THERMODYNAMICS
Revista Mexicana de Fı́sica S 59 (1) 224–229
FEBRUARY 2013
Qualitative and quantitative optimization of a standard irreversible Brayton cycle
G. Aragón-Camarasa
CVGG, School of Computing Science, University of Glasgow,
17 Lilybank Gardens, Glasgow G12 8QQ. Scotland, UK.
G. Aragón-González∗ , † A. Canales-Palma, A. León-Galicia.
PDPA. UAM- Azcapotzalco.
Av. San Pablo # 180. Col. Reynosa. Azcapotzalco, 02200, D.F. Teléfono y FAX: (55) 5318-9057.
Received 30 de junio de 2011; accepted 30 de noviembre de 2011
A Brayton cycle with, external and internal irreversibilities, is analyzed. Optimization of the dimensionless work with respect to two
parameters: the isentropic temperatures ratio and the allocation ratio of the heat exchangers, is performed. The qualitative and asymptotic
behaviour of the coupled optimal analytic expressions obtained are presented. Using realistic numerical values of the isentropic efficiencies,
optimal analytic expressions for both optimal parameters and for the efficiency to maximum work, by an approximation, are obtained
(quantitative behaviour). This approximation, from a thermodynamic point of view, maintains, combines, perturbs and extends the optimal
operation conditions of the non-isentropic and endoreversible models. A remarkable correlation between the optimal effectiveness from the
heat exchangers from hot and cold sides, is obtained, which has not appeared and is different to the one used in the relevant literature.
Keywords: Brayton cycle; effectiveness; qualitative and quantitative optimization; work.
Se analiza un ciclo Brayton con irreversibilidades externas e internas. Se realiza la optimización del trabajo adimensional respecto a dos
parámetros: razón de las temperaturas isentrópicas y la razón de las dimensiones de los intercambiadores de calores. Con base en valores
numéricos realı́sticos de eficiencias isentrópicas, se obtienen, mediante una aproximación, las expresiones analı́ticas para ambos parámetros
óptimos y para la eficiencia para trabajo máximo. Desde un punto de vista termodinámico, esta aproximación mantiene, combina, perturba
y extiende las condiciones de operación óptimas de los modelos no-isentrópico y endoreversible. Se obtiene una correlación notable de las
efectividades óptimas de los intercambiadores de calor de los lados caliente y frı́?o, la cual no ha aparecido y es diferente a la usada en la
literatura relevante.
Descriptores: Ciclo Brayton; comportamiento cualitativo y cuantitativo; efectividades; trabajo.
PACS: 44.60.+k; 44-90.+c
1. Introduction
The air standard Brayton cycle has been used as a model
for the gas turbine heat engine. Currently, researchers
have renewed the rationale of Brayton-like cycles by considering more practical aspects of the entropy generation,
power, power-density, efficiency optimization and so on.
Bejan [1] showed that, if the entropy generation is minimum, the efficiency corresponding to the endoreversible
model the Chambadal-Novikov-Curzon-Ahlborn (CNCA)
efficiency [2] and the optimal allocation (size) of hot- and
cold-side heat exchangers is balanced.
Formerly, Leff [3] focused on the totally reversible Brayton cycle and obtained that the efficiency to maximum work
corresponds to the CNCA efficiency. Wu et. al [4] studied
a non-isentropic model and found that the isentropic temperatures ratio (pressure ratio), that maximizes the work, is
the same as a CNCA-like engine ([5] and [6]). In [7] was
optimized the endoreversible model by log-mean temperature difference for heat exchangers from hot and cold sides
and assumed that it was internally a Carnot cycle. Likewise,
Blank [8] optimized the power for an open Brayton cycle.
Chen et. al [9] also conducted the numerical optimization for
density power and distribution of the heat exchangers operation for the endoreversible Brayton cycle. Other optimiza-
tions of Brayton-like cycle can be found in the following reviews [2] and [10]. Recent optimizations of several objective functions (power, density power, efficiency, ecological
function, entropy, ecological coefficient, and so forth) for extended Brayton cycles (regenerative, intercooled, reheating,
with variable-temperature heat reservoirs, with pressure drop
and finite size heat exchangers and so on) were made in [11][19].
The optimizations of these objective functions, where
some of the optimum performance parameters were replaced
(for instance, balanced allocation or same effectiveness [6]
for the heat exchangers into the hot and cold side) as in [6],
were also made numerically, and only for a typical set of operating conditions previously established [9-11] and [15-19].
Albeit, if the objective function depends of two or more variables; then the optimization must be total. Partial optimizations can provide results completely different as has been indicated in [12] in reference to [9]. In general, the obtained optimal analytical expressions cannot be uncoupled since they
are too cumbersome.
On the other hand, [6] established a criterion to maximize the irreversible efficiency when the efficiency depends
on only one parameter: the isentropic temperatures ratio. It
was applied for the non-isentropic model and for a Brayton model (standard irreversible Brayton cycle) with in-
QUALITATIVE AND QUANTITATIVE OPTIMIZATION OF A STANDARD IRREVERSIBLE BRAYTON CYCLE
225
ternal (given by the isentropic efficiencies) and external irreversibilities of the heat transfer by the effectiveness (ratio
between transferred heat and maximum transferable heat) or
number of heat transfer units for single-pass counterflow heat
exchangers (ε-NTU method [20]). Later, in [19] was obtained the maximum work and an unbalanced allocation by
a qualitative analysis and a numerical approximation for this
Brayton cycle.
In what follows, the same standard irreversible Brayton
cycle (Fig. 1 below) is chosen because of its simplicity to account for two of the main irreversibilities that usually arise
in real power plants: finite rate heat transfer between the
working fluid and internal dissipation of the working fluid.
Whence, we have focused in combining (perturbing) the optimal operation conditions of the non-isentropic and endoreversible cycles and, consequently, we proposed further extension of them.
In this paper, a qualitative and quantitative optimization of a standard irreversible Brayton cycle which extends
(perturbing) the optimal operation conditions of the nonisentropic and endoreversible models is discussed. As a result, a approximated correlation between the optimal effectiveness has been found and consequently the allocation is
always unbalanced for other Brayton cycle which do not correspond to the non-isentropic and endoreversible cycles.
F IGURE 1.A standard irreversible Brayton cycle.
And η1 η2 are the isentropic efficiencies of the turbine
and compressor [6]; T3 , T1 are the maximum and minimum
temperatures achieved in the reversible cycle; the parameter
µ = TTHL , corresponds to the ratio between the temperatures
of hot- and cold-side; and εH , εL are the effectiveness given
by [6]:
εH =
2.
T3 − T2
;
TH − T2
εL =
T4 − T1
T4 − TL
(4)
The relation for the dimensionless work
A standard irreversible Brayton cycle is shown in Fig. 1.
For this cycle the following temperature relations are satisfied [19]:
T1
; T4s = T3 x
(1)
x
µ
¶
1−x
T2 = T1 1 +
; T4 = T3 (1 − η1 (1 − x)) (2)
η2 x
´
£
¤³
εL µx−1 + εH (1 − εL ) 1−x
η2 + x
TH ;
T2 =
[εL + εH (1 − εL )]
(3)
³
´
η1
[εH x + εL µ (1 − εH )] x1 − (1−x)
x
T4 =
TH
[εL + εH (1 − εL )]
The two single-pass counterflow heat exchangers are coupled to the reservoirs TH and TL ; so the external irreversibilities are defined by the respective effectiveness. Thereby, the
dimensionless work, w of the standard irreversible Brayton
cycle in relation to the maximum energy by mass unit attained in the cycle will be optimized (see Fig. 1):
T2s =
w=
1−
1
(5)
where CW is the thermal capacity rate (mass and specific heat
product) of the working substance [6].
The dimensionless expressions, q = CWQTH , for the hotside and cold-side heat transfer and the work w of this cycle
are:
¶
µ
¶
µ
T4
T2
; qL = εL
−µ
1−
TH
TH
·
¸
·
¸
T2
T4
w = εH 1 −
− εL
−µ
TH
TH
qH = εH
where x = ε γ , and ε = pp21 the pressure ratio (maximum
c
pressure divided by minimum pressure) and γ = cvp , with
cp the constant-pressure specific heat and cv the constantvolume specific heat. Henceforth, x denotes the isentropic
temperatures ratio of the working substance.
W
CW TH
(6)
(7)
Substituting these equations in the equation (3), the following the relation for the work is obtained:
Rev. Mex. Fis. S 59 (1) (2013) 224–229
226
G. ARAGÓN-CAMARASA, G. ARAGÓN-GONZÁLEZ, A. CANALES-PALMA AND A. LEÓN-GALICIA
"
#
¡
¢
εL µx−1 + εH (1 − εL ) (1 − x (1 − η2 ))
w = εH 1 −
η2 (εL + εH (1 − εL ))
³ ³
´ ´


η1(εH x+εL µ (1−εH )) x1 η11 −1 +1
− εL
−µ (8)
εL + εH (1 − εL )
Thus, on the analysis of the optimal operating states will
be focused. There are three limit cases, although only two are
relevant [15] (B and C below).
For any heat exchanger N = UCA , where U is the overall heat-transfer coefficient, A the heat-transfer surface
and C the thermal capacity. The number of transfer
units in the hot-side and cold-side, NH and NL , are
indicative of both heat exchangers size. And their respective effectiveness is given by [20]:
εH = 1 − e−yN ;
εL = 1 − e−(1−y) N
(15)
A) Totally reversible [εH = εL = η1 = η2 = 1]
As CW TH = mcP T3 m (is the mass and cp the
constant-pressure specific heat) TH = T3 and TL = T1
(see cycle 1 − 2s − 3 − 4s in Fig. 1), then, the dimensionless work w = mcW
, is maximum if (equation
P T3
(8)):
xT R =
√
µ∗ ; ηCN CA = 1 −
√
µ∗
(9)
The behaviour of the effectiveness for a value realistic
of the total number of transfer units of both heat exchangers [1] (N = 3) is shown in the Fig. 2.
Optimizing the equation (11) with respect to the isentropic temperatures ratio x and the allocation (size) of
both heat exchangers inventory y we obtain the following:
where µ∗ = TT31 and ηCN CA corresponds to the CNCA
efficiency [1].
yE =
B) Non-isentropic εH = εL = 1, 0 < η1 , η2 < 1
Newly, CW TH = mcP T3 , TH = T3 andTL = T1 (see
cycle 1 − 2 − 3 − 4 in Fig. 1). Thus, the dimensionless
work w is maximal if (equation (8))
xN I =
p
Iµ∗
(10)
1
;
2
xCN CA =
√
µ
(16)
Finally, these equations denote that the hot- and coldside heat exchangers have the same size -allocation
balanced- if yE = 12 and it, also, corresponds to the
CNCA efficiency [2]:
√
Iη2 (1 − µ∗ ) + Iµ∗ − 1
η N I = 1 − √ ³√
√ ¡√ ∗ ¢´ (11)
I
Iη2 (1−µ∗ )+ µ∗
Iµ −1
ηCN CA = 1 −
√
µ
(17)
1
η1 η2
where I =
and ηN I corresponds to the efficiency to
maximum work.
In this case necessarily: εH = εL = 1.
C) Endoreversible η1 = η2 = 1, 0 < εH , εL < 1
This case corresponds to the endoreversible Brayton
cycle (see Fig. 1), where TH > T3 and TL < T1 .
The dimensionless worwk is:
w=
εH εL (1 − x) x − (1 − x) µ
x [εL + εH (1 − εL )]
(12)
If the total number of transfer units of both heat exchangers is N , then, the following parameterization of
the total inventory of heat transfer [1] can be included
in the equation (11):
NH + NL = N
NH = yN
(13)
and NL = (1 − y) N
(14)
F IGURE 2. The behaviour of the effectiveness εH versus εL , if
N = 3.
Rev. Mex. Fis. S 59 (1) (2013) 224–229
227
QUALITATIVE AND QUANTITATIVE OPTIMIZATION OF A STANDARD IRREVERSIBLE BRAYTON CYCLE
3.
where xN I is given by the equation (10), if µ∗ is replaced
with µ. The inequality (20) is satisfied because of:
Optimal analytical expression
Now, the optimization of the standard irreversible Brayton
shown in Fig. 1 is analyzed. The non-critical parameters are
[11]: η1 , η2 , µ = TTHL and N ; and the critical parameters:
x = 1 − η and y are, except N all positive and strictly less
than one (including in the equation (8) the same parameterization (equations (12) and (13)) of the limit case C of the
section 2). Then, w depends only of the characteristics parameters x and y, and reaches a global maximum as is shown
in the Fig. 3, if η1 = η2 = 95%; µ = 0.25; N = 3.
Applying the extreme conditions:
∂w
=0
∂x
and
∂w
=0
∂y
(18)
the following coupled optimal analytical expressions forx
and y, are obtained
1<I≤
xN E = xCN CA =
µ
(24)
1
(25)
2
which corresponds to the endoreversible cycle (limit case C
of the section 2). Therefore, the equations (18) and (19) are
one generalization of the equations (15).
The optimal allocation (size) of the heat exchangers has
the following asymptotic behaviour:
N →∞
(19)
1
2
and
lim yN E =
η1 ,η2 →1
1
2
(26)
Furthermore, the result obtained by Swanson [7] is incorrect. Also, xN E has the following asymptotic behaviour:
(20)
Qualitative and asymptotic behavior
The equations (18) and (19) for xN E and yN E cannot be uncoupled (see below). A qualitative analysis and its asymptotic behaviour of the coupled analytical expressions for xN E
and yN E (equations (18) and (19)) have been performed [19]
in order to establish the bounds for xN E and yN E and to
see their behaviour in the limit cases. In [19] the following
bounds for xN E and yN E were found
0 < xN I ≤ x N E < 1
(21)
1
2
(22)
0 < yN E <
√
yN E = yE =
lim yN E =
where z1 = eN ; z = eyN ; A = η1 η2 eN + 1 − η2 ;
B = eN (η1 η2 + 1 − η2 ) and C = eN − η2 + η1 η2 .
3.1.
(23)
If I = 1 (η1 = η2 = 100%), the following values are
obtained
s
(z1 − z) (Cz − B)
xN E =
µ
(z − 1) (Az1 − Bz)
µ
¶
1
1
Ax − Bµ
yN E = +
ln
2 2N
Bx − Cµ
(z1 − z) (Cz − B)
(z − 1) (Az1 − Bz)
lim xN E = xN I
N →∞
lim ηN E = xN I
and
N →∞
(27)
Thus, the non-isentropic and endoreversible cycles are
particular cases of the cycle herein presented. Also we conclude that the allocation is always unbalanced (yN E < 12 ).
3.2.
Optimal approximated analytical expressions and
numerical results
Combining the equations (18) and (19) we obtain the following equation as function only of z
√
µ
µ
Bz1 − Cz 2
Az1 − z 2 B
s
¶
(z1 − z) (Cz − B)
(z − 1) (Az1 − Bz)
=
(28)
which gives a polynomial of degree 6 which cannot be solved
in closed form. The variable z relates (in exponential form)
to the allocation (unbalanced, εH < εL ) and the total number of transfer units N of both heat exchangers. To obtain a
closed form for the effectiveness εH , εL , the equation (27)
can be approximated by
√
µ
µ
Bz1 − Cz 2
Az1 − z 2 B
¶
µ
=
1 1
+ H
2 2
¶
(29)
using the linear approximation
√
F IGURE 3. If η1 = η2 = 95%; µ = 0.25; N = 3, then, the
dimensionless work reaches a global maximum.
³
´
1
2
(H − 1) + O (H − 1)
2
(z1 − z) (Cz − B)
H=
(z − 1) (Az1 − Bz)
H =1 +
Rev. Mex. Fis. S 59 (1) (2013) 224–229
(30)
(31)
228
G. ARAGÓN-CAMARASA, G. ARAGÓN-GONZÁLEZ, A. CANALES-PALMA AND A. LEÓN-GALICIA
It is remarkable that the non-isentropic and endoreversible limit cases (limit cases B and C of the section 2, respectively) are not affected by the approximation and remain
invariant within the framework of the cycle herein presented.
Thus, this approximation maintains and combines the optimal operation conditions of these limit cases and, moreover,
they are included. Also a noteworthy correlation between optimal effectivenes εH and εL is obtained (see equation (32)
in the Conclusions section)
The equation (28) is a polynomial of degree 4 and it can
be solved in closed form for z with respect to the non-critical
parameters: µ or N , for realistic values for the isentropic efficiencies [4] of turbine and compressor: η1 = η2 = 0.8 or
0.9, but it is too large to be included here. Fig. 4 shows the
values of z(zmp ) with respect to µ
F IGURE 5. Behavior of ηN E , ηN I and ηCN CA versus µ, if
η1 = η2 = 0.9 and N = 3.
Using these numerical values and with a total number of
heat transfer units N = 3 [1] (so exists a finite difference of
temperatures). Then, Fig. 5 shows that the efficiency to maximum work ηN E , with respect to µ, can be well approached by
the efficiency of the non-isentropic cycle ηN I (equation(10))
for a value of N = 3 and isentropic efficiencies of 90%
The behavior of yN E with respect to the total number of
transfer units N of both heat exchangers, with the same numerical values for the isentropic efficiencies of turbine and
compressor η1 = η2 = 0.8 or 0.9 and a Carnot efficiency
of 70%, is presented in Fig. 6. The allocation for the heat
exchangers yN E is approximately 2 − 8% or 1 − 3%, respectively, less than the asymptotic value of 12 , if the number of
heat transfer units N is between 2 and 5. This result describes
that the size of the heat exchanger in the hot side decreases.
F IGURE 6. Behaviour of y versus N , when η1 = η2 = 0.8 or 0.9
and µ = 0.3.
4.
Conclusions
Relevant information about the performance of Brayton-like
cycles has been described in this work. The study performed
combines and extends the optimal operation conditions of endoreversible and non-isentropic cycles since the standard irreversible Brayton cycle provides more realistic values for
the optimal isentropic temperatures ratio, efficiency to maximum work and optimal allocation (size) for the heat exchangers than the values corresponding to the non-isentropic or the
endoreversible operations
Now, if η1 = η2 = 0.8 (I = 1.5625); yN E = 0.45 then
N∼
= 3.5 (see Fig. 6) and for the equations (14):
εH = 0.74076 and εL = 0.80795
F IGURE 4. Behaviour of z
N = 3.
zmp () versus µ, if η1 = η2 = 0.8 and
(32)
Therefore, our work [6] must be reviewed since we have
assumed that the effectiveness are the same εH = εL < 1;
whilst I > 1. Current literature on the Brayton-like cycles,
where have taken the same effectiveness less than one and
contains internal irreversibilities, would have to be reviewed
too. To conclude εH = εL , if and only if the allocation is
balanced (yE = 12 ) and the unique thermodynamic possibility is: optimal allocation balanced (equations (16); that is
Rev. Mex. Fis. S 59 (1) (2013) 224–229
QUALITATIVE AND QUANTITATIVE OPTIMIZATION OF A STANDARD IRREVERSIBLE BRAYTON CYCLE
εH = εL . And εH < εL if and only if I > 1 (there is internal
irreversibilities).
Furthermore, combining the equations (16) and as
z = eyN , the following remarkable correlation is obtained
εH =
1
zN E
zN E e−N
1−
1−
εL
(33)
where zN E is calculated by the equation (28) and shown in
∗. [email protected]
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