Introduction & Context

The calculation of power consumption for high-viscosity mixing is a critical task in process engineering, particularly for batch operations involving non-Newtonian fluids such as dough, polymers, or heavy pastes. In these systems, the unit operation is typically performed in a sigma-blade kneader where the flow regime is dominated by viscous shear rather than inertial forces.

Unlike low-viscosity mixing, where turbulent flow patterns prevail, high-viscosity mixing operates in the laminar regime (Re < 10). In this state, the power required is directly proportional to the fluid viscosity and the square of the impeller speed. Accurate estimation is essential for sizing motors and gearboxes to prevent mechanical failure during the high-torque peaks associated with material development or phase changes.

Methodology & Formulas

The following methodology translates the physical requirements of the mixing process into actionable engineering equations. All calculations assume a laminar flow regime where viscous drag is the primary resistance to impeller motion.

First, the rotational speed is converted from revolutions per minute to revolutions per second:

\[ N = \frac{N_{\text{rpm}}}{60} \]

The average shear rate (\(\dot{\gamma}\)) is determined by the impeller geometry constant (\(k_{s}\)) and the rotational speed:

\[ \dot{\gamma} = k_{s} \cdot N \]

The Reynolds number (Re) is calculated to confirm the validity of the laminar model:

\[ Re = \frac{\rho \cdot N \cdot D^{2}}{\mu_{\text{app}}} \]

The steady‑state power consumption (P) in kilowatts is derived from the laminar geometry constant (K_p), apparent viscosity, rotational speed, and impeller diameter, as explained in our laminar mixing power calculation guide.

\[ P = \frac{K_{p} \cdot \mu_{\text{app}} \cdot N^{2} \cdot D^{3}}{1000} \]

The steady-state torque (T) is calculated based on the power output and rotational speed:

\[ T = \frac{P \cdot 1000}{2 \pi N} \]

Finally, to account for the viscoelastic nature of materials like dough, an overload factor (\(F_{\text{overload}}\)) is applied to determine the peak torque requirement for motor and gearbox selection:

\[ T_{\text{peak}} = F_{\text{overload}} \cdot T \]

Parameter	Condition / Limit	Engineering Significance
Reynolds Number (Re)	Re < 10	Required for laminar flow model validity.
Overload Factor (\(F_{\text{overload}}\))	2.0 – 4.0	Accounts for peak torque during material development.
Geometry Constant (\(K_{p}\))	100 – 250	Specific to sigma blade geometry; not interchangeable.

For high‑viscosity mixing in the laminar regime, the power number (\(N_{p}\)) is not a constant but varies inversely with the Reynolds number. Follow these steps to obtain it from the provided methodology:

Determine the apparent viscosity (\(\mu_{\text{app}}\)) of the non‑Newtonian fluid at the impeller shear rate.
Calculate the laminar Reynolds number: \(Re = \frac{\rho N D^{2}}{\mu_{\text{app}}}\).
Using the geometry constant \(K_{p}\) for your sigma‑blade kneader (100–250), the power number is \(N_{p} = \frac{K_{p}}{Re}\).
Once \(N_{p}\) is known, total power can be found from \(P = N_{p} \, \rho \, N^{3} \, D^{5}\), or directly from the laminar power equation \(P = \frac{K_{p} \, \mu_{\text{app}} \, N^{2} \, D^{3}}{1000}\) (kW).

In high viscosity applications, the proximity of the impeller to the vessel floor significantly influences the flow pattern and torque requirements. Key considerations include:

Reduced clearance often leads to higher shear rates near the bottom head, increasing the local power draw.
Excessive clearance can result in stagnant zones, which may cause uneven torque distribution on the shaft.
Optimizing the clearance is essential to prevent motor overload while ensuring complete turnover of the viscous medium.

Fluctuations in power consumption are common during the startup phase due to several physical factors:

Thermal gradients within the vessel can cause localized viscosity variations.
The transition from a solid-like state to a flowable state requires overcoming the yield stress of the material.
Mechanical resistance is highest when the impeller is fully submerged in a high-viscosity, non-homogenous mass, leading to transient torque spikes.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle