Efficiency Knowledge-Based Study

Study notes

1. What is Efficiency?

   In any real physical process or machine, energy is transferred from one form to another or transmitted to perform work. However, not all of the supplied energy accomplishes the intended purpose.

   Efficiency is a measure of how successfully an energy converter transforms total input energy into useful output energy. An ideal device would convert all input energy into useful work, but practical devices always transfer a portion of the input into non-useful, or wasted, energy forms.

   • Useful Energy Output: Energy transferred in the specific form and direction required for the device's designated function (such as mechanical kinetic energy from an electric motor).
   • Wasted Energy: Energy transferred into unintended forms or dissipated into the surroundings (typically thermal energy caused by friction or electrical resistance).

   Quantifying Efficiency

   Efficiency can be calculated in terms of either energy transfers or the rate of energy transfer (power). Because power is energy transferred per unit time, the ratio remains mathematically identical whether expressed using energy or power values over the same time interval.

   Energy-Based Formulation

   When analyzing total quantities of energy transferred, efficiency is given by:

   $$\text{Efficiency} = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}}$$

   To express this value as a percentage, multiply the resulting decimal fraction by 100:

   $$\text{Percentage Efficiency} = \left( \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} \right) \times 100\%$$

   Power-Based Formulation

   When continuous rates of work or energy flow are given, efficiency is calculated using power:

   $$\text{Efficiency} = \frac{P_{\text{useful}}}{P_{\text{in}}} = \frac{\text{Useful Power Output}}{\text{Total Power Input}}$$

   Because both the numerator and the denominator share the same units (joules for energy or watts for power), efficiency is a dimensionless quantity (ratio) that possesses no physical units.

   The Law of Conservation of Energy

   The principle of conservation of energy states that energy cannot be created or destroyed; it can only be transferred between stores. For any closed system, the total energy supplied must equal the sum of all energy outputs:

   $$\text{Total Energy Input} = \text{Useful Energy Output} + \text{Wasted Energy Output}$$

   Why Efficiency Cannot Exceed 100%

   Because no new energy can be created within a passive machine, the useful energy output can never exceed the total energy input. Therefore, efficiency must always satisfy:

   $$0 \le \text{Efficiency} < 1 \quad \text{or} \quad 0\% \le \text{Percentage Efficiency} < 100\%$$

   In mechanical and electrical systems, irreversible processes such as friction between moving parts, air resistance, and resistive heating in conductors continually dissipate energy into internal thermal stores of the components and surroundings. Consequently, an efficiency of 100% is physically unattainable in continuous real-world machines.

   Sample Problems

   Example 1: Calculating Percentage Efficiency

   An electric crane lifts a shipping crate, performing 12,000 J of useful gravitational work while drawing 20,000 J of electrical energy from the power supply. What is the efficiency of the crane?

   1. Identify the known quantities: $\text{Useful Energy Output} = 12{,}000\text{ J}$, $\text{Total Energy Input} = 20{,}000\text{ J}$.
   2. Apply the efficiency ratio: $$\text{Efficiency} = \frac{12{,}000\text{ J}}{20{,}000\text{ J}} = 0.60$$
   3. Convert to percentage: $$0.60 \times 100\% = 60\%$$

   Example 2: Determining Wasted Power

   An electric motor operates with a total electrical power input of 500 W and has an efficiency of 82%. Calculate the rate at which energy is dissipated as waste heat.

   1. Determine useful power output: $$P_{\text{useful}} = \text{Efficiency} \times P_{\text{in}} = 0.82 \times 500\text{ W} = 410\text{ W}$$
   2. Use conservation of power to find wasted power: $$P_{\text{wasted}} = P_{\text{in}} - P_{\text{useful}} = 500\text{ W} - 410\text{ W} = 90\text{ W}$$

   Minimizing Dissipated Energy

   Engineering design focuses on reducing non-useful energy pathways to increase the proportion of input energy transferred to useful work.

   • Lubrication: Applying oil or grease between contacting surfaces reduces the coefficient of friction, minimizing thermal dissipation in moving mechanical parts.
   • Streamlining: Shaping vehicles and rotating elements aerodynamically reduces drag and turbulence caused by air resistance.
   • Low-Resistance Conductors: Using high-conductivity copper wiring with adequate cross-sectional area reduces resistive heating ($P = I^2R$) in electrical circuits.
   • Thermal Insulation: Reducing heat loss through conduction, convection, and radiation keeps thermal energy confined where it is required, such as inside industrial boilers and engines.