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Relative permeability relations: A key factor for a drying model

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Abstract

In the modelling of heat, mass and momentum transfer phenomena which occur in a capillary porous medium during drying, the liquid and gas flows are usually described by the generalised Darcy laws. Nevertheless, the question of how to determine experimentally the relative permeability relations remains unanswered for most materials that consist of water and humid air, and as a result, arbitrary functions are used in the drying codes. In this paper, the emphasis is on deducing from both numerical and experimental studies a method for estimating pertinent relations for these key parameters. In the first part, the sensitivity of liquid velocity and, consequently, of drying kinetics in the variation of the relative permeabilities is investigated numerically by testing various forms. It is concluded that in order to predict a realistic liquid velocity behaviour, relative permeabilities can be linked to a measurable quantity: the capillary pressure. An estimation technique, based on simulations coupled with experimental measurements of capillary pressure, together with moisture content kinetics obtained for low or middle temperature convective drying, is deduced. In the second part, the proposed methodology is applied to pine wood. It is shown that the obtained relations provide closer representation of physical reality than those commonly used.

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Abbreviations

AV:

averaging volume

(AV) j :

phase volume within the averaging volume AV

a w :

water activity

B f :

resistance factor in the effective diffusitivity coefficient of vapour in the medium

C :

mass fraction of the vapour in the gaseous phase

C p :

constant pressure heat capacity [J kg-1 K-1]

D :

diffusivity [m2 s-1]

D.C.:

convective drying condition assumed to remain constant during the overall process

F m :

total moisture mass flux [kg m-2 s-1]

g :

gravity vector [m s-2]

h a :

intrinsic averaged enthalpy of dry air [J kg-1]: h a = Cp a (T − T r )

h b :

specific averaged enthalpy of bound water [J kg-1]: h b = h l − H b

\(\bar h_b\) :

intrinsic averaged enthalpy of bound water [J kg-1]: \(\bar h_b = h_l - \frac{1}{{\bar \rho _b }}\int_0^{\bar \rho _b } {H_b d(\bar \rho _b )}\)

h l :

intrinsic averaged enthalpy of free water [J kg-1]: h l = Cp l (T − T r )

h s :

intrinsic averaged enthalpy of solid [J kg-1]: h s = CP s (T − T r )

h v :

intrinsic averaged enthalpy of vapour [J kg-1]: h v = H 0 v + Cp v (T − T r )

H b :

heat of desorption [J kg-1]

H 0 v :

latent heat of vaporisation at the reference temperature T r [J kg-1]

I.C.:

Initial conditions of the medium

J :

flux

k :

intrinsic permeability [m2]

K :

volumetric mass rate of evaporation [kg m-3 s-1]

k r :

relative permeability

L :

thickness of the medium [m]

n :

exterior normal unit vector

P :

pressure [Pa]

q :

source term

Q :

total heat flux [W m2]

RH:

external relative humidity [%]

T :

temperature [K or °C]

T infh :

wet bulb temperature [K or °C]

T r :

reference temperature [K]: T r = 273.16 K

t :

time [s]

S :

saturation

U :

conserved quantity

v :

velocity [m s-1]

W :

moisture content (in dry basis)

z :

space variable [m]

δz :

space step [m]

ε :

porosity

ε j :

volume fraction for the phase j: ε j = (AV)j/AV

Φ :

heat source [W m-3]

λ :

effective thermal conductivity [W m-1 K-1]

μ :

dynamic viscosity [kg m-1 s-1]

ϱ :

density [kg m-3]

τ m :

mass transfer coefficient [ms-1]

τ t :

heat transfer coefficient [W m-1 K-1]

σ :

surface tension [N m-1]

a :

dry air

atm:

tmospheric

b :

bound water

c :

capillary

eq :

equilibrium

g :

gas

inf:

drying air

ini:

initial

irr:

irreducible

l :

liquid

m :

macroscopic mean value

s :

solid

ssp:

solid saturation point

v :

vapour

vsat:

saturate vapour

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Authors and Affiliations

  1. Laboratoire Energétique et Phénomènes de Transfert, Unité de Recherche Associée au CNRS, URA 873, Esplanade des Arts et Métiers, 33405, Talence Cedex, France

    F. Couture & W. Jomaa

  2. Laboratoire de Génie des Procédés de Pau - Ecole Nationale Supérieure en Génie des Technologies Industrielles, Rue Jules Ferry, 64000, Pau, France

    J.-R. Puiggali

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  1. F. Couture
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  2. W. Jomaa
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  3. J.-R. Puiggali
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Couture, F., Jomaa, W. & Puiggali, JR. Relative permeability relations: A key factor for a drying model. Transp Porous Med 23, 303–335 (1996). https://doi.org/10.1007/BF00167101

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