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UBC Theses and Dissertations

Fibre motion in shear flow Wherrett, Geoffrey


The motion of a particle at low relative velocity to a shearing fluid is of importance in many areas of engineering. One such application is in the pulp and paper industry where pulp fibres may be separated by means of the fluid forces acting upon them. This process, known as fibre fractionation, divides a pulp fibre stock into fractions with higher percentages of fibres with certain properties. Experimental studies have shown that the motion of rigid fibres in shear may be described with an analytical model. The deformation that occurs in flexible fibres makes it difficult to model their motion in shear. In this thesis, a numerical model of a single fibre was formulated for both rigid and flexible motion. This research is part of a larger project aimed at modelling the hydrodynamic devices which could achieve efficient fractionation. The fibre was modelled as a cylindrical rod made up of elements of equal length and diameter. The inertial and structural properties of the model were based on the cylindrical geometry. The hydrodynamic force on each element was calculated for a sphere of the same diameter as the fibre. Equations of motion were formed for each element from the fluid and fibre forces as well as a continuity condition at the boundary of adjacent elements. These equations were simultaneously solved at each time step. The motion of the fibre was then followed by advancing the time step and recording the location of each element. Three series of trials were conducted with the model. The first determined if the model accurately simulated the motion of a rigid fibre. The second determined the effect of bending stiffness on the fibre's motion. The final series determined the effect of a localized weakness in the fibre. Trials were conducted with solid fibres of aspect ratio 10 and 60 in a Couette flow. The initial position of the fibre was aligned with the y axis either centred on or extending up from the origin. The first position resulted in purely rotational motion while the second position also included translation. Initial trials indicated that the model performed in accordance with previous results and within reasonable agreement with theory for both the rotational and translational motion of a rigid fibre. The second series of test was conducted with a 60 aspect ratio fibre of varying stiffness. This aspect ratio was chosen as it falls in the range of typical pulp fibres. The stiffness was measured by the ratio of elastic to shear forces, kei = E/ɳG. Here E is the elastic modulus of the fibre, ɳ is the absolute viscosity of the fluid and G is the rate of shear. The model performed rigidly when kei > 2 X 10⁶. Some deformation was seen in the fibre above 5 X 10⁵ but the period of rotation was nearly constant at a value close to that from the rigid case. As the stiffness was reduced below this value the fibre deformed significantly during rotation and its period of rotation decreased. The third series of trials was conducted with the same 60 aspect ratio fibre with kei — 5 x 10⁵. A localized weakness, at which the stiffness of the fibre was reduced by 80%, was placed in the fibre at 5 locations along its length. The results showed that the period of rotation did not change from the similar case in the previous series. However the fibre rotated in an asymmetric manner with one end of the fibre bending first. This differed from the previous series where both ends of the fibre behaved symmetrically. A fibre bending number B was derived to determine the type of motion in the fibre model depending on the fluid and fibre properties. This number is made up of three terms. The first is kei, the second is a function of the fibre's hollowness, and the third involves the aspect ratio. Comparisons with results showed that there existed a critical value for B, above which the fibre behaved rigidly. This is significant in that a simple rigid fibre model may be used when B is above the critical value. The trials conducted with the present model indicated that the model performed in a reasonable manner with respect to previous experimental and theoretical results. Several improvements could be made to the model. In particular, elements of larger aspect ratio would reduce the computational time.

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