Power number
The power number, denoted as N_p, is a dimensionless quantity in fluid mechanics and chemical engineering that characterizes the power consumption of rotating impellers in mixing and agitation processes within stirred tanks. It is defined by the relation N_p = \frac{P}{\rho N^3 D^5}, where P is the power input to the impeller, \rho is the fluid density, N is the rotational speed, and D is the impeller diameter.[1] This parameter enables the analysis of energy transfer in fluid systems by normalizing power requirements against fluid properties and operating conditions.[2] In practice, the power number correlates strongly with the impeller Reynolds number (Re = \frac{\rho N D^2}{\mu}, where \mu is the fluid viscosity), allowing prediction of flow regimes from laminar to turbulent. In the turbulent regime, typically at Re > 10^4, N_p becomes constant for a specific impeller geometry, facilitating scale-up of mixing operations from pilot to production scales without proportional changes in power draw.[3] Variations in N_p arise from impeller design factors, such as blade shape and number; for instance, radial-flow impellers like Rushton turbines exhibit higher N_p values (around 5–6) compared to axial-flow propellers (around 0.3–1.5), reflecting differences in shear and circulation efficiency.[4] Baffling in the tank and impeller-to-tank diameter ratios further influence N_p, with unbaffled systems showing oscillatory behavior at transitional Reynolds numbers.[5] The power number's utility extends to optimizing energy use in industries like pharmaceuticals, wastewater treatment, and food processing, where it informs impeller selection to balance mixing intensity against power costs. Experimental determination often involves torque measurements or calorimetry in controlled setups, with correlations compiled for common geometries to guide design.[6]Definition and Formulation
Definition
The power number, denoted as N_p, is a dimensionless parameter that characterizes energy dissipation rates in fluid systems, particularly in agitation and mixing processes where mechanical impellers impart energy to fluids. It quantifies the relationship between the power delivered to the fluid and the characteristic scales of the system, enabling consistent comparisons across varying equipment sizes and operating conditions.[7] The power number is defined as the ratio of the power input P to the product of the fluid density \rho, the cube of the impeller rotational speed N^3, and the fifth power of the impeller diameter D^5: N_p = \frac{P}{\rho N^3 D^5} where P is the power consumption in watts, \rho is the fluid density in kg/m³, N is the rotational speed in revolutions per second, and D is the impeller diameter in meters.[7] The power number was introduced in 1950 by J. H. Rushton, E. W. Costich, and H. J. Everett, providing a standardized approach to measuring and correlating power requirements in mixing operations and facilitating advancements in chemical engineering design.[7] The term originates from its explicit connection to the power transferred via mechanical agitation, emphasizing the energetic aspect of fluid motion induced by impellers.[7] In turbulent flow regimes, where inertial forces dominate over viscous effects, the power number remains constant for a specific impeller type and geometry, serving as a reliable indicator of energy input independent of fluid viscosity.[7] This property makes it essential for analyzing high-Reynolds-number conditions in industrial mixers. It complements the Reynolds number in delineating flow regimes during agitation.[7]Mathematical Formulation
The power number, denoted as N_p, arises from dimensional analysis applied to the power consumption in fluid agitation systems, utilizing the Buckingham π theorem to identify relevant dimensionless groups. Consider the power P delivered to an impeller as a function of the fluid density \rho, viscosity \mu, gravitational acceleration g, impeller rotational speed N, and impeller diameter D. These six variables involve three fundamental dimensions: mass (M), length (L), and time (T). According to the Buckingham π theorem, the number of dimensionless π groups is $6 - 3 = 3.[8] To derive the groups, select repeating variables that span the dimensions: \rho ([M L^{-3}]), N ([T^{-1}]), and D ([L]). The first π group incorporates the dependent variable P ([M L^2 T^{-3}])): \pi_1 = P \rho^a N^b D^c Balancing dimensions yields a = -1, b = -3, c = -5, so \pi_1 = N_p = \frac{P}{\rho N^3 D^5}. The second group uses \mu ([M L^{-1} T^{-1}])): \pi_2 = \mu \rho^d N^e D^f This gives d = -1, e = -1, f = -2, resulting in \pi_2 = \frac{\mu}{\rho N D^2} = \frac{1}{\text{Re}}, where \text{Re} = \frac{\rho N D^2}{\mu} is the impeller Reynolds number. The third group incorporates g ([L T^{-2}])): \pi_3 = g \rho^h N^i D^j Balancing yields h = 0, i = -2, j = -1, so \pi_3 = \frac{g}{N^2 D} = \frac{1}{\text{Fr}}, where \text{Fr} = \frac{N^2 D}{g} is the Froude number. Thus, the general relationship is N_p = f(\text{Re}, \text{Fr}).[8] In high Reynolds number flows, typical of fully turbulent regimes (\text{Re} > 10^4), inertial forces dominate over viscous forces, rendering N_p independent of \text{Re} and often approximately constant, provided free-surface effects (captured by \text{Fr}) are minimized, such as in baffled tanks. This simplifies to P = \rho N^3 D^5 N_p, where N_p depends primarily on impeller geometry.[7][7]Applications in Mixing and Agitation
Power Consumption in Stirred Tanks
In stirred tanks, the power consumption is analyzed through the dimensionless power number N_p, plotted against the impeller Reynolds number Re on a log-log scale, revealing three flow regimes that dictate energy requirements. In the laminar regime (typically Re < 10), viscous forces dominate, and N_p \propto 1/Re, resulting in higher relative power draw at low speeds. The transitional regime ($10 < Re < 10^4) features a sloping curve where N_p decreases gradually as inertial forces increase. In the turbulent regime (Re > 10^4), N_p stabilizes as a constant value, independent of Re, reflecting fully developed turbulence.[9][10] For common configurations in the turbulent regime, N_p \approx 5 for a standard six-bladed Rushton turbine in a baffled tank, providing a benchmark for power prediction in industrial mixing. Several factors influence this power draw, including the presence of baffles, which disrupt circumferential flow to suppress surface vortex formation and promote axial and radial circulation, thereby increasing effective power input compared to unbaffled setups. Additionally, the tank-to-impeller diameter ratio, typically J = D_T / D \approx 3 (or impeller diameter D about one-third of tank diameter D_T), optimizes flow patterns and power efficiency in standard designs.[11][12][13] The baseline equation for ungassed power consumption is P = N_p \rho N^3 D^5, where \rho is fluid density, N is impeller rotational speed, and D is impeller diameter; this formulation scales power requirements directly with operating conditions. In gassed systems, gas dispersion reduces effective density and alters hydrodynamics, yielding lower power draw expressed as P_g = N_{pg} \rho N^3 D^5, with the gassed power number N_{pg} typically less than N_p, often correlated empirically as the ratio P_g / P versus the gas flow number Q_g / (N D^3) to account for aeration effects.[14]Impeller-Specific Correlations
Empirical correlations for the power number vary significantly with impeller geometry, reflecting differences in flow patterns and energy dissipation. In the fully turbulent regime (Re > 10^4), the power number reaches a constant value characteristic of each impeller type, enabling straightforward comparisons of power efficiency across designs. These values are derived from experimental data in standard baffled stirred tanks, where the impeller diameter to tank diameter ratio (D/T) is typically 0.3–0.5.[15] Representative turbulent power numbers for common impellers are summarized below, highlighting the high power draw of radial-flow impellers like the Rushton turbine compared to low-power axial-flow options such as hydrofoils. Note that the anchor impeller, a close-clearance design for viscous fluids, operates primarily in the laminar regime, where its power number is notably lower than turbulent radial impellers but follows a different scaling.| Impeller Type | Regime | Power Number (N_p) |
|---|---|---|
| Rushton turbine | Turbulent | 5.0 |
| Pitched-blade turbine | Turbulent | 1.3 |
| Hydrofoil | Turbulent | 0.3 |
| Anchor | Laminar | ≈ 300 / Re |