Carnot Cycle :
A
Carnot gas cycle operating in a given temperature range is shown in the T-s
diagram in Fig. 4.1(a). One way to carry
out the processes of this cycle is through the use of state, steady-flow devices as shown in Fig.
4.1(b). The isentropic expansion process 2-3 and the isentropic compression
process 4-1 can be simulated quite well by a well-designed turbine and
compressor respectively, but the isothermal expansion process 1-2 and the
isothermal compression process 3-4 are most difficult to achieve. Because of
these difficulties, a steady-flow Carnot gas cycle is not practical.
The
Carnot gas cycle could also be achieved in a cylinder-piston apparatus (a reciprocating
engine) as shown in Fig. 4.2(b). The Carnot cycle on the p-v diagram is as shown
in Fig. 4.2(a), in which processes 1-2 and 3-4 are isothermal while processes
2-3 and 4-1 are isentropic. We know that the Carnot cycle efficiency is given
by the expression.
Fig.4.1.
Steady flow Carnot engine
Fig.4.3.
Carnot cycle on p-v and T-s diagrams
Fig.4.4.
Working of Carnot engine
Since
the working fluid is an ideal gas with constant specific heats, we have, for
the
isentropic
process,
Now,
T1 = T2 and T4 = T3, therefore
Carnot
cycle efficiency may be written as,
From
the above equation, it can be observed that the Carnot cycle efficiency
increases as ‘r’ increases. This implies that the high thermal efficiency of a
Carnot cycle is obtained at the expense of large piston displacement. Also, for
isentropic processes we have,
Since,
T1 = T2 and T4 = T3, we have
Therefore,
Carnot cycle efficiency may be written as,
From
the above equation, it can be observed that, the Carnot cycle efficiency can be
increased
by increasing the pressure ratio. This means that Carnot cycle should be
operated
at high peak pressure to obtain large efficiency.
Stirling Cycle (Regenerative Cycle)
The
Carnot cycle has a low mean effective pressure because of its very low work output.
Hence, one of the modified forms of the cycle to produce higher mean effective pressure
whilst theoretically achieving full Carnot cycle efficiency is the Stirling
cycle. It consists of two isothermal and two constant volume processes. The
heat rejection and addition take place at constant temperature. The p-v and T-s
diagrams for the Stirling cycle are shown in Fig.4.2.
Fig.4.2.
Stirling cycle processes on p-v and T-s diagrams
Stirling Cycle
Processes:
(a)
The air is compressed isothermally from state 1 to 2 (TL to TH).
(b)
The air at state-2 is passed into the regenerator from the top at a temperature
T1. The air passing through the regenerator matrix gets heated from TL to TH.
(c)
The air at state-3 expands isothermally in the cylinder until it reaches
state-4.
(d)
The air coming out of the engine at temperature TH (condition 4) enters into
regenerator from the bottom and gets cooled while passing through the
regenerator matrix at constant volume and it comes out at a temperature TL, at
condition 1 and the cycle is repeated. (
e)
It can be shown that the heat absorbed by the air from the regenerator matrix
during the process 2-3 is equal to the heat given by the air to the regenerator
matrix during the process 4-1, then the exchange of heat with external source
will be only during the isothermal processes.
Now
we can write, Net work done = W = Qs - QR
Heat
supplied = QS = heat supplied during the isothermal process 3-4.
Heat
rejected = QR = Heat rejected during the isothermal compression process,
1-2.
Now,
and
Thus
the efficiency of Stirling cycle is equal to that of Carnot cycle efficiency
when both are working with the same temperature limits. It is not possible to
obtain 100% efficient regenerator and hence there will be always 10 to 20 % loss
of heat in the regenerator, which decreases the cycle efficiency. Considering
regenerator efficiency, the efficiency of the cycle can be written as,
Where,
R η is the regenerator efficiency.
No comments:
Post a Comment