Mathematical models of the carding process
Carding is an essential pre-spinning process whereby masses of dirty tufted fibres are cleaned, disentangled and refined into a smooth coherent web. Research and development in this `low-technology' industry have hitherto depended on empirical evidence. In collaboration with the School of Textile Industries at the University of Leeds, a mathematical theory has been developed that describes the passage of fibres through the carding machine. The fibre dynamics in the carding machine are posed, modelled and simulated by three distinct physical problems: the journey of a single fibre, the extraction of fibres from a tuft or tufts and many interconnecting, entangled fibres. A description of the life of a single fibre is given as it is transported through the carding machine. Many fibres are sparsely distributed across machine surfaces, therefore interactions with other neighbouring fibres, either hydrodynamically or by frictional contact points, can be neglected. The aerodynamic forces overwhelm the fibre's ability to retain its crimp or natural curvature, and so the fibre is treated as an inextensible string. Two machine topologies are studied in detail, thin annular regions with hooked surfaces and the nip region between two rotating drums. The theoretical simulations suggest that fibres do not transfer between carding surfaces in annular machine geometries. In contrast to current carding theories, which are speculative, a novel explanation is developed for fibre transfer between the rotating drums. The mathematical simulations describe two distinct mechanisms: strong transferral forces between the taker-in and cylinder and a weaker mechanism between cylinder and doffer. Most fibres enter the carding machine connected to and entangled with other fibres. Fibres are teased from their neighbours and in the case where their neighbours form a tuft, which is a cohesive and resistive fibre structure, a model has been developed to understand how a tuft is opened and broken down during the carding process. Hook-fibre-tuft competitions are modelled in detail: a single fibre extracted from a tuft by a hook and diverging hook-entrained tufts with many interconnecting fibres. Consequently, for each scenario once fibres have been completely or partially extracted, estimates can be made as to the degree to which a tuft has been opened-up. Finally, a continuum approach is used to simulate many interconnected, entangled fibre-tuft populations, focusing in particular on their deformations. A novel approach describes this medium by density, velocity, directionality, alignment and entanglement. The materials responds to stress as an isotropic or transversely isotropic medium dependent on the degree of alignment. Additionally, the material's response to stress is a function of the degree of entanglement which we describe by using braid theory. Analytical solutions are found for elongational and shearing flows, and these compare very well with experiments for certain parameter regimes.