Indian Journal of Pure & Applied Physics
Vol. 42, January 2004, pp 31-37
Thermo-economic analysis of regenerative heat engines
Santanu Bandyopadhyay
Energy Systems Engineering, Department of Mechanical Engineering,
Indian Institute of Technology, Powai, Mumbai 400 076, India
e-mail: santanub@iitb.ac.in
Received 19 May 2003; accepted 8 September 2003
Analysis of ideal internally reversible regenerative heat engines is presented in this paper, from finite resources point of
view. Thermo-economic performances of a regenerative heat engine depend on the nature of polytropic processes and on the
efficiency of regeneration. The study of the effects of regeneration efficiency, on the resource allocation for optimal
performance of generalized regenerative engine, is also presented in this paper. Thermo-economic performances of an
internally reversible regenerative heat engine, with perfect regeneration, are equivalent to those of internally reversible
Carnot, Otto and Joule-Brayton engines. Concurrent employment of the first and the second laws ensure the optimal
allocation of finite resources simultaneously with minimization of entropy generation. The operating regions both for
operation and design of such regenerative engines are identified from the power-efficiency characteristic. The results are
derived without assuming any particular equation of state associated with the working fluid.
[Keywords: Heat engines; Regenerative heat engines; Reversible heat engines; Reversible regenerative heat engines;
Thermo-economic analysis; Carnot engine; Otto engine; Joule-Braydon engine]
1 Introduction
Carnot efficiency (ηC = 1 – Tmin/Tmax) is the
maximum possible efficiency of a heat engine with
which low-grade thermal energy may be reversibly
transformed into high-grade mechanical energy. Ideal
regenerative heat engines (such as Stirling and
Ericsson heat engines), with perfect regeneration, also
operate with the Carnot efficiency. To achieve Carnot
efficiency, thermal exchanges between the reservoirs
and the working fluid of the engine have to occur
through reversible isothermal processes. These
processes demand infinite heat exchanger surface
area. A heat engine with finite heat exchanger area,
result in zero power production. On the other hand,
the efficiency of an internally reversible Carnot
engine, deliver maximum power, given by ηMP = 1 –
(Tmin/Tmax)1/2 (Ref. 1).
The global need for fuel-efficient and
environmentally viable power production, with
thermodynamic reliability and economy, demands
moderation of the traditional energy conversion
processes with new approaches. Bera and
Bandyopadhyay2 have analyzed the effect of
combustion on the thermoeconomic performances of
Carnot, Otto and Joule-Brayton engines. Classical
reversible heat engines are never realizable in
practice, but the aim is to reach the highest limit of
power production within the constraints of finite
resources. With this end in view, regenerative heat
engine cycles have been studied and their design
philosophies have flourished.
Regenerative heat engines have other benefits
also. Exhaust emissions of a regenerative heat engine
are low and may be easily controlled as the
combustion is isolated from cyclic pressure and
temperature changes experienced by the working
fluid. Continuous complete combustion with 20−80 %
excess air replaces intermittent combustion occurring
in other piston engines. This is because quenching of
the flame does not take place at the ‘cold’ metal
surface. This leads to remarkably low noise levels3.
Regenerative engines are so thermally efficient that
they are prime contender for alternative power unit.
The mean effective pressure and the mechanical
efficiency of a regenerative engine are also quite
high4. Hence, generations of physicists and engineers
of past, focused on these types of engines. In this
paper, internally reversible regenerative heat engines
with imperfect regeneration are discussed and detailed
understanding for optimal design of such engines are
provided.
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INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004
The power-efficiency characteristics of a real
engine help a designer to identify the operating region
for optimal design of the heat engine and to realize
the upper bounds on power production and its
attainable
efficiency.
The
power-efficiency
characteristics for irreversible Carnot cycle,
irreversible Joule-Brayton cycle, and Rankine cycle
are equivalent to each other5. The power-efficiency
characteristics of a regenerative engine are expected
to be a strong function of regeneration efficiency. In
this paper, the power-efficiency characteristics of
regenerative heat engines are studied and the
operating regions are identified. Knowing the
governing equation of state related with any particular
working fluid, different design parameters may easily
be calculated. The results are derived without
assuming any particular equation of state associated
with the working fluid. However, for brevity, thermal
capacity rates are assumed to be independent of
temperature.
2 Regenerative Heat Engines
The ideal thermodynamic cycle corresponding to
regenerative heat engine consists of two isothermal
and two polytropic (of index n) processes. The
temperature-entropy diagram of a typical regenerative
cycle is shown in Fig. 1. Depending on the nature of
the polytropic process, regenerative cycle reduces to
Carnot (for adiabatic process), Ericsson (for isobaric
process) or Stirling (for isometric process) cycles. The
isothermal compression occurs between states 1 and
2. In the isothermal compression, heat is rejected by
the working fluid to the cold reservoir. The isothermal
expansion process, where injection of heat to the
working fluid from the external hot reservoir takes
place, occurs between states 3 and 4. The
temperatures of the hot and the cold reservoirs are
denoted by Tmax and Tmin, respectively. Processes from
states 2 to 3 and 4 to 1 are polytropic (of index n).
Portion of the heat rejected from the polytropic
process 4 to 1 is supplied partly to the process 2 to 3
through a regenerator. Condition of the working fluid
after the regeneration process is denoted by the states
2R and 4R (Fig. 1). Regenerator in the process, 2 to
2R supply heat, and external heat is provided by the
hot reservoir from 2R to 4. Similarly, heat is rejected
from 4R to 2. Therefore, total heat supplied to and
rejected with are given as
Qin = Q2R3 + Q34
… (1)
Qout = Q4R1 + Q12
… (2)
Assuming that the cycle is internally reversible
and all the irreversibility is associated with the finite
driving force of the heat transfer process (that is,
reversibility of the heat engine), the entropy balance
for the regenerative engine may be satisfied.
Q34/Th + Cn ln(Th/Tc) = Q12/Tc + Cn ln(Th/Tc)
… (3)
where Th (= T3 = T4) and Tc (= T1 = T2) are the
highest and the lowest temperatures attained by the
working fluid. In Eq. (3), Cn denotes the thermal
capacity rate of the polytropic process. Knowing the
equation of state that governs the working fluid and
the polytropic index, n, the polytropic thermal
capacity rate Cn, can be determined. For working fluid
obeying ideal gas laws, polytropic thermal capacity
rate may be calculated in terms of thermal capacity
rate at constant volume as Cn = Cv (n - γ)/(n - 1). For
brevity, Cn is assumed to be independent of
temperature. The temperatures of the working fluid
after the regeneration (QR) may be written from the
energy balance of the regenerator.
T4R = Th – QR/Cn
… (4)
T2R = Tc + QR/Cn
… (5)
With the help of Eqs (4) and (5), energy
exchange equations are rewritten as follows:
Fig. 1 ⎯ Temperature-entropy
regenerative heat engine
diagram
of
a
Qin = Cn (Th – Tc) – QR + Q34
… (6)
Qout = Cn (Th – Tc) – QR + Q12
… (7)
33
BANDYOPADHYAY : REGENERATIVE HEAT ENGINES
Heat exchangers are assumed to be countercurrent. Total thermal conductance in the hot side of
the engine is given as
Kh = Cn ln((Tmax – Tc – QR/Cn)/(Tmax - Th))
+ Q34/(Tmax – Th)
… (8)
Similarly, for cold side of the heat engine one
can get
… (9)
The maximum possible regeneration is Cn (Th –
Tc). The regeneration process may be modeled with an
efficiency of (1 - ε). Hence
QR = (1 - ε) Cn (Th – Tc)
… (10)
Defining the non-dimensional quantities such as
th = Th/Tmax, tc = Tc/Tmax, τC = Tmin/Tmax, k = Kh/(Kh +
Kc), s = Cn/(Kh + Kc), q = Q/(Tmax(Kh + Kc)), and w =
W/(Tmax(Kh + Kc)), above equations may be written in
dimensionless form. Note that, 1 ≥ th ≥ tc ≥ τC.
Combining these equations the energy input to and
rejected with are given by
qin = εs(th – tc) + (1 – th)[k – s ln(1 + ε(th – tc)/(1 – th))]
≈ εs(th – tc) + (1 – th)[k – εs (th – tc)/(1 – th))]
= (1 – th) k
… (11)
and similarly
qout = εs(th – tc) + (tc – τC)[(1 – k) – s ln(1
+ ε(th – tc)/( tc – τC))] ≈ (tc – τC) (1 - k)
… (12)
The reversibility [Eq. (3)] of the heat engine
translates to
(1 – th)[k – s ln(1 + ε(th – tc)/(1 – th))]/th = (tc – τC)
[(1 – k) – s ln(1 + ε(th – tc)/( tc – τC))]/tc
…
(14)
Denoting τ = tc/th, the power generated by the
internally reversible engine and its efficiency can be
expressed as
w ≈ (k(1 – k)(1 - τ)(τ - τC) - sε(1 – τ)2
(k + (1 – k)τC))/(τ - sε(1 – τ)2)
… (15)
and
Kc = Cn ln((Th – Tmin – QR/Cn)/(Tc – Tmin))
+ Q12/(Tc – Tmin)
(1 – k)/tc – sε(th – tc)/tc
… (13)
Neglecting higher terms of the expanded
logarithm function, this leads to
(1 – th)k/th – sε(th – tc)/th = (tc – τC)
η ≈ 1 - (k(1 – k)τ(τ - τC) + (1 – k)sετC
(1 – τ)2)/(k (1 – k) (τ - τC) – ksε(1 – τ)2)
… (16)
The power output [Eq. (15)] or the efficiency
[Eq. (16)] of the engine may be maximized to
optimize the performance of the regenerative engine.
Note that these approximations are reasonable, except
for very low working temperature ratio (τ → τC). For
a very low working temperature ratio original
equations have to be solved numerically.
Simultaneous solution of these equations will ensure
the concurrent employment of the first and the second
laws.
3 Power-Efficiency Characteristics
Internally reversible Carnot engine operates
between the limits of thermal ‘short-circuit’ and
thermal ‘open-circuit’ conditions5. No power is
produced either when the engine operates at thermal
short-circuit condition with zero efficiency or the
engine operates at thermal open-circuit condition with
maximum possible efficiency (Carnot efficiency, ηC).
The maximum power corresponds to CurzonAhlborn1 efficiency for internally reversible Carnot
engine.
Unlike
this,
the
power-efficiency
characteristic of a real heat engine corresponds to zero
power and zero efficiency at both the limits of thermal
short-circuit and open-circuit conditions5. The powerefficiency characteristics of most of the real heat
engines are akin to loop-like behaviour. Real heat
engines, with finite resources, exhibit possibility of
operation at maximum power or at maximum
efficiency.
The power-efficiency characteristics of an
internally reversible regenerative engine are shown in
Fig. 2 for different regeneration efficiencies. From
Fig. 2, it may be noted that the efficiency and the
power output both exhibit a maximum. These
34
INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004
material constraints. Therefore, the operating region
may not be fully accessible. In these cases, designer is
expected to select the best possible performance
subject to such constraints. In practice, neither
maximum power nor maximum efficiency can be the
sole objective of an energy conversion device.
However, better understandings of these limiting
cases are essential for the multiple-objective design
approach.
Maximizing power production of the
regenerative engine one can get
τMP = (b + (b2 – (a + b + d)
(b – d – aτC))1/2)/(a + b + d)
Fig. 2 ⎯ Power-efficiency characteristics of a regenerative heat
engine for different regeneration efficiencies
characteristics are similar to those of Carnot heat
engine with heat leak. It may be viewed from Eqs (6)
and (7) where the first term corresponds to equivalent
external heat leak. However, the power-efficiency
characteristic curve with perfect regeneration is
equivalent to that of an internally reversible Carnot
engine without thermal leakage (Fig. 2).
The characteristic curve passes through a
maximum power point (wmax) and a maximum
efficiency point (ηmax). The operating region is the
portion of the power-efficiency curve lying in
between maximum power and maximum efficiency
point. Beyond this range, the power production and
the operating efficiency, both deteriorate. Within the
operating range, as the operating efficiency of the heat
engine decreases from the maximum attainable limit,
the power production increases but the fuel utilization
decreases. These counteractive activities bring the
efficiency of the real engine to lie in between the
maximum power and maximum operating efficiency
points2. Therefore, the operating region may be
defined as the region that simultaneously satisfy both
the inequalities ηmax ≥ η ≥ ηMP and wmax ≥ w ≥ wME.
These can be summarized as a single criterion
1 ≥ τMP ≥ τ ≥ τME ≥ τC
… (17)
This is the criterion for optimal design of a real
heat engine and selection of optimal operating stages
for combined-cycle power generation6.
In some cases, full theoretical power-efficiency
curve cannot be measured due to mechanical or
… (18)
where a = k(1 - k), b = sεa(1 – τC), and d = sε(k
+ (1 – k)τC).
Maximum efficiency of the regenerative engine
corresponds to
2τME (a + f + d) = [2aτC + d(1 + τC) + f - h]
+ ((2aτC + d(1 + τC) + f – h)2
– 4(a + f + d)( aτC2 + dτC - h))1/2
… (19)
where f = sεk(1 – τC) and h = sε(1 – k)τC(1 – τC).
In Fig. 2, loci of maximum power and maximum
efficiency for different regeneration efficiencies are
also shown. From Fig. 2, it may be observed that the
operating region reduces and hence, the flexibility of
the engineer, as the regeneration efficiency
deteriorates.
Variations of maximum power, maximum
efficiency, power at maximum efficiency, and
efficiency at maximum power with the variations in
regeneration inefficiencies are shown in Fig. 3. The
performance of the heat engine deteriorates with
decreasing regeneration performances. Below a
certain regenerator efficiency (ε ≥ 0.2 as shown in
Fig. 3), the operating region collapses for all practical
purpose. It is interesting to note that unlike maximum
power, maximum efficiency, and efficiency at
maximum power, power at maximum efficiency does
not increase monotonically with increasing
regeneration efficiency. This implies that for any
practical regenerator with very high regeneration
efficiency (95% efficiency for the case shown in
Fig. 3), significant power may generate at maximum
efficiency of the engine. Therefore, for most of the
BANDYOPADHYAY : REGENERATIVE HEAT ENGINES
Fig. 3 ⎯ Variation of maximum power, power at maximum
efficiency, maximum efficiency and efficiency at maximum power
for different regeneration efficiencies (k = 0.5, s = 1.0, and τC = 0.3)
practical regenerator with very high regeneration
efficiency, designer has the significant design
flexibility without significant loss of efficiency and
power production from the engine.
Variations of maximum power, maximum
efficiency, power at maximum efficiency and
efficiency at maximum power with the thermal
capacity rate of polytropic process are shown in
Fig. 4. At the Carnot limit (Cn → 0) best performance
is observed, whereas Cn → ∞ indicates the isothermal
process and the engine becomes impossible to
operate. Again it may be observed that power at
maximum efficiency shows a maximum against heat
capacity rate of the polytropic processes. Therefore,
depending upon the efficiency of the regenerator, heat
capacity of the polytropic processes may be adjusted
to obtained significant power from the engine
operating at maximum efficiency.
Perfect regeneration (ε = 0) and/or Carnot heat
engine (s = 0) may be characterized by the relation sε
= 0. In either of these cases Eqs (18) and (19) reduce
to the following expressions.
τMP(sε = 0) = (τC)1/2
… (20)
and
τME(sε = 0) = τC
… (21)
35
Fig. 4 ⎯ Variation of maximum power, power at maximum
efficiency, maximum efficiency and efficiency at maximum
power for different thermal capacity rate of the polytropic process
(k = 0.5, ε = 0.01, and τC = 0.3)
Therefore, isothermal heat engines with perfect
regeneration operate with Curzon-Ahlborn1 efficiency
at maximum power point and with Carnot efficiency
at maximum efficiency point. Note that, internally
reversible Otto and Joule-Brayton engines also
operate with the same efficiency at maximum power
point2.
4 Distribution of Thermal Conductance
Power production or efficiency of a regenerative
heat engine can further be maximized subject to the
distribution of heat exchanger thermal conductance.
The optimum distribution for maximum power output
comes out to be
2kMP = 1 - sε(1 – τMP)(1 – τC)/(τMP – τC)
… (22)
and for maximum efficiency
((τME – τC)kME + τC)2 = τC(τME – sε(1 – τME)2)
… (23)
Eqs (22) and (23) suggest that, 1 ≥ 2kMP(or ME) or
1 - kMP(or ME) ≥ kMP(or ME). That is, in other words,
Kc(at MP or ME) ≥ Kh(at MP or ME)
… (24)
Therefore, the cold side exchanger requires more
thermal conductance. For imperfect regeneration, the
amount of heat rejection increases and a larger cold
side exchanger reduce the external entropy generation
by allowing the heat engine to reject energy at lower
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INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004
temperature. For perfect regeneration ( ε = 0 ) or for
Carnot heat engine ( s = 0 ), Eqs. (22) and (23) reduce
to the following expression.
τMO(sε = 0) = (δ τC)1/2
kMP(or ME)(sε = 0) = ½
kMO(sε = 0) = ½
… (25)
… (30)
and
… (31)
This is the well-known ‘equal distribution of
thermal conductance’ as reported by Bejan7 for
Carnot heat engine at maximum power.
These results are equivalent to the results
reported for internally reversible Carnot, Otto and
Joule-Brayton engines2.
5 Thermo-economics
The efficiency of heat engine operating at
minimum operating cost is always higher than the
efficiency corresponding to the maximum power
condition and less than the maximum efficiency of the
heat engine. At reversible operating limit the
efficiency of the engine is maximum but the engine
does not produce any power2. The cost optimal
operation should remain within the operating region
as indicated by Eq. (17). Let, g1 be the per unit cost of
input energy (proportional to fuel cost), g2 be the per
unit cost of heat rejection (depends cooling utility
cost), and g3 be the per unit selling price of power
produced. The operating cost may be written as
6 Conclusions
In
this
paper,
the
thermo-economic
performances of ideal regenerative heat engines have
been analyzed with finite resource constraints. The
thermo-economic performances of an internally
reversible Carnot, Otto and Joule-Brayton heat
engines are identical to those of a regenerative heat
engine with perfect regeneration. The procedure
concurrently employs the first and the second laws
simultaneously and it ensures optimal allocation of
thermal conductance at the hot and the cold end with
minimization of entropy generation8,9. Importantly,
results are derived without assuming any particular
equation of states associated with the working fluid
but with the assumption that thermal capacity rates
are independent of temperature. The power-efficiency
characteristic of a regenerative heat engine is
analogous to that of any real heat engine and more
closed to Carnot engine with external heat leak. From
the power-efficiency characteristic of the heat engine,
the operating region for optimal design has been
identified. The goal of reaching maximum power
production as well as the highest efficiency for
regenerative heat engine, with proper allocation of
resource, has also been discussed in this paper.
Σ = g1Qin + g2Qout – g3(Qin – Qout) …
… (26)
This may be written in dimension less form,
combining earlier equations, as
σ ≈ (k(1 – k)(τ - δ)(τ - τC) + sε(1 – τ)2
(δk + (1 – k)τC))/(τ - sε(1 – τ)2)
… (27)
with δ=(g3–g1)/(g3+g2) and σ=Σ/(Tmax(Kh+Kc)(g3+g2)).
This is equivalent to Eq. (15). Minimum
operating cost of the engine corresponds to
k(1 – k)(τMO2 - δτC) - sε(1 – τMO2)(δk + (1 – k)τC)
– sε(1 – τMO2)(1 – k)[2(τMO - δτC) –
(1 + τMO)(δ + τC)] = 0
… (28)
and
2kMO=1+sε(1–τMO)2(δ+τC)/((τMO–τC) (τMO–δ)) … (29)
For sε = 0 that is either for Carnot engine or
regenerative heat engine with perfect regeneration the
optimum operating cost corresponds to
Nomenclature
a, b, d, f, h
constants
C
thermal capacity rate
g
cost coefficients
K
thermal conductance
LMTD log mean temperature difference
n
index of polytropic process
Q
heat
q
non-dimensional heat flow
s
non-dimensional capacity rate
t
non-dimensional temperature
T
temperature
w
non-dimensional power
BANDYOPADHYAY : REGENERATIVE HEAT ENGINES
W
power generation
δ
non-dimensional cost coefficient
ε
γ
inefficiency of regeneration
ratio of specific heats
η
k
Σ
efficiency
non-dimensional conductance
operating cost
σ
non-dimensional operating cost
τ
temperature ratio
Subscripts
1,2,3… state points
c
cold
C
Carnot
h
hot
in
inlet
max
maximum
ME
maximum efficiency
min
minimum
MO
MP
n
out
R
v
37
minimum operating cost
maximum power
index of polytropic process
outlet
regeneration
volume
References
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2
3
4
5
6
7
8
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Gordon J M & Huleihil M, J Appl Phy, 72 (1992) 829.
Bandyopadhyay S, Bera N C & Bhattacharyya S, Energy
Convers & Manage, 42 (2001) 359.
Bejan A, Advanced engineering thermodynamics, (Wiley,
New York), 1988.
Bejan A, J Appl Phys, 79 (1996) 1191.
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