The general formula for efficiency. Maximum efficiency of heat engines (Carnot's theorem)

Efficiency (efficiency) - a characteristic of the efficiency of a system (device, machine) in relation to the conversion or transfer of energy. It is determined by the ratio of useful energy used to the total amount of energy received by the system; usually denoted η ("this"). η = Wpol/Wcym. Efficiency is a dimensionless quantity and is often measured as a percentage. Mathematically, the definition of efficiency can be written as:

X 100%

where BUT- useful work, and Q- spent energy.

By virtue of the law of conservation of energy, the efficiency is always less than unity or equal to it, that is, it is impossible to obtain more useful work than the energy expended.

Heat engine efficiency- the ratio of the perfect useful work of the engine, to the energy received from the heater. The efficiency of a heat engine can be calculated using the following formula

where - the amount of heat received from the heater, - the amount of heat given to the refrigerator. The highest efficiency among cyclic machines operating at given hot spring temperatures T 1 and cold T 2, have heat engines operating on the Carnot cycle; this limiting efficiency is equal to

Not all indicators characterizing the efficiency of energy processes correspond to the above description. Even if they are traditionally or erroneously called "efficiency", they may have other properties, in particular, exceed 100%.

boiler efficiency

Main article: Boiler thermal balance

The efficiency of fossil fuel boilers is traditionally calculated from the net calorific value; it is assumed that the moisture of the combustion products leaves the boiler in the form of superheated steam. In condensing boilers, this moisture is condensed, the heat of condensation is usefully used. When calculating the efficiency according to the lower calorific value, it can eventually turn out to be more than one. In this case, it would be more correct to consider it according to the gross calorific value, which takes into account the heat of steam condensation; however, the performance of such a boiler is difficult to compare with data from other installations.

Heat pumps and chillers

The advantage of heat pumps as a heating technique is the ability to sometimes receive more heat than the energy spent on their work; similarly, a refrigeration machine can remove more heat from the cooled end than is expended in organizing the process.

The efficiency of such heat engines is characterized by coefficient of performance(for chillers) or transformation ratio(for heat pumps)

where is the heat taken from the cold end (in refrigeration machines) or transferred to the hot end (in heat pumps); - the work (or electricity) spent on this process. The best performance indicators for such machines have the reverse Carnot cycle: in it the coefficient of performance

where , are the temperatures of the hot and cold ends, . This value, obviously, can be arbitrarily large; although practically it is difficult to approach it, the coefficient of performance can still exceed unity. This does not contradict the first law of thermodynamics, since, in addition to the energy taken into account A(e.g. electric), into heat Q there is also energy taken from a cold source.

Literature

• Peryshkin A.V. Physics. 8th grade. - Bustard, 2005. - 191 p. - 50,000 copies. - ISBN 5-7107-9459-7.

Notes

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Energy supplied to the mechanism in the form of work of driving forces A dv.s. and moments for a cycle of steady motion, is spent on useful work A p.s. , as well as to work A Ftr associated with overcoming the forces of friction in kinematic pairs and the forces of resistance of the medium.

Consider steady motion. The increment of kinetic energy is equal to zero, i.e.

In this case, the work of the forces of inertia and the forces of gravity are equal to zero A Ri = 0, And G = 0. Then, for a steady motion, the work of the driving forces is equal to

And dv.s. =A p.s. + A Ftr.

Consequently, for a full cycle of steady motion, the work of all driving forces is equal to the sum of the work of the forces of production resistance and non-production resistance (friction forces).

Mechanical efficiency η (efficiency)- the ratio of the work of the forces of production resistance to the work of all driving forces during the steady motion:

η = . (3.61)

As can be seen from formula (3.61), the efficiency shows what fraction of the mechanical energy brought to the machine is usefully spent on doing the work for which the machine was created.

The ratio of the work of the forces of non-productive resistance to the work of the driving forces is called loss factor :

ψ = . (3.62)

The mechanical loss factor shows what proportion of the mechanical energy supplied to the machine is ultimately converted into heat and wasted uselessly in the surrounding space.

From here we have a relationship between efficiency and loss factor

η =1- ψ.

It follows from this formula that in no mechanism the work of the forces of non-productive resistances can be equal to zero, therefore the efficiency is always less than one ( η <1 ). From the same formula it follows that the efficiency can be equal to zero if A dv.s \u003d A Ftr. The movement in which A dv.s \u003d A Ftr is called single . The efficiency cannot be less than zero, because for this it is necessary that A dv.s<А Fтр . The phenomenon in which the mechanism is at rest and at the same time the condition A dv.s is satisfied<А Fтр, называется the phenomenon of self-braking mechanism. The mechanism for which η = 1 is called perpetual motion machine .

Thus, the efficiency is in the range

0 £ η < 1 .

Consider the definition of efficiency for various ways of connecting mechanisms.

3.2.2.1. Determination of efficiency in series connection

Let there be n sequentially connected mechanisms (Figure 3.16).

And dv.s. 1 A 1 2 A 2 3 A 3 A n-1 n A n

Figure 3.16 - Scheme of series-connected mechanisms

The first mechanism is set in motion by driving forces that do work A dv.s. Since the useful work of each previous mechanism spent on production resistances is the work of the driving forces for each subsequent mechanism, the efficiency of the first mechanism will be equal to:

η 1 \u003d A 1 /A dv.s ..

For the second mechanism, the efficiency is:

η 2 \u003d A 2 /A 1 .

And, finally, for the nth mechanism, the efficiency will look like:

η n \u003d A n /A n-1

The overall efficiency is:

η 1 n \u003d A n /And dv.s.

The value of the overall efficiency can be obtained by multiplying the efficiency of each individual mechanism, namely:

η 1 n = η 1 η 2 η 3 …η n= .

Consequently, general mechanical efficiency in series connected mechanisms equals work mechanical efficiency of individual mechanisms that make up one common system:

η 1 n = η 1 η 2 η 3 …η n .(3.63)

3.2.2.2 Determining the efficiency in a mixed connection

In practice, the connection of mechanisms turns out to be more complicated. More often series connection is combined with parallel. Such a connection is called mixed. Consider an example of a complex connection (Figure 3.17).

The flow of energy from mechanism 2 is distributed in two directions. In turn, from the mechanism 3 ¢¢ the energy flow is also distributed in two directions. The total work of the forces of production resistance is equal to:

And p.s. = A ¢ n + A ¢ ¢ n + A ¢ ¢¢ n.

The overall efficiency of the entire system will be equal to:

η \u003d A p.s /A dv.s =(A ¢ n + A ¢ ¢ n + A ¢ ¢¢ n)/A dv.s . (3.64)

To determine the overall efficiency, it is necessary to isolate the energy flows in which the mechanisms are connected in series, and calculate the efficiency of each flow. Figure 3.17 shows the solid line I-I, the dashed line II-II and the dash-dotted line III-III three energy flows from a common source.

And dv.s. A 1 A ¢ 2 A ¢ 3 ... A ¢ n-1 A ¢ n

II A ¢¢ 2 II

A ¢¢ 3 4 ¢¢ A ¢¢ 4 A ¢¢ n-1 n ¢¢ A ¢¢ n

III 3 ¢¢ …

A ¢ dv.s \u003d A ¢ n / η ¢ 1n

A ¢¢ dv.s = A ¢ ¢ n /η ¢¢ 1 n (3.65)

A ¢¢¢ dv.s =A ¢ ¢¢ n /η ¢¢¢ 1 n

The total work of the driving forces of the entire system will be equal to the sum

A dv.s = A ¢ dv.s + A ¢¢ dv.s + A ¢¢¢ dv.s.

Or A dv.s=(A¢n / η ¢ 1n)+(A ¢ ¢ n /η ¢¢ 1n)+(A¢¢¢n /η ¢¢¢ 1n).

Substituting this expression into formula (3.64), we obtain efficiency equation for mixed connections

For parallel-connected mechanisms, the method for determining the efficiency is similar to the previous case.

The efficiency factor (COP) is a value that expresses in percentage terms the efficiency of a particular mechanism (engine, system) regarding the conversion of the received energy into useful work.

Read in this article

Why diesel efficiency is higher

The efficiency index for different engines can vary greatly and depends on a number of factors. have a relatively low efficiency due to the large number of mechanical and thermal losses that occur during the operation of a power unit of this type.

The second factor is the friction that occurs during the interaction of mating parts. Most of the useful energy consumption is driven by the pistons of the engine, as well as the rotation of the parts inside the motor, which are structurally fixed on the bearings. About 60% of the combustion energy of gasoline is spent only to ensure the operation of these units.

Additional losses are caused by the operation of other mechanisms, systems and attachments. It also takes into account the percentage of losses due to resistance at the time of the next charge of fuel and air, and then the release of exhaust gases from the internal combustion engine cylinder.

If we compare a diesel plant and a gasoline engine, a diesel engine has a noticeably higher efficiency compared to a gasoline unit. Power units on gasoline have an efficiency of about 25-30% of the total amount of energy received.

In other words, out of 10 liters of gasoline spent on the engine, only 3 liters are spent on useful work. The rest of the energy from the combustion of fuel was dispersed into losses.

With the same displacement indicator, the power of an atmospheric gasoline engine is higher, but is achieved at higher speeds. The engine needs to be “turned”, losses increase, fuel consumption increases. It is also necessary to mention the torque, which literally means the force that is transmitted from the motor to the wheels and moves the car. Gasoline ICEs reach their maximum torque at higher RPMs.

A similar naturally aspirated diesel achieves peak torque at low rpm, while using less diesel to do useful work, which means higher efficiency and fuel economy.

Diesel fuel generates more heat compared to gasoline, the combustion temperature of diesel fuel is higher, and the knock resistance index is higher. It turns out that a diesel internal combustion engine has more useful work done on a certain amount of fuel.

Energy value of diesel fuel and gasoline

Diesel fuel is made up of heavier hydrocarbons than gasoline. The lower efficiency of a gasoline plant compared to a diesel engine also lies in the energy component of gasoline and the features of its combustion. Complete combustion of an equal amount of diesel fuel and gasoline will give more heat in the first case. Heat in a diesel engine is more fully converted into useful mechanical energy. It turns out that when burning the same amount of fuel per unit of time, it is the diesel engine that will do more work.

It is also worth considering the features of injection and the creation of appropriate conditions for the full combustion of the mixture. In a diesel engine, fuel is supplied separately from air, it is not injected into the intake manifold, but directly into the cylinder at the very end of the compression stroke. The result is a higher temperature and the most complete combustion of a portion of the working fuel-air mixture.

Results

Designers are constantly striving to improve the efficiency of both diesel and gasoline engines. An increase in the number of intake and exhaust valves per cylinder, active use, electronic control of fuel injection, throttle valve and other solutions can significantly increase efficiency. To a greater extent this applies to the diesel engine.

Thanks to these features, a modern diesel engine is able to completely burn a portion of diesel fuel saturated with hydrocarbons in the cylinder and produce a large torque at low revs. Low RPMs mean less friction loss and the resulting drag. For this reason, a diesel engine is today one of the most productive and economical types of internal combustion engines, the efficiency of which often exceeds 50%.

Read also

Why it's better to warm up the engine before driving: lubrication, fuel, wear of cold parts. How to warm up a diesel engine in winter.

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 exceeding 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 rigorously.

Consider first a heat engine operating on a reversible cycle with a real gas. The cycle can be any, 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 work with a common heater and a common cooler. Let the Carnot machine work in a reverse cycle (like a refrigeration machine), and the other machine in a direct cycle (Fig. 5.18). The heat engine performs work equal, according to formulas (5.12.3) and (5.12.5):

The 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 performed on it

(5.12.12)

Since by condition η" > η , then A" > A. Therefore, the heat engine can drive the refrigeration engine, and there will still be an excess of work. This excess work is done at the expense of heat taken from one source. After all, heat is not transferred to the refrigerator under the action of two machines 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 Carnot's machine in a straight line. We again come to a contradiction with the second law of thermodynamics. Therefore, two machines operating on reversible cycles have the same efficiency: η " = η .

It is a different matter if the second machine operates in an irreversible cycle. If we allow η " > η , then we again come to a contradiction with the second law of thermodynamics. However, the assumption m|"< 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 of heat engines. It shows that the heat engine is more efficient, the higher the temperature of the heater and the lower the temperature of the refrigerator. Only when the refrigerator temperature is equal to absolute zero, η = 1.

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

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 its incomplete combustion, etc. The real opportunities for increasing efficiency are still large here. So, 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 value of the efficiency is:

The actual value of the efficiency due to various kinds of energy losses is approximately 40%. The maximum efficiency - about 44% - have 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.

It is known that a perpetual motion machine is impossible. This is due to the fact that for any mechanism the statement is true: the total work done with the help of this mechanism (including heating the mechanism and the environment, to overcome the friction force) is always more useful work.

For example, more than half of the work of an internal combustion engine is wasted on heating the components of the engine; some heat is carried away by the exhaust gases.

It is often necessary to evaluate the effectiveness of the mechanism, the feasibility of its use. Therefore, in order to calculate what part of the work done is wasted and what part is useful, a special physical quantity is introduced that shows the efficiency of the mechanism.

This value is called the efficiency of the mechanism

The efficiency of a mechanism is equal to the ratio of useful work to total work. Obviously, the efficiency is always less than unity. This value is often expressed as a percentage. Usually it is denoted by the Greek letter η (read "this"). Efficiency is abbreviated as efficiency.

η \u003d (A_full / A_useful) * 100%,

where η efficiency, A_full full work, A_useful useful work.

Among engines, the electric motor has the highest efficiency (up to 98%). Efficiency of internal combustion engines 20% - 40%, steam turbine about 30%.

Note that for increasing the efficiency of the mechanism often try to reduce the force of friction. This can be done using various lubricants or ball bearings in which sliding friction is replaced by rolling friction.

Efficiency calculation examples

Consider an example. A cyclist with a mass of 55 kg climbs a hill with a mass of 5 kg, the height of which is 10 m, while doing 8 kJ of work. Find the efficiency of the bike. The rolling friction of the wheels on the road is not taken into account.

Solution. Find the total mass of the bicycle and the cyclist:

m = 55 kg + 5 kg = 60 kg

Let's find their total weight:

P = mg = 60 kg * 10 N/kg = 600 N

Find the work done on lifting the bike and the cyclist:

Auseful \u003d PS \u003d 600 N * 10 m \u003d 6 kJ

Let's find the efficiency of the bike:

A_full / A_useful * 100% = 6 kJ / 8 kJ * 100% = 75%

Answer: Bicycle efficiency is 75%.

Let's consider one more example. A body of mass m is suspended from the end of the lever arm. A downward force F is applied to the other arm, and its end is lowered by h. Find how much the body has risen if the efficiency of the lever is η%.

Solution. Find the work done by the force F:

η % of this work is done to lift a body of mass m. Therefore, Fhη / 100 was spent on lifting the body. Since the weight of the body is equal to mg, the body has risen to a height of Fhη / 100 / mg.