Thermal efficiency. Efficiency of a heat engine - formula

Modern realities assume a wide exploitation of thermal engines. Numerous attempts to replace them with electric motors have been failing so far. The problems associated with the accumulation of electricity in autonomous systems, are solved with great difficulty.

The problems of the technology of manufacturing electric energy batteries are still relevant, taking into account their long-term use. The speed characteristics of electric vehicles are far from those of cars on internal combustion engines.

The first steps to create hybrid engines can significantly reduce harmful emissions in megacities, solving environmental problems.

A bit of history

The possibility of converting steam energy into energy of motion was known in ancient times. 130 BC: Philosopher Heron of Alexandria presented a steam toy - eolipil - to the audience. The sphere, filled with steam, came into rotation under the action of the jets emanating from it. This prototype of modern steam turbines in those days did not find application.

Long years and centuries of development of the philosopher were considered only an amusing toy. In 1629 the Italian D. Branci created an active turbine. Steam set in motion a disk equipped with blades.

From this moment, rapid development of steam engines began.

Thermal machine

The transformation of the internal energy of fuel into the energy of movement of parts of machines and mechanisms is used in thermal machines.

The main parts of the machines are a heater (a system for obtaining energy from the outside), a working body (it makes a useful effect), a refrigerator.

The heater is designed to allow the working body to accumulate a sufficient supply of internal energy to perform useful work. The refrigerator takes away excess energy.

The main characteristic of efficiency is the efficiency of thermal machines. This value shows how much of the energy expended for heating is spent on committing useful work. The higher the efficiency, the better the performance of the machine, but this value can not exceed 100%.

Calculation of efficiency

Let the heater acquire from the outside the energy equal to Q _{1 .} The working body did work A, while the energy given to the refrigerator was Q _{2 .}

Based on the definition, calculate the value of the efficiency:

Η = A / Q ₁ . We take into account that A = Q ₁ - Q _2.

Hence, the efficiency of a heat engine, whose formula has the form η = (Q ₁ - Q ₂ ) / Q ₁ = 1 - Q ₂ / Q _1, allows us to draw the following conclusions:

• The efficiency can not exceed 1 (or 100%);
• To maximize this value, either an increase in the energy received from the heater or a decrease in the energy given to the refrigerator is necessary;
• Increase the energy of the heater to achieve a change in fuel quality;
• Reduction of energy given to the refrigerator, allow to achieve design features of engines.

The ideal heat engine

Is it possible to create such an engine, the efficiency of which would be maximum (ideally - equal to 100%)? A French theoretical physicist and talented engineer Sadi Carnot tried to find an answer to this question. In 1824, his theoretical accounts of the processes taking place in gases were made public.

The main idea embedded in an ideal machine can be considered as reversible processes with an ideal gas. We start with the gas expansion isothermally at temperature T ₁ . The amount of heat required for this is Q _1. After Gas without heat exchange expands (adiabatic process). Having reached the temperature T ₂ , the gas is isothermally compressed, transferring the energy Q ₂ to the refrigerator. The return of gas to its original state is done adiabatically.

The efficiency of an ideal Carnot thermal engine for accurate calculation is equal to the ratio of the temperature difference between the heating and cooling devices to the temperature that the heater has. It looks like this: η = (T ₁ - T ₂ ) / T _1.

The possible efficiency of a thermal machine, the formula of which has the form: η = 1 - T ₂ / T ₁ , depends only on the temperature of the heater and cooler and can not be more than 100%.

Moreover, this relationship allows us to prove that the efficiency of thermal machines can be equal to unity only when the refrigerator reaches absolute zero temperatures. As you know, this value is unattainable.

Theoretical calculations of Carnot allow us to determine the maximum efficiency of a thermal machine of any design.

The theorem proved by Carnot is as follows. An arbitrary thermal machine under any conditions is not capable of having an efficiency greater than that of an ideal heat engine.

Example of problem solving

Example 1. What is the efficiency of an ideal thermal machine, if the temperature of the heater is 800 ^о С, and the temperature of the refrigerator is 500 ^о С lower?

T ₁ = 800 ^о С = 1073 К, ΔT = 500 ^о С = 500 К, η -?

Decision:

By definition: η = (T ₁ - T ₂ ) / T _1.

We are not given the temperature of the refrigerator, but ΔT = (T ₁ - T ₂ ), hence:

Η = ΔT / T ₁ = 500 K / 1073 K = 0.46.

Answer: Efficiency = 46%.

Example 2. Determine the efficiency of an ideal thermal machine if at the cost of the purchased one kilojoule of heater energy, a useful work of 650 J is made. What is the temperature of the heater of the heat engine if the cooler temperature is 400 K?

Q ₁ = 1 kJ = 1000 J, A = 650 J, T ₂ = 400 K, η -?, T ₁ =?

Decision:

In this problem we are talking about a thermal installation, the efficiency of which can be calculated from the formula:

Η = A / Q _1.

To determine the temperature of the heater, we use the formula for the efficiency of an ideal thermal machine:

Η = (T ₁ - T ₂ ) / T ₁ = 1 - T ₂ / T _1.

Performing mathematical transformations, we get:

T ₁ = T ₂ / (1 - η).

T ₁ = T ₂ / (1-A / Q ₁ ).

We compute:

Η = 650 J / 1000 J = 0.65.

T ₁ = 400 K / (1 650 J / 1000 J) = 1142.8 K.

The answer is: η = 65%, T ₁ = 1142.8 K.

Real conditions

The ideal heat engine is designed taking into account the ideal processes. The work is done only in isothermal processes, its magnitude is defined as the area bounded by the graph of the Carnot cycle.

In fact, it is impossible to create conditions for the process of changing the state of a gas without accompanying temperature changes. There are no materials that would rule out heat exchange with surrounding objects. Adiabatic process is impossible. In the case of heat exchange, the temperature of the gas must necessarily change.

The efficiency of thermal machines created in real conditions is significantly different from the efficiency of ideal engines. We note that the process in real engines is so rapid that the variation of the internal thermal energy of the working substance during the process of changing its volume can not be compensated by the influx of heat from the heater and the return to the refrigerator.

Other thermal engines

Real engines work on other cycles:

• Otto cycle: the process for an unchanged volume varies adiabatically, creating a closed cycle;
• Diesel cycle: isobar, adiabat, isochor, adiabat;
• Gas turbine: a process that takes place at a constant pressure, is replaced by an adiabatic process, closes the cycle.

To create equilibrium processes in real engines (in order to approximate them to ideal ones) in the conditions of modern technology is not possible. The efficiency of thermal machines is much lower, even taking into account the same temperature conditions as in an ideal heat installation.

But do not reduce the role of the design formula for the efficiency of the Carnot cycle, since it becomes the reference point in the process of working on increasing the efficiency of real engines.

Ways of changing efficiency

Making a comparison of ideal and real thermal engines, it is worth noting that the temperature of the refrigerator can not be any. Usually a refrigerator is considered an atmosphere. The temperature of the atmosphere can only be accepted in approximate calculations. Experience shows that the temperature of the cooler is equal to the temperature of the exhaust gases in the engines, as is the case with internal combustion engines (abbreviated to ICE).

ICE is the most common thermal machine in our world. The efficiency of the heat engine in this case depends on the temperature created by the burning fuel. A significant difference between ICE and steam engines is the fusion of the functions of the heater and the working body of the device in an air-fuel mixture. Burning, the mixture creates pressure on the moving parts of the engine.

Increases in the temperature of working gases are achieved, significantly changing the properties of the fuel. Unfortunately, it is impossible to do this without limit. Any material from which the combustion chamber of the engine is made has a melting point. The heat resistance of such materials is the main characteristic of the engine, as well as the possibility to significantly affect the efficiency.

Values of engine efficiency

If we consider a steam turbine with a working steam inlet temperature of 800 K and an exhaust gas of 300 K, the efficiency of this machine is 62%. In reality, this value does not exceed 40%. Such a reduction occurs due to thermal losses when the turbine housing is heated.

The highest value of the efficiency of internal combustion engines does not exceed 44%. Increasing this value is a matter of the near future. Changing the properties of materials, fuels is a problem over which the best minds of mankind work.