Update and clean up V&V documentation

(Ref. #12)

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

@@ -52,7 +52,7 @@ J = 2 A K_r C^2.
 
 The beam flux on the upstream side of the sample during the experiment is presented in [val-2a_flux_and_pressure_TMAP4], and only 75 % of the flux remain in the sample. Other case and model parameters used in TMAP8 are listed in [val-2a_set_up_values_TMAP4].
 
-!table id=val-2a_flux_and_pressure_TMAP4 caption=Values of beam flux on the upstream side of the sample during the experiment [!citep](longhurst1992verification).
+!table id=val-2a_flux_and_pressure_TMAP4 caption=Values of beam flux on the upstream side of the sample during the experiment [!citep](anderl1985tritium,longhurst1992verification).
 | time (s)      | Beam flux $F$ (atom/m$^2$/s)   |
 | ---------     | ---------------------------- |
 | 0 - 5820      | 4.9$\times 10^{19}$          |
@@ -63,7 +63,7 @@ The beam flux on the upstream side of the sample during the experiment is presen
 | 17678 - 20000 | 0                            |
 
 !alert warning title=Typo in [!cite](longhurst1992verification)
-The times listed in [!cite](longhurst1992verification) for TMAP8 for the start and end times of the beam are not accurate. instead, TMAP8 uses the times directly from [!cite](anderl1985tritium) to better correspond to experimental conditions.
+The times listed in [!cite](longhurst1992verification) for TMAP8 for the start and end times of the beam are not accurate. Instead, TMAP8 uses the times directly from [!cite](anderl1985tritium) to better correspond to experimental conditions.
 
 !table id=val-2a_set_up_values_TMAP4 caption=Values of material properties. Note that $K_d$ are currently not used in the input file since the upstream and downstream pressure do not noticeably influence the results.
 | Parameter | Description                          | Value                                                       | Units                 | Reference                 |

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

@@ -33,11 +33,11 @@ This experiment is modeled using a two-segment model in TMAP8 with the segments
 
 The diffusivity of deuterium in beryllium was measured by [!cite](abramov1990deuterium). They made measurements on high-grade (99$\%$ pure) and extra-grade (99.8$\%$ pure). The values used here are those for high-grade beryllium, consistent with Dr. Macaulay-Newcombe's measurements of the purity of his samples.
 
-Deuterium transport properties of the BeO are more challenging. First, it is not clear in which state the deuterium exists in the BeO. However, it has been observed [!cite](longhurst1990tritium) that an activation energy of -78 kJ/mol (exothermic reaction) is evident for tritium coming out of neutron-irradiated beryllium in work done by D. L. Baldwin of Pacific Northwest Laboratory. The same value of energy has appeared in other results (can be inferred from Dr. Swansiger's work cited by [!cite](wilson1990beryllium) and by [!cite](causey1990tritium), among others), so one may be justified in using it. Concerning the solubility, measurements reported by [!cite](macaulay1992thermal) and in follow-up conversations indicate about 200 appm of D in BeO after exposure to 13.3 kPa of D$_2$ at 773 K. That suggests a coefficient of only 1.88 x 10$^{18}$ d/m$^3$Pa$^{1/2}$. Since much of the deuterium in the oxide layer will get out during the cool-down process (and because it gives a good fit), the solubility coefficient is taken to be 5 $\times$ 10$^{20}$ d/m$^3$Pa$^{1/2}$.
+Deuterium transport properties of the BeO are more challenging. First, it is not clear in which state the deuterium exists in the BeO. However, it has been observed [!cite](longhurst1990tritium) that an activation energy of -78 kJ/mol (exothermic reaction) is evident for tritium coming out of neutron-irradiated beryllium in work done by D. L. Baldwin of Pacific Northwest Laboratory. The same value of energy has appeared in other results (can be inferred from Dr. Swansiger's work cited by [!cite](wilson1990beryllium) and by [!cite](causey1990tritium), among others), so one may be justified in using it. Concerning the solubility, measurements reported by [!cite](macaulay1992thermal) and in follow-up conversations indicate about 200 appm of D in BeO after exposure to 13.3 kPa of D$_2$ at 773 K. That suggests a coefficient of only  1.88 $\times$ 10$^{18}$ d/m$^3$Pa$^{1/2}$. Since much of the deuterium in the oxide layer will get out during the cool-down process (and because it gives a good fit), the solubility coefficient is taken to be 5 $\times$ 10$^{20}$ d/m$^3$Pa$^{1/2}$.
 
 Deuterium diffusion measurements in BeO were made by [!cite](fowler1977tritium). They found a wide range of results for diffusivity in BeO depending on the physical form of the material, having measured it for single-crystal, sintered, and powdered BeO. The model in [!citep](longhurst1992verification,ambrosek2008verification) uses one expression for the charging phase and another for the thermal desorption phase, believing that the surface film changed somewhat during the transfer between the two furnaces. For the charging phase diffusivity, the model uses 20 times that for the sintered BeO. Thermal expansion mismatches tend to open up cracks and channels in the oxide layer, so this seems a reasonable value. The same activation energy of 48.5 kJ/mol, is retained, however. For the thermal desorption phase, the diffusivity prefactor of the sintered material (7x10$^{-5}$ m$^2$/sec) and an activation energy of 223.7 kJ/mol (53.45 kcal/mol) are used. These values give good results and lie well within the scatter of Fowler's data. Exposure of the sample to air after heating should have made the oxide more like a single crystal by healing the cracks that may have developed. Diffusivities and solubilities used in the simulation are listed in [val-2b_parameters].
 
-!table id=val-2b_parameters caption=Model parameter values for the charging and the desorption phases [!citep](longhurst1992verification,ambrosek2008verification). T is the temperature in Kelvin.
+!table id=val-2b_parameters caption=Model parameter values for the charging and the desorption phases [!citep](longhurst1992verification,ambrosek2008verification). $T$ is the temperature in Kelvin.
 | Property of deuterium | Value for charging phase              | Value for desorption phase            | Units               |
 | --------------------- | ------------------------------------- | ------------------------------------- | ------------------- |
 | Diffusivity in Be     | $8.0 \times 10^{-9} \exp(-4220/T)$    | $8.0 \times 10^{-9} \exp(-4220/T)$    | m$^2$/s             |
@@ -55,7 +55,7 @@ The model applies 13.3 kPa of D$_2$ for 50 hours and 15 seconds followed by cool
        image_name=val-2b_comparison.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=val-2b_comparison
-       caption=Comparison of TMAP8 calculation with the experimental data, which shows TMAP8's ability to accurately model this validation case.
+       caption=Comparison of TMAP8 calculation against experimental data, which shows TMAP8's ability to accurately model this validation case.
 
 !alert note title=Experimental data from [!cite](macaulay1991deuterium).
 The experimental data used in this case comes directly from Figure (2) in [!cite](macaulay1991deuterium). This is in contrast with the data used in [!cite](longhurst1992verification,ambrosek2008verification), which, although the scale of the data is very similar, differs slightly. Note also that the units in Figure (2) from [!cite](macaulay1991deuterium) should be atoms/mm$^2$/s $\times 10^{10}$ instead of atoms/mm$^2$ $\times 10^{10}$, which is corrected in [val-2b_comparison].

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

@@ -64,17 +64,17 @@ where $c_{\text{H}_2\text{O}}^0$ is the concentration of H$_2$O in the incoming
 
 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},
+\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}},
+\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}},
+\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}}.
+\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:
@@ -111,8 +111,8 @@ The case and model parameters used in both approaches in TMAP8 are listed in [va
 | $K^0$                      | $\boldsymbol{1.5} \times 2.0 \times 10^{-10}$                | $\boldsymbol{2.8} \times 2.0 \times 10^{-10}$                | m$^3$/Ci/s | Adapted from [!cite](longhurst1992verification) |
 | $D^e$                      | 4.0 $\times 10^{-12}$                                        | Identical                                                    | m$^2$/s    | [!cite](Holland1986)                            |
 | $D^w$                      | 1.0 $\times 10^{-14}$                                        | Identical                                                    | m$^2$/s    | [!cite](Holland1986)                            |
-| $K_S^e$                    | $\boldsymbol{5.0 \times 10^{-2}} \times 4.0 \times 10^{-19}$ | $\boldsymbol{1.0 \times 10^{-3}} \times 4.0 \times 10^{-19}$ | 1/m$^3$/Pa | Adapted from [!cite](longhurst1992verification) |
-| $K_S^w$                    | $\boldsymbol{3.5 \times 10^{-4}} \times 6.0 \times 10^{-24}$ | $\boldsymbol{3.0 \times 10^{-4}} \times 6.0 \times 10^{-24}$ | 1/m$^3$/Pa | Adapted from [!cite](longhurst1992verification) |
+| $K_S^e$                    | $\boldsymbol{5.0 \times 10^{-2}} \times 4.0 \times 10^{19}$ | $\boldsymbol{1.0 \times 10^{-3}} \times 4.0 \times 10^{19}$ | 1/m$^3$/Pa | Adapted from [!cite](longhurst1992verification) |
+| $K_S^w$                    | $\boldsymbol{3.5 \times 10^{-4}} \times 6.0 \times 10^{24}$ | $\boldsymbol{3.0 \times 10^{-4}} \times 6.0 \times 10^{24}$ | 1/m$^3$/Pa | Adapted from [!cite](longhurst1992verification) |
 | $t_{injection}$            | N/A                                                          | 3                                                            | hr         |                                                 |
 
 ## Results and discussion
@@ -132,7 +132,7 @@ It is also possible to perform this optimization with [MOOSE's stochastic tools
        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.
+       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

doc/content/verification_and_validation/ver-1a.md

@@ -68,7 +68,7 @@ The expression in Eq. (1) of [!cite](longhurst1992verification) writes $-\alpha_
        image_name=ver-1a_comparison_analytical_TMAP4_release_fraction.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1a_comparison_analytical_TMAP4_release_fraction
-       caption=Comparison of TMAP8 calculation with the analytical solution for the release fraction from [!cite](longhurst1992verification).
+       caption=Comparison of TMAP8 calculation with the analytical solution for the release fraction on the outer surface of the SiC slab [!cite](longhurst1992verification).
 
 ## Verification of the release fraction on the inner surface of the SiC slab (TMAP7)
 
@@ -80,7 +80,7 @@ In [!cite](ambrosek2008verification), i.e., TMAP7, the verification test focuses
 \end{equation}
 where $P(t)$ is the pressure at the surface of the enclosure over time.
 
-To derive $FR(t)$, [!cite](ambrosek2008verification) first reference the analytical solution for an analogous heat transfer problem [!cite](Carslaw1959conduction), which provides the solute concentration profile in the membrane as
+To derive $FR(t)$, [!cite](ambrosek2008verification) first references the analytical solution for an analogous heat transfer problem [!cite](Carslaw1959conduction), which provides the solute concentration profile in the membrane as
 
 \begin{equation}
     C(x,t) = 2 S P_0 L' \sum_{n=1}^{\infty} \frac{\exp \left(-\alpha_{n}^2 D t\right) \sin (\alpha_{n} x)}{[l(\alpha_{n}^2 + L'^2)+L'] \sin (\alpha_{n} l)},
@@ -116,7 +116,7 @@ which leads to
        image_name=ver-1a_comparison_analytical_TMAP7_release_fraction.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1a_comparison_analytical_TMAP7_release_fraction
-       caption=Comparison of TMAP8 calculation with the analytical solution for the release fraction from [!cite](ambrosek2008verification).
+       caption=Comparison of TMAP8 calculation with the analytical solution for the release fraction on the inner surface of the SiC slab [!cite](ambrosek2008verification).
 
 ## Verification of the tritium flux at the outer surface of the SiC slab (TMAP7)
 
@@ -140,7 +140,7 @@ Again, be aware of the typos in [!cite](ambrosek2008verification) and the differ
        image_name=ver-1a_comparison_analytical_TMAP7_flux.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1a_comparison_analytical_TMAP7_flux
-       caption=Comparison of TMAP8 calculation with the analytical solution for the release fraction from [!cite](ambrosek2008verification).
+       caption=Comparison of TMAP8 calculation with the analytical solution for the tritium flux at the outer surface of the SiC slab [!cite](ambrosek2008verification).
 
 ## Input files
 

doc/content/verification_and_validation/ver-1b.md

@@ -3,28 +3,28 @@
 # Diffusion Problem with Constant Source Boundary Condition
 
 This verification problem is taken from [!cite](longhurst1992verification). Diffusion of tritium through a semi-infinite SiC layer is modeled with a constant
-source located on one boundary. No solubility or traping is included. The
-concentration as a function of time and position is given by:
-
+source located on one boundary. No solubility or trapping is included. The
+concentration as a function of time and position is given by
 \begin{equation}
-C = C_o \; erfc \left(\frac{x}{2\sqrt{Dt}}\right)
+C = C_0 \; erfc \left(\frac{x}{2\sqrt{Dt}}\right),
 \end{equation}
+where $C_0$ the constant source concentration, erfc is the error function, $x$ is the distance from the boundary, $D$ is the diffusion coefficient, and $t$ is the time.
 
 Comparison of the TMAP8 results and the analytical solution is shown in
 [ver-1b_comparison_time] as a function of time at
-x = 0.2 mm. For simplicity, both the diffusion coefficient and the initial
-concentration were set to unity. The TMAP8 code predictions match very well with
-the analytical solution.
+$x = 0.2$ mm. For simplicity, both the diffusion coefficient and the initial
+concentration were set to unity. The TMAP8 code predictions match
+the analytical solution very well.
 
 !media comparison_ver-1b.py
        image_name=ver-1b_comparison_time.png
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1b_comparison_time
-       caption=Comparison of concentration as function of time at x\=0.2m calculated
-       through TMAP8 and analytically
+       caption=Comparison of concentration as function of time at $x=0.2$m calculated
+       through TMAP8 and analytically.
 
 As a second check, the concentration as a function of position at a given time
-t = 25s, from TMAP8 was compared with the analytical solution as shown in
+$t = 25$ s calculated by TMAP8 was compared with the analytical solution as shown in
 [ver-1b_comparison_dist]. The predicted concentration profile from TMAP8 is in
 good agreement with the analytical solution.
 
@@ -33,20 +33,19 @@ good agreement with the analytical solution.
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1b_comparison_dist
        caption=Comparison of concentration as function of distance from the source
-       at t\=25sec calculated through TMAP8 and analytically
+       at $t=25$s calculated through TMAP8 and analytically.
 
 Finally, the diffusive flux ($J$) was compared with the analytic solution where the
-flux is proportional to the derivative of the concentration with respect to x and
-is given by:
-
+flux is proportional to the derivative of the concentration with respect to $x$ and
+is given by
 \begin{equation}
 \label{eq:flux}
-J = C_o \; \sqrt{\frac{D}{t\pi}} \; exp \left(\frac{x}{2\sqrt{Dt}}\right)
+J = C_0 \; \sqrt{\frac{D}{t\pi}} \; exp \left(\frac{x}{2\sqrt{Dt}}\right).
 \end{equation}
 
 The flux as given by [eq:flux] is compared with values calculated by TMAP8.
-The diffusivity, D, and the initial concentration, C$_o$, were both
-taken as unity, and the distance, x, was taken as 0.5 in this comparison.
+The diffusivity, D, and the initial concentration, $C_0$, were both
+taken as unity, and the distance, $x$, was taken as 0.5 in this comparison.
 TMAP8 initially under predicts but the results match well subsequently. Comparison
 results are shown in [ver-1b_comparison_flux] with a root mean square percentage
 error of RMSPE = 6.03 %. The error is calculated for $t \geq 10$ s due to infinite
@@ -57,11 +56,11 @@ value at small $t$.
        style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto
        id=ver-1b_comparison_flux
        caption=Comparison of flux as function of time at x\=0.5m calculated through
-       TMAP8 and analytically
+       TMAP8 and analytically.
 
 
 The oscillations in the permeation graph go away with increasing fineness in the
-mesh and in `dt`.
+mesh and in the time step `dt`.
 
 ## Input files