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on rheology. Indeed, many books have been written on the subject. However, due to the extreme difficulty of dispersing gases into viscous liquids, highly viscous liquids are not used or produced in industrially significant fermenters. (I know, some will object, saying they have viscous fermentations. But it is a matter of perspective. While a Gellan gum fermentation may seem viscous to someone operating a fermenter, it doesn’t seem so viscous to those in the polymer industry. But they do not disperse gas into plastic or synthetic rubber melts. The difference is that a very high fermentation viscosity may be 5000 cP at the impeller, whereas in the polymer industry, they may see hundreds of thousands or even several million cP.)

      Therefore, we will focus on a few simple models that do an adequate job of describing the behavior of fermentation broths. We will look at shear stress versus shear rate and assume that normal (i.e. at right angles to the shear plane) stresses, such as found in viscoelastic fluids, do not apply. We will also confine ourselves to time‐independent models, because the fluids in fermenters are always under shear and in a quasi‐steady state.

Newtonian Model

      Newtonian fluids are those which have a shear stress that is directly proportional to shear rate (also called velocity gradient). Mathematically, this can be represented as:

      (3.19)

      where μ, the coefficient on shear rate, is also known as viscosity. This can be arranged to:

      (3.20)

      Most low viscosity fluids (<50 cP) are Newtonian. Intermediate and even some high viscosity fluids can be Newtonian if they are true solutions, especially if they are solutions of small molecules. For example, honey and corn syrups are often Newtonian even at 30 000 cP. Some silicone solutions are Newtonian even at 20 000 000 cP, but this is quite rare. Most suspensions of more than 5–10% solids exhibit at least some non‐Newtonian behavior. High solids slurries can be very non‐Newtonian.

      Low‐solids fermenter broths operating in a water‐like fluid are usually Newtonian. High solids broths may not be, and those containing both an aqueous and an oil phase usually are not. But the overall viscosity for most fermenter broths is less than 1000 cP. The exception is some filamentous broths and those containing soluble gums, such as Xanthan. Chapter 12 will cover these in more detail.

Pseudoplastic or Shear Thinning, Model (Aka Power Law Fluid)

      These fluids are called shear thinning because the slope of the shear stress vs. shear rate curve decreases as the shear rate increases. The behavior may be described by a power law model:

      (3.21)

      The exponent n is called the power law exponent. When it has a value of 1, the equation reverts to Newtonian. When it is less than 1, it is shear thinning. While it can be greater than 1, which would mean a shear thickening behavior, I have never seen a shear thickening fermenter broth. The apparent viscosity can be expressed by dividing by the shear rate on both sides of the equation:

      (3.22)

      where M is the viscosity coefficient, or the viscosity at a shear rate of 1 (usually in units of 1/s).

      Many fermenter broths exhibit this behavior at high cell density loading, or if there is a significant solids content in the broth.

Bingham Plastic

      These fluids exhibit a yield stress. That is, until the fluid is exposed to a certain minimum shear stress, it will not move. Examples include some high solids slurries, toothpaste, and any fluid that can hold some of its shape after being disturbed. Above the yield stress, Bingham fluids exhibit a constant slope. Mathematically:

      (3.23)

      and

      (3.24)

Herschel–Bulkley

These fluids combine a yield stress with shear‐thinning behavior above the yield stress:

      (3.25)

      and

      (3.26)

      One may recognize that all three previous models are special cases of the Herschel–Bulkley model, making this the most versatile model commonly in use.

These four models are described visually in Figures 3.8 and 3.9

Figure 3.8 shear stress curve.

Figure 3.9 viscosity curve.

Impeller Apparent Viscosity

      In laminar and transition flow, the apparent viscosity at the impeller can be evaluated using the average shear rate:

      (3.27)

where K_M is the Metzner–Otto constant. There are some disagreements about what that constant is for various impellers. Table 3.1 is an average of some of the published numbers.

      In turbulent flow, the shear rates may be higher, so the apparent viscosity may be less for most fluids.

A Bit of Impeller Physics

      Though we will get into impeller details more in Chapter 5, since this chapter is on the fundamentals, we will describe the basic physics of impeller behavior.

      Some have said that some impellers invest their power more as flow, and others invest it as shear. As we will see below, this is an erroneous description.

      Basically, a rotating impeller moves fluid. Agitation and gas dispersion are side effects of how much fluid is moved and how it moves it. But the macro relationship is very simple: power = flow times head, same as for pumps. Theoretical velocity head is proportional to tip speed squared or discharge velocity squared, though actual head produced depends on many other factors. Head is related to shear as well, as maximum shear depends on tip speed. So, power is more like a product of flow times shear squared, rather than a sum of flow and shear.

Some talk about the ratio of flow to power as an impeller

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