Hagen-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. 

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 ... 

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... 

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