The effectiveness of mixing within a batch reactor or other stirred tank is determined mainly by the stirring power. The power Φ is the rate at which work is done on the liquid by the moving impeller. It is affected by the angular velocity ω and diameter d of the impeller, ρ, μ, and sometimes g. A baffled tank with a turbine impeller is shown in Fig. P1.10. Lengths less critical than d, but still influential, include the width wi of the impeller blades, the width wb of the baffles that project in from the wall, the height h of the impeller above the bottom, the liquid depth H, and the tank diameter D. Geometric parameters that are ignored here for simplicity include the number and pitch of the impeller blades and the number of baffles.

(a) Considering the complete list of quantities (Φ, ω, d, ρ, μ, g, w_i, w_b, h, H, D), what can you conclude about Φ from dimensional analysis? It is helpful to arrange the groups so that Φ, μ, and g each appear in only one group. Show that

where Po is the power number.

(b) Baffles improve mixing by preventing a purely swirling motion of the liquid. They also tend to flatten the gas–liquid interface and thereby minimize the influence of gravity on the flow pattern. If g is unimportant and only geometrically similar tanks are to be considered, how can the dimensional analysis be simplified?

(c) Bates et al. (1963) found that Po → A/Re_i for small Re_i and Po → B for large Re_i, where Re_i is the impeller Reynolds number and A and B are each constant for a given impeller type. Could you have anticipated the forms of these relationships?

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H d W₁ Wb h D Figure P1.10 Geometric parameters for a stirred cylindrical tank with baffles.