Correcting moisture diffusivity model (ref idaholab#245)

doc/content/bib/blackbear.bib

@@ -233,3 +233,14 @@ @article{OH2011124
 	keywords = {Ductile fracture simulation, Finite element analysis, Stress-modified fracture strain},
 	abstract = {This paper proposes a new method to simulate ductile failure using finite element analysis based on the stress-modified fracture strain model. A procedure is given to determine the stress-modified fracture strain as a function of the stress triaxiality from smooth and notched bar tensile tests with FE analyses. For validation, simulated results using the proposed method are compared with experimental data for cracked bar (tensile and bend) tests, extracted from API X65 pipes, and for full-scale burst test of gouged pipes, showing overall good agreements. Advantages in the use of the proposed method for practical structural integrity assessment are discussed.}
 	}
+
+	@article{poyet2009temperature,
+	  title={Temperature dependence of the sorption isotherms of cement-based materials: Heat of sorption and Clausius--Clapeyron formula},
+	  author={Poyet, St{\'e}phane and Charles, S{\'e}bastien},
+	  journal={Cement and Concrete Research},
+	  volume={39},
+	  number={11},
+	  pages={1060--1067},
+	  year={2009},
+	  publisher={Elsevier}
+	}

doc/content/source/materials/ConcreteThermalMoisture.md

@@ -37,7 +37,7 @@ The mass density and isobaric (constant pressure) heat capacity of liquid water
 !equation id=Cw
 C_w = \rho_{water}C_{water}
 
-where $\rho_{water}$ is mass density of liquid water in $kg/m^3$ which is given as [!cite](raznjevic1970tables)
+where $\rho_{water}$ is density of water (1000 kg/m$^3$) and $C_{water}}$ is heat capacity of water (= 4180 J/kg-$\degree C$)
 
 !equation id=rho_w
 \rho_{water} = 314.4 + 685.6\left[1-\left(\frac{T-273.15}{374.14}\right)^\frac{1}{0.55}\right]^{0.55}
@@ -207,25 +207,35 @@ These various heat transfer constitutive models can be conveniently chosen and s
 
 ### Moisture capacity
 
-[!cite](xi_moisture_1994_a,xi_moisture_1994_b) developed a concrete moisture capacity model based on the Brunauer-Emmett-Teller (BET) adsorption isotherm theory, which was implemented here. The total water content $W$ in concrete at a constant temperature $T$ is referred as water adsorption isotherm, which was proposed by [!cite](xi_moisture_1994_a) as:
+The following three models have been considered for moisture diffusion in concrete
+
+1. [!cite](xi_moisture_1994_a)
+2. [!cite](bazant1979pore)
+3. [!cite](mensi1988sechage)
+
+Details of these models are provided below:
+
+1. [!cite](xi_moisture_1994_a)
+
+  [!cite](xi_moisture_1994_a,xi_moisture_1994_b) developed a concrete moisture capacity model based on the Brunauer-Emmett-Teller (BET) adsorption isotherm theory, which was implemented here.  The total water content $W$ in concrete at a constant temperature $T$ is referred as water adsorption isotherm, which was proposed by [!cite](xi_moisture_1994_a) as:
 
 !equation id=adsorption_isotherm
 W = \frac{CkV_mH}{(1-kH)[1+(C-1)kH]}
 
-where
+    where
 
-$C$ = $\exp\left(\frac{C_0}{T}\right) \\
-C_0 = 855$ \\
-$H$ = relative humidity \\
-$T$ = absolute temperature in $K$ \\
-$W$ = quantity of vapor absorbed at pressure $p$ (g water/g cement) \\
-$V_m$ = monolayer capacity: mass of adsorbate required to cover \\
-       the adsorbent with a single molecular layer \\
-$k$ = empirical constant
+    $C$ = $\exp\left(\frac{C_0}{T}\right) \\
+    C_0 = 855$ \\
+    $H$ = relative humidity \\
+    $T$ = absolute temperature in $K$ \\
+    $W$ = quantity of vapor absorbed at pressure $p$ (g water/g cement) \\
+    $V_m$ = monolayer capacity: mass of adsorbate required to cover \\
+           the adsorbent with a single molecular layer \\
+    $k$ = empirical constant
 
-The monolayer capacity, $V_m$, is defined as the mass of adsorbate required to cover the surface of the adsorbent with a single molecular layer. To
-evaluate $W$ at a given relative humidity value $V_m$ and the empirical constant $k$ in the above equation need to be evaluated first. This is done separately
-for cement and aggregate materials as follows:
+    The monolayer capacity, $V_m$, is defined as the mass of adsorbate required to cover the surface of the adsorbent with a single molecular layer. To
+    evaluate $W$ at a given relative humidity value $V_m$ and the empirical constant $k$ in the above equation need to be evaluated first. This is done separately
+    for cement and aggregate materials as follows:
 
 - **Monolayer capacity, $V_m$**
 
@@ -340,6 +350,13 @@ $\left(\frac{\partial{W}}{\partial{H}}\right)_{cp}$ = moisture capacity of cemen
 The total moisture capacity ${\partial W}/{\partial H}$ (with the units of g water/g material) is a function of water content $W$, temperature $T$
 and relative humidity, $H$, and strongly depends on the concrete texture.
 
+2. [!cite](bazant1979pore)
+
+Moisture capacity is not defined for this model. Therefore using [ConcreteMoistureTimeIntegration](ConcreteMoistureTimeIntegration.md) is equivalent of using [TimeDerivative](TimeDerivative.md)
+
+3. [!cite](mensi1988sechage)
+
+Moisture capacity is not defined for this model. Therefore using [ConcreteMoistureTimeIntegration](ConcreteMoistureTimeIntegration.md) is equivalent of using [TimeDerivative](TimeDerivative.md)
 
 ## Moisture diffusion
 
@@ -350,7 +367,7 @@ A comprehensive set of constitutive models and parameters for moisture diffusion
 The governing equation for moisture diffusion in concrete is formulated by using relative humidity, $H$, as the primary variable:
 
 !equation id=moisture_governing
-\frac{\partial{W}}{\partial{H}} \frac{\partial{H}}{\partial{t}} = \nabla (D_h\nabla H) + \nabla (D_{ht}\nabla T) + \frac{\partial{W_d}}{\partial{t}}
+\frac{\partial{W}}{\partial{H}} \frac{\partial{H}}{\partial{t}} = \nabla (D_h\nabla H) + \nabla (D_{ht}\nabla T)
 
 where
 
@@ -360,14 +377,13 @@ $P_{vs}$ = saturate vapor pressure $= P_{atm}\exp\left(4871.3\frac{T-100}{373.15
 $P_{atm}$ = standard atmospheric pressure $= 101.325 Pa$ \\
 $D_h$ = moisture diffusivity (also referred as humidity diffusivity), $kg/(m^2\cdot s)$\\
 $D_{ht}$= coupled moisture diffusivity under the influence of a temperature gradient, $kg/(m^2\cdot s\cdot K)$\\
-$W_d$= total mass of free evaporable water released into the pores by dehydration of the cement paste\\
 $t$  = time, $s$
 
-The term on the left side of [!eqref](moisture_governing) represents time-dependent effects, and is provided by [ConcreteMoistureTimeIntegration](ConcreteMoistureTimeIntegration.md). The first term on the right side of [!eqref](moisture_governing) represents Fickian diffusion, and the second term represents Soret diffusion. These are both provided by [ConcreteMoistureDiffusion](ConcreteMoistureDiffusion.md). The third term on the right hand side of this equation represents a source due to dehydrated water, and is provided by [ConcreteMoistureDehydration](ConcreteMoistureDehydration.md).
+The term on the left side of [!eqref](moisture_governing) represents time-dependent effects, and is provided by [ConcreteMoistureTimeIntegration](ConcreteMoistureTimeIntegration.md). The first term on the right side of [!eqref](moisture_governing) represents Fickian diffusion, and the second term represents Soret diffusion. These are both provided by [ConcreteMoistureDiffusion](ConcreteMoistureDiffusion.md).
 
 Moisture diffusivity $D_h$ depends on the relative humidity, $H$. Thus the moisture diffusion governing equation is highly nonlinear. The following sections describes in detail the constitutive models for moisture diffusivity.
 
-### Moisture diffusivity
+### Coefficients for moisture diffusivity and coupling with heat diffusion
 
 The moisture diffusivity of concrete, $D_h$, is a complex function of temperature, $T$, relative humidity, $H$, 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
@@ -376,73 +392,98 @@ mechanisms separately. Instead, they tend to reproduce the general combined tren
 
 Calculation of $D_h$ starts with the calculation of a reference moisture diffusivity, $D_{h,0}$, at a given temperature, $T$, and relative humidity, $H$. Three reference moisture diffusivity $D_{h,0}$ models are implemented as:
 
-1. [!cite](mensi1988sechage)
+1. [!cite](xi_moisture_1994_a)
 
-!equation id=Dh0_mensi
-D_{h,0} = Ae^{BC}
+  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
+!equation id=Dh
+D_{h,0} = D_{Hcp}\left[1 + \frac{g_i}{\frac{1-g_i}{3} + \frac{1}{\frac{D_{Hagg}}{D_{Hcp}}-1}}\right],
 
-$D_{h,0}$ = humidity diffusion coefficient of concrete\\
-$A = 3.8 \times 10^{-13} m^2/s$ \\
-$B = 0.05$ \\
-$C$ = free water content in L/m$^3$
+  where
 
-$C$ is a function of relative humidity, $H$, in concrete as given by
+  $D_{h}$ = humidity diffusion coefficient of concrete (m$^2$/s)\\
+  $D_{Hcp}$ = humidity diffusion coefficient of cement paste (m$^2$/s)\\
+  $D_{Hagg}$ = humidity diffusion coefficient of aggregate (m$^2$/s) \\
+  $g_i$ = the volume fraction of aggregate
 
-!equation id=C_mensi
-C = HC_0
+  The humidity diffusion coefficient in cm$^2$/day for cement paste is expressed as [!cite](xi_moisture_1994_a):
+
+!equation id=Dhcp
+D_{Hcp} [m^2/s] = \alpha_h + \beta_h (1-2^{-10\gamma_h(H-1)}) * 1e-4 / 86400
+
+!equation id=alpha_h
+\alpha_h [cm^2/s] = 1.05 - 3.8 \frac{w}{c} + 3.56 \left(\frac{w}{c}\right)^2
 
-where $C_0$ is constant takes a value of 130 (in L/m$^3$).
+!equation id=beta_h
+\beta_h [cm^2/s] = -14.4 + 50.4 \frac{w}{c} - 41.8 \left(\frac{w}{c}\right)^2
+
+!equation id=gamma_h
+\gamma_h [cm^2/s] = 31.3 - 136 \frac{w}{c} + 162 \left(\frac{w}{c}\right)^2
+
+  where $\alpha_h$, $\beta_h$ and $\gamma_h$ are coefficients from test data. Since the value of the humidity diffusivity coefficient for aggregates, $D_{Hagg}$, typically
+  is negligible compared with the value of $D_{Hcp}$, 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 $D_{ht}$, $C_{a}$, are $C_{w}$ are set to zero.
 
 2. [!cite](bazant1982finite)
 
-!equation id=Dh0_bazant
-D_{h,0} = D_1f_H
+  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
+
+!equation id=Dh_bazant
+D_h = \left\{\begin{array}{lll}
+      D_1 f_{1H} f_{2T} & \text{for} & T \le 95 \degree\text{C} \\
+      D_1 f_{2T}(T=95 \degree C) f_{3T} & \text{for} & T > 95 \degree\text{C}
+      \end{array}\right.
 
-where $D_1 = 3.10 \times 10^{-10}m^2/s$ and
+where $D_1 = 3.10 \times 10^{-10}m^2/s$ and f_{1H}, f_{2T}, and f_{3T} are given by
 
 !equation id=fH
-f_H = \left\{\begin{array}{lll}
-        \alpha_D + \frac{1-\alpha_D}{1+\left(\frac{1-H}{1-0.75}\right)^n} & \text{for} & T \le 95\degree C\\
-        1 & \text{for} & T > 95 \degree C
+f_{1H} = \left\{\begin{array}{lll}
+        \alpha_D + \frac{1-\alpha_D}{1+\left(\frac{1-H}{1-0.H_c}\right)^n} & \text{for} & H \le 1\\
+        1 & \text{for} & H > 1
     \end{array}\right.
 
-Also, $f_H = 1$ when $H > 1$ and $\alpha_D$ is given by
+!equation id=f2T
+f_2(T) = \exp\left(\frac{Q}{R}\left(\frac{1}{(T_{ref} + 273.15)}-\frac{1}{(T + 273.15)}\right)\right)
+
+!equation id=f3T
+\begin{array}{lll}
+    f_3(T) = \exp\left(\frac{T-95}{0.881 + 0.214(T-95)}\right) & \text{for} & T > 95 \degree C
+\end{array}
+
+where $\alpha_D$ is given by
 
-!equation id=D_bazant
+!equation id=alfaD_bazant
 \alpha_D = \frac{1}{1+\frac{19(95-T)}{70}}
 
-where $T$ is in $\degree$C. $\alpha \in [0.037:1]$ from $T\in[0\degree C : 95\degree C]$ and $n\in[6:16]$.
+$Q$ is activation energy for water migration along the adsorption layers in the necks, and $R$ is gas constant with $Q/R$=2700 K, and $T$ is in $\degree$C. $\alpha \in [0.037:1]$ from $T\in[0\degree C : 95\degree C]$ and $n\in[6:16]$.
 
-It's obvious that all three reference moisture diffusivity models strongly depend on the value of humidity $H$, and indirectly on the temperature $T$. Once the value of
-reference moisture diffusivity $D_{h,0}$ is obtained, the actual concrete moisture diffusivity $D_h$ required by the moisture diffusion governing equation, [!eqref](moisture_governing),
-can then be calculated by
+It has been reported by [!cite](bazant1982finite) that the additional moisture diffusion due to thermal gradients included in the moisture governing equation is negligible. Thus the value of $D_{ht}$ is set to $1.0\times10^{-5} \times D_h$ by default. This parameter can, however, be set to an arbitrary value if desired.
 
-!equation id=Dh_coupled
-D_h = \left\{\begin{array}{lll}
-        D_{h, 0}f_1(T) & \text{for} & T \le 95 \degree C \\
-        D_{h, 0}f_1(95\degree C)f_2(T) & \text{for} & T > 95 \degree C
-    \end{array}\right.
+Since the values of parameters such as $C_{a}$ and $C_{w}$ are not provided in the numerical model, they are taken from different references. [!cite](poyet2009temperature) observed that $C_{a}$ 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. [!cite](mensi1988sechage)
+
+  This model is for for normal-strength concrete for ambient conditions.
+
+!equation id=Dh0_mensi
+D_{h} = Ae^{BC}
 
 where
 
-!equation id=f1T
-\begin{array}{lll}
-    f_1(T) = \exp\left(\frac{Q}{R}\left(\frac{1}{T_{ref}}-\frac{1}{T}\right)\right) & \text{for} & T \le 95° C
-\end{array}
+$A = 1.042 \times 10^{-13}  m^2/s$ \\
+$B = 0.05$ \\
+$C$ = initial water content in concrete (kg/m$^3$)
 
-in which $T$ is the absolute temperature ($K$), $Q$ is activation energy for water migration along the adsorption layers in the necks, and $R$ is gas constant
-with $Q/R$=2700 K, and
+$C$ is given by
 
-!equation id=f2T
-\begin{array}{lll}
-    f_2(T) = \exp\left(\frac{T-95}{0.881 + 0.214(T-95)}\right) & \text{for} & T > 95 \degree C
-\end{array}
+!equation id=C_mensi
+C = w/c \times M_{cement}
 
-### Coupled moisture diffusion by thermal gradient, $D_{ht}$
+where $M_{cement}$ mass of cement in concrete mix.
 
-It has been reported by Bazant et al. [!cite](bazant1982finite) that the additional moisture diffusion due to thermal gradients included in the moisture governing equation is negligible. Thus the value of $D_{ht}$ is set to $1.0\times10^{-5} \times D_h$ by default. This parameter can, however, be set to an arbitrary value if desired.
+No coupling between heat and moisture transfer is considered in this model, so the values of $D_{ht}$, $C_{a}$, are $C_{w}$ are set to zero.
 
 !syntax parameters /Materials/ConcreteThermalMoisture
 

include/kernels/ConcreteLatentHeat.h

@@ -43,7 +43,7 @@ class ConcreteLatentHeat : public TimeDerivative
 
   /// Material property of porosity
   const MaterialProperty<Real> & _ca;
-  const MaterialProperty<Real> & _moisture_capacity;
+  const MaterialProperty<Real> * _moisture_capacity; //  dW/dH
 
   /// Time derivatives of relative humidity (moisture)
   const VariableValue & _H_dot;