When determining the overall efficiency of a system composed of multiple components, it is necessary to multiply the efficiencies of the individual components rather than add them up. This is based on the concept of energy conservation and the way efficiencies are defined.
Efficiency is a measure of how effectively a system converts input energy to useful output energy. It is calculated by dividing the useful output energy by the input energy. When multiple components are combined, each component has its own efficiency, representing how efficiently it converts energy.
To understand why we multiply efficiencies, let's consider a simple example of a system with two components: Component A and Component B.
The efficiency of Component A is defined as the ratio of the useful output energy from Component A to the input energy into Component A:
Efficiency_A = Useful output energy from Component A / Input energy into Component A
Similarly, the efficiency of Component B is defined as:
Efficiency_B = Useful output energy from Component B / Input energy into Component B
To find the overall efficiency of the system when both components are combined, we need to determine the efficiency of the entire system. Since the output energy of Component A becomes the input energy for Component B, we multiply the efficiencies:
Overall Efficiency = Efficiency_A * Efficiency_B
Multiplying the efficiencies accounts for the fact that the output energy of Component A serves as the input energy for Component B. It captures the cumulative effect of each component's efficiency on the overall system efficiency.
In contrast, adding the efficiencies would imply that the output energy of Component A and Component B are used simultaneously and independently, which is not the case in a series combination of components.
By multiplying the efficiencies, we consider the energy conversion efficiency of each component in sequence, allowing us to determine the overall efficiency of the combined system accurately.