Poiseulle flow

The field of transport phenomena is the basis of modeling in polymer processing. This chapter presents the derivation of the balance equations and combines them with constitutive models to allow modeling of polymer processes. The chapter also presents ways to simplify the complex equations in order to model basic systems such as flow in a tube or Hagen-Poiseulle flow, pressure flow between parallel plates, flow between two rotating concentric cylinders or Couette flow, and many more. These simple systems, or combinations of them, can be used to model actual systems in order to gain insight into the processes, and predict pressures, flow rates, rates of deformation, etc. 

The flow in many extrusion dies can be approximated with one, or a combination, of simplified models such as slit flow, Hagen Poiseulle flow, annular flow, simple shear flow, etc. A few of these are presented in the following sections using non-Newtonian as well as Newtonian flow models. 

A frill analysis ofthe modified velocity field would, in fact, be quite complicated. Because /i must depend on z, as already explained, the exact velocity field is not even unidirectional. For present purposes, we therefore limit our analysis to the simpler limiting situation in which we assume that the changes in temperature are small enough that we can ignore these complicated changes and approximate the velocity field by using the isothermal Poiseulle flow solution ... 

FIGURE 10.4.2. Laminar flow in a pipe (Poiseulle flow). 

Assume that a fluid of mass density p flows through a pipe of diameter d =2a (a is the radius) as shown in Fig. 5.6 (Hagen-Poiseulle flow). When the velocity field is one-dimensional, the differential equation governing the pipe flow problem along with the boundary conditions (BC) is given in cylindrical polar coordinates (r, z) as follows ... 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

The relative importance of different transport contributions in a porous structure is given in Table 9.3 which shows that the contribution of Poiseulle (viscous) flow becomes important in larger pores (range 0.1-0.3 pm). At high pressure (10 bar) the Poiseuille flow is already important in pores with a radius of 10 nm. 

In principle, these values could be directly used as regulating variable in a feedback system to control the valves for flow adjustment in the different branches. However, from (21.2) and Hagen-Poiseulle s law, it follows that... 

In order to complete the macroscopic description we shall evaluate L /T and L22/T. If the pore size is much greater than the mean free path, this flow of fhe fluid through a circular pore of radius r would be governed by Poiseulle s equation [30]... 

Study of domain of validity of linear laws is of particular interest in the context of biological systems. For example, even Poiseulle s law, which has a much larger domain of validity otherwise, is reported to have significant deviations in physiological systems. The effect of increase of arterial pressure on flow through the various tissues of the body is greater than one would expect on the basis of Poiseulle s law [69],... 

Poiseulle s equation states that the flow rate in a tube is inversely related to the distance of the liquid movement. Based on Poiseulle s equation, Lucas [40] and Washburn [41] developed an equation for flow rates in capillaries ... 

Rodney J. Groleau outlined a method on how to find the relative viscosity using empirical means. The pressure drops and fill time were measured from the data displayed on the injection machine control panel. The volumetric flow rate can be determined by Poiseulle s equation [1]. 

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