Validation of porosity effects

The velocity profile for a porous zone is shown in figure 5.11. There is a marginal increase in maximum velocity.

Temperature contour is shown in figure 5.12. It is observed that there is a change in the contour developed due to the presence of porous media. As the temperature

Figure 5.9: Temperature profile of a non porous zone 2

320 325 330 335 340 345 350 355 360

0 0.001 0.002 0.003 0.004 0.005

Tfluid

0 0.05 0.1 0.15 0.2 0.25

0 0.001 0.002 0.003 0.004 0.005

U

Figure 5.10: Temperature and velocity variation (non porous zone 2) at x= 0.1m

gradient reduces with positive x-direction, the lesser it diffuses thus generating the contour shown.

Variation of velocity and temperature along y-axis at x= 0.1m is shown in 5.10. The reduced values of temperature can be observed. The temperature profile follows the velocity profile. The average velocity developed in case of a porous zone 2 is 0.16718 ms⁻1. Temperature is sampled at the same point considered for the previous case.

The value of temperature here is 323.223K. This is because of the block profile case of a porous zone with a porosity value of 0.9. The value of temperature gradient normal to the wall is 1737.4003Km⁻¹. The mean temperature is 326.95619 K. Nusselt number for this case is 9.13. The literature value of Nusselt number for a block flow profile is 9.80 [31].

Figure 5.11: Velocity profile for a non porous zone 2

Figure 5.12: Temperature profile for a porous zone 2

323 323.5 324 324.5 325 325.5 326 326.5 327 327.5 328

0 0.001 0.002 0.003 0.004 0.005

Tfluid

0 0.05 0.1 0.15 0.2 0.25

0 0.001 0.002 0.003 0.004 0.005

U

Figure 5.13: Temperature and velocity variation (porous zone 2) at x= 0.1m

5.3 Closure

The Loschmidt tube considered as the validation case for Maxwell-Stefan source terms was set up and explained. Abstraction of experimental set up for CFD

simu-lations were also shown. The behaviour patterns of a ternary mixture was explained.

Since it is a closed geometry, the pressure value used here was 101.325KPa. The large value of pressure significantly reduced the time step to obtain a stable simula-tion. The time step observed for the simulation was 10⁻¹⁰ seconds. The analytical solution developed as a Scilab program for a transient 1 dimensional case was used to verify the results. The phenomenon observed for comparison was the variation of phase fraction of all three gases in the left tube of the capillary. By observing the variation of phase fraction, it is indeed equivalent to the observation of variation of flux of each phase. The simulation results are in very good agreement with the analytical results over the complete observation period. Additionally, the various kinds of diffusion behavior were also captured in the simulation.

The second case of fully developed flow over parallel plates was used for the validation of temperature transport and porous drag effects. The case setup involves a sudden cooling of a mixture entering into a region cooled by the walls. Nusselt number was used as the validation criteria and compared with values available in the literature.

In order to validate both the stated effects, the cooling zone was considered non-porous for the first case and as non-porous zone with a porosity of 0.9 for the second.

For non-porous region, the Nusselt number calculated was 8.85 compared to the literature value of 7.54. For the porous case, the Nusselt number calculated was 9.13 compared to the literature value of 9.8. Since the solver deals with incompressible phases, instabilities were observed when total pressure was used due to the presence of large numbers which may increase cumulative errors. Hence only the relative pressure or pressure fluctuation values were used for this case. The terms related by ideal gas law were kept intact by adding a value 10⁵ Pa to the pressure terms. This may be the cause for the observed deviation. The temperature profile developed for the two cases are realistic and rational.

6. Conclusions

Scope of this work and its implications have been extensively discussed in the pre-vious chapters. A few important conclusion and further potential improvements are stated here:

• A circumspective attitude is generally taken towards the Eulerian approach to solve for transport of multicomponent gases. The reason being the complexity and computation costs expended for the approach. For flow scenarios which require slip velocities of individual components, the mixture approach which is the most chosen alternative to Eulerian approach cannot be used since it circumvents solving individual component velocities. Hence the Eulerian ap-proach is chosen and models the mulitcomponent flow scenario accurately.

• Maxwell stefan relations models diffusion dominated problems involving mass transfer. In multicomponent flows, inter-component and not intra-component momentum tranfer takes place which is established in section 3.1.2. The phe-nomena of osmotic diffusion, reverse diffusion and diffusion barrier is accurately modelled by Maxwell-Stefan relations which the Fick’s law fails to predict.

• The intercomponent momentum transfer implemented to the modified solver multicomponentporousFoam simulates ternary diffusion in a Loschmidt tube accurately. The results are in perfect agreement with the analytical solution for the case solved as transient one dimensional flow. The graph of variation of phase fractions can be seen in figures 5.4 5.5 and 5.3.

• Phase intensive momentum equations are not used in the solver. Hence flows involving sudden changes of phase fraction of a component must be avoided.

The presence of the gradient term of phase fraction is responsible for this short coming which cause floating point exceptions when it encounters large gradients. The use of upwind interpolation scheme to solve for phase fractions restricts the use of large time steps. It is not a major problem for transient flows due to the presence of a small time step. When a steady state solution is sought, the demand for a small time step may be a drawback.

• Volume averaged equations are implemented in the solver. The presence of porososity, porousPlug and tortuosity fileds provides a generic operating form for the solver. The porous drag terms and the effect of porosity can be switched on in a particular zone by setting the mentioned fields appropriately. Currently, isotropic and non deformable porous media have been modelled. Extensions to orthotropic media can be done by adding appropriate tensorial terms in the solver. The equations are inversed for this purpose. The results generated by the solver is in close agreement.

• Transport of temperature of the mixture assuming local thermodynamic equi-librium between components accurately predicts the transfer of temperature in the pseudo homogeneous media. The difference between the contour of tem-perature in a non porous domain and porous domains can be seen in figures 5.9 and 5.12.

• The deviation in Nusselt number is attributed to the use of only the relative pressure values for simulation. But it does not largely affect the simulation since all the terms which are related by ideal gas properties are properly ac-counted by adding a value of 10⁵Pa to the relative pressure solved for incom-pressible flows. The calculation of gradients at the wall in order calculate Nusselt number is also speculated to be the cause for the error. The probe location for calculation of gradient at a wall can vary continuously in the cell adjacent to the wall. The temperature also varies accordingly resulting in many possibilities to calculate temperature gradient at the wall.And even a small variation in its value affects the final solution quite largely.

• Temperature dependance of viscosity and diffusion coefficients have been mod-elled in the solver. Density is considered as a scalar in the solver. This restricts its dependance on temperature. It is also predicted to be a source of error in the final solution. A large dependance of variables exist on density in the solver. Hence to convert it into a scalarField may prove detrimental since the aspect of considering individual components as incompressible may be violated and cause mass conservation problems.

• Source terms which consists of pressure terms are modelled explicitly. Pro-jecting pressure dependant source terms out of the momentum equation and solving for them in the poisson equation which is set for pressure improves the accuracy and may even reduce the small time step requirement. But the presence of velocity terms simultaneously poses a problem in doing so since pressure-velocity is solved using a segregated approach.Thus, using a coupled solution procedure would be suitable for handling the source terms which con-tain pressure and velocity terms. The article [16] which implements a coupled pressure based solution using a block matrix structure may be taken as a cue for establishing a coupled solution procedure. Further, Maxwell Stefan terms are added explicitly in the solver. A semi-implicit treatment of the terms may improve accuracy in flows involving large pressure gradients.

• Reaction terms can be easily added to the phase continuity equation in the solver. The MULES algorithm which solves the equation contains explicit and implicit source terms as its arguments. Thus reaction can be modelled by passing the source term field to the function.