Agitator Design for Gas-Liquid Fermenters and Bioreactors. Gregory T. Benz

number is the ratio of the convective heat transfer coefficient times a reference length divided by the thermal conductivity of the fluid. The reference length dimension depends on the heat exchange surface. For a vessel jacket, that will normally be the tank diameter. For tubes inside the tank, it would normally be the tube OD. Expressed in terms of tank diameter, we have:

      (3.8)

      It is used in heat transfer calculations, to determine the convective coefficient, h. It is a function of impeller type, Reynolds number, Prandtl number, and various geometric factors, as well as the ratio of local viscosity to bulk viscosity.

Froude Number

      Froude number is the ratio of a reference rotational speed times a reference velocity divided by the local gravitational acceleration. For similar reasons that we used for Reynolds number, we use N for rotational speed and ND for velocity, resulting in:

      (3.9)

      Note that g is not a constant; it is the local gravitational acceleration. While it is almost uniform on the Earth’s surface, it will be quite different on other planets. So the same impeller operating at the same speed would have different Froude numbers on Earth, the moon, Mars, and Jupiter. This will be important if we ever build fermenters on another planet.

      Conceptually, the Froude number is the ratio of inertial forces to gravitational forces. High Froude numbers mean inertia dominates. This is associated with a choppy surface and vortex formation. Low Froude numbers mean gravity dominates, which is associated with a quiet, flat surface. Froude number, in combination with Reynolds number, impeller type, baffle number, and various geometric factors, can be used to predict mean vortex depth.

      In gas–liquid contacting, the impeller gassing factor is a function of Froude number, impeller type, Reynolds number, Aeration number, and geometry factors. That is its principal use for bioreactor design. It is also used in most impeller flooding correlations.

Prandtl Number

      Prandtl number is simply a physical property group used in heat transfer correlations:

      (3.10)

      The Nusselt number is a function of the Prandtl number. Care must be taken to make sure all units cancel. The most common units for the fluid properties, especially viscosity, do not cancel. SI units work well. If common English units are used, the viscosity must be converted to pounds mass/foot‐hour. The conversion is 1 cP = 2.419 lb/ft‐h.

Geometric Ratios

      Relative size and placement of impellers in a tank affects power draw, blend time, pumping capacity, solids suspension capability, vortex formation and maybe a few other things. Some parameters are more sensitive to geometry than others. The dimensionless geometric ratios used in calculations include D/T, Z/T, C/T, S/D, O/T, and W/D. As long as the same units are used for numerator and denominator, it does not matter what system of units is used.

Baffle Number

      Although this could be lumped in under geometric ratios, it affects things in different and important ways, so I decided to mention it separately. Essentially, it represents total baffle width (calculated normal to the tank wall) divided by tank diameter:

      (3.11)

      Power draw and vortex formation depend on baffle number. At a baffle number of zero, the tank is called unbaffled and power draw is at a minimum. Under such a condition, in turbulent flow, there is a lot of swirl, with mostly tangential motion rather than axial or radial. Gas dispersion is essentially impossible to do under such a condition. As baffle number increases, power draw increases to a maximum and then falls off. In the USA, “standard” baffling is 4 baffles at 90° to the tank wall, each one being 1/12 of the tank diameter, resulting in a baffle number of 0.33. In Europe, it is more common to use a baffle width of 1/10 of the tank diameter, giving a baffle number of 0.4. The power draw is pretty constant over this range and is basically at the maximum. D/T and impeller type also interact with baffle number.

Dimensionless Hydraulic Force

When an impeller operates in turbulent flow, the loads on each blade are fluctuating about a mean. To better visualize this, imagine riding along in a car with your hand outside the window. You will note that your hand is buffeted about, with a highly variable force. This is due to turbulence, involving various shedding of vortices, etc. The same thing happens to an impeller in turbulent flow. The load on each blade varies with time, and the loads on each blade are not synchronous with each other. The result is that there will be a fluctuating net side load on the impeller, creating a bending load on the shaft. This load, sometimes called an imbalanced hydraulic force, is not due to mechanical imbalance or any lack of manufacturing precision. It is entirely due to the nature of turbulence.

      The dimensionless hydraulic force equals the product of hydraulic force times impeller diameter divided by impeller torque:

      (3.12)

      Normally, the peak value of this ratio is correlated, so that a peak value of hydraulic force may be predicted and used in shaft and impeller design. This group is a function of Reynolds number and impeller type, as well as geometric ratios, aeration number, and Froude number, and direction of pumping for axial flow impellers.

Thrust Number

      This relates impeller thrust to impeller parameters and density:

      (3.13)

      It is primarily used to determine impeller thrust for mechanical purposes. It also is used to predict cavern size for shear‐thinning fluids, as will be discussed in Chapter 12. Qualitatively, it is a constant in laminar and turbulent flow, but goes through a minimum in transition flow.

Typical Dimensionless Number Curves

Figure 3.4 represents a typical log/log plot of Power number as a function of Reynolds number, with D/T as a parameter. (The D/T effect is primarily noticeable on axial impellers. Radial impellers show much less dependency on D/T, as we will see in Chapter 5.)

      Examining this curve, we can observe several things. One is that the power number becomes constant for a given D/T under turbulent conditions, i.e. at high Reynolds numbers (typically, above 20 000). When a manufacturer of an impeller states it power number, it is normally the turbulent power number at a D/T of 1/3 and a C/T of 1/3. Axial impellers will tend to have a decreasing power number at increasing D/T.

Also note that the curve becomes a 45° angle at laminar flow conditions (typically, N_Re <10). That means the product of Reynolds number and Power number is constant in that range. A simple derivation reveals that in the laminar range,

Figure