Power consumption evaluation

Power consumption evaluation in CSTRs is crucial to evaluate operation costs. In particular, the power required to maintain the motion when steady state conditions are achieved is of interest. In this work, the power consumption is calculated during the entire simulation and is used to assess convergence to steady state. A typical trend for power consumption is shown in Figure A.4. At each time step, power consumption is obtained as the torque on the impeller blades and shaft times the angular velocity:

$$P=\int_I \omega {\bf r} \times d {\bf F}$$ (7)

where I is impeller surface and $d~F$ is the infinitesimal force, given by the sum of shear and pressure forces:

$$d~\bf {F}=({\bf {\sigma}} + p \bf {I})$$ (8)

This represents the flux of energy entering the vessel from the impeller. Contributions to torque include pressure and shear forces, with the first giving the main contribution. The energy entering the vessel is spent to increase the mean kinetic energy and to balance viscous and turbulent dissipation in the vessel. At steady state, the power is equal to dissipation. At steady state, power consumption could be also obtained by integration on the vessel volume of dissipated energy [Armenante and Chou, 1996]. This method is not used in this work since it is not reliable: a very accurate distribution for kinetic energy and dissipation rate should be available by calculations to obtain power consumption values that are comparable to experimental measurements. From the power consumption, the power number is obtained as

$$Ne=\frac{P}{\rho N^3 d^5}$$ (9)

For each CSTR configuration, power number is calculated for different operating conditions, synthetically identified by the value of the Reynolds number ( $Re=\rho N d^2/\mu$), and is compared against analytical curves that may be derived from theory (see Appendix B).