for a given impeller size in turbulent flow, flow is directly proportional to shaft speed but power is proportional to shaft speed cubed. Therefore, flow/power diminishes as shaft speed increases. It approaches infinity as shaft speed approaches zero. Likewise, in laminar flow, power is proportional to shaft speed squared. So, there is no attribute of a flow/power ratio for any particular impeller. It depends on the specific application.
Table 3.1 Metzner–Otto constants.
| Impeller type | KM |
|---|---|
| Propeller | 10 |
| Low solidity hydrofoil | 10 |
| High solidity hydrofoil | 11 |
| Pitched blade turbine | 11 |
| Radial turbines | 11.5 |
| Chaos (reversing pitch) turbines | 40 |
So, if we were to want to primarily create flow and minimize power, how would we do it? Let’s start with impeller design, based on the flow vs. head relationship. We want to minimize the head term. We will look at several design attributes and the limitations imposed on them.
One attribute is the number of blades. Increasing the number of blades increases the flow, but it also increases the head, as the wake of each blade creates some degree of additivity for the net pressure, as the discharge flow will be moving at a higher velocity as we add blades. So, from a flow‐for‐power maximization standpoint, the optimum impeller would have only one blade. But there is a reason why we seldom see one‐bladed impellers: we prefer not to break the shaft. The hydraulic imbalance would be huge! Hydraulic forces are still quite large for two‐bladed impellers, so they are quite uncommon except in transition flow regimes, and even then, there will usually be several impellers, with the orientation staggered to minimize the bending moment on the shaft. Most “high flow” or hydrofoil impellers have three blades, because this is the minimum that has reasonable hydraulic stability. Using more than three blades improves hydraulic stability somewhat, but at the price of requiring more power for the flow produced.
Another attribute is blade chord angle from the horizontal plane. If the blades were infinitely thin, a zero degree angle would produce no flow and draw no power, save that created by skin drag on the blade surface. As the angle increases, flow increases and so does power, so a very shallow angle will create the most flow for the power. However, the absolute flow produced may be too small for a given diameter and speed. So, either the diameter would have to be increased or the shaft speed would have to be increased. But we are confined in diameter by the tank, so we cannot always increase the diameter; a D/T value of greater than 1 does not work very well. So, we would have to increase shaft speed in some cases in order to use a very shallow angle. But increasing shaft speed also increases head and power, negating the desired power savings from a low angle. The end result is that hydrofoil impellers available in the marketplace generally have chord angles between 15 and 30°, with 20–25° being most common. The power number range is about 0.2–0.4, with 0.3 being most common. The chord angle must never be greater than the stall angle, which is about 37° in water, or boundary layer separation will occur, greatly increasing power draw. For example, a 45° pitched blade turbine will require almost three times the power as a hydrofoil to create the same flow at the same shaft speed.
A third attribute is blade width or a related measure: impeller solidity (will be defined more in Chapter 5). As blade width increases, so does pumping, but power also increases due to higher discharge pressure. So, most high solidity impellers are used for applications that are not just about creating flow.
One of the most important attributes is shaft speed, and this applies to any impeller type. A small impeller rotating at a high speed can create the same flow as a larger impeller rotating at a slower shaft speed. The smaller impeller has a smaller discharge area; hence, its mean discharge velocity will be higher, and thus, the discharge head will be higher. Therefore, it requires more power to produce the same flow. For flow controlled applications, significant power savings can be achieved by using larger impellers at slower shaft speed. Sometimes, the capital cost is higher, but the payback time is usually a matter of months.
Of course, not all agitator design problems are flow controlled. The main application in this book, gas–liquid dispersion, is closer to being turbulent eddy dissipation controlled than anything else. So, in most cases, the power requirements will be governing, and impeller design will focus on other things besides flow.
Summary of Chapter
This chapter has presented basic material that all must know and understand in order to continue into the details required of fermenter design. It can be referred to as needed while reading the rest of the book.
List of Symbols
COff bottom clearanceCPHeat capacity at constant pressureDImpeller diameter, edge to edgeDSSwept diameter, or point‐to‐point diagonal diameterdv/dxShear rateFdDimensionless hydraulic forceFhHydraulic forceFTImpeller axial thrustgGravitational accelerationhConvective heat transfer coefficientkThermal conductivityKMMetzner–Otto constantMViscosity coefficientNShaft speedNAAeration number; also called gas flow numberNBDimensionless blend timeNPPower numberNQFlow or pumping numberNReReynolds numberNTThrust NumbernPower law exponentnbbaffle numberOImpeller offset from tank center linePPower drawPgGassed power drawPuUngassed power drawQImpeller discharge rate; also called pumping capacityQgActual gas flow at the impellerSImpeller spacingTTank DiameterTqImpeller torquetbBlade thicknessWActual blade widthWPProjected blade widthwbBaffle width, calculated normal to tank wallvLiquid velocityZLiquid height
Greek letters
θBlade angle from a horizontal planeτBlend timeμDynamic viscosityμaApparent viscosityρFluid densityσsShear stressσyYield stressγShear rate; dv/dx
References
Because the descriptions and use of dimensionless numbers in this chapter are found in many references, we chose not to do a line by line reference. Instead, one can find an introduction to these concepts in the references below, among others.
1 1 Dickey, D.D. and Fenic, J.G. (1976). Dimensional analysis for fluid agitation systems. Chemical Engineering Magazine: 7–13.
2 2 Paul, E. et al. (eds.) (2004). Handbook of Industrial Mixing. Various authors. Wiley Interscience (Look up by the name of the dimensionless number).
4 Agitator Behavior under Gassed Conditions
When an agitator is operated under gassed conditions (whether the gas is added or evolving from the process), it behaves differently from operating in liquid only. The presence of gas, in addition to possibly influencing mass transfer when the gas and liquid are not in equilibrium, can create several important effects. Among those are flooding of the impeller versus dispersing the gas, power draw effects, entrained gas, or holdup in the liquid and mechanical effects. We will go through each of these in turn, as well as the issue of what happens when there is a variable gas flowrate.
The above behaviors are mainly correlated by three dimensionless number groups: Reynolds number, Froude number, and Gassing factor. A fourth group, known as the dimensionless hydraulic force, is used for mechanical design and will be used in examples in Chapter 16.
Flooding