Atmosphere, Land , and Water
in a Lake are Examples
of Thermal Reservoirs

Source and Sink

A thermal reservoir is a specific kind of system with a large thermal energy capacity that can supply or absorb finite amounts of heat and always remains at constant temperature. Such a system can be approximated in a number of ways:

• Large land masses
• Earth's atmosphere
• Large bodies of water: oceans, lakes, or rivers
• Any physical body whose thermal energy capacity is large relative to the amount of energy it supplies or absorbs, for example, a large block of ice

A reservoir that supplies energy in the form of heat is called a source and one that absorbs energy in the form of heat is called a sink. For example, atmospheric air is a source for heat pumps and a sink for air conditioners.

When a system in a given initial state experiences a series of quasi-equilibrium processes and returns to its initial state, the system undergoes a cycle. The energy balance for any system undergoing a cycle takes the form

      ΔE_cycle = Q_cycle - W_cycle

where
      Q_cycle = the net amount of energy transferred
                  by heat for the cycle,
                  Q_cycle = Q_in- Q_out
      W_cycle = the net amount of energy transferred
                  by work for the cycle,
                  W_cycle = W_out - W_in

Notice that the directions of the heat and work are indicated by the subscripts in and out. Therefore, Q_in, Q_out, W_out, and W_in are all positive numbers.

Power Cycle and Refrigeration and
Heat Pump Cycle

Since the system is returned to its initial state after the cycle, there is no net change in its energy. Therefore,

      ΔE_cycle = 0

Then the equation reduces to

      Q_cycle = W_cycle

This expression can satisfy every thermodynamic cycle, regardless of the sequence of processes followed by the system undergoing the cycle or the nature of the substances making up the system.

If the system undergoing cycles delivers a net work to its surroundings during each cycle, the cycle is called a power cycle.

      W_cycle = Q_in - Q_out

On the other hand, if the system needs work input from the surroundings to run each cycle, the cycle is called a refrigeration and heat pump cycle.

      W_cycle = Q_out - Q_in

where W_cycle has a positive value.

What is a Heat Engine

Schematic of a Basic Power Plant

Most people understand that work can always be converted to heat directly and completely. But converting heat to work requires the use of special devices. These devices are called heat engines.

Heat engines operate on a cycle and receive heat from a high-temperature source, convert part of this heat to work, and then reject the remaining waste heat to a low-temperature sink during the cycle.

A steam power plant is an example of heat engine. The schematic of a basic steam power plant is shown on the left. The cycle is:

• Heat (Q_in) is transferred to the steam in the boiler from a furnace, which is the energy source.
• The turbine produces work (W_out ) when steam passes through it.
• A condenser transfers the waste heat (Q_out) from steam to the energy sink, such as the atmosphere.
• A pump is used to carry the water from the condenser back to the boiler. Work (W_in) is required to compress water to boiler pressure.

The net work output from this power plant is the difference between the work output and the work input.

      W_{net, out} = W_out - W_in

From the energy balance of the cycle, the net work output is

      W_{net, out} = Q_in - Q_out

A heat engine can only convert part of the energy it received from the source to work. A certain amount of heat is dissipated to the sink as waste heat. The fraction of the heat input that is converted to net work output is a measure of the performance of a heat engine and is called the thermal efficiency(η_th). In general, the efficiency (or performance) can be expressed in terms of the desired output and the required input as

      Performance = Desired output/ Required input

For heat engines, the desired output is the net work output (W_{net, out}) and the required input is the heat input( Q_in). Hence the thermal efficiency of a heat engine can be expressed as

      Thermal efficiency = Net work output/Heat input

or

      η_th= W_{net, out}/Q_in

Since W_{net, out} = Q_in - Q_out , it can be rewritten as

      η_th= (Q_in- Q_out)/Q_in = 1 - Q_out/Q_in

To bring uniformity to the treatment of heat engines, refrigerators, and heat pumps (wIll be introduced in the following paragraph), Q_H and Q_L are defined as

• Q_H equals the amount of heat transferred between the device (heat engines, refrigerators, and heat pumps) and a thermal reservoir of high temperature T_H .
• Q_L equals the amount of heat transferred between the device (heat engines, refrigerators, and heat pumps) and a thermal reservoir of low temperature T_L.

Note that Q_H and Q_L are all positive numbers.

Hence, the thermal efficiency for any heat engine is:

      η_th= W_{net, out}/Q_H = (Q_H- Q_L)/Q_H = 1 - Q_L/Q_H

Note that for heat engine, Q_L is always less than Q_H ,and η_th is always less than 1.

For refrigerators or heat pumps, the efficiency is in terms of the coefficient of performance (COP). A subscript R is used to denote refrigerators (COP_R) and HP for heat pumps (COP_HP).

A refrigerator is used to remove heat (Q_L) from a lower temperature space with an electric work input (W_net,in), then dissipates the total energy from the heat input and the electric work (Q_H) to a higher temperature thermal reservoir. Hence, the desired output is Q_L and the required input is W_{net, in}. So the COP_R can be expressed as

      COP_R = Heat removed/Net work input
                = Q_L / W_{net, in}
                = Q_L /( Q_H- Q_L)
                = 1/(Q_H/Q_L-1)

A heat pump is a device which transfers heat from a low-temperature medium to a high-temperature one. For example, a heat pump is used to heat a room in winter, which transfer heat from the low-temperature outdoor air to the high-temperature air inside the room. Hence, the desired output is the heat transferred to the room (Q_H). Also, a net work input (W_{net, in}) is necessary. The COP_HP can be expressed as

      COP_R = Heat delivered/Required work input
                = Q_H / W_{net, in}
                = Q_H /( Q_H- Q_L)
                = 1/(1-Q_L/Q_H)