Analytical solution of flow field and pressure drop for filters with rectangular fibers

In a previous paper, the general approach of forcing the truncated general solution of the biharmonic equation Δ2 (Δ2ψ) = 0 to satisfy appropriate boundary conditions on a fiber surface and at the cell boundary has been developed. In this paper, the method is applied to the specific case of rectangular fibers. Both the flow field and the dimensionless fiber drag have been calculated for various parameters, such as the fiber aspect ratio and packing density. Since the resulting expression is quite complicated, only the formula for the zeroth-order approximation is given for selected fiber aspect ratios. The range of applicable packing densities in the zeroth-order solution is given for several allowed error values at the fiber surface. The zeroth-order solutions are in good agreement with Fardi and Liuʹs numerical results based on an iterative, finite-difference scheme. In cases where the zeroth-order approximation does not provide sufficiently accurate results, higher-order approximation can be used. Since analytical solutions with variable coefficients are sometimes too complicated or time consuming to obtain, analytical solutions with constant coefficients have been obtained with considerable saving in computer time. This allows the dimensionless fiber drag for specific filter packing density and fiber aspect ratio to be obtained. In addition, using the slip boundary condition at the fiber surface, the method has been used to obtain the dimensionless drag of rectangular fibers in the slip flow regime. The advantages and limitations of the method have also been discussed.