Carnot Engine

Carnot engine is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824. Carnot states that a hot body is required that generates heat and a cold body to which the caloric is conveyed, which produces mechanical work in the process. It also states that said work is free of the material that is used to create heat and the construction and design material of the machine.

Modern Diagram

(Image will be Uploaded Soon)

The above figure displays a block diagram of a general heat engine, like the Carnot engine. In the diagram, the “working body”, a word presented by Clausius in 1850, can be any vapor or fluid body through which heat “Q” can be transmitted to yield work. Carnot had proposed that the fluid body could be any material capable of expansion, such as vapor of alcohol, the vapor of mercury, the vapor of water, a permanent air or gas etc. While, in these initial years, engines came in a number of patterns, usually QH was delivered by a boiler, wherein water was boiled over a heater; QC was usually delivered by a stream of cold flowing water in the form of a condenser situated on a separate part of the engine. The yield work, W, denotes the movement of the piston as it is used to rotate a crank-arm, which in turn was normally used to power a pulley so as to lift water out of submerged salt mines. Carnot states work as “weight lifted through a height”

Carnot Engine Principles

Carnot principles are just for cyclical devices such as heat engines, which state that:

The effectiveness of an irreversible heat engine is always less than the efficiency of a reversible one functioning between the similar two reservoirs.

The effectiveness of all reversible heat engines working between the similar two reservoirs is equal.

To escalate the thermal efficiency of a gas power turbine, it is essential to increase the temperature of the combustion room. such as, turbine blades cannot hold out the high-temperature gas which will eventually lead to early fatigue.

Carnot’s Theorem

(Image will be Uploaded Soon)

This theorem defines that no engine functioning between two known temperatures can be more effective than a reversible engine functioning between the similar two temperatures and that all the reversible engines functioning between the same two temperatures have the similar efficiency, whatever the working material might be. As per the Carnot theorem, the reversible engine will always have greater productivity than the irreversible one. The reversible heat engine works on a reverse cycle and behaves as a heat pump.

The Efficiency of Carnot’s Cycle

The Carnot cycle is reversible, signifying the upper limit on the efficiency of an engine cycle. Practical engine cycles are irreversible and therefore have inherently much lower efficiency than the Carnot efficiency when working at similar temperatures. One of the factors determining efficiency is the addition of the working fluid in the cycle and its removal. The Carnot cycle reaches maximum efficiency because all the heat is pushed to the working fluid at the maximum temperature.

Carnot Cycle

(Image will be Uploaded Soon)

Step 1: Isothermal Expansion 

Heat is passed reversibly from the high-temperature pool at fixed temperature TH (isothermal heat absorption). In this step (1 to 2 on image 1, A to B in image 2) the gas is allowed to expand, doing work on the surroundings by pushing up the piston (stage 1 figure, right). Even though the pressure drops from points 1 to 2 (image 1) the temperature of the gas does not alter in the process because it is in thermal contact with the hot pool at Th, and therefore the expansion is isothermal. Heat energy Q1 is absorbed from the high-temperature pool resulting in an increase in the entropy of the gas by the amount.

\[ \Delta S_1 = \frac {Q_1}{T_h}\]

Step 2: Isentropic (Reversible Adiabatic)

Expansion of the gas. In this step (2 to 3 on image 1, B to C in image 2) the gas in the engine is thermally shielded from both the hot and cold pools. Thus they cannot gain nor lose heat, so it is called an 'adiabatic' process. The gas rises to expand by the drop in pressure, doing work on the surroundings (raising the piston; stage 2 figure, above), and losing a volume of internal energy similar to the work done. The gas begins to expand without heat input causing it to cool to the "cold" temperature, Tc. The entropy remains the same.

Step 3: Isothermal Compression

Heat shifted reversibly to a low-temperature pool at constant temperature TC. (isothermal heat elimination) (3 to 4 on image 1, C to D on image 2) Now the gas in the engine is in thermal contact with the cold pool at temperature Tc. The surroundings do work on the gas, by pushing the piston down (stage 3 image, above), causing a volume of heat energy Q2 to leave the system to a low-temperature pool and the entropy of the system to decline by the amount. (This is the equal amount of entropy absorbed in step 1, as can be observed from the Clausius inequality.)

Step 4:  Adiabatic Reversible Compression

Once again the gas in the engine is thermally shielded from the hot and cold pools, and the engine is expected to be frictionless, therefore it is reversible. In this step, the surroundings do work on the gas, pushing the piston down more (stage 4 image, above), raising its internal energy, compressing it, and producing its temperature to rise back to The due only to the work added to the system, but the entropy remains the same. At this stage, the gas is in the same state as at the beginning of the step.

Therefore, the total work done by the gas on the environment in one complete cycle is shown by

\[ W =  \mu R\]

\[ W = \mu R T_1 ln \frac {v_2}{v_1} - \mu R T_2 ln \frac {v_3}{v_4} \]

Net efficiency= \[\frac{\text{Net work done by the gas}}{\text{Heat absorbed by the gas}}\]

So, the expression for a net efficiency of Carnot engine decreases to:

Net efficiency = \[ 1 - \frac {T_2}{T_1}\]

The Carnot engine cycle when behaving as a heat engine contains the following steps:

• Reversible or changeable isothermal compression of the gas at the “cold” temperature.

• Isentropic compression of the gas.

• Reversible or changeable adiabatic (Isentropic) expansion of the gas.

• Reversible or changeable isothermal expansion of the gas at the “hot” temperature.

Applications of the Carnot Cycle

Thermal machines or thermal devices are one of the applications of the Carnot cycle. The heat pumps to generate heating, the refrigerators to yield cooling, the steam turbines used in the ships, the combustion engines of the combustion vehicles and the reaction turbines of an airplane are some of the examples that we can give.

Limitations

This equation shows that the bigger the temperature range, the more effective the cycle is.

1. T3: In practice, T3 cannot be decreased below about 300 K (27ºC), equivalent to a condenser pressure of 0.035 bar. This is because of two tractors:

1. Condensation of steam needs a bulk supply of cooling water and such a continuous natural supply below the atmospheric temperature of around 15°C is unavailable.

2. If the condenser is to be of a rational size and cost, the temperature difference between the condensing steam and the cooling water must be minimum 10°C.

2. TI: The extreme cycle temperature T1is also limited to near 900 K (627°C) by the strength of the substance available for the extremely stressed parts of the plant, like boiler tubes and turbine blades. This upper limit is known as the metallurgical limit.

3. Critical Point: The steam of the Carnot cycle has a tremendous cycle temperature well beneath this metallurgical limit due to the properties of steam; it is limited to the critical-point temperature of 374°C (647 K). Therefore modern substances cannot be used to their best advantage with this cycle when steam is the functioning fluid. Also, because the saturated steam and water curves converge to the critical point, a plant working on the Carnot cycle with its extreme temperature near the critical-point temperature would have a very large s.s.c., i.e. it would be very big in size and very costly.

4. Compression Process: Compressing a very wet steam blend would need a compressor of cost and size comparable with the turbine. It would absorb work comparable to that advanced by the turbine. It would have a short life because of blade corrosion and cavitation problems. These reasons the Carnot cycle is not useful.
   
   

The Highest Level of Fuel Efficiency for a Heat Engine

All typical heat engines (steam, gasoline, diesel) operate by applying heat to a gas, which subsequently expands in a cylinder and pushes a piston to perform its function. So it's simple to see how to convert heat into work, but that's a one-time thing. To have a functioning engine, we need it to keep repeating. As a result, the heat and/or gas must be released out of the cylinder before the next cycle begins, or else all of the effort done by the gas on expanding will be spent compressing it back!

The simplest heat engine is known as a Carnot engine, and one complete heating/cooling, expanding/contracting cycle back to the original gas volume and temperature is known as a Carnot cycle, named after Sadi Carnot, who discovered the correct formula for the maximum possible efficiency of such a heat engine in terms of the maximum and minimum gas temperatures during the cycle in 1820.

Carnot's discovery was that if the greatest hot temperature achieved by the gas is TH and the lowest temperature during the cycle is TC (degrees kelvin, or simply kelvin, of course), the efficiency is the proportion of heat energy input that comes out as mechanical work.

Efficiency =  \[TH -\frac {TC}{TH}\]

This was an incredible finding since it was right while being based on a total misunderstanding of the nature of heat!

The Importance of Understanding Water Wheel Efficiency in Understanding the Heat Engine

Carnot thought that heat, like electricity, was a fluid that went from hot to cold (and somehow through space as radiation).

What prompted Carnot to attempt to determine the energy efficiency of steam in 1820? It was the Industrial Revolution, after all, and the efficiency of your power source controlled your profit margin.

Big engines were principally employed in the mass manufacturing of fabric in mills. Up until the late 1700s, these mills were built along fast-flowing rivers, and the power source was a giant water wheel that powered a long spinning rod that extended the length of the building.

 Ropes harnessed the force of pulleys on this rod to turn separate looms, which were handled by semi-skilled workers, often youngsters. The image shown is considerably later (1914), and steam-powered, but it depicts the powering system.

The steam engine provided an appealing option since it did not require proximity to a river. In contrast to the watermill, it required coal or wood for a fire.

Because the waterwheel was the primary source of industrial power until the late 1700s, a lot of effort went into making it as efficient as possible, and because Carnot believed heat was a fluid, he applied waterwheel thinking in studying the steam engine. So, how can a waterwheel be made as efficient as possible?

Because potential energy is lost as the water is dragged down by the wheel, the maximum energy attainable is mgh watts, where m is the mass of water flowing per second. (We're omitting potential kinetic energy contributions from fast-moving ingoing water – this is a very minor factor that doesn't relate to Carnot's heat engine calculations.)

How is energy squandered? Obviously, we want as low friction as possible in the wheel. There must be a steady flow of water with no splashing.

The water must flow into and out of the wheel without falling any considerable distance, otherwise, it will waste that much potential energy without doing work.

Carnot's Concept: a "Heat Water Wheel"

Carnot's conviction that heat was a fluid (we still conceive of it that way when we think of heat conduction or, for example, cooking) led him to compare the steam engine to a water wheel. Water falls via a gravitational potential difference in the water wheel, and that potential energy is converted into work by the wheel. The term "electric fluid" currently refers to a fluid that loses electrical potential energy while creating work or heat. So, how about the "caloric fluid" (as it was dubbed)? The analogy to gravitational potential is obviously just temperature! The gas in the cylinder does operate as it expands, but its temperature drops.

Carnot assumed that the steam engine was nothing more than a water wheel for this caloric fluid, so the most efficient engine would have minimal friction, but also, similar to how water enters and exits the wheel gently with no intermediate loss of height, heat would enter and exit the gas in the engine isothermally (remember the temperature is analogous to the gravitational potential, thus the height). By similarity with gh, the temperature drop TH-TC quantifies the potential energy given up by a unit amount of the "heat fluid."

The most efficient steam engine, like the most efficient water wheel, would have isothermal heat exchange (negligible temperature changes in heat exchange) (only a tiny drop as the water enters and leaves the wheel). This is, of course, the theoretical limit: some drop is required for functioning. The crucial point is that, at the limit of perfect efficiency, both the engine and the waterwheel are reversible — if given work, they can convert it into the same amount of heat that was required to create the work in the first place.

But how does this connect to the energy used to generate the heat in the first place? Carnot, on the other hand, was aware of something else: there existed an absolute zero of temperature. As a result, he reasoned, cooling the fluid to absolute zero would cause it to lose all of its heat energy. So, what proportion of the maximum amount of energy you can extract by cooling it from TH to TC is that of chilling it to absolute zero?

Getting the Most Out of a Hot Gas: Isothermal and Adiabatic Flows

Let's dive into the specifics of getting the most out of hot gas. We want the process to be as reversible as possible: there are two ways to move the piston reversibly: isothermally, which means heat gradually flows in or out of a reservoir at a temperature infinitesimally different from the temperature of the gas in the piston, and adiabatically, which means there is no heat exchange at all and the gas simply acts as a spring.

As a result, as heat is provided and the gas expands, the temperature of the gas must remain constant with the temperature of the heat source (the "heat reservoir"): the gas is expanding isothermally. Similarly, when it loses heat later in the cycle, it must compress isothermally.

To calculate efficiency, we must follow the engine through a whole cycle, noting how much work it performs, how much heat it absorbs from the fuel, and how much heat it expands in preparation for the next cycle. To understand the concept, look at the applet: the cycle comprises four steps: an isothermal expansion as heat is absorbed, an adiabatic expansion, an isothermal contraction as heat is lost, and finally an adiabatic contraction to the initial configuration. We'll take things slowly at first.