Tubular pinch effect

The tubular pinch effect is a phenomenon in fluid mechanics, which has importance in membrane technology.

Mark C. Porter first suspected that the pinch effect was responsible for the return of separated particles into the core flow by the membrane. This effect was first demonstrated in 1956 by G. Sergé and A. Silberberg. They had been working with dilute suspensions of spherical particles in pipelines. While the particle was flowing through the pipeline, it appeared to migrate away from the pipe axis and pipe wall and reach equilibrium in a radial eccentric position.

Radial distribution of the particle concentration c in a tube during the flow-through

If:

${\displaystyle v_{p}}$	radial velocity component of a particle
${\displaystyle d_{T}}$	tube diameter
${\displaystyle d_{p}}$	particle diameter
${\displaystyle r^{*}}$	equilibrium radius
${\displaystyle w^{*}}$	mean flow velocity
${\displaystyle Re}$	Reynolds number
${\displaystyle r}$	radius

then the pinch effect follows the relation:

${\displaystyle v_{\text{p}}=0.17\cdot w^{*}\cdot Re\cdot {\left({\frac {d_{\text{p}}}{d_{\text{T}}}}\right)}^{2.84}\cdot {\frac {2r}{d_{\text{T}}}}\cdot \left(1-{\frac {r}{r^{*}}}\right)}$

This effect is of importance in cross-flow filtration and especially in dialysis. It is significant especially for particles with a diameter of 5 μm and for particles which follow laminar flow conditions and slows down the process of filter cake formation, which prolongs the service life and the filtering stays permanently high.