Advection-modulated gaseous diffusion through an orifice

Abstract

We examine flow and transport through an orifice in a flat wall separating semi-infinite atmospheres of two dissimilar gases. The analysis assumes steady conditions and order-unity values of the Schmidt number Sc and Péclet number Pe, such that advection and diffusion contribute comparably to mass and momentum transport. Mixing between the two gases induces order-unity variations in viscosity and density, resulting in coupled concentration and velocity fields. The solution yields the mass transfer rates of both gases, expressed in terms of an appropriately defined Sherwood number, as well as the overpressure required to sustain the flow, all as functions of Sc and Pe. An explicit analytical solution is obtained in the limit of small Pe, while numerical integration is used to describe flows with Pe = O(1). The mixing of hydrogen and air is used as an illustrative example that serves to highlight the influence of large gas-molecular-weight differences on the flow structure and associated mixing rate, with additional selected results given for the case of hydrogen and water vapor.

Figures (8)

• Figure 1: A schematic representation of the orifice-flow configuration studied, including an indication of the cylindrical coordinates used in the description. The left half of the plot show streamlines $\psi=$constant (gray solid curves, $\Delta \psi = 0.025$ ) and isocontours of mass fraction $Y$ (black solid curves, $\Delta Y=0.1$) calculated numerically for $Pe=\alpha=\beta=0$. Dots are used to represent the analytical solutions given in Eqs. \ref{['psi_sampson']} and \ref{['Y_sampson']}.
• Figure 2: Streamlines $\psi=$ constant (gray curves, streamfunction increment $\Delta \psi = 0.025$) and isocontours of mass fraction $Y=$ constant (black curves, ${\Delta Y = 0.1}$) calculated for Air-H$_2$ with $Pe = 0$ (left) and $Pe=12$ (right). The dots on the left half represent the theoretically predicted mass-fraction isocontours given in Eq. \ref{['paraboloidal surfaces']}.
• Figure 3: Streamlines $\psi=$ constant (gray curves, streamfunction increment $\Delta \psi = 0.025$) and isocontours of mass fraction $Y=$ constant (black curves, ${\Delta Y = 0.1}$) calculated for H$_2$-Air with $Pe = 0$ (left) and $Pe=12$ (right). The dots on the left half represent the theoretically predicted mass-fraction isocontours given in Eq. \ref{['paraboloidal surfaces']}.
• Figure 4: Numerically computed distributions of axial velocity $v_z$ (left) and mass fraction $Y$ (right) along the symmetry axis $r=0$ for $Pe=0$ and different pairs of gases (indicated in the figure). The dots in the right-hand-side plot corresponds to the analytical solution given in Eq. \ref{['Yz_sampson']}, while those in the left-hand-side plot represent the constant-density solution obtained by Sampson sampson1891, given in Eq. \ref{['vz_sampson']}. The black circles in the left figure indicate the location at which the maximum velocity is achieved.
• Figure 5: Numerically computed distributions of axial velocity $v_z$ (left) and mass fraction $Y$ (right) along the symmetry axis $r=0$ for $Pe=12$ and different pairs of gases (indicated in the figure). The black circles in the left figure indicate the location at which the maximum velocity is achieved.