Water vapor permeability of porous media

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Canadian Journal of Physics, 37, 4, pp. 413-416, 1959-04-01

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Water vapor permeability of porous media

Woodside, W.

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WATER VAPOR PERMEABILITY

OF

POROUS MEDIA1

ABSTRACT

Followirlg the a~jalogy betwee11 the laws of heat conduction and vapor diffusion, lbvo theoretical expressiol~s for the thernial conductivity of a con~posite medium are applied to the water vapor permeability coefficient of certain porous media. I t is sllowll that both expressiolls reduce to a form very similar to the empirical relationships found Ily Penniat~ and Edenholm for soils, glass spheres. charcoal, and cellular concrete. The calculatio~l of the variation of water vapor permeability with density for a cellular lightweight concrete is illustrated.

granular matel-ial is made up of one or more types of particles surrounded homogeneous medium (usually air). All granular ltiaterials have associated them values of thermal conductivity, electrical conductivity, dielectric tarlt, magnetic permeability, atid other diffusion-type coefficients. In

y instances, it would he advantageous to be able to calculate the values ich coeficients from tile corresponding values for the various coristituents the \.olurne fractions of the constituents.

nce the mathematical theories of steady-state thermal and electrical uction, electric and magnetic fields, atid diflusion are all similar, ;l formula

red for the thermal conductivity of a granular material will also be icable to the calculation of electrical conductivity, dielectric constant, netic permeability, and diffusion coefficient (de Vries 1952). Two of these icie~its are of main interest here, 11arne1y thermal conductivity and water )r diffusion coefficient. Several equations have already been applied by author to the calculation of the thennal conductivity of porous media odside 1958). I t is the object of this note to describe the application of iar equations to the calci~latio~~ of the water vapor diffusion coefficient or !r vapor permeability of certain porous media.

he followi~lg equation of de Vries (1052) was derived by Maxwell for rical conductivity and applied by Euclcen to the calculation of theloma1 luctivi ty.

re P .is tile fractior~al porosity ol the porous nleclium, and k, k,, and

k,

the thermal conductivities of the porous medium, the gas phase (air), solid respectively. This equation applics to a granular porous material, . is, one in which discrete solid particles are distributed in a continuoos phase. For a cellular porous material, one i n wl~icll discrete gas cells ;ire -ibuted in a continuous solid phase, the equation must be modified by rchangixlg kg arid

k,

and by replaci~iy

P

by ( 1

-Pj.

fanuscript received November 24, 1958.

ontribntion from the Building Services Section, Division of Building Research, Niltio~ial arch Council, Ottawa, Canada.

,sled as N.R.C. No. 5071. J . Phys. Vol. 37 (1959)

For application of equatioll (1) to calculation of v;rpor perrt~cability porous media, the thermal coiiductivities k,

kg,

ancl

k,

must be rej)laccd 1: the corresponding vapor yerrrieabilities p, pn, and p,. Thus the equation fa

the vapor perrrleabilitp p of a gr;~nular n~aterial rends

where _P is i t g a i ~ ~ the porosit.y, and p, and p, are the vapor permeabilitics the gas (air) and solid respectively.

For most grarlular n~aterials, the solid particles are impervious to wat vapor, and therefore p, = 0. Also since ail- is the gas ~ ~ h a s e in most. granul materials, 12, = pRlr. 'I'herefore,

and equatio~i (2) reads

Maxwell's equation (cle Vries 1032) was derived for a rand0111 distributi of spl~erjcal inclusions. Russell (1935) has derived a thermal co~~d\~c.tivi equation for s cubic lattice of cubic inclusio~ls. Russell's equation for a gl--,tr~u. material is

Again, replacing k, kg, and k, by the corresponding vapor pern~eabilities, p,, ;111ri p,, and also placing p , = 0 ;iud p, = pa,,, this becotnes

By expanding the terms it1 brackets, this reduces to

For st~iall values of the porosity Y , the tcrrns of order two ;rnd higher may neglected, arid the equation is

This is identicill with equatiorl

(3,

wliicll was derived from A;Iaxwell's equati

This simple relationship between the vapor permeability of a granl material and its porosity is vcrv siinilnr to the relationship found exp

mentally B y Pennian (1040) :

(5) p = O.OGPp,lr.

and moist soils a r ~ d other grauular nlaterials satisfied tll-is rela tior,shiy ne case of rnoist soils,

P

represetits the air-filled porosity, that is, the total Ftj7 less the volumetric water corittrrit.

he dry-cup method for water vajmr permeability dete~mination illvolves a -imeLric measuremcrl t of the steatly-state water vapor transraissioi~ thro~tgh ate of the test material under isothernial co~lditiolls (7.3" F) with a relative iidi ty of 0% rnai~itained on one side (by means of n desiccant) and a. relative jidity of 50% maintained on the other. For most porous media the equili-

m m o i s t ~ ~ r e con ten1 corrcspo~~diug LV a. mcan relative hutnidit.)~ of 25%, low that moisture transmission by mechanisms other t h a n vapor diffusion tlatively small. A t higher mean relative Iiumiclities, the apparent vapor neability may be se\reral times larger than the dry-cup value, owing to 11. n~echallistns of inoisture transfer such as capillarity. The true vapor neability must decrease a t increasi~ly meirri relative humidities because he decrease in air-filled pore space available for diffusion. Since vapor neabilities dctermi~iecl by the dry-cup neth hod correspond most closely to vapor diffusion, only drp-cup K~lrres will be conip:~red \vith c:alculatecl ,r perrneabili tics.

denholm (1945) obsel-ved expet-irnerit;~lly tllat the 'dry-cup' \vstcr vapor leability p is given by

re k, is a constant for any given rrlaterial (named red~~ction factor by nholm) and

P

is the coarse-capill;lry prosit\. of the material. Ede~lllolrn ~d a value of 0.65 lor k, for nlatcrials such as sand, glass spheres. charcoal,

a cellular concrete. Thev;rlue of pd,, the \~irter vapor permeability of air.

oonl temperature is- 120 perm. .ill., the perm. in. being the presently pted unit lor vapor permeability, which is grains in./hr f t y i n . Hg). I n nholm's'ut~its, brlr = 0.069 kg/m at111 hr.

he ar~nlysis given above results in a porosity-dependent reductiorl factor, e from eqrration (3)

k,

= 2/(3-

P),

reas both Penman and Edenholtn found k, to be constant. However, the orrable agreemerl t between the theore tical and measured values of vapor neabiliry is illustrated by the folloa~ing calculation. Ederlholm measured

3r yerrneabilities r a ~ ~ g i n g betwee11 0.018 and 0.022 kg/rn atm hr for a bed lass spheres of porosity 0.395. S ~ ~ b s t i t u t i ~ ~ g P = 0.3115 and pal, = 0.069

cquatiort (3), givcs a value for p = 0.0209 kg/m atnz hr, which is between two measut-ed values.

ftrrther applicatiotl of the theory to the calc\~lation of the variation of l r permeability with density for a cellular corlcrete is now described. Since llular ~naterial is cssentiaIIy a continuous solid with distributed air cells, equation to be used is the converse of equation (2) :

41 ti CANADIAN JOURNAL OF PHYSICS. VOL. 37. 1959

Haller~ (6) reports a value of 5.6 perm. in. a t a density of 80 lb/fta fc particular cellular concrete being: considered. I t is assunicd that a t thi5 density there are no 'large' air cells iu the material (Edenholm's coarse lary porosity equal to zero). T h u s p, =I 5.6. The vapor permeability c cellular concrete a t 32 Ib/ft3 density will now be calculated. A t this dt the 'coarse' porosity, P, is

Taki~tg pal, = 120 perm. in. a s the value for p,, equation (7) results in a for ja of 24.0 perm. in. This compares favo,rably with Iaboratory 'drj

test resl~lts for rrlaterial of density 32 lb/ft" which show a value of 22.3

in.

The applicatiorl of a thermal conductivity equation for -porous media i

calculation of water vapor pernleabiIity of certain porous .materials :

warranted by the agreement wit11 the empirical equatior~s of Pe~irnai Edenholni and by the agreement of predicted a11d experimental values.

I t would appear that, because of the arialogy between Darcy's law fol flow in porous media and the heat conductior~ cqnation, the same tech a s described above cotlld be applied to the cal(:ulation of the saturate( meability (of porous media) for liquid water. The difficulty it1 this applic would appear to be the assignment of a value for

kg

(k, in most cases be zero), since the value of

kg

must obviously depend upon the meat! cize and hence upon the pore size distribution.

This is a contributiotl from the Division of Buiiding Research, Na Research Council of Canada, and is published with the approval (; Director of the Di.vision.

REFERENCES

EDENHOLM, H. 1945. hiledd. Statens Furskningskornmitte Lantmannabyggnader, 5,

HALLEN, W. 1957. Private communication, Decen~ber 11. PENMAN, H. L. 1!)40. J . Agr. Sci. 30, 438, 670.

RUSSELL, H. W. 1936. J. Am. Cerarn. Soc. 18. 1.

DE VRIES, D. A. 1952. Mededel. I~ndbouwhogeschool Wageni~~gen, 52, 1.