Journal of Fluid Flow, Heat and Mass Transfer (JFFHMT)

ISSN: 2368-6111

(4)

(5)

where G_k is the turbulent kinetic energy generation term due to the mean velocity gradients, is the scalar invariant of the strain rate tensor , ν is the kinematic viscosity, C_ε1 and C_ε2 are the model coefficients, and σk and σs are the turbulent Prandtl numbers. The turbulent viscosity μ_t is given as,

(6)

where C_μ depends on turbulent properties and mean flow [8].

Because of large variations of flow properties inside vortex tubes, the two-layer, all-Y+ (blended) option of the turbulence model is adopted. It relies on the wall functions at large Y+ values while resolving more detailed structure of the boundary layer at small Y+ values. Specifics of numerical methods in the program are provided in the STAR-CCM+ manual [9]. Properties of both hydrogen isomers are incorporated into simulations via tables obtained from REFPROP [10].

The vortex tube geometry in the numerical setup originates from the experimental study reported by Bunge [11]. A schematic of the experimental device and its CAD models used for CFD are shown in Fig. 2. The vortex tube has an inlet maintained at high pressure from which hydrogen enters tangentially into the tube via three nozzles of combined cross-sectional area of about 0.5 mm². There are two outlets at the opposite ends of the tube. The core outlet extracts fluid near the tube centreline close to the inlet, whereas the periphery outlet is located far from the inlet. While most vortex tubes have smooth surfaces on the inside, the current setup employed a rifled tube to maximize the surface area covered by catalyst while helping to maintain the vortex further along the tube away from the inlet. The length and averaged diameter of the rifled portion are 180 mm and 3.3 mm, respectively.

In the catalysed vortex tube experiments, the rifled portion of the internal surface was covered with hydrous ruthenium catalyst [11]. For simplified modelling of catalytic conversion between para- and orthohydrogen states, a thin zone (0.25-mm-thick) in the vicinity of the tube surface was chosen as the reaction zone (Fig. 3a). The rate of conversion was modelled inside this zone using a relaxation-type equation,

(7)

where ρ_ort and ρ_par are the densities of ortho- and parahydrogen, respectively, t is the time, y_{ort, actual} and y_{ort, equil} are the actual and equilibrium (at a local temperature) mass fractions of orthohydrogen. The reaction coefficient k was not known in advance. It was determined by matching the CFD prediction of the peripheral temperature to a value recorded in the experiment. According to Eq. (7), the orthohydrogen production rate is proportional to the local density of parahydrogen and depends on how far the mixture is from the equilibrium.

3. Verification and Validation

To establish a suitable numerical grid, a mesh verification study was accomplished using numerical results for the peripheral temperature obtained at three numerical grids of different density in the non-catalyzed tube (Fig. 3b). The experimental conditions included the inlet and core-outlet pressures 56.60 psia and 16.13 psia, respectively, the total mass flow rate of 0.159 g/s, and the inlet and peripheral-outlet temperatures of 53.71 K and 52.70 K, respectively. The constructed numerical grids were of polyhedral type with prism layers near solid surfaces. An illustration of a numerical mesh is shown in Fig. 3b. The computationally predicted reductions of the peripheral temperature with respect to the inlet state were found to be 1.19 K, 0.93 K, and 1.04 K for the coarse, medium and fine grids, respectively. These results demonstrate oscillatory numerical convergence. Using the Richardson extrapolation [7] and the factor of safety [12], the numerical uncertainty was estimated to be 0.10 K. Since this value is greater than the difference between test and numerical results for the peripheral temperature (0.03 K), the numerical approach has been validated.

4. Results and Discussion

To evaluate the effective conversion rate between ortho- and parahydrogen isomers, simulations were conducted for experimental conditions corresponding to a catalyzed tube [11]. In this case, the inlet and core-outlet pressures were 62.03 psia and 15.83 psia, respectively, and the inlet and peripheral-outlet temperatures were 52.81 K and 47.72 K, respectively. While the inlet boundary conditions and the pressure drop are similar to those in the non-catalyzed tube, one can notice much larger observed temperature drop due to cooling caused by the para-orthohydrogen conversion. To attain agreement with this exit temperature, the reaction coefficient k in Eq. (7) was varied, and the numerical value of 210 s^-1 was found to produce the best match with experimental data.

Illustrations of the pressure and velocity fields inside this tube are given in Fig. 4. The pressure is large at the inlet, whereas most of the pressure drop occurs across the nozzles that choke the flow (Fig. 4a). The pressure evolves little inside the tube due to minimal obstruction to the flow in this part of the device.

The velocity is very small at the inlet due to a large cross-sectional area, but it reaches very high values in the nozzles, where Mach number is about one (Fig. 4b). Due to tangential orientation of the nozzles with respect to the main body of the tube, a strong swirl is formed inside the larger portion of the tube. This vortex extends to significant distances away from the nozzles (Fig. 4b). Eventually, at several diameters of the tube away from the nozzles, this vortex dissipates, and the flow becomes primarily axial.

Temperature varies little upstream of the nozzles, as the flow is relatively slow (Fig. 5a). Temperature reaches minimum near the nozzle throats, where the flow reaches sonic speeds, and afterwards, temperature partially recovers due to flow deceleration in the tube section close to the nozzle. However, in the longer rifled portion of the tube, which is covered with a catalyst, temperature starts dropping again due to endothermic conversion of parahydrogen into orthohydrogen occurring near walls due to presence of catalysts.

The distribution of the orthohydrogen fraction in the fluid is illustrated in Fig. 5b. The ortho component in the flow entering the system is about 2.1% Once the fluid reaches the catalyzed portion of the main tube, the para-orthohydrogen conversion is initiated, and the ortho-fraction increases in the downstream direction, reaching about 11% at the peripheral exit.

5. Conclusion

High-speed compressible flow of cryogenic hydrogen was successfully modeled inside a vortex tube. Upon verification and validation study in the non-catalyzed tube, a reactive flow inside a catalyzed tube was simulated. By matching the numerical exit temperature to the experimental value, the effective rate coefficient of para-orthohydrogen conversion was determined to be 210 s^-1 for the specific relaxation-type reaction-rate form and the near-wall reaction zone of 0.25 mm. This conversion model can be suggested for approximate simulations of high-speed vortical flow of cryogenic hydrogen undergoing para-orthohydrogen transformation in the presence of ruthenium catalyst. Future extensions of this work can include optimization of practical systems for hydrogen cooling that employ vortex tubes, quantum physics-based modeling of conversion between hydrogen isomers, and validation in broader ranges of experimental conditions.

Acknowledgement

This work was supported by the US Department of Energy under contract No. DE-EE0008429.

References