Nick and Matas: 2006
Introduction: background information:
Rising bubbles in water are a familiar sight. Bubbles may be large and irregular in shape as above, or they may be tiny and spherical. Soda has carbon dioxide gas dissolved in it. When a soda bottle is opened, the pressure is released, causing tiny bubbles to form in the soda water.
Most of the small bubbles rise from a particular point in a steam of bubbles (Figure 1). Bigger bubbles rise from random points and do not form a steam of bubbles. As they rise the size of the bubbles increases due to a decrease in pressure as they approach the surface.
Research Question
How does the increase in size of rising bubbles in soda water
affect the shape of the position-time graph?
Hypothesis
Bubbles with larger volumes are expected to rise more quickly
but the terminal velocity versus diameter relationship is predicted
to be complex, not modeled by a simple linear relationship, because
the drag and buoyancy forces both depend on the cross sectional
area which increases as the bubble rises and the pressure is reduced,
and for larger bubbles the shape also changes.
Materials and procedure
A short length of wire was bent with two upward pointing ends
and placed in the bottom of a 500
ml cylinder. The wire was added to create a source of bubble
production. 325 ml of Singh soda-water was then poured carefully
down the side of the cylinder to avoid excessive release of carbon
dioxide gas. Bubbles were then filmed with a Sony 25 fps digital
video camera, so the distance-time relationships for the rising
bubbles could be plotted.
The film was exported to QuickTime and imported to Logger Pro.
Bubble motion was analyzed in two ways.
1 Distance vs. time graphs for isolated rising bubbles were plotted frame by frame at time intervals of 0.04 seconds.
2 A single image Figure
1 was analyzed by placing multiple points on the same frame
to plot the position-time graph for the rising stream of bubbles
(assuming a constant rate of bubble production).
Data: position-time graphs
Position-time graphs of three different bubbles were plotted
in Logger Pro. the bubble positions were marked on each frame
to within ±0.25 mm.
Graph 1: Position
vs. time for a single small bubble in the steam (~0.2 mm in diameter).
Graph 2: Position vs. time graph (Medium size bubble (~1 mm in diameter).
Graph 3: Position vs. time graph for a larger bubble (~ 2 mm in diameter).
Analysis and Discussion
Quadratic functions were fitted to each set of data points in Graphs 1-3. Within errors - over the limited ranges shown in each graph -the quadratic functions are a good fit to the data points. The corresponding velocity-time graphs are straight lines, showing that over the limited size ranges shown on each graph the rising bubbles have constant acceleration. It appears that, contrary to expectations, for small bubbles the increase in diameter and hence the buoyancy as they approach the surface, combined with the increased drag due to increasing diameter combine to give nearly constant acceleration.
Data: velocity-time graphs
Graph 1a - velocity versus time for the tiny bubbles rising in the stream shown in figure 1.
Graph 2a - velocity versus time for bubbles of approximately 1 mm diameter rising in soda water.
Graph 3a - velocity versus time for the larger bubbles of approximately 2 mm diameter rising in soda water.
That the linear relationship is approximate can be seen from the accelerations.
If the constant acceleration relationship were to hold exactly over the entire size range, the accelerations would be the same for each graph. It appears - from this limited set of data - that the acceleration is larger for 1 mm bubbles.
That the rising stream of bubbles shown in Figure 1 are released at regular time intervals can be seen from Graph 4. Graph 4 is, within errors another parabola, showing that the acceleration is constant - if bubbles of the same size are released at the same time intervals.
Evaluation
The radius of the bubbles could be estimated but not measured
accurately because the movie clips in Logger Pro were pixilated
which made the small bubbles appear to be square. Film with higher
resolution is required to overcome this limitation.
Graphs 1-3 do not cover the entire range from 0.1 mm to 2 mm diameters. Individual graphs appear to be parabolic but as can be seen from the accelerations the approximation does not hold over the entire range. To overcome this limitation film taken over a much larger distance (~2 meters) is required.
Suggestions for further work:
1 To overcome both limitations a much deeper cylinder is required and film of much higher resolution must be obtained. It is difficult to achieve both at the same time, but filming the ascent of a single bubble with three or four cameras placed at intervals would be a large improvement.
2 Alternatively a high resolution (12 Mpx) still image
could be made of a rising stream of bubbles (over 20 cm) at the
same time as a video was taken. Measuring the bubble diameters
as a function of depth from the still image, and plotting displacement-time
data from both the still image and the video would provide more,
more accurate, diameter-terminal velocity measurements.