4. Formalization of drying

This part refers to the drying development specification document in Code_Aster [Bib.3], as well as to L. Granger’s thesis [Bib.2].

4.1. Drying modeling and equations

The thermal or thermo-hydration and drying models are decoupled during the resolution. Drying is then presented as a chained thermal operation. As the equations for solving drying and nonlinear thermal are similar except for coefficients, this decoupling makes it possible to integrate the resolution of the drying calculation into Code_Aster, directly using the nonlinear thermal resolution module, without adding new phenomena, new types of elements or new calculation options, without adding new phenomena, new types of elements or new calculation options, and thus minimizing the volume of code added and duplicated.

The concentration or content of water \(C\), a calculation variable in drying modeling, is assimilated, in terms of the type of variable, to a temperature (type TEMP). The transient temperature field, which is involved in the drying equation, is only an auxiliary parameter on which the diffusion coefficient may depend.

Thermal and drying phenomena, in the context of a decoupled modeling between thermal and drying, are governed by the following equations:

• « classical » thermal equation:

\(\begin{array}{c}\rho {C}_{p}\frac{\text{dT}}{\text{dt}}+\text{div}\mathrm{q}\mathrm{=}Q\frac{d\xi (T)}{\text{dt}}+s(T)\\ \mathrm{q}\mathrm{=}\mathrm{-}\lambda (T)\text{grad}T\end{array}\mathrm{\}}\) eq 4.1-1

(\(\rho {C}_{p}\) the volume heat at constant pressure, \(\lambda\), the thermal conductivity,, the thermal conductivity, \(Q\) the heat of hydration and \(s\) the internal source).

• equation characterizing drying:

\(\frac{\partial C}{\partial t}-\text{Div}\left[D(C,T)\nabla C\right]=0\) eq 4.1-2

where

\(C\) (\({m}^{3}\mathrm{/}{m}^{3}\) or \(l\mathrm{/}{m}^{3}\)) is the calculation variable (concentration or content of water by volume), \(T\) is the calculation input variable (temperature), an auxiliary variable for the drying resolution, \(D\) (\({m}^{2}\mathrm{/}s\)) is a diffusion coefficient, characterizing the nonlinearity of the equation, and depending on both the calculation variable, \(C\), and the auxiliary variable, \(T\). This diffusion law is given in various forms, depending on the model chosen, (Bazant law, Granger law, Mensi law , cf. [§4.3] and [bib2]).

The equations [éq 4.1-1] and [éq 4.1-2] correspond to a thermal/drying chained calculation. We can therefore calculate \(T\) without knowing the water concentration, then calculate the latter, for which \(T\) is then a parameter, (assuming that the thermal conductivity \(\lambda\) does not depend on the water concentration \(C\) step). It should also be noted that the drying phenomenon is decoupled from the mechanical evolutions of concrete.

4.2. Diffusion coefficient

The material is described by the diffusion coefficient \(D\) , characteristic of the material, depending on both the temperature \(T\) and the water concentration \(C\). The equation for the migration of humidity in concrete is derived from those for the mechanics of porous media. Please refer to [bib2] for more details. Conventionally, a law of diffusion expresses a flow as the product of a quantity characteristic of the material by the gradient of an intensive quantity. The various quantities considered are defined by an average over the representative elementary volume, as long as this average can be defined for the material in question, in such a way that the derivation operators make sense. In general, the hypothesis is therefore made which consists in assuming that the liquid and gaseous phases are related:

for vapor diffusion, we start from the positivity of the dissipation associated with the transport of the gas phase, by differentiating between two phenomena, a permeation type phenomenon (**Darcy*), linked to pressure gradients, and a diffusion type phenomenon (Fick), linked to concentration gradients,

for the diffusion of liquid water, the positivity of the dissipation associated with the transport of liquid water, and with the law of**Darcy*, makes it possible to express the flow of liquid as a function of the pressure of the liquid. The**Kelvin law** describing the coexistence of the two liquid and gaseous phases by writing the equality of the free mass enthalpies leads to the expression of the flow as a function of the gradient of the degree of humidity.

From the two previous results, we obtain the expression of the total flow as a function of the gradient of the degree of water concentration. Conventional experimental methods in drying problems generally provide access to water concentration, and very rarely to relative humidity. It is therefore preferable to express the flux as a function of the water content, using classically the desorption isotherm of concrete, which relates the water content, \(C\), and the relative humidity, \(h\). Relative humidity is the ratio between vapour pressure and saturated vapour pressure.

The local state postulate states that the current state of a homogeneous system in any evolution can be characterized by the same variables as at equilibrium, and that it is independent of the rates of evolution. In other words, water content \(C\), and relative humidity \(h\), are well connected by the same relationship as with balance. This leads to the classical diffusion equation:

\(\frac{\partial C}{\partial t}-\text{Div}\left[D(C,T)\nabla C\right]=0\) eq 4.2-1

This equation highlights the non-linear nature of the diffusion of humidity in concrete. In industrial cases, the temperature is generally not uniform in the structure. It is therefore necessary to take into account a humidity diffusion coefficient that depends on the temperature. In practice, in the literature, the best known authors (Bazant cf. [bib2]) propose an expression for the diffusion coefficient of the type:

\(D(C,T)\mathrm{=}D(C,{T}_{0})(\frac{T}{{T}_{0}}){e}^{(\frac{\mathrm{-}{Q}_{s}}{R}(\frac{1}{T}\mathrm{-}\frac{1}{{T}_{0}}))}\) eq 4.2-2

with \({Q}_{s}\mathrm{/}R\mathrm{=}4700{K}^{\mathrm{-}1}\) and \(T\) in \(°K\) [1] _

Note:

From the way things are presented, it seems that we have not taken advantage of the fact that drying is a phenomenon coupled with mechanics, (that is, it is the cause of desiccation shrinkage). In reality, we hypothesized a decoupling of phenomena, when we used the sorption/desorption curve. In fact, when measuring weight loss at equilibrium as a function of h, the test body realizes a withdrawal. At the microscopic level, everything happens as if the shrinkage, modifying the porosity, would interact with the relative humidity inside the sample, since the vapor pressure and h increase. Since this desiccation shrinkage is very low, it is usual to neglect it in water content calculations. There is therefore only a link between the calculation of the water content and the mechanical calculation of desiccation shrinkage.

4.3. Common distribution laws

The diffusion law, a function of the two parameters, \(C\) and \(T\), can be freely defined by the user in the form of a sheet. However, the usual expressions of the law of diffusion, which are found in the literature, are as follows:

Law proposed by Granger:

\(D\left(C,T\right)=A\text{.}{e}^{\left(B\text{.}C\right)}\left(\frac{T}{{T}_{0}}\right){e}^{\left(\frac{-{Q}_{s}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{0}}\right)\right)}\) eq 4.3-1

\(A({m}^{2}\mathrm{/}s)\) , \(B\) , \({T}_{0}\) , * **, * \({Q}_{s}\), and \(R\) (\(\mathit{Qs}\mathrm{/}R\) in \(°K\)) are coefficients chosen by the user. \(D\) is a function of temperature and water concentration.

Mensi’s law:

\(D(C)=A\text{.}{e}^{(B\text{.}C)}\) eq 4.3-2

\(A\) and \(B\) are coefficients chosen by the user. \(D\) is a function of water concentration only.

Bazant’s law:

Bazant’s law is expressed from the humidity level \(h\), which is linked to the water concentration by the sorption/desorption curve. The form of this law is as follows:

\(D(h)={D}_{1}(\alpha +\frac{1-\alpha }{1+{(\frac{1-h(C)}{1-0\text{.}\text{75}})}^{n}})\) eq 4.3-3

Usually,

\({D}_{1}\mathrm{=}{3.10}^{\mathrm{-}10}{m}^{2}\mathrm{/}s\) \(\alpha\) is between \(0.025\) and \(0.1\) \(n\) is in the order of \(6\). \(h(C)\) is the humidity level, which is expressed as a function of the water concentration using the sorption/desorption curve.

The sorption/desorption curve can be introduced in the form of a tabulated standard function, knowing that in reality, this curve has hysteresis, but can be considered as being invertible, if only one direction of travel is taken into account.

4.4. Modeling boundary conditions

Boundary conditions are generally expressed by a non-linear relationship between water concentration flow (\(l\mathrm{/}{m}^{3}\mathrm{\times }{\mathit{ms}}^{\mathrm{-}1}\)) \({w}^{\text{fl}}\) and water concentration. These conditions are therefore analogous to the so-called heat exchange conditions. For example, we could use the formula proposed by L. Granger [bib2] page 181:

His expression is as follows:

\({w}^{\text{fl}}=\frac{0\text{.}5\beta }{{({C}_{0}-{C}_{\text{eq}})}^{2}}\left[C-(2\text{.}{C}_{0}-{C}_{\text{eq}})\right](C-{C}_{\text{eq}})\) eq 4.4-1

where	\({C}_{\text{eq}}\) is the water concentration for a humidity of 50% RH, \({C}_{0}\) is the water concentration for a humidity of 100% RH, \(\beta\) (\(l\mathrm{/}{m}^{3}\mathrm{\times }m\mathrm{/}s\)) is a coefficient, which can be defined experimentally and can change according to the cracking of the exchange surface ([bib2]),
and	\(C\) is the current (unknown) concentration on exchange surfaces.

[1]

The temperature is entered in degrees Celsius in the data entry. The conversion to degrees Kelvin is done in the sources.