Add ver-1kc-2

doc/content/verification_and_validation/index.md

@@ -33,7 +33,7 @@ TMAP8 also contains [example cases](examples/tmap_index.md), which showcase how
 | ver-1jb | [Radioactive Decay of Mobile Tritium in a Slab with a Distributed Trap Concentration](ver-1jb.md) |
 | ver-1ka | [Simple Volumetric Source](ver-1ka.md)                                                            |
 | ver-1kb | [Henry’s Law Boundaries with No Volumetric Source](ver-1kb.md)                                    |
-| ver-1kc | [Sieverts’ Law Boundaries with No Volumetric Source](ver-1kc.md)                                  |
+| ver-1kc-1 | [Sieverts’ Law Boundaries with No Volumetric Source](ver-1kc-1.md)                              |
 | ver-1kc-2 | [Sieverts’ Law Boundaries with Chemical Reaction and No Volumetric Source](ver-1kc-2.md)        |
 
 # List of benchmarking cases

doc/content/verification_and_validation/ver-1kc-2.md

@@ -1,23 +1,59 @@
 # ver-1kc-2
 
-# Sieverts’ Law Boundaries with No Volumetric Source
+# Sieverts’ Law Boundaries with Chemical Reactions and No Volumetric Source
 
 ## General Case Description
 
-Two enclosures are separated by a membrane that allows diffusion according to Sieverts' law, with no volumetric source present. Enclosure 2 has twice the volume of Enclosure 1.
+Two enclosures are separated by a membrane that allows diffusion according to Sieverts' law and chemical reactions, with no volumetric source present. Enclosure 2 has twice the volume of Enclosure 1.
 
 ## Case Set Up
 
 This verification problem is taken from [!cite](ambrosek2008verification).
 
-Unlike the ver-1kc-1 case, which only considers tritium T$_2$, this setup describes a diffusion system in which tritium T$_2$, dihydrogen H$_2$ and tritium hybride HT are modeled across a one-dimensional domain split into two enclosures. The total system length is $2.5 \times 10^{-4}$ m, divided into 100 segments. The system operates at a constant temperature of 500 Kelvin. Initial tritium T$_2$ and dihydrogen H$_2$ pressures are specified as $10^{5}$ Pa for Enclosure 1 and $10^{-10}$ Pa for Enclosure 2. Initially, there is no tritium hybride HT in either enclosures.
+Unlike the (ver-1kc-1)[ver-1kc-1.md] case, which only considers tritium T$_2$, this setup describes a diffusion system in which tritium T$_2$, dihydrogen H$_2$ and HT are modeled across a one-dimensional domain split into two enclosures. The total system length is $2.5 \times 10^{-4}$ m, divided into 100 segments. The system operates at a constant temperature of 500 Kelvin. Initial tritium T$_2$ and dihydrogen H$_2$ pressures are specified as $10^{5}$ Pa for Enclosure 1 and $10^{-10}$ Pa for Enclosure 2. Initially, there is no HT in either enclosure.
 
 The reaction between the species is described as follows
 
 \begin{equation}
 \text{H}_2 + \text{T}_2 \leftrightarrow 2\text{HT}
 \end{equation}
 
+The kinematic evolutions of the species are given by the following equations
+
+\begin{equation}
+\frac{d C_{\text{HT}}}{dt} = 2K_1 C_{\text{H}_2} C_{\text{T}_2} - K_2 C_{\text{HT}}^2
+\end{equation}
+
+\begin{equation}
+\frac{d C_{\text{H}_2}}{dt} = -K_1 C_{\text{H}_2} C_{\text{T}_2} + \frac{1}{2} K_2 C_{\text{HT}}^2
+\end{equation}
+
+\begin{equation}
+\frac{d C_{\text{T}_2}}{dt} = -K_1 c_{\text{H}_2} C_{\text{T}_2} + \frac{1}{2} K_2 C_{\text{HT}}^2
+\end{equation}
+
+where $K_1$ and $K_2$ represent the reaction rates for the forward and reverse reactions, respectively.
+
+At equilibrium, the time derivatives are zero
+
+\begin{equation}
+2K_1 C_{\text{H}_2} C_{\text{T}_2} - K_2 C_{\text{HT}}^2 = 0
+\end{equation}
+
+From this, we deduce
+
+\begin{equation}
+P_{\text{HT}} = \eta \sqrt{P_{\text{H}_2} P_{\text{T}_2}}
+\end{equation}
+
+where the equilibrium constant $\eta$ is defined as
+
+\begin{equation}
+\eta = \sqrt{\frac{2K_1}{K_2}}
+\end{equation}
+
+Similarly to TMAP7, the equilibrium constant $\eta$ has been set to a fixed value of $\eta = 2$.
+
 The diffusion process for each species in the two enclosures can be expressed by
 
 \begin{equation}
@@ -29,8 +65,8 @@ and
 \end{equation}
 
 where $C_1$ and $C_2$ represent the concentration fields in enclosures 1 and 2 respectively, $t$ is the time, and $D$ denotes the diffusivity.
+Note that the diffusivity may vary across different species and enclosures. However, in this case, it is assumed to be identical for all.
 
-This case is similar to the [ver-1kb](ver-1kb.md) case, with the key difference being that sorption here follows Sieverts' law instead of Henry's law.
 The concentration in Enclosure 1 is related to the partial pressure and concentration in Enclosure 2 via the interface sorption law:
 
 \begin{equation}
@@ -41,44 +77,55 @@ where $R$ is the ideal gas constant in J/mol/K, $T$ is the temperature in K, $K$
 
 ## Results
 
-We assume that $K = \frac{10}{\sqrt{RT}}$, which is expected to result in $C_1 = 10 \sqrt{C_2}$ at equilibrium.
-As illustrated in [ver-1kc-2_comparison_time_k10], similarly to ver-1kc-1, T$_2$ and H$_2$ pressures reach equilibrium in both enclosures. What is new, however, is that HT is produced in both enclosures following the sorption law. The concentration ratios for T$_2$, H$_2$, and HT between enclosures 1 and 2, shown in [ver-1kc-2_concentration_ratio_T2_k10], [ver-1kc-2_concentration_ratio_H2_k10], and [ver-1kc-2_concentration_ratio_HT_k10], demonstrate that the results obtained with TMAP8 are consistent with the analytical results derived from the sorption law for $K \sqrt{RT} = 10$.
+We assume that $K = 10/\sqrt{RT}$, which is expected to result in $C_1 = 10 \sqrt{C_2}$ at equilibrium.
+As illustrated in [ver-1kc-2_comparison_time_k10], similarly to ver-1kc-1, T$_2$ and H$_2$ pressures reach equilibrium in both enclosures. What is new, however, is that HT is produced in both enclosures following the sorption law.
+
+Thus, it is crucial to ensure that the chemical equilibrium between HT, T$_2$ and H$_2$ is achieved. This can be verified in both enclosures by examining the ratio between $P_{\text{HT}}$ and $\sqrt{P_{\text{H}_2} P_{\text{T}_2}}$, which must equal $\eta=2$.
+As shown in [ver-1kc-2_equilibrium_constant_k10], this ratio approaches $\eta=2$ for both enclosures, as observed in TMAP7. However, achieving this balance involves a compromise. On one hand, $K_1$ must be sufficiently large to ensure that the chemical kinetics in Enclosure 1 are significantly faster than other processes, such as diffusion and surface sorption. On the other hand, $K_2$ should not be excessively large, as this could hinder the diffusion of species into Enclosure 2, where no species are initially present.
+
+The concentration ratios for T$_2$, H$_2$, and HT between enclosures 1 and 2, shown in [ver-1kc-2_concentration_ratio_T2_k10], [ver-1kc-2_concentration_ratio_H2_k10], and [ver-1kc-2_concentration_ratio_HT_k10], demonstrate that the results obtained with TMAP8 are consistent with the analytical results derived from the sorption law for $K \sqrt{RT} = 10$.
 
 As shown in [ver-1kc-2_mass_conservation_k10], mass is conserved between the two enclosures over time for all species. The variation in mass is only $0.4$ % for T$_2$ and H$_2$. This variation in mass can be further minimized by refining the mesh, i.e., increasing the number of segments in the domain.
 
 !media comparison_ver-1kc-2.py
        image_name=ver-1kc-2_comparison_time_k10.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1kc-2_comparison_time_k10
-       caption=Evolution of species concentration over time governed by Sieverts' law with $K = \frac{10}{\sqrt{RT}}$.
+       caption=Evolution of species concentration over time governed by Sieverts' law with $K = 10/\sqrt{RT}$.
+
+!media comparison_ver-1kc-2.py
+       image_name=ver-1kc-2_equilibrium_constant_k10.png
+       style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
+       id=ver-1kc-2_equilibrium_constant_k10
+       caption=Equilibrium constant as a function of time.
 
 !media comparison_ver-1kc-2.py
        image_name=ver-1kc-2_concentration_ratio_T2_k10.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1kc-2_concentration_ratio_T2_k10
-       caption=Tritium concentrations ratio between enclosures 1 and 2 at the interface for $K = \frac{10}{\sqrt{RT}}$. This verifies TMAP8's ability to apply Sieverts' law across the interface.
+       caption=T$_2$ concentration ratio between enclosures 1 and 2 at the interface for $K = 10/\sqrt{RT}$. This verifies TMAP8's ability to apply Sieverts' law across the interface.
 
 !media comparison_ver-1kc-2.py
        image_name=ver-1kc-2_concentration_ratio_H2_k10.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1kc-2_concentration_ratio_H2_k10
-       caption=Dihydrogen concentrations ratio between enclosures 1 and 2 at the interface for $K = \frac{10}{\sqrt{RT}}$. This verifies TMAP8's ability to apply Sieverts' law across the interface.
+       caption=H$_2$ concentration ratio between enclosures 1 and 2 at the interface for $K = 10/\sqrt{RT}$. This verifies TMAP8's ability to apply Sieverts' law across the interface.
 
 !media comparison_ver-1kc-2.py
        image_name=ver-1kc-2_concentration_ratio_HT_k10.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1kc-2_concentration_ratio_HT_k10
-       caption=Tritium hybride concentrations ratio between enclosures 1 and 2 at the interface for $K = \frac{10}{\sqrt{RT}}$. This verifies TMAP8's ability to apply Sieverts' law across the interface.
+       caption=HT concentration ratio between enclosures 1 and 2 at the interface for $K = 10/\sqrt{RT}$. This verifies TMAP8's ability to apply Sieverts' law across the interface.
 
 !media comparison_ver-1kc-2.py
        image_name=ver-1kc-2_mass_conservation_k10.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1kc-2_mass_conservation_k10
-       caption=Total mass conservation across both enclosures over time for $K = \frac{10}{\sqrt{RT}}$.
+       caption=Total mass conservation across both enclosures over time for $K = 10/\sqrt{RT}$.
 
 ## Input files
 
 !style halign=left
-The input file for this case can be found at [/ver-1kc-2.i]. To limit the computational costs of the test cases, the tests run a version of the file with a coarser mesh and less number of time steps. More information about the changes can be found in the test specification file for this case [/ver-1kc-2/tests].
+The input file for this case can be found at [/ver-1kc-2.i].
 
 !bibtex bibliography

test/tests/ver-1kc-2/comparison_ver-1kc-2.py

@@ -157,10 +157,11 @@
 gs = gridspec.GridSpec(1,1)
 ax = fig.add_subplot(gs[0])
 
-ax.plot(TMAP8_time_k10, equilibrium_constant_encl_1, label = r"Equilibrium constant in Enclosure 1", c='tab:blue')
-ax.plot(TMAP8_time_k10, equilibrium_constant_encl_2, label = r"Equilibrium constant in Enclosure 2", c='tab:red', linestyle='--')
+ax.plot(TMAP8_time_k10, equilibrium_constant_encl_1, label = r"Enclosure 1", c='tab:blue')
+ax.plot(TMAP8_time_k10, equilibrium_constant_encl_2, label = r"Enclosure 2", c='tab:red')
+ax.axhline(y=2, color='tab:green', linestyle='--', label='TMAP7 Equilibrium Constant')
 ax.set_xlabel('Time (s)')
-ax.set_ylabel(r"Equilibrium constant in Enclosure 1")
+ax.set_ylabel(r"Equilibrium constant $P_{\text{HT}} / \sqrt{P_{\text{H}_2} P_{\text{T}_2}}$")
 ax.legend(loc="best")
 ax.grid(which='major', color='0.65', linestyle='--', alpha=0.3)
 fig.savefig('ver-1kc-2_equilibrium_constant_k10.png', bbox_inches='tight', dpi=300)