(Re > ~104),

Po = const. (6)

so that P a rL, independent of m. There is also a large tran- sition  region  (~10 < Re < ~104).  The  two  factors   which

Figure 3. Plot of Power number versus Reynolds number (power characteristic); logarithmic co-ordinates; Rushton turbine. (a) Po = Kp/Re, slope = –1; (b) Po = ~5 = const. (from [12] with permission).

mainly determine in practice the Reynolds number are the viscosity of the liquid phase and though often overlooked, the scale of the system. The power characteristic in Fig. 3 is

non-Newtonian characteristics, some of which will be intro- duced later.

Historically, the other major early and more theoretical contribution according to [11] was on turbulent flow by a Russian polymath, Kolmogorov [15]. Two papers were pub- lished in Russian in 1941 but were translated into English much later in 1991 [15]. However, his ideas were being used much earlier, largely due to the book Physicochemical Hydrodynamics by Levich, published in 1962 in its English translation [16]. Kolmogorov postulated that though large scale turbulent structures are anisotropic, the smaller eddy scales are isotropic and due to inertial forces, there is a con- served energy cascade range of sizes (the inertial subrange). Below this range, turbulent energy is converted to heat due to viscosity (the viscous subrange). The velocity of the ed- dies in the inertial subrange only depend on the local specif-

ic energy dissipation rate eT and the eddy size l, whilst in the viscous subrange, the kinematic viscosity n (= m/rL), plays a role. The eddy size at which this transition occurs is called the Kolmogorov microscale lK, where

1=4

for a Rushton turbine (Fig. 4), which is a radial flow impel-

lK ¼ ðn3=eTÞ

(8)

ler. This type gives a minimum Po number in the transi- tional region [12]. With axial flow impellers, as Re de- creases, there is a  slow  increase  in  slope  from  zero  at Re > ~104 to –1 at Re < ~10 [13].

Figure 4. Radial flow impellers. (a) Rushton turbine with six flat blades mounted vertically on a disc (Po = ~5.0); (b) Chemineer CD6 (Po = ~2.3 based on the swept diameter) (from [12] with permission).

With Newtonian fluids, the viscosity is constant, but there is an enormous variety of fluids (non-Newtonian fluids) where that is not so. The most common are shear-thinning, i.e., the faster the stirring rate, the lower the viscosity. The viscosity depends on the shear rate in the fluid developed by the stirrer, which varies spatially throughout the vessel. In 1957, Metzner and Otto [14], labelled as outstanding contribution in [11], showed that

g_ a  ¼ kS N (7)

where g_ a is the average shear rate and kS depends on the agitator (typically ~13). Thus, ma can be determined from a rheogram, and hence, Re and Po. Though developed for the laminar regime, this approach has been used successfully for many applications outside that range. There are other

The mean specific energy dissipation rate ¯eT can be easily calculated since all the energy imparted into the stirred liq- uid from the impeller must be dissipated as heat so that

¯eT  ¼ P=rL V ¼ PoN3D5=V (9)

Somewhat later in 1966, Cutter [17], another outstanding paper [11], showed experimentally using a photographic technique that the flow in stirred tank reactors is far from homogeneous on the macroscale and that the local specific energy  dissipation  rate  near  the  Rushton  impeller    was

~70 ¯eT. Copious later studies have shown similar high val- ues. However, it has been shown that the assumption of local isotropy over a small but finite volume enables these ideas to be utilized extensively. It is the cornerstone of many theoretical developments associated with bubble, drop and floc sizes, damage to microorganisms, suspension of solids,