COURSE OF ENERGY CONVERSION
SECOND-LAW ANALYSIS OF
POWER CYCLES
These class notes are for the students of the course "Energy conversion A" at Politecnico di Milano.
Anyone who finds inaccuracies or, anyhow, wishes to send comments to improve them is invited to
the lecturer (gianluca.valenti@polimi.it), who thanks in advance.
Second-law analysis of power cycles - Energy Conversion A – V7.0
Why these classnotes ............................................................................................................................ 5
1
Introduction ................................................................................................................................... 6
2
Thermodynamics references ......................................................................................................... 7
2.1
Definitions............................................................................................................................. 7
2.2
First reference: second law of thermodynamics (Carnot's theorem) .................................... 7
2.3
Second reference: "ideal cycle” and “Reversible cycle” ...................................................... 8
2.3.1 Example 1: how can a Rankine cycle be made reversible? .............................................. 8
1.1.1.1 Saturated vapor cycle ................................................................................................ 8
1.1.1.2 Vapor cycle with superheating................................................................................ 10
2.3.2 Example 2: how can a Joule cycle be made reversible? ................................................. 12
2.3.3 Example 3: Stirling cycle ................................................................................................ 15
2.4
Third reference: reversible heat pump between sources/sinks at constant temperature: .... 16
3
Energy sources for power plants ................................................................................................. 18
3.1
Energy Sources Definition .................................................................................................. 18
3.2
Non-reacting flows, maximum work, exergy ..................................................................... 18
3.2.1 Trilateral and trapezoidal cycles ..................................................................................... 20
3.3
Fossil and renewable fuels .................................................................................................. 23
3.3.1 Stoichiometry in combustion reactions ........................................................................... 23
3.3.2 Enthalpies of formation and enthalpy balance of combustion ........................................ 26
3.3.3 Adiabatic flame temperature ........................................................................................... 28
3.3.4 Heating value of the fuel ................................................................................................. 29
3.3.5 Recuperative preheating.................................................................................................. 31
3.3.6 Entropic balance for a reacting system ........................................................................... 33
3.3.7 Irreversibility generated in combustion .......................................................................... 35
3.3.8 Reversible work and chemical exergy ............................................................................ 35
3.3.9 Comparison of indexes ................................................................................................... 37
4
Irreversible processes and loss of useful work............................................................................ 39
4.1
Irreversibilities and useful work losses ............................................................................... 39
4.1.1 First example of irreversibility: heat transfer .................................................................. 39
4.1.2 Second example of irreversibility: Isenthalpic throttling ................................................ 41
4.2
General demonstration ........................................................................................................ 44
4.2.1 Heat transfer: replacement with a reversible engine + heat pump .................................. 45
4.2.2 Throttling: replacement with reversible isothermal expansion + heat pump .................. 46
5
The most common causes of irreversibility ................................................................................ 49
5.1
Losses in Heat transfer ........................................................................................................ 49
5.1.1 General definitions .......................................................................................................... 49
5.1.2 Entropy generated in an irreversible heat exchange ....................................................... 52
5.1.3 Considerations on the design of the components ............................................................ 54
5.2
Losses in Expansion and Compression ............................................................................... 55
5.2.1 Gas/vapor turbomachines: turbines and compressors ..................................................... 55
5.2.2 Hydraulic machines: Pumps............................................................................................ 58
5.2.3 Considerations on the design of the components ............................................................ 58
5.3
Losses due to Pressure drops .............................................................................................. 59
5.3.1 Pressure drops of a liquid ................................................................................................ 59
5.3.2 Pressure drops of a gas .................................................................................................... 59
5.3.3 Considerations on the design of the components ............................................................ 60
5.4
Losses due to Mixing .......................................................................................................... 60
5.5
Losses due to Chemical reactions ....................................................................................... 61
5.6
Other losses: self-consumptions, auxiliary systems............................................................ 61
6
Applying the general formula to analysis of power plants ......................................................... 62
2 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
6.1
Definition of the time interval ............................................................................................. 62
6.2
Definition of the boundaries of the system considered ....................................................... 62
6.3
Definition of the energy source and of material flows........................................................ 62
6.4
The "dead" state .................................................................................................................. 63
6.5
Definition of the efficiency of the power cycle .................................................................. 63
6.6
Number N of modeled processes ........................................................................................ 64
7
Second law analysis of a steam rankine cycle ............................................................................ 65
7.1
Initial definitions ................................................................................................................. 65
7.2
∆𝜼𝟏 irreversibility of heat transfer in the condenser .......................................................... 66
7.3
∆𝜼𝟐 fluid dynamic irreversibility in the pumps .................................................................. 70
7.4
∆𝜼𝟑 irreversibility of heat transfer in the preheating line ................................................... 72
7.4.1 Indirect preheaters ........................................................................................................... 72
7.4.2 Deaerator ......................................................................................................................... 74
7.5
∆𝜼𝟒 irreversibility of heat transfer in introducing heat into the cycle ................................ 75
7.5.1 ∆𝜼𝟒𝒂 Derating the thermal energy from 𝑻∞ to 𝑻𝒎𝒂𝒙 ................................................. 77
7.5.2 ∆𝜼𝟒𝒃 Introducing heat into the cycle starting from 𝑻𝒎𝒂𝒙 of the working fluid .......... 80
7.5.3 Binary cycles ................................................................................................................... 82
7.6
∆𝜼𝟓 fluid dynamic irreversibility in the turbine ................................................................. 83
7.7
∆𝜼𝟔 pressure drops in the liquid phase ............................................................................... 83
7.8
∆𝜼𝟕 pressure drops in the vapor phase ............................................................................... 83
7.9
∆𝜼𝟖 thermal losses .............................................................................................................. 84
7.10 ∆𝜼𝟗 mechanical/electrical losses ........................................................................................ 84
7.11 ∆𝜼𝟏𝟎 auxiliary losses ......................................................................................................... 84
7.12 Comparison of losses .......................................................................................................... 84
7.13 Other numerical examples .................................................................................................. 86
8
Effect of the thermodynamic properties of the working fluid on Rankine cycle performance .. 87
8.1
Sources at constant temperature - Non-regenerative cycle ................................................. 87
8.1.1 Effect of the molecular complexity ................................................................................. 88
8.2
Sources at constant temperature - Regenerative cycle ........................................................ 90
8.3
Sources at variable temperature .......................................................................................... 91
8.3.1 Simple molecules ............................................................................................................ 91
8.3.2 Complex molecules ......................................................................................................... 93
8.4
General conclusions ............................................................................................................ 95
8.4.1 Economics ....................................................................................................................... 95
8.4.2 Considerations concerning the choice of the working fluid ........................................... 96
8.4.3 Mixtures of fluids ............................................................................................................ 99
9
Second law analysis of an open-loop Brayton GAS cycle........................................................ 102
9.1
Criteria for choosing the compression ratio ...................................................................... 102
9.2
Simple cycle with non-cooled turbine .............................................................................. 104
9.2.1 𝚫𝜼𝟏 irreversibility due to pressure drops (in the air filtration system and in every other
part of the cycle) ....................................................................................................................... 106
9.2.2 𝚫𝜼𝟐 fluid dynamic irreversibilities in the compressor ................................................. 107
9.2.3 𝚫𝜼𝟑 combustion irreversibility ..................................................................................... 107
9.2.4 𝚫𝜼𝟒 fluid dynamic irreversibilities of the turbine (adiabatic assumption) ................... 108
9.2.5 𝚫𝜼𝟓 losses due to the discharge of exhaust gas into the atmosphere (stack losses) ..... 108
9.2.6 𝚫𝜼𝟔 thermal losses ........................................................................................................ 109
9.2.7 𝚫𝜼𝟕 mechanical/electrical/auxiliary losses .................................................................. 109
9.2.8 Observations ................................................................................................................. 109
9.3
Recuperated cycle with non-cooled turbine ...................................................................... 110
3 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
9.3.1 𝚫𝜼𝟖 losses relating to the heat transfer in the recuperator ........................................... 111
9.3.2 Observations ................................................................................................................. 112
9.4
Gas cycle with cooled expansion ...................................................................................... 113
9.5
Integrating cooled gas turbines in combined cycles ......................................................... 120
9.6
Effects of real gas in the gas cycles .................................................................................. 123
9.7
Other numerical examples ................................................................................................ 124
4 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
WHY THESE CLASSNOTES
Cycles are the most useful architecture for power generation. The definition of thermodynamic cycle
is a series of processes that involve energy transfer, both in form of heat and mechanical work, and
eventually returns the system to the initial state. The analysis of such systems is of paramount to
design, rate and size the equipment needed, to assess the critical aspects, and highlight which
components and/or processes can be improved in order to achieve higher outputs.
In this frame of reference, it is useful to anticipate that the second law analysis aims at studying how
a real cycle differs for the ideal, reversible cycle in the same condition, i.e. at studying the
differences between the real cycle and the ideal one working with the same energy input. Second law
analysis quantifies these differences and allows to understand which processes introduce the most
irreversibilities and are thus the processes where the most improvement could be made.
The scope of this class note is to provide the student with a solid theoretical basis of the effect of
entropy generation in thermodynamic cycles and the ability to perform a second law analysis, or
entropic analysis on the said thermodynamic cycles both in a preliminary and an accurate way.
The structure of the present document is as follows:
•
•
•
•
•
•
•
•
•
Chapter 1 introduces the second law analysis concepts
Chapter 2 shows the thermodynamic frame of second law analysis in general
Chapter 3 discusses the second law analysis of energy sources
Chapter 4 shows examples of irreversibilities
Chapter 5 lists all possible irreversibilities in power plants
Chapter 6 shows the thermodynamic frame of second law analysis for power plants
Chapter 7 presents the list of possible losses in Rankine cycles
Chapter 8 shows the effect of the working fluid on losses in Rankine cycles
Chapter 9 presents the list of possible losses in Joule-Brayton cycles
5 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
1
INTRODUCTION
The energy analysis of power systems requires a knowledge of technical and theoretical aspects in
order to define properly a system, the surrounding hypotheses and the parameters of merit. The
enormous variability of technologies, plant configurations and applications requires a methodology
for thoroughly comparing different solutions.
The objective of this chapter is to provide the student with the information and knowledge necessary
for the first and second law analysis applied to power plants. In particular, the second law analysis,
also called entropic analysis, allows identifying the main causes for loss of power output, and which
processes shall be modified for improving the performance of the plant. The purpose of this
methodology is to get the net work of the plant starting from the maximum reversible work, which can
be obtained with a reversible system, minus the sum of the works lost due to irreversibility and, hence,
to increases in entropy of the universe.
𝑊𝑢 = 𝑊𝑟𝑒𝑣 − ∑ ∆𝑊𝑖
(1.1)
𝑖
The definition and calculation of the wasted works ∆𝑊𝑖 passes through the second-law analysis of the
different processes. This method will be applied to the two main power production plants, i.e. the
Rankine cycle and the Brayton cycle.
6 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
2 THERMODYNAMICS REFERENCES
This chapter illustrates the main reference cycles that are used when analyzing different power cycles.
This is done in order to set a standard procedure that is thermodynamically consistent. It will also show
with examples the difference between ideal cycles and reversible cycles.
2.1 DEFINITIONS
It is useful to refer to some basic concepts of thermodynamics, which should be well known to those
students who have attended the courses preliminary to the energy conversion course.
•
Power cycle: a thermodynamic cycle whose purpose is to convert thermal energy into
mechanical work. In the temperature-entropy plane, a power cycle works clockwise.
•
Efficiency: is defined as the ratio between the useful work (net mechanical energy supplied
outside) and the heat entering the cycle
2.2 FIRST REFERENCE: SECOND LAW OF THERMODYNAMICS (CARNOT'S THEOREM)
Carnot's theorem states that “all reversible power cycles operating between a heat source at a
constant temperature 𝑇𝑚𝑎𝑥 and a heat sink at a constant temperature 𝑇𝑚𝑖𝑛 have the same efficiency,
equal to:
𝜂𝑐𝑎𝑟𝑛𝑜𝑡 = 1 −
𝑇𝑚𝑖𝑛
𝑇𝑚𝑎𝑥
(2.1)
Fig. 2.1 - Diagram of the Carnot cycle operating between two sources at a constant temperature
The three conditions necessary so that a thermodynamic cycle (not necessarily a Carnot cycle) has the
Carnot efficiency are:
1. Reversibility: all processes must be reversible, therefore devoid of any dissipative process that
causes an increase in the entropy of the universe;
2. Heat source (also called heat sink) with constant temperature equal to 𝑇𝑚𝑎𝑥 and therefore with
infinite heat capacity;
3. Cold source (also called cold sink) with constant temperature equal to 𝑇𝑚𝑖𝑛 and therefore with
infinite heat capacity.
If even one of these conditions fails, the Carnot efficiency cannot be achieved. Therefore, the Carnot
efficiency increases the more the source and heat sink move away, and asymptotically tends to 1 for
𝑇𝑚𝑎𝑥 tending to infinity.
7 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
2.3 SECOND REFERENCE: "IDEAL CYCLE” AND “REVERSIBLE CYCLE”
A cycle obtained with ideal components is an ideal cycle. This includes:
• machines that operate in the absence of friction and irreversible fluid dynamic processes
(turbulence, boundary layer separations, mixing);
• heat exchangers that work with infinite surface, and therefore with an infinitesimal temperature
difference at least in one section;
• components that have no pressure drops.
An ideal cycle can operate with “real” fluids, i.e. with fluids characterized by volumetric behavior
appreciably different from the ideal gas equation of state. The “reversible” cycle, on the other hand,
requires that all transformations are reversible, including the introduction and the rejection of energy
by heat interaction.
When applying the second law of thermodynamics to power cycles, the following recommendations
should be kept in mind:
• Do not confuse ideal cycles (ideal machines, infinite surface heat exchangers, no pressure
drops, real fluids) with reversible cycles (all processes must be reversible, including the heat
exchanges with external sources)
• The Carnot efficiency is achieved only if the source and heat sink are at constant temperature.
If the temperature is variable, it will be possible to define a new reversible cycle called Lorentz
cycle, presented later. The Lorentz cycle has efficiency lower than that of Carnot.
The following examples will clarify the meaning of the recommendations.
2.3.1
Example 1: how can a Rankine cycle be made reversible?
1.1.1.1 Saturated vapor cycle
The ideal cycle with saturated vapor shown in Fig. 2.2 is considered. The cycle will have a lower
efficiency than the Carnot cycle operating between the same temperatures even assuming that the heat
source is at a constant temperature equal to the evaporation temperature, and that the heat sink is at a
constant temperature equal to the condensing temperature.
CYCLE LAYOUT
4
Wout
Qin
• 1-2
isentropic
compression
considering incompressible liquid
• 2-3 introduction of heat to bring the
condensate from the condensation to
evaporation temperature
• 3-4 introduction of heat to bring the
saturated liquid to saturated vapor
conditions
• 4-5 isentropic expansion of the vapor
from the evaporation to condensation
pressure
• 5-1 condensation of the vapor which
brings the working fluid back to the
saturated liquid conditions
5
3
Qout
2
1
Win
3
1≡2
4
5
Fig. 2.2 - Ideal cycle with saturated vapor
8 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Despite being made with ideal components (pump and turbine without fluid dynamic losses, infinite
surface heat exchangers), the saturated Rankine cycle is in fact not reversible as there is an irreversible
process in introducing heat. The preheating of the water takes place between the hot source and the
working fluid of the cycle with heat transfer under finite difference in temperature, and therefore
causes irreversibility.
To make this cycle reversible, the irreversibilities have to be eliminated. This can be achieved with an
infinite series of infinitesimal bleeds that preheat the condensate as shown in Fig. 2.3 and Fig. 2.4 in
an infinite series of infinite surface heat exchangers. In such cases, all of the heat is introduced only in
the evaporation stage at constant temperature, and therefore the cycle reaches the Carnot cycle
efficiency. In practice, this is not feasible because it would require an infinite number of components
operating with infinitesimal flows, accompanied by unsustainable technological complications and an
economic burden.
In industrial practice, the regeneration is achieved with a finite number of regenerative bleeds, each of
which allows the pressurized fluid to be heated by a certain finite ∆𝑇. In each of these components the
heat transfer takes place under finite temperature differences. Therefore, there is a production of
irreversibility and a reduction in efficiency with 𝜂 < 𝜂𝑐𝑎𝑟𝑛𝑜𝑡 .
CYCLE LAYOUT
4
Wout
Qin
3
5
2
1
Qout
An infinite number of regenerative
bleeds (each with infinitesimal flow
rate) from the turbine is assumed:
each bleed goes to an infinite surface
heat exchanger, where it gives up the
heat of the vapor fraction phase
change to the condensate. The
preheaters are in cascade and the
condensed vapor coming out from a
preheater enters into the previous
preheater following the path of the
condensate, together with the vapor.
Win
3
4
1≡2
5
Fig. 2.3 - Ideal cycle with saturated vapor with ideal regenerative preheating.
T
T
T
Q
Q
9 of 124
Q
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 2.4 - Reduction of the mean temperature difference in the preheating process obtained by
adopting an increasing number of regenerators from left to right
1.1.1.2 Vapor cycle with superheating
If the cycle with superheated vapor in Fig. 2.5 operates with a heat source of infinite heat capacity at
the superheating temperature, the entire heat introduction phase takes place irreversibly.
CYCLE LAYOUT
5
4
Wout
Qin
3
6
2
1
• 1-2 isentropic compression of a liquid
• 2-3 introduction of heat to bring the
condensate from the condensation to
evaporation temperature
• 3-4 introduction of heat to bring the
saturated liquid to the superheated
vapor conditions
• 4-5 expansion of the vapor from the
evaporation to condensation pressure
• 5-1 condensation of the vapor which
brings the working fluid back to the
saturated liquid conditions
Qout
Win
5
3
1≡2
4
6
Fig. 2.5 - Ideal cycle with superheated vapor
To make the cycle reversible, it is necessary to imagine:
•
the adoption of an infinite number of reheatings: therefore endless turbine stages, each of
which performs an infinitesimal expansion, and an infinite number of heat exchangers for
introducing thermal energy into the cycle. Alternatively, it is possible to think of an isotherm
expansion;
• a recuperation that heats the working fluid from the condensing temperature up to the
superheating temperature, with a counter-current heat exchanger that cools the vapor leaving
the last turbine stage.
The resulting cycle under these assumptions is depicted in Fig. 2.6.
10 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
CYCLE LAYOUT
Qin
Wout
4
3
2
5
Qout
1
Win
4
3
An infinite series of reheatings
during the expansion that makes it
isotherm and a recuperation which
should take the condensate from the
condensing temperature up to the
state of superheated vapor are
introduced
1≡2
5
Fig. 2.6 - “Carnotized” diagram of an ideal cycle with superheated vapor plant
If on one hand an isothermal expansion is theoretically possible, a reversible recuperation is not
feasible even from the conceptual point of view. That is because the specific heat at constant pressure
of the pressurized side is different (much greater, moreover infinite during the phase change) than that
of the low pressure side. This is also represented by the slope of the isobaric curves in the Ts plane,
𝑇
which is equal to . Under these conditions, even if adopting an infinite surface exchanger, leading to
𝑐𝑝
the cancellation of ∆𝑇 at least in one point, the pressurized current could not be taken to the maximum
temperature of the cycle.
Fig. 2.7 shows the TQ diagram of the infinite surface recuperated heat exchanger. Remember that in
this plane the slope of the curves is inversely proportional to the heat capacity 𝐶 (defined as the product
of the flow rate on a mass basis 𝑚̇ and the specific heat at constant pressure 𝑐𝑝 of the flows affected
by the heat transfer). For the evaporation section, it is equal to zero given that 𝑐𝑝 tends to infinity,
while in the vapor phase it is always greater than in the liquid phase1.
Therefore, the process cannot be made reversible due to the different slope of the high and low pressure
isobaric lines, even in the presence of infinite surfaces. The heat available from the cooling of the
vapor is only sufficient to preheat the condensate and to evaporate the vapor partially, even assuming
1
Also remember that the difference in slope between the liquid and vapor phases depends on the
complexity of the fluid. The simpler the fluid, the greater, in relative terms, the correction of the
specific heat passing from liquid to saturated vapor. While for complex fluids characterized by a high
number of degrees of freedom, the specific heat of the vapor is about equal to that of the liquid.
The proportion between the heat introduced during phase change and that introduced during preheating
and superheating is instead mainly a function of the molecular weight. The greater the molecular
weight is, the less ∆ℎ𝑒𝑣𝑎 generally is. Therefore, the lower the relative weight of the evaporation phase
is with respect to the preheating phase.
11 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
cooling up to the condensation temperature. The rest of the heat necessary to get superheated vapor
must still be supplied from the outside.
Two irreversible heat transfer processes are always present in those systems:
• recuperation
• introduction of heat to complete evaporation and achieve superheating.
T
Q
Fig. 2.7 - Temperature-exchanged thermal power diagram for an ideal recuperation
2.3.2
Example 2: how can a Joule cycle be made reversible?
An ideal Joule cycle has a lower efficiency than a Carnot cycle operating between sources of infinite
heat capacity with same 𝑇𝑚𝑎𝑥 and 𝑇𝑚𝑖𝑛 . This is due to the presence of two irreversible processes: the
introduction and release of heat as shown in Fig. 2.8. This cycle was improved by introducing a
recuperator that uses the heat discharged by the turbine to preheat the compressed gas. Even if
reversible, this component however does not make the cycle totally reversible by itself because a
certain amount of heat must always be exchanged with the sink under finite temperature differences.
In addition, the recuperator may be reversible only if it has an infinite surface and if the two currents
have equal heat capacities throughout the temperature change. The first condition is not accessible
from a technical and economic point of view, but it is conceptually feasible. The second, on the other
hand, can be precluded a priori if the cycle shows an internal combustion system or if the gas presents
different real fluid effects on the high and low-pressure sides. In both cases, if the heat transferred is
equal, the temperature change on the two sides will be different. Thus, there will be certainly finite ∆𝑇
in some sections of the heat exchanger.
The first case is representative of a gas turbine cycle in which both the flow rate and the gas
composition (and the specific heat) are different on the two sides of the heat exchanger. The most
relevant factor is given by the difference in flow rate since downstream of the compressor a substantial
part of the compressed air is extracted for cooling high-pressure blades and for diluting exhaust gases
in the combustor. The mass flow at the outlet of the turbine is thus larger that the mass flow at the
outlet of the compressor. The second factor can be represented, for example, by a closed gas cycle
operating close to the saturation dome. In this case the real fluid effects become important on the high
pressure side, increasing the specific heat of the fluid which will therefore have a lower temperature
change, heat transferred being equal.
12 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
CYCLE LAYOUT
Qin
2
3
Wout
3
• 1-2 isentropic compression of an
ideal gas
• 2-3 introduction of heat to bring
the compressed gas to the
maximum temperature of the cycle
(coinciding with that of the source)
• 3-4 isentropic expansion of an
ideal gas
• 4-1 transfer of heat to bring the
expanded gas to the minimum
temperature (coinciding with that
of the heat sink)
4
1
Qout
4
2
1
Fig. 2.8 - Ideal Joule cycle
Fig. 2.9 - Diagrams of heat transfer for a recuperator of an open gas cycle (left) and for a closed
gas cycle with real fluid effects (right)
To eliminate the irreversibilities of heat transfer with the heat sinks, instead, three alternatives are
conceptually viable:
• Make an isotherm compression at the temperature 𝑇𝑚𝑖𝑛 and an isotherm expansion at the
temperature 𝑇𝑚𝑎𝑥 (Fig. 2.10). These processes are theoretically achievable with an infinite
plant cost and, at the same time, the adoption of a recuperator2. This cycle is called Ericsson
cycle.
2
In practice, in gas cycles the idioms regeneration and recovery are used without distinction. Originally, recuperator in
English indicates a surface heat exchanger, regenerator a heat accumulating device, a path alternatively from a hot current
and a cold current.
13 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
•
Decrease the compression ratio (up to the unit value) and simultaneously use a recuperator
(Fig. 2.11).
• Increase the compression ratio until all the gas is entirely heated in the compression phase.
All these solutions maximize the efficiency that becomes coincident with that of Carnot. However, the
last two lead to a useful work of the cycle equal to zero since, in both cases, the compression and
expansion works are coincident and the introduction of heat is zero.
CYCLE LAYOUT
Qin
3
Wout
1
4
3
2
Qout
4
2
1
• 1-2 isotherm compression of an
ideal gas at 𝑇𝑚𝑖𝑛
• 2-3 introduction of heat to bring
the compressed gas to the
maximum
cycle
temperature
(coinciding with that of the source)
by means of a reversible
recuperator
• 3-4 isotherm expansion of an ideal
gas at 𝑇𝑚𝑎𝑥
• 4-1 transfer of heat to bring the
expanded gas to the minimum
temperature (coinciding with that
of the heat sink) by means of a
reversible recuperator
Fig. 2.10 - Reversible Joule cycle obtained with isotherm compression and expansion (called
Ericsson cycle).
CYCLE LAYOUT
Qin
3
3
Wout
2
1
6
4
5
6
4
Qout
2
1
5
• 1-2 isentropic compression of an
ideal gas (T2= T1)
• 2-6 preheating of the compressed gas
at the expense of the expanded gas
• 6-3 introduction of heat to bring the
compressed gas to the maximum
cycle temperature (coinciding with
that of the source): since T3=T6, the
heat introduced is infinitesimal
• 3-4 isentropic expansion of an ideal
gas (T3= T4)
• 4-5 recuperative cooling of the
expanded gas
• 5-1 transfer of heat to bring the
expanded gas to the minimum
temperature (coinciding with that of
the heat sink); the heat transferred is
infinitesimal (T5=T1)
Fig. 2.11 - Reversible Joule cycle obtained with compression ratios tending to one and infinite
surface recuperator.
14 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
3
2
CYCLE LAYOUT
Qin
2
3
Wout
4
1
1
• 1-2 isentropic compression of an
ideal gas (T2 coincides with T3)
• 2-3 introduction of heat to bring the
compressed gas to the maximum
cycle temperature (coinciding with
that of the source): since T3=T2, the
heat introduced is infinitesimal
• 3-4 isentropic expansion of an ideal
gas (T3= T4)
• 4-1 transfer of heat to bring the
expanded gas to the minimum
temperature (coinciding with that of
the heat sink); the heat transferred is
infinitesimal (T4=T1)
4
Qout
NOTE: All the temperature increase
occurs due to isentropic compression
Fig. 2.12 - Reversible Joule cycle obtained with high compression ratios that lead to making the heat
contribution from the outside infinitesimal.
2.3.3
Example 3: Stirling cycle
The ideal (or limit, in this case the two definitions coincide) Stirling cycle is composed of two
isochores and two isotherms as shown in Fig. 2.13. The heat is introduced at constant temperature
𝑇𝑚𝑎𝑥 and is transferred at constant temperature 𝑇𝑚𝑖𝑛 . Both processes take place with infinitesimal
temperature differences, so they are reversible. The recuperation has the same limitations seen in the
previous example, and it does not present irreversibilities only if the gas is an ideal gas on both sides
of the exchanger. With this condition, the two curves in the TQ diagram are superimposed since the
specific heat depends only on the temperature. The efficiency of the Stirling cycle in this case is
identical to that of the Carnot cycle operating between the same temperatures.
3
2
• 1-2 reversible isotherm compression
of an ideal gas (T1= T2)
• 2-3 introduction of heat to bring the
compressed gas to the maximum
cycle temperature (coinciding with
that of the source) at constant
volume, at the expense of the
expanded gas (recuperation)
• 3-4 reversible isotherm expansion of
an ideal gas (T3=T4) with
introduction of heat at the maximum
cycle temperature (coinciding with
that of the source)
• 4-1 transfer of heat to the
compressed gas to bring the
expanded gas to the minimum
temperature (coinciding with that of
the heat sink) at constant volume
4
1
Fig. 2.13 - Representation in the temperature-specific entropy plane of the Stirling cycle
15 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
2.4
THIRD
REFERENCE: REVERSIBLE HEAT PUMP BETWEEN SOURCES/SINKS AT CONSTANT
TEMPERATURE:
The thermodynamic cycles can operate following:
• a clockwise path as in Fig. 2.14a: it is the case of the direct or power cycles. The heat 𝑄1 enters
the cycle (and therefore transfers from a heat source to the working fluid) in one or more
transformations at high temperature. The heat 𝑄2 exits the cycle (and therefore transfers from
the working fluid to the heat sink) at low temperature. The useful effect of a power cycle is the
conversion of part of the heat introduced into work 𝑊𝑢 . The parameter for identifying the
energy performance of a power cycle is the first law efficiency:
𝜂𝐼 =
•
𝑊𝑢
𝑄1
(2.2)
a counterclockwise path as in Fig. 2.14b: it is the case of the inverse cycles. The heat 𝑄2 enters
the cycle (and therefore transfers from a heat source to the working fluid) in one or more
transformations at low temperature. The heat 𝑄1 exits the cycle (and therefore transfers from
the working fluid to a heat sink) at a higher temperature. Depending on the application, the
useful effect of an inverse cycle is the removal of heat from the low temperature source
(refrigeration cycle) or the transfer of heat to the heat sink at a higher temperature (heat pump
cycle). In both cases these heat transfers are achieved at the expense of the work 𝑊𝑢 introduced
into the cycle.
In some cases, it may happen that both effects are to be considered "useful". One easy example
to be remembered is a cycle used by a machine that dispenses ice (using cold) and hot water
(using heat). Another case, more significant in terms of energy, is an inverse cycle designed
for the climate control of a building which simultaneously has areas where cooling is required
and areas where heating is required. In these cases, the use of heat and/or cold is often partial.
T
T
s
s
Fig. 2.14 - Direct reversible cycles (or power cycles) and indirect cycles (or refrigeration, or with
heat pump) in the temperature-specific entropy plane
The parameter to identify the energy performance of an inverse cycle is the coefficient of
performance (𝐶𝑂𝑃), defined as:
𝐶𝑂𝑃 =
𝑄𝑥
𝑊𝑢
16 of 124
(2.3)
Second-law analysis of power cycles - Energy Conversion A – V7.0
where 𝑄𝑥 is the heat taken from the cold source (𝑄2 ) in the case of a refrigeration cycle, or the heat
released to the heat sink (𝑄1 ) in the case of a heat pump. For the first law of thermodynamics:
𝐶𝑂𝑃𝐻𝑃 =
𝑄1 𝑄2 + 𝑊𝑢
=
= 𝐶𝑂𝑃𝐶𝐻𝐿 + 1
𝑊𝑢
𝑊𝑢
(2.4)
In the case of reversible cycles (and only for them), the ratios between the transferred heat flows (𝑄1
and 𝑄2 ) and 𝑊𝑢 are obviously independent from the direction of the cycle, therefore:
𝜂𝐼 =
1
1
=
𝐶𝑂𝑃𝐻𝑃 𝐶𝑂𝑃𝐶𝐻𝐿 + 1
(2.5)
If the reversible cycles operate with sources/heat sinks at a constant temperature as in Fig. 2.15:
𝑇𝑚𝑖𝑛
)
𝑇𝑚𝑎𝑥
𝑇𝑚𝑎𝑥
𝐶𝑂𝑃𝐻𝑃 =
𝑇𝑚𝑎𝑥 − 𝑇𝑚𝑖𝑛
𝑇𝑚𝑖𝑛
𝐶𝑂𝑃𝐶𝐻𝐿 =
𝑇𝑚𝑎𝑥 − 𝑇𝑚𝑖𝑛
𝜂𝐼 = 1 − (
(2.6)
(2.7)
(2.8)
For real cycles these relations no longer apply. In particular, the 𝐶𝑂𝑃𝐻𝑃 is less than the ratio 1/𝜂𝐼 .
T
Qin
Qout
Wout
Win
Qout
Qin
s
Fig. 2.15 - Reversible Carnot cycle in the T-S plane. It can be covered in a clockwise or
counterclockwise direction, without changes of the ratios between the work exchanged with the
outside and the heats exchanged with the two heat sources.
It must be also remembered that the area of the cycle in the TS plane has no physical meaning for real
cycles, unlike what takes place for ideal or limit cycles. Similarly, the area under a real transformation
(∫ 𝑇 𝑑𝑆) does not represent the heat transferred during the transformation.
17 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
3 ENERGY SOURCES FOR POWER PLANTS
This chapter introduces the calculation of the potential of different energy sources. A classification is
first given. Then the definition of exergy for an energy source is given. A chemical and entropic
analysis of fuels is performed, and combustion as a mean of transferring energy is analyzed.
3.1 ENERGY SOURCES DEFINITION
The power production plants can be powered by a wide variety of energy sources. Some of the main
ones are:
• non-reacting flows: recovery of waste heat, geothermal energy;
• fossil and renewable fuels: solid (like coal and biomass), liquid (like gasoline, diesel,
bioethanol, and biodiesel) and gaseous (like natural gas, syngas, biogas, and biomethane);
• solar energy, etc.
Each of these sources can be modelled in a different way, and some can be considered with infinite
heat capacity while others cannot. Other sources, although available at very high temperature
(sometimes modeled as an infinite temperature: solar radiation, combustion flue gases) require the use
of a heat-transfer fluid to transfer heat from the energy source to the working fluid, so they refer to the
first type.
3.2 NON-REACTING FLOWS, MAXIMUM WORK, EXERGY
The maximum work obtainable from a flow rate of a fluid in thermodynamic conditions different from
the environment is that obtained with any reversible path (or formed by one or more reversible
processes) that brings the fluid in equilibrium with the environment3. If all the processes are reversible,
there is no increase in entropy of the universe and, as a result, there is no loss of useful work.
An energy source constituted by a constant flow rate 𝑚̇ of air with pressure 𝑝𝑥 and temperature 𝑇𝑥 is
assumed. It is assumed that the air is an ideal gas with 𝑐𝑝 constant with the temperature. The purpose
of this hypothesis is the analytical steps, but the same considerations can be made for any fluid.
Let's consider the simplest reversible path4 in Fig. 3.1:
1. a reversible adiabatic expansion (isentropic) from point x up to condition A at temperature 𝑇0
and pressure 𝑝𝐴 < 𝑝0
2. a reversible isotherm compression from condition A up to ambient pressure p0
For all the transformations that will be considered here and in the following paragraphs, it will be
assumed that:
a) The flow is one-dimensional at the input and output sections, so it is permissible to identify the
kinematic and thermodynamic properties of the fluid with average values in these sections
b) The flow is stationary and each quantity is independent from time. In analytical
𝜕𝑋
terms, = 0
𝜕𝑡
3
Equilibrium of the fluid with the environment means that the source is brought to the same temperature and pressure of
the dead state. As we will see in the section dedicated to fuels, the maximum work can be obtained when the equilibrium
considers also the chemical contribution (i.e. same composition).
4
Another chain of reversible processes might be: (i) an isotherm compression up to a point z with a 𝑝𝑧 greater than 𝑝𝑥 such
as to get an entropy equal to 𝑠0 and (ii) an isentropic expansion up to 𝑝0 . In this case a Carnot cycle that reversibly recovers
the heat released by the compressor is necessary. On the other hand, an isentropic expansion up to 𝑝0 and the subsequent
isobaric cooling up to 𝑇0 is not a reversible path since the heat transfer with the cold sink would take place with the
production of irreversibility. In this case a trilateral cycle that reversibly recovers the heat released by the compressor
would be necessary.
18 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Adopting the sign conventions best suited to energy conversion (positive heat if entering the system,
positive work if going out of the system), the following equation is obtained:
𝑉1 2
𝑉2 2
ℎ1 +
+ 𝑔𝑧1 + 𝑄 = ℎ2 +
+ 𝑔𝑧2 + 𝑊
2
2
(3.1)
The further hypothesis of null (or in any case negligible) changes in kinetic (𝑉 2 /2) and gravitational
energy (𝑔𝑧) can be applied in most of the components used in conversion plants
Fig. 3.1 - Sequence of ideal and reversible transformations necessary for the production of the
maximum power extractable from a flow of gas not in equilibrium with the dead state, and
representation in the Ts diagram
The power obtained from the isentropic and adiabatic expansion (𝑄 = 0) from 𝑥 to A is equal to:
𝑊̇𝑥→𝐴 = 𝑊̇𝑇𝑅𝐵 = 𝑚̇(ℎ𝑥(𝑇𝑥,𝑝𝑥) − ℎ𝐴(𝑇0 ,𝑝𝐴) )
(3.2)
The power spent in the isotherm and reversible compression (transfer under infinitesimal temperature
differences and without friction) from A to 0 is instead equal to:
𝑊̇𝐴→0 = −𝑊̇𝐶𝑀𝑃 = −𝑄̇𝐶𝑀𝑃 = −𝑚̇𝑇0 (𝑠𝐴(𝑇0 ,𝑝𝐴 ) − 𝑠0(𝑇0 ,𝑝0) )
(3.3)
The resulting total power is therefore equal to:
𝑊̇𝑇 = 𝑊̇𝑇𝑅𝐵 − 𝑊̇𝐶𝑀𝑃 = 𝑊̇𝑥→𝐴 + 𝑊̇𝐴→0
= 𝑚̇(ℎ𝑥(𝑇𝑥,𝑝𝑥) − ℎ𝐴(𝑇0 ,𝑝𝐴 ) )−𝑚̇𝑇0 (𝑠𝐴(𝑇0 ,𝑝𝐴 ) − 𝑠0(𝑇0 ,𝑝0) )
(3.4)
where being 𝑠𝐴 = 𝑠𝑥 and ℎ𝐴 = ℎ0 it turns out
𝑊̇𝑇 = 𝑚̇[(ℎ𝑥 − 𝑇0 𝑠𝑥 ) − (ℎ0 − 𝑇0 𝑠0 )]
(3.5)
where the terms in round brackets are also known as specific exergy, a state function of the fluid.
𝑊̇𝑇 = 𝑚̇[𝑏𝑥 − 𝑏0 ]
19 of 124
(3.6)
Second-law analysis of power cycles - Energy Conversion A – V7.0
The result obtained for this particular example is generally valid. Therefore, the specific exergy has
the physical meaning of the capacity to produce mechanical work (or maximum obtainable work) for
unit mass of a fluid in the absence of changes in kinetic and potential energy, from a fluid that, as it is
able to exchange heat only with the external environment, evolves from an initial generic state to a
final state in thermodynamic equilibrium with the environment.
3.2.1
Trilateral and trapezoidal cycles
Now a look will be taken at how reversible work can be assessed with the assumption of a flow at a
temperature 𝑇 > 𝑇0 and pressure equal to that of reference, which is cooled up to the condition of
equilibrium with the environment. It is also assumed that the source is at constant heat capacity. It is
possible to imagine a reversible system with the following processes:
a) a "trilateral" reversible cycle, sometimes said Lorenz’ cycle, consisting of:
1. a step of heat introduction at variable temperature, from 𝑇0 to 𝑇𝑥 , in which the source
is reversibly cooled from the temperature 𝑇𝑥 to temperature 𝑇0 . The following has to be
assumed: (i) a counter-current heat exchanger in which the working fluid and the source
have the same thermal capacity at each temperature, (ii) of infinite surface, (iii)
adiabatic outwards and (iv) without pressure drops
2. an isentropic expansion step that cools the working fluid of the cycle up to the
temperature 𝑇0
3. a reversible isotherm compression step at temperature 𝑇0 , which reversibly releases
heat to the environment
b) a mixing step in thermodynamic equilibrium (equal 𝑇, 𝑝) of the source. The production of
entropy resulting from the irreversible phenomenon of a release into the environment of a
fluid in thermodynamic equilibrium, but with a chemical composition different from the
ambient, will be discussed in the chemical exergy topic.
T
reversible cycle
2
2
1
1
3
3
s
Fig. 3.2 - Use of a trilateral cycle for extracting the maximum mechanical power from the cooling of
a hot flow.
Therefore, the reversible power is equal to the exergy difference of the fluid in point x and in
equilibrium with the environment conditions
𝑊̇𝑟𝑒𝑣 = 𝑚̇[(ℎ𝑥 − 𝑇0 𝑠𝑥 ) − (ℎ0 − 𝑇0 𝑠0 )]
𝑊̇𝑟𝑒𝑣 = 𝑚̇[(ℎ𝑥 − ℎ0 ) − 𝑇0 (𝑠𝑥 − 𝑠0 )]
(3.7)
(3.8)
If a process is considered isobaric, then the terms of enthalpy difference and entropy difference are a
function of the temperature only.
20 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑑ℎ = 𝑐𝑝 (𝑇)𝑑𝑇
(3.9)
𝑐𝑝 (𝑇)
(3.10)
𝑑𝑠 =
𝑑𝑇
𝑇
If it is possible to consider constant specific heat (or at least to calculate an average value), the heat
capacity of the source is constant, and the following is found:
𝑇𝑥
𝑊̇𝑟𝑒𝑣 = 𝑚̇ [𝑐𝑝 (𝑇𝑥 − 𝑇0 ) − 𝑇0 𝑐𝑝 𝑙𝑛 ( )]
𝑇0
(3.11)
By gathering 𝑐𝑝 (𝑇𝑥 − 𝑇0 ) and observing that 𝑚̇𝑐𝑝 (𝑇𝑥 − 𝑇0 ) = 𝑄̇ 1 , it turns out that:
𝑊̇𝑟𝑒𝑣 = 𝑄̇1 [1 −
𝑇0
𝑇
]
(3.12)
(𝑇𝑥 − 𝑇0 )⁄𝑙𝑛 ( 𝑥 )
𝑇0
𝑇
where the term (𝑇𝑥 − 𝑇0 )⁄𝑙𝑛 ( 𝑥) is called mean logarithmic temperature of the hot source 𝑇𝑚𝑙𝑛(𝑇𝑥|𝑇0) .
𝑇0
The term in square brackets is called the efficiency for the trilateral cycle:
𝜂 𝑇𝑟𝑖𝑙 = 1 −
𝑇0
(3.13)
𝑇𝑚𝑙𝑛(𝑇𝑥|𝑇0)
The efficiency thus found can be considered as the efficiency of an equivalent Carnot cycle operating
between ambient temperature and the mean log temperature of heat introduction.
This relation may be used both for gaseous and liquid streams in which the cooling is isobaric and the
heat capacities can be considered constant.
A similar result is also obtained by limiting the cooling of the source up to a temperature 𝑇𝑦 higher
than the ambient temperature. The chain of reversible transformations to calculate the expression of
the maximum obtainable work requires: (i) isentropic expansion from 𝑇𝑥 to 𝑇0 , (ii) isothermal
compression at 𝑇0 up to the entropy 𝑠𝑦 , (iii) isentropic compression from 𝑇0 to 𝑇𝑦 .
T
reversible cycle
2
1
2
1
4
4
3
3
s
21 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 3.3 – Adoption of a trapezoidal cycle for extracting the maximum mechanical power from the
cooling of a hot flow with a minimum temperature limit.
It is therefore equal to
𝑊̇𝑟𝑒𝑣 = 𝑊̇𝑇𝑅𝐵 − 𝑊̇𝐶𝑀𝑃1 − 𝑊̇𝐶𝑀𝑃2
̇
𝑊𝑟𝑒𝑣 = 𝑚̇[(ℎ𝑥 − ℎ0 ) − 𝑇0 (𝑠𝑥 − 𝑠𝑦 ) − (ℎ𝑦 − ℎ0 )]
𝑊̇𝑟𝑒𝑣 = 𝑚̇[(ℎ𝑥 − ℎ𝑦 ) − 𝑇0 (𝑠𝑥 − 𝑠𝑦 )]
𝑊̇𝑟𝑒𝑣 = 𝑚̇[𝑏𝑥 − 𝑏𝑦 ]
(3.14)
(3.15)
(3.16)
(3.17)
With the hypothesis of constant heat capacity, the efficiency of the trapezoidal cycle that reversibly
receives heat from the heat source is obtained with a procedure similar to that used for the trilateral
cycle.
𝜂 𝑇𝑟𝑎𝑝 = 1 −
𝑇0
(3.18)
𝑇𝑚𝑙𝑛(𝑇𝑥|𝑇𝑦)
Therefore, it is clear that for heat capacities of the source tending to infinite, the source becomes
isothermal, 𝑇𝑦 → 𝑇𝑥 , and the trilateral cycle efficiency obviously tends to that of the Carnot cycle.
Finally, the most general case is the one with both hot and cold sources with variable temperature. The
reference cycle is the mixtilinear cycle shown in Fig. 3.4, whose efficiency can be calculated starting
from the variation in exergy of the hot and cold sources calculated according to:
𝑇0
∆𝐵̇ℎ = 𝑄̇𝑖𝑛 (1 −
)
𝑇𝑚𝑙𝑛(𝑇𝑚𝑎𝑥2|𝑇𝑚𝑎𝑥1)
𝑇0
∆𝐵̇𝑐 = 𝑄̇𝑜𝑢𝑡 (1 −
)
𝑇𝑚𝑙𝑛(𝑇𝑚𝑖𝑛2|𝑇𝑚𝑖𝑛1)
(3.19)
(3.20)
where exergy variations are calculated from the Lorentz efficiency (Eq. (3.13)). Then, the reversible
power that can be produced is given by the difference between the two sources exergy variations:
𝑊̇𝑟𝑒𝑣 = ∆𝐵̇𝑐 − ∆𝐵̇𝑐
(3.21)
With ∆𝐵̇ℎ and ∆𝐵̇𝑐 positive. From the first principle of thermodynamics, 𝑄̇𝑖𝑛 = 𝑄̇𝑜𝑢𝑡 + 𝑊̇𝑟𝑒𝑣 and the
𝑊̇
efficiency is defined as 𝜂 = 𝑟𝑒𝑣, it turns out that 𝑄̇𝑜𝑢𝑡 = 𝑄̇𝑖𝑛 (1 − 𝜂) and:
𝑄̇𝑖𝑛
𝑊̇𝑟𝑒𝑣 = 𝑄̇𝑖𝑛 (1 −
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑎𝑥2|𝑇𝑚𝑎𝑥1 )
) − 𝑄̇𝑖𝑛 (1 − 𝜂) (1 −
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑖𝑛2|𝑇𝑚𝑖𝑛1)
)
(3.22)
By gathering 𝑄̇𝑖𝑛 the following expressions can be obtained
𝜂 = (1 −
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑎𝑥2 |𝑇𝑚𝑎𝑥1 )
−1+
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑖𝑛2|𝑇𝑚𝑖𝑛1)
22 of 124
) + 𝜂 (1 −
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑖𝑛2|𝑇𝑚𝑖𝑛1)
)
(3.23)
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝜂(
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑖𝑛2 |𝑇𝑚𝑖𝑛1 )
)=(
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑖𝑛2|𝑇𝑚𝑖𝑛1 )
−
𝑇0
𝑇𝑚𝑙𝑛(𝑇
𝑚𝑎𝑥2 |𝑇𝑚𝑎𝑥1 )
𝑇𝑚𝑙𝑛(𝑇𝑚𝑖𝑛2|𝑇𝑚𝑖𝑛1)
𝑇𝑚𝑙𝑛,f
𝜂 = (1 −
) = (1 −
)
𝑇𝑚𝑙𝑛(𝑇𝑚𝑎𝑥2|𝑇𝑚𝑎𝑥1)
𝑇𝑚𝑙𝑛,c
)
(3.24)
(3.25)
T
𝑇𝑚𝑎𝑥
𝑇𝑚𝑎𝑥2
𝑇𝑚𝑙𝑛 𝑇𝑚𝑎𝑥1,2
𝑇𝑚𝑎𝑥1
𝑇𝑚𝑖𝑛2
𝑇0
𝑇𝑚𝑙𝑛 𝑇𝑚𝑖𝑛1,2
𝑇𝑚𝑖𝑛1
s
Fig. 3.4 – Quadlateral mixtilinear cycle
3.3 FOSSIL AND RENEWABLE FUELS
The energy source most common for power production processes are the fuels (identified by the
chemical composition, thermodynamic conditions, etc.) which after combustion reactions can make a
certain amount of heat available.
The energy content of the fuel can be described by four values specific to the mass unit5:
• Lower Heating Value
• Higher Heating Value
• Reversible Work
• Chemical exergy.
Their theoretical definition will be provided in this chapter, and the main differences between these
indexes will be highlighted. In addition, practical recommendations for their use in the analysis of
power cycles will be discussed.
3.3.1
Stoichiometry in combustion reactions
Combustion reactions are exothermic oxidation reactions that transform the reactants into combustion
products. The reactants are formed by the fuel (compound that contains chemical species able to
oxidize) and oxidant (compound that provides the oxygen necessary for oxidation). The products of
combustion are obtained at a temperature higher than that of the reagents, making useful heat available
to thermodynamic power production processes. A combustion reaction always respects the mass
balance and the atomic balance, while the number of moles may change in accordance with the
chemical species destroyed and formed during the reaction.
The atomic chemical species involved in the combustion reactions are mainly carbon and hydrogen,
which form typical fuels in the largest portion. There may be traces of sulfur and nitrogen, depending
on their composition.
5
Generally, reference is made to the Sm3 for gaseous fuels, which is always a value that identifies the mass. Indeed, in 1m3
at standard conditions (T=0°C, p=1bar) 44.03 moles are always contained. Therefore, once the gas composition is known,
the mass is as well.
23 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The basic combustion reactions for a mole of these elements with molecular oxygen 𝑂2 are:
𝐶 + 𝑂2 → 𝐶𝑂2
1
1
𝐻 + 𝑂2 → 𝐻2 𝑂
4
2
A combustion reaction of a fuel of chemical composition 𝐶𝑎 𝐻𝑏 is now analyzed. The total combustion
𝑏
of a mole of this compound will require 𝑎 + moles of 𝑂2 .
4
𝑎𝐶 + 𝑎𝑂2 → 𝑎𝐶𝑂2
𝑏
𝑏
𝑏𝐻 + 𝑂2 → 𝐻2 𝑂
4
2
This reaction is called stoichiometric since there is no trace of the reagents in combustion products as
they have been completely consumed. The ratio between oxidant mass and fuel mass for a combustion
reaction is called stoichiometric ratio and is expressed as:
𝛼=
𝑚𝑂2
𝑚𝐶𝑎 𝐻𝑏
(3.26)
For example, the combustion of a gas mixture with composition on a mass-basis 𝑦𝑖 (methane 60%,
ethane 30%, propane 10%) can be calculated with the following steps:
1. calculation of the composition on a molar-basis of the mixture: the kilomoles of each
compound are found considering a base of 1 kg of fuel as:
0.6
= 0.0375 kmol → 75.3%
16
𝑦𝑖
𝑚𝑖
0.3
𝑛𝑖 =
=
𝐶2 𝐻6 →
= 0.01 kmol → 20.1%
𝑀𝑀𝑖 𝑀𝑀𝑖
30
0.1
𝐶
𝐻
→
= 0.0023 kmol → 4.6%
3
8
{
44
𝐶𝐻4 →
2. the moles of oxygen required for complete combustion of the reacting compound moles are
calculated
𝑛𝑂2 ,𝑖 |
𝑠𝑡𝑜𝑖𝑐
𝐶𝐻4 → 0.0375 (1 + 0.25 ∙ 4) = 0.1125 kmol
𝑏𝑖
(2 + 0.25 ∙ 6) = 0.05
kmol
= 𝑛𝑖 (𝑎𝑖 + ) {𝐶2 𝐻6 → 0.01
4
𝐶3 𝐻8 → 0.0023 (3 + 0.25 ∙ 8) = 0.016 kmol
3. the flow rate on a mass-basis of oxygen required is calculated
𝛼𝑠𝑡 = 𝑚𝑂2 |
𝑠𝑡𝑜𝑖𝑐
= 𝑀𝑀𝑂2 ∑𝑛𝑂2 ,𝑖 |
𝑠𝑡𝑜𝑖𝑐
= 3.883 kg
𝑖
The flue gas produced shall consist solely of 𝐶𝑂2 and 𝐻2 𝑂 with a composition on a molar-basis given
by:
24 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝐶𝐻4 → 0.0375 (1) = 0.0375
𝑛𝐶𝑂2 |
= ∑ 𝑛𝑖 (𝑎𝑖 ) = ∑ {𝐶2 𝐻6 → 0.01 (2) = 0.02
= 0.0644 kmol
𝑠𝑡𝑜𝑖𝑐
𝐶3 𝐻8 → 0.0023(3) = 0.0069
𝑖
𝑖
→ 36.1%
𝐶𝐻4 → 0.0375 (0.5 ∙ 4) = 0.075
𝑏𝑖
𝑛𝐻2 𝑂 |
= ∑ 𝑛𝑖 ( ) = ∑ {𝐶2 𝐻6 → 0.01 (0.5 ∙ 6) = 0.03 = 0.1142 kmol
𝑠𝑡𝑜𝑖𝑐
2
𝐶3 𝐻8 → 0.0023 (0.5 ∙ 8) = 0.0092
𝑖
𝑖
→ 63.9%
And in mass-basis terms
𝑚𝐶𝑂2 |
𝑠𝑡𝑜𝑖𝑐
= 𝑀𝑀𝐶𝑂2 𝑛𝐶𝑂2 |
𝑚𝐻2𝑂 |
𝑠𝑡𝑜𝑖𝑐
= 𝑀𝑀𝐻2 𝑂 𝑛𝐻2𝑂 |
𝑠𝑡𝑜𝑖𝑐
𝑠𝑡𝑜𝑖𝑐
= 2.834
kg → 58%
= 2.0556 kg → 42%
That allows the mass balance of the reaction to be checked. It can be noted that the combustion
𝑎
products are always damp, and have a greater water content the greater the ratio in the fuel.
𝑏
If the reaction takes place with ambient air instead of pure oxygen in the combustion products, the
presence of nitrogen will also have to be considered. Assuming a simplified composition of ambient
air equal to 79% 𝑁2 and 21% 𝑂2 on a molar-basis, the value 𝛼𝑠𝑡 can be calculated as:
𝛼𝑠𝑡𝑜𝑖𝑐 = 𝑚𝑂2 |
𝑠𝑡𝑜𝑖𝑐
+ 𝑚𝑁2 |
𝑠𝑡𝑜𝑖𝑐
= 𝑀𝑀𝑂2 ∑𝑛𝑂2 ,𝑖 |
𝑠𝑡𝑜𝑖𝑐
𝑖
+ 𝑀𝑀𝑁2 ∑ 3.76 𝑛𝑂2,𝑖 |
𝑠𝑡𝑜𝑖𝑐
𝑖
where the factor 3.76 is given by the ratio between the mole fractions of nitrogen and oxygen in the
ambient air.
If the reaction is not stoichiometric, a flow rate of oxidant different from that calculated above is used,
and it is useful to define the combustion ratio as:
𝜆=
𝛼
𝛼𝑠𝑡𝑜𝑖𝑐
=1+𝑒
(3.27)
Two cases are possible:
• combustion in excess of oxygen 𝜆 > 1, 𝑒 > 0: more oxidant than the stoichiometric is
introduced and total combustion of the fuel is obtained. The combustion products will have the
same content on a mass-basis of the stoichiometric case for the species 𝐶𝑂2 and 𝐻2 𝑂, plus the
unreacted oxygen and non-reacting nitrogen
• combustion lacking oxygen 𝜆 < 1, 𝑒 < 0: less oxidant than the stoichiometric is introduced
and total combustion of the fuel is not obtained. The content 𝐶𝑂2 and 𝐻2 𝑂 in the combustion
products will be lower than that of the stoichiometric case. Added to this will be the unreacted
fuel, non-reacting nitrogen and the presence of incomplete combustion products, such as 𝐶𝑂.
Depending on the type of fuel and the technological solutions adopted, it will be more or less expedient
to use a certain excess of oxygen:
• In coal dust plants, the heterogeneous combustion between coal and combustion air requires a
certain degree of excess air, about 6%, in order to be certain that the reaction will be completed,
25 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
•
•
3.3.2
and to have no unburnt materials in the flue gas or in the ash collection hopper. For gas boiler
plants, the excess air drops to 3% since the combustion is in the homogeneous phase.
In gas turbines, on the other hand, an extremely high combustion ratio is used even if the fuel
is gaseous. That is because it is necessary to dilute the flue gas produced in the combustor, to
limit the formation of NOx and reduce the wall temperature of the combustion chamber.
In internal combustion engines a lack of oxygen is instead used to keep down the emissions.
In this case, there is the presence of CO in the flue gas since it is a not wholly oxidized species.
Enthalpies of formation and enthalpy balance of combustion
It is possible to write the energy balance for reacting systems considering that the system evolves from
the reactants to the products in a continuous reactor that, in the most general case, can exchange heat
and work with the external environment with
𝑄 − 𝑊 = 𝐻𝑃 − 𝐻𝑅
reagents
combustion
products
(fuel)
+1
(combustion products)
(comburent)
W
Fig. 3.5 - Enthalpy balance of the first law for a reacting system
or
𝑄 − 𝑊 = ∑ 𝑛𝑖 ℎ̃𝑖(𝑇,𝑝)𝑃 − ∑ 𝑛𝑗 ℎ̃𝑗(𝑇𝑗,𝑝𝑗)𝑅
𝑖∈𝑃
(3.28)
𝑗∈𝑅
where the enthalpies must be calculated as ℎ̃(𝑇,𝑝) = ℎ̃0 + Δℎ̃(𝑇,𝑝) , considering the enthalpy of reference
ℎ̃0 and Δℎ̃(𝑇,𝑝) due to the deviation from the thermodynamic state of reference. The second term Δℎ̃(𝑇,𝑝)
is the increase of enthalpy given by the difference in temperature and pressure for the pure fluid with
respect to the reference conditions. Its value can be derived from reliable thermodynamic tables or
equations of state since it must also consider effects of real fluid, phase transitions, etc.
In this discussion the assumption of ideal gas will be adopted since it can be applied to the combustion
products and reagents in gaseous form, without introducing excessive errors. In this specific case of
ideal gas, the term Δℎ̃(𝑇,𝑝) = Δℎ̃(𝑇) and it takes on the form:
𝑇
Δℎ̃(𝑇) = ∫ 𝑐𝑝0 (𝑇)𝑑𝑇
𝑇0
26 of 124
(3.29)
Second-law analysis of power cycles - Energy Conversion A – V7.0
In the study of a combustion reaction, however, special attention has to be paid to the reference values
adopted for the various thermodynamic quantities. Unlike what has been seen so far on enthalpy
balances of non-reacting systems, the value of the enthalpy of reference ℎ0 cannot be chosen arbitrarily
but must correspond to very specific properties of the compound. In fact, for non-reacting systems the
choice of ℎ̃0 was totally irrelevant because, if the difference in enthalpy between an initial and a final
state were to be calculated, this term always simplified and its choice was therefore arbitrary. On the
contrary, in the case of reacting systems in which chemical species disappear and form, this value
cannot be chosen without considering the type of compound, the reactions necessary for its formation
and the thermodynamic state of reference. In this case the reference enthalpy is referred as standard
enthalpy of formation ℎ̃°𝑓 .
The first step is to define the standard reference state, which by convention has 𝑇0 = 25°𝐶 and 𝑝0 =
1 atm. All the molecules formed by a single element in the stable form at this thermodynamic condition
have a ℎ̃°𝑓 = 0 by convention. The compounds that fall within this category are monatomic molecules,
such as graphite 𝐶 or the noble gases (He, Ar, etc.), and diatomic molecules such as 𝐻2 , 𝑂2 , 𝑁2 and
not, for example, their atomic species that are much more unstable in the standard reference conditions.
For all the other compounds, the reference enthalpy is equal to the standard enthalpy of formation
starting from the stable compounds, i.e. the extractable thermal power starting from reactants in
standard conditions and bringing the products back to the same conditions. The values of the enthalpies
of formation can be obtained experimentally through statistical thermodynamics or spectroscopy
methods.
Take, for example, the formation reaction of 𝐶𝑂2 starting from its constituents in a stable form (𝐶 and
𝑂2 ) under the standard conditions. Have the reaction take place in a reactor that then brings the product
to the initial standard conditions. A certain amount of heat, which in the specific case is equal to 393.52
kJ/kmol, must be released into the environment since the reaction is exothermic. The enthalpy of
formation of 𝐶𝑂2 is therefore equal to ℎ̃°𝑓 = −393.52 kJ/kmol, since for convention negative and
positive sign is adopted for exothermic and endothermic reactions, respectively.
Some compounds may have different aggregating forms at the reference conditions, and they are
generally provided with two different values. For example, for water two values of enthalpy of
formation are provided: (i) one referring to liquid water ℎ̃°𝑓,𝐻2 𝑂𝑙 which therefore considers the
possibility of recovering the enthalpy of condensation, and the other (ii) referring to water in the vapor
state at the reference conditions ℎ̃𝑓,𝐻2 𝑂𝑔 . It follows that ℎ̃°𝑓,𝐻2 𝑂𝑙 = ℎ̃°𝑓,𝐻2 𝑂𝑔 + ∆ℎ̃𝑐𝑜𝑛𝑑,𝐻2 𝑂(𝑇0 ) .
Once the reference enthalpy, i.e. the standard enthalpy of formation for a compound in a reacting
system, has been defined, the enthalpy in a certain thermodynamic state at temperature 𝑇 and pressure
𝑝 can be written, like:
ℎ̃(𝑇,𝑝) = ℎ̃°𝑓 + Δℎ̃(𝑇)
(3.30)
Using the assumption of ideal gas.
Referring to the enthalpy balance, the input conditions may be different from one compound to the
next (i.e. the different reagents may not be in thermodynamic equilibrium). While the conditions at the
discharge are considered common to all the species present in the exhaust gases. With the assumption
that also the reagents enter at the same conditions, the balance of the first law on a mass-basis can be
rewritten in the following manner.
27 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑄 − 𝑊 = ∑ 𝑚𝑖 (ℎ°𝑓,𝑖 + Δℎ𝑖,𝑇𝑃 ) − ∑ 𝑚𝑗 (ℎ°𝑓,𝑗 + Δℎ𝑗,𝑇𝑅 )
𝑖∈𝑃
(3.31)
𝑗∈𝑅
In order to be solved, this relation requires many considerations:
• combustor type, i.e. whether or not it is possible to extract/introduce heat and work.
• the possibility to neglect components of kinetic and potential energy.
• the thermodynamic state of the reagents, i.e. if they are in the liquid, gaseous or solid state.
• the thermodynamic state of the products. In particular, the aggregation state of the water
contained in the flue gas, that may be in the liquid, vapor or partially condensed state according
to the phase equilibrium at a certain temperature.
• whether or not the reactants are pre-mixed.
Once these assumptions are defined, it is possible to define quantities such as the adiabatic flame
temperature and the heating value of the fuel.
3.3.3
Adiabatic flame temperature
Let’s consider a complete adiabatic combustion of the unit of fuel mass, to which corresponds a mass
of oxidant air given by the value of the stoichiometric ratio 𝛼𝑠𝑡𝑜𝑖𝑐 . Starting from the equation and the
previous assumptions, the following is considered:
(i)
an adiabatic combustion chamber
(ii)
the absence of exchanged work
(iii) the perfect mixing between the reaction products at temperature 𝑇𝑃
(iv)
the perfect mixing between the reagents at temperature 𝑇𝑅
(v)
an isobaric process at pressure 𝑝0
reagents
adiabatic
combustor
combustion
products
(fuel)
+1
(combustion products)
(comburent)
Fig. 3.6 - First law enthalpy balance for an adiabatic reacting system and without extracting work to
determine the adiabatic flame temperature
It is possible to calculate the adiabatic flame temperature 𝑇𝐴𝐹 as that temperature of the combustion
products that guarantees the first law balance for an adiabatic system. It is found by means of an
iterative calculation, given the implicit bond between enthalpy and temperature for ideal polyatomic
gases.
∑ 𝑚𝑖 (ℎ°𝑓,𝑖 + Δℎ𝑖,𝑇𝐴𝐹 ) = ∑ 𝑚𝑗 (ℎ°𝑓,𝑗 + Δℎ𝑗,𝑇𝑅 )
𝑖∈𝑃
𝑗∈𝑅
28 of 124
(3.32)
Second-law analysis of power cycles - Energy Conversion A – V7.0
The adiabatic flame temperature therefore depends on:
• the temperature of the reactants 𝑇𝑅 : as the temperature of the reactants increases, the adiabatic
flame temperature increases, which is generally defined with 𝑇𝐴𝐹,𝑇𝑅 . As will be seen, it can be
raised at will by means of recuperative preheating.
• from the ratio 𝛼 between the air flow rate and the fuel flow rate: as the air flow rate increases
starting from the minimum flow rate (the one that corresponds to the stoichiometric ratio 𝛼𝑠𝑡
for which complete combustion can be achieved), the adiabatic flame temperature decreases.
That is because the heat generated by the combustion reaction becomes diluted over a greater
flow rate of exhaust gases.
3.3.4
Heating value of the fuel
With reference to the previous case, consider now an isobaric heat recovery downstream of the
combustion process of a unit of fuel mass at pressure 𝑝0 . It is also assumed that this recovery brings
the products back from a temperature 𝑇𝐴𝐹,𝑇𝑅 to that of the reagents 𝑇𝑅 not necessarily equal to 𝑇0 . The
obtainable thermal power will be given by the difference of enthalpy of the products and reagents at
the same conditions of 𝑇𝑅 , 𝑝0 .
Fig. 3.7 - Adiabatic combustion followed by isobaric cooling
The enthalpy balance expressed on a mass-basis, considering 𝑊 = 0 and introducing the hypothesis
of ideal gases, is therefore:
𝑄 = ∑ 𝑚𝑖 (ℎ°𝑓,𝑖 + Δℎ𝑖,𝑇𝑅 ) − ∑ 𝑚𝑗 (ℎ°𝑓,𝑗 + Δℎ𝑗,𝑇𝑅 )
𝑖∈𝑃
𝑄 = ∑ 𝑚𝑖 ℎ°𝑓,𝑖 − ∑ 𝑚𝑗 ℎ°𝑓,𝑗 +
𝑖∈𝑃
𝑗∈𝑅
𝑗∈𝑅
𝑇𝑅
0
(𝑇)𝑑𝑇
∑ 𝑚𝑖 ∫ 𝑐𝑝,𝑖
𝑇
0
𝑖∈𝑃
(3.33)
𝑇𝑅
0
− ∑ 𝑚𝑗 ∫ 𝑐𝑝,𝑗
(𝑇)𝑑𝑇
𝑗∈𝑃
𝑇0
(3.34)
where all the terms of Δℎ can in part be simplified by introducing the hypothesis that the sum of the
heat capacities of the reagents and products is coincident at every temperature in the interval 𝑇0 ÷ 𝑇𝑅 .
With these hypotheses, the previous relation is simplified as:
𝑄 = ∑ 𝑚𝑖 ℎ°𝑓,𝑖 − ∑ 𝑚𝑗 ℎ°𝑓,𝑗
𝑖∈𝑃
(3.35)
𝑗∈𝑅
That depends only on the enthalpies of formation of the species involved in the combustion reaction.
29 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The extractable thermal power is called heating value 𝐻𝑉. It is equal to – 𝑄 and is the thermal power
that can be obtained by cooling the flue gas from the adiabatic flame temperature up to 𝑇𝑅 :
𝐻𝑉 = −𝑄 = 𝐻𝑅 − 𝐻𝑃 = ∑ 𝑚𝑗 ℎ°𝑓,𝑗 − ∑ 𝑚𝑖 ℎ°𝑓,𝑖
𝑗∈𝑅
(3.36)
𝑖∈𝑃
𝑇𝐴𝐹,𝑇𝑅
𝐻𝑉 = 𝑚𝑓𝑢𝑒𝑙 (𝛼 + 1) ∫
𝑐𝑝0 (𝑇)𝑑𝑇
(3.37)
𝑇𝑅
This reaction heat depends on the assumptions on the thermodynamic state of the products and reagents
and, in particular, on the state of aggregation of the water in the exhaust gases. It does not depend on
the degree of dilution of the oxidant since the species not involved in the reaction give no net
contribution in terms of ℎ°𝑓 ; the dilution has an effect only on the adiabatic flame temperature, whose
reduction is balanced by an increase in 𝛼. Conceptually, the reaction heat depends on the temperature
𝑇𝑅 since the sum of the heat capacities of the reagents and products are not exactly the same, but this
change is modest.
Depending on whether or not the enthalpy of condensation of the water in the combustion products is
recovered, the following can be defined:
• Lower Heating Value (LHV), defined as the heat that can be obtained from the unit of mass
with a path that considers:
1. a complete combustion of the unit of mass of the fuel, without heat transfer (adiabatic
combustion), starting from reagents at a reference temperature 𝑇𝑅 (by convention, for
natural gas, 𝑇𝑅 = 𝑇0 =25°C6) until combustion products are obtained at the adiabatic
flame temperature 𝑇𝐴𝐹,𝑇𝑅 .
2. an isobaric cooling of the combustion products from the adiabatic flame temperature
𝑇𝐴𝐹,𝑇𝑅 up to the reference temperature 𝑇𝑅 , assuming that all the H2O in the combustion
products is at the vapor state
𝐿𝐻𝑉 = (𝛼 + 1) ∫
𝑇𝐴𝐹,𝑇𝑅
𝑐𝑝0 (𝑇)𝑑𝑇 = 𝐻𝑅 − 𝐻𝑃
(3.38)
𝑇𝑅
in which the coefficient 𝛼 must be greater than the stoichiometric one, and 𝐻𝑃 is calculated using
ℎ°𝑓,𝐻2 𝑂(𝑔) .
•
Higher Heating Value (HHV), defined as the heat that can be obtained with the same path
described above for the LHV, with the only difference that in this case it is assumed that all
the H2O in the combustion products is at the liquid state.
𝑇𝐴𝐹,𝑇𝑅
𝐻𝐻𝑉 = (𝛼 + 1) ∫
𝑇𝑅
𝑐𝑝0 (𝑇)𝑑𝑇 +
𝑚𝐻2 𝑂
∆ℎ𝑐𝑜𝑛𝑑,𝐻2 𝑂(𝑇0 )
𝑚𝐹
(3.39)
in which 𝐻𝑃 is calculated using ℎ°𝑓,𝐻2 𝑂𝑙 .
Both hypotheses at the base of the definition of LHV and HHV are unrealistic (and therefore
unreproducible in practice). If the combustion products are cooled to 25°C at the pressure of reference
6
Recent European convention: the reference was previously at 15°C.
30 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
(usually atmospheric), the content of H2O will be partially at the vapor state and partially at the liquid
state, with a mole fraction in gaseous mixture given by the condition, 𝑥𝐻2 𝑂𝑔 𝑝0 = 𝑝𝑠𝑎𝑡(𝑇0 ) . Both LHV
and HHV are usually calculated from the chemical composition of the fuel once the thermodynamic
properties of all components are known. For instance, to determine the heating value of natural gas,
the starting point is a gas chromatograph analysis that identifies its chemical composition. The LHV
is then calculated using the tabulated values of the enthalpies of formation.
As an alternative, if the composition of the fuel is unknown, these values can be obtained
experimentally with calorimetric experiments that use:
• Mahler bomb calorimeter for solid or liquid fuels. It entails the total combustion of a
measured amount of fuel in an airtight constant volume container immersed in water. The
reaction is triggered in an atmosphere of pressurized oxygen. When the system reaches the
balance, the water temperature is measured and the value of heat removed is obtained.
• Junker gas calorimeter for gaseous fuels. It involves the combustion of a certain measured
flow rate of fuel in an open system and counter-current heat exchange with a measured
flow rate of water.
These experiments obtain the LHV value since they are obtained with a large excess of fuel and,
therefore, the water produced has an extremely small partial pressure that is less than the saturation
pressure at temperature 𝑇0 . Therefore, there are no condensation phenomena. The HHV is obtained
from the LHV analytically by adding the latent enthalpy of vaporization of the water.
3.3.5
Recuperative preheating
It is now important to point out that the combustion heat transfer is not limited to the adiabatic flame
temperature 𝑇𝐴𝐹,𝑇0 achievable with 𝑇𝑅 = 𝑇0 . Because this thermal energy, whether it is the LHV or
HHV, can be made available at very high temperatures thanks to the recuperative preheating of the
reagents.
It is assumed that:
• the sum of the heat capacities of the two reactants is equal to that of the products of combustion
in the temperature range 𝑇0 ÷ 𝑇𝑅
• the heat exchanger (recuperator) that preheats the reactants by cooling the combustion products
has an infinite surface
• the heat released from the combustion does not change with the increase of 𝑇𝑅
The recuperative preheating of the reactants produces an increase in the adiabatic flame temperature,
and increases both the terms that define the mean temperature of the heat made available to a power
production plant, if any. So thanks to the possibility of the recuperative preheating, the energy of a
fuel can be generated at a desired high temperature. Therefore, the efficiency limits with which it is
converted into mechanical work are not tied to intrinsic irreversibilities of the combustion process, but
to technological limitations of the power cycle.
31 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 3.8 - Recuperative preheating of the reactants of an adiabatic combustion
It is assumed there is a reversible cycle capable of producing work by exploiting the cooling of the
exhaust gases from the temperature 𝑇𝐴𝐹,𝑇𝑅 up to 𝑇𝑅 . The maximum work that can be produced by a
source not in thermodynamic equilibrium with the reference state is the exergy of the flow, as
previously shown.
In the case where 𝑇𝑅 = 𝑇0 the reversible cycle is a trilateral cycle. With the hypothesis of constant heat
capacity and isobaric process, the reversible power that can be produced is therefore only a portion of
the thermal power available:
𝑊𝑟𝑒𝑣(𝑇𝑅=𝑇0 ) = 𝐻𝐻𝑉 (1 −
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝐴𝐹,𝑇0 |𝑇0 )
)
(3.40)
Instead, in the case where the reactants are preheated, the producible power will be higher. Indeed, the
HHV does not depend on 𝑇𝑅 , whereas the efficiency of the trapezoidal cycle increases, increasing the
mean log temperature of heat introduction
𝑊𝑟𝑒𝑣(𝑇𝑅) = 𝐻𝐻𝑉 (1 −
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝐴𝐹,𝑇
𝑅
)
|𝑇𝑅 )
(3.41)
For a preheating tending to infinity, a conversion efficiency equal to 100% is ideally obtained. In
reality, this does not happen because in the combustion reaction chemical species, whose absolute
entropies should be properly considered, were formed and destroyed, as will be explained in the next
chapter.
32 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 3.9 - Increase of the mean extraction temperature of the thermal power when the recuperative
preheating of the reactants increases
3.3.6
Entropic balance for a reacting system
Now an entropic balance of the general combustion reaction can be made considering a general
production of entropy by irreversibility and the fact that the heat released by the system to the
environment (negative) involves an increase in entropy of the same environment:
∆𝑆𝑖𝑟𝑟 + 𝑆𝑅 +
𝑄
− 𝑆𝑃 = 0
𝑇0
(3.42)
By applying the entropic balance on a mass-basis to a combustion reaction, the following will be
obtained:
𝑄
+ ∆𝑆𝑖𝑟𝑟 = ∑ 𝑚𝑖 𝑠𝑖𝑃 (𝑇, 𝑝) − ∑ 𝑚𝑗 𝑠𝑗 (𝑇, 𝑝)
𝑇0
𝑖∈𝑃
(3.43)
𝑗∈𝑅
where the entropies must be calculated as 𝑠(𝑇, 𝑝) = 𝑠0 + Δ𝑠(𝑇,𝑝) , taking into account the enthalpy of
reference 𝑠0 and Δ𝑠(𝑇,𝑝) due to the deviation from the thermodynamic state of reference.
What was said before referring to the enthalpy balance of reactant systems is also valid for the
entropies since also in this case the choice of the reference value cannot be arbitrary and should be
made according to well-defined considerations.
The reference entropy for a compound involved in a reaction must be defined in accordance with the
third law of thermodynamics. Based on this law, all substances that show a pure crystalline structure
at a temperature of zero Kelvin have null entropy. On the contrary, the entropy will be greater than
zero.The reference value for the entropy is called absolute entropy and is given by two contributions:
the increase in entropy of heating and that of the phase change starting from zero Kelvin up to standard
reference conditions 𝑇0 , 𝑝0 .
33 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 3.10 - Qualitative representation of the definition of reference entropy at zero Kelvin
So, for example, if the component can be considered an ideal gas at the reference state, it turns out:
𝑠 𝑎 (𝑇0 , 𝑝0 )
𝑇𝑓𝑢𝑠
𝑐𝑝 𝑠1 (𝑇)
𝑐𝑝 𝑠2 (𝑇)
∆ℎ𝑓𝑢𝑠
∆ℎ𝑡𝑐
=∫
𝑑𝑇 +
+∫
𝑑𝑇 +
𝑇
𝑇𝑡𝑐
𝑇
𝑇𝑓𝑢𝑠
0
𝑇𝑡𝑐
𝑇𝑒𝑣𝑎 𝑐 𝑙 (𝑇)
𝑇0 𝑐 𝑔 (𝑇)
∆ℎ𝑒𝑣𝑎
𝑝0
𝑝
𝑝
+∫
𝑑𝑇 +
+∫
𝑑𝑇 − 𝑅𝑔 𝑙𝑛 (
)
𝑇
𝑇𝑒𝑣𝑎
𝑇
𝑝𝑠𝑎𝑡(𝑇𝑒𝑣𝑎)
𝑇𝑓𝑢𝑠
𝑇𝑒𝑣𝑎
𝑇𝑡𝑐
(3.44)
where for greater clarity the specific heats of solid, liquid and gas, respectively, are indicated with
𝑐𝑝 𝑠1 , 𝑐𝑝 𝑠2 ,𝑐𝑝 𝑙 , 𝑐𝑝 𝑔 .
The absolute entropy of a general compound at state 𝑇, 𝑝 is therefore equal to:
𝑠(𝑇,𝑝) = 𝑠 𝑎 + Δ𝑠(𝑇,𝑝)
(3.45)
Since both the reactants and the products are mixed and considering it an ideal mixture of ideal gases,
it has to be considered that every species is not at pressure 𝑝, but at the partial pressure 𝑝𝑖 .
The corrective term in molar basis becomes Δ𝑠(𝑇,𝑝𝑖 ) , and will take the form of:
𝑇 𝑐 𝑔 (𝑇)
𝑝
Δ𝑠𝑖,(𝑇,𝑝𝑖 ) = ∫
𝑇0
𝑇
𝑇 𝑐 𝑔 (𝑇)
𝑝𝑖
𝑝
𝑝𝑖
𝑝
𝑑𝑇 − 𝑅𝑔 𝑙𝑛 ( ) = ∫
𝑑𝑇 − 𝑅𝑔 𝑙𝑛 ( ) − 𝑅𝑔 𝑙𝑛 ( )
𝑝0
𝑇
𝑝0
𝑝
𝑇0
(3.46)
𝑝
where the term −𝑅𝑔 𝑙𝑛 ( 𝑖 ) = −𝑅𝑔 𝑙𝑛(𝑥𝑖 ) is the mixing entropy.
𝑝
By applying the entropic balance in mass form to a combustion reaction in which the products are
brought back to the inlet condition of the reactants and with the hypothesis of ideal gases, the following
will be obtained:
𝑄
+ ∆𝑆𝑖𝑟𝑟 = ∑ 𝑚𝑖 𝑠 𝑎 𝑖 − ∑ 𝑚𝑗 𝑠 𝑎𝑗 +
𝑇0
𝑖∈𝑃
𝑗∈𝑅
34 of 124
(3.47)
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑇𝑅 𝑐 𝑔 (𝑇)
𝑝
+ ∑ 𝑚𝑖 ∫
𝑇
𝑇0
𝑖∈𝑃
− [∑ 𝑚𝑖
𝑖∈𝑃
𝑇𝑅 𝑐 𝑔 (𝑇)
𝑝
𝑑𝑇 − ∑ 𝑚𝑗 ∫
𝑇0
𝑗∈𝑅
𝑇
𝑑𝑇 −
𝑝𝑗
𝑅𝑢
𝑝𝑖
𝑅𝑢
𝑙𝑛 ( ) − ∑ 𝑚𝑗
𝑙𝑛 ( )]
𝑀𝑀𝑖
𝑝0
𝑀𝑀𝑗
𝑝0
𝑗∈𝑅
Moreover, with the assumption that the sum of the heat capacities is equal for the reactants and the
products in the field 𝑇0 ÷ 𝑇𝑅 , it is found that the entropy balance does not depend on the temperature
of the reactants 𝑇𝑅 .
3.3.7
Irreversibility generated in combustion
When applying the entropy balance to an adiabatic reactor, the reaction products are obtained at the
adiabatic flame temperature. The previous relation makes it possible to calculate the entropy variation
generated by the combustion process:
∆𝑆𝑐𝑜𝑚𝑏 = ∑ 𝑚𝑖 𝑠 𝑎 𝑖 − ∑ 𝑚𝑗 𝑠 𝑎𝑗 +
𝑖∈𝑃
+ ∑ 𝑚𝑖 ∫
𝑖∈𝑃
𝑇𝐴𝐹,𝑇𝑅
𝑇0
− [∑ 𝑚𝑖
𝑖∈𝑃
𝑗∈𝑅
𝑇𝑅
𝑐𝑝 𝑔 (𝑇)
𝑐𝑝 𝑔 (𝑇)
𝑑𝑇 − ∑ 𝑚𝑗 ∫
𝑑𝑇 −
𝑇
𝑇
𝑇0
𝑗∈𝑅
(3.48)
𝑝𝑗
𝑅𝑢
𝑝𝑖
𝑅𝑢
𝑙𝑛 ( ) − ∑ 𝑚𝑗
𝑙𝑛 ( )]
𝑀𝑀𝑖
𝑝0
𝑀𝑀𝑗
𝑝0
𝑗∈𝑅
That is to say, the difference in entropy is due to a term which depends only on the nature of the
chemical species, regardless of pressure and temperature, one term that depends on the composition 𝑥
and on the temperature, and one that depends on composition and pressure:
∆𝑆𝑐𝑜𝑚𝑏 = ∆𝑆𝑥𝑎 + ∆𝑆(𝑥,𝑇) + ∆𝑆(𝑥,𝑝)
(3.49)
Introducing the hypothesis of equal heat capacities, the term dependent on the temperature can be
written as:
𝑇𝐴𝐹,𝑇𝑅
∆𝑆(𝑥,𝑇) = ∑ 𝑚𝑖 ∫
𝑇𝑅
𝑖∈𝑃
𝑐𝑝 𝑔 (𝑇)
𝑑𝑇
𝑇
(3.50)
which tends to zero for 𝑇𝑅 → 𝑇𝐴𝐹,𝑇𝑅 , that is for very high recuperative preheating.
3.3.8
Reversible work and chemical exergy
After combining the enthalpy and entropy balance applied to a system that cools the reaction products
up to the temperature of the reactants 𝑇𝑅 , it is found that:
𝑄 − 𝑊 = 𝐻𝑃 − 𝐻𝑅 = ∑ 𝑚𝑖 ℎ𝑖,(𝑇𝑅,𝑝0 ) − ∑ 𝑚𝑗 ℎ𝑗,(𝑇𝑅,𝑝0 )
𝑖∈𝑃
𝑗∈𝑅
{
𝑄
+ ∆𝑆𝑖𝑟𝑟 = 𝑆𝑃 − 𝑆𝑅 = ∑ 𝑚𝑖 𝑠𝑖,(𝑇𝑅,𝑝0 ) − ∑ 𝑚𝑗 𝑠𝑗,(𝑇𝑅,𝑝0 )
𝑇0
𝑖∈𝑃
𝑗∈𝑅
35 of 124
(3.51)
Second-law analysis of power cycles - Energy Conversion A – V7.0
where in the case of a globally reversible process ∆𝑆𝑖𝑟𝑟 = 0 the following turns out:
𝑊𝑟𝑒𝑣 = (𝐻𝑅 − 𝐻𝑃 ) − 𝑇0 (𝑆𝑅 − 𝑆𝑃 )
(3.52)
where the term 𝐻𝑅 − 𝐻𝑃 is the maximum thermal power that can be extracted by the combustion
reaction and therefore it is equal to the 𝐻𝐻𝑉.
The maximum work is therefore different from the 𝐻𝐻𝑉 due to the change in entropy between
reactants and products. Indeed, it depends on the absolute entropies and the terms of pressure, and
represents the production of entropy due to the combustion process. Nevertheless, this correction is
small and the numerical value of the 𝑊𝑟𝑒𝑣 is, for most of the fuels in question, similar to the 𝐻𝐻𝑉.
This is because the reaction heat can be made available at very high temperature with a recuperative
preheating. Thus, it can be ideally converted into work with yields tending to the unit.
The reversible work is also equal to the difference of exergy between reactants and reaction products
obtained by bringing the products back to the inlet thermodynamic conditions of the reactants, and
without them changing their chemical composition or without them being reversibly remixed in the
environment. In some texts, the term 𝐻 − 𝑇0 𝑆 takes the name of physical exergy, in the sense that it
represents the exergy in a certain thermodynamic state without considering a chemical composition
change. The reversible work can therefore be written as:
𝑝ℎ
𝑝ℎ
𝑊𝑟𝑒𝑣 = 𝐵𝑅 − 𝐵𝑃
(3.53)
In literature, the common approach for reversible work calculation considers the contribution of each
reactant and product as if they are at standard pressure and physically separated from each other. In
this case, the difference in entropy due to mixing is null, and 𝑆𝑅 − 𝑆𝑃 can be calculated from the
standard entropy of formation of the species sa .
The results found up to this point are valid if the fact that both reactants and products are not in
chemical equilibrium with the environment is neglected. For example, in the case of stoichiometric
𝑂
combustion of methane with pure oxygen, the mole ratio 2 is equal to 2. Thus, the mole fraction of
𝐶𝐻4
the oxygen in the reactants is equal to 66% versus 21% in the ambient air. It is thus feasible to use an
ideal membrane to separate the oxygen from the environment at a pressure of 0.21𝑝0 and then
compress it with an isotherm compressor up to pressure 0.66𝑝0 . The work executed is equal to the
heat transferred to the environment that, considering the isothermal process, can be calculated on a
mass-basis as:
𝑊 = 𝑄 = 𝑇0 ∆𝑆 = −𝑇0 𝑚𝑂2
𝑝𝑂 ,𝑅
𝑝𝑂 ,0
𝑥𝑂 ,𝑅
𝑅𝑢
𝑅𝑢
[𝑙𝑛 ( 2 ) − 𝑙𝑛 ( 2 )] = −T0 𝑚𝑂2
𝑙𝑛 ( 2 )
𝑀𝑀𝑂2
𝑝0
𝑝0
𝑀𝑀𝑂2
𝑥𝑂2 ,0
(3.54)
which has a negative value and represents a spent work.
Likewise, it is possible to separate the combustion products through ideal membranes selective for a
single chemical species, and then using isothermal machinery (turbines and compressors) to take these
compounds from the condition of partial pressure in burnt gases to the partial pressure they have in
the atmosphere. In this way, it is possible to reversibly emit combustion products into the atmosphere.
36 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
By expressing the flow rate of each species as 𝑚𝑖 =𝑚𝐹 (𝛼 + 1)𝑦𝑖 , the specific additional power given
by the whole of the isothermal and reversible compressors and turbines obtained is:
∆𝑊𝑟𝑒𝑣,𝑚𝑖𝑥 = 𝑚𝑓 (𝛼 + 1) ∙ 𝑇0 [∑ 𝑦𝑖
𝑖∈𝑃
𝑝𝑗,𝑅
𝑅𝑢
𝑝𝑖,𝑃
𝑅𝑢
𝑙𝑛 ( ) − ∑ 𝑦𝑗
𝑙𝑛 (
)]
𝑀𝑀𝑖
𝑝𝑖,0
𝑀𝑀𝑗
𝑝𝑗,0
(3.55)
𝑗∈𝑅
where a positive work is achieved if 𝑝𝑖,𝑃 > 𝑝𝑖,0 (for the products) and if 𝑝𝑗,𝑅 < 𝑝𝑗,0 (for the reactants).
The sum of the reversible work and of these contributions is called chemical exergy change in the fuel:
𝑊𝑟𝑒𝑣 + ∆𝑊𝑟𝑒𝑣,𝑚𝑖𝑥 = ∆𝐵𝑐ℎ = 𝐵𝑅𝑐ℎ − 𝐵𝑃𝑐ℎ
(3.56)
It represents the maximum useful work that can be extracted from the combustion of a unit of mass of
fuel bringing components back into not only thermodynamic equilibrium (same temperature and
pressure), but also chemical equilibrium with the environment (same composition). It can be noticed
that the assumption of fuels and combustion products as perfectly mixed or as separated from each
other at standard pressure changes the value of 𝑊𝑟𝑒𝑣 but not of ∆𝐵𝑐ℎ .
Fig. 3.11 - Reversible mixing of the combustion products in the ambient air achieved by adopting
semipermeable membranes and reversible isothermal expanders/compressors.
3.3.9
Comparison of indexes
The values of LHV, HHV, 𝑊𝑟𝑒𝑣 and ∆𝐵𝑐ℎ are very similar to each other, and it can be noted that:
𝐻
• the difference between HHV and LHV is greater the higher the ratio of the fuel. The two
𝐶
indexes are equivalent for graphite.
• the difference between 𝑊𝑟𝑒𝑣 and ∆𝐵𝑐ℎ is small since it depends only on the modest contribution
of the powers that can be obtained from the reversible mixing of the different compounds with
the ambient air. This difference is less than 1% for methane.
• 𝑊𝑟𝑒𝑣 falls between the LHV and the HHV for the hydrocarbons. This is because, as already
stated, the combustion heat can be made available at as high a temperature as one likes through
recuperative preheating, thus exploitable at yields tending to the unit.
• Since the enthalpy of condensation of water is released at low temperatures, it has a low exergy
content, and hence generally ∆𝐵𝑐ℎ < 𝐻𝐻𝑉
37 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
•
•
The chemical exergy is hardly used since the technological solution required for taking the
combustion products to chemical equilibrium with the ambient air is extremely costly: infinite
surface membranes, infinite intercooled compression and infinite reheated expansion.
Furthermore, the membranes available are not similar to ideal components since they show
high pressure drops.
For the analysis of fossil fuel-powered thermodynamic plants, reference will therefore be made
to the HHV or LHV for a first law analysis, and to 𝑊𝑟𝑒𝑣 for a second law analysis.
38 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
4 IRREVERSIBLE PROCESSES AND LOSS OF USEFUL WORK
In this chapter the theorem of lost work is explained. The relationship between entropy generation and
loss of mechanical work is proved and analyzed.
4.1 IRREVERSIBILITIES AND USEFUL WORK LOSSES
In the last chapter the reversible works extractable from various energy sources such as the following
were defined:
• sources with infinite heat capacity and therefore isothermal: the reversible work is calculated
using the Carnot efficiency;
• flows not in equilibrium with the environment: the reversible work is calculated as the
difference of the source exergy between the inlet and outlet conditions without considering
variations in its composition. In the case of constant heat capacities, the efficiency of a trilateral
or trapezoidal cycle can be used to calculate the reversible work;
• reacting systems: the reversible work can be assumed to be similar to the HHV or the LHV,
depending on the aggregation state of water in the reaction products. This assumption passes
through the observation that the heating value can be extracted from the system at temperatures
ideally tending to infinity using the recuperative preheating method.
These indexes represent the thermodynamic limit that can ideally be reached using only reversible
processes. Thus, they are the term with which it is possible to compare when the performance of a
plant in a second law perspective has to be assessed. In fact, this limit can never be reached due to the
presence of non-ideal and non-reversible components that involve losses of useful work.
When there is an irreversibility, there is a loss of ability to produce useful work. To clarify this concept,
the following two examples should be examined. In both cases, a heat sink with infinite heat capacity
with which heat can be exchanged (taken or released) freely without variation in the ability to produce
heat must be defined. 𝑇0 is the temperature of this heat sink.
Starting from the ideal and reversible cases seen previously, now a cause of irreversibility in each of
them is introduced:
• an irreversibility of heat transfer in the exploitation of sources with infinite capacity
• a pressure drop in the exploitation of a source with finite heat capacity not in equilibrium with
the environment
4.1.1
First example of irreversibility: heat transfer
Consider the following sequence of transformations (all reversible):
• the heat transfer 𝑄1 between a source with constant temperature 𝑇𝑚𝑎𝑥 and a fluid that evolves
into a Carnot engine cycle between 𝑇𝑚𝑎𝑥 and 𝑇𝑚𝑖𝑛 .
• the working fluid converts part of the heat 𝑄1 into work 𝑊𝑢 and transfers heat 𝑄2 to a heat
sinks at constant temperature 𝑇𝑚𝑖𝑛 .
• The minimum temperature may be higher than that of the environment 𝑇0 .
Hence, the work obtained is equal to:
𝑊𝑢 = 𝑄1 (1 −
𝑇𝑚𝑖𝑛
)
𝑇𝑚𝑎𝑥
(4.1)
Now an irreversibility represented by a heat transfer is introduced. The heat 𝑄1 is transferred under
finite temperature difference between the heat source at 𝑇𝑚𝑎𝑥 and the working fluid of the Carnot cycle
that receives it at temperature 𝑇𝑥 .
39 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The comparison between these two systems will have to be made with thermal powers transferred to
the heat sinks being equal, with at the most the addition of another system consisting of a sink at
ambient temperature 𝑇0 .
In this case the work of the Carnot cycle is equal to:
𝑊𝑢 ′ = 𝑄1 (1 −
𝑇𝑚𝑖𝑛
)
𝑇𝑥
(4.2)
while the work loss caused by the irreversibility is:
∆W𝑢′ = 𝑊𝑢 − 𝑊𝑢 ′ = 𝑄1 (
𝑇𝑚𝑖𝑛 𝑇𝑚𝑖𝑛
−
)
𝑇𝑥
𝑇𝑚𝑎𝑥
(4.3)
𝑄1
𝑄1
−
)
𝑇𝑥 𝑇𝑚𝑎𝑥
(4.4)
which can be rewritten as:
∆𝑊𝑢 ′ = 𝑊𝑢 − 𝑊𝑢 ′ = 𝑇𝑚𝑖𝑛 (
The term in parentheses represents the total increase of entropy ∆𝑆 caused by the irreversible process
considered. It is equal to the difference between the entropy increase of the fluid that receives heat
𝑄
𝑄
∆𝑆𝑐 = 1 and the entropy decrease of the source that releases heat ∆𝑆ℎ = 1 :
𝑇𝑥
𝑇𝑚𝑎𝑥
∆𝑆 = ∆𝑆𝑐 + ∆𝑆ℎ = (
𝑄1
𝑄1
−
)
𝑇𝑥 𝑇𝑚𝑎𝑥
(4.5)
And therefore
∆W𝑢 ′ = 𝑇𝑚𝑖𝑛 ∆𝑆
(4.6)
However, this is not the real loss of useful work associated with the presence of an irreversible process.
In fact, compared to the reversible case, the heat sink at temperature 𝑇𝑚𝑖𝑛 now receives a heat 𝑄2 ′ >
𝑄2 . The difference is equal to the term ∆𝑊𝑢′ because it is equal to:
𝑄2 ′ = 𝑄1 − 𝑊𝑢 ′ = 𝑄1 − (𝑊𝑢 − Δ𝑊𝑢 ′ ) = 𝑄1 − 𝑊𝑢 + Δ𝑊𝑢 ′ = 𝑄2 + Δ𝑊𝑢 ′
(4.7)
If one wanted to make a correct comparison between the two systems, the additional heat discharged
to the sink at 𝑇𝑚𝑖𝑛 must be removed and discharged reversibly to a sink at temperature 𝑇0 .
If 𝑇𝑚𝑖𝑛 > 𝑇0 , the additional heat discharged has an energy potential and can be transformed into work
with a further reversible cycle between 𝑇𝑚𝑖𝑛 and 𝑇0 , getting a certain useful work that goes in part to
compensate for the loss of work due to the introduction of heat with an irreversible process:
𝑊𝑢 ′′ = Δ𝑊𝑢 ′ (1 −
𝑇0
)
𝑇𝑚𝑖𝑛
(4.8)
Then, if the comparison is made properly, keeping unaltered both the heat 𝑄1 transferred by the source
𝑇𝑚𝑎𝑥 and the heat 𝑄2 received from the sink at temperature 𝑇𝑚𝑖𝑛 , it is found that the actual useful work
is equal to:
40 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Δ𝑊𝑢 = ΔW𝑢 ′ − 𝑊𝑢 ′′ = Δ𝑊𝑢 ′ [1 − (1 −
𝑇0
𝑇0
)] = ΔW𝑢 ′
𝑇𝑚𝑖𝑛
𝑇𝑚𝑖𝑛
(4.9)
Remembering that ∆𝑊𝑢 ′ = 𝑇𝑚𝑖𝑛 ∆𝑆
Δ𝑊𝑢 = 𝑇0 ∆𝑆
(4.10)
Thus, the product of the sink temperature 𝑇0 at which the heat has no energy content for the increase
of entropy (in this case, the only one) of the universe ∆𝑆.
Irreversible heat transfer
MC
MC
a)
=
+
MC
b)
Fig. 4.1 - Irreversible path caused by a heat transfer under finite temperature differences
4.1.2
Second example of irreversibility: Isenthalpic throttling
Now an irreversibility is introduced in the reversible path used to draw the definition of exergy:
suppose interposing a valve upstream of the isentropic expansion, in which a process called
"isenthalpic throttling" is performed.
2
The total enthalpy, defined as the sum of the static enthalpy and the kinetic energy ℎ 𝑇 = ℎ𝑆 + 𝑉2 , is
conserved in an adiabatic duct without work exchange (𝑊 = 0 and 𝑄 = 0), neglecting the variations
in potential energy linked to the altitude.
41 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 4.2 - First law balance for an isenthalpic valve
Depending on the thermodynamic state of the fluid that experiences the isenthalpic throttling, different
phenomena can be observed:
• For an ideal gas, the isenthalpic process is also isotherm;
• For a real gas below the Joule-Thompson inversion temperature at the given pressure (e.g.
superheated steam or supercritical fluids), the throttling cools down the fluid;
• For real gas above the Joule-Thompson inversion temperature at the given pressure (in
particular for liquids) the throttling heats up the fluid.
The static temperature of the fluid can be higher or lower in the downstream section because of the
variations in kinetic energy due to the different pipe section and fluid density. The temperature is
different also in the smallest section of the valve (nozzle throat), typically lower. If the phenomenon
is analyzed more in detail considering the physical processes that take place, it can be observed that
when the flow, starting from thermodynamic equilibrium conditions (1), meets the narrowing caused
by the insertion of the valve plug in the duct, it undergoes an expansion. Consequently, the kinetic
energy increases and the static enthalpy decreases up to the minimum section of the duct, called
contracted section (C). This is valid for subsonic flows upstream of the valve, a situation that, however,
covers all cases in real energy plants.
The expansion takes place generally without large irreversible effects, so with modest increases in
entropy, since the acceleration of a fluid or its expansion is simple to achieve. Subsequently, the flow
encounters an abrupt enlargement in section, which causes important irreversible phenomena
(boundary layer separations, vortices, etc.), and large increases in entropy until a new condition of
thermodynamic equilibrium (2) is reached. In this phase, the fluid slows down and is compressed.
Fig. 4.3 - Representation of the irreversible phenomena due to turbulence in the transformation of
isenthalpic throttling for an ideal gas
By inserting a throttling in the example shown in point 3.2 upstream of the isentropic turbine, a process
that can be divided into three transformations can be obtained, as shown in Fig. 4.4.
42 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The transformations are:
1) the irreversible throttling from 𝑥 to 𝑦
2) the isentropic and adiabatic, and therefore reversible, expansion from 𝑦 to 𝐵
3) the isothermal and reversible compression from 𝐵 to 0 passing through point 𝐴
Fig. 4.4 - Irreversible path given by an isenthalpic throttling in the exploitation of a current of gas
not in equilibrium with the dead state
The powers that are obtained in the two reversible machines with the hypothesis of ideal gas are:
•
•
expansion power:
𝑊̇𝑇𝑅𝐵 = 𝑚̇ (ℎ𝑦 − ℎ𝐵 ) = 𝑚̇ (ℎ𝑥 − ℎ0 )
compression power: 𝑊̇𝐶𝑀𝑃 = 𝑚̇ 𝑇0 (𝑠𝐵 − 𝑠0 ) = 𝑚̇ 𝑇0 (𝑠𝑦 − 𝑠0 )
while the reversible work in the previous case was 𝑊̇𝑟𝑒𝑣 = 𝑚̇ [(ℎ𝑥 − ℎ0 ) − 𝑇0 (𝑠𝑥 − 𝑠0 )].
The expansion work does not change compared to the previous case. The compression work increases
because the compression starts with a lower pressure.
The loss of useful work (power) between the two cases is given by the difference between the
compression works:
∆𝑊̇ = 𝑚̇𝑇0 (𝑠𝐵 − 𝑠0 ) − 𝑚̇𝑇0 (𝑠𝐴 − 𝑠0 ) = 𝑚̇𝑇0 (𝑠𝐵 − 𝑠𝐴 ) = 𝑚̇𝑇0 (𝑠𝑦 − 𝑠𝑥 )
(4.11)
This is only due to the term of entropy increase of the universe given by the irreversible process:
̇
∆𝑊̇ = 𝑇0 ∆𝑆𝑥→𝑦
(4.12)
The result caused by the fluid dynamic irreversibility described above in terms of loss of useful work
(power) is thus identical to that caused by the irreversibility of heat transfer explained previously: the
product of the ambient temperature for the increase in total entropy (of the universe) caused by the
irreversibility.
43 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
4.2 GENERAL DEMONSTRATION
With reference to Fig. 4.5, the system S (consisting of fluids, machines, heat exchangers, etc.) is
considered. A single assumption is made in a demonstration that is entirely general in nature: the
system is initially in a condition of thermodynamic equilibrium I (with its thermodynamic properties
identified), and it evolves toward a new condition of thermodynamic equilibrium F with a real process
(irreversible).
It is assumed that the transformation is adiabatic. This does not limit the general validity of the
demonstration. If there is a heat transfer, it suffices to incorporate the elements that exchange heat with
each other in the system S and consider the system as adiabatic again. The same sign convention
previously defined (outgoing work and outgoing heat positive) is adopted. Moreover, the existence of
an environment (a heat sink with infinite thermal capacity) at temperature 𝑇0 is assumed.
If the first law of thermodynamics is applied to the isolated system, the work exchanged with the
outside is equal to the variation of internal energy of the system:
𝑈𝐹 = 𝑈𝐼 − 𝑊 → 𝑊 = 𝑈𝐼 − 𝑈𝐹
(4.13)
Whereas the second law of thermodynamics states that every real transformation of an isolated system
involves an increase in its entropy:
𝑆𝐹 ≥ 𝑆𝐼
(4.14)
Now consider bringing the system S from state I to state F with a reversible path. To do so, it is
inevitable to introduce a change in the properties of the system. The environment at temperature 𝑇0 is
introduced into the system considered. To make the path reversible, it is necessary to decrease the
entropy, assuming a heat transfer which involves the availability of the environment. A certain amount
of heat 𝑄∗ will be taken from the environment, which will decrease the entropy of the environment.
𝑄∗ will be equal to the additional work extracted from the system ∆𝑊𝑢 , since the initial I and final F
instants are not changed.
W
*
+
Fig. 4.5 - Diagram used in the general demonstration for calculating the power ideally recoverable
from replacing an irreversible process with a reversible process
The first law of thermodynamics applied to S is:
𝑊 + ∆𝑊𝑢 = 𝑈𝐼 − 𝑈𝐹 + 𝑄 ∗
44 of 124
(4.15)
Second-law analysis of power cycles - Energy Conversion A – V7.0
where 𝑄∗ is the heat taken for "free" from the environment at 𝑇0 . An additional work term exchanged
by the system 𝑆 appears to maintain the same initial and final conditions.
Having assumed a reversible path, the second law of thermodynamics leads to the immutability of the
entropy of the universe:
𝑄∗
=0
𝑇0
(4.16)
∆𝑊𝑢 = 𝑄∗ = 𝑇0 (𝑆𝐹 − 𝑆𝐼 ) = 𝑇0 ∆𝑆𝑎𝑚𝑏
(4.17)
𝑆𝐹 − 𝑆𝐼 + ∆𝑆𝑎𝑚𝑏 = 𝑆𝐹 − 𝑆𝐼 −
And therefore 𝑆𝐹 − 𝑆𝐼 = ∆𝑆𝑎𝑚𝑏
Thus, it turns out that:
The physical meaning of the equation is that, by replacing an irreversible process with a reversible
process, with the same initial and final states, an additional work ∆𝑊𝑢 is obtained. equal to the product
of the ambient temperature and the increase in entropy of the universe caused by the irreversibility
present in the irreversible process. In other words, every time the entropy of the universe increases,
because of an irreversible process, an ability to produce mechanical work equal to ∆𝑊𝑢 is lost.
Now this general demonstration will be applied to the two previous examples. In both cases, the
irreversible process will be replaced by one or more reversible processes and the increase of producible
work will be calculated.
4.2.1
Heat transfer: replacement with a reversible engine + heat pump
The first process described will be reconsidered. System S consists of sources 𝑇𝑚𝑎𝑥 and 𝑇𝑚𝑖𝑛 , and the
Carnot engine that operates between temperature 𝑇𝑥 and temperature 𝑇𝑚𝑖𝑛 . Initial state I switches to
final state F, where the source at 𝑇𝑚𝑎𝑥 has released the heat 𝑄1 , the heat sink at 𝑇𝑚𝑖𝑛 has received the
heat 𝑄2 , and the work 𝑊 is obtained. The process contains an irreversibility caused by the heat transfer
between the source at 𝑇𝑚𝑎𝑥 and the engine that receives it at 𝑇𝑥 .
A completely reversible path can be imagined by replacing the heat transfer between 𝑇𝑚𝑎𝑥 and 𝑇𝑥 . with
a reversible engine that receives the heat 𝑄1 and transforms one fraction of it into mechanical work:
𝑊𝑢 ′ = 𝑄1 (1 −
𝑇𝑥
)
𝑇𝑚𝑎𝑥
(4.18)
The heat discharged to the starting engine operating between 𝑇𝑥 and 𝑇𝑚𝑖𝑛 is less than that of the initial
diagram of the term 𝑊𝑢 ′ . To comply with the condition that the final state F is the same, it is necessary
to re-establish the heat 𝑄1 that enters the starting engine altogether, and assume a heat intake from the
environment at temperature 𝑇0 . To transfer this heat at temperature 𝑇𝑥 , a reversible heat pump is
needed, which requires a mechanical work 𝑊𝑢 ′′ equal to:
𝑊𝑢 ′′ = 𝑊𝑢 ′ (1 −
𝑇0
)
𝑇𝑥
(4.19)
The additional work achieved by replacing the irreversible process with a reversible process is equal
to:
45 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
∆𝑊𝑢 = 𝑊𝑢 ′ − 𝑊𝑢 ′′ = 𝑊𝑢 ′ − 𝑊𝑢 ′ (1 −
𝑇0
𝑇0
) = 𝑊𝑢 ′ ( )
𝑇𝑥
𝑇𝑥
(4.20)
Which can also be rewritten as
∆𝑊𝑢 = 𝑄1 (1 −
𝑇𝑥
𝑇0
𝑄1
𝑄1
) ( ) = 𝑇0 ( −
)
𝑇𝑚𝑎𝑥 𝑇𝑥
𝑇𝑥 𝑇𝑚𝑎𝑥
(4.21)
where the term in brackets is the difference of entropy variation of the heat sink at temperature 𝑇𝑚𝑎𝑥
and that of the tank at temperature 𝑇𝑥
∆𝑊𝑢 = 𝑇0 (
𝑄1
𝑄1
−
) = 𝑇0 (∆𝑆𝑇𝑥 − ∆𝑆𝑇𝑚𝑎𝑥 ) = 𝑇0 ∆𝑆
𝑇𝑥 𝑇𝑚𝑎𝑥
(4.22)
Which is what was set out to be proven.
MC’
’
W
MC
MC
PdC
Fig. 4.6 - Replacement of the irreversibility of heat transfer with a series of reversible processes that
preserve the initial and final conditions of the sources
4.2.2
Throttling: replacement with reversible isothermal expansion + heat pump
Now the irreversible process of isenthalpic throttling that occurs in the valve is considered. In this case
the system S is represented by the supply duct to the valve, the valve itself and the exhaust pipe. The
initial state I is with the fluid at a standstill, the final F state requires that a mass 𝑚 of air be passed
from the inlet section 𝑥 to the outlet section 𝑦.
46 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
To make the process from I to F reversible, it can be assumed to replace the irreversible adiabatic
expansion with a reversible isotherm expansion at temperature 𝑇𝑥 , which occurs in a reversible
machine producing a work equal to the absorbed heat, that is:
𝑊𝑢 ′ = 𝑄∗ = 𝑇𝑥 ∆𝑆𝑥→𝑦
(4.23)
This work is equal to the “dissipated work due to fluid friction”, or simply “work of fluid friction” in
the irreversible isenthalpic throttling process. It is equal to the work that would have been obtained
with an isentropic expansion between the 𝑝𝑥 and the 𝑝𝑦 .
𝑦
𝑦
𝑊𝑤 = ∫ 𝑇𝑥 𝑑𝑆 = ∫ 𝑉𝑑𝑝
𝑥
𝑥
Reversible
isothermal
expander
Fluid-dynamic irreversibility
(w
a )
HP
a)
b)
+
Fig. 4.7 - Replacement of the throttling irreversibility with a series of reversible processes that
preserve the initial and final conditions of the sources
The heat 𝑄∗ that must be supplied at temperature 𝑇𝑥 can be taken from the environment at temperature
𝑇0 by a heat pump requiring work equal to:
𝑊𝑢 ′′ = 𝑄∗ (1 −
𝑇0
𝑇0
) = 𝑊𝑢 ′ (1 − )
𝑇𝑥
𝑇𝑥
(4.24)
In terms of work, the net result is given by:
∆𝑊𝑢 = 𝑊𝑢 ′ − 𝑊𝑢 ′′ = 𝑊𝑢 ′ [1 − (1 −
𝑇0
𝑇0
)] = 𝑇𝑥 ∆𝑆𝑥→𝑦 = 𝑇0 ∆𝑆𝑥→𝑦
𝑇𝑥
𝑇𝑥
(4.25)
It is noticed that ∆𝑊𝑢 , obtainable by making the process reversible, is less than the 𝑊𝑤 dissipated by
𝑇
the isenthalpic throttling according to the ratio 0 . This is because not all the irreversibility produced
𝑇𝑥
by a valve can be recovered, and the recoverable portion depends on the temperature at which the
irreversibility takes place.
It follows that:
47 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
•
•
•
•
•
•
An isenthalpic throttling involves the dissipation of a mechanical work equal to 𝑊𝑤 = 𝑇𝑥 ∆𝑆;
This dissipated work remains within the fluid found at the end of throttling with a temperature
higher than it would have had following an isentropic expansion and, therefore, with a greater
possibility of providing work through downstream processes;
In both cases, by bringing the source back to conditions of equilibrium with the environment
through a sequence of reversible processes, the work lost forever is equal to 𝑇0 ∆𝑆, regardless
of the temperature at which throttling takes place;
A part of the work dissipated by friction can therefore ideally be recovered with a series of
reversible processes. Its extent depends on the temperature at which the irreversibility was
introduced, 𝑊𝑚𝑎𝑥,𝑟𝑒𝑐 = 𝑊𝑤 − ∆𝑊𝑢 = ∆𝑆(𝑇𝑥 − 𝑇0 );
It is always preferable to introduce irreversibility at high temperatures because the wasted work
can then be exploited more efficiently by the downstream processes;
𝑊𝑚𝑎𝑥,𝑟𝑒𝑐 can be fully recovered if the downstream processes are all reversible, otherwise only
a part of it will be recovered.
This result can be physically described according to two ways of thinking:
a) According to the diagram illustrated above: the replacement of the irreversible phenomenon
with one that is reversible requires an introduction of heat, which is not "free" because it is
required at a temperature 𝑇𝑥 > 𝑇0 . The work required to restore it must be subtracted from the
reversible expander. This work tends to zero the more 𝑇𝑥 tends to 𝑇0 , and it is larger the higher
the temperature where irreversibility takes place is.
b) According to the concept of “recovery”: the fluid "sees" the work dissipated due to friction as
heat introduced at temperature 𝑇𝑥 . This heat has an energy value: if a reversible process is
considered, a mechanical work can be recovered from it. This work is equal to a fraction of the
work of fluid friction expressed by the efficiency of a Carnot cycle between 𝑇𝑥 and 𝑇0 .
Obviously, the two ways have identical results: both clarify the physical meaning of the term ∆𝑊𝑢 ,
which is the loss of ability to produce work caused by an irreversibility, imagining that the rest of the
transformations (in this case: the heat pump or the Carnot recovery cycle) is reversible. In other terms,
when a work of fluid friction 𝑊𝑤 is dissipated, ∆𝑊𝑢 is always and in any case lost, while the difference
(𝑊𝑤 − ∆𝑊𝑢 ) can be recovered totally or in part, depending on the subsequent transformations the
fluid undergoes.
48 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
5 THE MOST COMMON CAUSES OF IRREVERSIBILITY
In this chapter, a list of irreversibilities is presented. In power production plants, multiple forms of
irreversibility can take place.
The most important are:
• heat transfer
• irreversible fluid dynamic phenomena
• combustion reactions
5.1
5.1.1
LOSSES IN HEAT TRANSFER
General definitions
Adiabatic efficiency
The adiabatic efficiency of the heat exchanger is defined as the ratio between heat introduced into the
cold flow and the heat transferred altogether from the hot flow.
𝜂𝑎𝑑 =
𝑄̇𝑑𝑖𝑠𝑝
𝑄̇𝑐 𝑄̇ℎ − 𝑄̇𝑑𝑖𝑠𝑝
=
=1−
𝑄̇ℎ
𝑄̇ℎ
𝑄̇ℎ
(5.1)
This efficiency depends on the temperature difference between the hot fluid and the environment, and
on the insulation of the component. High temperature heat exchangers have always an adiabatic
efficiency lower than one, while condensers can be considered adiabatic since they work at
temperatures similar to the one of the environment.
Effectiveness
The heat exchanger effectiveness is defined as the ratio between the actual heat exchanged and the
heat ideally exchangeable with an infinite surface as
ε=
𝑄̇
𝑄̇S≡∞
(5.2)
The effectiveness is also defined as the ratio between the actual temperature difference and that
obtainable with an infinite surface heat exchanger, referred to the fluid with lower heat capacity.
Excluding the cases in which the pinch point is inside the heat exchanger, if the heat capacity of the
cold flow is lower than that of the hot flow, the pinch point will be at the hot side of the heat exchanger.
If the surface of the heat exchanger is infinite, the cold flow will reach the inlet temperature of the hot
flow. Assuming that the heat capacity of the cold flow doesn’t change considerably with the
temperature, the effectiveness can be calculated as
𝜀=
𝑚̇𝑐 𝑐𝑝,𝑐 (𝑇𝑐,𝑜𝑢𝑡 − 𝑇𝑐,𝑖𝑛 )
(𝑇𝑐,𝑜𝑢𝑡 − 𝑇𝑐,𝑖𝑛 )
∆𝑇𝑐
=
=
𝑚̇𝑐 𝑐𝑝,𝑐 (𝑇ℎ,𝑖𝑛 − 𝑇𝑐,𝑖𝑛 )
(𝑇ℎ,𝑖𝑛 − 𝑇𝑐,𝑖𝑛 )
∆𝑇𝑐,𝑆≡∞
whose representation is depicted in the following figure
49 of 124
(5.3)
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 5.1 - Graphic representation of the effectiveness of heat exchangers with pinch point on the hot
side (left) and on the cold side (right)
If the pinch point is inside the heat exchanger, attention should be paid to the calculation of the
exchangeable heat with ε=1. This is the case of a Once-Though heat exchanger for preheating and
evaporation, or a supercritical heat exchanger, or those cases in which the mean heat capacities of the
fluids change drastically while adopting an infinite surface heat exchanger. The case of an O-T heat
exchanger is represented in the following figure
Fig. 5.2 - T-Q Diagram of a O-T heat exchanger for a simple (left) and complex (right) molecule
In the base case, only preheating and evaporation occurs. While in an infinite surface heat exchanger,
also superheating is realized. In the case of simple molecule, the heat capacity of the vapor is much
lower than that of the liquid phase. This makes the mean equivalent capacity of the cold side lower
than that of the hot side. Thus, the ∆𝑇=0 condition is at the hot side. On the other hand, in the case of
complex molecule, the heat capacities of liquid and vapor phase are similar, and the pinch point is
placed at the beginning of the evaporation phase. Note that even with ε=1, it is impossible to obtain a
reversible heat exchange.
Heat exchange surface calculation
Let’s consider a heat exchanger crossed countercurrent by two flows in which the heat is spontaneously
transferred from the hot fluid to the cold fluid. The heat transfer surface 𝐴 of a heat exchanger can be
calculated using the relation:
𝑄̇ = 𝑈𝐴∆𝑇𝑚𝑙𝑛
50 of 124
(5.4)
Second-law analysis of power cycles - Energy Conversion A – V7.0
where 𝑈 is the global heat transfer coefficient obtained from the transfer coefficients of the hot and
cold flow, and the thermal resistance of the metal wall and the fouling. Reference is generally made to
the internal or external surface of heat transfer, and the ratio between them must be taken into account,
especially in cases of finned tubes.
𝑅𝑓𝑜𝑢𝑙,𝑒𝑥𝑡
1
1
1
| =
+ 𝑅𝑓𝑜𝑢𝑙,𝑖𝑛𝑡 + 𝑅𝑤 + 𝐴𝑒𝑥𝑡 +
𝐴
𝑈 𝑖𝑛𝑡 ℎ𝑖𝑛𝑡
ℎ𝑒𝑥𝑡 𝑒𝑥𝑡
𝐴𝑖𝑛𝑡
(5.5)
𝐴𝑖𝑛𝑡
Instead, the mean log temperature difference in the case of two currents with constant heat capacity
can be calculated as:
∆𝑇𝑚𝑙𝑛 =
∆𝑇1 − ∆𝑇2
∆𝑇1
ln (
∆𝑇2
(5.6)
)
which tends to zero if the temperature difference is canceled out inside or at one of the two ends of the
exchanger. If the heat capacities are not constant, the parameter ∆𝑇𝑚𝑙𝑛 cannot be calculated this way,
but it is necessary to divide the exchanger into a number of subsystems in order to be able to consider
the heat capacities of the constant flows in each one. This approach is fundamental in the case in which
the pinch-point is inside the heat exchanger and not in one of the two ends. For each subsystem, the
surface required for heat transfer can be calculated and the total value of ∆𝑇𝑚𝑙𝑛 can then be calculated
as:
∆𝑇𝑚𝑙𝑛 =
𝑄̇
∑𝑖(𝑈𝑖 𝐴𝑖 )
(5.7)
One simple case is the primary heat exchanger of a non-regenerative saturated Rankine cycle, in which
the heat exchanger must be divided into an economizer and an evaporator. It must be considered that
in the two components the transfer coefficients may be different. A more complex case is represented
by the exchanger of heat introduction of a supercritical cycle. In this case, as shown in Fig. 5.3, the
division of the heat exchanger into a high number of sections is required to properly locate the pinchpoint and calculate the surface.
T
T
Q
Q
Fig. 5.3 - Division of a heat exchanger into several sections for the correct estimation of the transfer
surface
51 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
5.1.2
Entropy generated in an irreversible heat exchange
Whenever there is a heat transfer, a negative entropy variation of the hot current and a positive entropy
variation of the cold current take place. Considering the infinitesimal process of introducing heat into
the cold source at a certain temperature and assuming the process isobaric, it is found that:
𝑑𝑆̇ =
𝑚̇𝑐𝑝 (𝑇)
𝑑𝑄̇
𝑑𝑇 =
𝑇
𝑇
(5.8)
this represents an irreversibility, which gives rise to an increase in entropy.
Once the TQ diagram is plotted for adiabatic heat exchangers crossed by two fluids, the entropy
production can be calculated as:
𝑇𝑐,𝑜𝑢𝑡
∆𝑆̇ = ∆𝑆𝑐̇ + ∆𝑆ℎ̇ = + ∫
𝑇𝑐,𝑖𝑛
𝑇ℎ,𝑜𝑢𝑡
𝑑𝑄̇
𝑑𝑄̇
+∫
𝑇
𝑇
𝑇ℎ,𝑖𝑛
(5.9)
where ∆𝑆𝑐̇ is positive and ∆𝑆ℎ̇ negative.
Fig. 5.4 - General notation for modeling an adiabatic countercurrent heat exchanger
From which it is possible to extract the integral mean values 1/𝑇𝑚,𝑐 and 1/𝑇𝑚,ℎ by applying the mean
value theorem:
∆𝑆̇ =
𝑄̇
𝑄̇
(𝑇𝑚,ℎ − 𝑇𝑚,𝑐 )
−
= 𝑄̇
𝑇𝑚,𝑐 𝑇𝑚,ℎ
(𝑇𝑚,ℎ 𝑇𝑚,𝑐 )
(5.10)
The equation shows that the entropy production and the loss of useful work:
• are proportional to the transferred heat 𝑄̇ (if one refers to the power instead of to the energy,
the thermal power transferred)
• are proportional to the mean difference in temperature between the two currents. The smaller
the difference at each point of the TQ diagram for the two currents is, the lower the production
of irreversibility is. At the limit for temperatures coinciding across the heat transfer, the integral
mean values are coincident 𝑇𝑚,ℎ = 𝑇𝑚,𝑐 and ∆𝑆̇ = 0, but infinite transfer surfaces are
necessary (this would also be true if the temperatures of the two currents were equal at one
point only).
• are inversely proportional to the square of the mean temperature at which the heat is transferred.
To understand the physical meaning of this effect, one can think of a way to achieve the heat
52 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
transfer in a reversible manner by introducing an infinite number of reversible cycles
interposed between the two currents that exchange heat with a certain temperature difference.
Now, imagine to raise the mean temperature of the process, but keeping the same precise
temperature difference. Since the efficiency of each infinitesimal cycle is equal to that of
Carnot, its efficiency is inversely proportional to the mean temperature at which it receives
heat.
𝜂 =1−
𝑇𝑐
𝑇ℎ − Δ𝑇 Δ𝑇
=1−
=
𝑇ℎ
𝑇ℎ
𝑇ℎ
(5.11)
For 𝑇ℎ → ∞, 𝜂 → 0 and therefore it is impossible to recover reversible work from the heat transfer
process, so the process is already irreversible.
In practice, this aspect is little used since generally the heat source is at a certain temperature and it is
not possible to raise it at will. An exception is the case of preheating in the combustion of fossil fuels.
If the heat exchanger is not adiabatic, a further entropy production, linked to the heat exchange to the
environment that receives it at a constant temperature 𝑇0 , is taken into account. In this case the loss of
useful work is broken down into two terms: (i) one similar to what has been seen due to the heat
transfer between the hot flow and the cold flow, (ii) the other linked to the thermal loss.
The total increase in entropy is given by:
̇
∆𝑆̇ = ∆𝑆𝑐̇ + ∆𝑆ℎ̇ + ∆𝑆𝑎𝑚𝑏
= +∫
𝑇𝑐,𝑜𝑢𝑡
𝑇𝑐,𝑖𝑛
𝑇ℎ,𝑜𝑢𝑡
𝑑𝑄̇
𝑑𝑄̇ 𝑄̇𝑑𝑖𝑠𝑝
+∫
+
𝑇
𝑇
𝑇0
𝑇ℎ,𝑖𝑛
(5.12)
̇
where ∆𝑆𝑐̇ and ∆𝑆𝑎𝑚𝑏
are positive and ∆𝑆ℎ̇ is negative.
The heat exchanger is fictitiously divided into two different components by dividing the flow of hot
fluid: in the first a flow of hot fluid equal to 𝑚̇′ℎ = 𝑚̇ℎ 𝜂𝑎𝑑 that transfers heat 𝑄̇𝑐 to the flow of cool
fluid circulates, in the second a flow 𝑚̇′′ℎ = 𝑚̇ℎ (1 − 𝜂𝑎𝑑 ) that transfers heat 𝑄̇𝑑𝑖𝑠𝑝 to the environment
circulates.
Fig. 5.5 - General notation for modeling a non-adiabatic countercurrent heat exchanger
53 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The increase of entropy of the universe of the two sub-components can be calculated as:
(𝑇𝑚,ℎ − 𝑇𝑚,𝑐 )
(𝑇𝑚,ℎ 𝑇𝑚,𝑐 )
(𝑇𝑚,ℎ − 𝑇0 )
∆𝑆̇′′ = 𝑄̇𝑑𝑖𝑠𝑝
(𝑇𝑚,ℎ 𝑇0 )
∆𝑆̇′ = 𝑄̇𝑐
5.1.3
(5.13)
(5.14)
Considerations on the design of the components
These general considerations have a direct effect on the design sizing choices of heat exchangers of a
power production plant. A heat exchanger is sized following a technical-economic optimization
process that sees, on one hand, the desire to maximize the efficiency of the plant and, on the other, that
of curbing the investment costs. To reduce the irreversibility of heat transfer, and thus raise the
efficiency, it is necessary to minimize the temperature differences between the two currents. However,
this leads to an increase in the transfer surfaces or an increase in the number of components in which
the heat transfer process is divided (consider the regeneration of a saturated Rankine cycle).However,
larger surfaces involve higher costs. Thus, the optimal solution is given by the trade-off between the
increase in remuneration obtainable from greater efficiencies and the increase in the investment cost.
For example, the analysis of the cost of electricity produced (LCOE – Levelized Cost of Electricity)
is considered, and two plants based on the same technology with the same number of equivalent hours
are compared. As the surface of a heat exchanger increases, on the one hand there will be an increase
$
of the intercept
(investment cost by cooling capacity factor (CCF)), and on the other, a
𝑘𝑊∙𝑦𝑒𝑎𝑟
$
reduction in the slope of the variable costs
(increase of plant efficiency). The minimum LCOE
𝑘𝑊ℎ
can be obtained with one plant or another, depending on the proportion between the costs/benefits of
the intervention.
It should also be kept in mind that in many cases the increase in surface and cost reduces the
irreversibility of heat transfer only to a minimum degree. For example, for two currents with a large
difference in heat capacity, the reduction of the temperature difference on the cold side of the heat
exchanger reduces the irreversibilities only minimally, and the advantages in terms of efficiency in the
face of a very high increase in cost. In other cases, instead, the reduction of loss of heat transfer can
also lead to penalizing the power produced. For example, consider the condenser of an air-cooled
steam power plant. On the one hand, reducing the condensation pressure is expedient in terms of
efficiency. On the other hand, however, the heat exchangers are more expensive with a consumption
of the fan motors that can grow considerably, ending up reducing the actual benefit in terms of
efficiency.
54 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
2
1
0
h
Fig. 5.6 - Comparison in terms of LCOE of heat exchangers with larger surfaces. In case 1, the
operation is not economically advantageous. It is, instead, in case 2.
5.2 LOSSES IN EXPANSION AND COMPRESSION
Turbomachines will be handled differently depending on whether they process a fluid whose specific
volume depends on the temperature and pressure (gas and vapor) conditions, or an incompressible
fluid (pumps).
5.2.1
Gas/vapor turbomachines: turbines and compressors
The operation of a turbine or an expander is defined by an adiabatic efficiency 𝜂𝑖𝑠 on the adiabatic
expansion as a whole, or by a polytropic efficiency 𝜂∞ or 𝜂𝑦 that takes into account the fluid dynamic
quality of the infinitesimal process. It defines the ratio between the real and the ideal work for a whole
expansion or infinitesimal process.
Consider an adiabatic infinitesimal expansion between 𝑝𝑥 and 𝑝𝑥 − 𝑑𝑝. With the definition of
polytropic efficiency 𝜂𝑦 , it is seen that the real work produced will be given by the difference of that
ideally producible work along an isentropic expansion and that dissipated due to friction phenomena,
which cause a production of irreversibility and an increase in entropy of the fluid.
𝜂𝑦 =
𝑑𝑊̇𝑖𝑟𝑟
;
𝑑𝑊̇𝑟𝑒𝑣
𝜂𝑦 =
𝑑𝑊̇𝑟𝑒𝑣 − 𝑑𝑊̇𝑤
𝑑𝑊̇𝑟𝑒𝑣
(5.15)
where the ideal work is equal to 𝑑𝑊̇𝑟𝑒𝑣 = −𝑚̇𝑣𝑑𝑝 (considering that the 𝑑𝑝 < 0) and friction work is
equal to 𝑑𝑊̇𝑤 = 𝑇𝑑𝑆̇
𝑑𝑊̇𝑤 = 𝑇𝑑𝑆̇ = 𝑑𝑊̇𝑟𝑒𝑣 (1 − 𝜂𝑦 ) = −𝑚̇𝑣𝑑𝑝(1 − 𝜂𝑦 )
(5.16)
For polytropic efficiencies equal to zero (isenthalpic throttling), the ideal work and the friction work
are equal and there is the maximum production of irreversibility. By integrating the entire expansion,
it turns out that:
𝑝𝑜𝑢𝑡 −𝑚̇𝑣𝑑𝑝(1
∆𝑆̇ = ∫
𝑝𝑖𝑛
− 𝜂𝑦 )
𝑇
55 of 124
(5.17)
Second-law analysis of power cycles - Energy Conversion A – V7.0
The specific volume can now be expressed in terms of the compressibility factor like 𝑣 = 𝑍
∆𝑆̇ = −
𝑅𝑢 𝑝𝑜𝑢𝑡 𝑍𝑇 𝑚̇(1 − 𝜂𝑦 )
𝑚̇𝑅𝑢
𝑝𝑖𝑛
∫
𝑑𝑝 =
𝑍̅(̅̅̅̅̅̅̅̅̅
1 − 𝜂𝑦 )𝑙𝑛 (
)
𝑀𝑀 𝑝𝑖𝑛 𝑝
𝑇
𝑀𝑀
𝑝𝑜𝑢𝑡
𝑅𝑢 𝑇
𝑀𝑀 𝑝
(5.18)
Therefore, the loss of efficiency depends on the "fluid dynamic quality" (defined by 𝜂𝑦 ) of the turbine,
and grows as the expansion ratio increases. It is independent from the temperature at which the
expansion takes place since the infinitesimal increase of entropy 𝑑𝑆 is proportional to the infinitesimal
work, which in turn is proportional to 𝑣 and 𝑇, but it is divided by 𝑇, which therefore cancels out.
Another quantity called adiabatic efficiency 𝜂𝑖𝑠 is often preferred to the polytropic efficiency. It
considers only the inlet and outlet conditions of the fluid from the machine, but requires that the flow
rate does not vary between inlet and outlet. The gas that is ideal and at specific constant heat is
considered, and it is assumed that there is the condition for calculating the expansion temperature
change given by a certain polytropic efficiency. If the specific heat is constant, then it can be written
that
𝜂𝑦 =
𝑐𝑝 𝑑𝑇𝑖𝑟𝑟 𝑑𝑇𝑖𝑟𝑟
𝑑ℎ𝑖𝑟𝑟
=
=
𝑑ℎ𝑟𝑒𝑣 𝑐𝑝 𝑑𝑇𝑟𝑒𝑣 𝑑𝑇𝑟𝑒𝑣
(5.19)
If the isentropic 𝑑ℎ linked to each infinitesimal process is considered, the ideal work is found for 𝑑𝑠 =
0 and it is equal to 𝑑ℎ𝑟𝑒𝑣 = 𝑣𝑑𝑝
Then, it can be written for an ideal gas that
𝑑ℎ𝑖𝑟𝑟 𝑐𝑝 𝑑𝑇𝑖𝑟𝑟
=
𝑑ℎ𝑟𝑒𝑣
𝑣𝑑𝑝
𝑅𝑔 𝑑𝑝
𝑑𝑇
| = 𝜂𝑦
𝑇 𝑖𝑟𝑟
𝑐𝑝 𝑝
𝜂𝑦 =
(5.20)
(5.21)
Which integrated between the inlet and outlet conditions gives:
𝑅𝑔
𝑇𝑜𝑢𝑡
𝑝𝑜𝑢𝑡
)| = 𝜂𝑦 𝑙𝑛 (
)
(5.22)
𝑇𝑖𝑛 𝑖𝑟𝑟
𝑐𝑝
𝑝𝑖𝑛
𝑇𝑜𝑢𝑡
| = 𝛽−𝜂𝑦 𝜃
(5.23)
𝑇𝑖𝑛 𝑖𝑟𝑟
Then, the adiabatic efficiency can be defined as the ratio between the work produced and the isentropic
enthalpy change from the inlet conditions. Again, considering the specific constant heats, this means
that:
𝑙𝑛 (
𝜂𝑖𝑠 =
𝑐𝑝 𝛥𝑇𝑖𝑟𝑟 𝑇𝑖𝑛 (1 − 𝛽 −𝜂𝑦 𝜃 )
∆ℎ𝑖𝑟𝑟
=
=
∆ℎ𝑟𝑒𝑣 𝑐𝑝 𝛥𝑇𝑟𝑒𝑣
𝑇𝑖𝑛 (1 − 𝛽 −𝜃 )
(5.24)
For an expansion, the value of 𝜂𝑖𝑠 is always greater than that 𝜂𝑦 due to the recovery phenomenon.
This phenomenon can be explained with a discretization of the expansion process.
Consider carrying out an expansion in very small pressure changes such that each one has an efficiency
equal to the polytropic efficiency. The presence of an irreversibility in the expansion process leads to
56 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
an increase in entropy and to the dissipation of a certain work of fluid friction which remains within
the fluid. Thus, at the end of each expansion the enthalpy is greater than the corresponding isentropic
one, with ∆ℎ1 > ∆ℎ1′′ and 𝑇1 > 𝑇1′′ . Due to the divergence of the isobaric lines, it is evident that
∆ℎ𝑟𝑒𝑣,1→2′ > ∆ℎ𝑟𝑒𝑣,1′′ →2′′ in the second expansion , ∆ℎ𝑟𝑒𝑣,2→3′ > ∆ℎ𝑟𝑒𝑣,2′′ →3′′ in the third, and so on.
0
T
1
1’’=1’
2
2’’ 2’
3’’
3
3’
4
4’
4’’
5
5’
5’’
s
Fig. 5.7 - Graphic representation of the recovery work for a gas turbine
When extending this observation up to the discharge pressures, the isentropic enthalpy change
obtained from the sum of enthalpy changes along the real expansion ∆ℎ𝑟𝑒𝑣,𝑦 is greater than that
calculated simply between inlet and outlet conditions ∆ℎ𝑟𝑒𝑣,𝑎𝑑
∆ℎ𝑟𝑒𝑣,𝑦 = ∑ ∆ℎ𝑟𝑒𝑣,𝑖→(𝑖+1)′′
(5.25)
∆ℎ𝑟𝑒𝑣 = ∑ ∆ℎ𝑟𝑒𝑣,𝑖 ′′→(𝑖+1)′′ = ℎ0 − ℎ5′′
(5.26)
Since the specific work produced is:
𝑤 = 𝜂𝑦 ∆ℎ𝑟𝑒𝑣,𝑦
(5.27)
An adiabatic efficiency higher than the polytropic efficiency is obtained.
𝜂𝑖𝑠 =
where the term
∆ℎ𝑟𝑒𝑣,𝑦
∆ℎ𝑟𝑒𝑣,𝑖𝑠
𝑤
∆ℎ𝑟𝑒𝑣,𝑖𝑠
= 𝜂𝑦
∆ℎ𝑟𝑒𝑣,𝑦
∆ℎ𝑟𝑒𝑣,𝑖𝑠
(5.28)
> 1.
In compressors, the effect is exactly the opposite. Recovery plays against the process since the fluid
heats up, and it requires greater power to be further compressed due to the divergence of the isobaric
lines. The real work is the sum of the ideal work and the wasted work due to irreversible fluid-dynamic
phenomena. For this example, the sign notation for the work is changed: now it is positive if entering
the system.
57 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑑𝑊̇𝑖𝑟𝑟
𝑑𝑊̇𝑟𝑒𝑣 + 𝑑𝑊̇𝑤
1 − 𝜂𝑦
𝑑𝑊̇𝑤 = 𝑑𝑊̇𝑟𝑒𝑣 (
) = 𝑇𝑑𝑆̇
𝜂𝑦
𝜂𝑦 =
𝑑𝑊̇𝑖𝑟𝑟
;
𝑑𝑊̇𝑟𝑒𝑣
𝜂𝑦 =
(5.29)
(5.30)
The increase in entropy given by the compression process is therefore equal to:
∆𝑆̇ = ∫
𝑝𝑜𝑢𝑡
𝑝𝑖𝑛
5.2.2
(
𝑝𝑜𝑢𝑡
1 − 𝜂𝑦 𝑚̇𝑣𝑑𝑝
1 − 𝜂𝑦
𝑚̇𝑅𝑢 𝑇
)
=∫
(
)𝑍
𝑑𝑝
𝜂𝑦
𝑇
𝜂𝑦
𝑀𝑀𝑝𝑇
𝑝𝑖𝑛
̅̅̅̅̅̅̅̅̅
1 − 𝜂𝑦
𝑚̇𝑅𝑢
𝑝𝑜𝑢𝑡
=
𝑍̅ (
) 𝑙𝑛 (
)
𝑀𝑀
𝜂𝑦
̅̅̅
𝑝𝑖𝑛
(5.31)
Hydraulic machines: Pumps
The operation of a pump is defined by a polytropic efficiency 𝜂∞ or 𝜂𝑦 , which defines the ratio between
the ideal work and the real work for an infinitesimal compression. In the case of incompressible fluid,
the adiabatic efficiency 𝜂𝑎𝑑 is equal to the polytropic efficiency.
Considering that the temperature increase is negligible, the work of fluid friction is equal to:
𝑊̇𝑤 = ∫ 𝑇𝑑𝑆̇ = 𝑇𝑥 ∆𝑆̇
(5.32)
The ideal work is equal to 𝑊̇𝑖𝑑𝑒𝑎𝑙 = 𝑚̇𝑣∆𝑝. It is positive (∆𝑝 > 0) and entering the system. Therefore,
∆𝑆 can be calculated as:
∆𝑆̇ =
𝑊̇𝑤 𝑊̇𝑖𝑑𝑒𝑎𝑙 1 − 𝜂𝑦
𝑚̇𝑣∆𝑝 1 − 𝜂𝑦
=
(
)=
(
)
𝑇𝑥
𝑇𝑥
𝜂𝑦
𝑇𝑥
𝜂𝑦
(5.33)
Concerning the efficiency loss related to the fluid dynamic losses of the pumps, it can be highlighted
that:
• It is preferable to raise the mean temperature 𝑇𝑥 at which the pumping takes place. The physical
meaning of this result is clear: since a work of fluid friction 𝑊̇𝑤 has to be introduced (which
does not depend on the temperature), it is preferable that the introduction take place at the
highest possible temperature in order to have a lower ∆𝑆 and to subsequently be able to recover
a larger fraction of it.
• Even if the effects of real liquid start to become noticeable as the temperature rises, the
conclusions do not change since
• The greater the increase in pressure is, the greater the irreversibility production.
• As always when considering the fluid dynamic irreversibilities in machines, the losses are
linked to the fluid dynamic quality of the components.
• It is generally small since the specific volume is very small for all fluids
5.2.3
Considerations on the design of the components
Regarding the turbomachinery, the close relationship between design and execution costs and
efficiency of the machine is clear. For a turbine, high volumetric expansion ratios lead to supersonic
flows which in certain conditions may create shock waves that penalize the fluid dynamic efficiency
58 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
of the machine. Dividing the expansion into many stages certainly allows these effects to be reduced
and to obtain a greater polytropic efficiency on the face of a larger investment and higher machine
inertia. A similar argument can be made for distributing the enthalpy change over multiple stages in
order to limit the peripheral mean speed and stresses on the blades. The pumps instead should be
positioned at the high temperatures in order to be able to recover a larger fraction of the work of fluid
friction, but this entails pumps with a higher specific cost and an excellent compromise solution.
5.3 LOSSES DUE TO PRESSURE DROPS
They involve a reduction in pressure in a specific component or in piping, and they may be localized
(filters, valves, abrupt sections and direction variations, etc.) and distributed (ducts) pressure drops.
Considering the isenthalpic throttling, it suffices that 𝑑ℎ = 𝑇𝑑𝑠 + 𝑣𝑑𝑝:
𝑑𝑠 = −
𝑣𝑑𝑝
𝑇
(5.34)
Since 𝑑𝑝 < 0, the pressure drop involves an increase in entropy. It is more intuitive to refer to an
absolute pressure drop and use the relation:
𝑑𝑠 =
𝑣𝑑𝑝
𝑇
(5.35)
These drops will be treated differently for liquids and gases.
5.3.1
Pressure drops of a liquid
A pressure drop in liquid phase causes an infinitesimal change in temperature. Considering the specific
constant volume, it can be written that:
∆𝑆̇ = 𝑚̇
𝑣∆𝑝
𝑇
(5.36)
The equation shows that the pressure drops in liquid phase are paid more at low temperature and are,
of course, proportional to the flow rate which sustains the drop and to the absolute pressure drops
(regardless of pressure at which it takes place). Since the specific volume 𝑣 is small, the influence on
the efficiency of the cycle is small. In fact, large pressure drops (many dozens of bars) on the liquid
side are accepted. In designing thermodynamic plants, the absolute pressure drops on the liquid side
are generally kept constant.
5.3.2
Pressure drops of a gas
A pressure drop in the vapor phase instead causes an increase in entropy equal to:
𝑅𝑢 𝑝𝑖𝑛 𝑍𝑇 𝑑𝑝
∫
𝑀𝑀 𝑝𝑜𝑢𝑡 𝑝 𝑇
𝑅𝑢
𝑝𝑖𝑛
∆𝑆̇ = 𝑚̇
𝑍̅𝑙𝑛 (
)
𝑀𝑀
𝑝𝑜𝑢𝑡
𝑅𝑢
𝑝𝑜𝑢𝑡 + ∆𝑝
𝑅𝑢
∆𝑝
∆𝑆̇ = 𝑚̇
𝑍̅𝑙𝑛 (
𝑍̅𝑙𝑛 (1 +
) = 𝑚̇
)
𝑀𝑀
𝑝𝑜𝑢𝑡
𝑀𝑀
𝑝𝑜𝑢𝑡
∆𝑆̇ = 𝑚̇
where for small pressure drops 𝑙𝑛 (1 +
∆𝑝
𝑝𝑜𝑢𝑡
)~
∆𝑝
𝑝𝑖
59 of 124
(5.37)
(5.38)
(5.39)
Second-law analysis of power cycles - Energy Conversion A – V7.0
The equation shows that the pressure drops in the gas phase are more critical at low pressure since the
∆𝑝
relative pressure drops
count and are, of course, proportional to the flow rate which sustains the
𝑝𝑖
drop and to the coefficient of compressibility 𝑍̅. Since the specific volume is high, the influence on
the cycle efficiency is greater than in the case of liquid. Careful attention must therefore be paid to the
pressure drops, especially at low pressure. In designing thermodynamic plants, the relative pressure
drops on the gas side are generally kept constant in order to not penalize low pressure configurations
too much.
5.3.3
Considerations on the design of the components
The pressure drops in a heat exchanger are another typical example of a technical-economic
optimization. A shell and tube heat exchanger is considered. The analysis is done with the same
number of tubes. When the diameter of the tubes is reduced, the metal mass is decreased due to two
effects: (i) the first is that lower thicknesses can be used keeping constant the pressure and allowable
𝑃𝐷
stress difference as defined by the law of Mariotte
= 𝜎; (ii) the second is that for decreasing cross2𝑡
sections the fluid flows faster with higher transfer coefficients, thus, the surface required is reduced
with the same differences in temperature and transferred heat.
So, if on the one hand reducing the cost is possible, on the other, however, greater pressure drops are
obtained. Indeed, they are related to the square of the speed with a consequent effect of loss of useful
work due to production of irreversibility and increased consumption of the pump.
5.4 LOSSES DUE TO MIXING
Mixings are considered isobaric and are reversible only if the flows that are mixed have the same
temperature and equal chemical composition. One typical example is the mixing of liquid and vapor,
both at saturation conditions.
Three types can be outlined (Fig. 5.8):
a) Two currents with different pressure and at the same temperature and composition: the
irreversibility is given by the isenthalpic throttling sustained by the current at a higher pressure
before being mixed;
b) Two currents at different temperature: the increase of entropy of the universe is comparable to
that of a heat transfer with a surface of infinite transfer;
c)
Different chemical composition: the process is irreversible even if it takes place at the same
temperature and pressure since the partial pressures of the various fluids are considered. The
increase of entropy of fluids that are mixed together at the same temperature and pressure is
caused by the increase in specific entropy of both liquids, which is caused by a decrease of
their partial pressure. In order for the mixing to take place reversibly, therefore without a total
increase of entropy, it would be necessary to imagine semipermeable membranes, through
which the fluids can mix without decreasing their partial pressure, as already seen for the
definition of chemical exergy of fuels. A classic example is the desuperheating of the vapor
that is performed with sprayed water in order to reduce the temperature of the live steam
produced by the steam generator. From the entropic analysis point of view, it is better to carry
out this process with hot water instead of with cold water.
60 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
T
𝑝𝑦
𝑝𝑥
𝑦
𝑚̇𝑥 , 𝑃𝑥 , 𝑇𝑥
𝑚̇𝑥 + 𝑚̇𝑦 , 𝑃𝑥 , 𝑇𝑥
𝑃𝑥 < 𝑃𝑦
𝑥
𝑚̇𝑦 , 𝑃𝑦 , 𝑇𝑥
∆𝑆̇ = 𝑚̇𝑦 𝑆𝑥 − 𝑆𝑦
s
T
𝑇𝑥
𝑚̇𝑥 , 𝑃𝑥 , 𝑇𝑥
𝑚̇𝑥 + 𝑚̇𝑦 , 𝑃𝑥 , 𝑇𝑧
𝑇𝑧
𝑚̇𝑦 , 𝑃𝑥 , 𝑇𝑦
𝑚̇𝑥 + 𝑚̇𝑦 𝑆𝑧 > 𝑚̇𝑥 𝑆𝑥 ̇ + 𝑚𝑦 𝑆𝑦
𝑇𝑦
Q
Fluido A 𝑚̇𝑥 , 𝑃𝑥 , 𝑇𝑥
𝑚̇𝑥 + 𝑚̇𝑦 , 𝑃𝑥 , 𝑇𝑥 , 𝑆𝑧
Fluido B
𝑚̇𝑦 , 𝑃𝑥 , 𝑇𝑥
𝑚̇𝑥 + 𝑚̇𝑦 𝑆𝑧 > 𝑚̇𝑥 𝑆𝑥 ̇ + 𝑚𝑦 𝑆𝑦
Fig. 5.8 - Different types of irreversible mixing of two currents
5.5 LOSSES DUE TO CHEMICAL REACTIONS
The chemical reactions of combustion bring with them a certain production of irreversibility depending
on how the combustion itself takes place. Specifically, reference is made to the reaction of a unit of
fuel mass starting from reactants at temperature 𝑇𝑅 . The increase in entropy is due to the formation
and destruction of chemical species and absolute entropies associated with them, to the partial
pressures of the mixed components, and to the temperature. The possibility to minimize the increase
of entropy of the universe connected with the sole combustion reaction with the recuperative
preheating method has already been shown.
5.6 OTHER LOSSES: SELF-CONSUMPTIONS, AUXILIARY SYSTEMS
These losses are simply electrical or mechanical powers that are spent for the operation of the auxiliary
systems, such as indoor air conditioning, electrical consumption, safety and fire systems. These powers
are simply considered as lost works ∆𝑊̇𝑖 .
61 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
6 APPLYING THE GENERAL FORMULA TO ANALYSIS OF POWER PLANTS
In this chapter, the main framework for second law analysis in power systems is defined. This
framework is identified in terms of parameters needed to perform a second law analysis.
If a production plant is considered, and it is seen as broken down into an N series of components (or
subsystems), the following formula can be written:
𝑁
𝑁
̇
𝑊̇𝑖𝑟𝑟 = 𝑊̇𝑟𝑒𝑣 − ∑ ∆𝑊̇𝑖 = 𝑊̇𝑟𝑒𝑣 − 𝑇0 ∑ ∆𝑆𝑖𝑟𝑟,𝑖
1
where
•
•
•
•
𝑊̇𝑖𝑟𝑟 :
𝑊̇𝑟𝑒𝑣 :
∆𝑊̇𝑖
̇ :
∆𝑆𝒊𝒓𝒓,𝒊
(6.1)
1
is the real work;
is the work that would be obtained with a series of reversible processes;
are the N useful works lost due to irreversibility in the N components;
are the N productions of entropy that occur in the N components considered.
A series of elements has to be defined in order to properly apply this equation:
1. Definition of the time interval;
2. Definition of the physical (or conceptual) boundaries of the system;
3. Definition of the energy source;
4. Definition of the dead environment;
5. Definition of the efficiency of the power cycle;
6. Choice of the N processes producing irreversibility in which the system is schematized.
6.1 DEFINITION OF THE TIME INTERVAL
The equation can be applied to:
• instantaneous quantities (powers): useful for defining the specific rated powers of the single
components and of the plant. This analysis is used to set the performance to be tested,
performance to which important penalties/bonuses are tied when reached;
• quantities integrated in the time interval (energies): useful for energy, economic and
environmental balances, for technical-economic feasibility studies, and for final balance
sheets; the most common time interval is the year, but often monthly, weekly, daily and hourly
balances are carried out;
• average quantities during the time interval (powers).
6.2 DEFINITION OF THE BOUNDARIES OF THE SYSTEM CONSIDERED
For example, the following are identified:
• flanges: real or virtual inlet and outlet flanges of all fluids involved in the process;
• electricity inlets and outlets (gross power, pump and fan consumption);
• auxiliary systems;
• thermal loss to the environment (insulation, isolation).
6.3 DEFINITION OF THE ENERGY SOURCE AND OF MATERIAL FLOWS
Flows of matter can enter and exit the borders of the system, and thermal, mechanical and electrical
powers can be exchanged.
The main flow to define is that of the energy source. Added to this one, there are the flows of matter
entering and exiting the system, and the flows of electrical or mechanical power that are introduced or
62 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
generated by the system. One example is the consumption of the auxiliary systems, when it is not
considered a plant self-consumption itself.
As for the flows of matter, the work potentially obtainable between the inlet and outlet conditions must
instead be considered. Therefore, the following shall be defined:
• flow rate on a mass-basis;
• chemical composition;
• thermodynamic conditions at the inlet: 𝑝x ,𝑇x ;
• restrictions on the discharge conditions: 𝑇𝑚𝑖𝑛 may coincide with the dead state 𝑇0 . 𝑇𝑚𝑖𝑛 may
also be higher, as often occurs for gases containing sulfur oxides or nitrogen, for which there
is no desire to reach the dew point, or for geothermal fluids, for which there is a desire to
prevent the deposit of compounds that can obstruct the flow in the re-injection sink;
• Kinetic energy, 𝑉 2 /2 (the "total" quantities of the flow are usually considered, and in the case
of wind energy it is the prevailing term);
• Gravitational energy, 𝑔𝑧 (important only in hydroelectric applications, of little importance for
gases and vapors).
6.4 THE "DEAD" STATE
The most common assumptions are:
• ambient air at 𝑇0 , 𝑝0 : it is the most useful solution for open cycles that use the ambient air as
the working fluid or for air-condensed closed cycles.
• sea water, or gravitational water at 𝑇0 , 𝑝0 : it is a solution often adopted for closed cycles that
have to transfer heat to the environment through a coolant. It can be assumed at infinite heat
capacity or at finite heat capacity. As will be seen afterwards, these two assumptions are
equivalent, and an additional mixing loss will have to be introduced if a coolant at finite heat
capacity is used.
• minimum cycle temperature: this is also a solution often adopted for closed cycles that have to
transfer heat to the environment through a coolant; it coincides with the first for open cycles.
• a suitable mean temperature of the coolant: this is the case of cogeneration plants in which the
enthalpy of condensation is released to a thermal utility through a heat transfer fluid.
6.5 DEFINITION OF THE EFFICIENCY OF THE POWER CYCLE
An efficiency is defined for plants that generate mechanical energy or electricity. It is obtained by
dividing 𝑊̇𝑖𝑟𝑟 by an FE term that defines the energy input of the energy source. This FE parameter can
be arbitrarily chosen in accordance with the type of energy source considered and the type of analysis
one wants to make, and actually it does not affect the generality of the entropic analysis.
The FE parameter can be chosen, for example, equal to the thermal power available at the plant inlet
𝑄𝑖𝑛 , getting the expression of the entropic analysis applied to a thermodynamic efficiency of first law.
𝑁
𝑊̇𝑖𝑟𝑟
𝑊̇𝑟𝑒𝑣
∆𝑊̇𝑖
=
−∑
𝑄̇𝑖𝑛
𝑄̇𝑖𝑛
𝑄̇𝑖𝑛
(6.2)
𝜂𝐼 = 𝜂𝑟𝑒𝑣 − ∑ Δ𝜂𝐼,𝑖
(6.3)
𝑖=1
𝑁
i=1
One alternative is, on the other hand, to assume FE equal to 𝑊̇𝑟𝑒𝑣 of the system, getting an expression
that instead refers to the second law.
63 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑁
𝑊̇𝑖𝑟𝑟
𝑊̇𝑟𝑒𝑣
∆𝑊̇𝑖
=
−∑
𝑊̇𝑟𝑒𝑣
𝑊̇𝑟𝑒𝑣
𝑊̇𝑟𝑒𝑣
𝑁
(6.4)
i=1
𝜂𝐼𝐼 = 1 − ∑ Δ𝜂𝐼𝐼,𝑖
(6.5)
i=1
In this case the efficiency 𝜂𝐼𝐼 represents the thermodynamic quality of the system with respect to the
reversible thermodynamic limit, and it is an extremely useful index for comparing systems that exploit
very different energy sources.
6.6 NUMBER N OF MODELED PROCESSES
The choice of the N processes into which the system is divided is arbitrary and must be made based
on technical and theoretical considerations. In the general approach, the system is divided into
components, each of which is a source of an overall irreversibility that characterizes its operation.
Nothing, however, prohibits unifying multiple components and analyzing them as a single process.
One example is the steam generator of a combined cycle that is physically made up of different banks
of heat transfer tubes (economizer, evaporator, superheater) but that can be handled as a single heat
transfer process in which superheated vapor is produced by cooling a certain current of hot burnt gases
discharged by the gas turbine.
The opposite approach is instead that of dividing the processes as much as possible with the aim of
giving each cause of loss the right weight. Therefore, a heat exchanger can be broken down into four
different irreversible processes: (i) the actual heat exchange to transfer heat between the hot and cold
currents under non-infinitesimal temperature changes, (ii) the pressure drops on the hot side, (iii) the
pressure drops on the cold side modelled as isenthalpic throttling and (iv) the thermal loss to the
environment. The recommended approach is the latter, the only one able to point out on which causes
of irreversibility to focus in order to improve the efficiency of a thermodynamic cycle.
64 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
7 SECOND LAW ANALYSIS OF A STEAM RANKINE CYCLE
In this chapter, the list of losses in Rankine cycles is shown. The framework presented in the previous
chapter is first outlined, then the losses are evaluated and commented.
7.1 INITIAL DEFINITIONS
Boundaries of the system considered
Fluid inlet and outlet flanges, electricity inlets and outlets, auxiliary systems, etc.
The analysis is applied only to the vapor cycle considering that all the auxiliary systems are powered
inside the cycle, so the only electricity outlet is the interface with the net downstream of all the
electrical losses. The system receives heat from the steam generator where combustion takes place.
Dead state
Sea water (or ambient air) of infinite heat capacity at T0 is assumed. The choice is discussed by
illustrating the efficiency losses of the condenser.
Reference to the rated instantaneous quantities
The analysis is conducted referring to the instantaneous quantities, hence to the powers.
Energy source
Combustion of a given fossil fuel, whether it is solid (coal) or gaseous (natural gas), is considered as
the energy source. The heat ideally extractable from the combustion is the HHV or the LHV, which
can be converted into work with efficiencies ideally close to the unit thanks to regenerative preheating.
The reversible work is very similar to the LHV (or HHV), so the entropic analysis applied to the first
law efficiency is equivalent to that applied to the second law.
Definition of efficiency
The formula to be applied is:
𝑁
𝑊̇𝑢
𝑊̇𝑟𝑒𝑣
∆𝑊̇𝑖
=
−∑
𝑄̇𝐿𝐻𝑉
𝑄̇𝐿𝐻𝑉
𝑄̇𝐿𝐻𝑉
𝑁
(7.1)
1
𝜂𝐼 = ~1 − 𝑇0 ∑
𝑖=1
𝛥𝑆𝑖̇
𝑄̇𝐿𝐻𝑉
Number of modeled processes
The following ten processes are identified:
o ∆𝜂1 irreversibility of heat transfer in the condenser
o ∆𝜂2 fluid dynamic irreversibility in the pumps
o ∆𝜂3 irreversibility of heat transfer in the preheating line
o ∆𝜂4 irreversibility of heat transfer in introducing heat into the cycle
o ∆𝜂5 fluid dynamic irreversibility in the turbine
o ∆𝜂6 pressure drops in the liquid phase
o ∆𝜂7 pressure drops in the vapor phase
o ∆𝜂8 thermal losses
o ∆𝜂9 mechanical/electrical losses
o ∆𝜂10 losses tied to consumptions of the auxiliary systems
The Ts diagram of the general steam Rankine cycle is shown in Figure 7.1.
65 of 124
(7.2)
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑇𝑚𝑎𝑥
5
7
6
3
4
1≡2
8
𝑇0
Fig. 7.1 – General Ts diagram for a Steam Rankine Cycle
7.2 ∆𝜼𝟏 IRREVERSIBILITY OF HEAT TRANSFER IN THE CONDENSER
The condenser of a Rankine cycle with water vapor can use water or ambient air, or can transfer heat
to a current of heat transfer fluid for thermal heating systems or thermal users in a backpressure
cogeneration arrangement. In this latter case, the cold fluid is water at ambient pressure or pressurized
if temperatures close to or higher than 100°C have to be reached. Only the first case with a cooling
current in equilibrium with the environment is considered here. The current is remixed with the
environment itself after having been heated in the condenser. It is assumed that the condenser is
adiabatic. This assumption is reasonable since it operates at temperatures near ambient temperature.
Thus, the losses, which depend on the difference in temperature between the hot fluid and the
environment, are in fact negligible even without insulation. In the condenser, the fluid discharged by
the turbine is condensed. A discharge from the turbine in two-phase or in superheated vapor can be
achieved, depending on the type of plant, the evaporation pressure, the superheating temperature and
the reheating, if any. In this latter case, the first part of the condenser desuperheats the fluid to the
point of saturated vapor.
The condenser is built following different architectures, based on the type of coolant:
• water-based: it is built as a Shell&Tube heat exchanger (Figure 7.2), in which the cold water
flows in the tubes and the vapor in the shell. The vapor comes into contact with the cold tubes,
condensates and subcools. The drops of liquid fall onto the tubes underneath and meet with the
vapor, performing a partial heat and mass transfer. Sufficient volume is guaranteed under the
bank of tubes, in which the subcooled liquid has a residence time sufficient to return to
equilibrium with the vapor and reach a saturated liquid condition. This volume is called heat
sink of the condenser and its level is one of the control parameters of a Rankine cycle. There
is usually a normal operation zone defined by a minimum level that has to always be guaranteed
in order to prevent pump cavitation, and a maximum level that serves to prevent some tubes
66 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
from remaining submerged, which would result in an actual subcooling of the condensate. The
water discharge temperature is limited by regulations protecting sea7 and river ecosystems.
Fig. 7.2 - S&T condensers for vapor systems
•
Air: the vapor is distributed in different tubes finned on the outside to increase the heat transfer
coefficient of the ambient air. The air is usually moved by forced convection with fans (suction
or forced draft), while natural circulation towers can be used for large power plants. In the first
case, depending on the tube class, condensate conditions that range from subcooled liquid (first
class) to two-phase fluid (last classes) can be obtained at the outlet. In this case as well it is
necessary to build a heat sink to re-balance the vapor liquid equilibrium condition after mixing
the various flows. These condensers are commonly used in modern combined cycle plants and
are shown in the following Figure (7.3).
Fig. 7.3 - Air condensers for vapor systems
•
Evaporation tower: a water circuit is used as coolant and it is cooled in an evaporation tower,
that removes heat through heat and mass exchange between the ambient air and the water
sprayed by the nozzles. The evaporation towers can be open cycle (if the cooling water itself
is that sprayed) or closed cycle. In both cases the cooling water is taken to the wet bulb
temperature of the ambient air and thus it is possible to reduce the condensation pressure with
less water consumption (Figure 7.4).
7
For the sea, the discharge must never be higher than 35°C and must not involve changes in temperature higher than 3°C
at 1000 m from the outlet. For lakes, the discharge must never be higher than 30°C and must not involve changes in
temperature higher than 3°C at 50 m from the outlet. For rivers, the mean temperature difference between any section
upstream and downstream must be not higher than 3°C (1°C on the half-section). For canals, the maximum temperature at
the outlet is 35°C.
67 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 7.4 - Schematics of closed cycle (left) and open cycle (right) evaporation towers
Subcooling is harmful for the efficiency of a power production plant. Indeed, it involves a lower inlet
temperature in the preheating and, therefore, demands a higher flow rate bled from the turbine with
reduced obtainable power. It always has to be limited with a proper sizing of the heat sink. The
condensate extraction pump installed in the pit under the condenser controls the heat sink level.
Air and water-based condensation will be discussed, while the evaporation tower will not be analyzed
in detail since it considers not only heat transfer, but mass transfer as well The condensation phase
mainly involves two irreversibilities: that of heat transfer and that of mixing the coolant discharged by
the condenser into the environment.
Fig. 7.5 - TQ diagram of the heat transfer process to the environment and diagram of the
irreversibilities considered
•
the first is related to the heat exchange between the vapor that condenses and the coolant
̇ =∫
∆𝑆1𝑎
𝑇𝑐,𝑜𝑢𝑡
𝑇0
𝑇𝑜𝑢𝑡
𝑑𝑄̇
𝑑𝑄̇
−∫
𝑇
𝑇
𝑇𝑖𝑛
(7.3)
This formula is always valid (also in case of desuperheating), but if the fluid discharged by the
turbine is saturated vapor or in two-phase conditions (the most common condition in a vapor
cycle), then it is possible to use a simplified formula, that will be used in the following
calculations:
𝑇𝑐,𝑜𝑢𝑡
̇ =∫
∆𝑆1𝑎
𝑇0
𝑑𝑄̇ 𝑄̇𝑐𝑜𝑛𝑑
−
𝑇
𝑇𝑐𝑜𝑛𝑑
68 of 124
(7.4)
Second-law analysis of power cycles - Energy Conversion A – V7.0
•
the second is linked to mixing the coolant with the heat sink
̇ =+
∆𝑆1𝑏
𝑇0
𝑄̇𝑐𝑜𝑛𝑑
𝑑𝑄̇
+∫
𝑇0
𝑇𝑐,𝑜𝑢𝑡 𝑇
(7.5)
If there is no desuperheating, the total entropy increase is given by:
∆𝑆1̇ = +
𝑄̇𝑐𝑜𝑛𝑑 𝑄̇𝑐𝑜𝑛𝑑
𝑇𝑐𝑜𝑛𝑑 − 𝑇0
−
= 𝑄̇𝑐𝑜𝑛𝑑
𝑇0
𝑇𝑐𝑜𝑛𝑑
𝑇𝑐𝑜𝑛𝑑 𝑇0
(7.6)
And therefore:
∆𝜂1 =
𝑄̇𝑐𝑜𝑛𝑑 𝑇𝑐𝑜𝑛𝑑 − 𝑇0
𝑄̇𝐿𝐻𝑉 𝑇𝑐𝑜𝑛𝑑
(7.7)
It can be noted that the temperature 𝑇𝑐,𝑜𝑢𝑡 has no influence and so the share between ∆𝑇𝑚𝑖𝑛 and ∆𝑇𝑐 is
not important, but rather their sum. The choice of 𝑇𝑐,𝑜𝑢𝑡 has, however, obviously an effect on the flow
rate of cold fluid, the consumption of the pumps and the surface and cost of the exchanger.
The formula shows that the efficiency loss:
a) is proportional to the
𝑄̇𝑐𝑜𝑛𝑑
𝑄̇𝐿𝐻𝑉
= 1 − 𝜂 relation, so it is more important the less the cycle efficiency
is;
grows with the increase in the terms ∆𝑇𝑚𝑖𝑛 and ∆𝑇𝑐 ; both terms are set based on technicaleconomic considerations (cost of the exchanger, investments and consumption for the
circulation of the coolant, equivalent plant operation hours, etc.). Ecological and permit
limitations may also arise to limit ∆𝑇𝑐 . In the case of very low 𝑇0 values, also considerations
related to the higher cost of the LP vapor turbine and the difficulties in maintaining a too high
vacuum level affect the choice of 𝑇𝑐𝑜𝑛𝑑 . In the most common cases, the 𝑇𝑐𝑜𝑛𝑑 − 𝑇0 difference
takes on the following values:
• 12/15°C for cooling with flowing water (river, sea inlet)
• 15/20°C for cooling with air (reference: dry bulb temperature of the ambient air)
• 10/15°C for cooling with evaporation towers (reference: wet bulb temperature of the
ambient air)
Once set the condensation temperature, the division between ∆𝑇𝑚𝑖𝑛 and ∆𝑇𝑐 is decided based on
technical-economic assessments. As ∆𝑇𝑐 rises, there is a reduction of pump and fan consumption, but
there is an increase in the heat transfer surface and condenser costs.
The condenser of a Rankine cycle cooled with ambient air or water works at sub-atmospheric pressures
(𝑝𝑠𝑎𝑡(35°C) = 0.056 bar) and is, hence, subject to leakages. The ambient air that penetrates the
condenser is partially mixed in the liquid phase and is then drawn by the pump. It is important to
remove these gases, typically called “incondensable gases” or better said “supercritical species”, from
the working fluid before they reach the boiler, where they would create corrosion phenomena. Their
accumulation in the condenser also leads to increased pressure in the component and reduced
performance. The deaerator removes them. A vacuum pump is also connected to the condenser to
remove the air it contains during start-up following a shut-down.
69 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
In addition to the irreversibilities of heat transfer, it is then necessary to consider those due to the
pressure drops and the condensation auxiliary systems:
− The pressure drops are certainly on the cold side in the case of both water and air condensers.
These losses involve the use of pumps and fans that lead to power absorption, which in the
case of the air condensers, is indicatively 1% of the discharged heat, and therefore a nonnegligible amount of the gross power produced.
− On the condensate side, on the other hand, the pressure drops are significant only in the case
of air condensers in which the working fluid flows in the tubes, where concentrated
(distribution and collection) and distributed pressure drops occur. In this case, the condensation
is not isobaric, and therefore not at constant temperature. A pressure drop equivalent to 0.31°C of difference between the saturation temperatures is considered in the large plants.
− In the case of large water condensate plants, the pressure drops are instead negligible, and the
process is actually isothermal.
7.3 ∆𝜼𝟐 FLUID DYNAMIC IRREVERSIBILITY IN THE PUMPS
Pumps are installed in two points of the plant:
• condensate extraction pumps: the first group of pumps is located under the condenser and its
job is to extract the saturated liquid and to raise its pressure in order to overcome the pressure
drops in the low pressure preheaters up to the deaerator. The pump works at subatmospheric
pressure and the seals have to be designed to cut down the leakages of incondensable gases.
These pumps have to work under a proper head because the fluid is saturated and cavitation
phenomena are more likely.
• feed pumps: these are located under the deaerator and carry the working fluid up to the
maximum plant pressure, which in the case of ultra-supercritical plants has to be higher than
300 bar. This process is usually executed with two pumps in series: the first, called booster,
which raises the fluid pressure (5-7 bar) in order to guarantee an adequate head for the actual
feed pump that has to achieve high compression ratios. This pump can be driven by an electric
motor under an inverter to perform the adjustment, or be one of a group of turbine pumps in
which the pump is driven by a turbine that expands medium pressure vapor that is then
discharged to the condenser. This solution is preferable for the large plants in which the
efficiency of the turbine pump group may exceed that of the electric motor.
The pumps are generally multi-stage centrifugal pumps, and are never single-stage; for safety and
reliability reasons 2x100 or 3x50 configurations are recommended, as shown in Figure 7.6.
3 x 50
2 x 100
Fig. 7.6 - Arrangement of the pumps in a power plant to guarantee high reliability
The efficiency loss linked to the fluid dynamic irreversibilities that take place inside the pumps is
given by the truly lost work, which is not that of friction but 𝑇0 ∆𝑆:
70 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
∆𝜂2 =
𝑇0
𝑄̇𝐿𝐻𝑉
∑ ∆𝑆𝑖̇ =
𝑖
𝑇0
𝑄̇𝐿𝐻𝑉
∑[
𝑖
𝑚̇𝑖 𝑣𝑖 ∆𝑝𝑖 1 − 𝜂𝑎𝑑,𝑖
(
)]
𝑇𝑥,𝑖
𝜂𝑎𝑑,𝑖
(7.8)
As for the efficiency loss regarding the fluid dynamic losses of the pumps, the following notations can
be put forward:
a) It is preferable to raise the mean temperature 𝑇̅ at which pumping takes place. If one wants
to force the plant layout to the utmost to enhance this effect, it would be necessary to adopt a
preheating line set up with a series of direct (or surface) heat exchangers, each of which
preceded by a pump, moving the feed pump to the end of the preheating process as shown in
Fig. 7.7.
b) The relative importance of the losses in the pumps rises linearly with the maximum cycle
pressure (𝑝𝑚𝑎𝑥 − 𝑝𝑚𝑖𝑛 ~ 𝑝𝑚𝑎𝑥 8) , while the term 𝑄̇𝐿𝐻𝑉 varies little with the maximum cycle
pressure.
c) As always, when the fluid dynamic irreversibilities in the machines are considered, the losses
are related to the fluid dynamic performances of the components.
In view of the small specific volumes, the efficiency losses associated with the pumping phase are
small compared to those of other components, so economic efforts and plant complications to increase
the polytropic efficiency or raise the temperature at which the pumps work are not usually rewarded.
An exception to the rule are the ultra-supercritical plants, where in view of the high maximum
pressures of the cycle, the term of useful work loss (basically proportional to 𝑝𝑚𝑎𝑥 ) may be substantial.
It is generally always expedient to raise the maximum pressure of a vapor cycle since the increase in
specific work spent in the pumping is orders of magnitude lower than what can be obtained in
expansion (𝑣 𝑙 ≪ 𝑣 𝑣 ). Nevertheless, there is an optimum cycle pressure beyond which the increase
in power absorbed by the pump is higher than that obtainable by the turbine since the specific volume
in the gas phase is linked both to temperature and pressure. This pressure is found at many hundreds
of bar, for which technical-economic restrictions related to the materials used in the boiler intervene.
This is shown in Figures 7.7 and 7.8.
bleedings from steam turbine
8
deaerator
6
deaerator
4
7
10
9
4
8
3
2
5
high-pressure preheaters
low-pressure preheater
1
3
T
2
low-pressure
prehaters
5
feedwater
pump
11
7
6
1
direct heat exchangers
s
Fig. 7.7 - Configuration of a preheating line that minimizes the irreversibilities of the pumping
process
8
The minimum pressure of the vapor cycles is usually a few hundredths of a bar, four orders of magnitude lower than the
maximum pressures.
71 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Δ𝜂2
𝜂𝑦
𝑝𝑚𝑖𝑛 = 𝑝𝑐
𝑝𝑚𝑎𝑥
Fig. 7.8 - Change in the term ∆𝜂2 in relation to the maximum pressure of the cycle
7.4 ∆𝜼𝟑 IRREVERSIBILITY OF HEAT TRANSFER IN THE PREHEATING LINE
The preheating line of a modern steam power plant has a high number of preheaters, all of the surface
type except for the deaerator in which the liquid mixes with the vapor. By convention, the heat
exchangers arranged between the condenser and the deaerator are low pressure preheaters, and those
installed between the deaerator and the heat generator are high pressure preheaters. The aim is to raise
the mean heat introduction temperature by preheating the liquid going into the boiler up to a
temperature 𝑇𝑥 .
7.4.1
Indirect preheaters
Every preheater is built as a Shell&Tube exchanger, in which the liquid to be heated flows in the tubes
and the vapor condensates outside. Based on the bled vapor conditions, one may or may not get a
desuperheating section (typical of the high pressure preheaters), while the fluid is usually two-phase
for the low pressure preheaters. The condensing vapor flow is mixed with the flow coming from the
next previously throttled preheater, which is generally found in two-phase conditions with a small
vapor title. The condensate on the hot side is also sub-cooled in order to use less flow rate of fluid bled
from the turbine, reduce heat transfer ∆𝑇𝑚𝑙𝑛 and finally increase the efficiency of the plant.
The only substantial difference between low and high pressure preheaters is the pressure at which they
work. The two exchangers are recognizable from the shape of the headers for distribution and
collection of water in the tubes. The headers are flat for the low pressure exchangers and are domeshaped for the high pressure exchangers in order to better withstand the high pressures reducing the
thicknesses of the material used.
72 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 7.9 - TQ diagram of the cycle preheating phase.
Fig. 7.10 - Indirect preheater with dual passage on the water side
Therefore, these heat exchangers operate with three flows, as seen in Figure 7.10: two that transfer
heat (the vapor and the condensate recycled by the next preheater) and a fluid receiving it (the
pressurized water current). The two hot currents can also be thought as mixed at the unit inlet.
The loss of useful work associated with the production of irreversibility is therefore due to three
phenomena: (i) the heat transfer, (ii) the pressure drops and (iii) mixing of bled vapor and recycled
condensate. The last two losses will be discussed separately in the following chapters, while that of
heat transfer can be written as:
∆𝜂3,𝑖 =
∆Ẇ𝑖
𝑇0
𝑇0
𝑇0
𝑑𝑄̇𝑟𝑒𝑐
𝑑𝑄̇𝑟𝑒𝑐
=
∆𝑆𝑖̇ =
(∆𝑆ℎ̇ + ∆𝑆𝑐̇ ) =
(∫
−∫
)
𝑇
𝑇
𝑄̇𝐿𝐻𝑉 𝑄̇𝐿𝐻𝑉
𝑄̇𝐿𝐻𝑉
𝑄̇𝐿𝐻𝑉 ℎ
𝑐
(7.9)
where by introducing appropriate average quantities the following is achieved
∆𝜂3,𝑖 = 𝑇0
𝑄̇𝑟𝑒𝑐 𝑇𝑚,ℎ − 𝑇𝑚,𝑐
𝑄̇𝐿𝐻𝑉 𝑇𝑚,ℎ 𝑇𝑚,𝑐
73 of 124
(7.10)
Second-law analysis of power cycles - Energy Conversion A – V7.0
From which one notes that the loss of efficiency grows:
• With the increase in weight of regenerated heat compared to introduced heat, which rises in
turn as temperature 𝑇𝑥 of final preheating increases. Actually, the efficiency of the cycle
benefits from an increase in 𝑇𝑥 because the loss of efficiency related to the introduction of heat
in the cycle decreases, which certainly takes place under greater differences in temperature.
• With the increase in the mean temperature change between vapor and condensate. This change
in turn depends on two parameters:
− ∆𝑇𝑚𝑖𝑛 9, the temperature difference at the pinch-point that in turn is the result of a
technical-economic optimization in sizing the heat exchanger: (i) the smaller ∆𝑇𝑚𝑖𝑛 ,
the larger the transfer surfaces and therefore the dimensions, weights and costs of the
heat exchangers; (ii) the larger ∆𝑇𝑚𝑖𝑛 , the larger the productions of entropy, which turns
into a smaller plant efficiency. Since these are heat exchangers that operate with
extremely high heat transfer coefficients on both sides of the transfer surfaces (water in
forced convection and condensing vapor), the economic optimum leads to very small
∆𝑇𝑚𝑖𝑛 , in the order of 2-3°C.
− ∆𝑇𝑎𝑝 which in turn depends on the sub-cooling of the condensate: it is expedient to
reduce this difference in temperature to get a lower ∆𝑇𝑚𝑙𝑛 . Sub-cooling is achieved by
positioning a certain number of tubes under the level of the liquid and by checking this
level with the valve on the regenerative bleeding side; if the level drops, the valve is
choked and less vapor flow passes through.
• Number of heat exchangers 𝑁𝑅: it is the result of a technical-economic optimization. The
greater the 𝑁𝑅, the more complex and expensive the plant becomes, but the temperature
differences that can be adopted are fewer. Three low pressure and four high pressure preheaters
are generally used.
Δ𝜂3
Δ𝑇𝑚𝑖𝑛
𝑁𝑅
𝑇𝑐
𝑇𝑥
Fig. 7.11 - Trend of ∆𝜂3 as the final preheating temperature and number of preheaters change
7.4.2
Deaerator
Since the condenser and condensate extraction pump, and the final stages of the turbine, work at
subatmospheric pressures, there are some leakages of ambient air that partially mix with the liquid.
These incondensable gases must be eliminated by the deaerator in order to prevent them from
accumulating in the condenser (with increased condensation pressure and reduced performance), and
to prevent corrosion phenomena at high temperatures. The deaerator consists of a column containing
drilled plates at the top of which the pressurized liquid is distributed. The liquid falls downward,
breaking up into drops, and meets the superheated vapor injected at the base of the tower. A heat and
mass transfer occurs between vapor and liquid, in which the incondensable gases are stripped and
9
∆𝑇𝑚𝑖𝑛 and ∆𝑇𝑝𝑝 are used without distinction in the lecture notes.
74 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
removed from the liquid current due to the change in solubility of the incondensable gases (Henry's
law). The vapor and incondensable gases are drawn from the top of the tower. In some models, the
stripped gases are conveyed in a series of paths created with sheets so that the incondensable gases
can be stratified at the base of the tower as they are heavier than the vapor. The discharge into the
atmosphere is usually seen as a plume of condensed vapor. Demineralized water is added to restore
this loss and to restore the nominal salts content.
There is a plenum chamber under the tower to recreate a condition of liquid and vapor equilibrium.
The deaerator is always placed high up to allow the feed pump to have the correct head and not run
into cavitation, as can be seen in Figure 7.12.
incondensable gases
+ steam
deaerated water
bled steam
pressurized
liquid
Fig. 7.12 - Diagram of the deaerator and its operation
The deaerator is modelled as a direct heat exchanger that takes the pressurized liquid to the saturation
temperature. The ∆𝑇𝑚𝑖𝑛 is therefore null and the irreversibility loss is calculated like in the previous
exchangers.
7.5 ∆𝜼𝟒 IRREVERSIBILITY OF HEAT TRANSFER IN INTRODUCING HEAT INTO THE CYCLE
Combustion takes place in a boiler in which the power released by the reaction is transferred to the
working fluid in different heat transfer sections. The zone where there is the flame is surrounded by
membrane water tubes that mainly receive heat by radiative and partly convective heat transfer, where
the liquid evaporates to form saturated vapor. The maximum temperatures in this zone are very high,
but lower than the adiabatic flame temperature, and gradually decrease as they move toward the top
of the combustion chamber. It is impossible to define a real ∆𝑇𝑚𝑙𝑛 in this section since the heat transfer
coefficient is not only convective, but radiative as well.
Moreover, the definition of the ∆𝑆 is not univocal because in this section there are heat transfer
irreversibilities and combustion irreversibilities at the same time. The fluid enters with a certain degree
of sub-cooling in order to be certain that there is no beginning of vaporization in the economizer, that
would lead to an appreciable increase in pressure drops. The evaporation takes place in boilers having
cylindrical bodies, positioned at the top of the tube bundle where there is physical separation of vapor
and liquid in equilibrium conditions. The saturated liquid is taken from an insulated downcomer tube
outside the boiler to the base of the boiler in the lower cylindrical body. The fluid is then distributed
in numerous membrane water tubes surrounding the combustion chamber. Circulation is natural and
guaranteed by differences in density between the two columns of fluid. The liquid rises up toward the
upper cylindrical body while partially evaporating. The vapor title is always kept low with an
appropriate circulation flow rate to prevent burn-outs and reduced transfer coefficients. Vapor is
75 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
extracted from the upper cylindrical body and there is usually a demister installed to block any drops
of liquid.
Following the evaporation section are the superheating and reheating banks that cool the flue gas
produced in the combustion chamber. In this case it is instead legitimate to consider a well-defined ∆𝑆
of heat transfer once the conditions upstream and downstream of each component are known. The
maximum temperatures that the vapor can reach are about 600-620°C due to the technological
limitations on the materials.
The economizer is found at the lower temperatures, which takes the compressed liquid from final
preheating 𝑇𝑥 up to temperatures close to those of evaporation. Lastly, there is the regenerative
preheating of the reactants, or of the oxidant air, through a Ljungström exchanger. Preheating the air
to higher temperatures is not always advantageous since on the one hand a higher adiabatic flame
temperature, hence lower irreversibility of combustion, would be obtained. On the other hand, a greater
irreversibility of heat transfer would be obtained since heat at temperatures higher than the maximum
temperatures of the materials cannot be introduced. Moreover, the differences in heat capacity do not
allow high preheating and curbed stack temperature to be obtained at the same time. A complete
schematic of a coal dust boiler is seen in Figure 7.13.
ECO out (to EVA)
SH2 out (to turbine)
SH1 out
SH2
RH
SH1
EVA
ECO
SH1 inlet
Ljungstrom
Boiler
feedwater
inlet
Fig. 7.13 - Diagram of a coal dust boiler for a large steam power plant
76 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 7.14 - Temperature-thermal power diagram of a highly advanced ultra-supercritical (USC)
steam generator: the vapor temperatures (700/720°C) are much higher than those corresponding to
today's state of the art. Note that the introduction of heat into the vapor cycle corresponds to a
temperature Tx equal to about 340°C, to which a combustion product outlet temperature of around
380°C corresponds. Also note the regenerative preheating of the combustion air at very high
temperatures (about 360°C), which allows the low temperature combustion products to be
discharged, compatible with a high steam generator efficiency
The loss of useful work of a steam generator can be conceptually divided into:
1. Non-reversible preheating of the reactants
2. Combustion that takes the flue gas to the adiabatic flame temperature
3. Cooling of the flue gas and introduction of heat in the working fluid in the evaporator tubes
4. Further cooling of the flue gas and introduction of heat in the SH, RH and ECO banks
5. Mixing the flue gas with the environment
This choice does not represent the physical meaning of the problem because, as already stated, the
second and third losses occur at the same time. The combustion is not adiabatic since part of the heat
is transferred to the evaporative tube bundle and therefore the combustion products do not reach the
adiabatic flame temperature. Due to this difficulty, it is decided to consider the irreversibility of the
steam generator in only two contributions, referring to the thermal power entering the cycle in
agreement with the definition of efficiency provided in equation (6.2):
1.
2.
Derating of the exergy of the fuel to a heat introduction on an isotherm at the maximum cycle
temperature;
Heat transfer from the maximum cycle temperature to the working fluid.
7.5.1
∆𝜼𝟒𝒂 Derating the thermal energy from 𝑻∞ to 𝑻𝒎𝒂𝒙
Now the physical meaning of the choice of an intermediate sink at a constant temperature equal to the
maximum temperature of the cycle will be clarified. Today's state of the art of heat generators allows
maximum vapor temperatures of 600-620°C to be reached, as shown in Fig. 7.15. Also bear in mind
77 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
that the maximum temperature to which metal is taken is at least 30°C higher than the temperature of
the working fluid.
external
convection
conduction
internal convection
T
Fig. 7.15 - Temperature trend in the SH or RH banks
The many attempts made to overcome these limitations over the decades have been unsuccessful
because the sizing of the steam generator requires adopting:
• Enormous heat transfer surfaces: because of the not high global heat transfer coefficients,
especially in the convective part of the generator where on the one hand there are burnt gases that
have low exchange coefficients, like all low-density gases;
• These enormous surfaces not only transfer heat, but must also withstand high internal pressures.
This is the reason why the need to adopt large thicknesses arises, meaning large volumes and
metal material weights able to retain good mechanical properties at a high temperature;
• Therefore, it is not acceptable to use metal alloys with high specific costs (€/kg), which certainly
exist and are adopted in other plants (e.g. gas turbines), which would involve prohibitive costs.
The example shown in Table 7.1 and the following figures show these concepts.
Fig. 7.16 - Breakdown in percentage of the materials used in the steam power plants according to
technological advancement in terms of pressures and maximum temperatures of the working fluid
78 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 7.17 - Maximum temperatures (SH and RH) adopted in the coal plants, based on the metal
materials used
Table 7.1 - Comparison between unit costs of the superheater tubes of a steam generator operating
at the current state of the art (600°C) and those (prohibitive) required to raise the temperature by
100°C. In the solution with the highest temperature, the ratios (thickness to diameter) increase both
because the pressure rises and because the allowable stress of the material decreases.
Dimensions, mm (internal diameter – thickness)
Material
Material cost per kg
Material cost per meter
Cost per meter ratio, with the same passage area
𝑇𝑚𝑎𝑥 = 600°𝐶
221 – 32
P91
€ 5.5
€ 1100
1
𝑇𝑚𝑎𝑥 = 700°𝐶
175 - 60
Alloy A617 A130
€ 48.0
€ 16,600
24
This loss considers the derating of the fuel exergy modelled as a heat available at infinite temperature
in heat at the maximum cycle temperature. It should be emphasized that not even this loss has an actual
physical meaning since the flue gases are discharged at a lower temperature than 𝑇𝑚𝑎𝑥 and, therefore,
the flue gas cooling curve inevitably crosses the isothermal source. However, the approach is valid
considering that it is possible to raise the heat transfer temperature at will through recuperative
preheating.
The meaning of this term is, on the other hand, that of representing the loss given by technological
limitations that do not allow the live steam conditions at the turbine inlet to rise beyond a certain
temperature.
∆𝜂4𝑎 =
𝑇0
𝑄̇𝐿𝐻𝑉
(
𝑄̇𝐿𝐻𝑉
𝑇0
)=
𝑇𝑚𝑎𝑥
𝑇𝑚𝑎𝑥
That is, the complement at one of the Carnot efficiency between 𝑇0 and 𝑇𝑚𝑎𝑥 .
79 of 124
(7.11)
Second-law analysis of power cycles - Energy Conversion A – V7.0
7.5.2
∆𝜼𝟒𝒃 Introducing heat into the cycle starting from 𝑻𝒎𝒂𝒙 of the working fluid
While examining the quality of a vapor cycle, there is no particular interest in penalizing an
introduction of heat starting from temperatures higher than the assumed Tmax (e.g. 620°C), which
would in any case not be acceptable for the reasons explained above. Another point to clarify is the
assumption that the source has a constant temperature. Actually, the combustion products in a heat
generator are a source at variable temperature, with heat transfer temperatures even much lower than
the assumed 620°C. The introduction of heat is not penalized by the need to cool the combustion
products, not even for cycles in which the temperature at which the heat starts to transfer from the
source ventures to extremely high values.
The efficiency loss can be expressed as:
∆𝜂4𝑏 =
𝑇0
𝑄̇𝐿𝐻𝑉
(∫
𝑑𝑄̇ 𝑄̇𝐿𝐻𝑉
𝑇𝑚𝑎𝑥 − 𝑇𝑚
−
) = 𝑇0
𝑇
𝑇𝑚𝑎𝑥
𝑇𝑚𝑎𝑥 𝑇𝑚
(7.12)
where 𝑇𝑚 is the mean temperature of introduction of heat into the cycle (falling between 𝑇𝑚𝑎𝑥 and 𝑇𝑥 )
. The efficiency loss naturally decreases as 𝑇𝑚 rises. This can be obtained by:
• Increasing 𝑇𝑥 and, therefore, with greater liquid preheatings;
• Increasing the evaporation pressure (and hence the temperature), by switching to
supercritical pressure; the mean heat introduction temperature continues to rise, even if the
variation of 𝑇𝑚 is relatively low with the pressure increase (Fig. 7.18 ) since a significant
part of the heat in any case tends to enter at temperatures close to the critical temperature
of water (Fig. 7.19). In the gradual transition between the liquid and vapor states, the
supercritical isobaric line presents very high specific heat values in proximity of the critical
temperature (which result in a low isobaric slope in the TS plane). For very high pressures,
the benefits in terms of ∆𝜂4𝑏 can be canceled out by the increase in the pump losses, which
linearly grow with the maximum pressure.
600
450
400
Mean temperature, °C
Temperature, °C
550
500
500
400
300
450
400
220
200
350
350
300
250
100
subcritical
300
supercritical
200
0
0.2
0.4
0.6
0.8
1
Q, %
100
200
300
400
500
Pressure, bar
Fig. 7.18 - Variation of Tm (°C) as the pressure (bar) changes, for Tx =350 °C and Tmax = 600 °C. A
pressure increase from 300 to 500 bar leads to an increase in the mean heat introduction
temperature of just 26°C.
80 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
80
250 bar
70
cp, KJ/kgK
60
275
50
40
300
350
30
20
400
500
10
0
300
350
400
450
500
550
600
T, C
Fig. 7.19 - Variation of the specific heat of water with the temperature for different supercritical
pressures. The area under the dome ∫ 𝑐𝑝 𝑑𝑇 represents the heat introduced for the gradual transition
from the liquid state to the vapor state
•
Introducing one or more reheatings. The pressure at which reheating is carried out must be
optimized considering thermodynamic and technical-economic aspects. From the
thermodynamic viewpoint, when the reheating pressure decreases, (i) the incoming thermal
power increases (beneficial effect since 𝑄̇𝐿𝐻𝑉 is the denominator of all the ∆𝜂𝑖 ), but (ii) the
efficiency loss linked to the heat transfer in the reheater increases. The optimum RH
pressure is that for which the efficiency loss introduced by the reheater is equal to the sum
of the efficiency loss reductions of the other processes obtained due to an increase in 𝑄̇𝐿𝐻𝑉
with the reduction of 𝑝𝑅𝐻 as can be seen in Fig. 7.20.
Fig. 7.20 - Increase in the thermal power introduced into the vapor cycle as the RH pressure
decreases (left), and definition of the optimum RH pressure from the point of view of the second law
analysis (right)
From the technical and economic viewpoint, the RH is necessary in order to limit the liquid title in the
expansion, which would be too high for subcritical cycles at high pressures or, at most, USC cycles
with just SH. From the economic point of view, introducing a superheating is very costly since:
• It is necessary to interrupt the expansion;
• It is necessary to transport vapor at high temperatures to the steam generator (and back) with
relatively high thermal losses and large passage sections;
• High cost of the heat transfer section materials;
• Reduced plant reliability.
81 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The most recent plants tend to limit the number of reheatings to one, due to the extremely high costs
that adopting a second reheating would entail.
Even if the pressure is pushed to the maximum and one/two reheatings are adopted, the term ∆𝜂4𝑏 is
nevertheless high. This is the inevitable consequence of adopting a fluid with a critical temperature
very remote from the maximum heat introduction temperatures that the current "state of the art" allows.
7.5.3
Binary cycles
One alternative proposed to overcome this situation, is replacing water vapor with another working
fluid with higher critical temperatures. Liquid metals (Hg, Na, K) meet this requirement. Furthermore,
they are thermally stable at high temperature since they are monatomic. In theory, it is therefore
possible to hypothesize a saturated vapor cycle with evaporation temperature of about 600°C and
condensation temperature of about 30-40°C, as shown in Fig. 7.21a. Actually, when a fluid has a high
critical temperature, it inevitably has very low saturation pressures, unacceptable for a number of
reasons in a large power cycle10. In other words, every working fluid is suitable for carrying out a
saturated vapor cycle (the best thermodynamic cycle) in a relatively limited temperature range (200300°C), much lower than the range (about 600°C) that the state of the art of the materials and the
ambient temperatures allow. This is why "binary" cycles are often proposed. They are formed by two
overlapping cycles in which the enthalpy of condensation of the top cycle is introduced into the
evaporator of the bottom cycle. Although another inevitable irreversibility is introduced, represented
by the heat transfer between the two cycles, the total efficiency can be higher than a supercritical vapor
cycle. A series of technological problems linked to the adoption of liquid metals (Hg incompatible
with the environment, Na explosive with water) has, in fact, thwarted the commercial success of these
solutions. An example is seen in Fig. 7.21.
Another proposal studied, though never carried out commercially, is a water (topping) and ammonia
(bottoming) binary cycle to exploit very low condensation temperatures, smaller seal problems and a
smaller size of the turbomachinery.
Fig. 7.21 - a) Saturated vapor cycle with mercury: the condensation pressure at ambient
temperature is unacceptable (Torricellian vacuum!) b) Hg/H2O binary cycle, where the fluids work
at technically acceptable pressures. Assuming regenerative limit cycles, a total efficiency of the two
cycles equal to the Carnot efficiency between Tmax and T0 would be obtained.
10
Actually, pressures higher than 400 bar or lower than 0.01 bar are not adopted in power cycles. The first limitation is
related to the mechanical stresses in the primary heat exchanger, the second to the volumetric flow rates of the turbine and
the seal problems (air re-entering the cycle).
82 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
7.6 ∆𝜼𝟓 FLUID DYNAMIC IRREVERSIBILITY IN THE TURBINE
The expansion of a large steam power plant is divided into a high number of turbine stages and
cylinders.First of all, there is usually a very high-pressure stage followed by a high pressure cylinder.
This is generally followed by reheating and then one or more medium-pressure cylinders in parallel.
The flow at low pressures is divided into multiple cylinders (two or four, depending on the size of the
plant) through a crossover to reduce the size of the final turbine stages. Holes are drilled in the turbine
cylinders to draw regenerative bleeds; the only bleed controlled under pressure is the one of the
deaerator that has to always be at a pressure higher than the ambient pressure in order to guarantee
venting of the incondensable gases.
The following results from the relation previously obtained for the irreversibility produced due to
expansion in the turbine:
∆𝜂5 =
𝑇0
𝑄̇𝐿𝐻𝑉
[
𝑚̇𝑅𝑢
𝑝𝑖𝑛
̅̅̅̅̅̅̅̅̅
𝑍̅(1
− 𝜂𝑦 )𝑙𝑛 (
)]
𝑀𝑀
𝑝𝑜𝑢𝑡
(7.13)
This loss for a steam power plant is not easy to calculate because:
• the flow rate is not constant along the expansion due to the presence of regenerative bleeds.
The useful work loss must be calculated for every turbine section between two bleeds and the
inlet and outlet conditions;
• The polytropic efficiency is not constant along the machine and depends on the design and
construction constraints of each stage. The high pressure and low pressure cylinders generally
have an efficiency lower than the medium pressure casings because the first are penalized by
high profile losses and the others by high losses due to 3D effects;
• The presence of drops of liquid further penalizes the efficiency for turbines that expand a twophase fluid.
7.7 ∆𝜼𝟔 PRESSURE DROPS IN THE LIQUID PHASE
The total efficiency loss is equal to:
∆𝜂6 =
𝑇0
𝑄̇𝐿𝐻𝑉
∑(
𝑖
𝑣𝑖 𝑚̇𝑖 ∆𝑝𝑖
)
𝑇̅𝑖
(7.14)
𝑅𝑢
∆𝑝𝑖
𝑍𝑖̅
)
𝑀𝑀
𝑝𝑖
(7.15)
7.8 ∆𝜼𝟕 PRESSURE DROPS IN THE VAPOR PHASE
The total efficiency loss is equal to:
∆𝜂7 =
𝑇0
𝑄̇𝐿𝐻𝑉
∑ (𝑚̇𝑖
𝑖
The equation demonstrates that the pressure drops in the vapor stage are mostly paid at low pressure
∆𝑝
(the relative pressure drops 𝑖 count, and are obviously proportional to the flow rate that sustains the
𝑝𝑖
drop and to the compressibility coefficient 𝑍̅). Since the specific volume is high, the influence on the
cycle efficiency is greater than in the case of liquid. Careful attention must therefore be paid to the
pressure drops, especially at low pressure.
As a result, if a valve has to be placed inside a Rankine cycle, it is preferable to do so on the liquid
side and at the highest possible temperature: the best point is therefore at the outlet of the economizer
83 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
(maximum temperature of the liquid), while the worst is downstream of the turbine where the pressure
∆𝑝
is minimum and the 𝑖 is maximum
𝑝𝑖
7.9 ∆𝜼𝟖 THERMAL LOSSES
The thermal losses, related to the inefficient insulation of the hot components, cause an efficiency loss
given by:
∆𝜂8 =
𝑇0
𝑄̇𝐿𝐻𝑉
∑ [𝑄̇𝑑𝑖𝑠𝑝,𝑖 (
𝑖
1
1
− )]
𝑇0 𝑇𝑖
(7.16)
The loss is higher the greater the dissipated thermal power is and the higher the mean temperature of
the fluid releasing heat is.
There is a significant thermal loss in the pipeline connecting the steam generator to the turbine. This
section is particularly critical because the distances are far and the temperatures are the highest in the
plant. There is generally a temperature reduction of about 2-3°C in these sections.
7.10 ∆𝜼𝟗 MECHANICAL/ELECTRICAL LOSSES
The mechanical and electrical losses represent a dissipation of totally lost "valuable" energy. In
general, it is impossible to recover anything. They degrade into heat at ambient temperature
∑𝑖 𝑊̇ 𝑚𝑒,𝑖
(7.17)
∆𝜂9 =
𝑄̇𝐿𝐻𝑉
7.11 ∆𝜼𝟏𝟎 AUXILIARY LOSSES
If it is assumed that the auxiliary electrical consumption is drawn upstream of the measurement of
useful power 𝑊̇𝑢 , it represents a cycle efficiency loss equal to:
∑𝑖 𝑊̇𝑎𝑢𝑥,𝑖
𝑄̇𝐿𝐻𝑉
A schematic of the last losses presented is shown in Fig. 7.22.
∆𝜂10 =
(7.18)
Fig. 7.22 - Flows of power assumed for the auxiliary consumption
7.12 COMPARISON OF LOSSES
The result of the analysis described above regarding an USC coal dust power plant taken as reference
from the European Benchmark Task Force (𝑇𝑚𝑎𝑥 = 620 °C) is shown in Fig. 7.23. The benchmark for
84 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
the heat sink is 15°C. The power plant has a net electrical efficiency of 45.5%, which corresponds to
a cycle efficiency of 47.36%11
Fig. 7.23 - Example of a second law analysis applied to a supercritical vapor cycle (EBTF
reference). Note the importance of the losses linked to the introduction of heat into the cycle.
The most critical processes are also those on which it is preferable to intervene:
• By introducing heat: when raising 𝑇𝑚𝑎𝑥 the mean heat introduction temperature is raised, and
hence also the cycle efficiency;
• By improving the efficiency of the turbine;
• By increasing the preheating efficiency;
• By reducing the condensation temperature;
• Auxiliary systems (fans, circulation pumps, safety systems and self-consumptions).
A final consideration is that very often when comparing steam power plants, the loss due to combustion
and the transfer of heat at temperature 𝑇𝑚𝑎𝑥 is not considered, and the analysis of the losses starts from
a reversible work given by
𝑊̇𝑟𝑒𝑣 = 𝑄̇𝐿𝐻𝑉 (1 −
𝑇0
)
𝑇𝑚𝑎𝑥
(7.19)
The analysis of the efficiency losses therefore becomes equal to:
𝑁
𝑊̇𝑢
𝑊̇𝑟𝑒𝑣
∆𝑊̇𝑖
=
−∑
𝑄̇𝐿𝐻𝑉
𝑄̇𝐿𝐻𝑉
𝑄̇𝐿𝐻𝑉
(7.20)
𝜂𝐼 = 𝜂𝑟𝑒𝑣 − ∑ ∆𝜂𝑖
(7.21)
1
𝑁
1
11
The analysis does not consider the losses of conversion from coal to heat entering the cycle, so the resulting efficiency
refers to the heat entering the cycle.
85 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
7.13 OTHER NUMERICAL EXAMPLES
The student has to supplement all the qualitative considerations put forth in this chapter with the
numerical results obtained during Exercise 4 dedicated to the second law analysis for USC steam
power plants.
86 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
8
EFFECT OF THE THERMODYNAMIC PROPERTIES OF THE WORKING FLUID ON
RANKINE CYCLE PERFORMANCE
In this chapter the impact of the properties of fluids is discussed. The thermodynamic properties of the
working fluid have a significant impact on the maximum performance that can be obtained from a
Rankine cycle. In particular, the critical properties, the molecular complexity and the molar mass of
the fluid can significantly modify the Ts diagram, the TQ diagram and influence the design of the
turbine, making a vapor cycle more or less suitable to exploit certain types of heat sources. First of all,
the effect of molecular complexity will be analyzed because it is the most influent on the
thermodynamics of the cycle, while the impact of molecular mass and of critical properties will be
discussed at the end of the chapter.
8.1 SOURCES AT CONSTANT TEMPERATURE - NON-REGENERATIVE CYCLE
In order to exploit a source with infinite heat capacity, the limit thermodynamic cycle of reference is
the saturated Rankine cycle, operating between the source temperature 𝑇𝑚𝑎𝑥 and the cold sink at
temperature 𝑇0 with unitary machinery efficiency and without pressure drops. In this case it is
considered to not adopt regeneration. Two fluids with different molecular complexity are compared:
the first is water representing a simple molecule fluid, the other is MD4M, a siloxane with high
complexity (C14H42O5Si6). The two fluids differ in complexity and molecular weight (18 kg/kmol
versus 459 kg/kmol) while they have a very similar critical temperature (374°C versus 380°C).
The choice of using two fluids with the same critical temperature is justified by the need to simplify
the analysis by using the principle of corresponding states. Evaporation and condensation temperatures
being equal, it is therefore possible to consider the residual enthalpy and that of the specific heat on a
molar-basis equal for the two fluids. In addition, the evaporation temperature is 180°C, which
corresponds to a reduced temperature of 0.7, at which it is possible to consider the volumetric and
thermodynamic behavior of the saturated vapor equal to that of an ideal gas without introducing
excessive approximations.
The comparison will be performed on ideal cycles and therefore with adiabatic and isentropic
machines, negligible pressure losses and heat exchangers with infinite surfaces. Fig. 8.1 shows the Ts
diagrams for the two fluids for non-regenerative saturated cycles.
Fig. 8.1 - Ts diagrams for two saturated Rankine cycles operating with water (left) and with MD4M
(right)
87 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
8.1.1
Effect of the molecular complexity
As it is well known, the molecular complexity affects the shape of the saturation dome and, in
particular, the slope of the saturated vapor line. For the siloxane MD4M, the expansion takes place in
the superheated vapor field and with small temperature changes. For both fluids the introduction of
heat into the economizer, 𝑄̇𝐸𝐶𝑂 = 𝑛̇ 𝑐̃ 𝑙 (𝑇𝑚𝑎𝑥 − 𝑇0 ), is an irreversible process, while for the complex
fluid also the transfer of heat to the environment is irreversible in the desuperheating section, 𝑄̇𝐷𝐸𝑆 =
𝑛̇ 𝑐̃𝑝 0 (𝑇𝐷𝐸𝑆 − 𝑇0 ). In the case of the simple fluid, this drop is not present since the expansion takes
place in the two-phase field. By applying the second law analysis to the first law efficiency, what is
obtained in the most general case is:
̇
̇
𝑊̇𝑢 𝑊̇𝑟𝑒𝑣 𝑇0 ∆𝑆𝐸𝐶𝑂
𝑇0 ∆𝑆𝐷𝐸𝑆
=
−
−
𝑄̇𝑖𝑛
𝑄̇𝑖𝑛
𝑄̇𝑖𝑛
𝑄̇𝑖𝑛
𝜂𝐼 = 𝜂𝑟𝑒𝑣 − ∆𝜂𝐸𝐶𝑂 − ∆𝜂𝐷𝐸𝑆
𝑇0
𝜂𝐼 = (1 −
) − ∆𝜂𝐸𝐶𝑂 − ∆𝜂𝐷𝐸𝑆
𝑇𝑚𝑎𝑥
(8.1)
(8.2)
(8.3)
where the reversible efficiency is clearly equal to the efficiency of the Carnot cycle operating between
the evaporation and condensation temperatures. Considering the liquid having a constant specific heat,
the loss of efficiency in the economizer is:
̇
𝑇0 ∆𝑆𝐸𝐶𝑂
𝑇0
̇ + ∆𝑆𝑠𝑜
̇ )
=
(∆𝑆𝑓𝑙𝑑
𝐸𝐶𝑂
̇
𝑄𝑖𝑛
𝑄̇𝑖𝑛
𝑇0
𝑇𝑚𝑎𝑥
𝑛̇ 𝑐̃ 𝑙 (𝑇𝑚𝑎𝑥 − 𝑇0 )
=
(𝑛̇ 𝑐̃ 𝑙 𝑙𝑛 (
)
)−
𝑇0
𝑇𝑚𝑎𝑥
𝑄̇𝑖𝑛
∆𝜂𝐸𝐶𝑂 =
∆𝜂𝐸𝐶𝑂
(8.4)
(8.5)
That when collecting 𝑛̇ 𝑐̃ 𝑙 (𝑇𝑚𝑎𝑥 − 𝑇0 ) is:
∆𝜂𝐸𝐶𝑂
𝑐̃ 𝑙 (𝑇𝑚𝑎𝑥 − 𝑇0 )
𝑇0
𝑇0
= 𝑙
(
−
)
𝑐̃ (𝑇𝑚𝑎𝑥 − 𝑇0 ) + ∆ℎ̃𝑒𝑣𝑎 𝑇𝑚𝑙𝑛,𝐸𝐶𝑂 𝑇𝑚𝑎𝑥
(8.6)
if mole quantities are considered, then the ∆ℎ̃𝑒𝑣𝑎 and the correction of ∆𝑐̃𝑝 between liquid and saturated
vapor are independent from the properties of the fluid (complexity and molecular mass) given the same
reduced temperature. Therefore, the sole variable affecting the ∆𝜂𝐸𝐶𝑂 is the 𝑐̃𝑝 0 of ideal gas since 𝑐̃ 𝑙 =
𝑐̃𝑝 0 + ∆𝑐̃𝑝 . It follows that for simple fluids with small 𝑐̃𝑝 0 , the heat introduced in evaporation is greater
than the heat introduced in economizing. These fluids are therefore able to reduce the weight of the
loss of efficiency due to the introduction of heat and maximize efficiency. Complex fluids instead have
a high value of 𝑐̃𝐿 and an introduction of heat with a greater production of irreversibility.
Moreover, the loss in the desuperheating has to added for fluids with high complexity. It is equal to:
̇
𝑇0 ∆𝑆𝐷𝐸𝑆
𝑇0
̇ + ∆𝑆𝑎𝑚𝑏
̇
=
(∆𝑆𝑓𝑙𝑑
)𝐷𝐸𝑆
𝑄̇𝑖𝑛
𝑄̇𝑖𝑛
𝑛̇ 𝑐̃𝑝 0 (𝑇𝐷𝐸𝑆 − 𝑇0 )
𝑇0
𝑇𝐷𝐸𝑆
=
(−𝑛̇ 𝑐̃𝑝 0 𝑙𝑛 (
)
)+
𝑇0
𝑇0
𝑄̇𝑖𝑛
∆𝜂𝐷𝐸𝑆 =
∆𝜂𝐷𝐸𝑆
And therefore
88 of 124
(8.7)
(8.8)
Second-law analysis of power cycles - Energy Conversion A – V7.0
∆𝜂𝐷𝐸𝑆
𝑐̃𝑝 0 (𝑇𝐷𝐸𝑆 − 𝑇0 )
𝑇0
= 𝑙
(1 −
)
̃
𝑇𝑚𝑙𝑛,𝐷𝐸𝑆
𝑐̃ (𝑇𝑚𝑎𝑥 − 𝑇0 ) + ∆ℎ𝑒𝑣𝑎
(8.9)
The turbine discharge temperature 𝑇𝐷𝐸𝑆 is the higher the greater the complexity of the fluid: if the
complexity increases, 𝛾 → 1 and 𝜃 → 0, and the isentropic expansion tends to also be isothermal. The
loss of efficiency due to the transfer of heat into the environment is therefore dependent on the
molecular complexity of the fluid, both due to the effect of greater heat transferred and to the increase
of 𝑇𝑚𝑙𝑛,𝐷𝐸𝑆 . Considering vapor an ideal gas throughout the expansion, it turns out that:
0
𝑐̃𝑝 (𝑇𝑚𝑎𝑥 (1 − 𝛽
∆𝜂𝐷𝐸𝑆 =
𝑐̃ 𝑙 (𝑇𝑚𝑎𝑥
𝑅
𝑐𝑝 0
−
) − 𝑇0 )
− 𝑇0 ) + ∆ℎ̃𝑒𝑣𝑎
(1 −
𝑇0
𝑇𝑚𝑙𝑛,𝐷𝐸𝑆
)
(8.10)
The table shows the efficiency losses and the efficiency that can be achieve with the two fluids. It is
possible to see that in this case the water allows a cycle to be carried out with an efficiency greater
than 10 percentage points compared to the complex fluid MD4M.
Table 8.1 - Comparison of efficiency losses for two saturated Rankine cycles operating with water
and with MD4M
water MD4M
𝜂𝑟𝑒𝑣
30.89%
27.84% 17.12%
𝜂𝐼
∆𝜂𝐸𝐶𝑂 3.06% 8.64%
5.14%
∆𝜂𝐷𝐸𝑆
The TQ diagram for the two cycles is shown in Fig. 8.2 to highlight what is obtained analytically.
Fig. 8.2 - TQ diagrams for two saturated Rankine cycles operating with water and with MD4M to
exploit a source at constant temperature
89 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
8.2 SOURCES AT CONSTANT TEMPERATURE - REGENERATIVE CYCLE
Now a regenerative cycle is considered: as already mentioned, a saturated cycle with simple molecule
can reach the Carnot efficiency through a reversible regenerative preheating obtained with infinite
regenerative bleeds from the turbine. In this case the introduction of heat is totally reversible, since it
takes place at the temperature of the source. In practice, this is not possible, so a limited number of
preheaters (surface or direct) is used. Each of these involves a loss of useful work due to the heat
transfer under finite temperature differences, but it allows the loss due to heat introduction into the
cycle to be limited, with beneficial effects on the efficiency. Their number is defined by a technicaleconomic optimization, given the greater plant engineering complexity and higher plant cost of a
regenerative solution.
For a complex fluid, on the other hand, it can be noted that the regeneration can take place not only
through regenerative bleeds, but also (and above all) by using the hot vapor discharged by the turbine
to preheat the compressed liquid. To do this, it is used a surface recuperator that cools the vapor up to
temperatures close to that of condensation, and that reduces both the loss of introduction of heat into
the cycle in the economizing section and the loss of transfer of heat to the environment in the
desuperheating section. This heat exchanger is generally built as a finned coil placed at the turbine
outlet: the liquid flows in the tubes while vapor flows outside, where the transfer surface is larger to
compensate in part the low heat transfer coefficients characteristic of a gas in rarefaction conditions.
While significantly increasing the efficiency of the cycle by reducing the two efficiency losses that
characterize it, this component obviously results in losses of useful work due to:
• pressure drops on both sides of the heat exchanger, which involve an increase in consumption
of the pump and reduction of the work of the turbine;
• heat transfer under finite temperature differences. Even imagining a heat exchanger with
infinite surface, the different heat capacity of the vapor and liquid inevitably involves a change
in temperature on the hot side greater than that obtainable on the cold side, and thus a
cancellation of the temperature difference only on the cold end of the heat exchanger. A
reversible recuperation is obtainable only with an infinitely complex fluid, since the specific
heat of liquid and vapor in this case are equal, and the isentropic expansion is also isothermal.
See the TQ diagram in Fig. 8.3: in the first case recuperation is not adopted and the cycle
efficiency is about 17%; by adopting an ideal recuperation (recuperator with infinite exchange
surface), the energy input is reduced with a considerable increase in plant efficiency.
Table 8.2 - Comparison of efficiency losses for two saturated Rankine cycles operating with MD4M:
non-recuperative and recuperative for the exploitation of a source at a constant temperature
𝜂𝑟𝑒𝑣
𝜂𝐼
∆𝜂𝐸𝐶𝑂
∆𝜂𝐷𝐸𝑆
∆𝜂𝑟𝑒𝑐
non-REC
REC
30.9%
17.1%
27.5%
8.6%
2.1%
5.1%
0.0%
1.3%
90 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 8.3 - TQ diagrams for two saturated Rankine cycles operating with MD4M: non-recuperative
(right) and recuperative (left) for the exploitation of a source at a constant temperature
8.3 SOURCES AT VARIABLE TEMPERATURE
In the case of sources at variable temperature, the reversible cycle of reference is the trilateral cycle
(if the source can be cooled down to ambient temperature) or the trapezoidal cycle (if the minimum
temperature of the source is higher than the ambient temperature). These cycles allow heat to be
introduced reversibly and the source to be totally exploited. In real cycles, the critical temperature of
the fluid plays an important role in addition to the complexity. The reversible work obtainable with a
reversible trapezoidal cycle is:
𝑊̇𝑟𝑒𝑣 = 𝑄̇𝑖𝑛,𝑚𝑎𝑥 (1 −
𝑇0
𝑇𝑚𝑙𝑛(𝑇𝑚𝑎𝑥2|𝑇𝑚𝑎𝑥1)
)
(8.11)
where 𝑄̇𝑖𝑛,𝑚𝑎𝑥 = 𝐶(𝑇𝑚𝑎𝑥2 − 𝑇𝑚𝑎𝑥1 ) is the thermal power available that completely cools the hot
source. 𝑇𝑚𝑎𝑥1 can be equal to the 𝑇0 when there are no limits in cooling the source.
A non-reversible cycle will have a lower useful work because of losses linked to the introduction of
heat into the evaporator and the economizer, the transfer of heat in the desuperheating and the total
lack of exploitation of the source. The second law analysis applied to the second law balance gives:
𝑊̇𝑢
𝑊̇𝑟𝑒𝑣
𝑇0
̇
̇
̇
̇ )
=
−
(∆𝑆𝐸𝑉𝐴
+ ∆𝑆𝐸𝐶𝑂
+ ∆𝑆𝐷𝐸𝑆
+ ∆𝑆𝑚𝑖𝑥
𝑊̇𝑟𝑒𝑣 𝑊̇𝑟𝑒𝑣 𝑊̇𝑟𝑒𝑣
𝜂𝐼𝐼 = 1 − ∆𝜂𝐸𝑉𝐴 − ∆𝜂𝐸𝐶𝑂 − ∆𝜂𝐷𝐸𝑆 − ∆𝜂𝑚𝑖𝑥
(8.12)
(8.13)
In this numerical example, the source has a maximum temperature of 230°C and may be cooled up to
ambient temperature.
8.3.1
Simple molecules
A simple molecule fluid is considered, and a saturated cycle is used. Most of the heat is introduced at
constant temperature equal to that of evaporation. The efficiency is penalized due to the introduction
of heat into the cycle under finite temperature differences, and to the total lack of exploitation of the
source that will lead to a certain loss due to mixing with the environment. If a saturated limit cycle
91 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
with a reversible preheating is considered, it has the efficiency of the Carnot cycle. The optimum
evaporation temperature is identified considering that the producible power is given by:
𝑊̇ = 𝑄̇𝑖𝑛 (1 −
𝑇0
)
𝑇𝑒𝑣𝑎
(8.14)
where the incoming power is only a fraction of the maximum power extractable from the source.
Considering the infinite surface of the heat exchanger, it is found that 𝑄̇𝑖𝑛 = 𝐶𝑐 (𝑇𝑚𝑎𝑥,2 − 𝑇𝑒𝑣𝑎 ). By
differentiating from the variable 𝑇𝑒𝑣𝑎 , it is possible to find the maximum producible power:
𝑇0
𝜕 [𝐶𝑐 (𝑇𝑚𝑎𝑥,2 − 𝑇𝑒𝑣𝑎 ) (1 −
)]
𝜕𝑊̇
𝑇𝑒𝑣𝑎
=
=0
𝜕𝑇𝑒𝑣𝑎
𝜕𝑇𝑒𝑣𝑎
𝜕𝑊̇
𝑇0
𝑇0
= −𝐶𝑐 (1 −
=0
) + 𝐶𝑐 (𝑇𝑚𝑎𝑥,2 − 𝑇𝑒𝑣𝑎 )
𝜕𝑇𝑒𝑣𝑎
𝑇𝑒𝑣𝑎
𝑇𝑒𝑣𝑎 2
𝜕𝑊̇
𝑇0
= −𝐶𝑐 + 𝐶𝑐 𝑇𝑚𝑎𝑥,2
=0
𝜕𝑇𝑒𝑣𝑎
𝑇𝑒𝑣𝑎 2
(8.15)
(8.16)
(8.17)
and therefore
𝑇𝑒𝑣𝑎,𝑜𝑝𝑡 = √𝑇0 𝑇𝑚𝑎𝑥,2
(8.18)
The value of the optimum evaporation temperature is therefore not dependent on a limit on the cooling
temperature of the source until it is less than the 𝑇𝑒𝑣𝑎,𝑜𝑝𝑡
𝑇𝑒𝑣𝑎,𝑜𝑝𝑡 = max (√𝑇0 𝑇𝑚𝑎𝑥,2 , 𝑇𝑚𝑖𝑛,𝑐 )
(8.19)
Fig. 8.4 - Representation of the optimum temperature of a Carnot cycle for the exploitation of
sources at variable temperature
Fig. 8.5 shows the TQ diagram optimized for the case with water and non-regenerative saturated cycle.
Note that the optimum evaporation temperature is equal to 129°C versus the 124°C expressed in the
theoretical formula. The available heat is not fully exploited. The loss due to the irreversible mixing
92 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
of the source with the environment must also be added to the losses of useful work in the process of
heat introduction (both into the economizer and the evaporator).
A superheating, which increases the heat introduction mean temperature, improves the saturated cycle.
While a substantial increase in efficiency is obtainable using one or more evaporation levels, a
common practice in combined cycle plants. In this way, it is possible to minimize the ∆𝑇 of heat
transfer and introduce more heat into the cycle.
T
T
𝑄𝐸𝐶𝑂
250
𝑄𝐸𝑉𝐴
200
150
100
𝑊𝑢
50
𝑄𝑚𝑖𝑥
𝑄𝐶𝑂𝑁𝐷
0
-100%
-50%
0%
50%
100%
Q
Fig. 8.5 - TQ diagram for a water recovery cycle with optimized evaporation temperature (left), and
possibility of reducing the temperature difference in the heat introduction process with a cycle
having two levels of evaporation and superheating (right)
Adopting a simple fluid but with lower critical temperature, instead, the evaporation section compared
to the heat introduced always weighs less, and for critical temperatures lower than 𝑇𝑚𝑎𝑥,𝑐 , it is feasible
to implement a supercritical cycle with the objective of reducing the irreversibilities due to heat
introduction process. But in this case the large difference of specific heat between liquid and vapor
results in a highly marked point of inflection at the critical point and an limited increase in mean
temperature of the heat introduction, as seen previously for the water vapor cycle.
8.3.2
Complex molecules
Substantial advantages can instead be obtained with a complex molecule that shows a greater liquid
specific heat on a molar-bases. It is possible to raise the evaporation temperature for a subcritical
saturated cycle while keeping the same pinch-point, and at the limit for an infinitely complex cycle
there would be an introduction of heat only into the economizer with the possibility of minimizing the
temperature differences in the primary heat exchanger.
Fig. 8.6 shows the TQ diagram optimized for water and for MD4M. Note that a greater optimum
evaporation temperature is obtained for the fluid with greater complexity, and the different share
between heat introduced into the economizer and evaporator practically allows the ∆𝑇 of heat transfer
to be canceled out in the economizer. The losses due to introducing heat are still present in the
evaporator, while those due to mixing with the environment are severely limited. The available heat is
almost fully exploited and, despite the heavy loss due to the desuperheating process, the power
produced increases by 23% compared to the case with water.
93 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The loss due to the transfer of heat into the environment (desuperheating) can be limited by using a
recuperator according to the minimum temperature of the source.
Fig. 8.6 - Comparison between the TQ diagrams for the exploitation of a source at variable
temperature with water (left) and with MD4M (right)
Table 8.3 - Comparison of the efficiency losses for two saturated Rankine cycles operating with
water and MD4Mfor the exploitation of a source at variable temperature
𝜂𝑟𝑒𝑣
𝜂𝐼𝐼
∆𝜂𝐸𝐶𝑂
∆𝜂𝐸𝑉𝐴
∆𝜂𝑚𝑖𝑥
∆𝜂𝐷𝐸𝑆
water MD4M
100.0%
58.8% 72.5%
3.6%
0.4%
20.4% 10.2%
17.2%
0.1%
16.9%
For the complex fluids, the switch to supercritical cycles is instead highly expedient since the slope of
the isobaric lines is similar in the liquid and vapor fields with a supercritical transition characterized
by a less pronounced point of inflection. This is shown in Fig. 8.6.
94 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
T
T
simple
complex
s
s
Fig. 8.7 - Comparison between the TQ diagrams for the exploitation of a source at variable
temperature with supercritical cycles using a simple molecule (left) and a complex molecule (right)
8.4 GENERAL CONCLUSIONS
From this discussion, we can gather that water is not the only option, despite it is the most used working
fluid in Rankine cycles applications. Moreover, it is not the best solution when the available heat
source is characterized by low thermal power and/or low maximum temperature. For large fossil-fired
power plants, it is out of the question that water vapor cycles are the best option (apart from binary
cycles with I group metals as working fluid). Any other thermal source (geothermal, heat rejection,
biomass, solar plants) the optimal working fluid can be chosen among a large number of candidates,
considering thermodynamics, but also economics and technical issues. The cycles that uses organic
fluids (hydrocarbons, refrigerants, siloxane) as working fluid are called ORC (Organic Rankine
Cycles). These cycles are applied in a large number of energy systems related with renewable energy
and energy efficiency improvement.
8.4.1
Economics
The cost of a plant mainly consists of the cost of the heat exchangers and the turbine. The cost
distribution between heat exchangers and turbine is certainly influenced by the efficiency of the cycle:
the lower this parameter is, the greater the cost of the heat exchangers is (and especially the condenser)
compared to the cost of the turbine. The other aspect that affects the cost of the two components are
the different economies of scale that can be exploited.
Heat exchangers, Shell&Tube for example, are components that have a certain maximum size in both
diameter and length of the Shell (TEMA sheets), and consequently a modular solution is used for large
plants by placing multiple components in series and/or in parallel with a specific cost for the
approximately constant transfer surface and small economies of scale. The turbine on the other hand
(if the need to have a cross-over in the very large plants is ruled out) is a scalable component and has
high reductions in the specific cost per kW produced as the size increases. Using the same working
fluid for the same inlet and outlet conditions and same speeds, the number of stages and the optimal
geometry of the machine does not change. Therefore, its cost is proportional to the diameter of the
machine itself, which in turn is linked to the square root of the flow rate on a mass-basis with obvious
economic improvement for machines of greater power.
These considerations are not always valid and in particular do not apply if:
• the machines are very small and there are technological limitations in the machines themselves;
• for very large machines in which the crossover in multiple turbine cylinders is required;
• for expansions that do not work under the same inlet and outlet conditions;
95 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
•
for fluids with a different complexity and molecular weight. With the same expansion ratio 𝛽,
a greater molecular complexity involves greater volumetric expansion ratios and an expansion
that must be divided into a greater number of stages in order to ensure a high efficiency. As
already stated, the molecular weight affects the enthalpy change, and once again the number
of stages of the turbomachine that cannot be excessively filled.
Even if all these exceptions for plants with comparable efficiency are considered, it is correct to ponder
that the cost of the heat exchangers has more weight for large-scale plants, whereas the turbine is the
main cost for the small ones.
8.4.2
Considerations concerning the choice of the working fluid
The choice of the working fluid depends strongly on the size of the plant, on the maximum temperature
of the heat source and on its heat capacity (if it can be considered at constant or variable temperature),
on technical limitations on the components, and on the chemical stability of the working fluid itself.
Once again remember that the effects of the molecular weight and molecular complexity on the
thermodynamic properties of the fluids are separated. Although very often highly complex fluids are
also heavy fluids, it is recommended to not confuse these two properties and their effects on the
thermodynamic cycles.
Effect of the molecular complexity
If the heat source is at constant temperature and there are no technical limitations that force a minimum
vapor quality at the turbine outlet, it can be stated that is always theoretically possible to carry out a
subcritical saturated reversible cycle both with simple fluids and complex fluids. It is necessary to use
infinite regenerative bleeds in the first case and an infinitely complex fluid in the other case. In case
of real cycles, the efficiency will be reduced by the losses due to fluid-dynamics and heat exchange.
From the technological point of view, regeneration is quite different according to the case:
• in a single molecule cycle, it is more complex and costly, it requires regenerative bleeds and a
high number of additional components;
• in the case of complex molecule, a surface recuperator that does not considerably change the
architecture of the cycle and that attains high benefits in terms of efficiency is adopted.
The turbine design is also different when the type of fluid changes:
• For simple molecules, the expansion is in the two-phase field with problems related to the
presence of drops of liquid hammering of the surfaces, and reduction of the performance and
useful life of the component. The volumetric ratio of expansion, however, is slightly less with
the same 𝛽 compared to the complex fluids.
• Expansion is in the field of superheated vapor for complex molecules. The design of the
turbine, however, can be more critical due to the higher volumetric expansion ratios.
When the minimum vapor quality at the turbine outlet is a technical issue, the advantage in using
complex fluids is evident. In case of simple fluids, superheated cycles are required, reducing the
evaporation temperature and reducing the performances of the cycle because of the large temperature
differences and the consequent losses.
When the heat source has a finite heat capacity (variable temperature), complex working fluids exhibit
advantages because of the possibility to reduce the temperature differences in the heat introduction
process. In addition, it is possible to raise the maximum temperature of the cycle. In presence of
limitations on the minimum temperature of the heat source, a recuperator downstream the turbine can
96 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
be included, increasing the efficiency without reducing the heat source exploitation. When the heat
source changes its temperature, supercritical cycles with complex molecules are even more convenient
because of the better coupling between the fluid and the heat source thermal profiles.
In conclusion, a complex fluid can be convenient both in case of constant and variable temperature
heat sources, when the maximum temperature is not sufficient to design an high-efficiency water steam
cycle.
Effect of the molecular weight
The molecular weight of the fluid instead has no influence on thermodynamic considerations and on
the ideal cycles; on the other hand, it has a strong impact on the design of the components, in particular
on the turbine.
An increase in molecular weight has the following results :
• the flow rate on a mass-basis circulating in the cycle increases with the same thermal power
introduced because both evaporation heat and specific heat capacity decrease.
• the turbine enthalpy drop reduces: this has important consequences on the design of the
turbomachine, which can be made at the limit with only one slightly loaded stage. As will
be later seen, this is not entirely true since other limitations related to the maximum
volumetric ratio of expansion per stage are considered in the choice of the number of stages.
• for an ideal gas, sound speed decreases when molecular mass increases, leading to a
difficult design of the turbine blades, increasing the risk of shock waves and reducing offdesign performances.
The choice of a heavy fluid can be useful in order to improve the design of the turbomachinery that
are more economic than the corresponding ones in water vapor cycles.
Effect of the critical properties
As the critical temperature of the two fluids increases (or as the evaporation and condensation
temperatures decrease), the reduced temperatures within which the cycle evolves decrease, but the
results reported do not change significantly.
It is however pointed out that, independently from the complexity and the weight of the fluid:
• given the same evaporation temperature, ∆ℎ𝑒𝑣𝑎 increases as the critical temperature increases,
and so both the fluids witness a decrease in efficiency losses associated with the introduction
and transfer of heat at constant temperature. In case of variable temperature heat sources, it can
be convenient to choose a fluid with a low critical temperature to reduce the relative importance
of the evaporation phase, or to adopt a supercritical cycle.
• The higher the critical temperature, the smaller the pressure values of the cycle. It results in
higher expansion ratios of the turbine and lower condensation pressure. The design of the
turbine is more difficult (higher number of stages and higher volumetric flow rates at the
outlet), and the issue of leakages of incondensable gases increases. The effect of the critical
pressure is certainly present, but has a lower impact because of the exponential relation
between the saturation pressure and temperature.
• On the contrary, given the same evaporation and condensation temperatures, as well as the
same critical pressure, the lower is the critical temperature, the higher is the pressure of the
vapor at the turbine inlet and the lower the volumetric flow rate.
Moreover, once fixed the maximum temperature of the heat source, the choice of a fluid with an
optimal critical temperature can give some advantages:
• It is always possible to realize a supercritical cycle in presence of variable temperature heat
sources.
97 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
•
•
For large plants with low heat source temperature, it is convenient to use fluids with low critical
temperature, reducing issues related with the vacuum in the condenser and the volumetric flow
at turbine outlet, as well as the number of stages consequently. This is the typical application
for geothermal. Water as working fluid would lead to low efficiency cycles, and expensive and
large turbines.
For small plants with high temperatures (biomass, heat recovery, small concentrating solar
plants, domestic cogeneration), it is convenient to choose a fluid with high critical temperature,
but low critical pressure, resulting in low density vapor and high volumetric flow rates. Water
as working fluid (that has a high critical pressure) would lead to very small blades in the
turbine, with a strong decrease in expansion performances. This plants are adopted usually in
cogeneration architecture aiming at the reduction of expansion ratios in the turbine.
Other comments
These observations must necessarily be associated with others related to the safety and cost of the
working fluid. Certainly, water is an optimal fluid on many levels: it is cheap, non toxic, corrosive,
flammable, it has no impact on the ozone layer (ODP) and on the global warming (GWP), it is stable
up to very high temperatures, and has very good heat exchange properties. On the contrary, the organic
fluids are often flammable (hydrocarbons), toxic (methanol, benzene, toluene) or they present an high
ODP (chlorinated compounds now forbidden), or GWP (in particular, the refrigerants). In addition,
they have lower heat exchange coefficients in comparison with water, in particular in case of mixtures.
It is also mandatory to take into account the thermal stability of the fluids that beyond a certain
threshold tend to show dissociation phenomena with consequent modification of the thermodynamic
properties and formation of sludge which can damage the components.
Specifically, the fluids that can be used in power production plants are water (a simple and light fluid)
and organic fluids (certainly more complex and generally much heavier than water). Primarily for
safety reasons, liquid metals that have high 𝑀𝑀 and low complexity, and relatively simple molecules
but with a molecular weight higher than that of water, such as ammonia or methanol, are excluded
from the comparison.
For large power plants with high maximum temperature sources the advantage of using water as
working fluid is clear due to the lower costs, the higher maximum temperature of the cycle and the
lower safety risks. The reference examples are:
• the vapor section of a large combined cycle in which the choice to increase the plant complexity
with HRSGs having three evaporation levels and RH is economically advantageous.
• Fossil-fired power plants, in which the turbine has a high number of stages in view of the high
enthalpy change, and the plant engineering complication created by the regenerative bleeds
relatively impacts on the overall plant cost in which the heat exchangers, and in particular the
steam generator and condenser, weigh more.
• Nuclear plants in which the cost of nuclear reactor is the largest cost.
For small or low temperature plants, it is instead expedient to use an appropriate organic fluid (among
tens of candidate fluids) in saturated or supercritical cycles that can better connect with the heat source
with high plant engineering simplicity. Added to this, the advantages linked to a simpler and less
expensive turbine should then be considered. Despite having a high cost due to the poor transfer
98 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
coefficients, the recuperator is recommended always since it generally leads to a real reduction in the
cost of the electricity produced. In general, the choice must be made based on thermodynamic,
technical and economic considerations, calculating the benefits in terms of potential increase in
efficiency and cost.
Fig. 8.8 shows the field of application of the organic fluid engines and the applications where they can
be used.
STEAM
THERMAL STABILITY LIMIT
400
grey zone
Heat source temperature, C
340
3
7
260
5
2
1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
8
9
180
10
4
6
100
11
20
Domestic cogeneration
Remote applications
Automotive engines WHR
Rural solar energy
Small solar plants
Big solar plants
Biofuels engines WHR
Biomass combustion
WHR from industrial plants
High T. geothermal brines
Low T. geothermal brines
OTEC
12
1 kW
10 kW
100 kW
1MW
10 MW
power output
Fig. 8.8 - Field of application for the organic fluid engines
8.4.3
Mixtures of fluids
The discussion up to now has been focused on pure fluids, but the use of mixtures of fluids can be
interesting both for exploiting sources at variable temperature and for improving the properties of the
resulting fluid.
Use of a mixture in case of variable temperature heat sources
Mixtures can be advantageous when both the heat source and the heat sink are at variable temperature.
This is due to the fact that the phase change, evaporation for example, is not isothermal and takes place
with a rise in temperature. A mixture formed by two miscible fluids that at the same temperature have
two different saturation temperatures is considered. The fluid F1 has a higher saturation temperature,
a lower saturation pressure, less volatility and is called high-boiling; the other F2 is more volatile and
is called, on the contrary, low-boiling. As the mixture composition varies for a certain pressure, the
bubble and dew points for the mixture, i.e. the temperature at which the first bubble is formed due to
an evaporation process or the first drop due to a condensation process, can be defined. Except for
azeotropic compositions, in which the two points coincide, the temperature increases as evaporation
progresses for all the other compositions.
99 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 8.9 - Notation used for defining the temperature glide for a phase transition of a zeotropic
mixture
Starting from a mixture with composition of liquid 𝑥 𝑙1 , the first bubble forms at the temperature 𝑇1 ,
has the composition 𝑥 𝑣1 , and is obviously richer in the more volatile component. The liquid phase is
now enriched in the less volatile component and will have 𝑥 𝑙2 composition. The evaporation
temperature is 𝑇2 > 𝑇1 and the vapor formed has 𝑥 𝑣2 composition. The process continues until the
liquid has 𝑥 𝑙3 composition corresponding at a temperature 𝑇3 . The evaporation process therefore takes
place with a temperature variation called glide. To implement this process, once-through heat
exchangers in which there is no free open surface evaporation like in the pool boiling evaporators are
required. Then by choosing an appropriate mixture of fluids it is possible to get an evaporation that
follows the cooling of the source well, with a consequent reduction of the heat introduction into the
cycle losses.
The same conclusions apply, however, also to the condensation of the fluid and therefore very often
the advantage during the introduction of heat phase is balanced by an increase in the loss relating to
the transfer of heat to the environment. Therefore, fluid mixtures are particularly interesting when both
the hot sink and cold sink are at variable temperature. This is the case of a cogeneration plant in which
the cold sink has a certain current of water (possibly pressurized), which heats up to the conditions
required by the thermal load. In this case, the reversible cycle of reference is the mixtilinear cycle, in
which both heat sources change their temperatures in the process.
The use of mixtures makes it possible to approximate this cycle as shown in the TQ diagrams in Fig.
8.10, where the diagram of heat transfer of the saturated cycle obtainable with a pure fluid and that
obtainable with the optimum mixture is shown. In this specific case the increase in efficiency is higher
than 8.8 percentage points.
100 of 124
Tempe
150
100
150
transition1500
lead
differences in th
1000
temperature diff
pushed to 500
the lo
0
surface.
100
Second-law analysis of power cycles - Energy Conversion A – V7.0
50
50
0
5
10
15
0
Power, MW
300
500
1000
300
60004000
Entropy, J/(kg K)
300
PU
Temperature, °C
Pure fluid
250
250
200
200
150
150
100
100
50
300
300
55
10
10
15
15
00
300
300
Power, MW
250
250
250
250
200
200
200
200
150
150
150
150
100
100
100
100
50
50
500
500
10001000 1500
300 300
6000 6000
Entropy, J/(kg K)
En
Optimal mixture
5000
5000TOLUE
TOLUEN
A higher
temp
4000 4000
Ahighlighted
relatively lim
s
3000
involves
releva
the 3000
economiz
phase
transition
enhance
the cy
2000
larger
than
the
the 2000
reductio
colder
temperatu
temperature
di
1000 1000
50
50
00
55
10
10
results obtainable
15
for15
pure
0
0
and
500
500
mixture
1000
1500
1500
with 1000
optimal toluene
Fig. 8.10 - Numerical
Toluene,
a Entropy, J/(kg K)
300
300
Power, MW
MW
300
300
Power,
Entropy, J/(kg K)
ethanol composition for biomass-fired cogeneration applications
Temperature,
°C
Temperature, °C
inlet. Both evap
30002000
involves
releva
characterized
b
phase transition
transition
lead
1500
larger2000
thaninthe
differences
th
1000
colder1000
temperatu
temperature
diff
500
pushed to the lo
surface. 0 0
50
00
Temperature,
°C
Temperature, °C
3500
5000
The optimal
cyc
TOLUEN
3000
an almost
4000 satu
2500 s
A relatively
250
250
and
0
0
300 300
6000
En
TOLUE
5000
TOLUE
250
250
Mixtures, however, are used very little since in the case of leaks there is a change in the composition
A higher incre
temp
of the fluid and
proceeding with the make up. Another aspectAisfurther4000
200the need for chemical analyses before200
200
200
highlighted
lim
to small temp
related to the coefficient of heat transfer in the evaporation phase: for pure fluids it is very high, while
3000
the
economiz
with an incr
for mixtures it150
may be considerably lower. This fact combined
with the reduction of the temperature
150
150
150
enhance
the cy
temperature
di
difference in the heat exchanger involves a considerable increase in the heat transfer surface and the
2000
the
reductio
cost of the heat
exchanger.
higher second
100
100
100
100
temperature
1000di
Use of a mixture
to change the critical properties of the50fluid
50
50
50
00
55
10
10
15
15
0
0
5
5
10
10
15
15
00
500
500
1000
1000 1500
1500 2000
0
5000
300
Temperature,
°C
Temperature,°C
300
300
The typical application
is thePower,
closed CO
cycles are developed
solar
Power,
MW2 cycle. These300
Entropy, for
J/(kg
K) power plants and
300
MW
Entropy,
J/(kg
K)
for improving flexibility of coal power plants. Increasing the maximum pressure of the cycle, the
4000
250
250
distance from250
the saturation line is reduced yielding 250
a decrease of the compression work due to the
TOLUE
TOLUE
lower compressibility factor. A condensation cycle would be even more advantageous, because the
A further incre
3000
200
200
compression 200
phase could be managed with a pump. 200
Unfortunately, it is not possible to realize this
The critical
te
to small temp
cycle for solar applications because the availability of low-temperature (15-20°C) heat sinks is usually
cycle adopts
with an 2000
incr
150
150
limited in typical
solar plant locations. This temperature
requirement is fixed by the critical
150
150
point tempe
temperature of the CO2 that is 31°C. Nevertheless, it is possible to mix the CO2 with small amountstemperature
of
di
recuperator is
other substances
100 with a higher critical temperature, aiming
100 at increasing the critical temperature of the
higher second
100
100
economizer1000
on
mixture; then, condensation cycles become possible with the mere availability of cooling air at 3035°C.
50
50
50
50
mperature, °C
300
250
200
150
Power, MW
Power, MW
300
0
0
250
101 of 200
124
150
500
500
1000
1000
1500
1500
Entropy, J/(kg K)
Entropy, J/(kg K)
2000
2000
0
2
5000
4000
TOLUE
3000
The critical
te
cycle adopts
2000
point tempe
Second-law analysis of power cycles - Energy Conversion A – V7.0
9 SECOND LAW ANALYSIS OF AN OPEN-LOOP BRAYTON GAS CYCLE
In this chapter, the list of losses in open-loop cycles is shown. The framework presented in Chapter 6
is first outlined, then the losses are evaluated and commented.
Physical (or conceptual) boundaries of the system considered
The boundaries of the system are: (i) the section where the air enters the filter and fuel chamber and
(ii) the section where the combustion products exit to the turbine discharge (or of the recuperator, if
there is one). All the auxiliary systems are powered inside the system, from which the output power
then exits.
Choosing the dead state
In the case of internal combustion engines, such as the open cycle gas turbines, choosing the dead state
is mandatory, and the atmospheric air is chosen. The nominal conditions can refer to the conditions:
• “ISO”, meaning 15 C, 1.01325 bar, standard composition, U.R: 60%, or
• “site”, those of the site where the gas turbine is installed
• Real.
Choosing the energy source
For static applications, the most widely used energy source is natural gas, which is defined through:
• Chemical composition
• Temperature
• Pressure
Of the various energy indexes used for the fuel, the preferable one for the second law analysis is 𝑊𝑟𝑒𝑣 .
It is not tied only to the combustion reactions, but takes into account the temperature and pressure, in
addition to the composition. The temperature is often different from that of the gas pipeline due to
preheating of the fuel. The fuel pressure is considerably higher (often 10 or more bar) than that of the
combustor due to the presence of control valves upstream of the combustor.
Reference to the rated instantaneous quantities
Reference will be made to the powers under nominal conditions in the following analysis.
9.1 CRITERIA FOR CHOOSING THE COMPRESSION RATIO
There are two performance indexes for an open cycle gas turbine: efficiency and specific work.
𝜂=
𝑊̇𝑢
𝑊̇𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑊̇𝑢
𝑤𝑢 =
𝑚̇𝑎𝑖𝑟
(9.1)
(9.2)
The first accounts for how the fuel is used and may change numerically from one country to the next
depending on the convention taken for the 𝑊̇𝑟𝑒𝑣,𝑓𝑢𝑒𝑙 , or if it refers to the LHV, the HHV, whether or
not it takes into account the gas pressure in the gas pipeline upstream of the combustor. The second,
on the other hand, accounts for the power of the plant in connection with its physical size. It is not
calculated as a difference of the specific turbine and compressor works since the flow rates in a gas
turbine cycle are different and change both along the compression (drawing air to cool the blades) and
along the expansion (mixing with cold air).
The cost of the electricity produced depends on both the investment cost and the variable costs. The
latter term depends on the fuel cost and the efficiency of the plant. The former term, on the other hand,
102 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
depends on the machine cost related to the installed power and, therefore, the inverse of the specific
work. In fact, it can be considered that the machine cost depends on its mass, and therefore on its
volume. In considering the machine as a cylinder of size 𝐴 (frontal cross section) and 𝐿 (length) (see
Fig. 9.1), it mainly depends on the mass of aspirated air and on the pressure ratio.
It is considered that the intake sections at the compressor and the discharge sections of turbine 𝐴
depend on the volumetric flow rate that crosses them 𝑉𝑎̇ and on the axial speed of the flow 𝑉𝑎
𝐴=
𝑉̇𝑎𝑖𝑟 𝑚̇𝑎𝑖𝑟
=
𝑉𝑎
𝜌𝑉𝑎
(9.3)
where the axial speed cannot be changed at will since the pressure drops increase as the speed rises
and the velocity triangles have to be well proportioned. Therefore, this parameter is independent from
the compression ratio and the power of the machine. Moreover, in open gas cycles it is not possible to
pressurize the cycle and change the fluid density. Thus, also the term 𝜌𝑎 have any influence. As a
result, the passage area depends on only the flow rate on a mass-basis and, therefore, on the circular
crown defined by the blade height ℎ and the average diameter 𝐷𝑚 of the stage:
2
𝑚̇𝑎𝑖𝑟 ∝ 𝐴 = 𝜋𝐷𝑚 ℎ = 𝜋𝐷𝑚
ℎ
𝐷𝑚
(9.4)
As will be seen further on, not even the blade height can be chosen regardless of the need to have high
efficiency stages. In particular, it is decided to set a certain optimum ℎ/𝐷𝑚 . The machine length can
be considered a function of the number of compressor stages. This number depends on the compression
ratio and, considering a 𝛽𝑙𝑖𝑚 value per stage, it will be equal to
𝑛𝑆𝑇𝐺 = next whole [
𝑙𝑛 (𝛽)
]
𝑙𝑛(𝛽𝑙𝑖𝑚 )
(9.5)
The volume of the gas turbine is therefore proportional to the flow rate of aspirated air 𝑚̇𝑎 ∝ 𝐷𝑚 2 and
the logarithm of the pressure ratio. Of the two effects, the one linked to the front dimension
predominates, which is therefore the parameter of interest for calculating the specific cost of a gas
turbine. It is therefore correct to consider the specific work as an index of the specific power produced.
A
L
Fig. 9.1 - Diagram of the overall dimensions of a gas turbine in simple cycle configuration.
103 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Reasoning with TIT (Turbine Inlet Temperature) being equal in simple cycle and increasing the
compression ratio, both indexes initially rise up until the maximum 𝑤𝑢 is reached. For higher 𝛽, the
specific work decreases, and the maximum efficiency is reached. The trend on the 𝜂 − 𝑤𝑢 plane is
shown in Fig. 9.2 for two different TITs.
Fig. 9.2 - Change in efficiency and specific useful work as the compression ratio changes for a
simple cycle gas turbine
Choosing the optimum compression ratio is therefore based on the need on the one hand to maximize
efficiency and hence reduce the variable operating costs, and on the other that of having small
machines and hence with a lower investment cost.
There are two philosophies:
• Heavy Duty Machine or industrial machinery. These are machines that work very few hours a
year (certainly less than 1000 hours a year, sometimes less than a hundred) to cover the peaks,
and so it is necessary that they have the lowest possible investment costs. Work is carried out
in maximum 𝑤𝑢 conditions. The optimum compression ratios are low, 12-15, and the
efficiencies are between 35% and 38-40%. They are recommended in cogeneration
applications, where the discharged heat can be used.
• Jet Machines or aeroderivative machinery. They aim at maximizing efficiency and therefore at
minimizing the use of fuel. They are designed to work many hours a year, and the optimum
compression ratio is higher than the previous case (about 30). Special attention have to be paid
to the polytropic efficiency of the turbine and compressor since the efficiency loss linearly
depends on the 𝑙𝑛(𝛽)
Now, focus will be placed on defining the optimum compression ratio in the perspective of maximizing
the cycle efficiency, based on the second law analysis and applying the methodology to a simple cycle
turbine, a turbine in recuperated cycle, and providing some information on the cooled turbines usually
used in industrial practice.
9.2 SIMPLE CYCLE WITH NON-COOLED TURBINE
This is a not very realistic example because today all modern open cycle gas turbines are equipped
with turbine blade cooling, but it is helpful for teaching purposes in order to present the characteristic
pattern of the typical losses of a gas cycle. If the blades are not cooled, it means that flue gases have a
maximum temperature of about 800-900 °C and the combustion is carried out with a large excess of
air.
104 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 9.3 - Section plane of a modern gas turbine in simple cycle
The cycle is outlined with the following seven irreversible processes:
• Δ𝜂1 irreversibility due to pressure drops (in the air filtration system and in every other part of
the cycle)
• Δ𝜂2 fluid dynamic irreversibilities in the compressor
• Δ𝜂3 combustion irreversibility
• Δ𝜂4 fluid dynamic irreversibilities of the turbine (adiabatic supposition)
• Δ𝜂5 losses due to the discharge of exhaust gas into the atmosphere
• Δ𝜂6 thermal losses
• Δ𝜂7 mechanical/electrical/auxiliary losses
mf
3
2
4
3
5
1
4
2
0
0
5
1
Fig. 9.4 - Simple cycle of a gas turbine in the T-S plane. Adiabatic turbomachinery are hypothesized
and the thermal losses (present in various parts of the plant) are not represented
Every one of them involves an efficiency loss whose dependency on the compression ratio 𝛽 is
examined. The analysis is conducted keeping the maximum cycle temperature value constant.
Consistent with the hypotheses adopted above, the general formula is the following:
105 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑊̇𝑢
𝑊̇𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
7
7
𝑊̇𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
∆𝑊̇𝑖
𝑇0 ∆𝑆̇ 𝑖
=
−∑
=1−∑
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑊̇𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑊̇𝑟𝑒𝑣
1
7
𝜂 = 1 − ∑ ∆𝜂𝑖
(9.6)
1
(9.7)
1
where 𝑚̇𝑓𝑢𝑒𝑙 is the fuel flow rate.
The analysis will be presented with the maximum cycle temperature being equal, while changing the
compression ratio: as the 𝛽 rises, the temperature of the air discharged by the compressor increases,
so the flow rate of required fuel decreases, which can be calculated by the enthalpy balance of the
combustor. The parameter 𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 increases.
9.2.1 𝚫𝜼𝟏 irreversibility due to pressure drops (in the air filtration system and in every other
part of the cycle)
The ambient air in a gas cycle is aspirated by the compressor and through a filter chamber in which all
the impurities suspended in it are removed in order to protect the compressor blades and limit fouling
phenomena. The filters used are so efficient that the GT machines outlet air can contain less particulate
matter than that contained in the aspirated air. The intake process is adiabatic and preserves the total
enthalpy and total temperature. Thus, acceleration of the air up to the inlet condition of the compressor
entails a temperature reduction. To prevent the formation of frost, it is necessary to keep the
temperature at an appropriate value by using anti-icing systems. These systems heat the air before the
filter chamber to prevent ice crystals from depositing on the filtering surfaces (and consequently
increasing pressure drops) or on the compressor blades.
The heat can come from the following sources:
• heat exchanger with hot water at 30-40°C heated with the gases discharged by the turbine;
• by mixing the aspirated air with the hot air of the turbine enclosure.
In these cases the process is not adiabatic and requires the fluid to be heated; this is always penalizing
for the cycle performance since the mass flow rate aspirated by the compressor is always reduced
because the compressor works at constant volumetric flow rate, in absence of VGV as it will be
discussed later. The anti-icing system has to be enabled only if strictly necessary, or for temperatures
lower than 6-7°C and relative humidity of about 50%.
In other cases, the opposite process is carried out, meaning cooling the aspirated air to increase the
power produced by the cycle. Cooling can be achieved with:
• battery with cold water chilled by a refrigeration cycle, that however demands added
consumption;
• a spray of cold water in a thermal and mass exchange process. From the energy viewpoint, it
is interesting, but if the water is not perfectly demineralized, it can foul the surfaces. A measure
of this type is usually poorly tolerated by the GT manufacturers that provide very stringent
criteria on the flows entering the machinery.
In the case of GT cycles, it is absolutely legitimate to consider the aspirated air an ideal gas since the
ambient conditions are very far away from the saturation curve. The specific heat of the air does not
depend then on the pressure, but only on the temperature since the air is made up of diatomic and
triatomic molecules.
106 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑁
𝑇0
𝑅𝑢 ∆𝑝𝑖
∆𝜂1 =
∑ 𝑚̇𝑖
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑀𝑀 𝑝𝑖
𝑖=1
(9.8)
where 𝑁 is the number of pressure drops (finite, but relatively small in relative terms, otherwise the
∆𝑝
term 𝑖 is to be replaced with ∫ 𝑑𝑝𝑖 /𝑝𝑖 ). ∆𝜂1 is directly proportional to 𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 and, therefore,
𝑝𝑖
increases as β increases.
The following important elements in the equation that defines the loss of efficiency has to be noted:
• There is no dependency on the temperature at which throttling takes place; even if the
dissipated infinitesimal work 𝑑𝑊𝑤,𝑖 = 𝑣𝑑𝑝 is proportionate to the specific volume (and hence
to the temperature), part of this might be recovered in other processes, so the loss of useful
work calculates only the amount lost forever.
• The absolute pressure drop does not count, only the relative does. In other words, the same Δ𝑝
is paid to a different extent depending on the pressure at which it takes place (the higher it is,
the less is paid).
• The drop is tied to the ratio between the flow rate on a mass-basis that the throttling sustains
and the flow rate of fuel: flow rates being equal, the greater the flow rate of the fuel introduced
into the cycle, the less the efficiency loss is.
9.2.2
𝚫𝜼𝟐 fluid dynamic irreversibilities in the compressor
The compressor is supposed as an adiabatic machine, distinguished by a polytropic efficiency 𝜂𝑦,𝐶 .
The efficiency loss associated with this component is:
∆𝜂2 =
1 − ̅̅̅̅̅̅̅̅
𝜂𝑦,𝐶𝑀𝑃
𝑇0 𝑚̇𝑎𝑖𝑟 𝑅𝑢
𝑝𝑜𝑢𝑡
𝑍̅ (
) 𝑙𝑛 (
)
𝑀𝑀
𝜂𝑦,𝐶𝑀𝑃
̅̅̅̅̅̅̅̅
𝑝𝑖𝑛
(9.9)
One can note that ∆𝜂2 grows as the compression ratio increases and that it does not depend on the
temperature at which the compression takes place. The increase of the term 𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 further
contributes to the rise in ∆𝜂2 with 𝛽. The importance of the efficiency loss is naturally correlated with
the "fluid dynamic quality" of the component defined by 𝜂𝑦,𝐶𝑀𝑃 .
9.2.3
𝚫𝜼𝟑 combustion irreversibility
The combustion reaction takes place in the combustor, in which the fuel (usually natural gas) reacts
with the oxidant (compressed ambient air) to form combustion products at a temperature that is based
on the preheating of the reactants and the dilution of the combustion. The irreversibility of combustion
is based on:
• the temperature of the reactants, meaning the air and fuel. Since the heat capacity of the
combustion air is much greater than that of the fuel, the benefits obtainable as the air
temperature increases are much greater than those obtainable by increasing the fuel
temperature.
• the temperature of the combustion products, which is equal to the adiabatic flame temperature,
coinciding with the turbine inlet temperature.
107 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
𝑁𝑝
𝑁𝑟
𝑇0
𝑇0
̇
∆𝜂3 =
∆𝑆𝑐𝑜𝑚𝑏
=
[∑(𝑚̇𝑖 𝑠𝑖 )𝑃 − ∑(𝑚̇𝑗 𝑠𝑗 )𝑅 ]
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑖=1
(9.10)
𝑗=1
To calculate this formula, it is necessary to use thermodynamic tables to calculate the absolute
entropies and the trend of specific heat of ideal gas with the temperature. As the compression ratio
̇
increases, the combustor inlet air temperature increases: ∆𝑆𝑐𝑜𝑚𝑏
and also 𝑚̇𝑓𝑢𝑒𝑙 decrease. Finally, the
∆𝜂3 loss decreases. The same result is obtained by associating 𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙 at the thermal power
produced in the combustion, potentially at infinite temperature. The equation can be expressed in
approximated terms (the heat capacity of the fuel is neglected) as:
∆𝜂3 =
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙 𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑇0
𝑇0
(
−
)=
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙 𝑇𝑚𝑙𝑛|𝑇𝑚𝑎𝑥,𝑇2
𝑇∞
𝑇𝑚𝑙𝑛|𝑇𝐼𝑇,𝑇2
(9.11)
where 𝑇2 is the temperature of the reactant made up main of combustor inlet air.
The equation highlights that the efficiency loss decreases as the mean temperature between the
reactants and products increases. With 𝑇𝐼𝑇 the same, the temperature of the combustion products is
fixed, so the loss depends only on the combustor inlet temperature, and therefore decreases as the
compression ratio increases, always and in any case keeping itself at high values.
9.2.4
𝚫𝜼𝟒 fluid dynamic irreversibilities of the turbine (adiabatic assumption)
Consistently with the hypothesis of adiabatic expansion, the turbine is assumed a machine
distinguished by a polytropic efficiency 𝜂𝑦,𝑇𝑅𝐵 . The efficiency loss associated with this component is:
∆𝜂4 =
𝑇0
𝑚̇𝑒𝑥ℎ 𝑅𝑢
𝑝𝑖𝑛
𝑍̅(1 − ̅̅̅̅̅̅̅̅)𝑙𝑛
𝜂𝑦,𝑇𝑅𝐵
(
)
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙 𝑀𝑀
𝑝𝑜𝑢𝑡
(9.12)
One can note that as for the compressor, as the compression ratio rises, ∆𝜂4 increases both due to
greater 𝛽 and because of the increase of the term 𝑚̇𝑒𝑥ℎ ⁄𝑚̇𝑓𝑢𝑒𝑙 . As always in the case of losses linked
to fluid dynamics in the gas phase, it does not depend on the temperature at which the expansion itself
takes place. The importance of the efficiency loss is naturally correlated with the "fluid dynamic
quality" of the component defined by 𝜂𝑦,𝑇𝑅𝐵 . In the case of non-cooled machinery, the turbine is a less
critical component of the compressor since with polytropic efficiency being the same, the term
(1 − ̅̅̅̅̅̅̅̅)
𝜂𝑦,𝑇𝑅𝐵 will be lower than the term (
̅̅̅̅̅̅̅̅̅̅
1−𝜂
𝑦,𝐶𝑀𝑃
̅̅̅̅̅̅̅̅̅̅
𝜂
𝑦,𝐶𝑀𝑃
). It should also be noted that the turbine will have
a lower expansion ratio than that of the compressor due to the pressure drops.
9.2.5
𝚫𝜼𝟓 losses due to the discharge of exhaust gas into the atmosphere (stack losses)
The discharge of the combustion products downstream of the turbine (at temperature TOT) is
assimilable to a heat transfer (with infinite transfer surface) with the environment at temperature 𝑇0 .
𝑇0
̇
̇
(∆𝑆𝑒𝑥ℎ
+ ∆𝑆𝑎𝑚𝑏
)
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑇0
1
1
∆𝜂5 =
𝑚̇𝑒𝑥ℎ,𝑐 𝑐𝑝 (𝑇𝑂𝑇 − 𝑇0 ) ( −
)
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑇0 𝑇𝑚𝑙𝑛|𝑇𝑂𝑇,𝑇0
∆𝜂5 =
108 of 124
(9.13)
(9.14)
Second-law analysis of power cycles - Energy Conversion A – V7.0
∆𝜂5 =
𝑚̇𝑒𝑥ℎ,𝑐 𝑐𝑝 (𝑇𝑂𝑇 − 𝑇0 )
𝑇0
(1 −
)
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑇𝑚𝑙𝑛|𝑇𝑂𝑇,𝑇0
(9.15)
As the compression ratio increases, the discharge temperature decreases on the one hand, and the term
𝑚̇𝑒𝑥ℎ,𝑐 /𝑚̇𝑓𝑢𝑒𝑙 increases on the other. Of the two effects, the first is predominant and this efficiency
loss decreases as 𝛽 increases. It is also seen that the term in brackets at the bottom of the equation
expresses the efficiency of a trilateral cycle operating with the hot gases as the heat source and with
the environment.
9.2.6
𝚫𝜼𝟔 thermal losses
Assuming that the system, in a certain point of the plant, loses an infinitesimal thermal power
𝑑𝑄𝑑𝑖𝑠𝑝,𝑖 = 𝑚̇𝑎𝑖𝑟 𝑑ℎ𝑑𝑖𝑠𝑝,𝑖 originating starting from a temperature 𝑇𝑥 to the environment, the resulting
efficiency loss is equal to:
𝑁
𝑑𝑄̇𝑑𝑖𝑠𝑝,𝑖
𝑑𝑄̇𝑑𝑖𝑠𝑝,𝑖
𝑇0
∆𝜂6 =
∑ [∫
−∫
]
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑇0
𝑇𝑥,𝑖
𝑖=1
(9.16)
𝑁
𝑚̇𝑎𝑖𝑟
𝑇0
∆𝜂6 =
∑ 𝑑ℎ𝑑𝑖𝑠𝑝,𝑖 (1 −
)
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
𝑇𝑚𝑙𝑛,𝑖
(9.17)
𝑖=1
With specific thermal losses being the same as the fluid flow rate, the term increases with the ratio
𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 and therefore with the rise in 𝛽, with the same law already found for Δ𝜂1 .
9.2.7
𝚫𝜼𝟕 mechanical/electrical/auxiliary losses
The mechanical and electrical losses depend on the gross power of the compressor and turbine, the
electrical losses on the mechanical power at the generator inlet. The consumption of the auxiliary
systems is usually defined as a fixed portion of the electrical power produced.
All these quantities are therefore proportional to the power produced.
∆𝜂7 =
9.2.8
̇
∑𝑁
𝑖=1 ∆𝑊𝑚𝑒,𝑎𝑢𝑥
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙
(9.18)
Observations
The trends represented in Fig. 9.5 are based on 𝛽, and are obtained for 𝑇0 =15°C, 𝑇𝐼𝑇=900°C,
polytropic turbine and compressor efficiency of 90%, relative pressure drops equal to 10%,
concentrated thermal losses at the turbine outlet (𝑇𝑂𝑇) equal to 10°C, mech/el/aux losses equal to 3%
of the useful power.
109 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
0.6
1
0.9
0.5
0.8
0.7
0.4
0.6
0.3
0.5
0.4
0.2
0.3
0.2
0.1
0.1
0
0
0
10
20
30
40
50
0
10
20
30
40
50
Fig. 9.5 - Trend of the efficiency losses (left) and cumulative diagram (right) of a single gas cycle as
the compression ratio changes.
One can note:
• that for the losses linked to the pressure drops (1) and to the thermal losses (6), they rise
(although weakly) as the compression ratio increases.
• that the losses in the turbomachinery (2 and 4) rise (a lot) as the compression ratio increases.
• that the combustion losses (3) and discharge losses (5) are very significant and decrease (the
discharge losses very much, the combustion losses less) as the compression ratio increases.
• since some losses (above all those in the turbomachinery) increase as the compression ratio
increases and others (combustion and discharge of the combustion products) decrease, there is
an optimal compression ratio between 15 and 20, for which the sum of the losses is minimum
and the efficiency is maximum.
• The optimum compression ratio for the useful work is lower, as already noted previously.
200
40%
35%
150
25%
100
20%
15%
50
10%
efficiency
30%
5%
0
0%
0
10
20
30
40
50
Fig. 9.6 - Trend of the specific work and efficiency as the compression ratio varies. The maximum
values are at different 𝛽
9.3 RECUPERATED CYCLE WITH NON-COOLED TURBINE
Unlike the previous case, recuperated cycles with non-cooled turbine are used in practice, in particular
for industrial or domestic cogeneration applications, or for the exploitation of renewable energy
sources (in particular, concentrating solar power towers). They are usually small gas turbines (300 kW
110 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
- 5 MW) because the heat exchanger is a critical component due to the low heat exchange coefficients
and the high operating temperatures. It could be very expensive for sizes larger than 5 MW. Small
sizes and high plant engineering simplicity requirements lead to a system without blades cooling and
therefore with maximum temperatures (TIT) of about 800-900 °C.
Fig. 9.7 - Section plane of a modern gas turbine in recuperated cycle
Now a recuperated cycle will be considered. A schematic of such systems is shown in Fig. 9.7, where
a highlight on the recuperator is placed, while the Ts and architecture are shown in Fig. 9.8.
3
mf
4
2’
2’
3
4’
2
1
5
4
2
4’
0
0
5
1
Fig. 9.8 - Recuperated cycle of a gas turbine in the T-S plane. Adiabatic turbomachinery are
hypothesized and the thermal losses (present in various parts of the plant) are not represented.
In addition to the seven losses considered in the previous case, an eighth one is added to them, tied to
the thermal transfer irreversibilities in the heat exchanger.
9.3.1
𝚫𝜼𝟖 losses relating to the heat transfer in the recuperator
The flue gas discharged by the turbine cool down in the recuperator to preheat the compressed air.
111 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
This component is a site of irreversibility since the heat transfer takes place under finite temperature
differences. The heat capacities on the hot side and cold side are different since the specific heats of
ideal gas are different and the flow rate of exhaust gas is higher than that of air. In the case of a noncooled turbine, the heat exchanger can theoretically work with very small temperature differences.
However, the cost of the heat exchanger rises sharply as the ∆𝑇𝑚𝑖𝑛 drops because of the low heat
transfer coefficients. In this example a pinch-point of 50°C on the hot end of the exchanger was
assumed. The TQ diagram of the recuperator is shown in Fig. 9.9.
Fig. 9.9 - TQ diagram of the recuperator of a non-cooled gas turbine
4′
2′
𝑇0
𝑑𝑄̇
𝑑𝑄̇
∆𝜂8 =
(∫
+∫
)
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙 4 𝑇
𝑇
2
𝑇0 𝑄̇𝑟𝑒𝑐
𝑇𝑚,ℎ − 𝑇𝑚,𝑐
∆𝜂8 =
(
)
𝑚̇𝑓𝑢𝑒𝑙 𝑤𝑟𝑒𝑣,𝑓𝑢𝑒𝑙 𝑇𝑚,ℎ 𝑇𝑚,𝑐
(9.19)
(9.20)
The formula points out that the loss is the more significant the higher the ratio between recuperated
power and that introduced into the cycle is. This ratio decreases (up to the point of being canceled out)
as the compression cycle of the cycle increases. Above this value of pressure ratio, it is not possible to
build a recuperated cycle. As it is obvious, the loss is heavily influenced by the mean temperature
difference between the two currents in the heat transfer. This difference cannot be small because the
heat transfer coefficients of the two currents are low (low density gas), and also because high speeds
cannot be adopted to limit the pressure drops, which in the gas cycles are strongly penalizing.
9.3.2
Observations
The effects of the recuperator reflect on all seven terms ∆𝜂 considered previously. Unlike what has
been seen for the simple cycle, the heat that can be recovered is gradually lower as the compression
ratio rises, more fuel is needed and the ratio 𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 decreases. At the opposite limit for 𝛽 = 1
there is no introduction of fuel and 𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 → ∞. On the one hand, the denominator in the formula
increases, so ∆𝜂 tend to decrease, but on the other the irreversibilities in the combustion (the air is less
preheated) and in the transfer of heat to the environment (the combustion products exit warmer)
increase. The new trend of the various ∆𝜂 is shown in the figure. The calculations are connected with
a hypothetical cycle characterized by: 𝑇0 =15°C, 𝑇𝐼𝑇=900°C, polytropic turbine and compressor
efficiency 90%, relative pressure drops equal to 10%, concentrated thermal losses at the turbine outlet
(𝑇𝑂𝑇) equal to 10°C, mech/el/aux losses equal to 3% of the useful power, ∆𝑇𝑚𝑖𝑛 in the recuperator
equal to 50 °C.
112 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
0.6
1
0.9
0.5
0.8
0.7
0.4
0.6
0.3
0.5
0.4
0.2
0.3
0.2
0.1
0.1
0
0
10
20
30
40
50
0
0
10
20
30
40
50
Fig. 9.10 - Trend of the efficiency losses (left) and accumulative diagram (right) of a recuperated
gas cycle as the compression ratio changes.
As a result, the following is obtained (as shown in Fig. 9.10):
• the losses concerning pressure drops (1) and thermal losses (6) now decrease as the
compression ratio rises as they are linked only to 𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 and weigh more than in the
previous case.
• The losses concerning the turbomachinery (2 and 4) are rising since the effect of increased 𝛽
is greater than that linked to the reduced 𝑚̇𝑎𝑖𝑟 ⁄𝑚̇𝑓𝑢𝑒𝑙 . They are less important than the previous
case.
• The combustion losses (3) continue to be highly significant, and increase (little) as the
compression ratio increases.
• That the losses at the discharge (5) rise (a lot) as the compression ratio increases.
• The losses in the recuperator (8) are very large at the low compression ratios since the
recoverable heat grows. They decrease as the compression ratio increases up to being canceled
out when recuperation is no longer possible, or when the difference between TOT and 𝑇2 at the
end of compression becomes lower than ∆𝑇𝑚𝑖𝑛 : a situation that in this case occurs due to 𝛽~15.
• Since some losses (combustion and discharge) increase as the compression ratio increases and
others (above all those of the recuperator) decrease, there is an optimal relatively modest
compression ratio (around 3), for which the sum of the losses is minimum and the efficiency
is maximum. Recuperated gas turbines are designed with pressure ratios that are lower than for
large simple cycle gas turbines. Then, it is possible to use single stage radial turbomachinery
for both compression and expansion, yielding a strong reduction in costs.
• The resulting optimal efficiency is about 10 points higher than for the simple cycle.
9.4
GAS CYCLE WITH COOLED EXPANSION
As previously stated, it is difficult from the economic viewpoint to reach fluid temperatures higher
than 600-620°C in steam power plants in which combustion is external. The temperatures felt by the
materials are always higher than that of the fluid, and in view of the large transfer surfaces, the high
thicknesses of the tubes is not advantageous in economic terms. It is best to use superalloys to build
the SH and RH banks. Similar observations apply to closed gas cycle plants.
The use of expensive materials able to withstand higher temperatures is instead common in the gas
turbines, where the increased efficiency and specific work is strictly tied to the start of expansion
113 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
temperature. Investing in more expensive materials is economically advantageous in the case of a gas
turbine since the combustion is internal (metal temperature lower than the fluid temperature) and the
masses are small. In the last 50 years the maximum temperature has constantly increased. This result
was attained with (i) the use of materials able to withstand higher temperatures, (ii) single crystal blade
manufacturing methods, (iii) improved cooling techniques and (iv) use of protective ceramic coatings.
The last two points are of fundamental importance since even the best materials from this point of
view (NiCrMo superalloys and blades formed by growing a single crystal) are unable to tolerate
temperatures over 900°C. Therefore, cooling flows are necessary, because they allow the maximum
temperature of the metal to be limited adopting different technological solutions (internal convection,
impingement, film cooling or transpiration).
Fig. 9.11 - Technological solutions for cooling high temperature blading of a modern gas turbine
If the technological development of gas turbines is examined in terms of TIT (Turbine Inlet
Temperature), one notes the trend illustrated in Fig. 9.12: a constant increase over the years by about
12-13°C a year, due to improvement of the cooling technologies for about two-thirds, and to
improvement of the materials (4-5°C) for about one-third.
Fig. 9.12 - Left: historical trend of the turbine inlet temperatures over the years, with an increase of
about 12-13°C/year. Right: historical trend of the maximum allowable temperatures of the materials
used in the hot parts of the gas turbines. The increase is about 4-5°C/year for the industrial turbines.
114 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The use of cooling flows on the one hand allows higher temperatures to be reached and to improve
efficiency. On the other hand, added losses are inevitable due to (i) heat transfer, (ii) pressure drops of
the cold air in the convective channels, (iii) the subsequent mixing with the exhaust gas, and (v) the
modification of the flow in the channels and thus worsening of the fluid dynamics of the machine. The
technological limit is transpiration cooling, which requires the use of porous materials and the
formation of a low speed flow that marginally influences the main flow undergoing expansion. The
qualitative trend of the increased efficiency attainable with a technological improvement, whether tied
to the materials or to the cooling techniques, is shown in Fig. 9.13. The continuous lines represent a
conventional technology: as the TIT grows, the cycle efficiency increases, but also a greater cooling
air flow rate is necessary. The effect is positive up until when the benefits given by the increased
temperature are canceled out by the increased losses due to the cooling of the blades and of the
combustor surfaces. The optimum efficiency (𝜂0 ) for a certain turbine inlet temperature is identified.
cooling
simple cycle
If one wants to further increase the efficiency, it is necessary to use more efficient cooling methods or
materials able to withstand higher temperatures. In both cases a smaller air flow rate is required and a
switch is made to the efficiency 𝜂1 . Technological progress however makes it possible to go to higher
TITs with coolant flow rates being equal and therefore to greater efficiencies (𝜂2 ), until the loss due
to the cooling becomes once again dominant and a new maximum efficiency (𝜂3 ) at higher TITs is
identified.
The figure explains two important concepts:
• There is a TIT value that optimizes the efficiency of a gas turbine, beyond which the
irreversibilities tied to the cooling cancel the positive effects (in particular, but not only, the
combustion irreversibilities) linked to the increase in TIT.
• This value increases as the cooling technology improves. In addition to increasing TIT, also
the flow rate of the cooling flows increases, with an overall positive effect on the efficiency.
At the same time, the gap between the TIT and the temperatures of the cooled materials
increases.
Advanced
technology
Conventional
technology
Fig. 9.13 - Maximum efficiency and new TITs obtainable with a technological improvement
115 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The hot parts of the machine are cooled with flows of air drawn from the compressor, some at
intermediate pressures and some at the compressor delivery. The flows are classified in two broad
categories, as can be seen in Fig. 9.14:
• non-chargeable flows: are the compressed air flows (all drawn from the compressor delivery) that
are injected upstream of the first turbine rotor, thus to cool the combustor and/or the first stator
array. They contribute to the energy balance that defines the total temperature upstream of the first
rotor array (hence from where the extraction of work from the cycle begins). This temperature will
be called TIT (Turbine Inlet Temperature, sometimes called TRIT, Turbine Rotor Inlet
Temperature).
• chargeable flows: are the compressed air flows (independent from the compressor drawing points)
that are used to cool the parts downstream of the first stator array (therefore downstream of TIT).
They represent a thermodynamic penalization as they bypass the combustion. The chargeable flows
are drawn at the appropriate pressure to be mixed at a certain point of the expansion and prevent
further losses given by isenthalpic throttling processes.
2i
2ii
“chargeable”
cooling bleedings
2iii
3i
3
compressor
3ii
3iii
3iv
“NON chargeable”
cooling bleedings
1
Fig. 9.14 - Representation of the cooling flows drawn from the compressor of a gas turbine
The temperatures that characterize a gas turbine are:
• 𝐶𝑂𝑇 (Combustor Outlet Temperature): is the total temperature downstream of the combustor,
therefore downstream of the duct, called transition piece, which connects the combustor outlet
with the turbine inlet. This component reduces the speed up to the value required at the turbine
inlet, and makes the flow uniform. Naturally, 𝐶𝑂𝑇 > 𝑇𝐼𝑇, because the cooling flows (nonchargeable) mix with the main current in the first stator array. It is not measured but obtained
from a mass and energy balance that considers all the non-chargeable flows used to cool the
liner and transition piece. The configuration of the combustors and the solutions adopted have
evolved over the years, changing from single combustors to silos, to many ring combustors
able to guarantee a much more uniform flow at the turbine inlet and better control of the
combustion process in order to limit the formation of pollutants.
116 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 9.15 - Evolution of the combustors of gas turbine plants
combustor
walls
transition
piece
4
3
premixed
combustion
cooling bleedings
for the combustor
4T
COT~1600°C
Combustor
Outlet
Temperature
3
3T
TAF=COT
4
Fig. 9.16 - Diagram showing the distribution of the cooling flows in a gas turbine combustor and in
the transition piece. First law balance for calculating the COT
117 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The value of 𝐶𝑂𝑇 has an influence on the emissions of nitrogen oxides and, generally speaking,
the greater 𝐶𝑂𝑇 is, the greater the NOx emissions are. They are limited by using premixed
flames that give lower adiabatic flame temperatures and guarantee NOx values of 8-9ppm, an
extremely lower value than just 60 years ago when diffusion flames were used that went over
1000 ppm. This technique however tends to be inefficient at the low loads where it is difficult
to keep a premixed flame stable and switching to diffusion flames is required, with the resulting
increase in nitrogen oxide emission. This is why every power plant is connected with an
environmental protection agency EMS (Emission Monitoring System) that continuously
analyzes the machine operation.
• 𝑇𝐼𝑇 (Turbine Inlet Temperature). The value of 𝑇𝐼𝑇 is significant because it is tied to the flow
temperature that strikes the high temperature part of the gas turbine most highly stressed
mechanically, i.e. the first rotor. The 𝑇𝐼𝑇 is not a measurable quantity due to the very high
temperature and the need to implement multiple gauge wells between the first stator and the
first rotor array, with penalization of the fluid dynamics of the machine. It can be obtained with
a mass and energy balance, but requires that all the cooling flow rates be known.
4T
4
stator
5
3T
5T
Turbine (Rotor)
Inlet Temperature
4
T5T=TIT
5S
5
Fig. 9.17 - Mass and energy balance astride the first stator array and definition of TIT
•
𝑇𝐼𝑇𝐼𝑆𝑂 : is the temperature resulting from a mass and energy balance that considers all the
cooling flows, regardless of whether they are chargeable or non-chargeable. Obviously,
𝑇𝐼𝑇𝐼𝑆𝑂 < 𝑇𝐼𝑇, 𝑇𝐼𝑇𝐼𝑆𝑂 ~𝑇𝐼𝑇 − 200. It is not representative of a real temperature, but is useful
when testing the turbine because it allows the operating conditions to be defined from a balance
of measurable quantities. In particular, knowing the mechanical power of the turbine, it is
possible to apply the relation:
𝑇𝐼𝑇𝐼𝑆𝑂
𝑊̇𝑇𝑅𝐵 = (𝑚̇𝑎𝑖𝑟 + 𝑚̇𝑓𝑢𝑒𝑙 ) ∫
𝑐𝑝 (𝑇)𝑑𝑇
(9.21)
𝑇𝑂𝑇
that corresponds to a fictitious adiabatic expansion that starting from 𝑇𝐼𝑇𝐼𝑆𝑂 arrives at the
turbine outlet temperature 𝑇𝑂𝑇 of the entire exhaust gas flow. The 𝑇𝑂𝑇 is instead measured
with many measurements along the circular crown at the outlet.
118 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
This parameter is important for calculating the machine performance during testing. Machine
data such as compression ratio, mass of aspirated air and 𝑇𝑂𝑇 are usually available. While the
data regarding 𝐶𝑂𝑇 and 𝑇𝐼𝑇 are never available because it is covered by industrial secrecy
obligations. During testing, the supplier might misrepresent the performance test in his favor
by reducing the cooling flow rates, raising the 𝑇𝐼𝑇 and therefore the efficiency, but showing
the metal surfaces at excessive temperatures that if maintained would damage the reliability of
the turbine. This is why the 𝑇𝐼𝑇𝐼𝑆𝑂 is used as the variable for checking the operating conditions
during testing.
Cooled
turbine
5
6
Adiabatic
turbine
4T
7
3T
5T
a
6
“chargeable”
b
7
TITiso
4
iso
3
a) cooled expansion
b) non-cooled expansion
Fig. 9.18 - Simplified representation of the expansion and introduction of TITISO
This concludes with a few observations on the influence of the cooling flows on the calculation of the
irreversibilities in the cycle:
• The flows circulating in the plant vary significantly both in the compression phase and in the
expansion phase. The difference between the air flow rates depends on the materials used and
the cooling techniques, in addition to the compression ratio of the cycle and the maximum
temperature of the gases.
• The expansion in the portion of cooled turbine (usually the first 4-5 stages) takes place at
decreasing entropy, if reference is made to the main flow that expands. Therefore, the equation
𝑤 = ∆ℎ does not apply, which is applicable only to adiabatic processes.
• In addition to the fluid dynamic irreversibilities common to all the expansion processes in real
machines (identified by the polytropic efficiency due to an adiabatic expansion), many other
irreversibilities take place during the cooled expansion of a gas turbine, which are linked to:
(i) the heat transfer between the main flow and cooling flows, (ii) the mixing of the main flow
and cooling flows (which are introduced into the main flow at temperatures lower than the
flow itself for heat transfer purposes), and (iii) the fluid dynamic losses that occur due to the
introduction of the cooling flows (that have lower speeds) into the main current (at high speed).
119 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The image in Fig. 9.19 illustrates a refined modeling of cooled expansion in a modern gas turbine to
emphasize the complexity of the treatment necessary for proper simulation of these machines. The
expansion refers to the turbine OL-AC and takes place in four stages: the first two cooled, the last two
considered adiabatic. Every line connects the static conditions to the inlet and outlet of each stage, and
is divided considering first the heat transfer effect and then the mixing with the cooling flow. The
broken lines describe transformations that take place as the flow rate (cooled arrays) changes, while
the continuous lines refer to the adiabatic expansion. The last line with increasing pressure represents
the diffuser. For greater detail, the expansion of every single array is divided into another 10 intervals.
Fig. 9.19 - Detailed representation of the numerical model that can be used to calculate a cooled
expansion
Note: consult the two scientific articles for an in-depth study of the topics discussed in this chapter
• Predicting the ultimate performance of advanced power cycles based on very high temperature
gas turbine engines, P. Chiesa, S. Consonni, G. Lozza, E. Macchi
• A thermodynamic analysis of different options to break 60% electric efficiency in combined
cycle power plants, P. Chiesa, E. Macchi
or the book “Turbine a gas e cicli combinati” by G. Lozza, Progetto Leonardo, Bologna.
9.5 INTEGRATING COOLED GAS TURBINES IN COMBINED CYCLES
The high temperature of the exhaust gases released by a gas turbine and the large thermal powers
available make the execution of a steam recovery cycle interesting. The plant takes the name of
combined cycle and is made up of a gas turbine and a steam recovery cycle. Having already separately
120 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
considered all the components that make up a gas cycle and a vapor cycle, now focus should be placed
on the element of junction between the two plants, i.e. the HRSG (Heat Recovery Steam Generator),
where the water is preheated, evaporated and superheated in one or more pressure levels. The
component will be discussed from the viewpoint of the second law analysis, while the student is
referred to other texts for an exhaustive description of these plants and their operation in only an
electrical or cogenerative arrangement.
Fig. 9.20 - Evolution of the efficiency of power production plants. Note the impact of the improved
performance of the gas cycles compared to the vapor cycles in improving the efficiency of combined
cycles.
The major difficulty in a combined cycle plant is matching the cooling curve of the flue gas well with
the heating curve of the working fluid. In view of the size of these plants and the temperature levels,
water is used. Since it is a single molecule fluid, it is not the most advisable for exploiting a source
with variable temperature like the exhaust gases. The use of organic fluids is not advisable in these
application since their use is restricted by their thermal stability to temperatures of about 400°C. As
already seen previously, the use of a single evaporation level does not allow the available heat to be
fully exploited and involves high efficiency losses due to a process of introducing irreversible heat.
This is mainly due to the greater weight of the evaporation section compared to the economizer or the
superheating section.
Generally, the process works at fixed ∆𝑇𝑝𝑝 and this condition usually occurs at the point of saturated
liquid. An exception is the case of post-combustion, in which the 𝑇𝑂𝑇 is so high that the pinch-point
is moved to the economizer inlet. Bear in mind that the recuperative preheating by bleeds from the
turbine for recovery plants is never expedient because it might on the one hand lead to a reduction in
the flow rate in the turbine and hence a lower produced power, and on the other an incomplete
exploitation of the thermal power available. When using a single-level cycle, there are mainly two
causes of irreversibility:
• Heat transfer: it results very high even when greatly reducing the ∆𝑇𝑝𝑝 the ∆𝑇𝑚𝑙𝑛 ;
• Mixing the flue gas with the environment.
Both of these losses can be limited by using recovery cycles with multiple evaporation levels. It is
common to find HRSGs with three evaporation levels plus superheating in conventional plants.
121 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The interaction between vapor cycle and gas cycle depends on the architecture of the plant. The three
main ones are called Frame F, Frame G and Frame H. Three levels are used in Frame F. That with the
lower temperature has a pressure adequate for achieving deaeration, so the deaerator is built into the
top cylindrical casing. The arrangement of the heat transfer banks is selected to best follow the cooling
curve of the flue gas. For example, the lower pressure SH and part of the high pressure ECO are placed
in parallel with the medium pressure ECO. The fuel is preheated by drawing a water flow from the
low pressure ECO. Frame G is similar to Frame F, but the fuel is preheated with hot water drawn from
the medium pressure cylindrical casing at about 200°C. In both configurations there is little interaction
between vapor cycle and gas turbine, and the blades are cooled by open cycle compressed air.
Fig. 9.21 - Configuration with Frame F and Frame G for combined cycles, and OL-AC cooling
diagram for the gas turbine blades
GE proposed integrated cycles in the '90s called Frame H, in which the blades were cooled with vapor
superheated in closed cycle with the aim of reducing the convective heat transfer losses, limiting the
mixing losses and recovering more thermal energy. Using vapor to cool the rotor blades however
introduced enormous difficulties in limiting vapor leakage along the turbine shaft. The Mitsubishi
model ATS was placed on the same line. Its vapor cooling was restricted to just the stators in order to
prevent problems linked to vapor leakage. These latter solutions proved unsuccessful owing because
the current energy market demands that these plants operate in intermediate load (3000-4000h and 50100 start-ups a year), following that does not go well with the need to have long start-up periods for
the vapor portion and less flexible plant control.
122 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
Fig. 9.22 - Configuration with Frame H for combined cycles, and ATS and CL-SC cooling diagram
for the gas turbine blades
9.6 EFFECTS OF REAL GAS IN THE GAS CYCLES
For this purpose, a recuperated closed gas cycle plant that can operate with a fluid affected by real gas
effects since it is near the saturation curve is considered. One example might be a carbon dioxide
closed cycle that has a critical temperature of about 37°C, thus compatible with cooling by ambient
air or water. In some transformations the gas can be considered ideal because it is at high temperature
(expansion 3-4) or at low pressure (expanded side 4-5-1). In the compression phase (1-2) and in the
compressed side (2-6), as the fluid approaches the saturation curve it certainly shows effects of real
gas, that on the one hand lead to decreasing its specific volume and on the other to increasing the
specific heat. This is shown in Fig. 9.23.
It can be considered that as the real gas effects grow, considering cycles with 𝛽 and 𝑇𝐼𝑇 being equal:
• the compression phase requires less power and the final temperature is lower since 𝜃 decreases
as the specific heat increases. The production of entropy and the associated loss of efficiency
decreases since it is linked to the medium compressibility factor of the fluid, which is certainly
lower than one unit.
• If the cycle is not recuperated, the lower end of compression temperature results in a greater
incoming thermal power. As it is at the denominator of the terms Δ𝜂, it allows the irreversibility
losses linked to the expansion and transfer of heat to the environment to be reduced. An
exception to the rule is the loss of introduction of heat since a lower end of compression
temperature involves increased power entering the cycle on the one hand, but also decreased
mean log temperature on the other. The two effects are opposite, but the result is that on the
whole improvement in the efficiency is recorded as the effects of real gas increases.
• Instead, if the cycle is recuperated, the greater the mean pressure of the cycle and the real gas
effect, the greater is the heat capacity of the cold side with respect to that of the hot side. If
∆𝑇𝑝𝑝 being equal is considered, it will be at the cold end of the heat exchanger, and therefore
temperature 𝑇5 will be lower in the case of real gas effects. The heat transfer ∆𝑇𝑚𝑙𝑛 increases
because the ∆𝑇 increases on the hot end of the exchanger, but the recuperated heat decreases.
123 of 124
Second-law analysis of power cycles - Energy Conversion A – V7.0
The increased thermal power introduced has a beneficial effect on both the losses of
compression (already positively influenced by a lower compressibility factor), of expansion
and of heat transfer to the environment (further limited by a lower 𝑇5 )
Also in this case, an improvement in efficiency is noted as the real gas effects increase.
3
3
4
4
6
6
2
2
5
1
5
1
CO2
Fig. 9.23 - Variation of the carbon dioxide closed gas cycle as the saturation curve is approached
due to real gas effects.
9.7 OTHER NUMERICAL EXAMPLES
The student has to supplement all the qualitative considerations put forth in this chapter with the
numerical results obtained during Exercises 3 and 5, dedicated to the second law analysis for
combined cycles at multiple evaporation levels and gas turbines in different configurations.
124 of 124