Performance analysis of an endoreversible Heat engine with thermal
reservoir of Finite Heat Capacitance at maximum power density
conditions
GOVIND MAHESHWARI & A.I. KHANDWAWALA
Department of Mechanical Engineering,
Devi Ahilya University,
Institute of Engineering & Technology, Khandwa Road, Indore-452017
INDIA.
Abstract : - This paper presents the finite time thermodynamic optimization based on the maximum power density
criterion for an endoreversible Carnot heat engine model coupled to variable temperature heat reservoirs.
Expressions for the power density and optimal efficiency at maximum power density conditions are derived with
heat resistance losses in the hot and cold side heat exchangers. The obtained results are compared with those of
using the maximum power criterion. The influences of some design parameters on the maximum power density are
provided by numerical examples, and the advantages and disadvantages of maximum power density are analyzed.
The power plant design with maximum power density leads to a higher efficiency and smaller size.
Key-Words: - Finite time thermodynamics, endoreversible, power density, hot and cold side heat exchangers.
at maximum power(MP) conditions. The power
density objective was also applied to an ideal
reversible Ericsson cycle[5] and an Atkinson
cycle[6] free of any irreversibility. Sahin et al[7]
applied the MPD technique to the endoreversible
Carnot heat engine which can be considered as a
theoretical comparison standard for all real heat
engines in finite time thermodynamics.
In all the above references [3-7],
researchers applied MPD techniques to the heat
engines having thermal reservoirs of infinite heat
capacity. However, for practical applications it is
very important to investigate the performance of
heat engine when the effect of finite thermal
capacitance of thermal reservoirs is taken into
account. In this paper we are analyzing the
endoreversible Carnot heat engine performance
using the power density objective with
considerations of the heat transfer irreversibility
in the hot and cold side heat exchangers and the
effects of finite thermal capacitance rates. The
results are compared with those of obtained by
using the maximum power criterion.
1 Introduction
Since finite time thermodynamics (FTT) or
entropy generation minimization was advanced,
much work has been carried out for the
performance analysis and optimization of finite
time processes and finite size devices[1-2]. In
these studies, the power output, thermal
efficiency, entropy generation and the ecological
benefits are chosen for the optimization criteria.
However, the performance analyses based on the
above optimization criteria do not take the
effects of engine sizes related to the investment
cost into account. In order to include the effects
of engine size in the performance analysis, Sahin
et al introduced the maximum power density
(MPD) as a new optimization criterion. Using
the MPD criterion, they investigated optimal
performance conditions for reversible[3] and
irreversible[4] non regenerative Joule-Brayton
heat engines. In their study, they maximized the
power density (the ratio of power to the
maximum specific volume in the cycle) and
found design parameters at MPD conditions,
which led to smaller and more efficient JouleBrayton heat engines than those engines working
1
W  Q H  Q L   H C H TH 1  TW    L C L TC  TL1  (6)
2 System Description and Analysis
A Heat engine working between two thermal
reservoirs of finite heat capacitance is shown in
Figure 1. The processes 1-2 and 3-4 are
reversible adiabatic, and processes 2-3 and 4-1
are isothermal. Assuming that the heat
exchangers
are
counterflow,
the
heat
conductance (product of heat transfer surface
area and heat transfer coefficient) of the hot and
cold side heat exchangers are UH and UL. The
thermal capacity rate (product of mass flow rate
and specific heat) of the hot side heat reservoir is
CH, and the inlet and outlet temperature of the
heating fluid are TH1 and TH2, respectively. The
thermal capacity rate of the cold side heat
reservoir is CL, and the inlet and outlet
temperatures of the cooling fluid are TL1 and TL2,
respectively. Temperatures of hot and cold side
working fluid of the heat engine are TW and TC
respectively. The T-S diagram of the heat engine
is shown in Figure 2.
The rate of heat flow(QH) from high
temperature heat source to the system is given
by:
QH  C H (TH 1  TH 2 )   H C H (TH 1  TW )
The power density defined as the ratio of power
produced to the maximum volume in the cycle
then takes the form:
 C T  T    L C L TC  TL1 
(7)
Wd  H H H 1 W
V2
The second law requires that:
QH QL
(8)

TW
TC
 H C H TH 1  TW   L C L TC  TL1 
or,
(9)

TW
TC
Assuming an ideal gas, the maximum volume in
the cycle V2 can be written as
mRTC
(10)
V2 
p min
where m is the mass of the working fluid and R
is the ideal gas constant. In the analysis, the
minimum pressure ( p min ) in the cycle is taken to
be constant[3]. The power density then becomes
 p   C T  TW    L C L TC  TL1  (11)
Wd   min  H H H 1

TC
 mR 

One can maximize the power density given in
Eq.(11) with respect to TW and TC by using
Eq.(9). The results are:
(1)
*
Tw
Similarly, the rate of heat flow(QL) from system
to the low temperature heat sink is


1   H
 L




 H


 L
2
 CH

C
 L
 CH

 C
 L




2
 TH 1    L

 
 T 
 L1   H
 1  CH


T  C
 L1  L
2
 CL

 C
 H
  L

 
  H
 1  H


 T  
 L1  L




2
 CL

 C
 H
 CH

C
 L
2
 TL1 


 T  
 H 1 
 1

 T 
 L1
 H CH   LCL  H CH TH 1   LCLTL1 
 H 2CH 2TH 1
 H CH   LCL 3  H CH TH 1   LCLTL1 
 l 4CL 4TL14TH 1
(12)
(2)
QL  C L (TL 2  TL1 )   L C L (TC  TL1 )
where H and L are, respectively, the
effectiveness of the hot and cold side heat
exchangers, defined as:
(3)
 H  1  exp N H 
(4)
 L  1  exp N L 
The number of heat transfer units, NH and NL are
based on the minimum thermal capacitance rates,
that is:
U
U
(5)
NH  H , NL  L
CH
CL
The power produced by the engine according to
the first law is:
TC
*

1   L
 H





2
 CL

C
 H
 TL1    L  CL    L  CL  TL1 
 H CH   LCL  H CH TH 1   LCLTL1 

 

 



 T      C      C  T  
 H CH TH 1
 H 1   H  H   H  H  H 1 
2
2
  L  CL  1   L  CL  1   L   CL  1
1
1




 

2
 
 


  H  CH  TH 1   H  CH  TL1   H   CH  TH 1 TL1 TH 1




2
(13)
Substituting Eqs. (12) and (13) into Eq(11), the
maximum power density (MPD) can be found as



*
*
 p min   H C H TH 1  TW   L C L TC  TL1
Wd max  

*
TC
 mR 
(14)
The thermal efficiency of heat engine is given
as:
2


TC
(15)
TW
Using Eqs. (12) and (13) into Eq(15), the
efficiency at MPD becomes
*
TC
 m pd  1  *
TW
  1
 mpd 
 H  L C H C L   H 2 C H 2 2   H 2 C H 2 [ L 2 C L 2   H  L C H C L   H  L C H C L   H 2 C H 2 ]
 H C H  ( L C L   H C H   H C H  )
(16)
TH 1
is the cycle heat reservoir inlet
TL1
temperature ratio.
Similarly, one can maximize the power output
given in Eq.(6) with respect to TW and TC using
the condition given in Eq(9). The results are:
where  
Twmp 
 2 C 2T    C C T   2 C 2T T ( C   C ) 2  C T
H L H L L1
H
H
H 1 L1 H H
L L
 L L: L1
 H H H 1
( H C H   L C L )  H 2 C H 2TH 1TL1 ( H C H   L C L ) 2
(17)
Tcm p
 2 C 2T    C C T   2 C 2T T ( C   C ) 2 
H L H L L1
H
H
H 1 L1 H H
L L
 L L: L1


( H C H   L C L ) 2
(18)
Substituting Eqs(17) and (18) into Eq(6), the
maximum power can be found as
Wmax   H C H [TH 1  Twmp ]   L C L [Tcmp  TL1 ] (19)
The thermal efficiency at maximum power
is:
 mp 
 H  L C H C L   H 2 C H 2   H 2 C H 2 ( H C H   L C L ) 2
 H C H  ( H C H   L C L )
(20)
3 Results and Discussion
To see the advantages and disadvantages of
maximum power density design, detailed
numerical analysis are provided and are
compared with those for the maximum power
objective. During the variation of any one
3
parameter, all other parameters are assumed to
be constant as given below:
TH1=1000K , TL1 = 160K, CH = CL = 1 kW/K
and H= L = 0.9 .
The variations of the normalized power density
(Wd/Wdmax) and the normalized power
(W/Wmax) with respect to thermal efficiency are
shown in Figure 3. As one can see, the thermal
efficiency at maximum power density conditions
(mpd) is always greater than the thermal
efficiency at maximum power conditions (mp).
Variations of the two efficiencies (mp and mpd)
with  and H(=L) are shown in Figures 4 and 5
respectively. The following comments can be
summarized from the observations of these
figures: (i) For the chosen values of the
parameters, the thermal efficiency at MPD(mpd)
is greater than the thermal efficiency at MP
conditions (mp); (ii)The thermal efficiency
advantage of the MPD conditions with respect to
MP conditions (= mpd - mp) increases with
the increase in  (cycle heat reservoir inlet
temperature ratio). eg. for  = 2, mp = 0.293 and
mpd = 0.317, and for  = 5, mp = 0.553 and mpd
= 0.636. The following observations can also be
seen by evaluating Eqs. (16) and (20);
(iii)Thermal efficiency at maximum power
density conditions increases with an increase in
, H and L while thermal efficiency at
maximum power conditions increases with an
increase of  but it is independent of H and L.
The variations of the dimensionless power
at MPD conditions (Pdmax= Wdmax/(pmin/mR)HCH )
and dimensionless power at MP conditions
(Pmax=Wmax/HCHTH1) with  are shown in Figure
6. As can be seen from the figure, Pmax is always
lower than Pdmax, that is if the design parameters
are selected at MPD conditions instead of MP
conditions, the thermal efficiency increases as
much as by = mpd - mp, and the power
increases by P= Pdmax - Pmax. On the other hand,
as  decreases, mpd and mp, Pdmax and Pmax
decreases and get closer to each other and they
become zero when  = 0. In order to obtain
positive performance in terms of thermal
efficiency and power for both the MPD and MP
conditions, it is necessary that  should be
greater than zero.
The size of a heat engine can be
characterized by the maximum volume in the
cycle, ie. V2 . The ratio of the maximum volume
at MPD to the one at MP can be written as:V max mpd
Vmax mp


1   L
H



 
 L
  H

2



2
 CL

 CH
 CL

 CH
2
 1  L

 
  H
2
 1  L

 
  H
 C L

 C H
 C L

 C H
 1 
1   
  
1
 

conditions, engine sizes will be smaller but will
require higher maximum pressure than the one
operating at MP.
4 Conclusions
The performance of an endoreversible Carnot
Heat engine cycle coupled to variable
temperature heat reservoirs with heat transfer
irreversibility in the hot and cold side heat
transfer irreversibility in the hot and cold side
heat exchangers was analyzed by taking the
power density as the optimization objective.
Comparisons between maximum power density
performance and maximum power performance
were made. The maximum power density design
has the advantages of smaller size and higher
efficiency. All of the analytical results with H =
L =1 and CH = CL   of this paper replicate
the results given by Sahin et al[7] in which
Carnot cycle without any irreversibility was
examined. The future step of this paper will be
the optimization of Carnot heat engine cycle
with internal irreversibilities. That will be the
subject of another article.
 L C L   H C H  L C L   H C H  
 L C L   H C H 
 H 2 C H 2
 H 2 C H 2
2


 H CH
 1 

  LC L   H C H 

(21)
Similarly, the ratio of the maximum
pressures at MPD ((pmax)mpd ) and maximum
pressures at MP((pmax)mp) conditions is given
by :
 pmax mpd
 pmax mp
 LCL   H CH 


H CH

  H CH   LCL 1    


 L 2 CL 2
   LCL   H CH  LCL   H CH   
 H CH


 2 C 2

   LCL   H CH  LCL   H CH    L L   LCL    LCL   H CH  
  H CH

(22)
Variations of (Vmax)mpd / (Vmax)mp and (pmax)mpd /
(pmax)mp are plotted with respect to  in Figure 7.
As can be seen from the figures, the ratio of the
maximum volumes in the cycle is less than unity
and the ratio of the maximum pressures in the
cycle is greater than unity. As a result, at MPD
4
Wdmax =
Nomenclature
CH
=
Thermal capacity rate of the hot
side heat reservoir. (kW/K).
CL
=
Thermal capacity rate of the cold
side heat reservoir. (kW/K)
Pmax =
Dimensionless maximum power
output.
Pdmax =
Dimensionless maximum power
density.
QH
=
Heat transfer rate from high
temperature reservoir to the
system. (kW).
QL
=
Heat transfer rate from cold
working fluid to low temp.
reservoir. (kW).
TC
=
Temperature of working
substance in low temperature
reservoir side. (K)
TCmp =
Temperature of working
substance in low temperature
reservoir side at max power. (K)
*
TC
=
Temperature of working
substance in low temperature
reservoir side at maximum power
density. (K)
TH1
=
Inlet temperature of the heat
source fluid. (K).
TL1
=
Inlet temperature of the heat sink
fluid. (K).
TWmp =
Temperature of working
substance in high temperature
reservoir side at maximum power.
(K).
TW* =
Temperature of working
substance in high temperature
reservoir side at maximum power
density . (K).
UH
=
Heat conductance of the hot side
heat exchanger. (kW/K).
UL
=
Heat conductance of the cold side
heat exchanger. (kW/K).
Wmax =
Maximum power output from the
heat engine. (kW).
Wd
=
Power density of the heat engine.
(kW/m3).
H
=
L
=

=
mp
=
mpd
=

=
Maximum power density of the
heat engine. (kW/m3).
Effectiveness of source side heat
exchanger.
Effectiveness of sink side heat
exchanger.
Thermal efficiency of heat
engine.
Thermal efficiency of heat engine
at maximum power conditions.
Thermal efficiency of heat engine
at maximum power density
conditions.
Inlet heat reservoir temperature
ratio.
References :
[1] Curzon FL & Ahlborn B, Efficiency of
Carnot engine at maximum power output,
American Journal of Physics, Vol.43, No.1,
1975, pp.22-24.
[2] Wu C, Power optimization of an
endoreversible Brayton gas turbine heat engine,
Energy Convers. Mgmt, Vol.31, No.6, 1991,
p.561-565.
[3] Sahin B, Kodal A & Yavuz H, Efficiency of
Joule Brayton engine at maximum power density
conditions, J Phy D: App Physics, Vol.28, No.8,
1995, pp.1309-1312.
[4] Sahin B, Kodal A & Yavuz H, Maximum
power density analysis of an irreversible Joule
Brayton Engine, J Phy D: App Physics, Vol.29,
No.8, 1996, pp.1162-1171.
[5] Erbay LB, Sisman A & Yavuz H, Analysis of
Ericsson cycle at maximum power density
conditions, ECOS’96 Stockholm, 1996, pp.175178.
[6] Chen L, Lin J & Sun F, Efficiency of
Atkinson cycle at maximum power density
conditions, Energy Convers Mgmt, Vol.9,
No.3/4, 1998, pp.337-341.
[7] Sahin B & Kodal A, Maximum power
density analysis for irreversible combined Carnot
cycle, J Phy D: Appl Phys, Vol.22, No.32, 1999,
pp.2958-2963
5
Temperature (T)
Heat Source
TH1
TH2
TH1
TH2
4
Warm working fluid
1
TW
QH
Heat engine cycle
TW
TC
W
3
Heat Engine
Cold working fluid
2
TL2
TL1
TC
Entropy (S)
Figure 2 : - T-S diagram of an endoreversible heat engine cycle.
QL
TL2
TL1
Heat Sink
Figure 1 : - Schematic diagram of a Carnot heat engine.
6
1.2
W/Wmax
Wd/Wdmax
W/Wmax, Wd/Wdmax
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0

Figure 3 : - Variations of normalized power density and normalized power with thermal efficiency.
0.7
mpd
mp
0.6
mpd, mp
0.5
0.4
0.3
0.2
0.1
0.0
1
2
3
4
5
6

Figure 4 : - Variations of efficiency at maximum power density and efficiency at maximum power
conditions with heat reservoir inlet temperature ratio.
7
0.70
mp
0.68
mpd
mpd, mp
0.66
0.64
0.62
0.60
0.58
0.2
0.4
0.6
0.8
1.0
H( = L)
Figure 5 : - Variations of thermal efficiency at maximum power density and efficiency at maximum
power conditions with effectiveness of heat exchangers.
Pmax
Pdamx
Pmax, Pdmax
0.6
0.4
0.2
0.0
1
2
3
4
5
6

Figure 6 : - Variations of dimensionless maximum power and dimensionless maximum power density
with heat reservoir temperature ratio.
8
1.00
(pmax)mpd / (pmax)mp
0.98
1.06
0.96
0.94
1.04
0.92
0.90
1.02
0.88
0.86
1.00
1
2
3
4
5

Figure 7 : - Variations of ratio of maximum volume at maximum power density to maximum volume
at maximum power conditions and ratio of maximum pressure at maximum power density
to maximum pressure at maximum power conditions with heat reservoir inlet temperature
ratio
9
(pmax)mpd / (pmax)mp
(Vmax)mpd / (Vmax)mp
1.08
(Vmax)mpd / (Vmax)mp