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. 3.61.
– 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)?