Direct-flow filtration is a common technique for filtering impurities from a fluid using a porous-walled channel or a pipe, one end of which is closed off with a cap. Pure fluid flows out of the porous walls, while impurities are left in the channel. Such systems are composed of a series of individual porous channels or pipes stacked in close proximity. We develop a mathematical model for the flow in a 2D filtration channel and a 3D pipe, with a capped end, to describe the behaviour within a direct-flow device. We study the axial dependence of the transmembrane pressure (TMP) across the membrane walls on the imposed flux, wall permeability and the proximity of the neighbouring fibres. The mathematical models derived are used to predict the operating regimes of the device that maximize the spatial uniformity in the TMP and thus optimize the use of the entire membrane area. We show how a large portion of the available membrane area is not used when the fibres are packed too closely together, with the majority of the filtration behaviour being localized near to the impermeable capped end; this leads to inefficient filtration. We quantify the device performance by examining the uniformity of the filtration across the length of the device and the output of the filtered fluid for a given operating pressure.