Efficiency of an ideal machine formula. Thermal engine. Second law of thermodynamics. Maximum efficiency value of heat engines

The work done by the engine is:

This process was first considered by the French engineer and scientist N. L. S. Carnot in 1824 in the book “Reflections on driving force fire and about machines capable of developing this force."

The goal of Carnot's research was to find out the reasons for the imperfection of heat engines of that time (they had an efficiency of ≤ 5%) and to find ways to improve them.

The Carnot cycle is the most efficient of all. Its efficiency is maximum.

The figure shows the thermodynamic processes of the cycle. During isothermal expansion (1-2) at temperature T 1 , work is done due to a change in the internal energy of the heater, i.e. due to the supply of heat to the gas Q:

A 12 = Q 1 ,

Gas cooling before compression (3-4) occurs during adiabatic expansion (2-3). Change in internal energy ΔU 23 during an adiabatic process ( Q = 0) is completely converted to mechanical work:

A 23 = -ΔU 23 ,

The gas temperature as a result of adiabatic expansion (2-3) drops to the temperature of the refrigerator T 2 < T 1 . In process (3-4), the gas is isothermally compressed, transferring the amount of heat to the refrigerator Q 2:

A 34 = Q 2,

The cycle ends with the process of adiabatic compression (4-1), in which the gas is heated to a temperature T 1.

Maximum efficiency value of ideal gas heat engines according to the Carnot cycle:

.

The essence of the formula is expressed in the proven WITH. Carnot's theorem that the efficiency of any heat engine cannot exceed the efficiency of a Carnot cycle carried out at the same temperature of the heater and refrigerator.

« Physics - 10th grade"

To solve problems, you need to use known expressions for determining the efficiency of heat engines and keep in mind that expression (13.17) is valid only for an ideal heat engine.

Task 1.

In the boiler of a steam engine the temperature is 160 °C, and the temperature of the refrigerator is 10 °C.
What is the maximum work that a machine can theoretically perform if coal weighing 200 kg with a specific heat of combustion of 2.9 10 7 J/kg is burned in a furnace with an efficiency of 60%?

Solution.

The maximum work can be done by an ideal heat engine operating according to the Carnot cycle, the efficiency of which is η = (T 1 - T 2)/T 1, where T 1 and T 2 are the absolute temperatures of the heater and refrigerator. For any heat engine, the efficiency is determined by the formula η = A/Q 1, where A is the work performed by the heat engine, Q 1 is the amount of heat received by the machine from the heater.
From the conditions of the problem it is clear that Q 1 is part of the amount of heat released during fuel combustion: Q 1 = η 1 mq.

Then where does A = η 1 mq(1 - T 2 /T 1) = 1.2 10 9 J.

Task 2.

A steam engine with a power of N = 14.7 kW consumes fuel weighing m = 8.1 kg per 1 hour of operation, with a specific heat of combustion q = 3.3 10 7 J/kg.
Boiler temperature 200 °C, refrigerator 58 °C.
Determine the efficiency of this machine and compare it with the efficiency of an ideal heat engine.

Solution.

The efficiency of a heat engine is equal to the ratio of the completed mechanical work A to the expended amount of heat Qlt released during fuel combustion.
Amount of heat Q 1 = mq.

Work done during the same time A = Nt.

Thus, η = A/Q 1 = Nt/qm = 0.198, or η ≈ 20%.

For an ideal heat engine η < η ид.

Task 3.

An ideal heat engine with efficiency η operates in a reverse cycle (Fig. 13.15).

Which maximum amount heat can be taken away from the refrigerator by performing mechanical work A?

Since the refrigeration machine operates in a reverse cycle, in order for heat to transfer from a less heated body to a more heated one, it is necessary for external forces to do positive work.
Schematic diagram of a refrigeration machine: a quantity of heat Q 2 is taken from the refrigerator, work is done by external forces and a quantity of heat Q 1 is transferred to the heater.
Hence, Q 2 = Q 1 (1 - η), Q 1 = A/η.

Finally, Q 2 = (A/η)(1 - η).

Source: “Physics - 10th grade”, 2014, textbook Myakishev, Bukhovtsev, Sotsky

Fundamentals of thermodynamics. Thermal phenomena - Physics, textbook for grade 10 - Classroom physics

The working fluid, receiving a certain amount of heat Q1 from the heater, gives part of this amount of heat, equal in modulus |Q2|, to the refrigerator. Therefore, the work done cannot be greater A = Q1 - |Q2|. The ratio of this work to the amount of heat received by the expanding gas from the heater is called efficiency heat engine:

The efficiency of a heat engine operating in a closed cycle is always less than one. The task of thermal power engineering is to make the efficiency as high as possible, that is, to use as much of the heat received from the heater as possible to produce work. For the first time, the most perfect cyclic process, consisting of isotherms and adiabats, was proposed by the French physicist and engineer S. Carnot in 1824.

3) By ideal we mean a heat engine that has maximum efficiency. at given values ​​of the heater T1 and refrigerator T2.
From the second law of thermodynamics it follows that even an ideal heat engine operating without losses has efficiency. fundamentally below 100% and is calculated using the formula:

The working fluid in an ideal heat engine is an ideal gas, and it operates according to the Carnot cycle:

4) Concept entropy was first introduced by Clausius in thermodynamics to determine the measure of irreversible energy dissipation, measures of deviation of a real process from an ideal one. Defined as the sum of reduced heats, it is a function of state and remains constant in closed reversible processes, while in irreversible processes its change is always positive.

Mathematically, entropy is defined as a function of the state of the system, equal in an equilibrium process to the amount of heat imparted to the system or removed from the system, related to the thermodynamic temperature of the system:

where is the entropy increment; - minimum heat supplied to the system; - absolute temperature of the process.

Entropy establishes a connection between macro- and micro-states. The peculiarity of this characteristic is that it is the only function in physics that shows the direction of processes. Since entropy is a function of state, it does not depend on how the transition from one state of the system to another is carried out, but is determined only by the initial and final states of the system.

For example, at a temperature of 0 °C, water can be in a liquid state and, with little external influence, begins to quickly turn into ice, releasing a certain amount of heat. In this case, the temperature of the substance remains 0 °C. The state of a substance changes, accompanied by the release of heat, due to a change in structure.

Rudolf Clausius gave the quantity the name "entropy", which comes from the Greek word τρoπή, "change" (change, transformation, transformation). This equality refers to the change in entropy, without completely defining the entropy itself.

The main significance of the formula (5.12.2) obtained by Carnot for the efficiency of an ideal machine is that it determines the maximum possible efficiency of any heat engine.

Carnot proved, based on the second law of thermodynamics*, the following theorem: any real heat engine operating with a temperature heaterT 1 and refrigerator temperatureT 2 , cannot have an efficiency that exceeds the efficiency of an ideal heat engine.

* Carnot actually established the second law of thermodynamics before Clausius and Kelvin, when the first law of thermodynamics had not yet been formulated strictly.

Let us first consider a heat engine operating in a reversible cycle with a real gas. The cycle can be anything, it is only important that the temperatures of the heater and refrigerator are T 1 And T 2 .

Let us assume that the efficiency of another heat engine (not operating according to the Carnot cycle) η ’ > η . The machines operate with a common heater and a common refrigerator. Let the Carnot machine operate in a reverse cycle (like a refrigeration machine), and let the other machine operate in a forward cycle (Fig. 5.18). The heat engine performs work equal to, according to formulas (5.12.3) and (5.12.5):

A refrigeration machine can always be designed so that it takes the amount of heat from the refrigerator Q 2 = ||

Then, according to formula (5.12.7), work will be done on it

(5.12.12)

Since by condition η" > η , That A" > A. Therefore, a heat engine can drive a refrigeration machine, and there will still be an excess of work left. This excess work is done by heat taken from one source. After all, heat is not transferred to the refrigerator when two machines operate at once. But this contradicts the second law of thermodynamics.

If we assume that η > η ", then you can make another machine work in a reverse cycle, and a Carnot machine in a forward cycle. We will again come to a contradiction with the second law of thermodynamics. Consequently, two machines operating on reversible cycles have the same efficiency: η " = η .

It’s a different matter if the second machine operates on an irreversible cycle. If we assume η " > η , then we will again come to a contradiction with the second law of thermodynamics. However, the assumption t|"< г| не противоречит второму закону термодинамики, так как необратимая тепловая машина не может работать как холодильная машина. Следовательно, КПД любой тепловой машины η" ≤ η, or

This is the main result:

(5.12.13)

Efficiency of real heat engines

Formula (5.12.13) gives the theoretical limit for the maximum efficiency value of heat engines. It shows that the higher the temperature of the heater and the lower the temperature of the refrigerator, the more efficient a heat engine is. Only at a refrigerator temperature equal to absolute zero does η = 1.

But the temperature of the refrigerator practically cannot be much lower than the ambient temperature. You can increase the heater temperature. However, any material (solid body) has limited heat resistance, or heat resistance. When heated, it gradually loses its elastic properties, and at a sufficiently high temperature it melts.

Now the main efforts of engineers are aimed at increasing the efficiency of engines by reducing the friction of their parts, fuel losses due to incomplete combustion, etc. Real opportunities for increasing efficiency here still remain great. Thus, for a steam turbine, the initial and final steam temperatures are approximately as follows: T 1 = 800 K and T 2 = 300 K. At these temperatures, the maximum efficiency value is:

The actual efficiency value due to various types of energy losses is approximately 40%. The maximum efficiency - about 44% - is achieved by internal combustion engines.

The efficiency of any heat engine cannot exceed the maximum possible value
, where T 1 - absolute temperature of the heater, and T 2 - absolute temperature of the refrigerator.

Increasing the efficiency of heat engines and bringing it closer to the maximum possible- the most important technical challenge.