ConcreteThermalMoisture

Governing equation

The governing partial differential equation for heat transfer in concrete is given by Bažant et al. (1982) as:

where:

= density, kg/m
= specific heat of concrete, J/kgC
= temperature, C
= thermal conductivity of concrete, W/mC
= mass density and isobaric (constant pressure) heat capacity of liquid water
= moisture flux ()
= water (moisture) content in g/g (for unit volume of material, m)
= pore relative humidity
= heat absorption of free water,
= moisture capacity,
= rate of heat per unit volume generated within the body, W/m
= time,

The term on the left side of Eq. (1) represents time-dependent effects, and is provided by ConcreteThermalTimeIntegration. The first term on the right side of Eq. (1) represents the thermal conduction, and is provided by ConcreteThermalConduction. The second term represents the convective transport of heat due to fluid flow, and is provided by ConcreteThermalConvection. The third term represents adsorption heat due to adsorption of free water molecules in pores onto pore walls, and is provided by ConcreteLatentHeat. The last term is a volumetric heating from other sources that can be provided by general-purpose kernels provided by MOOSE.

The mass density and isobaric (constant pressure) heat capacity of liquid water is given by

where is density of water (1000 kg/m) and is heat capacity of water (= 4180 J/kg-)

and is the isobaric (constant pressure) heat capacity of liquid water in J/kgC. The values of are tabulated in Yunus and Afshin (2011). The adsorption heat usually can be neglected according to Bažant et al. (1982). Thus, a small fraction of the concrete specific heat capacity value is simply assigned to (i.e., ).

Thermal capacity

Four constitutive models are available for concrete thermal capacity (in MJ/mC ):

1. A user-supplied constant thermal capacity;

2. The ASCE (1992) model for normal-strength concrete;

3. Kodur et al. (2004) model for high-strength concrete and

4. The Eurocode (2004) model for both normal- and high-strength concrete.

Details of these models are provided below:

1. Constant

   In this model, the user provides a value of that remains constant during the simulation.

2. ASCE (1992)

• Siliceous aggregate concrete

• Carbonate aggregate concrete

3. Kodur et al. (2004)

• Siliceous aggregate concrete

• Carbonate aggregate concrete

4. Eurocode (2004)

Note that thermal capacity for above models at T 20C is assumed to be the thermal capacity at T = 20C.

Thermal conductivity

Four thermal conductivity models are available, all depending on the temperature and concrete texture, including

1. A user-supplied constant thermal conductivity;

2. The ASCE (1992) model for normal-strength concrete at high temperature;

3. Kodur et al. (2004) model for high-strength concrete;

4. The Eurocode (2004) model for both normal- and high-strength concrete

Details of these models are provided below:

1. Constant

   In this model, the user provides a value of that remains constant during the simulation.

2. ASCE (1992)

• Siliceous aggregate concrete

• Carbonate aggregate concrete

3. Kodur et al. (2004)

• Siliceous aggregate concrete

• Carbonate aggregate concrete

4. Eurocode (2004)

• Upper limit

• Lower limit

Note that thermal conductivity at T 20C is assumed to be the thermal capacity at T = 20C.

These various heat transfer constitutive models can be conveniently chosen and specified from input file.

Moisture capacity

The following three models have been considered for moisture diffusion in concrete

1. Xi et al. (1994)

2. Bažant and Thonguthai (1979)

3. Mensi et al. (1988)

Details of these models are provided below:

1. Xi et al. (1994)

   Xi et al. (1994) and Xi et al. (1994) developed a concrete moisture capacity model based on the Brunauer-Emmett-Teller (BET) adsorption isotherm theory, which was implemented here. The total water content in concrete at a constant temperature is referred as water adsorption isotherm, which was proposed by Xi et al. (1994) as:

where

=
= relative humidity
= absolute temperature in
= quantity of vapor absorbed at pressure (g water/g cement)
= monolayer capacity: mass of adsorbate required to cover
the adsorbent with a single molecular layer
= empirical constant

The monolayer capacity, , is defined as the mass of adsorbate required to cover the surface of the adsorbent with a single molecular layer. To evaluate at a given relative humidity value and the empirical constant in the above equation need to be evaluated first. This is done separately for cement and aggregate materials as follows:

• Monolayer capacity,

  • Cement Paste:

    where is the age of concrete material in ; represents the effect of cement types on the adsorption isotherm and is given by table below Table 1; at room temperature and remains constant during simulations, and

    represents the effects of concrete age and represents the effect of water to cement ratio on the adsorption isotherm, respectively.

    Table 1: for different types of concrete

    Concrete Type	1	2	3	4
    	0.9	1.0	0.85	0.6

  • Aggregates:

    The monolayer capacity of aggregates is determined by

    where depends on the pore structure of various aggregates as listed in Table 2.

    Table 2: of various pore structure of aggregate

    Pore structure of aggregate
    dense	0.05-0.1
    porous	0.1-0.04

• Empirical constant

  The empirical constant in Eq. (16) is related to the the number of layers of adsorbed water molecule, , under saturated state. is determined separately for cement and aggregate materials.

  • Cement Paste:

    is expressed in terms similar to those of :

    where is given by Table 3 and at room temperature and remains constant during the simulation.

    Table 3: for different types of concrete

    Concrete Type	1	2	3	4
    	1.1	1.0	1.15	1.5

  • Aggregates:

    For the aggregate, is expressed as:

    where is defined in Table 4.

    Table 4: of various pore structure of aggregate

    Pore structure of aggregate
    dense	1.0-1.5
    porous	1.7-2.0

    Once the number of adsorbed layers of molecule, , is obtained, can be obtained by

Finally, once the monolayer capacity and empirical constant are obtained, then using Eq. (16), the water content, , in cement and aggregate materials can be obtained. The moisture capacities for cement paste or aggregate material can also be determined by taking derivatives of both sides of Eq. (16) with respect to relative humidity, , as

The total moisture capacity of the concrete structure required by the heat transfer governing equation Eq. (1) is then simply the weight-average value between cement and aggregate materials as:

where

= weight percentage of the aggregate
= weight percentage of the cement paste
= moisture capacity of aggregate (g/g) (for the unit volume of material, cm)
= moisture capacity of cement paste (g/g) (for the unit volume of material, cm)

The total moisture capacity (with the units of g water/g material) is a function of water content , temperature and relative humidity, , and strongly depends on the concrete texture.

2. Bažant and Thonguthai (1979)

Moisture capacity is not defined for this model. Therefore using ConcreteMoistureTimeIntegration is equivalent of using TimeDerivative

3. Mensi et al. (1988)

Moisture capacity is not defined for this model. Therefore using ConcreteMoistureTimeIntegration is equivalent of using TimeDerivative

Governing equation

The governing equation for moisture diffusion in concrete is formulated by using relative humidity, , as the primary variable:

where

= total water content (g/g)
= pore relative humidity, and /
= saturate vapor pressure Bary et al. (2012) (where is the temperature in K))
= standard atmospheric pressure
= moisture diffusivity (also referred as humidity diffusivity),
= coupled moisture diffusivity under the influence of a temperature gradient,
= time,

The term on the left side of Eq. (19) represents time-dependent effects, and is provided by ConcreteMoistureTimeIntegration. The first term on the right side of Eq. (19) represents Fickian diffusion, and the second term represents Soret diffusion. These are both provided by ConcreteMoistureDiffusion.

Moisture diffusivity depends on the relative humidity, . Thus the moisture diffusion governing equation is highly nonlinear. The following sections describes in detail the constitutive models for moisture diffusivity.

Coefficients for moisture diffusivity and coupling with heat diffusion

The moisture diffusivity of concrete, , is a complex function of temperature, , relative humidity, , and pore structure of concrete. Various diffusion mechanisms often interact, such as molecular diffusion in large pores (usually 50nm - 10 microns and beyond) and microcracks, Knudson diffusion in mesopores (2.5nm - 50 nm) and micropores (2.5nm) and surface diffusion along pore walls. Most existing moisture diffusivity models typically do not account for individual diffusion mechanisms separately. Instead, they tend to reproduce the general combined trend.

Calculation of starts with the calculation of a reference moisture diffusivity, , at a given temperature, , and relative humidity, . Three reference moisture diffusivity models are implemented as:

1. Xi et al. (1994)

   This model is applicable for model for normal-strength concrete for w/c > 0.5. The model evaluate the moisture diffusivity for concrete according to

where

= humidity diffusion coefficient of concrete (m/s)
= humidity diffusion coefficient of cement paste (m/s)
= humidity diffusion coefficient of aggregate (m/s)
= the volume fraction of aggregate

The humidity diffusion coefficient in cm/day for cement paste is expressed as Xi et al. (1994):

where , and are coefficients from test data. Since the value of the humidity diffusivity coefficient for aggregates, , typically is negligible compared with the value of , it is assumed to be zero in the current implementation. Note that this model is only applicable for concrete with w/c >= 0.5.

No coupling between heat and moisture transfer is considered in this model, so the values of , , are are set to zero.

2. Bažant et al. (1982)

   This model for normal-strength concrete at high temperature. The model describe the moisture diffusivity as a function of temperature and relative humidity according to

where and f_{1H}, f_{2T}, and f_{3T} are given by

where is given by

is activation energy for water migration along the adsorption layers in the necks, and is gas constant with =2700 K, and is in C. from and .

It has been reported by Bažant et al. (1982) that the additional moisture diffusion due to thermal gradients included in the moisture governing equation is negligible. Thus the value of is set to by default. This parameter can, however, be set to an arbitrary value if desired.

Since the values of parameters such as and are not provided in the numerical model, they are taken from different references. Poyet and Charles (2009) observed that vary between 42 kJ/mol and 80 kJ/mol for concrete, therefore, an average value is considered for general concrete, i.e., 60 kJ/mol.

3. Mensi et al. (1988)

   This model is for for normal-strength concrete for ambient conditions.

where

= initial water content in concrete (kg/m)

is given by

where mass of cement in concrete mix.

No coupling between heat and moisture transfer is considered in this model, so the values of , , are are set to zero.