Vapor-Liquid Jump Conditions within a Porous Medium

Transport in Porous Media 40: 73-111,2000. Printed in the Netherlands.

© 2000 Kluwer Academic Publishers.

73

Vapor-Liquid Jump Conditions within a Porous Medium: Results for Mass and Energy 10RGEN HAGER] and STEPHEN WHITAKER 2 ] Department of Chemical Engineering I, University of Lund, PO Box 124, S-221 00 Lund, Sweden 2 Department of Chemical Engineering and Material Science, University of California at Davis,

Davis, CA 95616, U.S.A.

(Received: 14 September 1998; in final form: 12 January 1999)

Abstract. In this paper, we develop the mass and energy jump conditions for a vapor-liquid boundary within a porous medium. The analysis is restricted to a single fluid component and leads to the appropriate jump conditions for an evaporation front. The condition of local thermal equilibrium is assumed to exist in the homogeneous liquid and vapor regions, but not in the boundary region where rapid changes in the saturation occur. Key words: jump condition, evaporation front, boundary region, vapor-liquid interface.

Nomenclature AWI)(t) Aw(t) AI)(t)

Aoo A,Ba (t) A,By(t) Aya(t) C(t) cp (pcp)a «(p)Cpb «(p)Cp)1) (pcpv)s ha (ha)a

Al)w(t) area of the dividing surface between the w- and I)-regions, m 2 . bounding surface of the homogeneous part of the w-region, m 2 . bounding surface of the homogeneous part of the I)-region, m 2 . Aw(t) + AI)(t), bounding surface of the volume, Voo , m 2 . Aa,B(t), area of the {J-a interface contained in the averaging volume, m 2 . Ay,B(t), area of the {J-y interface contained in the averaging volume, m 2 . Aay (t), area of the y-(J interface contained in the averaging volume, m 2 . bounding curve for the area, AWl) (t), m. heat capacity, J/(kg K). volumetric heat capacity of the a-phase (a = {J, y, a), J/(m 3 K). c,Bw(pcp),B + caw(pcp)a, volumetric heat capacity for the w-region, J/(m 3 K). cyl)(pcp)y + Cal) (pcp)a, volumetric heat capacity for the I)-region, J/(m 3 K). excess surface convective heat capacity, J/(m s K). enthalpy per unit mass for the a-phase (a = {J, y, a), J/kg. intrinsic average enthalpy per unit mass for the a-phase (a = {J, y, a), J/kg. intrinsic average enthalpy per unit mass for the a-phase in the w-region (a = {J, y, a), J/kg. deviation enthalpy per unit mass for the a-phase (a = {J, y, a), J/kg. enthalpy per unit mass for the a-phase at the reference temperature (a = {J, y, a), J/kg. thermal conductivity of the a-phase (a = {J, y, a), J/(m 2 s K). effective thermal conductivity tensor for the a-phase in the homogeneous w-region, J/(m 2 s K).

74 K*

KZ, K~

K; Cf3

Cy CB L (ril) (ril)s

of3cr 0",

01) 0"'1)

P

To Ta (Ta)a

1'a

10RGEN HAGER AND STEPHEN WHITAKER

dynamic thermal dispersion tensor, J/(m2 s K), thermal dispersion tensor in the homogeneous w-region, J/(m2 s K), thermal dispersion tensor in the homogeneous I)-region, J/(m 2 s K). surface thermal dispersion tensor, J/(m s K). small length scale associated with the ,a-phase, m. small length scale associated with the y-phase, m. thickness of the boundary region, m. large length scale, m. mass rate of evaporation per unit volume, kg/(m 3 s). mass rate of evaporation per unit area, kg/(m2 s). -ocrf3 unit normal vector directed from the ,a-phase toward the a-phase. outwardly directed unit normal vector for the w-region. outwardly directed unit normal vector for the I)-region. -nl)'" unit normal vector directed from the w-region toward the I)-region. 1- of3 y of3y, projection tensor. heat flux vector in the a-phase (a = ,B, y, a), J/(m2 s). radius of the averaging volume, m. time, s. reference temperature, K. temperature in the a-phase (a =,B, y, a), K. intrinsic average relative temperature in the ,a-phase (a = ,B, y, a), K. Ta - (Ta)a, spatial deviation temperature for the ,B-phase (a = ,B, y, a), K.

(Tf3)~

intrinsic average temperature in the ,B-phase determined by the ,B-phase thermal energy equation valid in the homogeneous w-region, K.

(Tf3)~

intrinsic average temperature in the ,B-phase determined by the ,B-phase thermal energy equation valid in the homogeneous I)-region, K. intrinsic average temperature in the a-phase determined by the a-phase thermal energy equation valid in the homogeneous w-region, K. intriosic average temperature in the a-phase determined by the a-phase thermal energy equation valid in the homogeneous I)-region, K. average temperature at the W-I) boundary, K. spatial average temperature, K. saturation temperature, K. velocity in the a-phase (a =,B, y), mls. intrinsic average velocity for the a-phase (a = ,B, y), mls. Va - (va)a, spatial deviation velocity for the a-phase (a = ,B, y), mls.

(T)s (T) Tsat Va

(va)a

Va (Vf3)~

V Vf3(x, t)

Voo V",(t) VI)(t)

intrinsic average velocity for the ,B-phase determined by equations valid in the homogeneous w-region, mls. intrinsic average velocity for the y-phase that is determined by equations valid in the homogeneous I)-region, mls. superficial average velocity for the ,B-phase determined by equations valid in the homogeneous w-region, mls. superficial average velocity for the ,B-phase that is determined by equations that are valid in the homogeneous I)-region, mls. averaging volume, m3 . volume of the ,B-phase contained within the averaging volume, m3. large-scale volume, m3. volume of the entire w-region contained in Voo , m3 . volume of the entire I)-region contained in Voo , m3.

75

MASS AND THERMAL ENERGY JUMP CONDITIONS

w· ".By (w) . "wry X

speed of displacement of the f3-y interface, m1s. speed of displacement of the W-TJ boundary, m1s. position vector locating the centroid of the averaging volume, m.

Greek Symbols L1hvap

Ea E(5

E.Bw Eyry Pa (ph)s (pv)s

heat of vaporization per unit mass, J/kg. Va(t)/V, volume fraction of the f3-phase (a = f3, y). V(5/V, volume fraction of the a-phase. V.Bw/V, volume fraction of the f3-phase in the homogeneous (V-region. Vyl)/V, volume fraction of the y-phase in the homogeneous TJ-region. density of the a-phase (a = f3, y, a), kg/m 3. surface excess enthalpy, J/m2 . surface excess mass flux vector, kg/em s).

1. Introduction

The process under consideration is illustrated in Figure 1, where we have identified two homogeneous regions, the w-region and the 7]-region, and the boundary region having a thickness, lB. This illustration is essentially identical to that given by Kaviany (1991, Chap. 12) who describes a variety of special cases, one of which is the moving-interface regime that is the subject of this paper. In order to analyze the heat and mass transfer processes that take place in this system, we need to develop volume averaged transport equations for the wand 7]-regions that are based on the averaging volume, V, shown in Figure 1. In addition, we need to develop the appropriate mass and energy jump conditions for the boundary region. The use of jump conditions to describe the heat and mass transfer processes in the partially saturated region is appropriate when lB « L and it is necessary when lB '" roo On the other hand, if the radius of the averaging volume is small enough, that is r0 « lB, volume averaged transport equations also can be developed for the boundary region. Under those circumstances, there is no need for a jump condition and the analysis of the boundary region is carried out in terms of the volume averaged transport equations for heat, mass, and momentum transport for two-phase flow in porous media (Whitaker, 1998). This approach also requires that the radius of the averaging volume be large compared to the characteristic lengths for the fJand y-phases that are illustrated in Figure 2. To be precise, we note that volume averaged equations can be developed for the boundary region when the following length-scale constraints are valid: (1.1)

When these length scale constraints are not satisfied because r0 is of the order of or greater than lB, we are forced to develop mass and energy jump conditions. In this study, we first develop the total mass jump condition given by 8.BwP.B((v.B)~ - (w)) . nwry = 8YI)Py((Vy)~ - (w)) . nWl)'

at AWI)(t),

(1.2)

76

JORGEN HAGER AND STEPHEN WHITAKER

v

Figure I. Evaporation at a vapor-liquid boundary in a porous medium.

where AWI)(t) represents the dividing surface between the saturated and unsaturated regions. In many applications, the porosity will be constant and Equation (1.2) can be simplified by imposing the condition that E{3w = EYI). On the other hand, the evaporation front may occur where there is an abrupt change in the porosity and E{3w will be different than EYI). For example, in the case of transpiration cooling (Kaviany, 1991) at the interface between a saturated porous medium and a homogeneous fluid, E{3w will be equal to the porosity and EYI) will be equal to one. In addition to developing the total mass jump condition, we derive the liquid-phase mass jump condition (1.3) in which (liz)8 represents the mass rate of evaporation per unit area of the dividing surface, AWl) (t). Both these conditions play an important role in the development of the thermal energy jump condition that is used to predict the speed of displacement,

77

MASS AND THERMAL ENERGY JUMP CONDITIONS

Figure 2. Averaging volume and length scales.

(w) . DW/l' of the dividing surface. When convective and conductive surface excess enthalpy contributions are negligible, the jump condition takes the special form given by

b..hvapC:f3wPf3((vf3)~ - (w)).

= -Dw (K:. V(T)w 1} •

DW1}

K~. V(T)1})

+