Modeling of Moisture Transport in Porous Building ...

Annex 41 MOIST-ENG, Working meeting, April 3-5, 2006, Kyoto, Japan

Modeling of Moisture Transport in Porous Building Materials by Gravimetric Sorption-desportion Tests Rafik Belarbi*, Menghao Qin, Abdelkarim Aït-Mokhtar LEPTAB, University of La Rochelle, La Rochelle, France. *Email: [email protected]

Abstract: A simple gravimetric method for determination of moisture diffusivity inside porous building materials is presented in this paper. The method allows calculating the moisture diffusion coefficient and moisture distribution inside building materials purely based on gravimetric measurements. The method was also applied to research into the properties of some porous materials. The results indicate that (i) the moisture diffusion coefficient during absorption is higher than during desorption process due to the absorption hysteresis; (ii) the increase of water-cement ratio in the cement pastes will result in an increase of moisture diffusion coefficient. Furthermore, the experiment also points out that the moisture coefficient of concrete with cement with fly ash additions is lower than that of concrete with OPC. Key words: Modeling; Humidity; Concrete and cement paste; Adsorption; Gravimetric method

1 Introduction In the past decades, durability of materials, service life of building constructions and lifecycle costs have been important topics in building research. Many researches show that moisture damage is one of the most important factors limiting a building’s service life [1]. High moisture level can cause metal corrosion, wood decay and structure deterioration. In addition, human perception of air quality and the emission of pollutants from paint and varnish depend largely on the relative humidity of the material. Therefore, the investigation of moisture transfer in porous building materials is important not only for the characterization of behavior in connection with durability, waterproofing, and thermal performance, but also avoiding health risk due to the growth of microorganisms. Most methods for determining moisture diffusivity are based on the analysis of moisture profiles [2–4]. For this analysis, a sufficient number of measured points forming the moisture profile are necessary. These experiments are always complicated and time, energy consuming, normally employ enough deep samples for the moisture profile measurements. Their application for thin materials in the direction through the plate is mostly not feasible because even advanced and precise methods for moisture measurements such as NMR [5, 6], microwave reflection or transmission [7, 8] or ?-ray attenuation [9, 10] have a limited space resolution. In addition, in the case of highly inhomogeneous materials even a better space resolution would not be of any sense because it would only copy the inhomogeneities. Therefore, a logical solution is using simplified methods. Most of these methods [11, 12] make it possible to determine an average value of moisture diffusivity. In this paper, we

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Annex 41 MOIST-ENG, Working meeting, April 3-5, 2006, Kyoto, Japan present a method, which allows determining the moisture distribution inside porous building materials only based on gravimetric experiments.

2 Mathematical modeling 2.1 Background of moisture transport in non-saturated porous media The moisture transport mechanisms relevant to calculations in non-saturated porous media are water vapor diffusion and liquid transport through capillary forces. Firstly, the following simplifying assumptions are made for the description of maintain [5]: (1) the solid matrix is rigid, macroscopically isotropic and inert. (2) the 'no-slip' condition at the water/solid matrix is valid. (3) the friction force, due to momentum transfer at the solid-water interface, is much larger than both the inertial force of the water and its internal viscous friction due to flow. (4) There are two immiscible fluids present; a wetting fluid (water) and a non-wetting fluid (gas). This gas consists of a binary ideal gas mixture of water vapor and air. (5) a local thermodynamic equilibrium between water and vapor exists throughout the porous media. (6) macroscopically, the air is at atmospheric pressure throughout the porous media or the pressure gradient can be neglected.

2.1.1 Liquid transport At the microscopic level, the liquid transport in non-saturated media can be given by the Darcy’s law with the pore water pressure pc as the driving potential: where the liquid transport coefficient Dl is a function of pore water pressure pc. q l = − Dl ∇ pc

(1)

Under isothermal conditions, the suction only varies with the liquid water content, and Eq. 1 can be rewritten as: (2) q l = − D m, l ∇wl Where (3) k ( w ) ∂p ( w ) Dm,l = l l ⋅ c l µl ∂wl is called the isothermal liquid diffusivity. k l(wl) is the effective permeability for liquid water. It is dependent on the moisture content and is related to parameters describing the microscopic geometrical configuration of the pores occupied by liquid water. wl is the volumic liquid water content. µl is the dynamic viscosity of liquid water. The macroscopic conservation of mass for liquid water expressed in volumic quantities can be written as [5]: (4) ∂wl = ∇ ⋅ ( Dm ,l ∇wl ) − E l→ v ∂t where El? v is the evaporation rate.

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Annex 41 MOIST-ENG, Working meeting, April 3-5, 2006, Kyoto, Japan 2.1.2 Vapor transport At the microscopic level, the vapor transport under isothermal conditions is given by Fick's law: (5) 1 M qv = − Dv ∇ pv ρl RT In this equation Dv is the diffusion coefficient of water vapor in air, M is the molecular mass of water, R is the gas constant, T is the absolute temperature, and pv is the vapor pressure. The term M/RT results from the ideal gas law, whereas ?l results from the transformation of the mass flux for vapor into an equivalent volumic liquid water flux. By volume averaging of Eq. 5, the macroscopic volumic flux through a porous media with only air and vapor is obtained. In case of a non-saturated porous media, the macroscopic vapor transport is given by [5]: (6) (n − wl ) M * qv = − T ( wl ) Dv ∇p v ρ1 RT * where T is the part of tortuosity related to the void-space available for vapor diffusion and is a function of the liquid water content. n is the porosity of the porous media. Under isothermal conditions, the relative humidity only varies with the suction and thereby with the moisture content wl. Therefore, the gradient of the vapor pressure can be rewritten as: (7) ∂h ∇p v = pvs ( ) T ∇wl ∂wl where h is the relative humidity and pvs the saturation pressure of water vapor above a flat surface. Combining equations 6 and 7 and using the ideal gas law, the mass balance equation on the macroscopic scale for water vapor in volumic quantities can be rewritten as:

∂wv = ∇ ⋅ ( Dm, v ∇wl ) − E v→ l ∂t

(8)

Where

(9) M ∂h T * ( wl ) Dv ( ) ⋅α ρ l RT ∂wl is called the isothermal vapor diffusivity. a is correction factor for vapor diffusivity enhancement by liquid islands and surface diffusion, respectively. And Ev? l is the rate of condensation (Ev? l = -El? v). D m, v = −( n − w l )

2.1.3 Moisture transport Combining equations 4 and 8, describing the liquid and vapor transport respectively, the moisture transport can be written as: (10) ∂( wv + wl ) = ∇ ⋅ ( Dm ∇wl ) ∂t Where (11) D m = D m, v + D m,l is called the isothermal moisture diffusivity. Assuming an equilibrium between water and vapor, the mass density of the vapor in the pores is given by: -3-

Annex 41 MOIST-ENG, Working meeting, April 3-5, 2006, Kyoto, Japan (12) wv ρl n − wl Hence, the moisture content can be written as: (13) ρ ρ w = wv + wl = ( v ) n + (1 − v ) wl ρl ρl -5 Since ?v is of the order of 10 ?l, the right hand side of Eq. 13 is to a good approximation equal to wl for liquid water contents larger than 10-3, hence, the moisture transport in a porous media can be approximated by a non-linear diffusion equation: ∂w = ∇ ⋅ ( D m ∇w) (14) ∂t

ρv =

The proposed method for determination of isothermal moisture transfer inside building materials is based on the assumption that the moisture diffusion coefficient Dm can be considered as piecewise constant with respect to the ambient relative humidity. Given the experimental setup (Fig. 1) (see Experimental section); the problem can be reduced to only one dimension. Thus, for each sorption or desorption stage, the transport equation can be written in the form as: (15) ∂w ∂ 2w = Dm 2 ∂t ∂x Where, Dm is the isothermal moisture diffusivity, which can be expressed as the sum of vapor diffusivity and liquid diffusivity [5]. The initial condition is: w t =0 = wb The boundary conditions can be defined as: ∂w − Dm = β ( w x =l − wo ) ∂x x =l ∂w =0 ∂x x =0 Where, ß is the Convective moisture transfer coefficient.

(16) (17)

Porous material

Fig 1. The schematic diagram of the test set-up

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2.2 Resolution Introducing a new variable ψ=w-wo , then the above model can be rewritten as: ∂ψ ∂ 2ψ = Dm 2 ∂t ∂x The initial condition and boundary conditions become: ψ t =0 = ψ b = wb − wo ∂ψ − Dm = β ⋅ψ x =l ∂x x =l ∂ψ =0 ∂x x = 0

(18)

(19) (20) (21)

Applying the method of separate variable, we can get the analytical solution of Eq. (18): (22) 4 sin µ n D t ψ ( x, t ) ∞  x  =∑ cos µ n  exp  − µ n 2 m2  ψb l   l n =1 2 µ n + sin( 2µ n )  where, µn is eigenvalue, which is the solution of the following equation: βl µ n tan µ n = Dm βl The value of µn under different can be found in [14]. Thus, we can get the expression of Dm moisture content inside the porous material as: ∞ 4 sin µ n D t  x  w( x, t ) = ∑ cos µ n  exp  − µ n 2 m2  ( wb − wo ) + wo (23) l   l n =1 2µ n + sin( 2 µ n )  D mt ) is larger l2 than 0.2, the solution can be simplified by taking only the first term into consideration, which still can provide enough accurate results for most engineering applications. In most sorption and desorption processes, the duration to reach the Fourier number of 0.2 is negligibly small compared to the total time of the experiment. Thus, Eq. (23) can be simplified as: (24) D t 4 sin µ1  x  w( x, t ) = cos µ 1  exp  − µ 12 m2  ( wb − wo ) + wo 2µ 1 + sin( 2 µ1 ) l   l  For a given stage of sorption or desorption test, the amount of water, which transported between the sample and the ambient during the time interval [0, t], can be expressed by the relation l (25) M w = ∫ ( wb − w( x, t )) dx

Eq. (23) is an infinitive progression. When the value of Fourier number ( Fo =

−l

The total mass of water transported during the considered stage of sorption or desorption process can be obtained by: l (26) M b − M e = ∫ ( wb − wo ) dx −l

Then, we can get the dimensionless moisture content of the material, which is expressed as follows:

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Annex 41 MOIST-ENG, Working meeting, April 3-5, 2006, Kyoto, Japan Mw 4 sin 2 µ1  2 D t = 1− ⋅ exp  − µ1 m2  2 Mb − Me l  2µ 1 + µ 1 sin( 2µ 1 ) 

(27)

W − We 4 sin 2 µ 1  2 D t = ⋅ exp  − µ 1 m2  2 Wb − We 2µ 1 + µ1 sin( 2 µ 1 ) l  

(28)

or:

2.3 Determination of the moisture diffusion coefficient The moisture content inside the material can be non-dimensionalized using the following equation: (29) W − We Φ= Wb − W e And the dimensionless moisture distribution can be expressed as [14]: Φ = G exp( − St) (30) Integrating with Eq. (25), G and S can be expressed as: (31) 4 sin 2 µ 1 G= 2 2µ 1 + µ1 sin( 2µ 1 ) 2 (32) µ1 D m S= l2 Then Dm can be expressed as: (33) S ⋅l2 Dm = 2 µ1 Where, G represents lag factor (dimensionless) and S (s-1) is a coefficient related to the desorption/sorption process. The profiles of the dimensionless moisture content of the whole material can be obtained by gravimetric measurements. The values of G and S can thus be determined by Eq. (31) correspondingly. Eq. (32) can be solved by some iterative methods such as Newton method. Then the value of Dm is obtained. 3 Experimental study 3.1. Materials and conservation Two high strength concretes and six cement-pastes were manufactured by using three cement ratios and two types of cements: an OPC of type of CEM I – 52.5 and a mixed cement-fly ash of type of CEM V – 42.5A according to European norms EN 197-1. Their chemical compositions are given in Table 1. Tables 2 and 3 give the composition and some properties of the concretes and the cement pastes, respectively. After manufacture, samples were kept in a saturated solution of whitewash during 90 days.

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Annex 41 MOIST-ENG, Working meeting, April 3-5, 2006, Kyoto, Japan Table 1 Chemical compositions of the cements used Chemical analysis (%) SiO 2 Al2 O3 Fe2 O3 CaO MgO K2O Na2 O SO3 S2ClTiO 2 MnO P2 O5

CEM - I 21.25 3.47 4.23 64.95 0.93 0.27 0.11 2.71 0.01 0.05 0.17 0.10 0.44

CEM - V 29.78 11.30 3.46 46.71 2.68 1.20 0.24 2.86 0.18 0.03 0.61 0.10 0.50

Table 2 Composition of the concretes Material Concrete (BI) Cement (kg/m3 ) CEM-I (400) 3 Sand 0/5 (kg/m ) 858 Grave 5/12,5 (kg/m3 ) 945 3 Water (l/m ) 168 Hyperplasticcizer (kg/m3 ) 10 (2.5 %/cement) Air (%) 0.9 W/C 0.42 Slump (cm) 22/23/24/22 Volume mass (kg/m3 ) 2460 Rc* at 3 days (MPa) 42.5 Rc at 7 days (MPa) 54.0 Rc at 28 days (MPa) 72.0 * Re is the compressive strength.

Concrete (BV) CEM-V (430) 800 984 180 10.75 (2.5 % / cement) 1.2 0.42 19/19.5/21.21 2415 36.0 46.5 69.5

Table 3 Composition of the cement pastes Material Cement W/C Hyperplasticcizer Water retentive agent

PI 30 0.30 2% 1%

PI 42 CEM I 0.42 1% 1%

PI 55

PV 30

0.55 0% 1.33 %

0.30 2% 1%

PV 42 CEM V 0.42 1% 1%

PV 55 0.55 0% 1.33

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3.2. Isothermal sorption/desportion tests The specimens were sawed in order to obtain thin discs of 110 mm diameter and 3 mm thickness. This size of samples was chosen in the aim of reducing the experimental duration required to reach balance between the material and its ambience during sorption/desorption tests. A climatic chamber, which allows a control of both temperature and relative humidity, was used for isothermal desorption and sorption tests. These tests were carried out at a constant temperature of 25 °C (± 0.1) and for four stages of relative humidity: - Firstly a desorption process: 100 %-90 %, 90 %-75 %, 75 %-65 % and 65 %-50 %, - Secondly a sorption process: 50 %-65 %, 65 %-75 %, 75 %-90 % and 90 %-100 %. Given the adopted protocol, all samples were saturated in water before the beginning of the tests. For each desorption or sorption stage, the test consists in measurements of sample’s mass according to time. In this way, the variation of water content of the sample is determined as a function of time. 4 Results and discussion The relative variation of weight vs. time for concrete BI is given in Fig 2. for desorption and sorption tests.

(a)

(b)

Fig 2. Relative variation of weight vs. time during desorption process (a) and sorption process (b) for concrete BI The moisture diffusion coefficients obtained by Eq. (33) for concrete BI and BV vs. relative humidity are summarized in Figure 3. The value of the moisture diffusion coefficient increases with the relative humidity. A similar distribution can be found in literature [13]. For the concrete, the value of moisture diffusivity varies from 10-13 to 10-11 m2 s-1. Fig. 3 also shows that the moisture diffusion coefficient of sorption process is larger than that of desorption process, highlighting the sorption hysteresis. As measuring results in Fig. 4 show that the hysteresis between sorption and desorption isotherms is distinct. These results make it possible to characterize in an intrinsic way various materials used compared to the

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Annex 41 MOIST-ENG, Working meeting, April 3-5, 2006, Kyoto, Japan kinetics of transfer of moisture. Furthermore, Fig. 3 indicates that the moisture diffusion coefficient of BV is lower than that of BI.

BI (desorption)

3E-11

BI (sorption)

2,5E-11

2

-1

Moisture diffusion coefficient (m s )

BV (desorption)

BV (sorption) 2E-11 1,5E-11 1E-11 5E-12 0 50

55

60

65

70

75

80

85

90

95

100

Relative humidity (%)

Fig 3. Moisture diffusion coefficient of BI and BV during desorption and sorption processes

0.9 0.8

0.9 0.8 0.7 Sr

0.7 Sr

1

Sr (sorp) BI Sr (desorp) BI Sr (sorp) BV Sr (desorp) BV

1

0.6

0.5

0.4

0.4

0.3

0.3

0.2

Sr (sorp) PI 30 Sr (desorp) PI 30

0.6

0.5

Sr (sorp) PV 30 Sr (desorp) PV 30

0.2 50

60

70

80

90

100

50

60

70

RH (%)

0.9

0.9

0.8

0.8

0.7

0.7 Sr

Sr

1

Sr (sorp) PI 42 Sr (desorp) PI 42 Sr (sorp) PV 42 Sr (desorp) PV 42

0.5 0.4 0.3

80

90

80

90

100

(b)

(a) 1

0.6

RH (%)

Sr (sorp) PI 55 Sr (desorp) PI 55 Sr (sorp) PV 55 Sr (desorp) PV 55

0.6 0.5 0.4 0.3 0.2

0.2 50

60

70

80 RH (%)

90

100

50

60

70

100

RH (%)

(c)

(d) Fig 4. Sorption and desorption isotherms

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100

100

95

95

90

90

85

85

80

80

0 5d 10 d 15 d 20 d

75

25 d 30 d 40 d

70

70

55 d

65

65

60

60

55

55

RH (%)

RH (%)


75

50

65 d 70 d 75 d 80 d 85 d

50

0

0,5

1

1,5

0

0,5

L(mm)

1,5

90 d 95 d

(b) BV (desorption) 0 5d

100

100

95

95

10 d

90

90

15 d 20 d

85

85

80

80

RH (%)

RH (%)

(a) BI (desorption)

75

25 d 30 d 40 d

75

70

70

65

65

60

60

55

55

55 d 65 d 70 d 75 d 80 d 85 d

50

50 0

0,5

1

0

1,5

0,5

L(mm)

1

1,5

(c) PI 42 (desorption)

(d) PV 42 (desorption) 100

95

0 1d 2d

95

90

90

85

5d 10 d 20 d

85

80

RH (%)

80

75

70

65

65

60

60

55

55

50

30 d 40 d

75

70

45 d 50 d 55 d 60 d 70 d 80 d 90 d 95 d

50 0

0,5

90 d 95 d

L(mm)

100

RH (%)

1 L(mm)

1 L(mm)

(e) BI (sorption)

1,5

0

0,5

1 L(mm)

(f) BV (sorption)

1,5

100 d 105 d 110 d 115 d 120 d 125 d

-10-

100

100

95

95

90

90

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85

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80 RH (%)

RH (%)


75

75

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70

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65

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55

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0

0,5

1 L(mm)

(g) PI 42 (sorption)

1,5

0

0,5

1

1,5

L(mm)

(h) PV 42 (sorption)

Fig 5. Profiles of the relative humidity inside the material during the desorption and sorption process Figure 5 shows the moisture distribution inside different materials during desorption and sorption processes. All curves are calculated by the modeling presented above (Eq. 24). The change of the moisture distribution inside the material at the first few hours can be clearly observed in the figure. Fig. 6 highlights that the effect of the W/C ratio on the moisture diffusion coefficients is quite small in the range of low moistures (RH