Thermo-economic analysis of regenerative heat engines Notes | EduRev

: Thermo-economic analysis of regenerative heat engines Notes | EduRev

 Page 1


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 (?
C
 = 1 – T
min
/T
max
) 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 ?
MP
 = 1 – 
(T
min
/T
max
)
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 
Bandyopadhyay
2
 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 levels
3
. 
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 
high
4
. 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. 
Page 2


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 (?
C
 = 1 – T
min
/T
max
) 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 ?
MP
 = 1 – 
(T
min
/T
max
)
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 
Bandyopadhyay
2
 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 levels
3
. 
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 
high
4
. 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. 
INDIAN J PURE & APPL PHYS,  VOL 42,  JANUARY 2004 
 
 
 
 
 
32 
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 other
5
. 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 T
max
 and T
min
, 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 
Q
in
 = Q
2R3
 + Q
34
 …  (1) 
Q
out
 = Q
4R1
 + Q
12
 …  (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. 
Q
34
/T
h
 + C
n
 ln(T
h
/T
c
) = Q
12
/T
c
 + C
n
 ln(T
h
/T
c
) …  (3) 
where T
h
 (= T
3
 = T
4
) and T
c
 (= T
1
 = T
2
) are the 
highest and the lowest temperatures attained by the 
working fluid. In Eq. (3), C
n
 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 C
n
, 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 C
n
 = C
v
 (n - ?)/(n - 1). For 
brevity, C
n
 is assumed to be independent of 
temperature. The temperatures of the working fluid 
after the regeneration (Q
R
) may be written from the 
energy balance of the regenerator. 
T
4R
 = T
h
 – Q
R
/C
n
 …  (4) 
T
2R
 = T
c
 + Q
R
/C
n
 …  (5) 
With the help of Eqs (4) and (5), energy 
exchange equations are rewritten as follows: 
Q
in
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
34
 …  (6) 
Q
out
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
12
 …  (7) 
 
Fig. 1 ? Temperature-entropy diagram of a 
regenerative heat engine 
Page 3


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 (?
C
 = 1 – T
min
/T
max
) 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 ?
MP
 = 1 – 
(T
min
/T
max
)
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 
Bandyopadhyay
2
 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 levels
3
. 
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 
high
4
. 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. 
INDIAN J PURE & APPL PHYS,  VOL 42,  JANUARY 2004 
 
 
 
 
 
32 
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 other
5
. 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 T
max
 and T
min
, 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 
Q
in
 = Q
2R3
 + Q
34
 …  (1) 
Q
out
 = Q
4R1
 + Q
12
 …  (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. 
Q
34
/T
h
 + C
n
 ln(T
h
/T
c
) = Q
12
/T
c
 + C
n
 ln(T
h
/T
c
) …  (3) 
where T
h
 (= T
3
 = T
4
) and T
c
 (= T
1
 = T
2
) are the 
highest and the lowest temperatures attained by the 
working fluid. In Eq. (3), C
n
 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 C
n
, 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 C
n
 = C
v
 (n - ?)/(n - 1). For 
brevity, C
n
 is assumed to be independent of 
temperature. The temperatures of the working fluid 
after the regeneration (Q
R
) may be written from the 
energy balance of the regenerator. 
T
4R
 = T
h
 – Q
R
/C
n
 …  (4) 
T
2R
 = T
c
 + Q
R
/C
n
 …  (5) 
With the help of Eqs (4) and (5), energy 
exchange equations are rewritten as follows: 
Q
in
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
34
 …  (6) 
Q
out
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
12
 …  (7) 
 
Fig. 1 ? Temperature-entropy diagram of a 
regenerative heat engine 
BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 
 
 
 
 
 
33
Heat exchangers are assumed to be counter-
current. Total thermal conductance in the hot side of 
the engine is given as 
K
h
 = C
n
 ln((T
max
 – T
c
 – Q
R
/C
n
)/(T
max
 - T
h
))  
 + Q
34
/(T
max
 – T
h
) …  (8) 
Similarly, for cold side of the heat engine one 
can get 
K
c
 = C
n
 ln((T
h
 – T
min
 – Q
R
/C
n
)/(T
c
 – T
min
)) 
  + Q
12
/(T
c
 – T
min
) …  (9) 
The maximum possible regeneration is C
n
 (T
h
 – 
T
c
). The regeneration process may be modeled with an 
efficiency of (1 - e). Hence  
Q
R
 = (1 - e) C
n
 (T
h
 – T
c
) …  (10) 
Defining the non-dimensional quantities such as 
t
h
 = T
h
/T
max
, t
c
 = T
c
/T
max
, t
C
 = T
min
/T
max
, k = K
h
/(K
h
 + 
K
c
), s = C
n
/(K
h
 + K
c
), q = Q/(T
max
(K
h
 + K
c
)), and w = 
W/(T
max
(K
h
 + K
c
)), above equations may be written in 
dimensionless form. Note that, 1 = t
h
 = t
c
 = t
C
. 
Combining these equations the energy input to and 
rejected with are given by 
q
in
 = es(t
h
 – t
c
) + (1 – t
h
)[k – s ln(1 + e(t
h
 – t
c
)/(1 – t
h
))] 
 ˜ es(t
h
 – t
c
) + (1 – t
h
)[k – es (t
h
 – t
c
)/(1 – t
h
))]  
= (1 – t
h
) k …  (11) 
and similarly 
q
out
 = es(t
h
 – t
c
) + (t
c
 – t
C
)[(1 – k) – s ln(1  
 + e(t
h
 – t
c
)/( t
c
 – t
C
))] ˜ (t
c
 – t
C
) (1 - k) …  (12) 
The reversibility [Eq. (3)] of the heat engine 
translates to 
(1 – t
h
)[k – s ln(1 + e(t
h
 – t
c
)/(1 – t
h
))]/t
h
 = (t
c
 – t
C
) 
[(1 – k) – s ln(1 + e(t
h
 – t
c
)/( t
c
 – t
C
))]/t
c
  …  (13) 
Neglecting higher terms of the expanded 
logarithm function, this leads to 
(1 – t
h
)k/t
h
 – se(t
h
 – t
c
)/t
h
 = (t
c
 – t
C
) 
(1 – k)/t
c
 – se(t
h
 – t
c
)/t
c  …  
(14) 
Denoting t = t
c
/t
h
, the power generated by the 
internally reversible engine and its efficiency can be 
expressed as 
w ˜ (k(1 – k)(1 - t)( t - t
C
) - se(1 – t)
2
  
(k + (1 – k) t
C
))/( t - se(1 – t)
2
) …  (15) 
and 
? ˜ 1 - (k(1 – k) t( t - t
C
) + (1 – k)se t
C 
(1 – t)
2
)/(k (1 – k) ( t - t
C
) – kse(1 – t)
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 ( t ? t
C
). 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’ conditions
5
. 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, ?
C
). 
The maximum power corresponds to Curzon-
Ahlborn
1
 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 conditions
5
. The power-
efficiency 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 
Page 4


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 (?
C
 = 1 – T
min
/T
max
) 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 ?
MP
 = 1 – 
(T
min
/T
max
)
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 
Bandyopadhyay
2
 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 levels
3
. 
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 
high
4
. 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. 
INDIAN J PURE & APPL PHYS,  VOL 42,  JANUARY 2004 
 
 
 
 
 
32 
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 other
5
. 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 T
max
 and T
min
, 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 
Q
in
 = Q
2R3
 + Q
34
 …  (1) 
Q
out
 = Q
4R1
 + Q
12
 …  (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. 
Q
34
/T
h
 + C
n
 ln(T
h
/T
c
) = Q
12
/T
c
 + C
n
 ln(T
h
/T
c
) …  (3) 
where T
h
 (= T
3
 = T
4
) and T
c
 (= T
1
 = T
2
) are the 
highest and the lowest temperatures attained by the 
working fluid. In Eq. (3), C
n
 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 C
n
, 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 C
n
 = C
v
 (n - ?)/(n - 1). For 
brevity, C
n
 is assumed to be independent of 
temperature. The temperatures of the working fluid 
after the regeneration (Q
R
) may be written from the 
energy balance of the regenerator. 
T
4R
 = T
h
 – Q
R
/C
n
 …  (4) 
T
2R
 = T
c
 + Q
R
/C
n
 …  (5) 
With the help of Eqs (4) and (5), energy 
exchange equations are rewritten as follows: 
Q
in
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
34
 …  (6) 
Q
out
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
12
 …  (7) 
 
Fig. 1 ? Temperature-entropy diagram of a 
regenerative heat engine 
BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 
 
 
 
 
 
33
Heat exchangers are assumed to be counter-
current. Total thermal conductance in the hot side of 
the engine is given as 
K
h
 = C
n
 ln((T
max
 – T
c
 – Q
R
/C
n
)/(T
max
 - T
h
))  
 + Q
34
/(T
max
 – T
h
) …  (8) 
Similarly, for cold side of the heat engine one 
can get 
K
c
 = C
n
 ln((T
h
 – T
min
 – Q
R
/C
n
)/(T
c
 – T
min
)) 
  + Q
12
/(T
c
 – T
min
) …  (9) 
The maximum possible regeneration is C
n
 (T
h
 – 
T
c
). The regeneration process may be modeled with an 
efficiency of (1 - e). Hence  
Q
R
 = (1 - e) C
n
 (T
h
 – T
c
) …  (10) 
Defining the non-dimensional quantities such as 
t
h
 = T
h
/T
max
, t
c
 = T
c
/T
max
, t
C
 = T
min
/T
max
, k = K
h
/(K
h
 + 
K
c
), s = C
n
/(K
h
 + K
c
), q = Q/(T
max
(K
h
 + K
c
)), and w = 
W/(T
max
(K
h
 + K
c
)), above equations may be written in 
dimensionless form. Note that, 1 = t
h
 = t
c
 = t
C
. 
Combining these equations the energy input to and 
rejected with are given by 
q
in
 = es(t
h
 – t
c
) + (1 – t
h
)[k – s ln(1 + e(t
h
 – t
c
)/(1 – t
h
))] 
 ˜ es(t
h
 – t
c
) + (1 – t
h
)[k – es (t
h
 – t
c
)/(1 – t
h
))]  
= (1 – t
h
) k …  (11) 
and similarly 
q
out
 = es(t
h
 – t
c
) + (t
c
 – t
C
)[(1 – k) – s ln(1  
 + e(t
h
 – t
c
)/( t
c
 – t
C
))] ˜ (t
c
 – t
C
) (1 - k) …  (12) 
The reversibility [Eq. (3)] of the heat engine 
translates to 
(1 – t
h
)[k – s ln(1 + e(t
h
 – t
c
)/(1 – t
h
))]/t
h
 = (t
c
 – t
C
) 
[(1 – k) – s ln(1 + e(t
h
 – t
c
)/( t
c
 – t
C
))]/t
c
  …  (13) 
Neglecting higher terms of the expanded 
logarithm function, this leads to 
(1 – t
h
)k/t
h
 – se(t
h
 – t
c
)/t
h
 = (t
c
 – t
C
) 
(1 – k)/t
c
 – se(t
h
 – t
c
)/t
c  …  
(14) 
Denoting t = t
c
/t
h
, the power generated by the 
internally reversible engine and its efficiency can be 
expressed as 
w ˜ (k(1 – k)(1 - t)( t - t
C
) - se(1 – t)
2
  
(k + (1 – k) t
C
))/( t - se(1 – t)
2
) …  (15) 
and 
? ˜ 1 - (k(1 – k) t( t - t
C
) + (1 – k)se t
C 
(1 – t)
2
)/(k (1 – k) ( t - t
C
) – kse(1 – t)
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 ( t ? t
C
). 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’ conditions
5
. 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, ?
C
). 
The maximum power corresponds to Curzon-
Ahlborn
1
 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 conditions
5
. The power-
efficiency 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 
INDIAN J PURE & APPL PHYS,  VOL 42,  JANUARY 2004 
 
 
 
 
 
34 
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 (w
max
) and a maximum 
efficiency point (?
max
). 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 
points
2
. Therefore, the operating region may be 
defined as the region that simultaneously satisfy both 
the inequalities ?
max
 = ? = ?
MP
 and w
max
 = w = w
ME
. 
These can be summarized as a single criterion 
1 = t
MP
 = t = t
ME
 = t
C
 …  (17) 
This is the criterion for optimal design of a real 
heat engine and selection of optimal operating stages 
for combined-cycle power generation
6
. 
In some cases, full theoretical power-efficiency 
curve cannot be measured due to mechanical or 
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 
t
MP
 = (b + (b
2
 – (a + b + d) 
 (b – d – a t
C
))
1/2
)/(a + b + d) …  (18) 
where a = k(1 - k), b = sea(1 – t
C
), and d = se(k 
+ (1 – k) t
C
).  
Maximum efficiency of the regenerative engine 
corresponds to 
2 t
ME
 (a + f + d) = [2a t
C
 + d(1 + t
C
) + f - h]  
+ ((2a t
C
 + d(1 + t
C
) + f – h)
2
  
– 4(a + f + d)( a t
C
2
 + d t
C
 - h))
1/2
 …  (19) 
where f = sek(1 – t
C
) and h = se(1 – k) t
C
(1 – t
C
). 
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 ( e = 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 
 
Fig. 2 ? Power-efficiency characteristics of a regenerative heat 
engine for different regeneration efficiencies 
Page 5


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 (?
C
 = 1 – T
min
/T
max
) 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 ?
MP
 = 1 – 
(T
min
/T
max
)
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 
Bandyopadhyay
2
 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 levels
3
. 
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 
high
4
. 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. 
INDIAN J PURE & APPL PHYS,  VOL 42,  JANUARY 2004 
 
 
 
 
 
32 
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 other
5
. 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 T
max
 and T
min
, 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 
Q
in
 = Q
2R3
 + Q
34
 …  (1) 
Q
out
 = Q
4R1
 + Q
12
 …  (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. 
Q
34
/T
h
 + C
n
 ln(T
h
/T
c
) = Q
12
/T
c
 + C
n
 ln(T
h
/T
c
) …  (3) 
where T
h
 (= T
3
 = T
4
) and T
c
 (= T
1
 = T
2
) are the 
highest and the lowest temperatures attained by the 
working fluid. In Eq. (3), C
n
 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 C
n
, 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 C
n
 = C
v
 (n - ?)/(n - 1). For 
brevity, C
n
 is assumed to be independent of 
temperature. The temperatures of the working fluid 
after the regeneration (Q
R
) may be written from the 
energy balance of the regenerator. 
T
4R
 = T
h
 – Q
R
/C
n
 …  (4) 
T
2R
 = T
c
 + Q
R
/C
n
 …  (5) 
With the help of Eqs (4) and (5), energy 
exchange equations are rewritten as follows: 
Q
in
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
34
 …  (6) 
Q
out
 = C
n
 (T
h
 – T
c
) – Q
R
 + Q
12
 …  (7) 
 
Fig. 1 ? Temperature-entropy diagram of a 
regenerative heat engine 
BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 
 
 
 
 
 
33
Heat exchangers are assumed to be counter-
current. Total thermal conductance in the hot side of 
the engine is given as 
K
h
 = C
n
 ln((T
max
 – T
c
 – Q
R
/C
n
)/(T
max
 - T
h
))  
 + Q
34
/(T
max
 – T
h
) …  (8) 
Similarly, for cold side of the heat engine one 
can get 
K
c
 = C
n
 ln((T
h
 – T
min
 – Q
R
/C
n
)/(T
c
 – T
min
)) 
  + Q
12
/(T
c
 – T
min
) …  (9) 
The maximum possible regeneration is C
n
 (T
h
 – 
T
c
). The regeneration process may be modeled with an 
efficiency of (1 - e). Hence  
Q
R
 = (1 - e) C
n
 (T
h
 – T
c
) …  (10) 
Defining the non-dimensional quantities such as 
t
h
 = T
h
/T
max
, t
c
 = T
c
/T
max
, t
C
 = T
min
/T
max
, k = K
h
/(K
h
 + 
K
c
), s = C
n
/(K
h
 + K
c
), q = Q/(T
max
(K
h
 + K
c
)), and w = 
W/(T
max
(K
h
 + K
c
)), above equations may be written in 
dimensionless form. Note that, 1 = t
h
 = t
c
 = t
C
. 
Combining these equations the energy input to and 
rejected with are given by 
q
in
 = es(t
h
 – t
c
) + (1 – t
h
)[k – s ln(1 + e(t
h
 – t
c
)/(1 – t
h
))] 
 ˜ es(t
h
 – t
c
) + (1 – t
h
)[k – es (t
h
 – t
c
)/(1 – t
h
))]  
= (1 – t
h
) k …  (11) 
and similarly 
q
out
 = es(t
h
 – t
c
) + (t
c
 – t
C
)[(1 – k) – s ln(1  
 + e(t
h
 – t
c
)/( t
c
 – t
C
))] ˜ (t
c
 – t
C
) (1 - k) …  (12) 
The reversibility [Eq. (3)] of the heat engine 
translates to 
(1 – t
h
)[k – s ln(1 + e(t
h
 – t
c
)/(1 – t
h
))]/t
h
 = (t
c
 – t
C
) 
[(1 – k) – s ln(1 + e(t
h
 – t
c
)/( t
c
 – t
C
))]/t
c
  …  (13) 
Neglecting higher terms of the expanded 
logarithm function, this leads to 
(1 – t
h
)k/t
h
 – se(t
h
 – t
c
)/t
h
 = (t
c
 – t
C
) 
(1 – k)/t
c
 – se(t
h
 – t
c
)/t
c  …  
(14) 
Denoting t = t
c
/t
h
, the power generated by the 
internally reversible engine and its efficiency can be 
expressed as 
w ˜ (k(1 – k)(1 - t)( t - t
C
) - se(1 – t)
2
  
(k + (1 – k) t
C
))/( t - se(1 – t)
2
) …  (15) 
and 
? ˜ 1 - (k(1 – k) t( t - t
C
) + (1 – k)se t
C 
(1 – t)
2
)/(k (1 – k) ( t - t
C
) – kse(1 – t)
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 ( t ? t
C
). 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’ conditions
5
. 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, ?
C
). 
The maximum power corresponds to Curzon-
Ahlborn
1
 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 conditions
5
. The power-
efficiency 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 
INDIAN J PURE & APPL PHYS,  VOL 42,  JANUARY 2004 
 
 
 
 
 
34 
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 (w
max
) and a maximum 
efficiency point (?
max
). 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 
points
2
. Therefore, the operating region may be 
defined as the region that simultaneously satisfy both 
the inequalities ?
max
 = ? = ?
MP
 and w
max
 = w = w
ME
. 
These can be summarized as a single criterion 
1 = t
MP
 = t = t
ME
 = t
C
 …  (17) 
This is the criterion for optimal design of a real 
heat engine and selection of optimal operating stages 
for combined-cycle power generation
6
. 
In some cases, full theoretical power-efficiency 
curve cannot be measured due to mechanical or 
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 
t
MP
 = (b + (b
2
 – (a + b + d) 
 (b – d – a t
C
))
1/2
)/(a + b + d) …  (18) 
where a = k(1 - k), b = sea(1 – t
C
), and d = se(k 
+ (1 – k) t
C
).  
Maximum efficiency of the regenerative engine 
corresponds to 
2 t
ME
 (a + f + d) = [2a t
C
 + d(1 + t
C
) + f - h]  
+ ((2a t
C
 + d(1 + t
C
) + f – h)
2
  
– 4(a + f + d)( a t
C
2
 + d t
C
 - h))
1/2
 …  (19) 
where f = sek(1 – t
C
) and h = se(1 – k) t
C
(1 – t
C
). 
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 ( e = 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 
 
Fig. 2 ? Power-efficiency characteristics of a regenerative heat 
engine for different regeneration efficiencies 
BANDYOPADHYAY : REGENERATIVE HEAT ENGINES 
 
 
 
 
 
35
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 (C
n
 ? 0) best performance 
is observed, whereas C
n
 ? 8 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 ( e = 0) and/or Carnot heat 
engine (s = 0) may be characterized by the relation s e 
= 0. In either of these cases Eqs (18) and (19) reduce 
to the following expressions. 
t
MP
(s e = 0) = ( t
C
)
1/2
 …  (20) 
and 
t
ME
(s e = 0) = t
C
 …  (21) 
Therefore, isothermal heat engines with perfect 
regeneration operate with Curzon-Ahlborn
1
 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 
point
2
. 
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 
2k
MP
 = 1 - se(1 – t
MP
)(1 – t
C
)/( t
MP
 – t
C
) …  (22) 
and for maximum efficiency 
(( t
ME
 – t
C
)k
ME
 + t
C
)
2
 = t
C
( t
ME
 – se(1 – t
ME
)
2
)  …  (23) 
Eqs (22) and (23) suggest that, 1 = 2k
MP(or ME)
 or 
1 - k
MP(or ME)
 = k
MP(or ME)
. That is, in other words, 
K
c(at MP or ME)
 = K
h(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 
 
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 t
C 
= 0.3)
 
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, e = 0.01, and t
C 
= 0.3) 
 
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