Moisture transport in concrete takes place due to changes in the environmental condi- tions. A disturbance to the equilibrium of the moisture content within a concrete member leads depending on the prevailing conditions to a moisture absorption or release. Dur- ing the moisture exchange between concrete and the environment, transport of water in liquid and/or gaseous state is carried out depending primarily on the structure of the hardened cement paste [106]. There have been numerous significant publications on the moisture transport processes of porous material. They were generally first documented by the end of the 19th century and applied to concrete by the middle of the 20th. A very extensive literature review on the moisture movement and moisture properties of building materials was published in 1980 by Kießl [82] comprising 650 publications.
2.2.1 Associated mechanisms
Concerning the movement of fluids through concrete, three main processes are distin- guished: permeation, diffusion and sorption. Permeation refers to flow of fluid when pressure is applied, diffusion is the movement of ions, atoms, or molecules under a dif- ferential in concentration and sorption is the capillary attraction of a liquid into empty or partially empty pores [45]. Fig. 2.8 shows an schematic overview of the main trans- port mechanisms in porous materials illustrating the variety and complex relationships between them.
Liquid Gaseous
Tempe- rature Partial
pressure Concen-
tration
Thermo- diffusion Vapour
diffusion Diffusion
Solution diffusion
Sorption
Surface diffusion Total
pressure Capillary
forces Electric
field
Hydraulic flow Capillary
suction Electro-
migration
laminar turbulent Capillary
condensation Physical stateDriving potentialTransport mechanism
Moisture transport in porous materials
Figure 2.8: Schematic overview of the phenomena of moisture transport in porous materials [82]
2.2 Moisture transport in concrete
Besides the three main processes named before, the fluids can also move if an electric or a temperature field is applied across the concrete. Due to an electric field the nega- tive ions will move towards the positive electrode in a process known as electromigra- tion [39] and through thermodiffusion the molecules of the fluid drift along a temperature gradient [46].
The dimensions of the pores are decisive to determine whether water is presented as free or bound water in liquid or gaseous state, and consequently also the mode of moisture transfer [79]. Due to the very complex geometry of the concrete microstructure, differ- ent mechanisms of moisture transport with different driving potentials perform unevenly according to the geometry of the pores in which the water is contained. Certain forms of moisture transport dominate depending on the frequency range of the pore sizes, the temperature level of the substance, and as shown in Fig. 2.5, the moisture content of the material. Rose [126, 127] described the process of wetting of a porous material in four stages according to the dominant water transport mechanisms by means of an idealized porous system consisting of a pore with a neck at each end. These stages are schemati- cally illustrated in Fig. 2.9.
Stage 1 Stage 2
Effusion, adsorption Vapour diffusion, mono and multimolecular adsorption Stage 3
Vapour diffusion, capillary condensation, capillary suction
Vapour diffusion, surface diffusion, capillary suction
Capillary suction, unsaturated flow
Capillary suction, saturated flow Stage 4
Vapour phase Adsorbed phase Liquid phase
Figure 2.9: Water movement in various stages in the wetting of a porous material [127]
In the first stage, water enters the microstructure of the material not as vapour but as iso- lated molecules that are adsorbed by the pore surfaces. This type of transport is termed
effusion or Knudsen diffusion [48]. In the second stage, unimpeded vapour transfer takes place. The water vapour moves from high to low concentration regions through diffu- sion while the surfaces of the pores are covered by water molecules placed in mono or multimolecular layers. In the third stage, water closes the necks shortening the effective path length for diffusion of vapour. This process is described as liquid-assisted vapour transfer. On the walls, depending on the thickness of the water film, water molecules can either be fixed to the surface through adsorption or diffuse on the surface after the film reaches a significant thickness. The fourth stage describes hydraulic flow in unsaturated or saturated conditions. The component of the fourth stage describing unsaturated con- ditions differs from the component of the third state where surface diffusion occurs in the fact that the air voids in stage 4 have the same curvature everywhere, which does not allow the development of concentration gradients within it, and therefore no vapour transfer takes place.
2.2.2 Describing moisture transport in porous materials
For the description of the moisture transport in porous materials a vast variety of models have been proposed. In general, one can differentiate between three different approaches to derive models for moisture transport: general linear, thermodynamic, and microme- chanical approach [58].
The general linear approach has been most widely used in the literature, probably be- cause of its simplicity and generality. According to the general linear approach, the moisture flow in porous media depends on a set of independent state variables and space- dependent material properties and their gradients. The moisture flow is calculated as the sum of the products of conductivities and gradients. Examples of the general linear ap- proach are the models proposed by Luikov [93], de Vries [43], or Garrecht [60] which consider only the transport of moisture. Models considering moisture transport coupled with heat transfer have also been developed, like those from Krischer [90], Kießl [80], or Künzel [85]. The models differ fundamentally in the structure of the moisture transport equation which defines which transport mechanisms, physical states of the fluid, and in- teractions between the driving potentials and the physical states are taken into account.
As the models do not necessarily use the same driving potentials, different conductivi- ties are used and therefore, each model needs its own material parameters, which limits the comparability of the models.
The thermodynamic and micromechanical approaches are more complex and even less general. In the thermodynamic approach the entropy production is investigated. An entropy increase is responsible for the dynamics of the system. The entropy produc- tion delivers information about the moisture flow and its respective driving forces [57].
Grunewald [65] derivated the entropy production rate per volume for a heat, air mois- ture, and salt transport model. He concluded that the thermodynamic driving force of the diffusive water vapour flow is the gradient of the chemical potential. Hassanizadeh
2.2 Moisture transport in concrete
and Gray [75] developed a model to describe two-phase flow in porous media taking into account the effects of the surfaces separating the phases. In the micromechanical approach the heat and mass transport processes are calculated on the microscopic scale, which requires the use of a microphysical model of the porous system. Bednar [19]
used micromechanics to investigate water vapour flow based on the theory of ideal gas mixtures. According to Bednar the driving potential for the water vapour flow is the va- pour pressure with a small contribution of thermodiffusion. Whitaker [151] obtained the liquid water flow and liquid conductivity from the Navier-Stokes equations in the pore system. According to Whitaker the driving potential of the liquid water flow is liquid pressure and the effect of thermodiffusion can be neglected.
2.2.3 Description of concrete drying based on diffusion
Bažant and Najjar [10, 11] proposed a simple but accurate model to describe the process of drying of concrete. The model predicts the distribution of relative humidity in the concrete pores based on the Fick’s second law. It considers the gradient of relative humidity as the driving potential for the moisture transport and the conductivity is given by a non-linear diffusion coefficient dependent on the temperature and the pore rela- tive humidity of the material. The model does not include the influence of temperature gradients but considers the effect of a temperature change through its influence on the diffusion coefficient and on the relative humidity of the concrete pores (see Eq. 2.3).
Assuming constant temperature, the model predicts how diffusion causes the relative humidity of the concrete pores to change with time by solving the following partial differential equation:
∂h(x,t)
∂t =div(D(h,T)·gradh(x,t)) (2.4) where his the relative humidity of the concrete pores,xis the vector of position,t is time,Dis the diffusion coefficient, andT is the temperature. The diffusion coefficient is calculated as the product of the diffusion coefficient at reference conditionsD1 (i.e.
h=1.0 andT =20 °C) and two functions accounting for the influence of the relative humidity f(h)and the temperature g(T).
D(h,T) =D1·f(h)·g(T) (2.5)
f(h) =α0+ 1−α0
1+
1−h 1−hc
n (2.6)
g(T) = T+273 Tref+273·exp
Q RD
1
Tref+273− 1 T+273
(2.7) In Eqs. 2.6 and 2.7 the relative humidityhis defined between 0 and 1.0 and the temper- atureTin centigrades. The influence of the relative humidity on the diffusion coefficient f(h)is approximated by a s-shaped function defined by three parameters as illustrated in the left diagram of Fig. 2.10. The parameterhcdefines the location of the inflexion point of the function half-way between the maximum and minimum value of the diffu- sion coefficient. The parameterα0represents the ratio between the minimum and the maximum value of the diffusion coefficient, andncharacterizes the spread of the drop in the diffusion coefficient. In Eq. 2.7, illustrated in the right diagram of Fig. 2.10,Tref
corresponds to the temperature of reference, i.e. 20 °C. The diffusion coefficient in- creases exponentially with the influence of temperature depending on the ratio between the activation energy of diffusionQand the specific gas constant of water vapourRD.
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0
2 0 4 0 6 0 8 0 1 0 0
125
1 0 2 0 4 0
0 n = 1 6
Influence of temperature, g(T) [-]
Influence of rel. humidity, f(h) [-]
P o r e r e l a t i v e h u m i d i t y , h [ - ] n = 6
hc
T e m p e r a t u r e , T [ ° C ]
Figure 2.10: Influence of the pore relative humidity and the temperature on the diffusion coefficient of concrete according to Bažant and Najjar [11]
Bažant and Najjar calibrated the model based on data on drying reported in the lit- erature. According to this calibration the parameter α0, ranges between 0.025 and 0.10, nbetween 6 and 16, hc lies around 0.75, and the activation energy of diffusion is roughly equal to 2160 kJ/(kg·K). Despite the limitations of the model, it has been very well accepted by the concrete scientific community which led to its inclusion in the CEB-FIP Model Code 1990 [N14] and it has been further maintained in the new release,
f ibModel Code 2010 [N15].