Pressure-Loss Correlations for Designing Foam Proportioning Systems

Abstract

Aspirating and compressed-air foam systems incorporate proportioners or pumps designed to draw a sufficient rate of foam concentrate into the flowing stream of water. The selection of relevant piping follows from the engineering correlations linking the pressure loss with the flow rate of a concentrate, for specified temperature (usually 20°C) and pipe size. Undocumented reports exist in industry that such correlations are often inaccurate, resulting in the design of underperforming suppression systems. To illustrate the problem, we introduce two correlations, sourced from industry and developed for the same alcohol-resistant concentrate, which describe, respectively, effective viscosity and pressure loss as functions of flow rate and pipe diameter. We then investigate the internal consistency of each data set. We demonstrate that neither of the data sets displays internal consistency, and suggest possible errors in data processing and unit conversion that might have led to the observed discrepancies. We correct the original data to produce correlations that are internally consistent and are characterised by realistic rheology. We then investigate three other pressure-loss correlations (one for Newtonian and two for non-Newtonian concentrates), obtained from a different manufacturer, to demonstrate the widespread existence of inaccurate correlations in industry.

Appendix

The steady-state volumetric flow rate of time independent fluid in a pipe with no slip at the wall can be calculated from

$$ Q = 2\pi \int\nolimits_O^R {urdr} = \pi \int\nolimits_O^{R^2 } {ud(r^2 ) = } -\!\pi \int\nolimits_{u(r = 0)}^0 {r^2 du} $$

(A1)

where the last equality follows from the integration by parts. In Equation (A1), u denotes the local velocity.

Changing the variable of integration from velocity to shear stress using

$$ \dot \gamma = - \frac{{du}}{{dr}}\quad {\rm and}\quad r = \frac{\tau }{{\tau _w }}R $$

(A2)

(with the latter expression obtained from the force balance on a cylindrical element of fluid in pipe flow) yields the acclaimed Rabinowitsch-Mooney-Schofield equation [8]

$$ \frac{Q}{{\pi R^3 }} = \frac{1}{{\tau _w^3 }}\int\nolimits_0^{\tau _w } {\tau ^2 } \dot \gamma d\tau $$

(A3)

Replacing Equation (1) into (A3) and integrating results in the so-called Buckingham equation for a power-law fluid [8]

$$ \frac{{32Q}}{{\pi D^3 }} = \frac{{4n}}{{3n + 1}}\left( {\frac{{\tau _w }}{m}} \right)^{1/n} $$

(A4)

Equation (A4) is obviously equivalent to Equation (3), with n′ and m′ defined in Equation (5), and the nominal shear rate at the wall in Equation (6).