Wetted-Area Minimum and Inlet-Outlet Reciprocity in Optimal Manifolds of Rarefied Gas Flows

TL;DR

This paper tackles the design of 2D intake manifolds for rarefied gas flows by applying adjoint-based topology optimization to a Boltzmann-BGK framework. A fictitious porosity model enables continuous optimization over geometry and topology while accurately capturing gas–solid interactions, boundary conditions, and flow regimes across $Kn$ numbers. It reveals a wetted-area minimum in the slip regime and an inlet–outlet reciprocity in the free-molecular regime, with detailed insights into curvature, compressibility, and the role of flow uniformity constraints. The findings provide practical design guidelines for manifolds operating in vacuum environments and illuminate the mechanisms governing rarefied gas transport.

Abstract

While flow optimization has been extensively studied in the continuum regime, its extension to rarefied gas flows remains less explored. Here, based on the Boltzmann model equation, an adjoint topology optimization method is employed to design two-dimensional single inlet multi outlet manifolds, aiming to maximize the total mass flow rate while maintaining outflow uniformity. Two key findings are revealed. (1) analogous to the Knudsen minimum in mass flow rate in the transition regime, a wetted-area minimum is identified, but in the slip flow regime. This phenomenon arises from the competition between flow bend loss and surface friction loss, with the latter being affected by velocity slip at the solid surface. (2) the inlet outlet reciprocity emerges in the free molecular flow regime, where the optimal design becomes invariant to inlet outlet orientation and pressure ratio. Additional insights are gained regarding the channel curvature, compressibility effects, and the constraint of outflow uniformity. These findings elucidate the mechanisms governing rarefied gas transport and offer design guidance for manifolds operating in vacuum environments.

Figures (6)

• Figure 1: Schematic of the topology optimization of a 2D manifold based on the material density $\theta$ (the gas and solid regions are represented by $\theta=1$ and 0, respectively). The cyan regions are fixed as gas. $\Omega$ is the $5\times5$ square design domain. The green edge $\Gamma _{\rm d1}$ is the inlet, the blue edges $\Gamma _{\rm d2}$ are three outlets, and the red edges $\Gamma _{\rm w}$ are solid walls. The coordinates are normalized by the inlet/outlet channel height $H$.
• Figure 2: Optimization of a 2D intake manifold for MFR under flow uniformity constraint. From left to right are the results for ${\rm Kn}=0.001,0.1,$ and 10, respectively.
• Figure 3: The general geometric configuration of the optimal manifold. $B_1$ and $B_2$ are the two bifurcation points, $O_1 \sim O_3$ are the central points of outlet boundaries. $\angle {O_1}{B_1}{B_2}$ and $\angle {O_2}{B_2}{O_3}$ are used to approximately characterize the magnitudes of the bifurcation angles.
• Figure 4: Optimization of a 2D intake manifold for MFR under flow uniformity constraint: the variations of the wetted area, the degree of $\angle {O_1}{B_1}{B_2}$, the slip velocity, and the mean MFR with the Knudsen number. The mean MFR is normalized by $\rho_{\rm in} a_\infty H$. The wetted area is obtained by measuring the perimeter of the flow passage (in the design domain the solid wall can be represented by the contour line $\theta=0.5$). The slip velocity is measured at point $S$ on the $x_1=4$ cross-section (shown in figure \ref{['fig:case1a']}) and is normalized by the maximum velocity along that cross-section.
• Figure 5: Optimization of the 2D manifold under reversed inlet--outlet conditions. From left to right are the results for ${\rm Kn}=0.001,0.1,$ and 10, respectively.