# The Geometry of Industrial Mixing

The term we use to describe a “family” of impellers is geometric series. A geometric series of impellers is one where all dimensions are maintained at constant ratios.

Fig. 7 demonstrates this principle for the pitched blade turbine and the Rushton turbine.

All impellers in a geometric series, whether 6” diameter or 60″ diameter, will have the same performance characteristics.

We also define a set of “standard conditions” (Fig. 8) that are used for measuring impeller performance. This figure appears in many books and articles and is the source of the common misconception that the impeller must always be located one diameter off bottom.

For our purposes, it is sufficient that we be cognizant of some of the basic principles, the dimensionless group relationships, and how we can use them.

The characteristic quantities we use in fluid mixing are:

- Length = Impeller Diameter = D
- Time = Reciprocal of Rotational Speed = 1/N
- Mass = Liquid Density X Length
^{3}=*ρ*D^{3} - Velocity = Length
*I*Time = N D - Pressure = Velocity Head x Density = N
^{2}D^{2}*ρ*/g_{c}

Returning to our analogies to pumps, friction loss in pipe flow is related to a dimensionless group called the Reynolds Number. This is the product of the pipe diameter times the fluid velocity times the density, divided by the viscosity;

N_{Re }= d v *ρ*

* µ * * *

In mixing, we use the impeller diameter, D, the characteristic velocity, ND, along with the fluid properties:

N_{Re }= (D)(ND)* ρ* = ND^{2}* ρ*

* µ µ*

Dimensionless Groups: The dimensionless groups we will encounter in fluid mixing are summarized in Fig. 9.

- Reynolds Number is defined as the ratio of inertial to viscous forces.

- Froude Number is defined as the ratio of inertial to gravity forces.

- Power Number is defined as the ratio of imposed forces to inertial forces, and is a function of the Reynolds Number and the Froude Number. Where the liquid surface is essentially flat, as in baffled tanks, gravity forces are negligable and the Froude Number can be eliminated.

- Pumping Number is defined as the ratio of actual velocity to characteristic velocity. Here, the actual velocity is the volumetric flow rate from the impeller divided by the discharge area.

- Weber Number is defined as the ratio of inertial to surface tension forces. The Weber Number is encountered in liquid-liquid and gas-liquid dispersions

- Prandtl Number is defined as the ratio of momentum to thermal diffusivity. It is encountered in heat transfer along with the – –

- Nusselt Number, which is the ratio of the mixer-side film coefficient to the fluid thermal conductivity, with a characteristic dimension added. The equation relating the Nusselt Number to the Reynolds and Prandtl Numbers is also known as the Sieder-Tate Equation.

REYNOLDS NO., N_{Re} = = ND^{2}ρ/µ

FROUDE NO., N_{Fr} = = N^{2}D/g

POWER NO., N_{P} = = Pg_{c}/ρN^{3}D^{5} = f(N_{Re}, N_{Fr})

(if gravity force is negligible) = f(N_{re})

PUMPING NO., N_{Q} = = (Q/D^{2})/ND = Q/ND^{3}

WEBER NO., N_{We} = = N^{2}D^{3} ρ/σ

PRANDTL NO., N_{Pr} = = C_{ρ} µ/k

NUSSELT NO., N_{Nµ} = h D / K = f(N_{Re}, N_{Pr})

SCHMIDT NO., N_{Sc} = . = µ/ρ δAB

DIMENSIONLESS BLEND TIME = = θN

VISCOUS POWER NO., N_{P}N_{Re} = Pg_{c}/µN^{2}D^{3}

REYNOLDS NO., N_{Re} = = ND^{2}ρ/µ

FROUDE NO., N_{Fr} = = N^{2}D/g

POWER NO., N_{P} = = Pg_{c}/ρN^{3}D^{5} = f(N_{Re}, N_{Fr})

(if gravity force is negligible) = f(N_{re})

PUMPING NO., N_{Q} = = (Q/D^{2})/ND = Q/ND^{3}

WEBER NO., N_{We} = = N^{2}D^{3} ρ/σ

PRANDTL NO., N_{Pr} = = C_{ρ} µ/k

NUSSELT NO., N_{Nµ} = h D / K = f(N_{Re}, N_{Pr})

SCHMIDT NO., N_{Sc} = . = µ/ρ δAB

DIMENSIONLESS BLEND TIME = = θN

VISCOUS POWER NO., N_{P}N_{Re} = Pg_{c}/µN^{2}D^{3}

Fig. 9 Dimensionless Groups

- Schmidt Number is defined as the ratio of momentum to molecular diffusivity. Encountered in chemical reactions.

- Dimensionless Blend Time is defined as the ratio of actual time to characteristic time, and is sometimes used as a correlating parameter.
- Viscous Power Number represents the relationship at low Reynolds Numbers (laminar flow). This relationship allows us to use impeller power draw as a measure of endpoint in many polymerizations.

Impeller Power Response: Having defined geometric series, standard conditions, Reynolds Number and Power Number, we now consider actual experimental correlations of Power Number versus Reynolds Number for different geometric series of impellers.

These correlations all have the general shape predicted by theory (Fig. 10): a turbulent range where Power Number is constant; a transition range; a viscous range where Power Number has a linear inverse relationship to Reynolds Number with a slope of minus one.

Power Number, Pgc/ρN^{3}D^{5}

Power Number, Pgc/ρN^{3}D^{5}

Reynolds Number, D2Nρ/µ

Fig. 10 N_{p} vs. N_{Re}

Because these curves are based on standard conditions and true geometric series, we must usually apply correction factors to account for variations from these conditions. These corrections will consist of:

- Proximity Factors, which deal with impeller off-bottom distance, impeller spacing, impeller coverage, and number of impellers.

- Blade Factors, which deal with non-standard blade sizes. For convenience in manufacturing, the same blade width will often be used for a range of impeller diameters, usually with pitched blade or flat blade turbines where the blades are made from plate. For example, we use 6″ wide blades on 25″ through 34″ diameters.

Impeller Pumping Capacity: We know from theory that the Pumping Number will also relate to the Reynolds Number (Fig. 11), is constant in the turbulent zone, and decreases in the transition zone.

Pumping number Q/ND^{3}

Pumping number Q/ND^{3}

Reynolds Number, D^{2}Nρ/µ

Fig. 11: N_{Q} vs. N_{Re}

Turbulent Pumping Numbers have been determined experimentally for a large number of impellers. Direct velocity measurements become very difficult as viscosity increases, so the shape of the curves are based more on theory and experience than on experiment.

The Pumping Number relationship applies to impeller flow, which we define as the direct discharge from the impeller. The impeller discharge stream induces additional flow in the surrounding fluid so the total flow in the tank is equal to the sum of the impeller flow and the induced flow (Fig. 12).

Impeller Head or Shear: Returning to our analogies to pumps and pipe flow, the head produced by a pump is proportional to the velocity, squared.

H α v^{2}

This “velocity head” is dissipated as friction loss in the pipe and this energy loss is reflected as a pressure drop.

With mixing impellers, the head produced is also proportional to a velocity, squared;

H α N^{2}D^{2 }

Since the fluid is not confined, the velocity head results in velocity gradients, macro-scale and micro-scale turbulence, and ultimate dissipation as heat.

Newton has defined velocity gradient as shear rate. Hence, our relationship of impeller head and impeller shear rate. What is a high shear impeller? Very simply, one that produces high velocity gradients.

Tank Baffles: A discussion of impeller fluid mechanics would not be complete without an explanation of the role of tank baffles.

At high Reynolds Numbers (low viscosity ), both radial and axial flow impellers will produce the same flow pattern in an unbaffled tank (Fig. 13)- a rotary circulation pattern with a deep vortex and no appreciable vertical component of flow.

When baffles are added, the vortex and swirl disappear. The radial impeller produces its characteristic dual circulation pattern, the axial impeller produces a single circulation pattern, and both produce excellent top to bottom mixing.

Suffice it to say here, we need baffled conditions at high Reynolds Numbers in order to have top to bottom mixing and predictable impeller loading.