How to Calculate Power and Efficiency
Learn how to calculate mechanical, electrical, and thermal power and system efficiency. Covers work-energy theorem, motor efficiency, and heat engine efficiency limits.
What Is Power?
Power is the rate of doing work or transferring energy: P = W / t (watts = joules per second). Equivalently, for a constant force F moving at velocity v, P = F · v. For rotation, P = τ · ω where τ is torque and ω is angular velocity in rad/s. A machine that lifts 500 N at 0.1 m/s requires P = 500 · 0.1 = 50 W of mechanical power at the load. Understanding power — not just force — is essential because it determines motor size, fuel consumption, and heat generation in any system.
Mechanical Power Formulas
Linear mechanical power: P = F · v · cos(θ), where θ is the angle between force and velocity vectors. For rotation: P = τ · ω = τ · 2π·N/60, where N is shaft speed in RPM. Converting: 1 horsepower = 745.7 W. A motor producing 50 N·m at 1500 RPM: P = 50 · 2π · 1500/60 = 7,854 W ≈ 10.5 hp. Instantaneous power for a body under acceleration is P = F · v = m·a · v, which increases with velocity even at constant force, explaining why a car requires more power to maintain acceleration at high speed.
Efficiency Definition and Calculation
Efficiency η = P_output / P_input × 100%, a dimensionless ratio expressed as a percentage. It measures how effectively input energy is converted to useful output. Losses appear as heat, sound, friction, or vibration. An electric motor consuming 5 kW and delivering 4.25 kW of mechanical output has η = 4.25/5 = 85%. For multi-stage systems, overall efficiency is the product of individual efficiencies: η_total = η₁ · η₂ · η₃. A 90% pump driven by an 85% motor driven by a 95% VFD has system efficiency 0.90 · 0.85 · 0.95 ≈ 72.7%.
Electrical Power
Electrical power for DC circuits: P = V · I (watts), where V is voltage and I is current. Ohm's law gives P = I²·R = V²/R. For AC single-phase: P = V·I·cos(φ), where cos(φ) is the power factor (ratio of real to apparent power). For three-phase AC: P = √3 · V_L · I_L · cos(φ), where V_L is line voltage. Apparent power S = V·I (volt-amperes, VA) and reactive power Q = V·I·sin(φ) (VAR) complete the power triangle. Poor power factor (cos(φ) < 0.85) means drawing more current than necessary, increasing cable and transformer losses.
Thermal Efficiency and Heat Engines
Heat engines convert thermal energy to mechanical work. Thermal efficiency: η_thermal = W_net / Q_in, where W_net is net work output and Q_in is heat input. The Carnot efficiency sets the theoretical maximum for any heat engine operating between temperatures T_H (hot source) and T_C (cold sink) in Kelvin: η_Carnot = 1 − T_C/T_H. A steam turbine with T_H = 800 K and T_C = 300 K has η_Carnot = 1 − 300/800 = 62.5%. Real turbines achieve 35–45% due to irreversibilities. This limit is fundamental and cannot be exceeded regardless of engineering improvements.
Coefficient of Performance for Refrigeration
Refrigerators and heat pumps are characterized by coefficient of performance (COP) rather than efficiency, since COP can exceed 1. For a refrigerator: COP_ref = Q_C / W_in, where Q_C is heat removed from cold space and W_in is work input. For a heat pump: COP_hp = Q_H / W_in = COP_ref + 1. The Carnot COP for a refrigerator: COP_Carnot = T_C / (T_H − T_C). A refrigerator cooling to 5°C (278 K) and rejecting heat at 35°C (308 K) has COP_Carnot = 278/30 ≈ 9.3; real systems typically achieve COP of 2–5.
Power and Efficiency in Motor Selection
When selecting an electric motor, first calculate the required mechanical output power from the load's force/torque and speed requirements. Add a service factor (typically 1.15–1.5) for intermittent overloads, then divide by the motor's expected efficiency to find the required input power. Motor efficiency ratings (IE2 = standard, IE3 = premium, IE4 = super-premium per IEC 60034-30) indicate energy losses over the motor's life, with IE4 motors offering 2–4% better efficiency than IE2 — significant for 24/7 industrial operations where motor energy cost far exceeds purchase price.
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