## Monday, March 09, 2009

### Abusing the Poiseuille equation

The abuse has always existed, but now it seems to be on the uptick and it's got to stop.

The eponymic Poiseuille equation (don't even begin to ask me about pronouncing the name) shows the relationship between the important variables when a fluid is flowing through a pipe. The volumetric flow rate, Q, is related to the pressure drop, DP, the pipe diameter, d, the pipe length, l, and the viscosity, h, by

Q =(p d4 DP)/(8 h l)

In getting to this result, there are TWO (2) assumptions. Violate either assumption and the results you get from the equation are useless.

1) The first assumption is that the flow field is fully developed, meaning that the liquid has been moving for a long enough time period in the tube that its velocity profile is not changing. Fully developed flow typically requires that l/d is at least 20.

2) The second is that the viscosity, h, is constant at all shear rates. This is a good assumption for low molecular weight fluids, but not for polymers and other non-Newtonian fluids where the viscosity typically decreases as the shear rate increases, although numerous other possibilities exist.

Given these two assumptions, it is clear that if you are looking at polymer flow through a short pipe (l/d < 20), YOU CANNOT USE THE POISEUILLE EQUATION. Instead, you need to rederive the equation using an appropriate function for the viscosity. (Power law or 8-constant Oldroyd or whatever.)

And yet this abuse is getting more and more common all the time. I saw this misuse in a major trade magazine for the plastics magazine where the author, who really should have known better, was attempting to quantitate melt-flow-index data. A melt flow index has a short die (l/d ~ 4) so that the flow profile never fully develops. All the numbers the author generated (and there were a lot) are meaningless.

I also ran across it on another blog. The worst part was that the blog seemed to be well read (based on the number of comments) and that the readers also were being lead down the wrong path.