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The Reynolds Number is the most widely used dimensionless number in fluid agitation. Many other dimensionless numbers are functions of it, as we will see in many subsequent chapters. Conceptually, the Reynolds number represents a ratio of inertial forces to viscous forces. When the Reynolds number is high, inertial forces dominate. This is the turbulent flow regime. When the Reynolds number is low, viscous forces dominate. This is the laminar regime. Intermediate Reynolds numbers constitute the transition flow regime and may exhibit attributes of both turbulent and laminar flow. In fact, in an agitated tank, there can be regions of laminar flow and regions of turbulent flow in the same tank when operating in the transition flow range of Reynolds numbers.

Mathematically, the Reynolds number is the product of a reference dimension times a reference velocity times the fluid density, divided by the fluid viscosity. The chosen reference dimension and velocity depend on the system being studied. For example, for a pipe, the typical reference dimension is the pipe inside diameter, and the reference velocity is the bulk velocity in the pipe.

For an agitated tank, it is customary to use the impeller diameter, D, as the reference length dimension. Likewise, it is customary to use a form of the impeller tip speed, πND, as the reference velocity. However, to avoid building a constant in a dimensionless number, π is dropped, so ND is used for velocity. The resulting expression is as follows:

(3.1)

Because all units must cancel, it is best to use a consistent set of units. SI (Systeme Internationale) units work well. Normally, this is no problem, except for viscosity. Most of the time, viscosity is stated as cP. However, to use SI units, viscosity should be in units of kg/m‐s, also known as Pa‐s. Fortunately, the conversion is simple. 1 cP = 1 mPa‐s = 0.001 Pa‐s = 0.001 kg/m‐s. English units become very peculiar. If the lengths are in feet and the density is in lb/ft³, with time in seconds, the viscosity unit to use is pound mass/foot‐second. The conversion from cP is 1 cP = 6.72 E‐4 pound mass/foot‐second. (I have yet to see a viscometer that reads in units of pound mass/foot‐second.)

For those who prefer to use inches, rpm, specific gravity, and cP, we can build in a conversion factor using those units:

(3.2)

Caution: do not use the above expression without using the prescribed units!