Nonlinear isotropic material for moisture transport - NlIsoMoisture

This is a more general model for nonlinear moisture transport in isotropic porous materials, based on a nonlinear sorption isotherm (relation between the pore relative humidity $ h$ and the water content $ w$) and on a humidity-dependent moisture permeability. The governing differential equation reads

(237)

where [kg/m$ ^3$] is the humidity-dependent moisture capacity (derivative of the moisture content with respect to the relative humidity), and [kg/m$ \cdot$s] is the moisture permeability.

So far, six different functions for the sorption isotherm have been implemented (in fact, what matters for the model is not the isotherm itself but its derivative--the moisture capacity):

  1. Linear isotherm ( ) is characterized only by its slope given by parameter .

  2. Piecewise linear isotherm ( ) is defined by two arrays with the values of pore relative humidity and the corresponding values of moisture content . The arrays must be of the same size.

  3. Ricken isotherm [15] ( ), which is widely used for sorption of porous building materials. It is expressed by the equation

    (238)

    where [kg/m$ ^3$] is the water content at and [m$ ^3$/kg] is an approximation coefficient. In the input record, only must be specified ( is not needed). Note that for this isotherm gives an infinite moisture content.

  4. Isotherm proposed by Kuenzel [15] ( ) in the form

    (239)

    where $ w_f$ [kg/m$ ^3$] is the moisture content at free saturation and is a dimensionless approximation factor greater than 1.

  5. Isotherm proposed by Hansen [13] ( ) in the form

    (240)

    characterizes the amount of adsorbed water by the moisture ratio [kg/kg]. To obtain the moisture content $ w$, it is necessary to multiply the moisture ratio by the density of the solid phase. In (240), is the maximum hygroscopically bound water by adsorption, and and $ n$ are constants obtained by fitting of experimental data.

  6. The BSB isotherm [4] ( ) is an improved version of the famous BET isotherm. It is expressed in terms of the moisture ratio

    (241)

    where is the monolayer capacity, and $ C$ depends on the absolute temperature $ T$ and on the difference between the heat of adsorption and condensation. Empirical formulae for estimation of the parameters can be found in [27]. Note that these formulae hold quite accurately for cement paste only; a reduction of the moisture ratio is necessary if the isotherm should be applied for concrete.

The present implementation covers three functions for moisture permeability:

  1. Piecewise linear permeability ( ) is defined by two arrays with the values of pore relative humidity and the corresponding values of moisture content . The arrays must be of the same size.

  2. The Bazant-Najjar permeability function ( ) is given by the same formula (236) as the diffusivity in Section 2.3. All parameters have a similar meaning as in (236) but is now the moisture permeability at full saturation [kg/m$ \cdot$s].

  3. Permeability function proposed by Xi et al. [27] ( ) reads

    (242)

    where , and are parameters that can be evaluated using empirical mixture-based formulae presented in [27]. However, if those formulae are used outside the range of water-cement ratios for which they were calibrated, the permeability may become negative. Also the physical units are unclear.

Note that the Bajant-Najjar model from Section 2.3 can be obtained as a special case of the present model if is set to 1 and is set to 0. The ratio then corresponds to the diffusivity parameter $ C_1$ from Eq. (235).

The model parameters are summarized in Tab. 55.

Table 55: Nonlinear isotropic material for moisture transport - summary.
Description Nonlinear isotropic material for moisture transport
Record Format NlIsoMoistureMat num(in) # d(rn) # isothermType(in) # permeabilityType(in) # [ rhodry(rn) #] [ capa(rn) #] [ iso_h(ra) #] [ iso_w(h)(ra) #] [ dd(rn) #] [ wf(rn) #] [ b(rn) #] [ uh(rn) #] [ A(rn) #] [ nn(rn) #] [ c(rn) #] [ k(rn) #] [ Vm(rn) #] [ perm_h(ra) #] [ perm_c(h)(ra) #] [ c1(rn) #] [ n(rn) #] [ alpha0(rn) #] [ hc(rn) #] [ alphah(rn) #] [ betah(rn) #] [ gammah(rn) #]
Parameters - num material model number
  - d material density
  - isothermType isotherm function as listed above (0, 1, ...5)
  - permeabilityType moisture permeability function as listed above (0, 1, 2)
  - rhodry [kg/m$ ^3$] density of dry material (for and 5)
  - capa [kg/m$ ^3$] moisture capacity (for )
  - iso_h [-] humidity array (for )
  - iso_w(h) [kg/m$ ^3$] moisture content array (for )
  - dd [-] parameter (for )
  - wf [kg/m$ ^3$] is the moisture content at free saturation (for )
  - b [-] parameter (for )
  - uh [kg/kg] maximum hygroscopically bound water by adsorption (for )
  - A [-] parameter (for )
  - n [-] parameter (for )
  - Vm (for )
  - k (for )
  - C (for )
  - perm_h [-] humidity array (for )
  - perm_c(h) [kg m$ ^{-1}$ s$ ^{-1}$] moisture permeability array (for )
  - c1 [kg m$ ^{-1}$ s$ ^{-1}$] moisture permeability at full saturation (for )
  - n [-] exponent (for )
  - alpha0 [-] ratio between minimum and maximum diffusivity (for )
  - hc [-] relative humidity at which the diffusivity is exactly between its minimum and maximum value (for )
  - alphah [kg m$ ^{-1}$ s$ ^{-1}$] (for )
  - betah [kg m$ ^{-1}$ s$ ^{-1}$] (for )
  - gammah [-] (for )
Supported modes _2dHeat


Borek Patzak
2018-01-02