Question

How do you figure out the efficiency of a system?

Answer 1

“Efficiency” tells you how much of the energy (or power) you put into a system is delivered as the useful output. The basic idea and steps are:

1. Define the system boundaries and what “useful output” means.
- Example: for an electric motor useful output = mechanical shaft power; for a boiler useful output = heat delivered to water.

2. Choose whether to compare energies (over a process or time interval) or powers (instantaneous or steady‑state).
- Energy form: Efficiency = useful energy output / total energy input.
- Power form: Efficiency = useful power output / total input power.
- Multiply the ratio by 100% if you want percent.

3. Measure or calculate input and useful output.
- Electrical: P_in = V × I (or measured with a wattmeter).
- Mechanical: P_out = torque × angular speed or F × v.
- Thermal: Q_in from fuel flow × heating value or from calorimetry; Q_out the heat delivered.
- For a heat engine: thermal efficiency η = W_out / Q_in = 1 − Q_out / Q_in.

4. Compute efficiency:
- η = (useful output) / (input). Example: motor with 1,000 W in and 850 W out → η = 850/1000 = 0.85 = 85%.

5. For multi‑stage systems, overall efficiency = product of stage efficiencies:
- η_total = η1 × η2 × ...

6. Check units (they cancel) and account for time if using energies (e.g., J/J) vs powers (W/W).

Notes and caveats
- Efficiency ≤ 100% (more than 100% implies measurement error or external energy source).
- Refrigerators/heat pumps are usually characterized by COP (coefficient of performance), not efficiency.
- For thermal engines the theoretical maximum is the Carnot efficiency η_Carnot = 1 − T_cold/T_hot (temperatures in K).
- If you want to account for work quality and irreversibilities use exergy efficiency instead of simple energy efficiency.

Quick examples
- Light bulb: P_in = 60 W electrical, radiant + visible useful output maybe 6 W → η = 6/60 = 10%.
- Heat engine: Q_in = 1000 J, W_out = 300 J → η = 300/1000 = 30%.

If you tell me what kind of system you have (mechanical, electrical, thermal, multi‑stage), I can give a more specific method and the measurements you’ll need.