Theory of liquids mixing
Many mixing situation involve a rotary impellor, so we speak in terms
of radial (outward from the axis of rotation) and longitudinal (along the direction
of the rotor shaft) component velocities in liquids being mixed.
For successful mixing, both the radial and the longitudinal velocities
are maximised by the use of baffles or by the positioning or orientation
of the agitator.
Calculation of power requirement for mixing
Liquid flow is defined by a series of dimensionless numbers:
The Reynold's Number: Re = (D2 N 
m)
/ 
m
The Froude Number: Fr = (D N2) / g
The Power Number: Po = P / (m
N3 D5)
where
P = power transmitted to the agitator (kW)
m = density of the mixture (kg / m3)
m = viscosity of the mixture (N s / m2)
D = agitator diameter (m)
N = agitator speed (rev / s)
g = acceleration due to gravity (9.81 m / s2)
These dimensionless numbers are related by the equation:
Po = K (Re)n (Fr)m
where K, n, and m are constants related to the geometry of the agitator,
which are found by experiment.
The Froude number accounts for the effects of gravitational forces
and is only relevant when the liquid surface is disturbed by the propeller.
It can be ignored for Reynold's Number below about 300.
The density of a mixture may be computed by the addition of the component densities:
m = V1 1
+ V2 2
where
1 and 2 are the densities of the components
V1 and V2 are the volume fractions of the components
The viscosity of a mixture may be computed from the viscosities
of the ingredients using the following equations for baffled and unbaffled mixers:
Baffled: m = 1V1
2V2
Unbaffled: m = m
(1 + 1.5 2 V2 / [V1 (1
+ 2)]
Full use of this theory is very long and involved, and not commonly used in practice.
What is more commonly required is the "scale up" of a mixing process from pilot to full commercial scale.
Scaling a mixing operation
When changing the scale of equipment with the goal of obtaining the same
mixing performance at the new scale:
(i) Geometric similarity should be preserved -- dimensional ratios should
be the same in the large tank as in the small.
(ii) Dynamic similarity should be preserved -- Reynolds numbers should
be the same in the large tank as in the small.
Note: The above analysis is correct for Newtonian fluids only
-- it becomes more complex when the non-Newtonian behaviour of food materials
is taken into account.
Mixing of low and moderate viscosity liquids
Turbulence should be induced to entrain slow-moving parts within faster-moving parts.
Turbulence is highest near the impeller, and liquid should be circulated
through this region as much as possible.
A vortex should be avoided because adjoining layers of circulating liquids travel
at a similar speed and entrainment does not take place
-- the liquids simply rotate around the mixer.