Optimal Heat Engines

[11.1] Heat Engine : The Model

The engines constructed over the industrial revolution used the heat produced by burning a fuel to produce useful mechanical work. The efficiency of a heat engine is the ratio of the work output t, to the heat input, such that

E= Wout/Qin

The concept of a heat engine is a theoretical model introduced to explain the process of

engines in terms of the laws of thermodynamics. The thermodynamic model of a heat engine

assumes a cyclic process of an ideal working uid that operates between two heat reservoirs

at xed temperatures TH and TC (TH>TC). Over each cycle of the engine, the working uid

acquires a QH amount of heat from the heat reservoir at the temperature TH (heat source),

and releases a QC amount of heat to the heat reservoir at the temperature TC (heat sink).

The schematic diagram (see the image) of a heat engine symbolizes all the basic assumptions

of this model,

In the schematic diagram, the central circle (not necessary to be a circle though) represent the working uid that undergoes the cyclic process. The arrows represent the heat and work flow for the working fluid. According to the conventions, the heat given to the working uid, and the work done by the working uid are considered positive (+). Then, the heat removed from the working uid and the work done on the working uid becomes negative(-). Therefore from the rst law of thermodynamics,

W = (QH - QC)

And therefore the thermodynamic efficiency of a heat engine become,

[11.1.1] Idealization of Real Engines

Different types of engines (steam engine, four strokes internal combustion engine, gas turbines,

etc.) brought into the theoretical idealization of a heat engine through approximating

their processes to a thermodynamic quasi-static cyclic process. For example, the Otto cycle

approximates the process of a four stroke spark ignited engine (See Sec. 10.3) to a thermodynamics

cycle containing who isochoric processes, two adiabatic processes, and two isobaric

processes.

The theoretical model does not account for the frictional losses that can occur in the real

engine, Therefore thermodynamic efficiency calculated using the specified thermodynamic

processes tends to be greater than the actual efficiency of the engine, such that

"thermodynamic > "actual"

[11.2] A Perfect Engine and the Second Law of Thermodynamics

A perfect engine would would convert the entire heat input into useful work such that the eciency of the engine is 1.

[11.2.1] Perfect Heat Engine: Impossible!

A perfect heat engine is a device that converts thermal energy into mechanical work at 100% efficiency while taking a working uid along a cyclic process. Such a device is impossible even in principle, as it is impossible to carry a working substance of a heat engine through a cyclic process without releasing a certain amount of heat into a low temperature body.

[11.2.2] Kelvin-Planck Statement of the Second law of Thermodynamics

The Kelvin- Plank statement of the second law of thermodynamics states that it is impossible to construct a heat engine that, operating in a cycle, produces no effect other than the absorption of energy from a heat reservoir and the performance of an equal amount of work.

[11.3] Heat Pump

Devices such as refrigerators, air conditioners, interior heaters move heat from a cold heat source to a warmer heat sink though the aid of mechanical work input (see. Sec. 10.4). Such devices are commonly called heat-pumps. The coefficient of performance of a heat-pump (C.O.P.H:P:) is the ratio of the heat output into the warmer heat sink QH and the work input to the device Win, such that,

When a heat-pump is used as a refrigerator, the coefficient of performance (C.O.P.Ref:) is the ratio of the heat acquired from the colder heat source QC and the work input to the device Win, such that,

The thermodynamic model of a heat-pump assumes a cyclic process of an ideal working fluid that operates between two heat reservoirs at fixed temperatures TH and TC (TH>TC). Over each cycle of the heat-pump, the working uid acquires a QC amount of heat from the heat reservoir at the temperature TC (heat source), and releases a QH amount of heat to the heat reservoir at the temperature TH (heat sink). The schematic diagram (see the image) of a heat-pump symbolizes all the basic assumptions of this model.

As implemented for a heat-engine the, the central circle of the schematic diagram represents

the working uid that undergoes the cyclic process. The arrows represent the heat and work flow for the working uid. For a heat pump the work is done on the working uid while it acquire a QC amount of heat from the cold heat reservoir and releases a QH amount of heat of the warmer heat sink. Then within the sign conventions for heat and work flow the first law of thermodynamics imply for a single cycle of the working uid that,

[11.4] Perfect Heat Pump

A perfect heat pump would operate in a cyclic process to acquire a certain amount of heat from a cold heat source at the temperature TC and transfer that exact amount of heat into a heat sink at a higher temperature TH with no input of work.

[11.4.1] Perfect Heat Pump

A perfect heat pump is a device that carry a certain working fuld in a cyclic process enabling the transport of heat from a cold heat source to a warm heat source with no input of work. Such a device is not possible as the process violates the second law of thermodynamics. A different version of the second law of thermodynamics known as the Clausius statement has been given to identify this phenomenon.

[11.4.2] The Clausius Statement of the Second Law of Thermodynamics

The Clausius statement of the second law of thermodynamics states that it is impossible to construct a device which operates on a cycle and produces no other effect than the transfer of heat from a cooler body to a hotter body.

[11.5] Maximum Efficiency of a Heat Engine: Theoretical Limit

The construction of a perfect heat engine or a heat-pump violates the second law of thermodynamics and therefore will not be a physically achievable. However, it would be interesting to nd if there is a theoretical upper bound for the performance of an engine.

[11.6] Carnot Theorem: Maximum Efficiency of an Engine

Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, is a principle that impose a theoretical limit on the maximum efficiency obtained by any heat engine. The theorem states that the maximum performance of an engine that is operating between two heat reservoirs is achieved when the engine operates through reversible pro- cesses. This maximum performance would depend solely on the difference between the hot and cold temperatures of the heat reservoirs . The particular eciency (performance) acquired by an engine meeting these conditions is known as the Carnot efficiency (performance).

[11.6.2] Carnot Cycle

Introducing an upper bound for the performance of an engine, Carnot also suggest a method to achieve this ideal eciency and the process Carnot proposed became into knowledge as the Carnot engine or the Carnot cycle. The Carnot engine involves four reversible processes performed on a system of an ideal gas.

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