AP2002_01.vp 1 Introduction In the national thermal standards of various countries [1], [2] there are diffusion data that were measured under some standard isothermal and isobaric conditions, e.g. for temper- ature Ta � 283 K and atmospheric pressure pa � 98 066.5 Pa. The diffusion data determined under such conditions are often used in cases for which neither isothermal nor isobaric conditions are fulfilled. For example, when the condensation of water vapour in building envelopes is estimated, an iso- thermal state can rarely be assumed. The condensation is usually estimated for winter conditions, when the temper- atures of interior T1 and exterior T2 are quite different. In this case the effective temperature T* � (T1+T2)/2 is introduced and is assumed to be close to the standard temperature Ta � 283 K. It is assumed that the effective temperature T* is common for the whole envelope, i.e., it is the isothermal approximation with the single temperature T* that enables numerical estimation of the non-isothermal condensation of water vapour diffusing through the envelope to be perform- ed. For this purpose the coefficient of diffusion permeability � is defined as follows � �� T g d p p * � � d 1 2 (1), T T T * � �1 2 2 (2) where d is the thickness of the envelope (wall), gd is diffusion flux and p1, p2 are partial pressures of water vapour at both surfaces of the wall. Some researchers have attempted to “improve” the mentioned isothermal approximation (1, 2) by measuring gd not in the isothermal state T T* � a but in non- -isothermal conditions T T T1 2� � *. It is clear that in such a situation relations (1, 2) becomes incorrect and must be replaced by some different relations holding for the real non-isothermal state. The aim of this paper is to derive the corresponding relations that govern the non-isothermal diffusion of water vapour through porous materials. The starting point for this derivation will be Fick’s laws of diffusion. 2 Non-isothermal diffusion of water vapours If there are no sources of diffusing particles and unidirect- ional steady state diffusion through a porous medium with diffusion constant D has been established, then Fick’s general equations can be rewritten in the simpler form q D c x d d d � � , (3) d d d dx D c x � � � � 0, (4) where concentration c of diffusing particles, i.e. molecules of water vapour, can be replaced by partial pressure p and absolute temperature T, according to the equation of thermo- dynamic state pV mRT c m V c R p T R� � � � � �, , , [ ] 1 462 1 1Jkg K . (5) Following the classical work of Schirmer [3] and Krischer [4] the diffusion constant D of a porous building material is dependent on atmospheric pressure pa, temperature Ta � T and type of porous material which is represented by the diffusion resistance factor � being a purely material constant. For “standard” pressure pa � 98 066.5 Pa, the diffusion con- stant D can be expressed as follows D T k T n k � � � � � � � � 8 9718 10 181 8 9718 10 10 1 81 10 2 . , . , . [ . � � n m s K� �1 1 81. ]. (6) Inserting (5), (6) into (3), (4) Fick’s equations will be modified � � � � � � q D x R x p T x x d d d � � � � � , (7) � � � � � � d d d d n x k R T x x p x T x� � � � � � � � � � � 0 , (8) with the following boundary conditions belonging to a non- -isothermal wall of thickness d � � � � � � � � p T p T p T p T 0 0 1 1 2 2 � �, d d . (9) A linear temperature profile T( x) inside the wall can be assumed, as follows � �T x T T T d x a bx� � � � �1 1 2 . (10) Inserting (10) into Fick’s equations (7), (8) and taking into account the first boundary condition (9), the corresponding solution can be found � � k R a bx x p T gn � � � � � � � � d d constd ., (11) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 25 Acta Polytechnica Vol. 42 No. 1/2002 Non-isothermal Diffusion of Water Vapour in Porous Building Materials T. Ficker, Z. Podešvová Non-isothermal diffusion is analysed using Fick’s laws. The exact relations for non-isothermal diffusion flux and partial pressure profiles in porous building materials are derived and discussed. Keywords: Fick’s laws, non-isothermal diffusion, partial pressure profile. � � � � � � d dd p T g R k a bx xn x p T p x T x � � � � � � ��� � 01 1 , (12) � � � � � � � � � g p T p T kb n R a a bx x x n nd � � � � � � � � � �� � 1 1 1 1 1 � . (13) Inserting the second boundary condition (9) into (11), we can express the steady state diffusion flux gd going through the non-isothermal wall with the linear temperature pro- file (10) � � � � � g p T p T kb n R a a bdn n d � � � � � � � � �� � 1 1 2 2 1 1 1 � . (14) The symbols a, b in (14) can be specified using (10) � � � � g k n Rd p T p T T T T Tn n d � � � � � � � � �� � 1 1 1 2 2 1 2 1 1 2 1� . (15) Relation (15) can be rearranged and effective diffusion resistance Rd * and “conductivity” Deff * may be introduced g c c R d d �1 2 * [kg�m�2s�1], R d D d eff * * [m �1s], � �� � � � D k n T T T Tn n eff * � � �� � 1 1 2 2 1 1 1 � [m2s�1] (16) where c p RT 1 1 1 [kg m�3], c p RT 2 2 2 [kg m�3]. By means of relations (16) non-isothermal diffusion flux is easily available. As can be seen, the relevant potentials re- sponsible for diffusion movement are not the partial pres- sures p1, p2 but concentrations c1, c2 of the water vapour at both wall surfaces. The differences between the non-isother- mal diffusion fluxes determined according to rigorous rela- tion (16) and that of isothermal approximation (1, 2), utiliz- ing effective temperature T*, ranges from several per cent to several tens of per cent, depending on the boundary conditions. For completeness, the partial pressure profile p(x) in- side the wall should be analyzed. Function p(x) is given by Eqs. (10), (13) and (14) � �p x T T T d x p T p T p T T Tn � �� � � � � � � � � � � � � � � �� 1 1 2 1 1 1 1 2 2 1 1 1 T T d x T T n n n 1 2 1 1 1 2 1 �� � � � � � � � � � � � � � � � � � � � � , (17) for n � 1.81 relation (17) will read � �p x T T T d x p T p T p T T T � �� � � � � � � � � � � � � � �� 1 1 2 1 1 1 1 2 2 1 0 81. 1 1 2 0 81 1 0 81 2 0 81 � �� � � � � � � � � � � � � � � � � � � � � T T d x T T . . . . (18) At first sight it is obvious that the p(x) profile is not lin- ear. Nevertheless, for usual temperature and partial pressure differences between outdoor and indoor spaces in our clima- tic region the graph of p(x) will closely follow the linear behaviour as depicted in Fig. 1. 3 Conclusion This paper has presented the exact procedure (16) for cal- culation of the non-isothermal diffusion flux of water vapour going through porous materials. It has presented a compari- son between the approximative procedure (1, 2) based on the 26 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 1/2002 p [Pa] d [m] 0 0.05 0.1 0.15 0.20 150 450 750 1050 Fig. 1: Partial pressure profile p(x) inside a wall of thickness d � 0.2 m with winter boundary conditions: �1 � 40 % RH, T1 � 293 K, p1 � 934,8 Pa, �2 � 60 % RH, T2 � 263 K, p2 � 156 Pa concept of effective temperature T* and the exact procedure respecting the real non-isothermal conditions. The corresponding non-isothermal pressure profile p(x) inside the wall proved to be almost linear under climatic conditions that are normal in the Central European region (viz Fig. 1). References [1] Czech Thermal Standard ČSN 73 0540, Čs. norma- lizační institut, Praha 1994 [2] German Thermal Standard DIN 4108, Deutsches Institut für Normung, Berlin 1999 [3] Schirmer, R.: Diffusionszahl von Wasserdampf–Luftgemis- chen und die Verdampfungsgeschwindigkeit. Z.VDI–Beil., Verfahrenstechnik, No. 6/1938, pp. 170–177 [4] Krischer, O.: Grundgesetze der Feuchtigkeitsbewegung in Trockengütern. Kapillarwasserbewegung und Dampfdif- fusion. Z.VDI–Beil. Verfahrenstechnik, No. 6/1938, pp. 373–380 Assoc. Prof. RNDr. Tomáš Ficker, DrSc. phone: +420 5 4114 7661 e-mail: fyfic@fce.vutbr. cz Department of Physics Ing. Zdenka Podešvová Department of Building Structures Faculty of Civil Engineering University of Technology Žižkova 17, 662 37 Brno, Czech Republic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 27 Acta Polytechnica Vol. 42 No. 1/2002