Question

Explain the heat engine and obtain its efficiency.

Answer 1

Heat Engine: In the modem technological world, the role of automobile engines plays a vital role in transportation. In motorbikes and cars, there are engines which take in petrol or diesel as input and do work by rotating wheels. Most of these automobile engines have efficiency not greater than 40%. The second law of thermodynamics puts a fundamental restriction on the efficiency of engines. Therefore understanding heat engines is very important.

Reservoir: It is defined as a thermodynamic system which has a very large heat capacity. By taking in heat from a reservoir or giving heat to the reservoir, the reservoir’s temperature does not change.

Example: Pouring a tumbler of hot water into the lake will not increase the temperature of the lake. Here the lake can be treated as a reservoir.

When a hot cup of coffee attains equilibrium with the open atmosphere, the temperature of the atmosphere will not appreciably change. The atmosphere can be taken as a reservoir.

We can define a heat engine as follows: A heat engine is a device which takes heat as input and converts this heat into work by undergoing a cyclic process.

A heat engine has three parts:

1. Hot reservoir
2. Working substance
3. Cold reservoir

A Schematic diagram for a heat engine is given below:

              Heat Engine

1. Hot reservoir (or) Source: It supplies heat to the engine. It is always maintained at a high-temperature T_H.

2. Working substance: It is a substance like gas or water, which converts the heat supplied into Work.

3. Cold reservoir (or) Sink: The heat engine ejects some amount of heat (Q_L) into the cold reservoir after it doing work. It is always maintained at a low-temperature T_L.

The heat engine works in a cyclic process. After a cyclic process, it returns to the same state. Since the heat engine returns to the same state after it ejects heat, the change in the internal energy of the heat engine is zero.

The efficiency of the heat engine is defined as the ratio of the work done (output) to the heat absorbed (input) in one cyclic process.

Let the working substance absorb heat Q_H units from the source and reject Q_L units to the sink after doing work W units.

           Heat engine

We can write,

Input heat = Work done + ejected heat

Q_H = W + Q_L

W = Q_H – Q_L

Then the efficiency of the heat engine

η = `"output"/"input" = "W"/"Q"_"H" = ("Q"_"H" - "Q"_"L")/"Q"_"H"`

η = `"output"/"input" = "W"/"Q"_"H" = 1 - "Q"_"L"/"Q"_"H"`

Note here that Q_H, Q_L and W all are taken as positive, a sign convention followed in this expression.

Since Q_L < Q_H, the efficiency (η) always less than 1. This implies that heat absorbed is not completely converted into work. The second law of thermodynamics placed fundamental restrictions on converting heat completely into work.