## Power Requirement

The general expression for power consumed by an impeller in an unaerated mixing vessel is given by (202):

where Np is the power number, which is a constant depending on the impeller type and size, and number of baffles present, NRe and NFr are the Reynolds and Froude numbers respectively, which describe the kinematic conditions.

The Power: Reynolds and Froude numbers are expressed as:

where Po is the power input during unaerated fermentation, D is the impeller diameter, N is the impeller speed, u is the viscosity, and p is the density of the liquid and g is the acceleration due to gravity.

The Reynolds number accounts for the effect of viscosity when viscous forces prevail, while the Froude number accounts for the force of gravity when it plays a part in the motion of the fluid.

The interrelation of impeller speed and diameter, fluid density and viscosity, and physical configuration of the tank and fittings; and consideration of the power necessary to move the impeller have resulted in the formulation of a Rushton plot (203) as shown in Figure 3.11. The Rushton plot is a logarithmic plot of power number (Np) against the Reynolds number (NRe).

For a particular fermentation system with constant impeller speed, fluid characteristics, and vessel geometry, the Reynolds number for the system, extrapolated to the Y axis in a Rushton Plot can be used to determine the value of Np. From Np, Pg/ Po, the ungassed power (Po) can be calculated.

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