Ind. Eng. Chem. Res. 1993,32, 2866-2872
2866
Heat-Transfer Model Assessment of Chill Rolls for Polymer Film Extrusion Enio Kumpinsky R&D Department, Ashland Chemical, Inc., P.O. Box 2219, Columbus, Ohio 43216
Eight heat-transfer models with diverse levels of lumping and distribution were developed for the casting of polymeric films on chill rolls. It is shown that some degree of distribution is necessary to adequately represent the process. Three models were found to be suitable to describe this operation: fully distributed, distributed in the film and lumped in the roll shell, and having uniform temperature in the film and distributed in the shell. The most convenient model to use is the one having temperature distribution in the film and lumping in the shell, because it does not require iterations to match roll conditions a t the beginning and a t the end of a turn and it allows for the easy incorporation of the contact heat-transfer coefficient. extruder
Introduction In cast film extrusion, the polymer melt drops onto a chill roll in a tangential contact, where it freezes into a film. The alignment of this roll is critical in relation to the falling film (Frados, 19761, and if there is a tendency to form air pockets between the film and the roll an air knife may be used (Richardson, 1974). Heat is indirectly transferred from the polymer to the cooling fluid inside the roll through the metal shell, as seen in Figure 1. The flow and inlet temperature of the cooling fluid, generally water, are controlled to meet the heat-transfer needs of the process. Rotary union joints allow the cooling fluid to circulate through the roll, and the mode of fluid circulation can vary considerably from roll to roll. It can be internally sprayed against the shell, be circulated through a straight jacket, or be forced to flow through spiral channels along the length of the roll. In general, the cooling fluid circulates a t high rates to produce a uniform temperature across the surface of the roll (Richardson, 1974). According to Richardson (1974) the chill roll (also known as cast roll) is the most important part of the polymer extrusion process, as controlled cooling yields an attractive film. Heat transfer is of paramount importance for chill roll design, and a few papers have been published on this subject. Aldheid (1974,1975) proposed a trans-film model and a lumped-parameter model, both of which are related to two models that have been developed in the present study. Parsons (1979) carried out an analysis of key parameters for chill roll operation, through the use of an overall heat-transfer coefficient. Haberstroh and Menges (1980) and later Haberstroh (1987) discussed the concept of contact coefficient, which takes into account the heattransfer resistance between the exterior of the roll and the film surface touching it. Chill rolls are also used in other sectors of technology. Quadracci and Modi (1992) developed a model that is relevant to the printing industry, and it applies to the cooling of paper on a chill roll after passing through an oven, where the ink hardens. In this work the heat transfer on chill rolls is studied by means of mathematical models having diverse degrees of lumping and distribution. The focus here is to test the ability of the various approaches to predict heat-transfer rates, compared to the fully distributed mechanism. The goal is to find the simplest possible model that can reasonably describe the heat-transfer phenomena associated with polymeric film casting on a chill roll.
chill
,4 . .
polymer film or
Figure 1. Schematic representation of an extruded polymer film in contact with a chill roll. Table I. Heat-Transfer Assumption model shell film A uniform temperature uniform temperature B uniform temperature lumped with he C lumpedwithh, uniform temperature D distributed temperature uniform temperature E uniform temperature distributed temperature lumped with h, F distributed temperature G lumped with h, distributed temperature H distributed temperature distributed temperature
The Models Eight models with different levels of lumping and distribution were developed, and Table I summarizes the basis for each of them. According to Figure 1, the roll turns at a certain frequency 9 with a cooling fluid flowing inside the shell. A compositeenergy balance is needed for the cooling section (between 0 and a rad), and a singlelayer balance is required for the return section (between a and 27r rad). Since the roll movement is cyclic, its temperature at 277 rad must match the initial temperature at 0 rad. The following assumptions were made for the development of the models. The cooling is fast, so there is no phase change in the film that involves heat release or absorption (e.g., crystallization) . The temperature of the cooling fluid and environment(air) are constant. There is a perfect contact between the film and the roll, with no air pockets trapped between the two layers. The physical properties are constant. The film coefficient on the cooling
08S8-5885/93/2632-2866$04.00/00 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2867 side takes fouling into account. The shell thickness is much smaller than the roll diameter, so the curvature effects can be disregarded for the distributed models. Thicker edges due to polymer neck-in are neglected. The operation is in steady state, but the shell and film locally undergo a transient change in temperature. The assumption of perfect contact between the roll and the film can be contested (see Haberstroh and Menges (198011, but it will be considered to be valid for the purpose of model scrutiny. This simplification can be lifted once an adequate model is identified. Each individual model will now be presented. It is desired to obtain the mean temperature of the shell in contact with the cooling fluid along the circumference of the roll and also the mean temperature along the thickness of the polymeric film at the point it disengages from the roll. In this manner one can verify the ability of each model to yield similar values for the heat gained by the cooling fluid and the heat loss by the film. Model A. This is the simplest of all models studied here and it assumes a uniform temperature throughout the shell and film composite. It can be mathematically stated as follows. For the cooling section T = T m = Tp,
-
dT [pmCP,,(6 - a ) + ppCp,p(b- 811 dt hc(Tc- T ) + he(Te- T ) (1) where
the shell, T m ( a )is readily calculated for the cooling section from
Tm(a)=
kmTe+ [(km/he) + 6 - alhCTc km + [(km/he) + 6 - alh,
Equations 1-4 have obvious solutions, and they will be omitted in this development. Model D. The cooling section requires an energy balance with uniform temperature through the film thickness and a distributed profile in the shell. It can be stated as
with
Tm(x,O)= f ( x > ;
Tp= Tp(0)at t = 0;
Tm(6,t)= Tp(t)
Note that the boundary condition at x = 6 contains a time derivative. An analytical solution for eq 5 and associated conditionscan be obtained,based on the method developed by Ramkrishna and Amundson (1974a,b, 1986a):
(6 - a)Tm(0)+ ( b - 6)Tp(0) b-a For the return section,
T(0)=
( Tm(x,t)- Te,un(x))
Tm(x,t)= Te+
( u n ( x ) ,u n ( x ) )
n=l
un(x)
(6)
where pmCp,m(6- a) dT,= hc(Tc- Tm)+ he(Te- Tm); dt T,(t,) = T(t,) (2)
(Tm(x,t) - Te, u n ( x ) )
Model B. For the cooling section the temperature in the shell is uniform and there is instant thermal equilibrium in the film, so ita resistance to heat transfer is lumped with the air side film coefficient. It can be written as
dT m pmCP,,(6 - a ) dt = h&TC- Tm)+ h P V e- Tm);
Tm= Tm(0)at t = 0 (3) whose integration is used to calculate the mean temperature in the film:
The energy balance on the return section is identical to that of model A. Model C. For the cooling section, the temperature in the film is uniform and there is instant equilibrium in the shell, so its resistance to heat transfer is lumped with the cooling fluid film coefficient. dTP p C ( b - 6) - hm(Tc- Tp)+ he(Te- Tp); P P.P dt Tp= Tp(0)at t = 0 (4)
sin[2zm,.,(6- a ) ]
p)'
+ [(
- l]cos[2r,,(6
1
- a)]
The eigenvaluesare found from the characteristicequation and the definition of Zm,n:
from which the temperature of the shell in contact with the cooling fluid, Tm(a,t),is obtained:
[km/(6 - a)lTp(t)+ hcTc [km/(6 - all + hc Since the assumption of instant equilibrium was made for Tm(a,t)=
For the return section the energy balance is the same as
2868 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
eq 5, but with the following boundary condition a t x = 6:
The solution may be obtained from the work of Ramkrishna and Amundson (1985a) by finite Fourier transform:
- (Tm(x,t),
u,(x))
Tm(x,t)=
n=l
u,(x)
(9)
(u,(x), u,(x))
where the eigenfunctions are formally the same as eq 7 and
+ An
1 e-xnt -[k z
T m,n
+ heu,(6)Te]
replaces he and the time lies between 0 and t,. The mean temperature in the polymer f i b is the same as in model B, with T m ( t ) replaced by Tm(6,t). The energy balance on the return section is the same as that of model D. Model G. This is the reverse of model F, in that at the cooling section the resistance to heat transfer through the shell is lumped with the resistance to heat transfer to the cooling fluid. The solution is analogous to that of model F, with the same change in nomenclature to transform the solution of model D to model E (except for the /? terms, which do not appear in model G). For the return section, the solution is identical to the return section of model C. Model H. This is the fully distributed model for the problem, and it is used as a measure of accuracy for the other models. However, it is necessary to keep in mind that drastic assumptionswere made, such as constant heattransfer coefficient along the roll and constant physical properties for metal and polymer. For the cooling section, the energy balance is
with the following conditions: with the eigenvalues calculated from the characteristic equation
Tm(x,O)= f ( x ) ,
a
< x < 6; T,(x,O) = T,(O),
km
6
