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.**