The principle of operation of heat engines. Coefficient of performance (COP) of heat engines

Coefficient of performance (COP) - a term that can be applied, perhaps, to every system and device. Even a person has an efficiency, though, probably, there is no objective formula for finding it yet. In this article, we will explain in detail what efficiency is and how it can be calculated for various systems.

efficiency definition

Efficiency is an indicator that characterizes the efficiency of a particular system in relation to the return or conversion of energy. Efficiency is a measureless quantity and is represented either numerical value in the range from 0 to 1, or as a percentage.

General formula

Efficiency is indicated by the symbol Ƞ.

General mathematical formula finding the efficiency is written as follows:

Ƞ=A/Q, where A is the useful energy/work done by the system, and Q is the energy consumed by this system to organize the process of obtaining a useful output.

The efficiency factor, unfortunately, is always less than one or equal to it, since, according to the law of conservation of energy, we cannot get more work than the energy spent. In addition, the efficiency, in fact, is extremely rarely equal to one, since useful work is always accompanied by losses, for example, for heating the mechanism.

Heat engine efficiency

A heat engine is a device that converts thermal energy into mechanical. In a heat engine, work is determined by the difference between the amount of heat received from the heater and the amount of heat given to the cooler, and therefore the efficiency is determined by the formula:

Ƞ=Qн-Qх/Qн, where Qн is the amount of heat received from the heater, and Qх is the amount of heat given to the cooler.

It is believed that the highest efficiency is provided by engines operating on the Carnot cycle. In this case, the efficiency is determined by the formula:

Ƞ=T1-T2/T1, where T1 is the temperature of the hot source, T2 is the temperature of the cold source.

Electric motor efficiency

An electric motor is a device that converts electrical energy into mechanical energy, so the efficiency in this case is the efficiency ratio of the device in relation to conversion electrical energy into mechanical. The formula for finding the efficiency of an electric motor looks like this:

Ƞ=P2/P1, where P1 - supplied electric power, P2 - useful mechanical power generated by the engine.

Electrical power is found as the product of system current and voltage (P=UI), and mechanical power is found as the ratio of work to unit time (P=A/t)

transformer efficiency

A transformer is a device that converts alternating current one voltage into an alternating current of another voltage, keeping the frequency. In addition, transformers can also convert AC to DC.

The efficiency of the transformer is found by the formula:

Ƞ=1/1+(P0+PL*n2)/(P2*n), where P0 - mode loss idle move, PL - load losses, P2 - active power delivered to the load, n - relative degree of loading.

Efficiency or not efficiency?

It is worth noting that in addition to efficiency, there are a number of indicators that characterize the efficiency of energy processes, and sometimes we can find descriptions of the type - efficiency of the order of 130%, however, in this case, you need to understand that the term is not used quite correctly, and, most likely, the author or the manufacturer understands a slightly different characteristic by this abbreviation.

For example, heat pumps are distinguished by the fact that they can give off more heat than they consume. Thus, the refrigerating machine can remove more heat from the cooled object than is spent in energy equivalent for the organization of the removal. The efficiency indicator of a refrigerating machine is called the coefficient of performance, denoted by the letter Ɛ and is determined by the formula: Ɛ=Qx/A, where Qx is the heat removed from the cold end, A is the work expended on the removal process. However, sometimes the coefficient of performance is also called the efficiency of the refrigeration machine.

It is also interesting that the efficiency of boilers running on fossil fuels is usually calculated on the basis of the lower calorific value, while it can turn out to be more than one. However, it is still traditionally referred to as efficiency. It is possible to determine the efficiency of the boiler by the gross calorific value, and then it will always be less than one, but in this case it will be inconvenient to compare the performance of the boilers with the data of other installations.

In life, a person is faced with the problem and the need to transform different types of energy. Devices that are designed to convert energy are called energy machines (mechanisms). For example, power machines include: an electric generator, an internal combustion engine, an electric motor, a steam engine, etc.

In theory, any kind of energy can completely transform into another kind of energy. But in practice, in addition to energy transformations in machines, energy transformations occur, which are called losses. The perfection of power machines determines the coefficient of performance (COP).

DEFINITION

The efficiency of the mechanism (machine) called the ratio of useful energy () to the total energy (W), which is supplied to the mechanism. Usually, the efficiency is denoted by the letter (this). AT mathematical form The definition of efficiency is written as follows:

Efficiency can be defined in terms of work, as the ratio of (useful work) to A ( full work):

It can also be found as a power ratio:

where is the power supplied to the mechanism; - the power that the consumer receives from the mechanism. Expression (3) can be written differently:

where is the part of the power that is lost in the mechanism.

From the definitions of efficiency, it is obvious that it cannot be more than 100% (or cannot be more than one). The interval in which the efficiency is located: .

The efficiency factor is used not only in assessing the level of perfection of the machine, but also in determining the effectiveness of any complex mechanism and all kinds of devices that are energy consumers.

They try to make any mechanism so that useless energy losses are minimal (). For this purpose, they try to reduce the friction forces (various kinds of resistance).

Efficiency of connections of mechanisms

When considering a structurally complex mechanism (device), the efficiency of the entire structure and the efficiency of all its nodes and mechanisms that consume and convert energy are calculated.

If we have n mechanisms that are connected in series, then the resulting system efficiency is found as the product of the efficiency of each part:

At parallel connection mechanisms (Fig. 1) (one engine drives several mechanisms), useful work is the sum of useful work at the output of each individual part of the system. If the work expended by the engine is denoted as , then the efficiency in this case is found as:

Efficiency units

In most cases, efficiency is expressed as a percentage.

Examples of problem solving

EXAMPLE 1

Exercise

What is the power of the mechanism that lifts a hammer having a mass m to a height h n times per second if the efficiency of the machine is ?

Solution

Power (N) can be found from its definition as:

Since the frequency () is specified in the condition (the hammer rises n times per second), we find the time as:

The job will be found as:

In this case (taking into account (1.2) and (1.3)) expression (1.1) is transformed into:

Since the efficiency of the system is , we write:

where is the desired power, then:

Answer

EXAMPLE 2

Exercise

What will be the efficiency of an inclined plane if its length, height h? The coefficient of friction when a body moves about a given plane is equal to .

Solution

Let's make a drawing.

As a basis for solving the problem, we take the formula for calculating the efficiency in the form:

Useful work will be the work of lifting the load to a height h:

The work done during the delivery of cargo by moving it along a given plane can be found as:

where is the traction force, which we find from Newton's second law, considering the forces that are applied to the body (Fig. 1):

Publication date 28.01.2013 13:48

Not a single action is carried out without losses - there are always losses. The result obtained is always less than the effort that has to be expended to achieve it. About how great the losses in the performance of work, and evidenced by the coefficient of performance (COP).

What is hidden behind this abbreviation? In fact, it is the coefficient of efficiency of the mechanism or indicator rational use energy. The value of efficiency does not have any units of measurement, it is expressed as a percentage. This coefficient is defined as the ratio of the useful work of the device to the work spent on its operation. To calculate the efficiency, the calculation formula will look like this:

Efficiency \u003d 100 * (useful work performed / work expended)

In various devices, to calculate this ratio, they use different meanings. For electric motors, the efficiency will look like the ratio of the useful work done to the electrical energy received from the network. For heat engines, the efficiency will be defined as the ratio of the useful work performed to the amount of heat consumed.

To determine the efficiency, it is necessary that all different types energy and work were expressed in the same units. Then it will be possible to compare any objects, such as nuclear power plants, power generators and biological objects, in terms of efficiency.

As already noted, due to inevitable losses during the operation of mechanisms, the efficiency is always less than 1. Thus, the efficiency of thermal plants reaches 90%, for internal combustion engines the efficiency is less than 30%, the efficiency of an electrical transformer is 98%. The concept of efficiency can be applied both to the mechanism as a whole and to its individual nodes. In a general assessment of the effectiveness of the mechanism as a whole (its efficiency), the product of the efficiency of individual constituent parts this device.

Problem effective use fuel did not appear today. With a continuous increase in the cost of energy resources, the issue of increasing the efficiency of mechanisms is turning from a purely theoretical into a practical issue. If the efficiency of a conventional car does not exceed 30%, then we simply throw away 70% of our money spent on fueling a car.

Consideration of the efficiency of the internal combustion engine (internal combustion engine) shows that losses occur at all stages of its operation. So, only 75% of the incoming fuel burns in the engine cylinders, and 25% is released into the atmosphere. Of all the burned fuel, only 30-35% of the released heat is spent on useful work, the rest of the heat is either lost with exhaust gases or remains in the car's cooling system. Of the received power, about 80% is used for useful work, the rest of the power is spent on overcoming friction forces and is used by the auxiliary mechanisms of the car.

Even on such simple example analysis of the efficiency of the mechanism allows you to determine the directions in which work should be carried out to reduce losses. Yes, one of priority areas– ensuring complete combustion of fuel. This is achieved by additional spraying of fuel and increasing pressure, which is why engines with direct injection and turbocharging are becoming so popular. The heat removed from the engine is used to heat the fuel to improve its volatility, and mechanical losses are reduced through the use of modern synthetic oils.

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

Not a single action is carried out without losses - they are always there. The result obtained is always less than the effort that has to be expended to achieve it. About how great the losses in the performance of work, and evidenced by the coefficient of performance (COP).

What is hidden behind this abbreviation? In essence, this is the coefficient of efficiency of the mechanism or an indicator of the rational use of energy. The value of efficiency does not have any units of measurement, it is expressed as a percentage. This coefficient is defined as the ratio of the useful work of the device to the work spent on its operation. To calculate the efficiency, the calculation formula will look like this:

Efficiency \u003d 100 * (useful work performed / work expended)

Different devices use different values to calculate this ratio. For electric motors, the efficiency will look like the ratio of the useful work done to the electrical energy received from the network. For will be defined as the ratio of the useful work done to the amount of heat consumed.

To determine the efficiency, it is necessary that all are different and the work is expressed in the same units. Then it will be possible to compare any objects, such as electricity generators and biological objects, in terms of efficiency.

The problem of efficient use of fuel did not appear today. With a continuous increase in the cost of energy resources, the issue of increasing the efficiency of mechanisms is turning from a purely theoretical into a practical issue. If the efficiency of a conventional car does not exceed 30%, then we simply throw away 70% of our money spent on fueling a car.

Even on such a simple example, the analysis of the effectiveness of the mechanism allows you to determine the directions in which work should be carried out to reduce losses. Thus, one of the priorities is to ensure the complete combustion of fuel. This is achieved by additional spraying of fuel and increasing pressure, which is why engines with direct injection and turbocharging are becoming so popular. The heat removed from the engine is used to preheat the fuel in order to improve its volatility, and mechanical losses are reduced through the use of modern grades

Here we have considered such a concept as it is described what it is and what it affects. The efficiency of its work is considered on the example of an internal combustion engine and the directions and ways of increasing the capabilities of this device, and, consequently, the efficiency, are determined.