 1.5 mm diameter. The shape of the porous medium is that of a parallelepiped whose longitudinal cross section is presented in Fig.2.

A vertically adjustable graduated bottle is used to supply the porous medium with water in a permanent manner (Fig.2); the scale

Fig. 2 Schematic diagram of water supply of the porous medium and the vertical distribution of the water saturation: i = imbibition; d = drainage; S = saturation.

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542 Nicholas K. Gidas & Tryfon Constantinou

marked on the container permits direct reading on the flow of water vapour. The water level in this supply bottle is kept constant during an experiment through the system (no. 11) shown in Fig.1(b). The water flows in the porous medium and rises by capillary action to a higher point than that occupied in the bottle; there, it evaporates. In the beginning of the drying process and under the condition that there is a vertical distribution of water saturation (S) as a function of depth (y) (Fig.2), the porous medium contains both air and liquid; the air is saturated with water vapour. During this period, the total mass flux transferred to the surface of the porous medium is composed of a flux in the liquid phase and a flux in the vapour phase while the rate of drying is constant. As the drying continues, the liquid flux in the upper zone diminishes and at some later time the liquid no longer reaches the surface; thus the permanent regime of evaporation is established. During this regime, the only vertical flux which crosses the dry zone towards the surface is that of the vapour phase.

The measurements and results of the present study are applicable in this permanent regime and therefore the only flux considered is that of the vapour phase.

In the steady-state regime of evaporation three different zones (Fig.1(b)) in the porous medium can be identified: the water saturated zone (no. 7), the evaporation zone (no. 4) whose thickness is so small that the term evaporation surface is used, and finally the water vapour zone (no. 1).

MATHEMATICAL MODEL

The two-dimensional, steady laminar flow of dry air is again considered. This is parallel to the horizontal surface of the porous medium (Fig.l). The heat transfer between the evaporative surface and the external surface of the porous medium is assumed to take place under one-dimensional conduction. The effect of the wind speed (V ) of water vapour on the boundary layer of the air flow is negligible and the physical properties of gas flow in the boundary layer are constant.

According to the hypotheses of Luikov (1966), the global equation of energy conservation is as follows:

3T 3T 32T u ^— + v — = a -r—2- (1)

3x 3y 3y

The s o l u t i o n of e q u a t i o n (1 ) g i v e s :

= e r f ( y H / 2 K ) + exp(Hy + K^) e r f c [ ( y H / 2 K ) + K] (2 ) T ( x , y ) - T v *,..„,„„ , „ „ „ , „ „ , T,2.

(T (x) - T v ) / ( T o o - T v ) = e x p ( l O e r f c ( K ) (3 )

where

K = / (8 /5 îT) H x / / P e (4 )

H = A * / ( A g h v ) (5 )

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Evaporation from a capillary porous medium 543

a is thermal diffusivity; T(x,y) is temperature at some point in the axis system (x,y), T is temperature at the surface of vaporization, T p is temperature at the surface of the porous medium, Pe is the Pec let number, X* is thermal conductivity of the dry zone of the porous element, Ag is thermal conductivity of wet air, hy is depth of surface of vaporization (Fig.2). According to a table of densities of heat flux, the thermal conductivity As can be calculated as Xs = 0.274 W m_1 °C_1 (De Vries, 1975).

Under the experimental conditions considered, the transfer of humidity in the porous medium would take place in the form of water vapour. The density of mass flux of water vapour Jv is expressed by Le Fur (1965):

Jv = - Xmv VWe - Xmtv VT (6)

where the coefficient of mass conductibility (Xmv) in the vapour phase is :

Xmv = f(We) D | ^ Pys <|J e) T (7)

Similarly the coefficient of mass conductibility Xmtv due to a temperature gradient is:

Xmtv = f(W e) ï L / * Fvs D R * F r r N + n T ( 8 )

cf. = exp (-mPc/RTpe) (9)

P„ = [2Y(T)cos6]/r (10)

respectively of water vapour in air; f(We) is the non-dimensional coefficient which depends on the porous medium and the degree of humidity; We is the degree of humidity; P v s is the saturation pressure of water vapour; p e, p v are densities of water and water vapour; m is molecular mass; R is the gas constant; Pc is capillary pressure. It was assumed that the junction angle (9 = 0°) and the average radius of the pores (r s 0.75 mm) of the medium do not vary with temperature. The surface tension (y) in dynes cm- is calculated as a function of the critical temperature (Tc = 374.2 C) using Eotvos1 formula (Gidas & Gidas 1974).

The local Nusselt number (Nu)p which describes the heat transfer between the air flow (Fig.l) and the surface of the porous medium is expressed by equation (11). However, the local Nusselt number (N u) v

which denotes the heat transfer between the air flow and the vaporization surface is expressed by equation (12):

(Nu)p = T P

3T(x,y) 3y

(11)

<Nu>v = ̂ T ^ f (N u) p (12)

Taking into account equations (2) and (3), it can be deduced from equations (11) and (12):

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544 Nicholas K. Gidas & Tryfon Constantinou

( N u ) p = /(578TT) / P e f (K) {1 - [ f (E)ZTrK)]} - 1 ( 13 )

( N u ) v = / ( 5 / 8 T T ) / P e f (K) (14 )

By definition the relative Nusselt numbers are:

<Nu)p = (N u) p/(N u) i s (15)

(Nu>* = <Nu>v/(Nu>is <16>

<Nu>R = <Nu>p/(Nu>v <17>

where

f(K) = A K exp(K2) erfc(K) (18)

<Nu>is = °- 3 3 2 Rx* P r 1 / 3 <19>

Rx = Reynolds number, Pr = Prandtl number.

The blow number (B) is defined by equation (20). The discharge coefficient (C„) and the friction coefficient without blowing (Cf) are given by equations (21) and (22) respectively.

B = 2 Cq/Cf (20)

Cq = Pp Vp/pa Va (21)

_ A

Cf = 0.664 Rx 2 (22)

Substituting equations (6), (21) and (22) in (20) it follows:

B = -3.012 55LJL. (Xmv VWe + Amtv VT) ( 2 3 )

pa va

The Damkohler number (Da) which describes the depth of vaporization surface hv is given by:

1 °a V ^ C r I* s

Là—£L_Tpa f 2 4 >

D„ - A* ( 2 4 )

where P , Va, C p a are the density, the average speed and specific heat of air above the porous medium; pp is the density of water vapour at the surface of the porous medium.

The capillarity number (N ) representing the relationship of capillary forces to the force due to the longitudinal component of gravity is:

A(T) cos9 a(£) (25) N CS K' (pe - py) g sina

3 2

e"d K, = _ p _ _ T (26)

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Evaporation from a capillary porous medium 545

where a(e) is a function of the porosity (e = 0.37) of the porous medium (s/e); g is acceleration due to gravity; pe and pv are densities of liquid water and water vapour. Since the flow is vertical sin a = 1. The coefficient of permeability K' (single-phase) was calculated from equation (26) of Kozeny-Karman; d is the average diameter of grains (= 1.5 mm) and t = tortuosity (S/TT/2) .

The number (1/G.) represents the height of capillary elevation (hc) with respect to the depth of vaporization surface (hv) and the number Nx represents the relationship of the abscissa (x) with respect to the length of the porous medium (L):

1/Gi = hc/hv (27)

Nx = x/L (28)

Since the diameter of the grains of this porous medium is constant, it follows that the function a(r) of the volume distribution of capillary radii is also constant. Consequently the height hc is determined by hc = Pc/(pe - pv)g and equation (10). However, when the liquid and vapour phases are present in a heterogeneous porous medium whose function a(r) is not constant, the liquid-vapour distribution will be at equilibrium if the two phases are separated by a surface of nearly uniform curvature. This form is extremely complicated to be determined analytically. It is possible never­theless to envisage a formula of the form:

ne = «S») ̂ ^ ^ ^ f l (29)

/K (pe - pv) g

provided that the function J(Sm) is known. This is the function of the saturation of the wetting fluid that determines the family of the porous medium.

The depth hv is a function of several factors which can be associated with the porous medium (granulometry, permeability, capillarity, saturation, thermal conductivity, temperature, etc.) and also with the thermodynamic state of the applied air flow (velocity, pressure, boundary layer, thermal flux, humidity, etc.)

Considering the random character of capillary height in a hetero­geneous porous medium whose function a(r) is not constant, as well as the double flux (liquid and vapour) in the evaporation zone, it is believed that the depth hv will vary bidimensionally along the upper surface of the porous medium. It is also possible that the evaporation zone can no longer be compared to a flat surface. Measurements taken have shown hv to vary slightly with x even for a porous medium with practically uniform granulometry.

EXPERIMENTAL AND CALCULATED RESULTS

During deep evaporation, the controlled parameters are: the heat flux injected in the porous medium-air-water system, the average speed (Va) , the pressure (Pa) of the air flow and the temperature (T) in the boundary layer of air in the porous medium. Temperature measurements permitted the determination of the profiles shown in

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546 Nicholas K. Gidas & Tryfon Constantinou

Fig.3. The water temperature in the supply tank (Fig.l) was maintained constant at 15 C whereas the measured temperature at the vaporization surface was below 15 C. For every test, the minimum temperature of the profiles (Fig.3) corresponds to the depth (hv) of

120--

-75 -50 -25 0 25 50 75 y (mm)

I porous medium J. air flow 1

Fig. 3 Vertical distribution of temperature along the axis x: without radiation (N.R.j; with radiation (W.R.); Pa = pressure of air flow; X i , x 2 , x3 = values of x.

the vaporization surface. * *

The variation of Nusselt numbers (N„) and (N ) as a function " P u V

of (hv) is established in Figs 4 and 5 respectively. The divergence of results (Fig.5) between experimental and theoretical values of Luikov for shallow depth (hv) is attributed to the simplified hypotheses of the theory. The numbers (Nu) and (Nu) take a maximum value for a certain depth (hv) near the surface of the porous medium. The variation of numbers (N u) p and (N u) v as a function of B is presented in Figs 4 and 5 respectively. The relative height of the capillarity (1/G^) is a hyperbolic function of the number (Ncg) whereas the same relative height (1/G^) is a decreasing function of the relative number (N U) R (Fig.6). It is observed (Fig.7) that the number (1/Da) is a decreasing hyperbolic function of the capillarity number (N c g); it is also observed that B increases when 1/D„ decreases and reaches a maximum for 1/D <10.

a All the results tend to indicate that approximate equations can be

written in the form:

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Evaporation from a capillary porous medium 547

1/D„ = A-, + Ao B^l + Ao N -a -n 3 "eg

(Nu) = A4 + ( N x / G i ) B"3

(30)

(31)

0

», (Nu)p

3-

2"

1.

,

\

\

20 40

(Nulp^lB)

(Nu)p = f2(hv)

x=x„*l53mm

y* xa+53mm

60

Q ^

h„lm

0 0.10 0.20 0,30 B

Fig. 4 Variation of the number (Nu)* as a function of B and hv

0 0.10 0.20 0.30 B

Fig. 5 Variation of trie number (N )* as a function of B and h

In order to determine accurately the coefficients Aj,..., A4 and the exponents Z 1 ( Z2, Z3, it is necessary to carry out measurements and computations with different porous media.

CONCLUSIONS

The experimental results and the mathematical model have permitted the establishment of certain correlations between the principal non-dimensional numbers associated with the process of evaporation of moisture from a porous medium. According to this study certain

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548 Nicholas K. Gidas & Tryfon Constant!nou

Fig. 6 Variation of the number G; as a function of N and (NU)R .

Fig. 7 Variation of the number (Da) as a function of N and B.

important parameters l i ke the depth of the vapor i za t ion surface and the heat and mass exchange c o e f f i c i e n t s can be determined as a funct ion of the phys ica l and geometrical c h a r a c t e r i s t i c s of the porous medium-air-water system.

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Evaporation from a capillary porous medium 549

REFERENCES

Bouchet , R.J. (1965) Evapotransp ira t ion r é e l l e , e v a p o r a t i o n p o t e n t i e l l e e t product ion a g r i c o l e . Dans: L'Eau et la Production Végétale, 1 5 1 - 2 3 2 . Centre N a t i o n a l e de Recherches Agronomiques, INRA, P a r i s .

B r u t s a e r t , W. & S t r i e k e r , H. (1979) An a d v e c t i o n - a r i d i t y approach to e s t i m a t e a c t u a l r e g i o n a l é v a p o t r a n s p i r a t i o n . Wat. Resour. Res. 15 ( 2 ) , 4 4 3 - 4 5 0 .

De V r i e s , D.A. (1975) Heat t r a n s f e r in s o i l s . In: Heat and Mass Transfer in the Biosphere, part I : Transfer Processes in the Plant Environment, 5 - 2 8 . John Wiley , New York.

D i n u l e s c u , H.A. & Eckert E.R.G. (1980) A n a l y s i s of the one-dimensional mois ture m i g r a t i o n caused by temperature g r a d i e n t s in a porous medium. Int. J. Heat Mass Transfer 2 3 , 1069 -1078 .

Gidas , N.K. (1971) Champ de v i t e s s e e t de température à l ' i n t é r i e u r d'un c a l o d u c . Revue Gên. de Therm. X ( 1 1 8 ) , 8 4 3 - 8 6 3 .

Gidas , N.K. (1979) Recherche sur l e s nombres sans d imens ions a s s o c i é s à 1 ' é v a p o r â t i o n au s e i n d'un m i l i e u poreux . Proc. IAHR, XVIIIe Congress ( C a g l i a r i , I t a l i e ) , v o l . 3 , 3 6 5 - 3 7 4 .

Gidas , N.K. & Gidas , B.K. (1974) Nondimensional numbers a s s o c i a t e d with heat t r a n s f e r in heat p i p e s . Proc. Int. Conf. on Computat­ional Methods in Nonlinear Mech. (Univ. o f Texas a t A u s t i n , USA), 7 0 5 - 7 1 4 .

Godin, A. & Gidas , N.K. (1973) In f luence du c h a u f f a g e par rayonnement sur l e nombre de N u s s e l t e n t r e un écoulement d ' a i r e t un m i l i e u poreux s a t u r é d ' e a u . C.R. IVe Congrès Canadien de Mécanique Appliquée ( M o n t r é a l ) , 5 0 3 - 5 0 4 .

Le Fur, B. (1965) Transport de cha l eur e t de mat i ère dans l e séchage des matériaux poreux. Génie Chimique 94 ( 6 ) .

Luikov, A.V. (1966) Heat and Mass Transfer in Capillary-Porous Bodies, 132-149 , 4 8 7 - 4 8 5 . Pergamon P r e s s .

Luikov, A.V. (1975) Corps p o r e u x - c a p i l l a i r e s e t t r a n s f e r t de c h a l e u r ou d ' é n e r g i e . Revue Gên. de Therm. VI ( 1 5 7 ) , 9 - 1 5 .

Royer , J.M. & Vachaud, G. (1974) Determinat ion d i r e c t e de l ' é v a p o ­t r a n s p i r a t i o n e t de l ' i n f i l t r a t i o n par mesure des t e n e u r s en eau e t des s u c c i o n s . Hydrol. Sci. Bull. 19 ( 3 ) , 39 , 3 1 9 - 3 3 3 .

Received 30 December 1982; accepted 29 April 1983.

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