qmN  ¼ const: (13)

Since on scale-up, constant ¯eT leads to constant N, obtaining equal mixing time appears to be feasible. How- ever, because Po and D are large, N is limited by the practi- cal level of ¯eT that can be used economically and hence, times can be many orders of magnitude longer than in tur- bulent flow. The optimum helical ribbon configuration, which determines the constant in Eq. (13), is dependent on the precise geometry [23].

An approach, which offers even more geometrical possi- bilities and appears to be particularly flexible, combines two co-axial stirrers on the same shaft. One of these is of large diameter rotating slowly, typical of those used for high vis- cosity blending, and the other of small diameter rotates much faster, either co-rotating (which gives the most ener- gy-efficient blending) or counter-rotating (which is most effective where a particulate phase needs to be broken up and dispersed). A wide variety of combinations of this type have recently been studied in depth  both experimentally and using advanced CFD techniques [30, 31]. Again, such complex co-axial stirrer systems are expensive.

A particularly difficult blending problem arises with shear thinning fluids exhibiting a yield stress ty. These fluids may be found wherever structure can develop, either in the fluid (food viscosifiers such as soluble Xanthan gum) or due to a second phase (high-concentration fine particle or mycelial suspensions, small drop size emulsions, etc). In such fluids,

a cavern develops [32] (a region of motion near the impeller while outside it, the fluid is stagnant (Fig. 7)). Generally, the Re determined via Eq. (7) shows that flow is transitional or turbulent. Thus, the STR should be baffled with the cavern size determined by [32] (another influencial paper [11])

ðDC=DÞ3  ¼ f1=p2ðHC=DC  þ 0:33ÞgPoðrL N2D2=ty Þ       (14)

where DC and HC are the diameter and height of the cavern (assumed cylindrical, HC /DC = ~0.4 to ~0.6) [30]. Eq. (14) suggests that dual impellers, with high Po and large D/T, are much more energy efficient in achieving bulk motion throughout a vessel and that scale-up at constant tip speed is an appropriate rule. If, with the cavern filling the vessel on small scale, scale-up is done at constant W kg–1 as it often is (see below), full vessel motion is obtained.

Figure 6. Schematic of the Sumitomo Maxblend' impeller (modified from [27]).

Figure 7.  Streak  photo  of  a  cavern  in  a  0.17 %  Carbopol    solution,

ty = ~5 Pa.

4.1.2 Chemical Reactions

Because reaction times are scale-independent, can vary by many orders of magnitude from milliseconds or less to hours and each reaction scheme is unique, devising experimental protocols to help the practitioner design STRs is most impor- tant. When the mixing time is fast compared to the reaction time, agitation intensity is unimportant and the overall reac- tion is entirely dominated by the kinetics. On the other hand, since the bulk mixing time increases with scale, in some cases, the time is comparatively short at the small scale, but becomes longer at the large. This phenomenon is called mac- romixing. For such cases, very few experimental or modeling studies have been undertaken. Two experimental studies backed by some theoretical concepts have shown that scale- up of semi-batch competitive reactions requires constant agitation speed N. Practically, this is not feasible and any experimental scale-down protocol should be undertaken in geometrically similar equipment and must use agitation speeds at the bench and pilot scale similar to those giving viable ¯eT values at the commercial scale [33, 34].

If the reaction time is very short, then it is the mixing