Читать книгу Agitator Design for Gas-Liquid Fermenters and Bioreactors - Gregory T. Benz - Страница 52

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 3.4 Typical power number curve.

Figure 3.5 Typical pumping number curve.

(3.14)

The range between laminar and turbulent is called the transition range. It is a very broad range. That is because turbulence commences first near the impeller blade tips somewhere in the region of N_Re = 10, but does not reach all of the far corners of the tank until N_Re > 20 000. The power number gradually levels out in this range. For some impellers, it goes through a minimum in the transition range.

Figure 3.5 represents a typical pumping number curve, as a function of Reynolds number and having D/T as a parameter.

In laminar flow, the pumping number becomes a constant. This means that under such conditions, the impeller pumping is independent of viscosity, though the power draw is directly proportional to viscosity.

In turbulent flow, the pumping number again becomes constant at a given D/T, though it is much higher than the laminar pumping number. As D/T increases, the pumping number generally decreases. This is because recirculating flow within the tank creates a velocity opposite the impeller discharge, impeding its flow. Nonetheless, larger impellers pump more for a given power input than smaller ones.

When vendors quote a pumping number, it is usually in turbulent flow and at a D/T of 1/3 and a C/T of 1/3. Many consider this geometry to be “standard”, but there is no fundamental reason to adhere to this geometry when designing agitation equipment.

Figure 3.6 represents a typical generic blend time curve.

We can observe that blend time becomes constant in both laminar and turbulent flow. However, the laminar dimensionless blend time is often several orders of magnitude greater than the turbulent blend time. Specialized impellers have been developed for laminar flow mixing. We need not delve into these in this book, as fermenters never operate in the laminar range, because it is basically impossible to both incorporate gas into highly viscous liquids and have the gas exit in a reasonable amount of time after the gas is depleted.

Some versions of the dimensionless blend time curve incorporate the D/T effect into the Y‐axis expression, as stated under the dimensionless blend time definition given earlier.

Figure 3.7 depicts gassing factor as a function of Aeration number for three different impeller types.

Gassing factor depends on Aeration number, Froude number, D/T, Reynolds number, and impeller type. Therefore, it is impossible to show the entire relationship on a simple two dimensional graph. Figure 3.7 is based on turbulent flow, a D/T of 1/3, and a Froude number of 0.5.

We can readily see that the traditional impeller for gas dispersion, the Rushton (or disk) turbine, has a rather severe drop in power at high gas flow rates. As we will see later, this is a problem for maintaining maximum mass transfer. That is why it has been pretty much replaced by modern, deeply concave impellers such as the BT‐6 by Chemineer, the Phasejet by Ekato, and several others that have less effect of gas flow on power draw.

Figure 3.6 Dimensionless blend time.

Figure 3.7 Gassing factors.

We will cover other dimensionless parameters, such as for heat transfer, as needed in subsequent chapters. For now, we will show some example calculations.