## Pneumatic Foams

### Quick Start

A pneumatic foam is one where gas-liquid foam continuously rises up a vertical column due to gas bubbles be sparged to the bottom. The liquid flowrate due to this rising foam, jf, and the volumetric liquid fraction, ε, which are fundamental to the design of processes such as foam fractionation and foam gas-absorption, are dependent upon gas flowrate, bubble diameter and liquid dynamic viscosity and density.

### Credits

The app implements the method of Prof Paul Stevenson in Hydrodynamic theory of rising foam, Minerals Engineering 20 (2007) 282–289.

### Pneumatic Foams

Bubble d mm
Viscosity η cP
Density ρ g/cc
m
n
jg* mm/s

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Bubbles of diameter d are continuously sparged at a superficial gas velocity of jg into a column of liquid of dynamic viscosity η and density ρ. If a foam layer is formed, and the bubble size is invariant in height, a foam with a volumetric liquid fraction of ε will form, and the liquid superficial velocity up the column will be jf . The volumetric liquid fraction is determined by numerical solution of:

(4ηj_g)/(nmρgd^2)=ε^(n-1)(1-ε)^2

whence the liquid superficial velocity is determined from:

j_f=(εj_g)/(1-ε)-(ρgd^2)/(4η)mε^n

The first term on the right-hand side would be the liquid superficial velocity if there were no slip between liquid and gas phases, with the second term being a correction for liquid drainage.

The liquid superficial velocity increases monotonically with gas superficial velocity, until the threshold of ‘gross bubbling’ is reached, which occurs at a critical gas superficial velocity of:

j_g"*"=(ρgd^2)/(η)mn(1/(n+1))^2((n-1)/(n+1))^(n-1)

In the above expressions, there appear two adjustable constants, m and n, which arise from the empirical description of liquid drainage and are dependent on the stress-state at the gas-liquid interface for a particular surfactant at a particular concentration [maybe you could link to your foam drainage app]. They must be determined experimentally, either by forced drainage or PFG-NMR. However, a survey of the data1 has shown that 0.0011<m <0.063 and 1.45<n<2.13. For the commonly used surfactant SDS at approximately the critical micelle concentration, it has been measured that m=0.016 and n=2.00, for instance.

### Units

Note that inputs are in "common" units and are converted to SI units within the app for the calculations.

1Paul Stevenson, On forced drainage of foam, Coll Surf A 305, 1-9, 2007