Modeling a turbulent fibre suspension flowing in a planar contraction: The one-dimensional headbox

An Eulerian model of a turbulent fibre suspension flowing through a planar contraction is proposed to predict the fibre orientation distribution. The model specifically applies to the flow of pulp suspensions through a papermachine headbox. The model accounts for the convection of orientation distribution of the mean fluid velocity and the dispersion caused by the turbulent velocity fluctuations and describes each by a dimensionless dispersion coefficient and a dimensionless velocity gradient characterized by the contraction ratio (ratio of duct inlet height to outlet height) of the duct. A numerical solution of the model equations was shown to compare well to two different experimental studies available in the literature. From comparison with experiment the rotational dispersion coefficient was estimated to be 2 s−1 for both studies. Using the experimental value of the rotational dispersion coefficient and assuming it remains constant, the fibre orientation distributions was calculated for a range of dimensionless dispersion coefficients and contraction ratios. It was demonstrated that increasing the contraction ratio, over the industrial range of 5–50, significantly increases the alignment of fibres exiting the headbox. It was also shown that varying the inlet fluid velocity, over a practical range, provides only small changes in the fibre orientation distribution. Both predicted effects agree with experimental observation. Furthermore, this study suggests that fibre orientation is approximately dependent on a single dimensionless Peclet number that is a function of the fluid contraction ratio, inlet velocity, contraction length and rotational dispersion coefficient and that this parameter can be used to guide the design and operation of commercial headboxes.