Acta Polytechnica Vol. 43 No. l/2003

Models for Non-isothermal Steady-State
Diffiision in Porous Buildirg Materials

T. Ficker, Z. Pode5vov6

Two modek for non-isothermal difusion of uater aapour in building m.aterials haae been deaeloped and their numcrical outputs compared

wi,th a snndard isothermnl approximation.

Keywords: Fick's lnus, non-isothertnal difusion, parthl pressure profile.

I Introduction
The structure of most building materials of silicate origin

consists of mutually interconnected microscopic pores. The
diameters of the pores show wide dispersion in their values
ranging from nanometers to millimeters. Since the mean free

path of water vapour molecules at normal room conditions
Iies below lnm, almost all pores are open for vapour diffu-
sion. Howeveq this is not the only transport mechanism that
enables moisture to enter a building material. At higher
relative humidity (approx. above 50 %) intensive capillary
condensation occurs on the surface of pores and the pores

are subsequently filled with liquid water which can migrate
into a material. Liquid transport in pores takes place both
by surface diffusion and by capillary flow. Howeve4 the pur-
pose of this paper is to deal with diffusion of the gas phase

only, i.e., of water vapour, without the liquid transport. This
simplification is commonly applied, e.g., within the Glaser

condensation model It], widely used in thermal building
technology.

A common practice in thermal building technology is

to calculate the vapour diffusion flux and pressure profile
p(x) inside building envelopes within an isothermal model,
although the envelopes normally experience non-isother-
mal conditions throughout the year. The temperature of the

envelope within the isothermal model is represented by a

one-year average for a given climate region (e.g., l0 "C for
Central Europe). Such an approximation has been incorpo-
rated into the national thermal standards ofvarious countries

t2l, t3l and has been in service for many years. Although
this procedure might provide satisfactory results for one-year
assessment of envelopes, it is clear that that the results for a
shorter period can show severe deviations from the experi-
mental data.

2 Non-isothermal diffusion
Let us suppose that the building envelope through which

the diffusion flux goes is represented by a plain brick wall of
thickness d = 44 cm. The wall separates a heated room with a

usual environment (surface temperature and relative humid-
ity: ?n,=293.15 K, er=60 7o RH) from an outdoor space

(Tz=255.15 K, Q:= 84 % RH). The atmospheric pressure of
dry air is usually considercd to be approximately the same on

both sides (1"=98066.5 Pa - used in [a]). The atmosPheric
pressure p consists of partial pressures of water vapor p* and

dry air p^

F=f*+f^, !*<F^, F"Fu' (l)

Similar relations also hold for mass concentrahsns aswell

c=cw+c^, cw<ca, c=ca (2)

A reasonable choice [5] for the temperature profile (x) of
a wall is the linear function

T, -T^T@)=rt--' ,-'*, (Tt>Til ' (3)
&

Frck's first law for non-isothermal diffusion assumes the
following form [12]

9* = )*(9* + 8u) - cD*^(T)Y1* , ,* =T =

where g* and q^ are diffiLrsion fluxes of water vaPor and
air, respectively, and D*u(T) is the temperature-dependent
di{fusivity. The diffusion flow should fulfil the condition of
continuous flow

d\l

s lt! =YO (5)

which is ri.r.t ,!lo.,a law. Assuming unidirectional steady-
-state diffusion @yl0t=0) along the x-axis and a neglig-
ibly small dif;hrsion flow (g" -+0) of heary air molecules
(Nr+Or+... ) as compared with lighter H2O molecules

@* > q), Fick's nro equations can be rewritten as follows

,. = - ,D*u(T) d/* , lL =, < l, (6)Yw- l-1,(l+r)dx' q*

d^
frf* =o - 4w =conSt. (7)

with the boundary conditions

l*(o) = )r* , y*(d) = yz* . (8)

Following the work of Schirmer [4] and Krischer [6] the
diffusivity of water vapor in the air pores of a material can be

expressed as the function of temperature
L

o =Lr". z =181. ft =8.9718r 10-10 m2s-IK-1'81 (9)
tl

where ;r is the 'diffusion resistance factor'- a purely material
constant corresponding to the wall material. According to the
gas law, mass concentration c is a function of pressure and
[emperature

,= f =Fu, &= R 
=297 lkg-lK-t,R*^T RuT " Ma

R = 83 I .J mol-]K-I.

From Eqs. (3) - (10) it follows

c\9

"w '"4
(4)

(10)



Acta Polytechnica Vol. 43 No. 112003

ru(x) xy f___g&_ =_q*[lr,4p Jl-y1*(l+r) 'JL-
)r- o

I

r*(x) = *1 t -[ -1*(r + r)].

t

(t7710.19

| 'l'n-ln -t:r.jI T' _r* 
I ao.is_fro is-

d,(*) =1 t -(t -r'*) 
Li_r_,

I

Inserting the second boundary condition (8) into (11) we

obtain the diffusion flux

z =181

which, together with ( I I ), gives the vapor profile 1* ( x)

.(15)

r,r-^ _(r,_4 -te*)t-' I' \' d ) |

.[r -rz*(1+ r)1---4;4- 
L.n =rsr

[l -y1*(l + r)l 
I

I

)

Thking into account the difinim through an immobil;izcd air
lnyer (DIAL model), i.e., q^ -+ 0 (r -+ 0), relations (12), (13)

can be simplified

n _o.r9p^h rt -rz _ ,"[f:fa, 
.l 

, (r4)c- - nlr, iliE:}Fn "'Lr-)lwl'

r!#,]'-"0, (r)

Relation (14) may be formally rewritten as follows

,-[t -r*lurt 

-|

* ll-)r*l )
lw =7, &" =-{- 1kg-rmrsl,

&n Deff

where

oio =or9k4^ . =Tt-72= =-en P& rf- -ri-"
5629x l0-8 Tt -Tc -, -r -r,=- :nffioiO [t(gm -s -1.

p a"'^" - ti ^"

Since 11.< I and y2*< l, further simplification can
formed

r, -l

hl I2aL 
| = ),, -)* , qi- *)tw- )zw 

[kg m-2s-,]. (t7)
Lr -)r* l &n

At

3 DRAL model
Relations (14) - (17) hold within the framework of rhe

DIAL approximation, when the air layer embedded in a po-
rous material is only slightly perturbed by the diffusion of
water vapor molecules that possess smaller mass and much
lower concentration than those of air. It seems to be natural
that not only the pressure of dry air remains constant but also
the concentrations of dry and wet air vary only slightly across
the wall. This can be nicely illustrated when the concentration
ratios are calculated for a particular case, e.9., for the internal
and external conditions defined at the beginning ofthe previ-
ous Section

ofner/router =16.23,

,lnntt/r:"ttt = 0870,

,innerr/router = 0gg3

From ( I 8) it can be seen that the variations of air concen-
trations ca, c across the wall are negligibly small and, thus,
the profiles c^(x), c(x) may be considered as approximately
horizontal, i.e., constant for common climatic conditions.
This does not hold at all for water vapor whose concentration
varies considerably and, thus, its profile shows clear func-
tional dependence on x

c* = f(x), ca !Corst.,

, rconst., c*(x) < c^ . (19)

If these relations are included into Fick's Eq.(6), we can

obtain a more simplified transport equation
Ar(Ta\qa=-D*u(T):tr, l-)*=l' (zo)

r x0, c !const.

An analogous transport equation holds for non-isother-
mal diffusion of a gas in a solid compact body, so present ap-
proximation might be termed as Diffusion through a 'Rigid'
Air Layer (DRAL model). At first sight this approximation
might seem rather unrealistic, but the final account of all
physical factors and their approximate befuruior leads to such a

conclusion. Eq. (20) is quite analogous to those presented in
technical literature for cases when the total concentration is

constant (see e.g. Eq. 16.2-3 in [7]). The assumPtion c = const.

does not necessarily mean the condition for the isothermal
state. Nearly constant total concentration c can be expected
not only with non-isothermal diffusion of a gas in a solid cotn-

pact body which does not contain any air pores but also in
solid materials containing closed pores (cavities) filled with air.
The diffirsion flux in such materials goes either through the
airless solid structures whose concentration (density) is almost
unaffected by temperature and through voids with a constant
air content (constant concentration), provided the walls of
the voids are hardly penetrable for heary air molecules in
conrast to lighter water molecules. Foam building materials
such as foam polyethylene approach this type of material. It
seems probable that the DRAL model might be more applica-
ble to such materials. However, it is necessary to stress that if a
strong non-isothermal state causes essential variations in the
total concentration profile, i.e., c=f(x), the DRAL model will
fail to determine a realistic diffusion flux. Briefly, the trans-
port equation (20) should enable the diffusion flux of water

(18)

( l3)

(16)

be per-



Acta Polytechnica Vol. 43 No. l/2009

vapor in various materials to be approximated only under
usual climatic conditions, since such conditions represent
a weakly non-isothermal state which does not essentiilly dis_
turb the total concentration profile. Nevertheless, for a highty
non-isothermal state, the DRAL model can yield correct
results only with quite special materials. Numerical compari-
son of the results obtained by various models is presented in
Section 5.

Let us continue determining the solution of differential
equation (20). Frrst, it is necessary to specifi the complete set
of Frck's equations within the DRAL approximation

q* =-++fg+], ,* =]*-,R* =a621r<g-rx-r, 1zr;R*dx[ rG) )' * R*T' * ---J-'o

, T / , .\-l
914-.,r")9[44+]l=0.,=rar e2)drfufi, ' 'd*[ r@) )]
and the boundary conditions belonging to a non-isothermal
wall of thickness d

F*(o) _ h. p*(d) _ p2*

r(o) r;' ;@) = ,, (23)

Since the non-linear temperarure profile (x) appears in
common walls only under special conditions [10], a linear
profile is assumed here as well

T(x)=rr-\ ,Tz x=a-bx.
a

Inserting (24) into Frck's equations (21),
into account the first boundary condition
sponding solution can be found

o 
( o - b*\" !( !!-l = -n, = consr. .prR*' dx\T/

p*(x)i?(x) x

frf+) = f - zu{.r', a - bx)-n dx,J \T ) J K

A*;Tr o

o"u =h(: ,t)(rt ,+) [-2,-rrF(4-' -ri-" )
where

Pt* ., -1, bq-,r* =fit [ks--r], ,r* =ffi [ks--3].

By means of relations (30) the non-isorhermal diffirsion
flux g" expressed within the DRAI approximation can be
easily calculated.

For completeness, rhe partial pressure profile p*(x) inside
the wall should be presented. Functions p*(x) and c*(x) are
given by Eqs. (27) and (28)

p*(d=(r,-a ,e ').\.- d )

Trt-n -71-n

,*(*)=-+ [# Hy

q,=fu#

&.=* rm

., -, -1-[Kgm -s '1,

-lr], (30)

( Pr* P2*-1. tt - rt(26) [.+
L

-(n - '';? .)'-"Trt-n

-(n-'f .)'-"

(31)
Pru ( Pt* /2* )
T-1. r, -T)

(24)

(22) and taking
(23), the corre-

Trl-n -7t-n
(32)

For n = l.8l the profiles p,(x) and c*(x) will read
( T,_T^ \(25) p*(*)=l Tr-'' ;'' *1.
\- d )

(27)

Inserting the second boundary condition (23) into (27),
we can express the steady-state diffusion flux q*

(t,* _f*("))ur,, _,.,
l.4 rQ))4*=# (28)rw 
*["t-' -(a - bd)t-'f

which goes through a non-isothermal wall with a linear
temperature profile (24). The symbols a, b in (28) can be
specified using (24)

, n =1.81 (29)

Relation (29) can be rearranged and the eflective diffirsion
resistance R.n and effective coefficient D"n may be introduced

(33)

7r-o.ar -(r, -Tt 
-,Tz *)-o'tt

c*(x) = rrw -(.rw -rz*)#. (3+)
11 - 12

At first sight it is obvious that thep*(x) and c,(x) profiles are
not linear. Nevertheless, for usual temperature and partial
pressure differences between outdoor and indoor spaces in
the Central European climatic region the graphs ofp*(x) and
c*(x) will closely follow linear behaviou4 as can be easily
verified.

4 IM-TDR and IM-TIR models
Glaser's standard condensation model [l] is based on

isothermal diffusion, i.e., the temperature of a wall is con-
sidered to be a constant T- and equal to the mean value of the
surface temperatures

t T1+T2/t" = -: ., (35)

Fick's equations for diffusion in an isothermal structure
can be obtained from (21) and (22) after inserting T= Z.

-0.8 r )-0 
8

*)Tl -Tz
d-;:0.st-

-t2I

(^
I lrU
:03

)4
)- Fl



Acta Polytechnica Vol. 43 No. ll200g

where

D=LrA, n =1.81.
p

Using the following boundary conditions

f*(o)=fw, F*(d)=P*" Q?)
the solution of (36) can be found easily

r'21t

o^ =Pw - h, . R^ =t!.7q 
& ' "q 6'

u=uH,n =1.8r (38)

(36) 4x10'

3x'10'7

^1otqt
kg m 

2s'

1x'l0 
7

-lxl07

(3e)

n, =_ D .dF* dqd 
=0.sd--&4,'dtr, A

F@)= Pw -Pw -Fxr , .

As can be seen, the pressure profile p(x) is a linear func-
tion of x in conrast to non-isothermal profiles (15) and
(34), which are nonlinear. In spite of the fact that Fick's
Eqs. (36) hold exactly for isothermal strucrures only, i.e., for
constant temperature, the thermal resistance ,Ro is actually
temperature dependent^Ro(T*). Such a model may be termed
as an 'isotfurmal modzl with tcmperature dcpendsnt resistanre', i.e.,
IM,JTDR.

Nevertheless, in civil engineering practice Fick's iso-
thermal equations are combined with thermal resistance that
is not temperature dependent, because the average temper-
ature is fixed to the one-year average, i.e., T.=!$1.15 11,

regardless of the actual value T- of a particular stnrcture.
Such a model may be called an'isothermal modcl with temper-

ature inlepmdmt resistanc.e', i.e., IM.:TIR.

'*o ,, 
" 

to* 4ooo Sooo

Fig. l: Diffusion fluxes in dependence on temperature

5 Results and conclusions

In order to analyse the behaviour of all the models discus-
sed, their results for the structure given at the beginning
of Chapter 2 are presented in Fig. I and Thble l. The
calculations were done under the assumption that the
external partial pressure of water vapour remained constant
while the external temperature was successively increas-
ing, i.e., the external relative humidity was decreasing. Such
conditions enable us to study the influence of remperature
di{Ierence LT =Tt -72 on the numerical compatibility of the
models.

Table l: Diffusion fluxes through the brickwall (p=9; separating external space (q= 84VoR.H., Tz=255.18 K,Frn= 105 Pa) and an
internal room with increasing temperature (20 "C-5000 "C)

Quantity 293.15 K
(20 "c)

303.15 K
(30'c)

323.t5K
(50'c)

373.15 K
(100'c)

773.15 K
(500'c)

t273.t5 K
(1000 "c)

5273.15 K
(5000 "c)

P* [Pa] 1402.2 t402.2 1402.2 1402.2 1402.2 1402.2 1402.2

c* x 103 [kg rn-u] 10.350 10.000 9.392 8.1340 3.9300 2.3800 5.7600

y* x 103 9.1 Ll 9.1 9.1 9.1 9.1 ol

IMJIIR
Ro x l0-e 1m s-11

20.956 20.956 20.956 20.956 20.956 20.956 20.956

IMJTDR
Rox lo-e1msr1

21.61 21.61 2r.29 19.35 12.98 9.42 3.32

DIAL

{ffx toa
lkgm-ls-11

14. l8 13.98 t3.61 12.8 9. l3 7.07 J.IJ

DRAL
R"nx 104 [*tr-t]

17.t3 16.62 15.68 t3.75 7.00 4.38 l. l5

IMJTIR
q x lo8 [kgm-2s-r]

6. l9 6. l9 6.19 6.19 6.19 6.19 6. l9

IMJIDR
g x lo8 Jkgm-2s-rl

6.00 6.00 6. l0 b. /u 10.00 t3.77 39.07

DIAL
4 x 108 Jt<gm-2fli

5.93 6.02 6.18 o.5 / 9.21 l1.9 26.87

DRAI
q x l081kgm-2s-r1

5.52 5.48 5.42 5.27 4.34 3.40 -2.83



Acta Polytechnica Vol. 43 No. 112003

Thble I and Fig. I show that at 'small' temperature dif-
ferences (A? < 40 K) the heat fluxes of all four models differ
only by a few percent and, thus, under such'quasi-isothermal'
conditions all the models give nearly identical results.

Since the temperature difference AT in the Central European
climatic region does not usually exceed fifty Kelvin during a
year, all the models seem to be applicable within civil engine-
ering practice.

Nevertheless, if strongly non-isothermal conditions are
established (AT>40 K), it is necessary to distinguish carefully
behveen computational models. While the IMjIIR scheme is
not applicable at all under such conditions, the applicability
of the remaining three models will depend on the material
structure. For materials with macroscopic open pores filled
with air (such as silicate building materials or mineral wool)
the most convenient models seem to be DIAL and IM:TDR,
the first of which should be given priority in practical
calculations. The applicability of the DRAL model under
strongly non-isothermal conditions is more problcmatic since
it requires a constant total concentration profile to be es-

tablished, which is fulfilled only with special materials.

References
tl] Glaser, H.: Craphisches Verfahren a.u Untersuchlmg aon

Difisionsuorg(mgm. Kiiltetechnik, 1959, Vol. I I, No. 10,

p.345-3a9.

t2l Cuch Therrutt Snndnrd: iSN Zg 0540, iesky normali-
zaini institut, Praha, 1994.

Germnn Thermnl Standard: DIN 4108, Deutsches Institut
{lir Normung, Berlin, 1999.

Schirmer, R: Difiniansznhl aon Wasserdampf-Lufige-
mischm und dtic Vnda,mpfungsguchwindigkeit. Z. VDI-Beil.,
Verfahrenstechnik, 1938, Vol. 6, p. 170_177.

Ficker, T., Myslin, J., Pode5vovi{, Z.: Nryn-linear ternpera-

ture lnofles. Acta Polytechnica, 2001, Vol. 41, No. 6,

p.6G68.
Krischer, O: Gru.nd,gesetzz dn Feucht'igheitsbewegtng in
Tro chengiitem. Kapilln ruass erban e gung m.d D ampfdifinion.
Z. VDI - Beih. Verfahrenstechnik, 1938, Vol. 82,
p. 373-380.

Bird, R. B., Stewart, W. E., Lightfoot, E. N.: Trarcport
Phznomm.a. New York: J. Wiley & Sons, 1965 (in Czech
translation).

t3l

l4l

IJJ

t6l

t7l

Assoc. Prof. RNDr. TomiiS Ficker. DrSc.
Department of Physics
phone: +420 541 147 661
e-mail: fyfic@fce.vutbr.cz

Ing. Zdenka PodeSvov:i
Department of Building Structures

Faculty of Civil Engineering
University of Technology
;..rLtzKova I I
662 37 Brno, Czech Republic


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