Counting Bubbles with Laplace Pressure

Counting bubbles might seem to be a pointless activity but it is actually a reliable way to measure very small gaseous flows. We have seen in a [»] previous post that we could read flow rates by measuring the pressure drop due to the friction of the fluid on a tiny hole. However, because gas have much smaller viscosity than liquids, measuring flow rate for gas require very large flow rates to compensate for the small viscosity. Also, such measurements are not linearly proportional to the actual flow rate and some processing of the data should be done before inferring any figures.

On the other hand, bubble counting allows precise and accurate measurements of small gaseous flows. The idea is to bubble the gas using a capillary (or simply tubing) into a solution where it is insoluble and to rely on the assumption that all bubbles have roughly the same volume. There are different ways to count bubbles, such as using an optical bridge, but here I will present a method based on Laplace pressure. I have developed the technology during my PhD and, to my knowledge, there have been no apparatus that use Laplace pressure to count bubbles so far. I would also like to acknowledge my colleague, Carlo, who introduced me to the theory of Laplace pressure which led me to the development of the counting procedure presented here.

Because of the surface tension between two immiscible fluids, every bubble has an internal pressure slightly above the surrounding medium. This phenomenon is known as the Laplace pressure (∆P) and is proportional to the surface tension constant between the two fluids (γ) and inversely proportional to the radius of the bubble (R):

The effect is therefore greater for small bubbles than for larger ones. To give some scale, a 1 mm air bubble in water has an internal pressure about 2.8 mbar above the surrounding water.

A classical way to measure the pressure is to grow a bubble on the mouth of a capillary tube by slowly injecting gas into the capillary as illustrated on Figure 1. When the bubble is flat, its curvature is extremely large (almost infinite) and the Laplace pressure inside the capillary is null. As the bubble grows the pressure keep increasing until it reach a maximum where the bubble has a curvature equal to the capillary mouth radius. Then, and this is not always mentioned in the theory but often observed experimentally, the bubble tends to tear off a fraction of air from the capillary, leaving a negative curvature and we therefore observe a small negative Laplace pressure.

Repetitive bubbles lead to a record such as on Figure 2. The three successive pressure discussed above clearly appear on the graph.

There is, however, one small trick that ought to be discussed. When we are measuring the pressure in the capillary we are not only measuring the Laplace pressure but also the pressure drop due to friction in the capillary and the hydrostatic pressure due to the column of water above the capillary mouth. This has two major consequences as large flows will add a pressure term that we would like to be smaller than the measured signal (if it were larger then we could use the pressure drop as in the [»] flow sensor post) and varying water level also produces small pressure changes. To avoid interaction between water-height motions due to bubbles with the measurement, it is often a good idea to use a second capillary placed at about the same depth and to record the pressure difference between the two capillaries. Hopefully, our [»] 24PCEFA6D sensor is a good match for this. If you don't find capillaries on e-bay, you can use medical syringe needles or small silicon tubing with 1 mm internal diameter or less (the results given on Figure 2 were actually obtained using 1 mm silicon tubing in a first place).

To count the number of bubbles we have to track every peak in the pressure graph. But because of the varying pressure due to friction, we cannot simply use a threshold. The idea is to first pass the pressure into a band-pass filter to select an approximated time-scale of a peak event. From Figure 2 we see that a peak can be roughly assimilated to a sine wave of period 0.25 sec so I have used a second order high-pass filter with a cut-off of about 4 Hz. Also, because we do not want noise from the sensor output, we should remove all frequencies above 1 kHz. The band-pass filtered signal can then be thresholded to detect the peaks and send the signal to a microprocessor (or you may also send the output to a low-pass filter to have an analog reading of the flow rate). A typical result is given on Figure 3.

So far we have been measuring the number of bubbles, to convert it to a flow rate we should count the number of event during a finite time period. This may be done either through a computer or using a RC low pass filter. In the experiments I have been doing, I was reading the number of bubbles produced per minute using a computer. This gives a rather interesting tool to study kinetics phenomenon that you can read [»] here.

As a conclusion, the sensor works quite well and is relatively cheap (less than 100 euros for the digital version with a micro-controller). The only down-side I found was that it required a bit of tuning for each kind of experiment you are running. Also, take care to corrosion of your needles (if you are using them) as clogging induced over-pressure in one of my experiment which could have ended in dramatic conditions with culture medium splashing all over the place... Hopefully, I manage to save everything before the apocalypse took place :) I would now recommend always operating using some kind of safety valve to avoid over-pressure with that kind of systems.

Don't forget to have a look at some [»] experimental results!