Reference ID: MET-E292 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Flow Number (Nq) is a dimensionless parameter that characterises the pumping capacity of an impeller relative to its rotational speed and size. Together with the Power Number (Np), it allows the process engineer to predict the bulk circulation generated by a given mechanical power input. Typical applications include:
Scale-up of stirred reactors and blending vessels.
Comparison of impeller geometries for energy-efficient mixing.
Estimation of mixing time and suspension quality in solid–liquid systems.
Methodology & Formulas
Impeller Selection
Empirical correlations for turbulent flow give constant values of Np and Nq for standard geometries. The code maps an integer flag to the following pairs:
Impeller Type
Power Number Np
Flow Number Nq
Rushton turbine
5.0
0.72
Pitched-blade turbine
1.3
0.60
Hydrofoil
0.35
0.40
High-solidity hydrofoil
0.70
0.55
Marine propeller
0.50
0.50
Rotational Speed from Power
The mechanical power P dissipated by the impeller is related to the rotational speed N (in s−1) by:
\[
P = N_p\,\rho\,N^3 D^5
\quad\Rightarrow\quad
N = \left(\frac{P}{N_p\,\rho\,D^5}\right)^{1/3}
\]
where ρ is fluid density and D is impeller diameter.
Flow Rate from Flow Number
By definition:
\[
N_q = \frac{Q}{N D^3}
\quad\Rightarrow\quad
Q = N_q\,N\,D^3
\]
yielding the volumetric circulation rate Q in m3 s−1.
Reynolds Number
The impeller Reynolds number:
\[
Re = \frac{\rho\,N\,D^2}{\mu}
\]
indicates the flow regime. A minimum value is enforced to avoid division by zero.
Regime
Re Range
Implication for Nq
Laminar
Re ≲ 10
Nq not constant; viscous drag dominates
Transitional
10 < Re < 10,000
Nq weakly dependent on Re
Fully turbulent
Re ≥ 10,000
Nq assumed constant for given geometry
If Re < 10,000, the code issues a warning that the tabulated Nq may not apply.
Pumping Efficiency Indicator
A dimensionless figure of merit compares circulation to power demand:
\[
\text{efficiency indicator} = \frac{N_q}{N_p^{1/3}}
\]
Higher values denote more effective conversion of mechanical power into bulk flow.
Flow Number (Nq) is a dimensionless parameter that relates the actual volumetric flow rate through a pump or compressor to the theoretical flow rate based on impeller speed and geometry. It is critical for pump selection because it allows engineers to:
Normalize performance curves across different sizes and speeds
Predict cavitation limits by comparing Nq to suction-specific speed (Nss)
Select hydraulically similar machines when scaling from test data
Ensure the chosen pump operates in the preferred efficiency zone
Use the definition Nq = Q / (n D³) with the following steps:
Convert flow rate Q to m³/s
Convert rotational speed n to rev/s
Estimate impeller diameter D from the specific diameter Ds = D (g H)0.25 / Q0.5, using typical Ds ≈ 1.0 for radial pumps
Plug values into Nq = Q / (n D³)
Check that the resulting Nq is between 0.05 and 0.60 for single-stage centrifugal pumps
Industry practice assigns impeller types based on Nq as follows:
Radial flow: Nq < 0.25
Mixed flow: 0.25 ≤ Nq ≤ 0.80
Axial flow: Nq > 0.80
These ranges help process engineers quickly narrow hydraulic selections during preliminary sizing.
Yes. Plotting measured Nq against the manufacturer’s Nq curve reveals:
Left-shifted Nq: indicates throttled or clogged suction, causing recirculation and vibration
Right-shifted Nq: suggests cavitation or impeller wear, reducing head and efficiency
Deviations > ±10% from design Nq warrant inspection of impeller, wear rings, and suction conditions
Worked Example – Verifying the Flow Number of a 400 mm Pitched-Blade Impeller in Water
A process engineer is checking the hydraulic performance of a 400 mm diameter, 3-blade pitched impeller operating in a 1.2 m diameter cylindrical vessel filled with water to a height of 1.2 m. The impeller is driven at 331 rpm and the measured shaft power is 600 W. Confirm that the calculated flow number Nq is consistent with the expected value for this impeller type.
Knowns
Impeller diameter, D = 0.400 m
Rotational speed, N = 331 rpm (5.511 rps)
Liquid density, ρ = 1000 kg/m³
Liquid viscosity, μ = 0.001 Pa·s
Measured shaft power, P = 600 W
Impeller type: 3-blade pitched blade (power number Np = 0.35)
Step-by-Step Calculation
Convert rotational speed to rps for consistency:
\[
N = \frac{331}{60} = 5.511\ \text{rps}
\]
Calculate the theoretical flow rate from the power number and the efficiency indicator:
\[
\eta_e = \frac{N_q}{N_p^{1/3}} \Rightarrow N_q = \eta_e \cdot N_p^{1/3}
\]
Using the computed efficiency indicator ηe = 0.568:
\[
N_q = 0.568 \cdot (0.35)^{1/3} = 0.568 \cdot 0.705 = 0.400
\]
Compute the volumetric flow rate using the definition of the flow number:
\[
Q = N_q \cdot N \cdot D^3
\]
\[
Q = 0.400 \cdot 5.511 \cdot (0.400)^3 = 0.400 \cdot 5.511 \cdot 0.064 = 0.141\ \text{m}³/\text{s}
\]
Check the Reynolds number to confirm turbulent regime:
\[
Re = \frac{\rho N D^2}{\mu} = \frac{1000 \cdot 5.511 \cdot (0.400)^2}{0.001} = 882,000
\]
Since Re ≫ 10,000, the flow is fully turbulent and the quoted power and flow numbers are valid.
Final Answer
The calculated flow number for the 400 mm pitched-blade impeller operating at 331 rpm in water is:
\[
N_q = 0.400
\]
This value lies within the expected range for a 3-blade pitched impeller and corresponds to a pumping capacity of 0.141 m³/s (507 m³/h).
"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
