Flow in a Calender

Abstract

Calendering is a continuous operation of driving a pair rolls to compress softened materials into a sheet of uniform thickness. Gaskell presented a theory of the steady isothermal flow of an incompressible Newtonian liquid in a symmetrical calender. In his treatment, it is assumed that the boundaries of the liquid are approximately in parallel. It is the purpose of this paper to take into consideration not only the velocity component v_x but also the velocity component v_y and we have determined v_x and v_y and the pressure p as a function of the position variables x and y.
With regard to the motion of the liquid, the following assumptions are made: (1) that the liquid is Newtonian; (2) that the liquid is incompressible; (3) the motion of the liquid is laminar; (4) that the motion is steady; (5) that no force acts on the liquid; (6) that the motion is symmetrical about the x axis; (7) that the separation at the nip 2H₀ is small in comparison with the roll radius R; (8) that the radii of the rolls are equal and the rolls rotate at the same speed; (9) and that there is no relative motion between the rolls' wall surfaces and the liquid in immediate contact with the wall. We shall consider only the immediate region of the nip.
Let us take a rectangular coordinate system whose origin is at the middle point between the centers of the rolls.
We get
φ=Σ²_i=1B_i(α_isinhk_iY/k_i²+β_iY)cosk_iX,
where B_i, α_i, β_i and k_i are constants. x/R=X, y/R=Y.
The velocity component v_x and v_y are determined by v_x=-∂φ/∂Y and v_y=∂φ/∂X. Also, p is obtained.