Polymer flows

The melted plastic enters the mold block through the sprue, then typically flows through a system of runners, and finally enters the mold cavity through the gate (Fig. 11.2). Most often, the mold block will contain several mold cavities, sometimes as many as 64, so several objects are molded at the same time. 

When the mold contains more than one identical mold cavity, it is important that the cavities fill equally. The usual approach to accomplishing this is to balance the flow paths for the plastic, so that distances and geometry, and thus pressure and flow, are equalized. Where nonidentical objects are being produced, this job is even more complex, but that seldom applies in packaging applications. It is also important to design the runner geometry to avoid dead spots, where plastic can accumulate and be subjected to an excessive heat history. 

A key to efficient production of components by injection molding is the ability to rapidly fill the mold. Therefore, polymers used for injection molding usually have relatively low viscosities (high melt flow rates). Injection rates of up to 2 kg/s (4.4 Ib/s) are common. 

Temperature of the melt is an important variable. Higher temperatures produce lower viscosities, and hence more rapid mold filling. On the other hand, high tern- 

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Computer modelling provides powerful and convenient tools for the quantitative analysis of fluid dynamics and heat transfer in non-Newtonian polymer flow systems. Therefore these techniques arc routmely used in the modern polymer industry to design and develop better and more efficient process equipment and operations. The main steps in the development of a computer model for a physical process, such as the flow and deformation of polymeric materials, can be summarized as ... 

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

The convection term in the equation of motion is kept for completeness of the derivations. In the majority of low Reynolds number polymer flow models this term can be neglected. 

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The majority of polymer flow processes involve significant heat dissipation and should be regarded as nou-isothermal regimes. Therefore in the finite element modelling of polymeric flow, in conjunction with the equations of continuity... 

Pearson, J. R. A., 1979. Polymer flows dominated by high heat generation and low heat transfer. Polym. Eng. Sci. 18, 1148-1154. 

Stabilization of the Cellular State. The increase in surface area corresponding to the formation of many ceUs in the plastic phase is accompanied by an increase in the free energy of the system hence the foamed state is inherently unstable. Methods of stabilizing this foamed state can be classified as chemical, eg, the polymerization of a fluid resin into a three-dimensional thermoset polymer, or physical, eg, the cooling of an expanded thermoplastic polymer to a temperature below its second-order transition temperature or its crystalline melting point to prevent polymer flow. 

Well above polymers flow in the viscous manner we have described already. When this happens the strength falls steeply. 

This example illustrates the simplified approach to film blowing. Unfortunately in practice the situation is more complex in that the film thickness is influenced by draw-down, relaxation of induced stresses/strains and melt flow phenomena such as die swell. In fact the situation is similar to that described for blow moulding (see below) and the type of analysis outlined in that section could be used to allow for the effects of die swell. However, since the most practical problems in film blowing require iterative type solutions involving melt flow characteristics, volume flow rates, swell ratios, etc the study of these is delayed until Chapter 5 where a more rigorous approach to polymer flow has been adopted. 

With the widespread use of software packages to assist with computational fluid dynamics (CFD) of polymer flow situations, other types of viscosity relationships are also used. For example, the regression equation of Klien takes the form... 

Most polymer processing methods involve heating and cooling of the polymer melt. So far the effect of the surroundings on the melt has been assumed to be small and experience in the situations analysed has proved this to be a reasonable assumption. However, in most polymer flow studies it is preferable to consider the effect of heat transfer between the melt and its surroundings. It is not proposed to do a detailed analysis of heat transfer techniques here, since these are dealt with in many standard texts on this subject. Instead some simple methods which may be used for heat flow calculations involving plastics are demonstrated. 

Fig. 6-7. Asymmetry factor (AJ of the L-enantiomer versus sample load (A) and versus flow rate (B) on L-PA-imprinted polymers. Flow rate 1.0 ml min . Mobile phase MeCN/[potassium phosphate 0.05 M, pH 7] (7/3, v/v).

Nobile et al. [3] reported that viscosity of a polycar-bonate-TLCP blend can increase or decrease in the same system at the same temperature, depending on the shear condition. At very low shear rates the viscosity was found to increase with TLCP loading, whereas at high shear rates a significant drop was observed. But in all of these cases, the way in which the TLCPs alter the bulk polymer flow is not yet well understood. 

Though the accuracy of description of flow curves of real polymer melts, attained by means of Eq. (10), is not always sufficient, but doubtless the equation of such a structure based on the idea of relaxation mechanism of non-Newtonian polymer flow, correctly reflects the main peculiarities of viscous properties. Therefore while discussing the effect a filler has on the viscosity properties of polymer melts, besides the dependences Y(filler modifies the characteristic time of relaxation. According to [19], a possible form of the X versus

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Forming the hinge cross-section by using an extruder die results in a hinge with poor flex life. Because hinges are formed in the direction of the polymer flow, they cannot be sufficiently oriented when flexed. However, if an extruded hinge is formed by the take-off mechanism while the polypropylene... 

The following qualitative picture emerges from these considerations in weak flow where the molecular coils are essentially undeformed, the polymer solution should behave approximately as a Newtonian fluid. In strong flow of a highly dilute polymer solution where the macroscopic velocity field can still be approximated by the Navier-Stokes equation, it should be expected, nevertheless, that in the immediate proximity of a chain, the fluid will be slowed down because of the energy intake to stretch the molecular coil thus, the local velocity field may deviate from the macroscopic description. In the general case of polymer flow,... 

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