In the next several sections, we will cover the major dimensionless numbers used in fermenter agitator design, and then we will show how some of them are used.
Reynolds Number
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:
Caution: do not use the above expression without using the prescribed units!
Power Number
Power number is conceptually the ratio of power draw to impeller parameters, speed, and density. It is defined as:
The main use of power number is to calculate power draw. It is a function of impeller type, Reynolds number, and various geometric factors. When using SI units, the power will be expressed in watts. If one chooses to use inches, rpm, Hp, and specific gravity, we can insert the conversion factor and rearrange for power:
(3.4)
Same caution as for Eq. (3.2)
Note that power draw is quite sensitive to both impeller diameter and shaft speed.
Pumping Number
Pumping number is the ratio of impeller discharge rate (aka primary pumping capacity) to the cube of its diameter and the shaft speed:
It is used to calculate the flow created by the impeller, which can be used to determine a characteristic velocity in the tank. It is a function of impeller type, Reynolds number, and geometric parameters.
The units used for N and D determine the resultant units for Q. For example, if D is in feet, and N is in rpm, Q will be expressed as cubic feet per minute. Likewise, if D is in m and N is in revolutions per second, Q will be in cubic meters per second.
There are many different ways to measure impeller pumping, and they do not all give the same results. The most widely accepted methods define a discharge area around the impeller and measure flow through it, usually by use of either laser Doppler or particle image velocimeters. It is not within the scope of this book to discuss such methods. The above methods measure what is commonly called primary impeller flow or discharge. There can also be entrained flow, which can be several times as high as the primary flow. Any claimed impeller pumping capacity should state whether it is primary flow or total flow.
Dimensionless Blend Time
Dimensionless blend time is the product of blend time (defined as the time to reach some degree of concentration variance reduction after an assigned starting time) times the shaft speed. In other words:
(3.6)
This group is used to determine the blend time to some degree of attenuation of concentration differences. It is the product of blend time and shaft speed. Essentially, it is how many revolutions of the impeller are required to achieve a certain degree of blending. It is a function of Reynolds number, impeller type, and geometric factors. Rarely is blend time a limiting factor in fermenters.
Sometimes, people will add a factor of (D/T)α to the right‐hand side of the equation to correct for geometric effects. The value of α depends on the impeller type, but is usually about 2.3 for pitched blade and straight blade turbines, and about 1.73 for propellers and hydrofoils.
Aeration Number
Aeration number, also called gas flow number, is the ratio of actual gas flow rate at the impeller (corrected for absolute pressure and temperature) divided by the impeller diameter cubed and the shaft speed:
(3.7)
This group is used for power draw calculations in the gassed condition, along with other dimensionless groups. It can be thought of, in a way, as being proportional to the ratio of gas flow rate to the impeller pumping capacity.
Gassing Factor
When gas is introduced into an impeller or is present in the tank, it affects the impeller power draw. Usually, it reduces the power draw, but under some circumstances, it may increase it. The ratio of gassed power to ungassed power is called the gassing factor, and it does not usually have a special symbol for it. Instead, the ratio is simply expressed as a ratio: Pg/Pu.
It is a function of impeller type, Reynolds number, Aeration number, Froude number, and geometric factors.
Nusselt Number
The