Study of heat and moisture migration properties in porous building materials

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1 Applied Thermal Engineering 25(2005) Study of heat and moisture migration properties in porous building materials Z.Q. Chen *, M.H. Shi Department of Power Engineering, Southeast University, No. 2 Si Pai Lou, Nanjing , Jiangsu Province, PR China Received 25December 2003; accepted 6 May 2004 Available online 13 August 2004 Abstract Based on the non-equilibrium thermodynamic theory, the thermal driving forces and the fluxes in heat and moisture migration process for unsaturated porous building materials are analyzed. The mechanisms of heat and moisture migration in unsaturated porous building materials are discussed and the phenomenological equations to describe the immigrating process in unsaturated porous building materials are established. By means of the diffusion law and the equation of state for ideal gas, the expressions of coefficients in the phenomenological equations are deduced. The effects of temperature, water content or partial vapour pressure on the phenomenological coefficients are also discussed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Heat and moisture migration; Phenomenological coefficients; Non-equilibrium thermodynamics; Porous building materials 1. Introduction Problems involving heat and moisture migration in porous building materials arise in a number of engineering interests, such as wall drying, the solar house designing, cooling load calculating of air conditioning, etc. Affected by porous structure, temperature gradients, moisture gradients and environmental characteristic, heat and moisture transport in the porous building materials is quite complex. Richards [1] firstly established the equation of unsaturated flow in porous materials, which is on the basis of Darcy s law and the principle of continuous motion. Philip and Vries [2] * Corresponding author. Tel.: ; fax: address: zqchen@seu.edu.cn (Z.Q. Chen) /$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi: /j.applthermaleng

2 62 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) developed a moisture migration model at inhomogeneous temperature profiles, in which moisture migration affected by temperature gradient was taken over. In the process of heat and moisture migration, the gradients of temperature, moisture and pressure, the main driving forces of heat and moisture migration in unsaturated porous media, influence one another [3]. Due to the complicated structure of porous building materials, it is difficult to consider micro-phenomena. In order to describe the continuity of porosity and other parameters in unsaturated porous media approximately, unsaturated models were developed by using the method of representative elementary volume [4,5]. Because of water evaporation, phase-change is taking place in unsaturated porous materials. To decide the transport properties, migration coefficients vs. temperature and volume moisture content were studied by Shah et al. [6]. The steady-flux measurements of moisture diffusivity in unsaturated porous media were studied by Richards [7]. In a paper series Wilson, Hall and coworkers [8 11] studied the water movement in porous building materials by use of unsaturated flow theory and obtained some experimental verifications of capillary absorption of water for different building materials and structures. The different moisture transport mechanisms and some interfacial phenomena in porous materials were investigated by Freitas et al. [12] using the theory of Luikov [3] and Philip and Vries [2]. Based on a onedimensional steady model, the dynamic and diffusive behavior of a three-layer building envelope element was investigated by Taylor et al. [13]. More recently, real-time thermal and moisture parameters in buildings were studied theoretically and experimentally [14,15]. All these previous works are very instructive, but the migration phenomenological coefficients are still not to be decided owing to the complexity of heat and moisture migration processes in porous materials. This paper will present a method to predict the phenomenological coefficients with consideration of heat and moisture migration mechanisms, in which a non-equilibrium approach is employed to explain combined heat transfer and water, including liquid and vapour, movement. 2. Linear phenomenological equations of heat and moisture migration Heat and moisture migration in porous building materials is the process affected by the interactions of temperature field, moisture field, and partial vapour pressure field. Generally, the coupled heat and moisture migration can be described in linear non-equilibrium thermodynamic theory [16]. According to the Curie s principle, the thermal driving forces and the fluxes in heat and mass transfer process are coupled. So the linear phenomenological equations of heat and moisture migrations in porous building materials can be described as ~J q ¼ L qq T 2 rt L ql T rh L qv T rp V ~J L ¼ L Lq T 2 rt L LL T rh þ q LK~g ~J V ¼ L Vq T 2 rt L VV T rp V ð1þ ð2þ ð3þ where, ~J q, ~J L and ~J V are heat flux, liquid mass flux and vapour mass flux, respectively. T, h and P V are the temperature, water content and partial vapour pressure. q L and K are liquid water density

3 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) and unsaturated hydraulic conductivity. We assume, k ¼ L qq, apparent thermal conductivity; T 2 k 1 P ¼ L qv, migration coefficient; k T P ¼ L VV, infiltration coefficient; k11 T P ¼ L Vq, thermal infiltration T 2 coefficient; k m ¼ L LL, mass diffusivity; T k1 m ¼ L ql, mass thermal diffusivity; k11 T m ¼ L Lq, thermomass T 2 diffusivity. So, Eqs. (1) (3) can be written as ~J q ¼ krt k 1 m rh k1 P rp V ð4þ ~J L ¼ k 11 m rt k mrh þ q L K~g ~J V ¼ k 11 P rt k PrP V ð5þ ð6þ 3. Analysis of thermodynamics flux In order to describe the linear phenomenological coefficients, the mechanisms of heat, moisture migration and thermodynamic flux are analyzed below Mechanism of heat migration Heat transfer in unsaturated building materials under temperature gradient covers heat conduction, infiltration convection heat transfer, radiation heat transfer and phase-change heat transfer. They are interactive. For there is no large temperature difference in unsaturated building materials, so radiation heat transfer is ignored. According to the analysis above, the heat flux is consisted of heat conduction heat flux ~J qd and convection heat flux ~J qc caused by infiltration fluid flow. ~J q ¼ ~J qd þ ~J qc ¼ k e rt þð~j L h L þ ~J V h V þ ~J a h a Þ ð7þ 3.2. Mechanism of moisture migration Moisture migration in unsaturated porous building materials includes diffusive migration and infiltration fluid flow. The diffusive migration has the forms of molecular diffusion, Knudsen diffusion and surface diffusion Liquid water mass flux Liquid flux in porous building materials consists of infiltration flow and surface diffusion caused by the gradients of moisture and temperature. According to Darcy law [17], the infiltration liquid flow ~J LC can be described as ~J LC ¼ q L ðd hl rh þ D TL rt K~gÞ ð8þ where, D hl ¼ K o/ oh, DTL ¼ K o/ ot, / is hydraulic potential. In non-isothermal conditions, surface diffusion fluid flow ~J LD caused by adsorption and desorption is [18]

4 64 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) ~J LD ¼ q L D TD rt ð9þ where, D TD ¼ C q L is the coefficient of adsorption-diffusion. T So, the liquid flux ~J L is ~J L ¼ ~J LC þ ~J LD ¼ q L ½D hl rh þðd TL þ D TD ÞrT K~gŠ ð10þ Vapour flux The total vapour flux includes the vapour convection flux, general molecular diffusion and Knudsen diffusion flux under the gradients of temperature and vapour partial pressure. Assuming that the vapour convection velocity is ~V, so vapour flux ~J V is [19] ~J V ¼ q V ~V D e rq V ð11þ where, D e and q V are effective diffusion coefficient and vapour density, respectively Air flux The total air flux also includes air convection flux and diffusion flux. For the convection velocity of air is the same as that of vapour [19], so the air flux is ~J a ¼ q a ~V D e rq a ð12þ where, q V is air density. 4. Phenomenological coefficients According to the analysis of thermodynamic flux and ideal gas state equation, the phenomenological coefficients are deduced below Liquid water The phenomenological coefficients include mass diffusivity k m and thermomass diffusivity k 11 m. Comparing Eqs. (5) and (10), they are respectively described as k m ¼ q L D hl ¼ q L K o/ oh ð13þ k 11 m ¼ q LðD TL þ D TD Þ¼q L K o/ ot þ D TD ð14þ 4.2. Vapour In the unsaturated porous building materials, comparing to vapour flux, air flux is very small [19], so the net transfer of air is zero ~J a ¼ ~ 0 ð15þ

5 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) Using Eq. (15) to (12) the vapour convection velocity ~V is described as ~V ¼ D e rq a =q a ð16þ Combined Eqs. (11) and (16), the vapour flux is rq ~J V ¼ D e q a V rq V q a q V ð17þ Assuming that the air and vapour obey the ideal gas law in the unsaturated porous building materials, therefore the vapour flux is described as [19] D e P ~J V ¼ R V T ðp P V Þ rp V ð18þ Therefore, by using Eqs. (12) and (18), the phenomenological coefficients of vapour flux is described as D e P k P ¼ ð19þ R V T ðp P V Þ k 11 P ¼ 0 ð20þ where, D e is the effective diffusion coefficient, R V is general gas constant, P and P V are total gas pressure and partial vapour pressure, respectively Heat The enthalpies of liquid water, vapour and air in unsaturated porous building materials are h L ¼ C L T h V ¼ C PV T þ H C h a ¼ C Pa T ð21þ ð22þ ð23þ Combining Eq. (7) and Eqs. (21) (23) the heat flux is ~J q ¼ ðk e þ k 11 m C LT ÞrT k m C L T rh k P ðh C þ C PV T ÞrP V þ q L C L TK~g ð24þ Compared Eqs. (4) and (24), the phenomenological coefficients of heat flux apparent thermal conductivity k, migration coefficient k 1 P and mass thermal diffusivity k1 m are k ¼ k e þ k 11 m C LT ð25þ k 1 P ¼ k PðC PV T þ H C Þ k 1 m ¼ k mc L T ð26þ ð27þ

6 66 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) Effects of temperature and moisture on the phenomenological coefficients According to the analysis above, the phenomenological coefficients of heat and moisture migrations are functions of temperature and moisture or partial vapour pressure. For sandy building block, the effects of temperature and moisture on the phenomenological coefficients are analyzed below Thermodynamic and thermal physical parameters Hydraulic potential / and conductivity K The hydraulic potential and conductivity are functions of temperature and water content. They can be described by Eq. (28) [20] and Eq. (29) respectively [23]. 4 h / ¼ / S expð ct Þ ð28þ e 2:75 / K ¼ K S S ð29þ / where, e, / S, K S and c are porosity, saturated hydraulic potential, saturated hydraulic conductivity and surface extended coefficient, respectively. For sandy block with mean pore diameter of 0.3 mm, e, / S, K S and c are 0.39, )0.0315m, m/s and C 1 [20] Partial vapour pressure P V In the unsaturated porous building materials, vapour partial pressure obeys the following thermodynamic relation [19]: P V ¼ P S exp g/ R V T where, P S is the saturated vapour partial pressure and g is the acceleration to gravity. ð30þ Effective thermal conductivity k e The effective thermal conductivity in unsaturated porous building materials can be described approximately as k e ¼ k S ð1 eþþk L h þ k g ðe hþ ð31þ where, k S, k L and k g are the thermal conductivity of solid, liquid and gas (vapour and air), respectively Effective diffusion coefficient D e The effective diffusion is the combination of general modular diffusion and Knudsen diffusion. So, its coefficient is D atmd Kn D e ¼ D atm þ D Kn ð32þ

7 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) General modular diffusion coefficient [21] D atm ¼ 4: et 1:5 =ðpf 0 Þ ð33þ Knudsen diffusion coefficient [22] 0:5 D Kn ¼ 8e2 2R V T ð34þ 3f 0 S g pm V where s, S and M are tortuosity factor, specific area for BET and molecular weight of water vapour. For sandy block with mean particle diameter of 0.3 cm, the coefficient of adsorptiondiffusion is very small and it can be neglected Effects of temperature and moisture on phenomenological coefficients For sandy block with mean particle diameter of 0.3 mm and porosity of 0.39, the effective curves of temperature and moisture on phenomenological coefficients apparent thermal conductivity k, mass thermal diffusivity k 1 m, mass diffusivity k m and thermomass diffusivity k 11 m are showed in Figs. 1 4 respectively. As shown, the phenomenological coefficients increase with the increasing of temperature and water content. But the mass thermal diffusivity k 1 m, mass diffusivity k m and thermomass diffusivity k 11 m are closed to zero when the water content is very small. So when the water content is below 0.1, the liquid water immigration, the effect of heat on liquid water migration and effect of liquid water migration on heat transfer are very small. The relations of migration coefficient and infiltration coefficient with temperature and vapour partial pressure are in Figs. 5and 6. As shown, the effects of temperature on migration coefficient and infiltration coefficient are larger than the effects of vapour partial pressure. Apparent thermal conductivity λ x10 4 (kw/m C) a T= Water content θ Apparent thermal conductivity λ x10 4 (kw/m C) 14.0 b θ= Temperature T ( C) Fig. 1. Apparent thermal conductivity curve.

8 68 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) Mass thermal diffusivity λ 1 m (kw/m) a T=10 Mass thermal diffusivity λ 1 m (kw/m) b θ= Water content θ Temperature T ( C) Fig. 2. Mass thermal diffusivity curve a 11.0 b Mass diffusity λ m x10 3 (kg/ms) T=10 Mass diffudivity λ x10 3 m (kg/ms) Water content θ θ= Temperature T ( C) Fig. 3. Mass diffusivity curve. These calculated values are used in the numerical simulation of heat and moisture transfer in unsaturated soil. The predicted results are compared with the experimental data obtained from one-dimensional column of soil and the agreement is satisfactory [18].

9 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) Thermo-mass diffusivity λ x 10 6 (kg/ms C) m " a T= Watercontent θ Thermo-mass diffusivity λ x 10 6 (kg/ms C) m " b θ= TemperatureT( C) Fig. 4. Thermomass diffusivity curve. Migration coefficient λ 1 x 10 8 (kw/m C) p a T=10 Migration coefficient λ 1 p x108 (kw/m C) b θ= Partialvapor pressure P v (Pa) Temparature T ( C) Fig. 5. Migration coefficient curve. 6. Conclusions The phenomenological equation to describe the immigration process of heat and mass in unsaturated porous building material is established. The phenomenological coefficients of heat and moisture are deduced. The phenomenological coefficients of heat and moisture migration are

10 70 Z.Q. Chen, M.H. Shi / Applied Thermal Engineering 25 (2005) Infiltration coefficient λ p x (kg/mpa s) a T=10 Infiltration coefficient λ p x (kg/mpa s) b θ= Partrial vapor pressure Pv (Pa) Temperature T ( C) Fig. 6. Infiltration coefficient curve. very useful for simulation of the heat and moisture migration process in porous building materials. The effects of temperature, water content or partial vapour pressure on the phenomenological coefficients are also discussed. For sandy block with mean particle diameter of 0.3 mm and porosity of 0.39, it is concluded that the liquid water immigration, the effect of heat on liquid water migration and effect of liquid water migration on heat transfer in the porous building materials are very small when the water content is below 0.1. Acknowledgements The authors thank the support from National Basic Research Project of China, no. G References [1] L.A. Richards, Capillary conduction of liquids through porous media, Physics 1 (3) (1931) [2] J.R. Philip, D.A. De Vries, Moisture movement in porous materials under temperature gradients, Transaction of American Geophysical Union 38 (2) (1957) [3] A.V. Luikov, System of differential equation of heat and mass transfer in capillary-porous bodies, International Journal of Heat Mass Transfer 18 (1) (1975) [4] S. Whitaker, Simultaneous heat, mass and momentum transfer in porous media: A theory of dry, in: J.P. Hartnett, T.F. Irvine (Eds.), Advances in Heat Transfer, Vol. 13, Academic Press, New York, [5] J. Bear, J. Bensabat, Heat and mass transfer in unsaturated porous media at a hot boundary: I. One-dimensional analytical model, Transport in Porous Media 6 (2) (1991) [6] D.J. Shah, An experimental determination of the heat and mass coefficients in moist, unsaturated soils, International Journal of Heat Mass Transfer 27 (6) (1984)

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