Reference ID: MET-DE1F | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Flow Number (Nq) is a dimensionless parameter essential in process engineering for characterizing the pumping capacity of an impeller. It serves as a critical metric for evaluating how effectively an impeller moves fluid relative to its rotational speed and physical dimensions. In industrial mixing applications, Nq is used alongside the Power Number (Po) to determine the overall pumping efficiency of a system. This calculation is particularly vital in the turbulent regime, where dimensionless numbers remain constant, allowing engineers to scale mixing processes reliably across different vessel sizes.
Methodology & Formulas
The calculation of the Flow Number relies on the relationship between the volumetric flow rate, the rotational speed of the impeller, and the cube of the impeller diameter. The following algebraic framework defines the process:
First, the rotational speed must be normalized to revolutions per second:
\[ N = \frac{N_{RPM}}{60} \]
The Reynolds Number (Re) is calculated to verify the flow regime, ensuring the validity of the constant Nq assumption:
\[ Re = \frac{\rho \cdot N \cdot D^2}{\mu} \]
The swept volume factor is determined by the cube of the impeller diameter:
\[ D^3 = D \cdot D \cdot D \]
Finally, the Flow Number (Nq) is derived from the ratio of the volumetric flow rate to the product of the rotational speed and the swept volume factor:
\[ N_q = \frac{Q}{N \cdot D^3} \]
Parameter
Condition / Threshold
Requirement
Flow Regime
Re > 10,000
Turbulent (Valid for constant Nq)
Flow Regime
Re < 10,000
Laminar/Transitional (Correlation invalid)
Geometry
D > 0
Impeller diameter must be positive
Viscosity
μ ≥ 1e-9
Prevent division by zero in Re calculation
The Flow Number, often referred to as the flow coefficient, is a dimensionless parameter used to characterize the performance of centrifugal pumps. It provides engineers with a standardized way to compare the hydraulic performance of different impeller designs regardless of their physical size. Key aspects include:
It relates the volumetric flow rate to the impeller rotational speed and diameter.
It helps in determining the optimal operating point for a specific pump geometry.
It serves as a critical input for scaling laws when transitioning from prototype testing to full-scale industrial applications.
To calculate the Flow Number, you must ensure consistent units are used across all variables. The standard calculation involves the following steps:
Identify the volumetric flow rate (Q) at the best efficiency point.
Determine the rotational speed (n) of the impeller in revolutions per unit time.
Measure the impeller diameter (D).
Apply the formula Nq = Q / (n * D3), ensuring that the units for flow and speed are dimensionally compatible with the diameter cubed.
The Flow Number is intrinsically linked to the velocity triangles at the impeller inlet. When an engineer operates a pump outside of its design Flow Number, the following issues often arise:
The angle of attack of the fluid entering the impeller vanes deviates from the design intent.
Local pressure drops occur at the vane leading edges, which can trigger cavitation.
Increased turbulence and flow separation reduce the Net Positive Suction Head Required (NPSHr) margin, leading to premature mechanical failure.
Worked Example: Evaluating Impeller Performance for a Mixing Application
In a water treatment facility, two candidate impellers for a new baffled mixing tank must be evaluated for their liquid pumping capacity. The fluid is water at standard conditions. The goal is to calculate and compare the Flow Number (\(N_q\)) for each impeller to determine which provides superior pumping per revolution under turbulent flow conditions.
Impeller A Rotational Speed, \(N_{RPM,A} = 600.0\ \text{rpm}\)
Impeller A Volumetric Flow Rate, \(Q_A = 0.25\ \text{m}^3/\text{s}\)
Impeller B Diameter, \(D_B = 0.5\ \text{m}\)
Impeller B Rotational Speed, \(N_{RPM,B} = 600.0\ \text{rpm}\)
Impeller B Volumetric Flow Rate, \(Q_B = 0.10\ \text{m}^3/\text{s}\)
Step-by-Step Calculation:
Convert rotational speed from rpm to rev/s (Hz). The conversion factor is 60 s/min.
For Impeller A: \(N_A = N_{RPM,A} / 60 = 600.0 / 60 = 10.0\ \text{rev/s}\).
For Impeller B: \(N_B = N_{RPM,B} / 60 = 600.0 / 60 = 10.0\ \text{rev/s}\).
Calculate the impeller diameter cubed, \(D^3\), which relates to swept volume.
For Impeller A: \(D_A^3 = (0.5)^3 = 0.125\ \text{m}^3\).
For Impeller B: \(D_B^3 = (0.5)^3 = 0.125\ \text{m}^3\).
Verify the flow is fully turbulent by ensuring the Reynolds number, \(Re = (\rho N D^2) / \mu\), exceeds 10,000.
For Impeller A: \(Re_A = 2500000.0\).
For Impeller B: \(Re_B = 2500000.0\).
Both values are significantly greater than 10,000, confirming turbulent flow and the validity of using a constant \(N_q\).
Calculate the Flow Number using its definition: \(N_q = Q / (N \cdot D^3)\).
For Impeller A: \(N_{q,A} = Q_A / (N_A \cdot D_A^3) = 0.25 / (10.0 \times 0.125) = 0.200\).
For Impeller B: \(N_{q,B} = Q_B / (N_B \cdot D_B^3) = 0.10 / (10.0 \times 0.125) = 0.080\).
Final Answer:
The calculated Flow Numbers are:
Impeller A: \(N_{q,A} = 0.200\) (dimensionless).
Impeller B: \(N_{q,B} = 0.080\) (dimensionless).
Interpretation: Impeller A, with the higher \(N_q\), demonstrates a superior pumping capacity per revolution compared to Impeller B under these specific operating conditions.
"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
