Aeration is a common industrial process and yet one in which there is often serious disagreement about correlations (Cussler, 1997). This is especially true for deep-bed fermentors and for sewage treatment, where the rising bubbles can be the chief means of stirring. We expect that the rate of oxygen mass transfer through the bed can be calculated in terms of a mass-transfer coefficient, \(k_{c}\), which will depend on the average bubble velocity \(v\), the solution's density \(ho\) and viscosity \(\mu\), the bubble diameter \(d\), the depth of the bed \(z\), and the diffusivity \(\mathrm{D}_{A B}\). Use the Buckingham method to determine the dimensionless groups to be formed from the variables significant to this problem. Choose \(\mathrm{D}_{A B}, d\), and \(ho\) as core variables.