12.3.3 Crevice Model Theory

ANSYS FLUENT solves the equations for mass conservation in the crevice geometry by assuming laminar compressible flow in the region between the piston and the top and bottom faces of the ring, and by assuming an orifice flow between the ring and the cylinder wall. The equation for the mass flow through the ring end gaps is of the form

where $C_d$ is the discharge coefficient, $A_{ij}$ is the gap area, $\rho$ is the gas density, $c$ is the local speed of sound, and $\eta_{ij}$ is a compressibility factor given by

where $\gamma$ is the ratio of specific heats, $p_i$ the upstream pressure and $p_j$ the downstream pressure. The equation for the mass flow through the top and bottom faces of the ring (i.e., into and out of the volume behind the piston ring) is given by

where $h_{ij}$ is the cross-sectional area of the gap, $W_r$ is the width of the ring along which the gas is flowing, $\mu_{\rm gas}$ is the local gas viscosity, $T$ is the temperature of the gas and $R$ is the universal gas constant. The system of equations for a set of three rings is of the following form:

$$\frac{d p_1}{d t} = \frac{p_1}{m_1} \left(\dot{m}_{01} - \dot{m}_{12} \right)$$ (12.3-4)

$$\frac{d p_2}{d t} = \frac{p_2}{m_2} \left(\dot{m}_{02} + \dot{m}_{12} - \dot{m}_{23} - \dot{m}_{24} \right)$$ (12.3-5)

$$\frac{d p_3}{d t} = \frac{p_3}{m_3} \left(\dot{m}_{23} - \dot{m}_{34} \right)$$ (12.3-6)

$$\frac{d p_4}{d t} = \frac{p_4}{m_4} \left(\dot{m}_{24} + \dot{m}_{34} - \dot{m}_{45} - \dot{m}_{46} \right)$$ (12.3-7)

$$\frac{d p_5}{d t} = \frac{p_5}{m_5} \left(\dot{m}_{45} - \dot{m}_{56} \right)$$ (12.3-8)

where $p_0$ is the average pressure in the crevice cells and $p_6$ is the crankcase pressure input from the text interface. The expressions for the mass flows for numerically adjacent zones (e.g., 0-1, 1-2, 2-3, etc.) are given by Equation  12.3-3 and expressions for the mass flows for zones separated by two integers (e.g., 0-2, 2-4, 4-6) are given by Equations  12.3-1 and 12.3-2. Thus, there are $2n_r -1$ equations needed for the solution to the ring-pack equations, where $n_r$ is the number of rings in the simulation.