Add val 2c to TMAP8 for Test Cell Experiment

doc/content/bib/tmap8.bib

@@ -186,3 +186,13 @@ @book{Incropera2002
    title = {Fundamentals of Heat and Mass Transfer},
    year = {2002},
 }
+
+@inproceedings{Holland1986,
+   author = {D F Holland and R A Jalbert},
+   city = {Austin, TX},
+   booktitle = {Eleventh Symposium on Fusion Engineering},
+   month = {11},
+   publisher = {IEEE Cat. No. CH2251-7, Vol 1. pp. 638-643},
+   title = {A model for tritium concentration following tritium release into a test cell and subsequent operation of an atmospheric cleanup system},
+   year = {1986},
+}

doc/content/verification_and_validation/figures/comparison_val-2c.py

@@ -0,0 +1 @@
+../../../../test/tests/val-2c/comparison_val-2c.py

doc/content/verification_and_validation/index.md

@@ -45,3 +45,4 @@ TMAP8 also contains [example cases](examples/tmap_index.md), which showcase how
 | Case   | Title                                          |
 | ------ | ---------------------------------------------- |
 | val-2b | [Diffusion Experiment in Beryllium](val-2b.md) |
+| val-2c | [Test Cell Release Experiment](val-2c.md)      |

doc/content/verification_and_validation/val-2c.md

@@ -0,0 +1,148 @@
+# val-2c
+
+# Test Cell Release Experiment
+
+## Case Description
+
+This validation problem is taken from [!cite](Holland1986) and was part of the validation suite of TMAP4 [!cite](longhurst1992verification) and TMAP7 [!cite](ambrosek2008verification).
+Whenever tritium is released into a fusion reactor test cell, it is crucial to clean it up to prevent exposure.
+This case models an experiment conducted at Los Alamos National Laboratory at the tritium systems test assembly (TSTA) to study the behavior of tritium once released in a test cell and the efficacy of the emergency tritium cleanup system (ETCS).
+
+The experimental set up, described in greater details in [!cite](Holland1986), can be summarized as such:
+the surface of an enclosure of volume $V$ was covered with an aluminum foil and then covered in paint with an average thickness $l$.
+A given amount T$_2^0$ of tritium T$_2$ is injected in the enclosure, which initially contained ambient air.
+This represents the tritium release.
+A flow rate $f$ through the enclosure represents the air replacement time expected for large test cells.
+The purge gas is ambient air with 20% relative humidity.
+A fraction of that amount is diverted through the measurement system to determine the concentrations of the enclosure.
+
+Several phenomena are taking place and need to be captured in the model.
+First, The following chemical reactions occur inside the enclosure:
+\begin{equation} \label{eq:chemical_reaction_T2}
+\text{T}_2 + \text{H}_2\text{O} \longleftrightarrow{} \text{HTO} + \text{HT}
+\end{equation}
+\begin{equation} \label{eq:chemical_reaction_HT}
+\text{HT} + \text{H}_2\text{O} \longleftrightarrow{} \text{HTO} + \text{H}_2
+\end{equation}
+mostly as a consequence of the tritium reactivity. The reaction rates of these reactions are $K_1$ and $K_2$, respectively, where
+\begin{equation} \label{eq:chemical_reaction_K_1}
+K_1 = 2 K^0 c_{\text{T}_2} \left( 2 c_{\text{T}_2} + c_{\text{HT}} + c_{\text{HTO}} \right)
+\end{equation}
+and
+\begin{equation} \label{eq:chemical_reaction_K_2}
+K_2 = K^0 c_{\text{HT}} \left( 2 c_{\text{T}_2} + c_{\text{HT}} + c_{\text{HTO}} \right)
+\end{equation}
+where $c_i$ represents the concentration of species $i$, and $K^0$ is a constant.
+
+Second, the different species will permeate in the paint.
+The elemental tritium species, T$_2$ and HT, have a given solubility $K_S^e$ and diffusivity $D^e$, while the tritiated water HTO and water H$_2$O have a solubility $K_S^w$ and diffusivity $D^w$.
+It is expected that species will initially permeate into the paint and later get released as the purge gas cleans up the enclosure air.
+
+The objective of this case is to determine the time evolution of T$_2$ and HTO concentrations in the enclosure and match the experimental data published in [!cite](Holland1986) and reproduced in [val-2c_comparison_T2] and [val-2c_comparison_HTO].
+
+## Model Description
+
+To model the case described above, TMAP8 simulates a one-dimensional domain with one block to represent the air in the enclosure, and another block to represent the paint.
+In each block, it tracks the local concentration of T$_2$, HT, HTO, and H$_2$O.
+Note that this case can easily be extended to a two- or three-dimensional case.
+
+In the enclosure, to capture the purge gas and the chemical reactions, the concentrations evolve as
+\begin{equation} \label{eq:enclosure:T2}
+\frac{d c_{\text{T}_2}}{dt} = - K_1 - \frac{f}{V}c_{\text{T}_2},
+\end{equation}
+\begin{equation} \label{eq:enclosure:HT}
+\frac{d c_{\text{HT}}}{dt} = - K_2 - \frac{f}{V}c_{\text{HT}},
+\end{equation}
+\begin{equation} \label{eq:enclosure:HTO}
+\frac{d c_{\text{HTO}}}{dt} = K_1 + K_2 - \frac{f}{V}c_{\text{HTO}},
+\end{equation}
+and
+\begin{equation} \label{eq:enclosure:H2O}
+\frac{d c_{\text{H}_2\text{O}}}{dt} = - K_1 - K_1 + \frac{f}{V} \left(c_{\text{H}_2\text{O}}^0 - c_{\text{H}_2\text{O}} \right),
+\end{equation}
+where $c_{\text{H}_2\text{O}}^0$ is the concentration of H$_2$O in the incoming purge gas.
+
+In the paint, TMAP8 captures species diffusion through
+\begin{equation} \label{eq:paint:T2}
+\frac{d c_{\text{T}_2}}{dt} = - \nabla D^e \nabla c_{\text{T}_2},
+\end{equation}
+\begin{equation} \label{eq:paint:HT}
+\frac{d c_{\text{HT}}}{dt} = - \nabla D^e \nabla c_{\text{HT}},
+\end{equation}
+\begin{equation} \label{eq:paint:HTO}
+\frac{d c_{\text{HTO}}}{dt} = - \nabla D^w \nabla c_{\text{HTO}},
+\end{equation}
+and
+\begin{equation} \label{eq:paint:H2O}
+\frac{d c_{\text{H}_2\text{O}}}{dt} = - \nabla D^w \nabla  c_{\text{H}_2\text{O}}.
+\end{equation}
+
+At the interface between the enclosure air and the paint, sorption is captured in TMAP8 with Henry's law thanks to the  [InterfaceSorption.md] object:
+\begin{equation} \label{eq:sorption}
+c_{i,enclosure} = K_S c_{i,paint} R T,
+\end{equation}
+where $c_{i,enclosure}$ and $c_{i,paint}$ are the concentrations of species $i$ in the enclosure and in the paint, respectively, and $K_S$ is the solubility (either $K_S^e$ or $K_S^w$).
+The boundary conditions are set to no flux since no permeation happens at the interface between the paint and the aluminum foil and the only flux leaving the enclosure is already captured by the purge gas [!citep](Holland1986).
+
+One of the assumptions made in the original paper and TMAP4 V&V case is that the tritium is immediately added to the enclosure [!citep](Holland1986,longhurst1992verification).
+However, this leads to an early HTO peak concentration, which does not exactly match the experimental data.
+In [!cite](ambrosek2008verification), TMAP7 introduces a new enclosure to account for a slower injection of tritium.
+Here, we model this case with two different approaches.
+The first approach, like [!citep](Holland1986,longhurst1992verification), assumes that the entire tritium inventory is immediately injected in the enclosure at the beginning of the experience.
+The second approach assumes that the tritium inventory is being injected into the enclosure at a linear rate during a period of time $t_injection$ until the entire tritium inventory is injected.
+The results of these two approaches are presented and discussed below.
+
+
+## Case and Model Parameters
+
+The case and model parameters used in both approaches in TMAP8 are listed in [val-2c_parameters]. Some of the parameters are directly leveraged from [!cite](Holland1986,longhurst1992verification,ambrosek2008verification), but others were adapted to better match the experimental data.
+
+
+!table id=val-2c_parameters caption=Case and model parameters values used in both approaches in TMAP8 with $R$, the gas constant, and $N_A$, Avogadro's number, as defined in [PhysicalConstants](source/utils/PhysicalConstants.md). When values are the same for both approaches, they are noted as identical. Units are converted in the input file.
+| Parameter                  | Immediate injection approach                    | Delayed injection approach                      | Unit       | Reference                          |
+| -------------------------- | ----------------------------------------------- | ----------------------------------------------- | ---------- | ---------------------------------- |
+| $V$                        | 0.96                                            | Identical                                       | m$^3$      | [!cite](Holland1986)               |
+| $l$                        | 0.16 (between 0.1 and 0.2)                      | Identical                                       | mm         | [!cite](Holland1986)               |
+| T$_2^0$                    | 10                                              | Identical                                       | Ci/m$^{3}$ | [!cite](Holland1986)               |
+| $c_{\text{H}_2\text{O}}^0$ | 714                                             | Identical                                       | Pa         | [!cite](longhurst1992verification) |
+| $f$                        | 0.54                                            | Identical                                       | m$^3$/hr   | [!cite](Holland1986)               |
+| $T$                        | 303                                             | Identical                                       | K          | [!cite](Holland1986)               |
+| Total time                 | 180000                                          | Identical                                       | s          | [!cite](Holland1986)               |
+| $K^0$                      | $1.5 \times 2.0 \times 10^{-10}$                | $2.8 \times 2.0 \times 10^{-10}$                | m$^3$/Ci/s |                                    |
+| $D^e$                      | 4.0 $\times 10^{-12}$                           | Identical                                       | m$^2$/s    | [!cite](Holland1986)               |
+| $D^w$                      | 1.0 $\times 10^{-14}$                           | Identical                                       | m$^2$/s    | [!cite](Holland1986)               |
+| $K_S^e$                    | $5.0 \times 10^{-2} \times 4.0 \times 10^{-19}$ | $1.0 \times 10^{-3} \times 4.0 \times 10^{-19}$ | 1/m$^3$/Pa |                                    |
+| $K_S^w$                    | $3.5 \times 10^{-4} \times 6.0 \times 10^{-24}$ | $3.0 \times 10^{-4} \times 6.0 \times 10^{-24}$ | 1/m$^3$/Pa |                                    |
+| $t_{injection}$            | N/A                                             | 3                                               | hr         |                                    |
+
+## Results and discussion
+
+[val-2c_comparison_T2] and [val-2c_comparison_HTO] show the comparison of the TMAP8 calculations (both with immediately injected and delayed injected T$_2$) against the experimental data for T$_2$ and HTO concentration in the enclosure over time.
+There is reasonable agreement between the TMAP8 predictions and the experimental data.
+In the case of immediate T$_2$ injection,  the root mean square percentage errors are equal to RMSPE = 58.98 % for T$_2$ and RMSPE = 139.10 % for HTO, respectively.
+When accounting for a delay in T$_2$ injection, the TMAP8 predictions best match the experimental data, in particular the position of the peak HTO concentration. The RMSPE values decrease to RMSPE = 58.05 % for T$_2$ and RMSPE = 74.77 % for HTO, respectively.
+Note that the model parameters listed in [val-2c_parameters] are somewhat different from [!cite](Holland1986,longhurst1992verification,ambrosek2008verification) to better match the experimental data.
+In particular, [!cite](longhurst1992verification,ambrosek2008verification) did not validate the TMAP predictions against T$_2$ concentration, which we do here in [val-2c_comparison_T2].
+This affects some of the model parameters.
+
+The agreement between modeling predictions and experimental data could be further improved by adjusting the model parameters.
+It is also possible to perform this optimization with [MOOSE's stochastic tools module](https://mooseframework.inl.gov/modules/stochastic_tools/index.html).
+
+!media comparison_val-2c.py
+       image_name=val-2c_comparison_TMAP8_Exp_T2_Ci.png
+       style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
+       id=val-2c_comparison_T2
+       caption=Comparison of TMAP8 calculations against the experimental data for T$_2$ concentration in the enclosure over time. TMAP8 matches the experimental data well, with an improvement when T$_2$ is injected over  a given period rather than immediately.
+
+!media comparison_val-2c.py
+       image_name=val-2c_comparison_TMAP8_Exp_HTO_Ci.png
+       style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
+       id=val-2c_comparison_HTO
+       caption=Comparison of TMAP8 calculations against the experimental data for HTO concentration in the enclosure over time. TMAP8 matches the experimental data well, with an improvement when T$_2$ is injected over a given period rather than immediately.
+
+## Input files
+
+!style halign=left
+The input files for this case can be found at [/val-2c_immediate_injection.i] and [/val-2c_delay.i]. Note that both input files utilize the base file [/val-2c_base.i] with the line `!include val-2c_base.i`. The base input file [/val-2c_base.i] contains all the features common to both cases, and [/val-2c_immediate_injection.i] and [/val-2c_delay.i] focus on what is specific to each case. This approach reduces duplication. Note that [/val-2c_immediate_injection.i] and [/val-2c_delay.i] and also used as test in TMAP8 at [/val-2c/tests].
+
+!bibtex bibliography

test/tests/val-2b/comparison_val-2b.py

@@ -73,5 +73,5 @@ def numerical_solution_on_experiment_input(experiment_input, tmap_input, tmap_ou
 RMSPE = RMSE*100/np.mean(expt_flux)
 ax.text(870,3e16, 'RMSPE = %.2f '%RMSPE+'%',fontweight='bold')
 ax.minorticks_on()
-plt.savefig('val-2b_comparison.png', bbox_inches='tight')
+plt.savefig('val-2b_comparison.png', bbox_inches='tight', dpi=300)
 plt.close(fig)