## 2 OpenFOAM Tritium Transport Model Upgrades
### Porosity Model  
Many CFD codes, including OpenFOAM, use porosity when the presence of two phases is to be simplified to a description in the frame of single-phase equations. In our case, helium purge flow through a ceramic breeder pebble bed is regarded as porous flow. Typical literature descriptions of porous flows, e.g., the terms of Darcy (slow/creep flow) or Forchheimer (basically this is the Darcy term + an inertia term) are based on the \"apparent\" velocity, i.e., a \"Darcy velocity\" \(\mathrm{v_{d}}\) and a \"seepage velocity\" \(v=\mathrm{v_{d}}/\varepsilon\) where \(\varepsilon\) is the fluid fraction or porosity. Seepage velocity and also the density are averaged data over the complete cell volume. OpenFOAM includes a model like this (to be activated by the \"fvOptions\" mechanism), too. Nevertheless, it is not suitable for our application for the following reasons:  
1. As the seepage velocity \(v\) is not a physical speed, things happening at the begin of the porous zone will affect the end of the zone due to downstream advective transport only retarded by a factor of \(\varepsilon\). 2. The total inventory of the purge gas--including a possible tritium inventory-- will be overestimated by a factor of \(\varepsilon\) if the fluid density and the carried tritium concentrations are not corrected. 3. The tortuosity \(\delta=\frac{\mathrm{effective\ path}}{\mathrm{cell\ dimension}}\) quantifies the deviation from the straight path of a particle moving through, e.g., a packed bed. The true particle velocity is enhanced by a factor of the tortuosity \(\delta\); however, the direction of this particle velocity will be the flow direction only as a mean value (because the particles are forced to meander, e.g., around the pebbles). Another effect of this elongated meander path is that the fluid diffusivity--already reduced by the porosity factor of \(\varepsilon\)--will be additionally reduced by the inverse tortuosity factor of \(1/\delta\). A general solution of these porosity-related transport and inventory issues will require quite different transport equations and new variables. If, however, the porosity and tortuosity are assumed to be isotropic and constant inside a region, there is a quite simple and practicable solution. The variable of the fluid concentration of hydrogen in each fluid region cell contains the actual hydrogen concentration in the fluid part of the porous fluid region cell and is no average value over the cell that contains a fluid and solid fraction. This way the system variables always contain the correct absolute concentration or partial pressure numeric values as they are required for the evaluation of key data like diffusion coefficients or interface transport. Obviously, this is achieved at the cost that the total region hydrogen species inventory is overestimated by a factor of the porosity \(\varepsilon\). Furthermore, a tortuosity \(\delta\) is introduced, which is set by default to \(\delta=1/\varepsilon\) (which is justified in this case because (1) the path elongation along a half-sphere circumference compared to its diameter as the hypothetical direct flow path is approximately \(\Pi/2=1/0.64\) while (2) a typical packing factor of a bed of spherical pebbles is close to 0.64). These model presumptions require that:  
* all volumetric source terms for hydrogen have to be divided by a factor of \(\varepsilon\). * the interface mass transfer has to be corrected by a factor of \(1/\varepsilon\) on the fluid side. * advective transport enhances by a factor of \(1/\varepsilon\). * diffusive transport reduces by a factor of \(\varepsilon/\delta\). Taking all this into account, porosity and tortuosity change the original fluid regions transport equation to:  
\[\frac{\partial c}{\partial t}-\Delta\Big{(}\frac{\varepsilon}{\delta}Dc\Big{)}+ \
abla(vc/\varepsilon)-q/\varepsilon=0 \tag{3}\]  
In summary, this simplified model will describe transients with diffusive and advective transport correctly.