Mixing

#bioreactor-design#scale-up

1.0

Equations

Power number (Newton number, $N_p$ )

N_p = \frac{P}{\rho N_i^3 D_i^5}

$N_p$ — power number (Newton number — dimensionless power drawn to agitate the fluid at a given speed and impeller diameter, analogous to a drag coefficient), dimensionless
$P$ — power, in watts ( $\text{W}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

Power in laminar flow

P = K_L \mu N_i^2 D_i^3

$P$ — power, in watts ( $\text{W}$ )
$K_L$ — laminar power constant, a dimensionless constant depending on the geometry of the impeller and tank (blade pitch, blade shape, number of blades, vessel geometry)
$\mu$ — dynamic viscosity of the fluid, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

Impeller tip speed

v_{tip} = \pi N_i D_i

$v_{tip}$ — impeller tip speed (maximum linear velocity at the blade edge — a proxy for the shear intensity cells experience near the impeller), in metres per second ( $\text{m/s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

Flow number ( $F_l$ / $F_Q$ )

F_l = \frac{Q}{N_i D_i^3}

$F_l$ — flow number (also written $F_Q$ — dimensionless measure of pumping capacity: how much fluid volume the impeller circulates per revolution), dimensionless
$Q$ — volumetric flow rate of fluid leaving the impeller blade, in cubic metres per second ( $\text{m}^3/\text{s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

If power is given

t_m = 5.9\, D_T^{2/3} \left(\frac{\rho V_L}{P}\right)^{1/3} \left(\frac{D_T}{D_i}\right)^{1/3}

$t_m$ — mixing time (time required for an added pulse of tracer/chemical to become homogeneously distributed through the vessel), in seconds ( $\text{s}$ )
$D_T$ — tank diameter, in metres ( $\text{m}$ )
$\rho$ — density of the fluid, in kilograms per cubic metre ( $\text{kg/m}^3$ )
$V_L$ — volume of liquid, in cubic metres ( $\text{m}^3$ )
$P$ — power, in watts ( $\text{W}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

If power number is given

t_m = \frac{5.4}{N_i} \left(\frac{1}{N_p}\right)^{1/3} \left(\frac{D_T}{D_i}\right)^2

$t_m$ — mixing time (time required for an added pulse of tracer/chemical to become homogeneously distributed through the vessel), in seconds ( $\text{s}$ )
$N_i$ — impeller rotational speed, in reciprocal seconds ( $\text{s}^{-1}$ )
$N_p$ — power number (dimensionless power drawn to agitate the fluid), dimensionless
$D_T$ — tank diameter, in metres ( $\text{m}$ )
$D_i$ — impeller diameter, in metres ( $\text{m}$ )

06 formulas