Extended PCT capabilities to low pressure for YHx

doc/content/source/interfacekernels/ADMatInterfaceReactionYHxPCT.md

@@ -4,31 +4,59 @@
 
 ## Description
 
-The composition of metal hydrides like yttrium hydride (YHx) is described by pressure-composition-temperature (PCT) data.
+The composition of metal hydrides like yttrium hydride (YH$_{x}$) is described by pressure-composition-temperature (PCT) data.
 In TMAP8, the PCT curve can be imposed by an interface kernel that then dictates the material composition.
-Hence, the `ADMatInterfaceReactionYHxPCT` interface kernel imposes the surface concentration of H in YHx based on the input pressure (Pa) and temperature (K). `ADMatInterfaceReactionYHxPCT` is related to [ADMatInterfaceReaction.md], but includes the YHx PCT curves.
+Hence, the `ADMatInterfaceReactionYHxPCT` interface kernel imposes the surface concentration of H in YH$_{x}$ based on the input pressure (Pa) and temperature (K). `ADMatInterfaceReactionYHxPCT` is related to [ADMatInterfaceReaction.md], but includes the YH$_{x}$ PCT curves.
  At the interface between a solid (main) and a gas (neighbor), it imposes:
 \begin{equation} \label{eq:test_interfacereactionYHxPCT}
-\frac{d C_s}{dt} = 0 = K_b f_{at}(T,P) \rho -  K_f C_s ,
+\frac{d C_s}{dt} = 0 = K_b f_{at}(T,P) \rho -  K_f C_s,
 \end{equation}
 where $C_s$ is the surface H concentration in mol/m$^3$,
 $K_b$ and $K_f$ are the backward and forward surface reaction rate in 1/s, respectively,
-$f_{at}(T,P)$ is the composition in atomic fraction of H in YHx given a gas temperature $T$ and gas pressure $P$,
-and $\rho$ is the yttrium atomic density in mol/m$^3$.
+$f_{at}(T,P)$ is the composition in atomic fraction of H in YH$_{x}$ given a gas temperature $T$ and gas pressure $P$,
+and $\rho$ is the yttrium molar density in mol/m$^3$.
+
+[YHx_PCT_fit_2D] shows the data used in this interface kernel along with TMAP8 simulation results and analytical fits. The experimental data originates from [!cite](Lundin_1962).
 
-[YHx_PCT_fit_2D] shows the data used in this interface kernel. The experimental data originates from [!cite](Lundin_1962), and the fit is from [!cite](Matthews2021SWIFT).
 
 !media comparison_YHx_PCT.py
        image_name=YHx_PCT_fit_2D.png
        style=width:80%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=YHx_PCT_fit_2D
-       caption=PCT data for YHx from [!cite](Lundin_1962) with fits from [!cite](Matthews2021SWIFT) implemented in TMAP8, and test results.
+       caption=PCT data for YH$_{x}$ sampled from [!cite](Lundin_1962) with TMAP8 simulation results and analytical fits.
+
+
+To incorporate the entire PCT curve in TMAP8, the curve was divided into three regions: the low-pressure (LP) region, the plateau region (PR), and the high-pressure (HP) region. The fitting procedures and the modeling approach for each region are described below.
+
+\begin{equation}\label{eq:atomic_fraction_HP}
+f_{at,LP}(T,P) = f_{at,max,LP}(T)-10\left[1 \times 10^{-3}+\exp(-50.0 + 5.73 \times 10^{-2} T + ( 8.30 \times 10^{-1} - 2.69 \times 10^{-3} T) (\log\left(P_{lim}(T) - P \right)))\right]^{-1},
+\end{equation}
+
+\begin{equation}\label{eq:atomic_fraction_PR}
+f_{at,PR}(T,P) = 1.33 - 2.18 \times 10^{-4}, T
+\left( 10.6 - 4.35 \times 10^{-3}, T \right)
+\log\left( \frac{P}{\gamma P_{\text{lim}}} \right),
+\end{equation}
+and
+\begin{equation}\label{eq:atomic_fraction_LP}
+f_{at,HP}(T,P) = 2.00-1.0015\left[f_{at,min,HP}(T)+\exp(24.89 - 2.53 \times 10^{-2} T + ( -3.98 \times 10^{-1} + 1.00 \times 10^{-3} T) (\log\left(P - P_{lim}(T)\right)))\right]^{-1}
+\end{equation}
+
+where $\gamma$ is a tolerance factor set to 1.15. While ${f_{max,LP}(T)}$ and ${f_{min,HP}(T)}$ represents maximum and miniumum atomic ratio per temperature in the low pressure and high high region, respectively. These formulas are expressed as:
 
-The plateau representing phase transition is captured as [!citep](Matthews2021SWIFT):
+\begin{equation}\label{eq:atomic_fraction_LP_Max}
+f_{at,max,LP}(T)=1.01\times10^{-6}T^{2}-2.56\times10^{-3}T+2.16
+\end{equation}
+and
+\begin{equation}\label{eq:atomic_fraction_HP_Min}
+f_{at,min,HP}(T)=-1.01\times10^{-6}T^{2}+2.55\times10^{-3}T-5.61\times10^{-1}.
+\end{equation}
+
+The plateau ($P_{lim}$) representing phase transition is captured as [!citep](Matthews2021SWIFT):
 \begin{equation} \label{eq:pressure_plateau}
-P_{lim} = \exp\left(-26.1+3.88 \times 10^{-2} T - 9.7 \times 10^{-6} T^2 \right),
+P_{lim} = \exp\left(-26.1+3.88 \times 10^{-2} T - 9.7 \times 10^{-6} T^2 \right).
 \end{equation}
-with $P_{lim}$ being the hydrogen partial pressure limit delineating the plateau in Pa and $T$ being the temperature in K.
+
 This fit is shown in [YHx_PCT_plateau_pressure_fit].
 
 !media comparison_YHx_PCT.py
@@ -37,16 +65,37 @@ This fit is shown in [YHx_PCT_plateau_pressure_fit].
        id=YHx_PCT_plateau_pressure_fit
        caption=Fit phase transition region as pressure as a function of temperature.
 
-The model currently only captures the high pressure region of the data, i.e., for pressure and temperatures above the plateau.
-In this region, the atomic fraction $f_{at}$ is determined as [!citep](Matthews2021SWIFT):
-\begin{equation} \label{eq:atomic_fraction}
-f_{at}(T,P) = 2-\left[1+\exp(21.6 -0.0225 T + (-0.0445 + 7.18 \times 10^{-4} T) (\log\left(P - P_{lim}(T)\right)))\right]^{-1},
+These fits are applied within the following conditional statement for entire PCT modelling capabilities
+
+\begin{equation}
+\text{If} \left (\frac{P}{P_{\text{lim}}}\right) > 1.15:
+\quad f_{at,HP}(T,P)
+\end{equation}
+
+\begin{equation}
+\text{Else-if} \left (\frac{P}{P_{\text{lim}}}\right) <1.05:
+\quad f_{at,LP}(T,P)
+\end{equation}
+
+\begin{equation}
+\text{Else: }
+\quad f_{at,PR}(T,P).
 \end{equation}
-where $P$ is the hydrogen partial pressure in Pa. This fit is plotted in [YHx_PCT_fit_2D].
 
-The [!param](/InterfaceKernels/ADMatInterfaceReactionYHxPCT/silence_warnings) option can be used to dictate how TMAP8 reacts when the pressure gets out of bounds.
-If `silence_warnings = false`, which is the default behavior, then TMAP8 will print a warning stating that the pressure and temperature are outside the bounds of the atomic fraction correlation.
-If `silence_warnings = true`, then TMAP8 will let the simulation continue without issuing any warnings.
+The validity of this present fit is between:
+\begin{equation} \label{eq:bounds}
+100 < P\,\text{[Pa]} < 10^{5}
+\end{equation}
+
+
+The ${f_{at,max,LP}(T)}$ and ${f_{at,min,HP}(T)}$ are quadratic fits that were verified by plotting the fit against the PCT data shown in the figure below. Evidently,the ${f_{at,min,HP}(T)}$ suffer slight losses due to non-symmetry of plateau region. Nonetheless, the fits are suitable for modelling purposes.
+
+!media comparison_YHx_PCT.py
+       image_name=YHx_PCT_Plateau_EndPoints_comparison.png
+       style=width:80%;margin-bottom:2%;margin-left:auto;margin-right:auto
+       id=YHx_PCT_Plateau_EndPoints_comparison
+       caption=Boundary of atomic ratio fit vs. PCT data from [!cite](Lundin_1962).
+
 
 ## Test
 
@@ -57,9 +106,36 @@ To model the interface, the input file employs the [InterfaceDiffusion.md] objec
 \begin{equation} \label{eq:test_interfacereaction}
 \frac{d C_s}{dt} = 0 = K (f_{at}(T,P) \rho - C_s),
 \end{equation}
-where $\rho$ is the yttrium atomic density.
 
-The results of the test for ($T$, $P$) = (1173.15 K, $1 \times 10^{3}$ Pa), (1173.15 K, $1 \times 10^{4}$ Pa), (1173.15 K, $5 \times 10^{4}$ Pa), and(1473.15 K, $5 \times 10^{4}$ Pa) are shown in [YHx_PCT_fit_2D] and show good agreement with [eq:atomic_fraction].
+
+The results of the high pressure test for ($T$, $P$) = (1173.15 K, $1 \times 10^{3}$ Pa), (1173.15 K, $1 \times 10^{4}$ Pa), (1173.15 K, $5 \times 10^{4}$ Pa), and(1273.15 K, $3 \times 10^{3}$ Pa),
+and the results for the the low pressure test for ($T$, $P$) =(1273.15 K, $3 \times 10^{2}$ Pa), (1473.15 K, $3 \times 10^{3}$ Pa), (1573.15 K, $6 \times 10^{2}$ Pa) and (1573.15 K, $6 \times 10^{2}$ Pa)
+
+
+The [YHx_PCT_fit_2D] figure compares the analytical fits, the independent low‑pressure model, and the independent high‑pressure model against the experimental data. Overall, the analytical fits show reasonable agreement with the measurements, yielding average root‑mean‑square errors of 0.19 for the low‑pressure region and 0.04 for the high‑pressure region. The low‑pressure fit, however, exhibits noticeable deviations near the tail ends of the dataset. This behavior is likely influenced by clusters of data points at higher temperatures, which can lead to poorer fitting performance. In contrast, the independent low‑ and high‑pressure TMAP8 simulations align well with the analytical trends, demonstrating consistency between the analytical approach and the model-based predictions.
+
+
+### Overall PCT testing
+
+[!file](/YHx_PCT.i) also tests the entire PCT modelling curves in TMAP8. The simulation follows the same structure as before, but has an arbitrarily high diffusion value to acheive steady-state quickly. The model also includes a linear pressure increase to cover the entire PCT curve. The [YHx_PCT_fit_2D] shows the PCT fit against the experimental data. Evidently, the fit exhibits moderate error, but the deviations remain within an acceptable range for modeling purposes
+
+\begin{equation}
+P = P_{\text{initial}} + t \frac{P_{\text{max}} - P_{\text{initial}}}{t_{\text{end}}}.
+\end{equation}
+
+The testing conditions include
+($T$, $P_{initial}$) = (1173.15 K, $2 \times 10^{2}$ Pa),
+(1223.15 K, $2 \times 10^{2}$ Pa),
+(1273.15 K, $2 \times 10^{2}$ Pa),
+(1323.15 K, $2 \times 10^{2}$ Pa),
+(1373.15 K, $2 \times 10^{2}$ Pa),
+(1423.15 K, $2 \times 10^{2}$ Pa),
+(1473.15 K, $2 \times 10^{2}$ Pa),
+(1523.15 K, $2 \times 10^{2}$ Pa),
+and (1573.15 K, $2 \times 10^{2}$ Pa).
+
+The [YHx_PCT_fit_2D] shows the PCT fit against the experimental data. Evidently, the fit exhibits moderate error with an averaged value of 15%, but the deviations remain within an acceptable range for modeling purposes.
+
 
 ## Example Input File Syntax
 

src/interfacekernels/ADMatInterfaceReactionYHxPCT.C

@@ -47,36 +47,68 @@ ADMatInterfaceReactionYHxPCT::ADMatInterfaceReactionYHxPCT(const InputParameters
 ADReal
 ADMatInterfaceReactionYHxPCT::computeQpResidual(Moose::DGResidualType type)
 {
+
+  // Variables
+  ADReal r = 0;
+
+  // tolerance for the pressure being closed to the plateau region
+  const Real tolerance = 1.15; // %
   using std::exp;
   using std::log;
   using std::max;
   using std::pow;
-  ADReal r = 0;
 
-  // Calculate the equilibrium concentration value based on PCT curve
-  // (/2 because two atoms for a molecule) (pressure in Pa)
+  // Gas pressure (Pa): R * T * c / 2 (two atoms per molecule)
   auto neighbor_pressure =
       PhysicalConstants::ideal_gas_constant * _neighbor_temperature[_qp] * _neighbor_value[_qp] / 2;
 
   // Calculate the value of the pressures for the phase transition plateau (pressure in Pa)
-  auto limit_pressure = exp(-26.1 + 3.88e-2 * _neighbor_temperature[_qp] -
-                            9.7e-6 * Utility::pow<2>(_neighbor_temperature[_qp]));
+  auto Plim = exp(-26.1 + 3.88e-2 * _neighbor_temperature[_qp] -
+                  9.70e-6 * Utility::pow<2>(_neighbor_temperature[_qp]));
+
+  // define atomic fraction variable
+  ADReal atomic_fraction = 0.0;
+
+  // define atomic ratio maximum fit variable for the low and high pressure region, respectively
+  ADReal At_Max_LP_fit = 1.01e-6 * Utility::pow<2>(_neighbor_temperature[_qp]) -
+                         2.56e-3 * _neighbor_temperature[_qp] + 2.16;
+
+  ADReal At_Min_HP_fit = -1.01e-6 * Utility::pow<2>(_neighbor_temperature[_qp]) +
+                         2.55e-3 * _neighbor_temperature[_qp] - 5.6e-01;
 
   // return warning if the PCT curves is used out of bounds (pressure in Pa)
-  if (!_silence_warnings && ((neighbor_pressure < limit_pressure) || (neighbor_pressure > 1.e6)))
+  if (!_silence_warnings && ((neighbor_pressure < 1.e2) || (neighbor_pressure > 1.e6)))
     mooseDoOnce(mooseWarning("In YHxPCT: pressure ",
                              neighbor_pressure,
                              "Pa and temperature ",
                              _neighbor_temperature[_qp],
                              "K are outside the bounds of the atomic fraction correlation. See "
                              "documentation for YHxPCT material."));
-
   // Calculate the atomic fraction based on the PCT curve
-  auto atomic_fraction =
-      2. - pow(1. + exp(21.6 - 0.0225 * _neighbor_temperature[_qp] +
-                        (-0.0445 + 7.18e-4 * _neighbor_temperature[_qp]) *
-                            (log(max(neighbor_pressure - limit_pressure, 1e-10)))),
-               -1);
+  if (neighbor_pressure / Plim > 1.15)
+  {
+    atomic_fraction =
+        2. - 1.0015 * pow(At_Min_HP_fit + exp(24.89 - 2.53e-02 * _neighbor_temperature[_qp] +
+                                              (-3.98e-01 + 0.001 * _neighbor_temperature[_qp]) *
+                                                  log(max(neighbor_pressure - Plim, 1.e-10))),
+                          -1);
+  }
+  else if (neighbor_pressure / Plim < 1.05)
+  {
+    // Low pressure region
+    atomic_fraction =
+        At_Max_LP_fit - 10 * pow(1.e-03 + exp(-50.0 + 5.73e-2 * _neighbor_temperature[_qp] +
+                                              (0.830 - 2.69e-3 * _neighbor_temperature[_qp]) *
+                                                  log(max(Plim - neighbor_pressure, 1.e-10))),
+                                 -1);
+  }
+  else
+  {
+    // Plateau Region
+    atomic_fraction = 1.33 - 2.18e-04 * _neighbor_temperature[_qp] +
+                      (1.06e01 - 4.35e-03 * _neighbor_temperature[_qp]) *
+                          log(neighbor_pressure / (tolerance * Plim));
+  }
 
   // Convert to concentration
   auto _surface_equilibrium_concentration = atomic_fraction * _density[_qp];

test/tests/yttrium_hydrogen_system/PCT_data/1000.csv

@@ -14,22 +14,22 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 0.5003812713236798, 9.661704954868043
 0.5248298036256153, 11.417576632627881
 0.5468295520651542, 13.19528722851941
-0.5663805166422968, 14.913806704747937
-0.5980103139789005, 14.90591251598713
-0.6345062339826738, 14.896809028798605
-0.6783091992516077, 15.052940279008315
-0.7196633806581451, 14.875589174160138
-0.7513010392591545, 15.034559330727195
-0.7877969592629279, 15.025377275076085
-0.8194267565995315, 15.01742402968676
-0.8461825700045595, 14.84411833474094
-0.8778123673411633, 14.836261033487002
-0.9143161486093423, 14.993589543517993
-0.943512884612361, 14.986263468670607
-0.9775757432825496, 14.977720905469436
-1.011638601952738, 14.969183211753325
-1.0432683992893417, 14.961259711053058
-1.0481345219565115, 14.960041083256012
+0.5663805166422968, 14.836261033487002
+0.5980103139789005, 14.84411833474094
+0.6345062339826738, 14.875589174160138
+0.6783091992516077, 14.896809028798605
+0.7196633806581451, 14.90591251598713
+0.7513010392591545, 14.913806704747937
+0.7877969592629279, 14.960041083256012
+0.8194267565995315, 14.961259711053058
+0.8461825700045595, 14.969183211753325
+0.8778123673411633, 14.977720905469436
+0.9143161486093423, 14.986263468670607
+0.943512884612361, 14.993589543517993
+0.9775757432825496, 15.01742402968676
+1.011638601952738, 15.025377275076085
+1.0432683992893417, 15.034559330727195
+1.0481345219565115, 15.052940279008315
 1.0798114868795496, 15.987540632145134
 1.1212285584013335, 17.27453057131089
 1.1455906167948053, 18.05573840599329
@@ -62,7 +62,4 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 1.9119538386554096, 524.2537323663885
 1.9291896608650534, 700.5286849730287
 1.9365517349810544, 765.8515855528643
-0.5712466393094666, 14.912591942103536
-0.5761127619766364, 14.911377278404247
-1.0384022766221719, 14.962478438118127
-1.0651895350848233, 15.464944999627683
+

test/tests/yttrium_hydrogen_system/PCT_data/1050.csv

@@ -26,20 +26,20 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 0.5399077087558763, 22.547710391605136
 0.5594586733330189, 25.48426484330247
 0.5716790088517838, 27.54919779324486
-0.5790017766457557, 28.483627797847237
-0.5960332059808501, 28.475508433196833
-0.6325291259846234, 28.45811758738781
-0.6763242299891515, 28.437262591711786
-0.7176862726600947, 28.417580240273985
-0.761473515400217, 28.08162556388147
-0.7931111740012262, 28.38172395731534
+0.5790017766457557, 28.021074736663728
+0.5960332059808501, 28.08162556388147
+0.6325291259846234, 28.276729535554693
+0.6763242299891515, 28.282488362300167
+0.7176862726600947, 28.298619317459913
+0.761473515400217, 28.317065960779317
+0.7931111740012262, 28.351685833131302
 0.82474097133783, 28.366700919219717
-0.8563707686744337, 28.351685833131302
-0.8904257660802164, 28.021074736663728
-0.9293626086819804, 28.317065960779317
-0.9682915900193385, 28.298619317459913
-1.0023544486895273, 28.282488362300167
-1.0145197553574516, 28.276729535554693
+0.8563707686744337, 28.38172395731534
+0.8904257660802164, 28.417580240273985
+0.9293626086819804, 28.437262591711786
+0.9682915900193385, 28.45811758738781
+1.0023544486895273, 28.475508433196833
+1.0145197553574516, 28.483627797847237
 1.0388975362797352, 30.222550814781602
 1.068133578604783, 31.9411986552508
 1.102243604861406, 34.13362375347649
@@ -72,7 +72,3 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 1.8658671760766001, 641.3779884427049
 1.878111095388583, 716.9523738731084
 1.8854653082401773, 775.1085545978324
-0.5741277927141802, 28.169828844760538
-0.5838678993129256, 28.4813077431683
-1.0096536326902819, 28.279032925527343
-1.0267086458185937, 29.233455068878477

test/tests/yttrium_hydrogen_system/PCT_data/1100.csv

@@ -30,16 +30,16 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 0.5427259720453438, 38.954870591321125
 0.5573950914265051, 43.05998403229226
 0.5769460560036477, 48.668003010889905
-0.5916151753848089, 53.796698608449915
-0.635410279389337, 53.75727471073713
-0.684071506061035, 53.71350426360323
-0.7278666100655631, 53.67414133329971
-0.7619294687357515, 53.643545665102174
-0.8154568180746191, 53.59550199070021
-0.8714172287470716, 53.54532051887158
-0.91034621008443, 53.510439382866785
-0.9444090687546185, 53.479937029074186
-0.976038866091222, 53.45162898363805
+0.5916151753848089, 53.45162898363805
+0.635410279389337, 53.479937029074186
+0.684071506061035, 53.510439382866785
+0.7278666100655631, 53.54532051887158
+0.7619294687357515, 53.59550199070021
+0.8154568180746191, 53.643545665102174
+0.8714172287470716, 53.67414133329971
+0.91034621008443, 53.71350426360323
+0.9444090687546185, 53.75727471073713
+0.976038866091222, 53.796698608449915
 1.002841847082685, 56.49353181939432
 1.0345188120057232, 60.373673467170235
 1.0661957769287613, 64.52031463660713
@@ -71,8 +71,3 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 1.7757810166187131, 587.4849499543402
 1.7977729037938461, 671.4214149099407
 1.8173395908998007, 775.9929751479735
-0.5867254689244219, 52.029732360862724
-0.5964812980519788, 53.792316747993
-0.6013474207191485, 53.787935244448335
-0.9711727434240522, 53.45598309206024
-0.9882277565523638, 55.260131561070004

test/tests/yttrium_hydrogen_system/PCT_data/1150.csv

@@ -30,17 +30,17 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 0.5455442353348113, 67.30093284112601
 0.5699770451079351, 77.77648448455122
 0.5870870870870871, 86.93390519466453
-0.6041892678018332, 96.0911962129842
-0.6504174331399463, 96.01686695668666
-0.6917794758108895, 95.9504105699897
-0.7307084571482477, 95.88790540075408
-0.7672043771520212, 95.82934378519286
-0.8037081584202002, 96.8455491378044
-0.837763155825983, 95.71622605709229
-0.8791251984969264, 95.64997775370932
-0.9229203025014545, 95.57988246837148
+0.6041892678018332, 95.57988246837148
+0.6504174331399463, 95.64997775370932
+0.6917794758108895, 95.71622605709229
+0.7307084571482477, 95.82934378519286
+0.7672043771520212, 95.88790540075408
+0.8037081584202002, 95.9504105699897
+0.837763155825983, 96.01686695668666
+0.8791251984969264, 96.0911962129842
+0.9229203025014545, 96.60524527949393
 0.9423926544345393, 96.62098462559449
-0.9521248997688787, 96.60524527949393
+0.9521248997688787, 96.8455491378044
 0.9813688033583321, 103.24458202382107
 1.0106127069477855, 110.34021689437361
 1.0398644718016445, 119.2468362939784
@@ -62,9 +62,3 @@ Atom Ratio (-),Partial Pressure (mm Hg)
 1.5738054808735438, 563.7401394031765
 1.6006477681870352, 630.0123853764603
 1.6323640394321024, 711.9185511096462
-1.6470095750200464, 761.0321426711316
-0.5968665000078613, 92.9388415495683
-0.611488451802588, 96.0794561903707
-0.6236537584705124, 96.05989267336636
-0.9472587771017089, 96.61311463202999
-0.9667468515636057, 99.86975603349624