PERFORMANCE OF A RECIPROCATING
ENDOREVERSIBLE BRAYTON CYCLE WITH
VARIABLE SPECIFIC HEATS OF WORKING FLUID
Yanlin GE, Lingen CHEN, and Fengrui SUN
POSTGRADUATE SCHOOL, NAVAL UNIVERSITY OF ENGINEERING,
WUHAN, P. R. CHINA
Rezumat. Folosind metode ale termodinamicii în timp finit se analizează performantele unui ciclu Brayton standard,
cu aer, ţinând cont de pierderile de căldură şi de faptul ca agentul de lucru are căldura specifică variabilă cu
temperatura. Printr-o analiză detaliată se obţin relaţii între lucrul mecanic produs şi raportul de comprimare, între
randamentul termic al ciclului şi raportul de comprimare, între lucrul mecanic produs şi randamentul termic al ciclului.
De asemeni, sunt analizate efectele pierderilor de căldură şi ale căldurii specifice variabile a agentului de lucru asupra
performanţelor. Rezultatele arată că efectele pierderilor de căldură şi ale căldurii specifice variabile cu temperatura a
agentului de lucru sunt evidente şi că trebuie ţinut cont de ele în practică. Rezultatele acestei lucrări pot furniza sugestii
care să fie luate în considerare în cursul proiectării şi exploatării motoarelor cu ardere internă.
Cuvinte-cheie: termodinamica în timp finit, ciclu Brayton, agent de lucru cu căldură specifică variabilă cu
temperatura, optimizare termodinamică generalizată
Abstract. Performance of an air-standard reciprocating Brayton cycle with heat transfer loss and variable
specific heats of working fluid is analyzed by using finite-time thermodynamics. The relations between the work
output and the compression ratio, between the thermal efficiency and the compression ratio, as well as the
optimal relation between work output and the efficiency of the cycle are derived by detailed numerical examples.
Moreover, the effects of heat transfer loss and variable specific heats of working fluid on the cycle performance
are analyzed. The results show that the effects of heat transfer loss and variable specific heats of working fluid on
the cycle performance are obvious, and they should be considered in practice cycle analysis. The results obtained
in this paper may provide guidance for the design of practice internal combustion engines.
Keywords: finite-time thermodynamics; Brayton cycle; reciprocating cycle; variable specific heats of working
fluid; generalized thermodynamic optimization.
1. INTRODUCTION
A series of achievements have been made since
finite-time thermodynamics was used to analyze and
optimize the performance of real thermodynamic
processes, devices and cycles [1-10]. A number of
authors have examined the finite-time thermodynamic
performance of open [11-13] and closed, simple [11,
12], regenerated [13], intercooled, intercooled and
regenerated, intercooled regenerative modified, and
regenerative reheating Brayton cycles with steady flow.
Qin et al [14] and Ge et al [15] derived the performance characteristics of universal reciprocating cycle
models, which included reciprocating the performance
characteristics of simple Brayton cycle, with heat
transfer loss [14] and with heat transfer and frictionlike term losses [15], respectively. The work mentioned
above was done without considering the variable
specific heats of working fluid, so Rocha-Martinez et al
[16] studied the effect of variable specific heats on Otto
an Diesel cycle performance, but they only took the
experiential expressions of specific heats of working
fluid into the final expressions of cycle power output
and efficiency without considering the effects of variable
specific heats on cycle process equations. While Ghatak
et al [17] analyzed the effects of variable specific heats
and external irreversibilities on Dual cycle performance
with considering the effect of variable specific heats on
cycle process equations. Ge et al [18, 19] studied the
effects of variable specific heats of working fluid on the
performances of Otto cycle with heat transfer loss [18]
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and with the heat transfer and friction losses [19], respectively. This paper will study the effects of the variable
specific heats of working fluid and heat transfer loss on
the performance of reciprocating simple Brayton cycle.
2. CYCLE MODEL
An air standard reciprocating Brayton cycle model
is shown in figure 1. The compression process is an
isentropic process 1 → 2 ; the heat addition is an
isobaric process 2 → 3 ; the expansion process is an
isentropic process 3 → 4 ; and the heat rejection is an
isobaric process 4 → 1 .
Fig. 1. T-s diagram for a Brayton cycle.
In practice cycle, specific heats of the working
fluid are variable and these variations will have great
influence on the performance of the cycle. According
to Refs. [17-19], it can be supposed that the specific
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Yanlin GE, Lingen CHEN, Fengrui SUN
heats of the working fluid are functions of temperature
alone and over the temperature ranges generally
encountered for gases in heat engines (300K to 2200K)
the specific heat curve is nearly a straight line which
may be closely approximated in the following forms:
(1)
C pm = a p + k1T
C vm = bv + k1T
(2)
where a p , bv and k1 are constants, C pm and C vm are
molar specific heats with constant pressure and volume,
respectively. Accordingly, one has
R = C pm − C vm = a p − bv
(3)
where R is the gas constant of the working fluid.
The heat added to the working fluid during process
2 → 3 is
T2
T3
T3
T2
Qin = M ∫ C pm dT = M ∫ (a p + k1T )dT =
(4)
= M [a p (T3 − T2 ) + 0.5k1 (T32 − T22 )]
where M is the molar number of the working fluid.
The heat rejected by the working fluid during
process 4 → 1 is
T4
T4
T1
T1
Qout = M ∫ C pm dT = M ∫ (a p + k1T ) dT =
(5)
The work output of the cycle is
+0.5k1 (T32 + T12 − T22 − T42 )]
adiabatic exponent k = C pm C vm will vary with temperature as well. Therefore, the equation often used in
reversible adiabatic process with constant k can’t be
used in reversible adiabatic process with variable k.
However, according to Refs. [17-19], a suitable
engineering approximation for reversible adiabatic
process with variable k can be made, i.e. this process
can be broke up into infinitesimally small processes,
and for each of these processes, adiabatic exponent k
can be regarded as constant. For example, for any
reversible adiabatic process between states i and j can
be regarded as consisting of numerous infinitesimally
small process with constant k. For any of these
processes, when small changes in temperature dT and
in volume dV of the working fluid take place, the
equation for reversible adiabatic process with variable k
can be written as follows
20
= (T3T1 T2T4 γ ) . Therefore equation (11) becomes
k1 (T3 − T4 ) + bv ln(T3 T4 ) −
− R ln(T2T4 T1T 3 ) = R ln γ
(7)
(12)
For an ideal reciprocating Brayton cycle model,
there are no losses. However, for a real reciprocating
Brayton cycle, heat transfer irreversibility between
working fluid and the cylinder wall is not negligible.
One can assume that the heat loss through the cylinder
wall is proportional to average temperature of both the
working fluid and the cylinder wall and that the wall
temperature is constant T0 . If the released heat by
combustion for one molar working fluid is A1 , the heat
leakage coefficient of the cylinder wall is B1 , one has
the heat added to the working fluid by combustion in
the following linear relation [19-22]
Qin = M { A1 − B1[(T2 + T3 ) / 2 − T0 ]}
(13)
= M [ A − B(T2 + T3 )]
where A = A1 + B1T0 and B = B1 / 2 are two constants
related to combustion and heat transfer.
The efficiency of the cycle is
ηbr =
(6)
Since C pm and C vm are dependent on temperature,
TV k −1 = (T + dT )(V + dV )k −1
In addition, for cycle processes, one has (V3 V4 ) =
= M [ A1 + B1T0 − (T2 + T3 ) B1 / 2] =
= M [a p (T4 − T1 ) + 0.5k1 (T42 − T12 )]
Wbr = Qin − Qout = M [a p (T3 − T2 − T4 + T1 ) +
Therefore, equations for processes 1 → 2 and
3 → 4 are as follows
k1 (T2 − T1 ) + bv ln(T2 T1 ) = R ln γ
(10)
k1 (T3 − T4 ) + bv ln(T3 T4 ) = − R ln(V3 V4 ) (11)
=1−
Wbr
=
Qin
a p (T4 − T1 ) + 0.5k1 (T42 − T12 )
(14)
a p (T3 − T2 ) + 0.5k1 (T32 − T22 )
When γ and T1 are given, T2 can be obtained from
equation (10); then, substituting equation (4) into equation
(13) yields T3 ; and at last, T4 can be worked out by
equation (12). Substituting T1 , T2 , T3 and T4 into
equations (6) and (14) yields the work and efficiency.
Then, the relations between the work output and the
compression ratio, between the thermal efficiency and the
compression ratio, as well as the optimal relation between
work output and the efficiency of the cycle can be derived.
3. NUMERICAL EXAMPLES
AND DISCUSSION
According to Refs. [17-19], the following parameters
are used in the calculations:
A = 60000 − 70000 J / mol , B = 20 − 30 J / mol.K ,
a P = 28.182 − 32.182 J / mol.K , T1 = 350 K ,
From equation (7), one gets
k1 (T j − Ti ) + bv ln(T j Ti ) = − R ln(V j Vi )
(8)
The compression ratio is defined as
γ = V1 V2
M = 1.57 × 10 －5 kmol ,
k1 = 0.003844 − 0.009844 J / mol.K 2 .
(9)
Using the above constants and range of parameters,
the characteristic curves of Wbr – γ, ηbr – γ and Wbr –ηbr
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PERFORMANCE OF A RECIPROCATING ENDOREVERSIBLE BRAYTON CYCLE WITH VARIABLE SPECIFIC HEATS
can be plotted. A and B are two constants related to
combustion and heat transfer. Where A reflects the
magnitude of heating value of fuel and the heat transfer
loss, the bigger A is, the higher heating value is, and the
less of the heat transfer loss will be. While B reflects
the magnitude of heat transfer loss, the bigger B is, the
more heat transfer loss is. Figures 2-7 show the effects
of the heating value and the heat transfer loss on the
cycle performance. One can see that the work versus
compression ratio characteristic and the work versus
efficiency characteristic are parabolic-like curves, i.e.,
there is a maximum work output during the range of
compression ratio γ. Heat transfer loss has great effect
on cycle work output, while has a little effect on cycle
efficiency, but the potential is that cycle efficiency
will increase with the increase in heat transfer loss.
From figures 2-7, one can see that when heat transfer
loss decreases, i.e. A increases or B decreases, the
maximum work output and the corresponding compression ratio, as well as the efficiency at maximum
work output will increase. In this case, if A increases
by about 17%, the maximum work output of the cycle
will increase by about 38.5%, the efficiency at the
maximum work output point will increase by about
11.1%. If B decreases by about 33%, the maximum
work output will increase by about 94.3%, and the
efficiency at the maximum work output will increase
by about 27.5%.
Fig. 2. The influences of A on cycle work output.
Fig. 3. The influences of A on cycle efficiency.
Fig. 4. The influences of A on cycle work output versus efficiency.
Fig. 5. The influences of B on cycle work output.
Fig. 6. The influences of B n cycle efficiency.
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Fig. 7. The influences of B on cycle work output versus efficiency.
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Yanlin GE, Lingen CHEN, Fengrui SUN
Figures 8-13 show the effects of the variable
specific heats of working fluid on cycle performance.
Figures 8-10 reflect the effect of ap on cycle
performance. One can see that the maximum work
output and the compression ratio at the maximum
work output will increase with the increase of ap,
while the efficiency at the same compression ratio
will decrease with the increase of ap. And ap has a
little effect on the efficiency at the maximum work
output point.
Fig. 8. The influences of ap on cycle work output.
Fig. 10. The influences of ap on cycle work output versus efficiency.
Fig. 12. The influences of k1 on cycle efficiency.
22
Fig. 9. The influences of ap on cycle efficiency.
Fig. 11. The influences of k1 on cycle work output.
Fig. 13. The influences of k1 on cycle work output
versus efficiency.
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PERFORMANCE OF A RECIPROCATING ENDOREVERSIBLE BRAYTON CYCLE WITH VARIABLE SPECIFIC HEATS
k1 reflects the variation degree of specific heat with
temperature. The bigger k1 is, the more acutely
variation degree of the specific heat with temperature is.
Figures 11-13 show the effect of k1 on cycle
performance. It can be found that with the increase of k1,
Wbr − γ curves move toward right, the maximum work
output decreases, while the compression ratio at the
maximum work output increases. Therefore, the effect
of k1 on the performance of the cycle is related to
compression ratio γ . If γ less than certain value, the
increase of k1 will make the work output less, on the
contrast, if γ exceed certain value, the increase of k1
will make the work output bigger. One also can see that
the maximum work output, the efficiency at the same
compression ratio, as well as the efficiency at the
maximum work output point will decrease with the
increase of k1, while the compression ratio at the
maximum work output point will increase with the
increase of k1. For this case, when k1 increases by about
156%, the maximum work output will decrease by
about 9.8%, the efficiency at the maximum work output
point will decrease by about 10.2%
According to above analysis, it can be found that
the effects of the heat transfer losses and the variable
specific heats of the working fluid on the cycle performance are obvious, and they should be considered in
practice cycle analysis in order to make the cycle
model be more close to practice.
4. CONCLUSION
In this paper, an air standard reciprocating simple
Brayton cycle model with the considerations of the heat
transfer and the variable specific heats of working fluid
was presented. The performance characteristic of the
cycle was obtained by detailed numerical examples.
The results show that the effects of the heat transfer and
variable specific heats of working fluid on the cycle
performance are obvious. Therefore, it is necessary
and important to take into account the features of the
variable specific heats of working fluid and heat
transfer loss in practice cycle analysis. The results
obtained in this paper may provide guidelines for the
design of practice internal combustion engines.
Acknowledgments
This paper is supported by Program for New Century Excellent
Talents in University of P. R. China (Project No. NCET-041006) and The Foundation for the Author of National Excellent
Doctoral Dissertation of P. R. China (Project No. 200136)
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