Precoat filtration with body-feed and variable pressure. Part I: Mathematical modelling

The precoat filtration with body-feed is an unit operation of agricultural and food engineering. Mostly it is implemented by using centrifugal pump, which pump curve has a partial horizontal trend. Classically, in filtration theory, this prerogative of the centrifugal pumps leads to the simplifying assumption that filtration occurs with constant pressure. Because of this, it is easy to integrate the Darcy’s differential equation [1, 2 and 3] for the precoat filtration with body-feed, obtaining the well known Carman equation [4]. This is the equation which relates the filtration time with the filtrate volume, the operating pressure, the filter area, and the solid-liquid suspension characteristics. The Carman equation is the start point for the subsequent optimization of the filtration cycles, e.g. by establishing the relationship between the filtration time and the filter cleaning time [5]. A better optimization of the precoat filtration with body-feed could be obtain, with some economic benefits, if an integration of the Darcy ODE was developed starting from actual trend of the pressure produced by the centrifugal pump, that is if a variable pressure was considered, as expected from the pump curve. In this sense a proposal was done by Tiller and Crump [6] many years ago in accordance with a graphic method of integration of the Darcy ODE. However the graphic procedure is tedious since it is iterative and not computerizable. For this reason the aim of this work was to find an analytical solution to the Darcy ODE for the filtration with variable pressure in order to obtain a quick and easy-to-use equation for the subsequent optimization calculations of filtration cycles, even if more complex of the Carman equation.