Dimensionless Numbers

Dimensionless numbers often correlate with some performance parameter and greatly aid engineering analysis and design. One of the best such dimensionless numbers is the Reynolds number defined as:

Do not confuse µ representing viscosity here with the specific growth rate coefficient very frequently found in biological equations. Unfortunately, this is common notation, and we will not deal with viscosity too often in presentations related to bioprocessing. In equations for calculating head losses in pipes, diameter is the I.D. of the pipe, V is the fluid velocity, rho is the fluid density, and µ is the viscosity of the fluid. Reynolds number has been so valuable for dealing with flow in pipes that analogous numbers are desirable for other flow situations such as mixing in tanks and transfer from gas bubbles. The same properties used in pipes will give a dimensionless number in another system if the units are consistent. The problem is assigning which property, and the decisions can seem arbitrary. The logical choices for a Reynolds number for a rising bubble are bubble diameter, relative velocity of the bubble versus the fluid, fluid density, and fluid viscosity.

Another Reynolds number, that for a stirred tank, is less obvious. The fluid properties seem to be good choices for rho and µ, but there are problems in selecting D and V. The impeller diameter is used for D even though an impeller also has height and usually several blades that must affect mixing or else engineers would not experiment with impeller designs. In the quest for something to use for V, the revolutions per second of the impeller is chosen even though it not a velocity. There is no length dimension in this term, so the dimensionless number is created by using the impeller diameter once more. The mixing Reynolds number is:

Despite its flimsy bases, this illogical Reynolds number is highly useful and correlates quite well when comparing systems that employ roughly similar impellers.

Scale up of mixing uses a correlation of Reynolds number with Power number defined as:

This equation is notable because the exponents are unusually large; Power number is very strongly affected by rpm and by impeller diameter. A correlation of Power number and Reynolds number is shown in the next figure. There are families of curves for different impellers (propellers, turbine impellers, etc.) and baffle configurations. If you use the same type of impellers in the large and small tanks, the graph is useful for scale up. Experiments are performed in one tank to find the Reynolds number that gives the desired process results. Usually the same Reynolds number will give about the same results in the other tank. The corresponding Power number is read from the graph. Substituting fluid properties and reasonable values for rpm leave power input as the only unknown in the equation. Of course, the power required per unit volume is the key to selecting the motor for the mixer.

Example of calculation.

While on sabbatical leave at ESB, Porto 1996