Non-isothermal flows

If the flow is non-isothermal, it may be possible to represent the relationship between pressure and volume by an equation of the form  

Allowing for a discharge coefficient, the mass rate of flow is given by  

This equation also gives G = 0 for P2 = 0 and for P2 = P. Differentiating both sides of equation 4.20 with respect to PilPp. 

Putting dG/d(p2/f i) = 0 for the maximum value of G(Gmax) Substituting into equation 4.19  

The velocity = /kP is shown to be the velocity of a small pressure wave if the pressure-volume relation is given by Ft/ = constant. If the expansion aj roximates to a reversible adiabatic (isentropic) process k y, the ratio of the specific heats of the gases, as indicated in equation 2.30. 

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As already mentioned, the present code corresponds to the solution of steady-state non-isothennal Navier-Stokes equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains ... 

A polymer melt is injected into a circular section channel under constant pressure. What is the ratio of the maximum non-isothermal flow length to the isothermal flow length in the same time for (a) a Newtonian melt and (b) a power law melt with index, n = 0.3. 

Little error is introduced if this expression is applied to the flow of a compressible fluid provided that the velocity is not greater than about 60 m/s. When the velocity is high, the equation of state must be used to give the relation between the pressure and the volume of the gas. For non-isothermal flow, Pvk = a constant,... 

For an ideal gas in non-isothermal flow. If the pressure and volume are related bv Pi — constant, then a similar analysis gives ... 

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]. 

The melt flow under isothermal conditions, when it is described by the rheological equation for the Newtonian or power law liquid, has been studied in detail63 66). The flow of the non-Newtonian liquid in the channels of non-round cross section for the liquid obeying the Sutterby equation have also been studied 67). In particular, the flow in the channels of rectangular and trigonal cross section was studied. In the analysis of the non-isothermal flow, attention should be paid to the analysis 68) of pseudo-plastic Bingham media. 

One of the most interesting and fruitful approaches to the theoretical study of the non-isothermal flow of polymer melts has been suggested by Merzhanov and coworkers 82 84), which reduced the set of equations for the flow to the well-known equation of thermal explosion. The authors obtained the profiles of temperature and velocity for the case of stationary flow. However, the thermal explosion was founda5) to become feasible at too high velocities v lOm/s which could hardly be attained at low-pressure injection moulding of polymers. 

When analyzing non-isothermal flow problems, we often assume that the viscosity decays exponentially with temperature following the relation... 

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. 

Figure 9.25 Flow front shape after isothermal (left) and non-isotheramal (right) flows. Non-isothermal flows are a result of a 20 second one-sided heating [12].

O.A. Estrada, I.D. Lopez-Gomez, C. Roldan, M. del P. Noriega, W.F. Florez, and T.A. Osswald. Numerical simulation of non-isothermal flow of non-newtonian incompressible fluids, considering viscous dissipation and inertia effects, using radial basis function interpolation. Numerical Methods for Heat and Fluid Flow, 2005. 

Fig. 10.48 Numerical simulation results of nonisothermal flow of HDPE, Melt Flow Index MFI = 0.1 melt obeying the Carreau-Yagoda model for a typical FCM model wedge of e/h — 3 and =15. (a) Velocity (b) shear rate and (c) temperature profiles [Reprinted by permission from E. L. Canedo and L. N. Valsamis, Non Newtonian and Non-isothermal Flow between Non-parallel Plate - Applications to Mixer Design, SPE ANTEC Tech. Papers, 36, 164 (1990).]...

E. L. Canedo and L. N. Valsamis, Non-Newtonian, Non-isothermal Flow between Nonparallel Plates in Relative Motion. Applications to Mixer Design, SPE ANTEC Tech. Papers, 36, 164-167 (1990). 

However, the experimental studies relate to spatial growth of disturbances as the flow system is always excited by fixed frequency sources. Hence a spatial theory is preferred to study the stability of non-isothermal flows. Despite the distinction between temporal and spatial methods, the neutral curve, however, is identical. Iyer Kelly (1974) reported results using linear spatial theory under parallel flow approximation for free-convection flow past heated, inclined plates. Tumin (2003) also reports the spatial stability of natural convection flow on inclined plates providing the eigen spectrum. 

The situation observed in PS-Keroplast melts non-isothermic flow is different. In these compositions, viscosity depends substantially on the temperature, the rj value decreases by two orders, when the temperature changes from 453 K to 413 K. In Fig. 3, pressure profiles along the channel length are shown. It is clear... 

Now let us discuss the case of non-isothermic flows when the channel walls temperature is lower than the composition phase transfer temperature. This is a common situation for filling moulding forms. In [18, 19, 32] this is realized by filling the snail spiral form. Before filling, the form was kept at room temperature. After filling, the form was taken apart. The length of the melt L was measured. 

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