Reference ID: MET-0B6C | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Reynolds number for mixing systems is a dimensionless group that quantifies the ratio of inertial to viscous forces within an agitated vessel. It is the primary correlating parameter used to scale-up or scale-down mixing operations, predict power draw, select impeller geometry, and ensure that laboratory or pilot-plant data can be reliably translated to industrial scale. Typical applications include blending of high-viscosity polymers, fermentation broths, slurries, and non-Newtonian fluids where the flow regime strongly affects heat/mass-transfer rates and mixing times.
Methodology & Formulas
Convert rotational speed
Impeller speed is usually supplied in revolutions per minute (RPM). Convert to radians per second (s-1) via:
\[ N = \frac{N_{\text{RPM}}}{60} \]
Convert dynamic viscosity
Viscosity is frequently reported in centipoise (cP). Convert to pascal-seconds (Pa·s) while preventing non-physical zero values:
\[ \mu = \max\left( \mu_{\text{cP}} \times 0.001,\; 1\times10^{-9} \right) \]
Compute the Reynolds number
The general definition for pipe flow is adapted to agitated systems by replacing the characteristic length with the impeller diameter:
\[ Re = \frac{\rho\,N\,D^{2}}{\mu} \]
where:
\( \rho \) = fluid density (kg·m-3)
\( N \) = impeller rotational speed (s-1)
\( D \) = impeller diameter (m)
\( \mu \) = dynamic viscosity (Pa·s)
Determine flow regime
The calculated Reynolds number is compared against standard thresholds for mechanically agitated systems:
Reynolds Number Range
Flow Regime
\( Re < 10 \)
Laminar
\( 10 \le Re < 10\,000 \)
Transitional
\( Re \ge 10\,000 \)
Turbulent
The Reynolds Number (Re) is a dimensionless quantity used to predict flow patterns in fluid dynamics. In mixing systems, it determines whether the flow is laminar, transitional, or turbulent. This is critical because the flow regime directly impacts mixing efficiency, energy consumption, and shear stress on the fluid. A low Re (laminar) results in smooth, predictable flow, while a high Re (turbulent) enhances mixing but may increase energy use.
The Reynolds Number for mixing is calculated using the formula: Re = (N * D2 * ρ) / μ, where:
N = impeller rotational speed (s-1)
D = impeller diameter (m)
ρ = fluid density (kg/m³)
μ = fluid viscosity (Pa·s)
Ensure all units are consistent. This calculation helps determine the flow regime and select appropriate mixing parameters.
The flow regimes in mixing systems are categorized as follows:
Laminar: Re < 10
Transitional: 10 ≤ Re ≤ 10,000
Turbulent: Re > 10,000
These ranges guide engineers in selecting impeller types and operating conditions to achieve desired mixing outcomes.
Worked Example: Determining the Reynolds Number for a Small‑Scale Mixing Vessel
Scenario: An engineer is designing a laboratory‑scale mixing system for a high‑viscosity polymer slurry. The vessel has a 0.5 m diameter impeller that will be driven at 100 rpm. The slurry density is 1300 kg m⁻³ and its viscosity is 100 000 cP. The engineer must calculate the Reynolds number to confirm that the mixing will occur in the laminar regime.
Knowns
\(\rho = 1300\) kg m⁻³ (density)
\(N_{\text{rpm}} = 100\) rpm (rotational speed)
\(D = 0.5\) m (impeller diameter)
\(\mu_{\text{cP}} = 100{,}000\) cP (dynamic viscosity in centipoise)
Step‑by‑Step Calculation
Convert viscosity from centipoise to pascal‑seconds:
\(\mu = \mu_{\text{cP}} \times 10^{-3} = 100{,}000 \times 10^{-3} = 100\) Pa·s
Convert rotational speed from rpm to revolutions per second:
\(N = \dfrac{N_{\text{rpm}}}{60} = \dfrac{100}{60} = 1.667\) rev s⁻¹
Apply the Reynolds number formula for mixing:
\[ \text{Re} = \frac{\rho \, N \, D^{2}}{\mu} \]
Insert the known values:
\[ \text{Re} = \frac{1300 \times 1.667 \times (0.5)^{2}}{100} \]
