Isothermal Newtonian Model

Despite the non-isothermal nature of the film blowing process we will develop here an isothermal model to show general effects and interactions during the process. In the derivation we follow Pearson and Petrie s approach [20], [19] and [21]. Even this Newtonian isothermal model requires an iterative solution and numerical integration. Figure 6.21 presents the notation used when deriving the model. 

In the following analysis we first present the Newtonian isothermal model, which leads to an analytical solution. Then we discuss the Newtonian nonisothermal model, which gives insight into the complexities of the coupled heat and momentum transfer equations. PET, Nylon, and polysiloxanes are three typical polymers which are almost Newtonian at spinning conditions. Finally, we introduce the non-Newtonian isothermal model together with its associated difficulties. High-density polyethylene (HDPE), LDPE, polypropylene (PP), and polystyrene (PS) are all pseudoplastic and viscoelastic and fall into the latter category. 

In summary, the steady-state Newtonian isothermal model is able to provide the axial velocity profile as well as the filament radius profile, and it is based on the following additional assumptions (1) slowly changing radial profile with axial distance, (2) negligible inertial and gravitational forces, (3) nonexistent radial velocity profile, (4) circular filament, (5) axial velocity profile not dependent on the radial coordinate, and (6) negligible surface tension and air drag forces. (See also Problem 9A.1 for the validity of some of the above assumptions and Schultz (1987) for a challenge of these assumptions.)... 

FIGURE 9.22 Calculated isoparametric curves of blowup versus thickness ratio, for the Newtonian isothermal model and X = 20. (Reprinted by permission of the publisher from Pearson and Petrie, 1970c.)... 

The Newtonian and Isothermal Model as a Special Case of the Newtonian and Nonisothermal Model. Prove that the solution of the Newtonian isothermal model, Eq. 9.26, can be deduced from Eqs. 9.66 to 9.68 with the appropriate simplifications. 

Newtonian-isothermal Hele-Shaw model. A special form of the Hele-Shaw type flow governing equations is the isothermal Newtonian case where r/(z) = //,. This simplification leads to flow a conductance given by... 

Newtonian flow models for fully filled elements were developed by several authors (57-61). Here we derive the simplest isothermal Newtonian model for calculating the flow rate in a channel, which is identical to that of SSEs, as discussed in detail in Section 6.3. The volumetric flow rate in each channel is Qch = Q/m, where Q is the total volumetric flow rate. 

The first milestone in modeling the process is credited to Pearson and Petrie (42—44). who laid the mathematical foundation of the thin-film, steady-state, isothermal Newtonian analysis presented below. Petrie (45) simulated the process using either a Newtonian fluid model or an elastic solid model in the Newtonian case, he inserted the temperature profile obtained experimentally by Ast (46), who was the first to deal with nonisothermal effects and solve the energy equation to account for the temperature-dependent viscosity. Petrie (47) and Pearson (48) provide reviews of these early stages of mathematical foundation for the analysis of film blowing. 

The time-temperature superposition principle can be incorporated into isothermal constitutive equations (including the generalized Newtonian fluid model) to solve non-isothermal flow problems. If we define... 

The non-Newtonian rheological model briefly discussed above was used with the assumption of isothermal behaviour of the oil. The test case used by Glovnea and Spikes [1] is moderately loaded in numerical terms having Moes and Bosma non-dimensional parameter values of M= 211 and L = 5.02 at their stated conditions. The computing domain used is -2.5a [Pg.80]

Example 9.1. Newtonian and Isothermal Model for Melt-Spun Nylon 6,6... 

Analyses of the Newtonian isothermal and nonisothermal models can be even further complicated by the introduction of the viscoelastic nature of the polymers in the melt state. The viscoelasticity of the polymer is important in cases where the relaxation time. A., is of the same order of magnitude or slower than the characteristic time constant of the process, which might be taken to be equal to vo/L. The ratio of these two time constants is called the Deborah number (Eq. 3.90), and it is equal to... 

It has been shown that an isothermal Zatloukal-Vlcek model employing Newtonian constitutive model does not predict the experimentally observed non-monotonic trend for melt strength vs. processibility. It has been suggested that incorporation of the more explicit extensional viscosity/hardening parameters in the Zatloukal-Vlcek model helps to understand such non-monotonic behavior. 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

A first model of the calender nip flow has been presented by ArdichviUi. Further on Gaskefl presented a more precise and well-known model. Both models are very simplified, which yields that the flow is Newtonian and isothermal, and they predict that the nip force is inversely proportional with the clearance. Since mbber materials show a shear thinning behavior Ardichvilli s model seems not to be very realistic. The purpose of this section is to present a calender nip flow model based on the power law. The model is stiU being considered isothermal. Such a model was first presented by McKelvey. ... 

It is possible to derive an expression for the pressnre profile in the x direction using a simple model. We assnme that the flow is steady, laminar, and isothermal the flnid is incompressible and Newtonian there is no slip at the walls gravity forces are neglected, and the polymer melt is uniformly distribnted on the rolls. With these assnmptions, there is only one component to the velocity, v dy), so the equations of continuity and motion, respectively, reduce to... 

CFD methods are used for incompressible- and compressible-, creeping-, laminar- and turbulent-, Newtonian- and non-Newtonian-, and isothermal- and non-isothermal flows. Most commercial CFD codes include the k-z turbulence model [10]. More accurate models are also becoming available. The accuracy of the solution depends on how the mesh fits the true geometry, on the convergence of the solution algorithm, and also on the model used to describe the turbulent flow [11]. 

Although all polymer processes involve complex phenomena that are non-isothermal, non-Newtonian and often viscoelastic, most of them can be simplified sufficiently to allow the construction of analytical models. These analytical models involve one or more of the simple flows derived in the previous chapter. These back of the envelope models allow us to predict pressures, velocity fields, temperature fields, melting and solidification times, cycle times, etc. The models that are derived will aid the student or engineer to better understand the process under consideration, allowing for optimization of processing conditions, and even geometries and part performance. 

Figure 6.23 Predicted film blowing process using an isothermal Newtonian model for a...

For this non-isothermal flow consider a Newtonian fluid between two parallel plates separated by a distance h. Again we consider the notation presented in Fig. 6.58, however, with both upper and lower plates being fixed. We choose the same exponential viscosity model used in the previous section. We are to solve for the velocity profile between the two plates with an imposed pressure gradient in the x-direction and a temperature gradient in the y-direction. 

The Isothermal Newtonian Standard Mathematical Pumping Model... 

The mathematical model of isothermal flow of a Newtonian fluid in shallow-screw channels results in a simple design equation, which gives excellent insight into the flow mechanism and is very useful for first-order calculations. This model serves as the classic... 

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