Page 132 - Advanced Thermodynamics for Engineers, Second Edition

CHAPTER

FINITE TIME

(OR ENDOREVERSIBLE) 6

THERMODYNAMICS

6.1 GENERAL CONSIDERATIONS

The thermal efficiency of a Carnot cycle operating between high-temperature (T H ) and low-
temperature (T C ) reservoirs is given by

T C
h ¼ 1 (6.1)
th
T H
This cycle is extremely idealised. It requires an ideal, reversible heat engine (internally reversible)
but in addition the heat transfer from the reservoirs is also reversible (externally reversible). To achieve
external reversibility it is necessary that the temperature difference between the reservoirs and the
engine is infinitesimal (see Chapter 4, Section 4.9.1.1), which means that the heat exchanger surface
area must be very large or the time to transfer heat must be long. The former is limited by size and cost
factors whilst the latter will limit the actual power output achieved for the engine. It is possible to
evaluate the maximum power output achievable from an internally reversible (endoreversible) heat
engine receiving heat irreversibly from two reservoirs at T H and T C . This will now be done, following
an approach used by Bejan (1988).
This analysis is referred to as finite time thermodynamics (FTT) because it considers the rates at
which energy can be transferred through the systems involved. It was stated in Chapter 2 that the
Carnot cycle was internally and externally reversible. It is possible to conceive of internal reversibility
because this relies on the processes inside the heat engine being reversible, e.g. the isentropic effi-
ciency of the devices is 100%, and there are no heat or pressure losses in the system; such an engine is
referred to as endoreversible. However, external reversibility requires that the heat transfer takes place
across an infinitesimal temperature difference – which will result in a very slow rate of heat transfer. In
FTT, the requirement of external reversibility is removed and the heat transfer takes place across a
significant temperature drop – the rate of heat transfer can then be calculated.
Assume that the engine is an endoreversible one operating at steady state, and therefore steady flow
(e.g. like a steam turbine or closed cycle gas turbine); a similar analysis is possible for an intermittent
device (e.g. like a Stirling engine). A schematic of such an engine is shown in Fig. 6.1.
The reservoir at T H transfers heat to the engine across a resistance and it is received by the engine at
temperature T 1 . In a similar manner, the engine rejects energy at T 2 but the cold reservoir is at T C .
It can be assumed that the engine itself is reversible and acts as a Carnot cycle device with
T 2
h ¼ 1 : (6.2)
th
T 1
Advanced Thermodynamics for Engineers. http://dx.doi.org/10.1016/B978-0-444-63373-6.00006-X 119
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