The Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and
expanded by others in the 1830s and 1840s. It can be shown that it is the most
efficient cycle for converting a given amount of thermal
energy into work, or conversely, creating a temperature difference
(e.g. refrigeration) by doing a given amount of work.
Every single thermodynamic system exists in a particular
state. When a system is taken through a series of different states and finally
returned to its initial state, a thermodynamic cycle is said to have occurred.
In the process of going through this cycle, the system may perform work on its
surroundings, thereby acting as a heat engine.
A system undergoing a Carnot cycle is called a Carnot heat engine, although such a
"perfect" engine is only a theoretical limit and cannot be built in
practice.
Stages of the Carnot Cycle
The Carnot cycle when acting as a heat engine
consists of the following steps:
- Reversible isothermal expansion of the gas at the "hot" temperature, T1 (isothermal heat addition or absorption). During this step (1 to 2 on Figure 1, A to B in Figure 2) the gas is allowed to expand and it does work on the surroundings. The temperature of the gas does not change during the process, and thus the expansion is isothermal. The gas expansion is propelled by absorption of heat energy Q1 and of entropy from the high temperature reservoir.
- Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (2 to 3 on Figure 1, B to C in Figure 2) the mechanisms of the engine are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy. The gas expansion causes it to cool to the "cold" temperature, T2. The entropy remains unchanged.
- Reversible isothermal compression of the gas at the "cold" temperature, T2. (isothermal heat rejection) (3 to 4 on Figure 1, C to D on Figure 2) Now the surroundings do work on the gas, causing an amount of heat energy Q2 and of entropy to flow out of the gas to the low temperature reservoir. (This is the same amount of entropy absorbed in step 1, as can be seen from the Clausius inequality.)
- Isentropic compression of the gas (isentropic work input). (4 to 1 on Figure 1, D to A on Figure 2) Once again the mechanisms of the engine are assumed to be thermally insulated. During this step, the surroundings do work on the gas, increasing its internal energy and compressing it, causing the temperature to rise to T1. The entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.
The pressure-volume graph
When the Carnot cycle is plotted on a pressure volume diagram, the isothermal
stages follow the isotherm lines for the working fluid, adiabatic stages move
between isotherms and the area bounded by the complete cycle path represents
the total work that can be done during one cycle.
Properties and significance
A generalized thermodynamic cycle taking place between a hot
reservoir at temperature TH and a cold reservoir at temperature TC.
By the second law of thermodynamics, the
cycle cannot extend outside the temperature band from TC to TH.
The area in red QC is the amount of energy exchanged between the
system and the cold reservoir. The area in white W is the amount of work energy
exchanged by the system with its surroundings. The amount of heat exchanged
with the hot reservoir is the sum of the two. If the system is behaving as an
engine, the process moves clockwise around the loop, and moves
counter-clockwise if it is behaving as a refrigerator. The efficiency of the
cycle is the ratio of the white area (work) divided by the sum of the white and
red areas (heat absorbed from the hot reservoir).
The behaviour of a Carnot engine or refrigerator is best
understood by using a temperature-entropy diagram (TS
diagram), in which the thermodynamic state is specified by a point on a graph
with entropy
(S) as the horizontal axis and temperature (T) as the vertical axis. For a
simple system with a fixed number of particles, any point on the graph will
represent a particular state of the system. A thermodynamic process will
consist of a curve connecting an initial state (A) and a final state (B). The
area under the curve will be:
which is the amount of thermal energy transferred in the
process. If the process moves to greater entropy, the area under the curve will
be the amount of heat absorbed by the system in that process. If the process
moves towards lesser entropy, it will be the amount of heat removed. For any
cyclic process, there will be an upper portion of the cycle and a lower
portion. For a clockwise cycle, the area under the upper portion will be the
thermal energy absorbed during the cycle, while the area under the lower
portion will be the thermal energy removed during the cycle. The area inside
the cycle will then be the difference between the two, but since the internal
energy of the system must have returned to its initial value, this difference
must be the amount of work done by the system over the cycle. Referring to
figure 1, mathematically, for a reversible process we may write the amount of
work done over a cyclic process as:
Since dU is an exact differential, its integral over any closed
loop is zero and it follows that the area inside the loop on a T-S diagram is
equal to the total work performed if the loop is traversed in a clockwise
direction, and is equal to the total work done on the system as the loop is
traversed in a counterclockwise direction.
The Carnot cycle
Evaluation of the above integral is particularly simple for
the Carnot cycle. The amount of energy transferred as work is:
This definition of efficiency makes sense for a heat engine,
since it is the fraction of the heat energy extracted from the hot reservoir
and converted to mechanical work. A Rankine
cycle is usually the practical approximation.
The Reversed Carnot cycle
The Carnot heat-engine cycle described is a totally
reversible cycle. That is, all the processes that comprise it can be reversed,
in which case it becomes the Carnot refrigeration cycle. This time, the cycle
remains exactly the same except that the directions of any heat and work
interactions are reversed. Heat is absorbed from the low-temperature reservoir,
heat is rejected to a high-temperature reservoir, and a work input is required
to accomplish all this. The P-V diagram of the reversed Carnot cycle is the
same as for the Carnot cycle except that the directions of the processes are
reversed.
Carnot's theorem
A real engine (left) compared to the Carnot cycle (right).
The entropy of a real material changes with temperature. This change is
indicated by the curve on a T-S diagram. For this figure, the curve indicates a
vapor-liquid equilibrium (See Rankine
cycle). Irreversible systems and losses of heat (for example, due to
friction) prevent the ideal from taking place at every step.
Carnot's theorem is a formal statement of this fact: No
engine operating between two heat reservoirs can be more efficient than a
Carnot engine operating between those same reservoirs. Thus, Equation 3
gives the maximum efficiency possible for any engine using the corresponding
temperatures. A corollary to Carnot's theorem states that: All reversible
engines operating between the same heat reservoirs are equally efficient.
Rearranging the right side of the equation gives what may be a more easily
understood form of the equation. Namely that the theoretical maximum efficiency
of a heat engine equals the difference in temperature between the hot and cold
reservoir divided by the absolute temperature of the hot reservoir. To find the
absolute temperature in kelvin, add 273.15 degrees to the Celsius temperature.
Looking at this formula an interesting fact becomes apparent. Lowering the
temperature of the cold reservoir will have more effect on the ceiling
efficiency of a heat engine than raising the temperature of the hot reservoir
by the same amount. In the real world, this may be difficult to achieve since
the cold reservoir is often an existing ambient temperature.
In other words, maximum efficiency is achieved if and only
if no new entropy is created in the cycle. Otherwise, since entropy is a state
function, the required dumping of heat into the environment to dispose of
excess entropy leads to a reduction in efficiency. So Equation 3 gives the
efficiency of any reversible heat engine.
In mesoscopic heat engines, work per cycle of operation
fluctuates due to thermal noise. For the case when work and heat fluctuations
are counted, there is exact equality that relates average of exponents of work
performed by any heat engine and the heat transfer from the hotter heat bath.
This relation transforms the Carnot's inequality into exact equality that is
applied to an arbitrary heat engine coupled to two heat reservoirs and
operating at arbitrary rate.
Efficiency of real heat engines
Carnot realized that in reality it is not possible to build
a thermodynamically reversible engine, so
real heat engines are less efficient than indicated by Equation 3. In addition,
real engines that operate along this cycle are rare. Nevertheless, Equation 3
is extremely useful for determining the maximum efficiency that could ever be
expected for a given set of thermal reservoirs.
Although Carnot's cycle is an idealisation, the
expression of Carnot efficiency is still useful. Consider the average
temperatures,
at which heat is input and output, respectively. Replace TH
and TC in Equation (3) by 〈TH〉
and 〈TC〉
respectively.
For the Carnot cycle, or its equivalent, the average value 〈TH〉
will equal the highest temperature available, namely TH, and 〈TC〉
the lowest, namely TC. For other less efficient cycles, 〈TH〉
will be lower than TH, and 〈TC〉
will be higher than TC. This can help illustrate, for
example, why a reheater or a regenerator can improve the thermal
efficiency of steam power plants—and why the thermal efficiency of
combined-cycle power plants (which incorporate gas turbines operating at even
higher temperatures) exceeds that of conventional steam plants.
SUBSCRIBERS - ( LINKS) :FOLLOW / REF / 2 /
findleverage.blogspot.com
Krkz77@yahoo.com
+234-81-83195664
No comments:
Post a Comment