Finite-Length Solutions for Rotordynamic Coefficients of Turbulent Annular Seals

Expressions are derived which define dynamic coefficients for high-pressure annular seals typical of wear-ring and interstage seals employed in multistage centrifugal pumps. Completely developed turbulent flow is assumed in both the circumferential and axial directions, and is modeled by Hirs’ turbulent lubrication equations. Linear zeroth and first-order perturbation equations are developed by an expansion in the eccentricity ratio. The influence of inlet swirl is accounted for in the development of the circumferential flow. The zeroth-order momentum and continuity equations are solved exactly, while their first-order counterparts are reduced to three ordinary, complex, differential equations in the axial coordinate Z. The equations are integrated to satisfy the boundary conditions and define the pressure distribution due to seal motion. Integration of the pressure distribution defines the reaction force developed by the seal and the corresponding dynamic coefficients. Finite-length solutions for the coefficients are compared to two “short-seal” solutions.