Numerical Examination of Non-Equilibrium Condensation in Supersonic Micronozzles

Abstract

The Micropropulsion Group of the department of Space Systems Engineering of the Delft University of Technology is developing resistojet thrusters for small satellites operating on the vapourisation of liquid water via an electric heater and its subsequent expansion via a nozzle into vacuum for the generation of thrust. As the water vapour expands in the nozzle, its pressure and temperature drop, leading it to cross the saturation limit of water and raising questions on whether the vapour will transition back to liquid in the nozzle’s divergent section. The spontaneous condensation of flowing steam in supersonic nozzles is a well-known phenomenon that results in the release of heat, in turn downgrading the nozzle’s performance and efficiency.

This has motivated the present numerical study to assess the likelihood of the phenomenon’s occurence inside the resistojet micronozzles. To that end, a condensation model is implented in the open-source platform OpenFOAM, by heavily modifying an existing compressible solver to account for the mild degrees of rarefaction in the nozzle, the real-gas thermodynamics of water and the phase-change phenomenon itself. The condensation is modelled as a two-step process, whereby liquid clusters first nucleate out of the vapour and then proceed to grow by gathering further vapor molecules. A baseline micronozzle case is defined as one with a 100 μm-depth nozzle with an expansion angle of 30 deg and stagnation pressure and temperature of 3 bar and 473 K. The simulations are run for three stagnation pressures (1, 3 and 5 bar), three stagnation temperatures (473, 573 and 673 K), three expander angles (15, 30, 45 degrees) and two nozzle depths (100 and 200 μm). The outputs are in each case compared to the baseline case, to extract conclusions on the effect of nozzle geometric and flow topological features on the occurence of condensation.

The results show that the onset of condensation is in most instances a likely scenario, but not necessarily a consequential one. The extremely fast cooling rates (10^8-10^9 K s−1) drive the vapour to unusual degrees of supersaturation (10^3 or more) before condensation occurs. When the birth of droplets eventually
takes place, it does so in impulsive fashion, with large numbers of miniscule liquid clusters (radii in the order of 10^(−10) m) appearing simultaneously in mass fractions in the vicinity of 2%. Even so, the macroscopic result is typically small, with the thrust and specific impulse changing only by 1-2% at most. This holds true for the variation of all above design parameters except for the nozzle depth: it is shown that doubling the depth decreases the wall’s heat influence and leaves room for substantial condensation to occur, that degrades the thrust output by more than 5%.

The general conclusion to be drawn is that assessing the impact of condensation on this type of nozzles is not as straightforward as in their conventional-scale counterparts and it is difficult to note consistent trends. If it does occur, the condensation process will enter a complex interaction mechanism with the heat supplied from the walls, the viscous layers developing on these walls and the degree to which the expansion can overcome either or both. While in conventional scale nozzles the occurence of the phenomenon typically guarantees a reduction in thrust and efficiency, here there is no general tendency and the nature of the influence depends on the extent to which the release of latent heat and its by-products can match in severity the rest of the phenomena inside the micronozzles. Overall, the analysis indicated that the geometric configuration of a 100 μm-deep thruster with a 30-deg expander angle strikes a good balance in generally avoiding the occurence of substantial phase change (that would cause a ≥ 2% change in macroscopic performance metrics) and also containing other contributing effects, such as the growth of viscous layers and expansion losses. The choice of stagnation conditions is mission-dependent and no recommendation can be given, but some suggestions are provided on approximate methods to select them such that condensation is avoided.