Chemical Engineering Journal 111 (2005) 199–204 Physico-mathematical modeling of crossflow filtration A.S. Ferreira a , G. Massarani b,∗ a Centro de Pesquisas e Desenvolvimento/Petrobras, Cidade Universit´ aria Q. 7, 21949-900 Rio de Janeiro, RJ, Brazil b PEQ/COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 689502, 21945-970 Rio de Janeiro, RJ, Brazil Abstract A phenomenological model that allows the correlation of pressure fields, filtration rate and cake thickness in the process that results from the axial flow of a suspension in a duct that is provided with a porous wall is presented. The model is based in the motion and the continuity equations for the phases and incorporates the following constitutive information: • Rheological properties of the suspension and of the fluid that makes up this suspension. • Rheological properties of the cake including the effects of the compressibility and the relation between the shear stress of mobilization and the normal stress associated to the Coulomb criterion in the stabilization of the thickness of the cake. • Properties of the filter medium. The experiments performed with an aqueous suspension of calcium carbonate (average particle diameter in the order of 0.5 m) confirms the common knowledge that the properties of the cake depend on the mode of filtration, making it of capital importance to perform specific assays for each case that is studied. © 2005 Published by Elsevier B.V. Keywords: Physico-mathematical modeling; Crossflow filtration; Non-Newtonian 1. Introduction Crossflow filtration results from the axial flow of a suspen- sion through a duct which walls are made of filter material, leading to the formation of a deposit on the filter surface and the production of a filtrate that percolates through it. Char- acteristically, the flow of filtrates drops with time, and may stabilize, in a longer operation, as a consequence of the action of the mechanisms that limit the growth of the cake. Crossflow filtration is used in the industry for the clarifi- cation of effluents and in the concentration of suspensions in a wide range of applications that employ the technology of membranes [1]. The operation of drilling and preparation of inclined oil wells involves essentially the same phenomena that prevail in the crossflow filtration with micro-membranes. It is the axial flow of a non-Newtonian suspension in the annular space between a cylinder in rotation and the formation of petroleum. ∗ Corresponding author. Tel.: +55 21 2562 8345; fax: +55 21 2562 8300. E-mail address: gmassa@peq.coppe.ufrj.br (G. Massarani). In this situation, the properties and the thickness of the deposit that is formed allow for the control of the damaging invasion of the drilling fluid in the petroleum formation [2]. This work is restricted to the study of the crossflow fil- tration that results from the axial flow of a suspension in a hose that was built with the filter medium used in industrial filtration. The modeling is made based on the continuity and motion equations for the phases, and has as its objective to establish the relation between the pressure fields, the flow of filtrate, and the thickness of the cake along the process [3]. The model considers the period of cake growing, with the attendant reduction in the rate of filtration, and the stage in which the thickness of the cake and the flow of filtrate may stabilize in a longer operation, in which the filter works as a thickener. The constitutive information aggregated to the proposed model are to be established, whenever it is possible, in assays that do not use the results from crossflow filtration itself: rheometry for the survey of the properties of the suspension and of the fluid that makes up this suspension, dead-end fil- tration for the characterization of the cake and of the filter 1385-8947/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.cej.2005.02.002