The investigation of gas bubbles rising in vertical, cylindrical tubes filled with liquid is of technical interest, for example for the design of air-lift pumps and circulation evaporators among others. However the calculation of this process is difficult, even when it is highly idealised.
With the help of dimensional analysis, a number of far-reaching statements can be made about the form of the solution in this case. The problem will be idealised as follows:
– It will be based on a very large air bubble in water, which virtually fills the entire cross section of the tube, see Fig. 3.57.
– Air and water will be taken as frictionless and incompressible.
– Capillary forces will be neglected.
– The bubble moves at a steady velocity w up the tube.
Under these preconditions the physical process can be described by the following influencing quantities:
w, g, d, W, A .
g is the acceleration due to gravity, d the tube diameter, A the density of air, W the density of water.
a) Determine the dimensionless groups which describe the process π1, π2, ..., πn.
b) What statements can be made from the relationship π1 = f(π2, π3, ..., πn) between the dimensionless groups, about the form of the equation w = f(g, d, W, A)?