# Power Number

on . Posted in Dimensionless Numbers

Power number, abbreviated as $$N_p$$, a dimensionless number, represents the proportional drag force and inertial force.  It is used in momentum transfer and power consumption by fans, mixers and pump impellers at a specified rotational speed.  The power number formula depends on the geometry of the impeller, the Reynolds number of the flow, and the physical properties of the fluid.  The value of the power number depends on the impeller design and the physical properties of the fluid, such as viscosity and surface tension.  The power number is commonly used in the design and optimization of mixing processes, such as in chemical and biochemical reactors, fermentation vessels, and wastewater treatment systems.

### Key points about the Power Number

• Performance Indicator  -  The power number is used to assess the performance of mixing equipment.  It quantifies the amount of power required to mix or agitate a fluid at a given speed and impeller size.
• Scale-Up and Design  -  Engineers use the power number to scale up mixing processes from laboratory to industrial scales.  It helps in the design of mixing equipment by selecting appropriate impeller sizes and speeds.
• Relationship to Flow Regime  -  The power number is related to the flow regime in the mixing vessel. It can be used to determine whether the flow is laminar, transitional, or turbulent.  Different flow regimes have different power number correlations.
• Efficiency and Energy Consumption  -  A low power number indicates that the mixing equipment is more energy efficient because it requires less power to achieve the desired mixing or agitation.  A high power number implies higher energy consumption.
• Impeller Design  -  The choice of impeller design and geometry can significantly affect the power number.  Different impeller shapes and sizes can result in different power numbers for the same mixing conditions.
• Empirical Data  -  Power numbers are often determined experimentally for specific mixing equipment and fluid combinations.  They can be found in reference materials and databases for various impeller types.
• Dimensionless Parameter  -  The power number is a dimensionless parameter, which means it does not depend on the units used for power, density, speed, or diameter, as long as consistent units are used.

Understanding the power number is essential for optimizing mixing processes, ensuring efficient energy use, and avoiding issues like inadequate mixing or excessive power consumption in industrial applications such as chemical processing, food production, and wastewater treatment.

### Power number formula

$$N_p \;=\; P \;/\; (\rho \; n^3 \; d^5)$$
Symbol English Metric
$$N_p$$ = Power number $$dimensionless$$
$$P$$ = power input to the fluid $$ft-lbf \;/\; sec$$  $$W$$
$$\rho$$  (Greek symbol rho) = density of the fluid $$lbm \;/\; ft^3$$ $$kg \;/\; m^3$$
$$n$$ = rotational speed of the impeller  $$rev \;/\; sec$$  $$rev \;/\; s$$
$$d$$ = impeller diameter $$in$$ $$mm$$