Page 333 - Plastics Engineering

3 16 Processing of Plastics

Considering unit width of the calender rolls the total throughput, Q, is given by
hI2
Q=2/Vdy

0
1
1 dP
= 27bd + --(3 - (h/2)2) dy
2q dx
0
(4.34)

Since the output is given by VdH

then (4.35)

dP
From this it may be seen that - = 0 at h = H.
dx
To determine the shape of the pressure profile it is necessary to express h as
a function of x. From the equation of a circle it may be seen that

h = Ho + 2(R - (R2 - x2)'I2) (4.36)
However, in the analysis of calendering this equation is found to be difficult
to work with and a useful approximation is obtained by expanding (R2 - x2)'/*
using the binomial series and retaining only the first two terms. This gives

(4.37)

Therefore as shown earlier dP/dx will be zero at
( L?)
H=Ho l+-

x = fJH - Ho)R' (4.38)
This gives a pressure profile of the general shape shown in Fig. 4.58. The
value of the maximum pressure may be obtained by rearranging (4.35) and
substituting for h from (4.37)

-- - (4.39)
dx (H0+;)3