Plug and Poiseuille Flow

TimeDomain CVD, Inc.

Two Simple Flows

Small Re for Small Minds

Two extremely simple flow configurations, valid for simple channels and small values of Re, can provide the basis for a lot of simple approximations for real transport...

Plug Flow

The simplest flow one could think of is Plug Flow: constant velocity of flow in every part of a system. Amazingly, such a simple flow is a fair approximation to actual flow in a simple channel or pipe when the Reynolds numbers (in this case, based on the pipe diameter D) are small, and when the distance from the entrance to the channel is small compared to the Entry Length, Le.

Plug flow allows us to define an unambiguous residence time which is the same for every streamline, since velocity is constant and the channel length is the same for all streamlines .

The velocity U can be expressed in terms of the volumetric flow F and cross-sectional area A. If we ignore temperature differences (often OK) F in cc/second can be found easily from the molar flow in sccm.

Finite Viscosity: Poiseuille (fully developed) Flow

If we consider the case of flow in a pipe or channel when Re is low BUT after the flow has been in the pipe for a distance much longer than the entry length, the fluid velocity will vary with radial position. The velocity must be zero exactly at the walls, and viscosity causes the velocity to be small in the vicinity of the walls. Therefore the flow in the center is actually faster for the same volumetric flow.

In the case of a cylindrical pipe with flow along the axis the velocity distribution is a simple quadratic, known as Hagen-Poiseuille or simply Poiseuille flow. A residence time and volumetric flow can again be defined in terms of the average velocity, but in this case the residence time is an average: streamlines near the center spend less time in the channel, and streamlines near the walls have very long actual residence times.

When large pressure differences exist across a long pipe (so that Re stays fairly small) one can integrate the Poiseuille formula to find the molar flow through the pipe (which of course is the same at all locations along the pipe in steady state)

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If we consider the case of flow in a pipe or channel when Re is low BUT after the flow has been in the pipe for a distance much longer than the entry length, the fluid velocity will vary with radial position. The velocity must be zero exactly at the walls, and viscosity causes the velocity to be small in the vicinity of the walls. Therefore the flow in the center is actually faster for the same volumetric flow.
In the case of a cylindrical pipe with flow along the axis the velocity distribution is a simple quadratic, known as Hagen-Poiseuille or simply Poiseuille flow. A residence time and volumetric flow can again be defined in terms of the average velocity, but in this case the residence time is an average: streamlines near the center spend less time in the channel, and streamlines near the walls have very long actual residence times.
When large pressure differences exist across a long pipe (so that Re stays fairly small) one can integrate the Poiseuille formula to find the molar flow through the pipe (which of course is the same at all locations along the pipe in steady state)