entropy
Article
Thermoeconomic Optimization of an Irreversible
Novikov Plant Model under Different Regimes
of Performance
Juan Carlos Pacheco-Paez 1, *, Fernando Angulo-Brown 1 and Marco Antonio Barranco-Jiménez 2
1
2
*
Escuela Superior de Física y Matemáticas del Instituto Politécnico Nacional, Ciudad de México 07738,
Mexico; [email protected]
Escuela Superior de Cómputo del Instituto Politécnico Nacional, Ciudad de México 07738, Mexico;
[email protected]
Correspondence: [email protected]; Tel.: +52-55-5527-6000 (ext. 52069)
Academic Editor: Milivoje M. Kostic
Received: 1 February 2017; Accepted: 10 March 2017; Published: 15 March 2017
Abstract: The so-called Novikov power plant model has been widely used to represent some actual
power plants, such as nuclear electric power generators. In the present work, a thermo-economic
study of a Novikov power plant model is presented under three different regimes of performance:
maximum power (MP), maximum ecological function (ME) and maximum efficient power (EP). In this
study, different heat transfer laws are used: The Newton’s law of cooling, the Stefan–Boltzmann
radiation law, the Dulong–Petit’s law and another phenomenological heat transfer law. For the
thermoeconomic optimization of power plant models, a benefit function defined as the quotient of
an objective function and the total economical costs is commonly employed. Usually, the total costs
take into account two contributions: a cost related to the investment and another stemming from
the fuel consumption. In this work, a new cost associated to the maintenance of the power plant is
also considered. With these new total costs, it is shown that under the maximum ecological function
regime the plant improves its economic and energetic performance in comparison with the other
two regimes. The methodology used in this paper is within the context of finite-time thermodynamics.
Keywords: thermo-economics optimization; irreversible heat engine; Novikov model
1. Introduction
During more than four decades, a branch of irreversible thermodynamics called finite-time
thermodynamics (FTT) has been developed [1–5]. Within the context of FTT, it has been possible to
elaborate detailed models of heat engines working at finite-time and with entropy production [6–10].
One of the applications of FTT has been in the field of thermoeconomics [11–14]. In 1995, Alexis De Vos
introduced for the first time a thermoeconomic study for the Novikov plant model [15]. In his study,
De Vos takes into account two costs: a cost related to the investment and another cost associated to the
fuel consumption. From his thermoeconomic study, De Vos found that the optimal thermoeconomic
efficiency under maximum power conditions is between the Curzon–Ahlborn (CA) efficiency and
Carnot efficiency [16]. Here, De Vos refers to the power production rate W of the Novikov plant model.
Later, Barranco-Jiménez and Angulo-Brown [17,18] extended the De Vos thermoeconomic approach,
but using the so-called ecological function [19,20], and they also showed that the economical efficiency
under this regime of performance lies between the CA efficiency and the Carnot efficiency. In 2009,
Barranco-Jiménez [21] studied the thermoeconomics of a nonendoreversible Novikov engine. He used
different heat transfer laws: The Newton’s law of cooling, the Stefan–Boltzmann radiation law [22],
the Dulong–Petit’s law of cooling [23] and a phenomenological heat transfer law [24,25] (the usage
Entropy 2017, 19, 118; doi:10.3390/e19030118
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Entropy 2017, 19, 118
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engine model to describe [22–27]). He calculated the thermoeconomic optimal efficiencies under two
regimes of
performance,
Entropy
2017, 19, 118 namely, the maximum power regime and the ecological regime, and
2 of 15he found
that when the Novikov model maximizes the ecological function, it reduces the rejected heat to the
environment
down to about 50% with respect to the heat rejected under maximum power conditions.
of each transfer law depends on the kind of heat engine model to describe [22–27]). He calculated
Recently,the
Pacheco-Paez
et al.
[28,29]
made under
a thermoeconomic
analysis ofnamely,
a Novikov
plant following
thermoeconomic
optimal
efficiencies
two regimes of performance,
the maximum
power
regime and
the ecological
and he found
that
when
the Novikov
model
maximizes
the to the
the De Vos
approach
(with
a linearregime,
heat transfer
law)
but
including
a new
cost
associated
ecological function, it reduces the rejected heat to the environment down to about 50% with respect
maintenance of the power plant. The aim of this paper is to extend the thermoeconomic analysis of the
to the heat rejected under maximum power conditions. Recently, Pacheco-Paez et al. [28,29] made a
Novikovthermoeconomic
plant model analysis
(see Figure
1) by considering different heat transfer laws. Besides, in the
of a Novikov plant following the De Vos approach (with a linear heat transfer
optimization
of the
benefit
functions,
we take
into
account aofnew
cost associated
to the
maintenance
of
law) but
including
a new
cost associated
to the
maintenance
the power
plant. The aim
of this
paper
the power
In our study,
we of
also
considerplant
three
regimes
of performance:
maximum
is toplant
extend[28,29].
the thermoeconomic
analysis
the Novikov
model
(see Figure
1) by considering
different
heat
transfer
laws.
Besides,
in
the
optimization
of
the
benefit
functions,
we
take
into
account
power (MP) [3], maximum efficient power (EP) [30,31] and maximum ecological function (ME) [19,20].
a new cost associated to the maintenance of the power plant [28,29]. In our study, we also consider
The methodology
used in this paper is within the context of finite-time thermodynamics. Apart from
three regimes of performance: maximum power (MP) [3], maximum efficient power (EP) [30,31] and
the De Vos
method, there exist other approaches for the thermoeconomic analysis of power plant
maximum ecological function (ME) [19,20]. The methodology used in this paper is within the context
models. Interestingly,
the thermoeconomic
analysis
based
on exergy
hasother
played
an important
of finite-time thermodynamics.
Apart from
the De Vos
method,
there exist
approaches
for the role in
this discipline
[32–36]. The
work
organized
as follows:
In Section
2, we present analysis
the thermoeconomic
thermoeconomic
analysis
of is
power
plant models.
Interestingly,
the thermoeconomic
based
on
exergy
has
played
an
important
role
in
this
discipline
[32–36].
The
work
is
organized
as
follows:
analysis of the Novikov power plant under different criteria of performance. In Section 3, we analyze
In Section 2, we present the thermoeconomic analysis of the Novikov power plant under different
the environmental
impact of the thermoeconomic model, and finally in Section 4, we present our
criteria of performance. In Section 3, we analyze the environmental impact of the thermoeconomic
conclusions.
model, and finally in Section 4, we present our conclusions.
Figure 1. Novikov’s
model
for a thermal
power
plant.QQ H,, Q
W are
per unit per
time.unit time.
Figure 1. Novikov’s
model for
a thermal
power
plant.
and
Wquantities
are quantities
QLLand
H
2. Thermoeconomic Analysis
2. Thermoeconomic Analysis
Applying the first law of thermodynamics to Figure 1, the power output is given by,
Applying the first law of thermodynamics to Figure 1, the power output is given by,
W = Q −Q ,
(1)
W = QH − HQL , L
(1)
QH Q
and Q
theheat
heat transfers
between
the thermal
baths and
the working
fluidworking
(W, Q H and
where Qwhere
and
are
the
transfers
between
the thermal
baths
and the
fluid ( W ,
L are
H
L
Q L are quantities per cycle’s period). On the other hand, the internal efficiency of the engine is given
are quantities per cycle’s period). On the other hand, the internal efficiency of the
QH andbyQ[21],
L
engine is given by [21],
η=
W
τ1
= 1−
,
QH
Rθ
(2)
W
τ 1
η =the parameter
= 1 − R , = ∆S1 /|∆S2 | is the non-endoreversibility
where τ = TL /TH , θ = T3 /TH and
QH
Rθ
where
(2)
parameter [37,38] (which characterizes the degree of internal irreversibility that comes from the
, θ = T3∆ST1Hbeing
and
parameter
R =entropy
ΔS1 Δalong
S2 the
is hot
theisothermal
non-endoreversibility
the the
change
of the internal
branch
τClausius
= TL THinequality),
parameter [37,38] (which characterizes the degree of internal irreversibility that comes from the
Clausius inequality), ΔS1 being the change of the internal entropy along the hot isothermal branch
and ΔS2 the entropy change corresponding to the cold isothermal compression. This parameter in
principle is within the interval 0 < R ≤ 1 ( R = 1 for the endoreversible limit). If we consider that the
Entropy 2017, 19, 118
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and ∆S2 the entropy change corresponding to the cold isothermal compression. This parameter in
principle is within the interval 0 < R ≤ 1 (R = 1 for the endoreversible limit). If we consider that the
heat transferred between the hot source and the working fluid obeys a generalized law of the type,
Q H = gTH n (1 − θ )n ,
(3)
where the exponent n can be any real number different from zero and it takes the values n = 1 for a
Newtonian heat transfer law (N) and n = 5/4 for a Dulong–Petit heat transfer law (DP) [23,27] and g
is the thermal conductance (see Figure 1). From Equations (2) and (3), the power output (W = ηQ H )
results in
n
τ
n
(4)
W = gTH
η 1−
R (1 − η )
The De Vos thermoeconomical analysis considers a profit function F, which is maximized [11].
This profit function is given by the quotient of the power output W (that is, the power production rate
W of the Novikov plant model) and the total costs involved in the performance of the power plant,
that is
W
F=
(5)
C
Apart from this De Vos method, there exist other approaches for the thermoeconomic analysis
of power plant models. Significantly, the thermoeconomic analysis based on exergy has played an
important role in this matter [32–36]. In his study, De Vos assumed that the running costs of the plant
consist of two parts: a capital cost which is proportional to the investment and, therefore, to the size of
the plant, and a fuel cost that is proportional to the fuel consumption and, therefore, to the heat input
rate. In this work, we also take into account a cost associated to the maintenance of the power plant
which is taken as proportional to the power output of the plant. This choice is based on an intuitive
idea that has to do with the wheatering of the engine depending on its excessive usage. Therefore,
the running costs of the plant exploitation are now defined as,
C = aQmax + bQ H + cW,
(6)
where the proportionality constants a, b and c have units of $/Watt, and Qmax = gTH n (1 − τ )n is the
maximum heat that can be extracted from the heat reservoir without supplying work in the same
manner previously considered by De Vos [11] (see Figure 1). Equation (6) is taken as a linear relation
inspired in the linear character of the first law of thermodynamics [39]. Using Equations (3) and (4),
the running costs of the plant are given by,
C = agTH n (1 − τ )n + β 1 −
τ
R (1 − η )
n
+ γη 1 −
τ
R (1 − η )
n ,
(7)
where β = b/a and γ = c/a (0 ≤ β < ∞ and 0 ≤ γ). After substituting Equations (4) and (7) into
Equation (5), the dimensionless objective profit function is given by,
N − DP
aFMP
=
η 1−
(1 − τ ) n + β 1 −
τ
R (1− η )
n
τ
R (1− η )
n
+ γη 1 −
τ
R (1− η )
n ,
(8)
which is valid for the transfer laws of Newton (N), and Dulong–Petit (DP). Now, we define
two additional objective functions, one in terms of the so-called modified ecological function
(E ≡ W − ∈ TL σ, with σ the rate of total entropy production of the Novikov model and ∈ a parameter
which depends on the heat transfer law used in the Novikov model [19,20]). This function is a slight
modification of the former ecological function [19], which preserves the original physical meaning of
that function; that is, it represents a good trade-off between high power output and low dissipation.
C
C
respectively. Analogously to Equation (8), by using Equations (4), (5) and (7), FEP and FME are
given by,


τ
η 1 −
R(1 − η ) 

Entropy 2017, 19, 118
n
4 of 15
2
N − DP
=
aFEP
,
(9)
n
n




τ
τ
On the other hand, we also use
a
profit
function
based
on
the
efficient
power
[30,31]
P
≡
ηW
,
(
)
EP
(1 − τ ) + β  1 − R(1 − η )  + γη  1 − R(1 − η ) 
which is an objective function of the class
of compromise
for a good trade-off


 functions looking

and,between power output and efficiency. Both objective functions are divided by the total costs involved
ηW
T σ
n L , respectively. Analogously to
in the performance of the plant, that is, FEP = C and FME = W −∈

C
τ
( 1 (5)
η − ε(7),
+ ε )and
(1 −FτEP) and
Equation (8), by using Equations (4),
are given by,
 1 − FRME
(1 − η ) 

N − DP
.
aFME =
(10)
n
n
n
τ
2
n



τη 1 −
τ
(1−
N − DP( 1 − τ ) + β  1 −
 + Rγη
 1η )− R(1
(9)
aFEP
=
− η ) 
n ,
n R(1 − η )  τ n
(1 − τ ) + β 1 − R(1−η ) + γη 1 − R(1τ−η )
The maximization of the objective functions given by Equations (8)–(10) by means of
n
d ( aFand,
)
dη
n
τ
= 0 gives the corresponding[(optimal
taking the derivatives of
1 + ε)η − efficiencies.
ε(1 − τ )] 1 − Therefore,
R (1− η )
N − DP
aFME
=
η∗
(1 − τ ) n + β 1 −
τ
R (1− η )
n
+ γη 1 −
τ
R (1− η )
n .
(10)
N − DP
N − DP
N − DP
aFMP
, aFPE
and aFME
with respect to η and setting them equal to zero, we obtain the
( )
d( aF ) The maximization of the objective functions given by Equations (8)–(10) by means of dη ∗ = 0
optimal efficiency η ∗ that maximizes Equations (8)–(10), respectively. In Figure 2, for
the
η case of a
gives the corresponding optimal efficiencies. Therefore, taking the derivatives of aF N − DP , aF N − DP
MP threePE
Newton heat
transfer law ( n = 1 in Equation (3)), we show the behavior of the
objective
N − DP
and aFME
with respect to η and setting them equal to zero, we obtain the optimal efficiency (η ∗ )
functions
versus the internal efficiency η for the following arbitrary values R = 1 (endoreversible
that maximizes Equations (8)–(10), respectively. In Figure 2, for the case of a Newton heat transfer
( )
in Equation
show the
the three
functions
versus
the internal
η ∗ which
0 1and
τ = 0.5 .(3)),
Wewe
observe
in behavior
Figure 2ofthat
thereobjective
exists an
efficiency
value
case),lawβ (n= =
efficiency η for the following arbitrary values R = 1 (endoreversible case), β = 0 and τ = 0.5.
depends
on the parameter γ . The benefits function diminishes as the parameter γ increases.
We observe in Figure 2 that there exists an efficiency value (η ∗ ) which depends on the parameter
aF Nofismaximum
γ. The
function
diminishes
as the conditions,
parameter γ the
increases.
Besides,
for the
obtained for
Besides,
forbenefits
the case
of maximum
power
maximum
value
of case
N is obtained for practically the same value of η for the
power
conditions,
the
maximum
value
of
aF
practically the same value of η for the two values of γ considered.
two values of γ considered.
Figure 2. Comparison between the three thermoeconomic objective functions with respect to the
internal efficiency for a Newton heat transfer law for two values of the parameter γ (F represents the
internal
for a Newton
heat transfer
law for
threeefficiency
objective functions
with a Newtonian
heat transfer
law).two values of the parameter γ (
Figure 2. Comparison between the three thermoeconomic objective functions with
respect to the
N
FN
represents the three objective functions with a Newtonian heat transfer law).
In Figure 3, we can observe for the three benefit functions how the optimal efficiency tends to the
In Figurevalue
3, we(Carnot
can observe
for the
three
benefit
functions
howinthe
efficiency
tends to the
reversible
efficiency)
when
β→
∞ , such
as it occurs
theoptimal
De Vos approach
[11,21].
reversible value (Carnot efficiency) when β → ∞ , such as it occurs in the De Vos approach [11,21].
Entropy 2017, 19, 118
Entropy 2017, 19, 118
5 of 15
5 of 15
Figure 3.
3. Comparison
thermoeconomic objective
to the
the
Figure
Comparison between
between the
the three
three thermoeconomic
objective functions
functions with
with respect
respect to
internal efficiency,
efficiency,taking
taking
into
account
the maintenance
costs
of theforplant
for two
values
of the
internal
into
account
the maintenance
costs of
the plant
two values
of the
parameter
β . ( Fthe three
represents
the three
objective
withheat
a Newtonian
heat transfer law).
parameter
β.
(F N represents
objective
functions
with functions
a Newtonian
transfer law).
N
The optimal
optimalefficiencies
efficiencies
obtained
in terms
the dimensionless
parameter
, but this
The
areare
obtained
in terms
of theof
dimensionless
parameter
β, but thisβ parameter
is
not easy is
to not
evaluate
[11]
and it is
more
to work in to
terms
ofinthe
so-called
fractional
parameter
easy to
evaluate
[11]
andconvenient
it is more convenient
work
terms
of thefuel
so-called
fuel
f ) [11,17],
cost
( f ) [11,17],
as the
ratio between
thebetween
fuel costthe
andfuel
the cost
totaland
costs
oftotal
the plant,
is,plant,
fractional
cost (defined
defined
as the ratio
the
costs that
of the
that is,
bQ H
f =
,
(11)
bQH
aQ
+
bQ H +, cW
f=
max
(11)
aQmax + bQH + cW
which is in the interval 0 ≤ f < 1. From the previous equation and by using Equations (2)–(4), we get
which is in the interval 0 ≤ f < 1 . From the previous equation and by using Equations (2)–(4), we get

n 
(1 − τ )n + γη 1 − τ
n


f
R (1− η )


β=
(12)
.
 ( 1− τ )n + γη 1 − τ  n 


1
−
f
τ

 f 
R
(1
)
η
−
1
−

R (1− η ) 
β =
(12)

.
n
 1− f 



τ

 efficiencies by solving polynomial
From Equations (8)–(10), we can numerically
 1 − Rfind
(1 − ηthe
)  optimal



equations of the form: H (τ, f , γ, R, η ) = 0. However, some analytical expressions can be obtained
we can numerically
the optimal
bypreviously
solving polynomial
for γFrom
= 0 inEquations
the cases (8)–(10),
of both Newton
(n = 1) andfind
Dulong–Petit
(nefficiencies
= 5/4) laws
reported
in
[21]. Forofthe
of maximum
power
the optimal
efficiencies
are given can
by [20]
equations
thecase
form:
H (τ , f , γ , R
,η ) = 0output,
. However,
some analytical
expressions
be obtained for
p
γ = 0 in the cases of both Newton ( n = 1 ) and Dulong–Petit
( n = 5 4 ) laws previously reported in [21].
4(1 − f ) Rτ + f 2 τ 2
f
N
η
τ,
f
,
R
=
1
−
τ
−
.
(13a)
(
)
MP
For the case of maximum power output, the optimal
efficiencies
2R
2Rare given by [20]
4(1 −10f()R
f τ−+1)f τ2τ 2
f
N
DP
(13a)
q
η MP
f, R
=
1
−
.
(13b)
(τ,ηMP
)
τ
,
f
,
R
=
1
−
τ
−
.
(
)
(5 f 2−R1)τ + 80(21R− f ) Rτ + (1 − 5 f )2 τ 2
10 ( f − 1)τ
DP
ηMP
f , R ) = 1 − under the maximum power conditions,
. the optimum value(13b)
(τ ,mentioned,
As it was previously
of η
2 2
5
f
−
1
τ
+
80(1
−
f
)
R
τ
+
1
−
5
f
τ
(
)
(
)
does not appreciably change the maximum of the benefit function, that is, the optimal efficiency is
practically independent of the parameter γ, which is associated to the maintenance costs of the plant
As it was previously mentioned, under the maximum power conditions, the optimum value of
(see Figure 2). Interestingly, the inclusion of this new parameter has a remarkable effect on the optimal
η does not appreciably change the maximum of the benefit function,
the optimal efficiency
∗ andthat
∗ is,
efficiencies for the cases of ME and EP maximizations. In fact, η ME
ηEP
increase with respect to
γ , which
is is,
associated
to the maintenance
costs
of the
is practically
independent
of thecosts
parameter
the
results without
maintenance
(see Figure
2); that
better maintenance
is reflected
as better
plant (see Figure
Interestingly,
theefficient
inclusion
of this
new
parameter
has a remarkable
effectγon=the
efficiency.
For the2).
case
of maximum
power
and
maximum
ecological
function (with
0),
*
*
the
optimal
efficiencies
optimal
efficiencies
for are
the given
cases by,
of ME and EP maximizations. In fact, ηME and ηEP increase with
respect to the results without maintenance costs (seeq
Figure 2); that is, better maintenance is reflected
8(1 power
− f ) Rτand
+ (1maximum
+ f )2 τ 2 ecological function
as better efficiency. For
the
case
of
maximum
efficient
1
+
f
(
)
N
ηEP
τ−
,
(14a)
(τ, f , R) = 1 −
(with γ = 0 ), the optimal efficiencies are given
4R by,
4R
Entropy 2017, 19, 118
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10( f − 1)τ
DP
ηEP
(τ, f , R) = 1 −
q
160(1 − f ) Rτ + (3 + 5 f )2 τ 2
q
4(1 − f ) Rτ 3/2 + f 2 τ 2
f
N
η ME
τ−
,
(τ, f , R) = 1 −
2R
2R
(3 + 5 f ) τ +
DP
η ME
(τ, f , R) = 1 −
where Λ = τ +
p
,
100(1 − f )τ
(1 − 5 f ) Λ +
q
800Λ(1 − f ) R + [(1 − 5 f )Λ]2
(14b)
(15a)
.
(15b)
τ (80 + τ ). For the cases f = 0 and R = 1, from Equations (13a), (14a)
[8τ (1+τ )]1/2
and (15a), we obtain the optimal efficiencies ηCA = 1 − τ 1/2 , ηEP = 1 − τ4 −
and
4
η ME = 1 − τ 3/4 previously reported by Curzon-Ahlborn [16], Yilmaz [30] and Arias-Hernández and
Angulo-Brown [20], for the Curzon-Ahlborn heat engine under maximum power output, maximum
efficient power and maximum ecological function conditions, respectively.
Now, we consider two other heat transfer laws between the hot source and the working fluid
as follows,
Q H = gTH n (1 − θ n )Sign(n),
(16)
Qmax = gTH n (1 − τ n )Sign(n),
(17)
Here, the exponent n takes the values of n = −1 for a phenomenological heat transfer law
(Ph) [24,25] and n = 4 for a Stefan–Boltzmann (SB) radiation law ([22,26]) and the Sign function leads
to Sign(n) = 1 if n > 0, and Sign(n) = −1 if n < 0. In this case, the objective functions under the
three regimes considered result in,
SB− Ph
aFMP
h
n i
η 1 − R(1τ−η )
=
h
n i
h
n i ,
+ γη 1 − R(1τ−η )
(1 − τ n ) + β 1 − R(1τ−η )
(18)
SB− Ph
aFEP
h
n i
η 2 1 − R(1τ−η )
=
h
n i
h
n i ,
+ γη 1 − R(1τ−η )
(1 − τ n ) + β 1 − R(1τ−η )
(19)
SB− Ph
aFME
n i
h
[2η − (1 − τ )] 1 − R(1τ−η )
=
h
n i
h
n i .
+ γη 1 − R(1τ−η )
(1 − τ n ) + β 1 − R(1τ−η )
(20)
and,
SB− Ph
SB− Ph
SB− Ph
In Figures 4 and 5, we show the behavior of aFMP
, aFEP
and aFME
for the case of a
phenomenological (Ph) heat transfer law (n = −1 in Equations (18)–(20)) versus the internal efficiency
for R = 1, β = 0 and τ = 0.5 (a similar behavior is found for the benefit functions for the case of
the Stefan–Boltzmann law). We can see in Figures 5 and 6 that these profit functions have a similar
behavior to the Newton and the Dulong–Petit cases; that is, the benefits function also diminishes as the
parameter γ increases. In [11], it was shown that the maxima points of the MP profit function tend to
the reversible limit (Carnot point) when the parameter β tends to infinite. This same behavior occurs
for the EP and ME profit functions. In Figure 6 we only show, as an example, the case of the EP profit
function for several increasing values of the parameter β, and it is clear how the curve that contains
the maxima points tend to the Carnot efficiency (ηC = 0.5, in this case).
In a similar way to the Newton and the Dulong–Petit cases, we can obtain the optimal efficiencies
in terms of the parameter β (or f ).
In a similar way to the Newton and the Dulong–Petit cases, we can obtain the optimal
efficiencies in terms of the parameter β (or f ).
Analogously to Newton and Dulong–Petit laws, we can also obtain some analytical expressions
for γ = 0 for both Ph ( n = −1 ) and SB radiation ( n = 4 ) laws. Therefore, under maximum power
Entropy 2017, 19, 118
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output, maximum efficient power and maximum ecological function conditions, we obtain,
respectively,
Analogously to Newton and Dulong–Petit
laws,Rwe
− τcan also obtain some analytical expressions for
Ph
η radiation
,
(τ , f , R(n) ==R4)(2laws.
(21a)
γ = 0 for both Ph (n = −1) and SB MP
− f ) Therefore, under maximum power output,
maximum efficient power and maximum ecological function conditions, we obtain, respectively,
Ph
ηEP
(τ , f , R ) =
τ ( f − 3)
,
R( f − 1) R− −
2τ τ
Ph
η MP
(τ, f , R) =
Ph
η ME
(τ , f , R ) =
R ( f − 3) − ( f − 1)τ  + 2τ
Ph
ηEP
(τ,
and,
R (2 − f )
(21b)
,
2R( fτ−( f2)− 3)
f , R) =
R( f − 1) − 2τ
(21a)
,
,
R[( f −
f τ− 1)τ ] + 2τ
8(3f)−−1)(R
Ph
S− B
,
= 1(−τ, f , R) =
η MP
,
(τ , f , R η) ME
2R( f − 2)
(4 f − 3)τ + (4 f − 3)2τ 2 + 16 R(1 − f )τ
and,
(
)
2( f − 1)τ − 8(1 −8(f f)τ−R1+) Rτ
(1 − 2 f )τ 2
−B
S −SB
, R)==1 −
1−
ηηEP
MP τ(,τ,f ,f R
,
,
2(R
4 f − 3)2 τ 2 + 16R(1 − f )τ
p2τ 2 + 8 R( f − 1)(1 + τ )τ
(4 f − 3)τ + (4 f − 3)
S− B
2
(
f
−
1
)
τ
−
8(1 − f )τR + (1 − 2 f )τ.2
ηME
τ
,
f
,
R
=
1
−
( ηEP
S− B )
,
(τ, f , R) = 1 −
2R 2R
(4 f − 3) τ +
q
(21c)
(21b)
(21c)
(22a)
(22a)(22b)
(22c)
(22b)
q
Similarly as for the Newton heat(4transfer
law,(4for
f − 3) τ +
τ )τ transfer law when
f −a3phenomenological
)2 τ 2 + 8R( f − 1)(1 +heat
S− B
η
τ,
f
,
R
=
1
−
.
(
)
f = 0 and R = 1 , ME
from Equation (21a–c), we obtain 2R
the optimal efficiencies
ηMP = (22c)
(1 − τ ) 2
Similarly as for the Newton heat transfer
3τ law, for a phenomenological heat transfer law when
η
=
and η ME =efficiencies
3 (1 − τ ) 4η (these
last two expressions
previously
reported
by
De
Vos
[24],
EP
f = 0 and R = 1, from Equation (21a–c), we obtain
MP = (1 − τ ) /2 previously
1 + 2τ the optimal
3τ
by Depublished
Vos [24], ηEPby=Barranco-Jiménez
− τ )/4
(these
two expressions were
1+2τ and η ME = 3(1[21]),
werereported
previously
for
the last
Curzon-Ahlborn
heat previously
engine under
published by Barranco-Jiménez [21]), for the Curzon-Ahlborn heat engine under maximum power
maximum power output, maximum efficient power and maximum ecological function conditions,
output, maximum efficient power and maximum ecological function conditions, respectively.
respectively.
Figure
4. Comparison
between
thethree
threethermoeconomic
thermoeconomic objective
with
respect
to the
Figure
4. Comparison
between
the
objectivefunctions
functions
with
respect
to the
internal efficiency for two values of the parameter γ. F Ph represents
the three objective functions with
internal
efficiency for two values of the parameter γ . F Ph represents
the three objective functions
a phenomenological heat transfer law.
with a phenomenological heat transfer law.
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Figure
5. Comparison
between
thermoeconomic
objectivefunctions
functions
with
respect
to the
Figure
5. Comparison
betweenthe
thethree
three thermoeconomic
thermoeconomic objective
with
respect
to the
Figure
5. Comparison
between
the
three
objective functions
with
respect
to the
internal
efficiency,
taking
into
account
the
maintenance
costs
of
the
plant
for
two
values
of
the
internal
efficiency,
taking
into
account
the
maintenance
costs
of
the
plant
for
two
values
of
the
internal efficiency, taking into account the maintenance costs of the plant for two values of the parameter
Ph Ph
Frepresents
represents
three
objective
functions
with
heatheat
transfer
law. law.
parameter
β . βF . the
thethe
three
objective
functions
witha aphenomenological
phenomenological
β. F Phparameter
represents
three
objective
functions
with
a phenomenological
heat transfer
law.transfer
Figure 6. Thermoeconomic objective function under efficient power (EP) conditions with respect to
the internal efficiency, taking into account the maintenance costs of the plant for several values of the
Figure
6. Thermoeconomic
Thermoeconomic
objective
under
efficient
(EP)
conditions
respect
β . We observe
how thefunction
maxima points
to thepower
reversible
that is, with
the Carnot
parameter
Figure
6.
objective
function
undertend
efficient
power
(EP)value;
conditions
with
respect to
to
the
efficiency,
taking
costs
of of
thethe
plant
forfor
several
values
of the
the internal
internal
efficiency,
takinginto
intoaccount
accountthe
themaintenance
maintenance
costs
plant
several
values
of
efficiency
( ηC ).
β
.
We
observe
how
the
maxima
points
tend
to
the
reversible
value;
that
is,
the
Carnot
parameter
the parameter β. We observe how the maxima points tend to the reversible value; that is, the Carnot
).
efficiency
( ηCC).Results:
(η
3. Numerical
Environmental Impact
d ( aFi )
, for i = MP , EP , ME ,
*
d( aFii) aFi ∗ , for i = MP, EP, ME,
When we calculate the derivative
of Equations (8)–(10), that is, d dη
(
)
∗
ηto
When
we equations
calculate of
the
that is,we have
for inumerically
= MP , EP , ME ,
i ,solve
we obtain
thederivative
form
= ( , , , (8)–(10),
, ). Therefore,
of Equations
Environmental
WhenResults:
we calculate
the derivative
of Equations (8)–(10), that is,
3. Numerical
Environmental
Impact
dη
η
( )
dη *
η
= H τ, β, γ, R, ηi∗ . Therefore, we have
to solve numerically
i
polynomial
equationsofofthe
theform
form, ( ) = ( , , , , ∗ ). Therefore, we have to solve numerically
we
obtain equations
HiN − DP = HiN − DP (τ , β , γ , R ,ηiN − DP ) = 0 ,
(23)
N
−
DP
N
−
DP
N
−
DP
polynomial equations of the form,
Hi
=H
τ, β, γ, R, ηi
= 0,
(23)
These previous expressions
are not ishown in explicit form
because they are very large. Finally,
we obtain equations of the form
d( aFi ) dη
polynomial equations of the form,
ni
(
)
N − DP
using the expression for H
theN − DP
fractional
= HiN − DPfuel
τ , βcost
, γ , Rf,ηi[11,17]
= (see
0 , Equation (12)), we obtain the (23)
i are not
These previous expressions
shown in explicit
form because they are very large. Finally,
numerical optimal efficiencies in terms of the parameter R , the parameter τ and the economic
usingparameters
the expression
for
fractional
fuelshow
cost the
f [11,17]
(see
Equationefficiencies
(12)), we obtain
the numerical
f andexpressions
γ .the
In Figure
we shown
optimal form
numerical
under
These previous
are7, not
in explicit
because they are
very maximum
large. Finally,
optimal efficiencies in terms of the parameter R, the parameter τ and the economic parameters f and γ.
maximum efficient
and maximum
ecological
function
In this the
f [11,17]
usingpower,
the expression
for thepower
fractional
fuel cost
(seeconditions,
Equationrespectively.
(12)), we obtain
In Figure
7, (see
we show
the optimal
numerical
efficiencies
under given
maximum
power, (13a),
maximum
efficient
Figure
solid
lines),
we
also
show
the
optimal
efficiencies
by
Equations
(14a)
and
numerical optimal efficiencies in terms of the parameter R , the parameter τ and the economic
power
and
maximum
ecological
function
conditions,
respectively.
In
this
Figure
(see
solid
lines),
(15a).
parameters f and γ . In Figure 7, we show the optimal numerical efficiencies under maximum
we also show the optimal efficiencies given by Equations (13a), (14a) and (15a).
power,
maximum
ecological
function
conditions,
respectively.
Inf this
Onmaximum
the other efficient
hand, inpower
Figureand
8 we
depict the
corresponding
optimal
efficiencies
against
for
Figure
(see
solid
lines),
we
also
show
the
optimal
efficiencies
given
by
Equations
(13a),
(14a)
and
the Newtonian case, but with R = 0.9 < 1 (this case is a representative one of any nonendoreversible
(15a).
Entropy 2017, 19, 118
9 of 15
On the other hand, in Figure 8 we depict the corresponding optimal efficiencies against f for the
9 of 159 of 15
Newtonian case, but with R = 0.9 < 1 (this case is a representative one of any nonendoreversible
situation).On
Asthe
weother
can hand,
see ininthis
case,8 we
thedepictoptimal
values are optimal
all smaller
than those
corresponding
Figure
the corresponding
efficiencies
against
f for the
N
Newtonian
case,
but
with
R
=
0.9
<
1
(this
case
is
a
representative
one
of
any
nonendoreversible
to Figure 7.
behavior
for R
< 1 the
is present
for any
heat
lawthan
and those
any operating
regime
situation).
AsThis
we can
see in this
case,
η optimal
values
aretransfer
all smaller
corresponding
to
situation).
As we canfor
seeRin<this
the for optimal
values
are all
smaller
thanoperating
those corresponding
[21,40].
Figure
7. This behavior
1 iscase,
present
any heat
transfer
law
and any
regime [21,40].
Entropy
19, 118
Entropy
2017, 2017,
19, 118
to Figure 7. This behavior for R < 1 is present for any heat transfer law and any operating regime
[21,40].
N represents
N
Optimal
efficiencies
versus
fractional
fuel ηcost.
cost.
represents
the
optimal
Figure
7.7.Optimal
efficiencies
versus
fractional
fuel cost.
the three
efficiencies
Figure
7. Optimal
efficiencies
versus
fractional
fuel
ηη
represents
theoptimal
threethree
optimal
with
a
Newtonian
heat
transfer
law
(For
R
=
1).
efficiencies
withwith
a Newtonian
heat
R ==1).
1).
efficiencies
a Newtonian
heattransfer
transferlaw
law (For
(For R
N
N represents
Figure
8. Optimal
efficiencies
versus
fractional
fuel ηcost.
η represents
theoptimal
three optimal
Figure
8. Optimal
efficiencies
versus
fractional
fuel cost.
the three
efficiencies
efficiencies
with
a
Newtonian
heat
transfer
law
(For
R
=
0.9).
N
with
a
Newtonian
heat
transfer
law
(For
R
=
0.9).
Figure 8. Optimal efficiencies versus fractional fuel cost. η
represents the three optimal
N
efficiencies
Newtonian
heat transfer
law (For
R =the
0.9).maximum ecological function regime (for
We canwith
alsoasee
that the optimal
efficiencies
under
We
can also
see
that the optimal
the maximum
function
all values
of the
parameters
, and efficiencies
for the caseunder
of Newton
heat transferecological
law, see Figures
7 and regime
8)
(for all
values
of
the
parameters
R,
γ
and
f
for
the
case
of
Newton
heat
transfer
law,
see
Figures
7 and
8)
We
can also
seethe
that
the optimal
efficiencies
under
the maximum
ecological
functionefficient
regime
(for
are greater
than
optimal
efficiencies
under both
maximum
power output
and maximum
are
greater
the optimal
efficiencies
output
andsee
maximum
efficient
powerofthan
conditions,
respectively.
For theunder
casethe
ofboth
bothmaximum
phenomenological
and Stefan–Boltzmann
laws,
all values
the parameters
, and
for
case
of
Newtonpower
heat transfer
law,
Figures
7 and 8)
the
optimal
efficiencies
have
a
similar
behavior
with
respect
to
the
parameters
,
and
.
Besides,
alllaws,
power
conditions,
respectively.
For
the
case
of
both
phenomenological
and
Stefan–Boltzmann
are greater than the optimal efficiencies under both maximum power output and maximum efficient
of
these
optimal
efficiencies
satisfy
the
following
inequality:
the
optimal
efficiencies
have
a
similar
behavior
with
respect
to
the
parameters
R,
γ
and
f
.
Besides,
power conditions, respectively. For the case of both phenomenological and Stefan–Boltzmann laws,
all
these optimal
efficiencies
satisfy
the
inequality:
theof
optimal
efficiencies
have a similar
and . Besides,
η behavior
> η following
> η with
> η respect
> η , to the parameters ,
(24) all
C
ME
EP
MP
CA
of these optimal efficiencies satisfy the following inequality:
The previous inequality was
Barranco-Jiménez
et al. for the case of both (24)
ηCrecently
> η MEobtained
> ηEP >byη MP
> ηCA ,
endoreversible and nonendoreversible
models
ηC > ηME
> ηEP of
> ηheat
>thermal
ηCA , engines [21,40]. In addition, we can (24)
MP
see inprevious
Figures 2inequality
and 3 howwas
the optimal
the maximum ecological
regime
The
recentlyefficiency
obtainedunder
by Barranco-Jiménez
et al. function
for the case
of both
The
previous
inequality
was
recently
obtained
by Barranco-Jiménez
et al. for
the case
of both
leads
to
a
high
reduction
on
the
profits
in
comparison
with
those
at
maximum
power
output.
endoreversible and nonendoreversible models of heat thermal engines [21,40]. In addition, we can see
endoreversible
andreduction
nonendoreversible
models
heatefficiency
thermal for
engines
[21,40]. the
In addition,
However, this
is concomitant
with aof
better
each value
parameterwe can
in Figures
2 and 3 how
the optimal
efficiency
under
the maximum
ecological of
function
regime leads
Table 12 in
[11]
fractional
fuel costs
for several
kinds
fuels). Thisecological
value of the
efficiency
see in(see
Figures
and
3 for
how
the optimal
efficiency
under
theofmaximum
function
regime
to a high reduction on the profits in comparison with those at maximum power output. However,
leads to a high reduction on the profits in comparison with those at maximum power output.
this reduction is concomitant with a better efficiency for each value of the parameter f (see Table 1
However, this reduction is concomitant with a better efficiency for each value of the parameter
in [11] for fractional fuel costs for several kinds of fuels). This value of the efficiency provides a
(see Table 1 in [11] for fractional fuel costs for several kinds of fuels). This value of the efficiency
diminution of the energy rejected to the environment by the power plant. Therefore, the ecological
criterion becomes an appropriate procedure for the search for insight into an engine’s performance for
Entropy 2017, 19, 118
10 of 15
providing a less aggressive interaction with the environment. Therefore, we propose the following
simplified analysis:
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10 of 15
From the first law of thermodynamics we get,
provides a diminution of the energy rejected to the environment by the power plant. Therefore, the
Q L procedure
= Q H − W,
ecological criterion becomes an appropriate
for the search for insight into an engine’s (25)
performance for providing a less aggressive interaction with the environment. Therefore, we
where
Q L isthe
thefollowing
heat rejected
to theanalysis:
environment by the power plant. For the cases of heat transfer of
propose
simplified
the Newton
and
Dulong–Petit
types,
when the
plant works under maximum power conditions
From the first law of thermodynamics
we power
get,
by using Equations (3), (4) and (25), we get
=
−
,
(25)
!n
where
is the heat rejected
bynthe power plant.τFor the cases of heat transfer of
∗ to the environment
n
.
Q L (η MP
, τ, R) = gTH
(26)
(1 − η MP
) 1−
∗ under
the Newton and Dulong–Petit types, when the power plant
works
maximum power
R 1−
η MP
conditions by using Equations (3), (4) and (25), we get
In a similar way, the expressions when the power plant works under maximum ecological function
∗
(26)
, ,
= are,
1 respectively,
−
1−
.
and maximum efficient power conditions
1− ∗
In a similar way, the expressions when the power plant works undern maximum ecological
τ
∗
n
n
,
Qefficient
R) =conditions
gTH
1−
(27)
(1 − ηare,
function and maximum
power
L ( η ME , τ,
ME )respectively,
∗
!
R 1 − η ME
∗
, ,
=
1−
1−
1− ∗
,
τ
∗
n
n
Q L (η∗EP
, τ, R) = gTH
(1 − ηEP
) 1−
∗
, ,
=
1−
1−
R∗ 1 − .ηEP
(27)
!n
.
1−
(28)
(28)
The
solutions
were obtained
obtainedinina anumerical
numerical
manner
similarly
to the
The
solutionsofofEquations
Equations (26)–(28)
(26)–(28) were
manner
similarly
to the
equations
inin
Section
we consider
considerdifferent
different
values
maintenance
of the
equations
Section2.2.InInour
ouroptimization,
optimization, we
values
forfor
thethe
maintenance
of the
power
plant
(parameter
for each
eachthermodynamic
thermodynamic
regime
while
and we
we can
can observe
observe for
regime
thatthat
while
the the γ
power
plant
(parameterγ)) and
parameter
increases,
thethe
heat
rejected
to the
environment
slightly
diminishes,
as itasis itshown
in Figure
parameter
increases,
heat
rejected
to the
environment
slightly
diminishes,
is shown
in 9.
Figure 9.
(a)
(b)
(c)
Figure 9. Heat rejected for two values of the parameter
under: (a) maximum power conditions; (b)
Figure 9. Heat rejected for two values of the parameter γ under: (a) maximum power conditions;
maximum ecological function conditions; (c) maximum efficient power conditions
(b) maximum ecological function conditions; (c) maximum efficient power conditions.
In Figure 9, we show the amount of heat rejected to the environment for each one of the three
regimes of performance. As we can see, the heat rejected to the environment under the maximum
Entropy 2017, 19, 118
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2017,
118
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19,19,
118
In2017,
Figure
9, we
11 of 15
of 15
show the amount of heat rejected to the environment for each one of the11three
regimes
of performance.
As weless
canaggressive
see, the heat
rejected
to environment
the environment
under thewith
maximum
ecologicalfunction
functionpresents
presents aa less
impact
to
the
inincomparison
thethe
ecological
aggressive
impact
to
the
environment
comparison
with
ecological
function
a less10a,b).
aggressive
impact
to the environment
comparison
other
other two
regimespresents
(see Figure
We can
also observe
in Figure 10inthat
when thewith
cost the
of the
other
two regimes
(see10a,b).
FigureWe
10a,b).
Weobserve
can also
observe
inthat
Figure
10
that
when
the
cost of the
two
regimes
(see
Figure
can
also
in
Figure
10
when
the
cost
of
the
maintenance
maintenance is taken into account, the rejected heat slightly diminishes in the three regimes of is
maintenance
is taken
into account,
the rejected
heatinslightly
diminishes
the three regimes
of
taken
into account,
the rejected
heat slightly
diminishes
the three
regimes of in
performance
considered.
performance
considered.
performance considered.
(a)
(b)
(b) maximum power
Figure 10. Ratio of(a)
heats rejected between: (a) ecological function conditions and
Figure 10. Ratio of heats rejected between: (a) ecological function conditions and maximum power
conditions
for two
valuesrejected
of the parameter
; (b)
efficientfunction
power conditions
and maximum power
Figure
10. Ratio
of values
heats
between: (a)
ecological
conditions
conditions
for two
of the parameter
γ; (b)
efficient power conditions
and maximum power
conditions
for
two
values
of
the
parameter
.
conditions for two
two values
valuesof
ofthe
theparameter
parameterγ. ; (b) efficient power conditions and maximum power
conditions for two values of the parameter .
In Figure 11, we show the ratio between the amounts of heat rejected for a heat transfer law of
Figure 11, we
show the ratio between the amounts of heat rejected for a heat transfer law of the
theIn
Dulong–Petit
type.
In Figure 11, we show the ratio between the amounts of heat rejected for a heat transfer law of
Dulong–Petit type.
the Dulong–Petit type.
Figure 11. Ratio of heat rejected between efficient power and maximum power conditions and
ecological function and maximum power conditions for a heat transfer of the Dulong–Petit type and
for two
theheat
parameter
Figure
11.values
Ratioofof
rejected. between efficient power and maximum power conditions and
Figure 11. Ratio of heat rejected between efficient power and maximum power conditions and
ecological function and maximum power conditions for a heat transfer of the Dulong–Petit type and
ecological
function
and maximum
powerlaw
conditions
for a heat transfer
of the
Dulong–Petit
type
Furthermore,
by applying
the second
of thermodynamic
(see Figure
1) we
get,
for
values
of the
. γ.
andtwo
for two
values
ofparameter
the parameter
=
− .
(29)
Furthermore, by applying the second law of thermodynamic (see Figure 1) we get,
Furthermore,
by applying
the (4)
second
law of
thermodynamic
(see Figure 1)
Using Equations
(1), (3) and
and after
some
algebraic manipulations,
forwe
theget,
cases of heat
=
−
.
(29)
transfer of the Newton and Dulong–Petit types, the entropy production under maximum power
QH − W
QH
conditions can be calculated by the following
σ = expression:
−
.
(29)
Using Equations (1), (3) and (4) and after Tsome
algebraic
manipulations, for the cases of heat
TH
L
transfer of the Newton and Dulong–Petit
∗
∗the entropy production under maximum power
(30)heat
, , (4) and
= 1after
−types,
−some
1−
.
∗
Using Equations (1), (3) and
algebraic
for the cases of
1− manipulations,
conditions can be calculated by the following expression:
transfer of the Newton and Dulong–Petit types, the entropy production under maximum power
Similarly to Equation (30), the expressions when the power plant works under maximum
conditions can be calculated by
∗ the following expression:
∗
(30)
, , efficient
= 1−
− conditions
1 − are, respectively,
.
ecological function and maximum
power
1− ∗
!n
τ
∗∗
∗ ∗
Similarly to Equation
theR)=expressions
under maximum
(31)(30)
, , ,τ,
1 −the power ,plant
works
σMP ((30),
η MP
= 1(1−− τ−− ηwhen
.
MP ) 1 −1− ∗
∗
R
1
−
η
ecological function and maximum efficient power conditions are, respectively,
MP
∗
, ,
= 1− −
∗
1−
1− ∗
,
(31)
Entropy 2017, 19, 118
12 of 15
Similarly to Equation (30), the expressions when the power plant works under maximum
ecological function and maximum efficient power conditions are, respectively,
τ
∗
∗
σME (η ME
, τ, R) = (1 − τ − η ME
) 1−
∗
R 1 − η ME
Entropy 2017, 19, 118
τ
∗
∗
σEP (ηEP
, τ, R) = (1 − τ − ηEP
) 1−
∗
∗
∗
R
1
−
ηEP
, ,
= 1− −
1−
.
∗
!n
,
(31)
12 of 15
!n
.
1−
(32)
(32)
It is
thatthat
all of
were made
a numerical
form. The
behavior
is important
importanttotomention
mention
allthe
of calculations
the calculations
were in
made
in a numerical
form.
The
of the above
equations
can be observed
in Figurein12.Figure
As can
the 12b,
entropy
behavior
of the
above equations
can be observed
12.be
Asseen
can in
be Figure
seen in12b,
Figure
the
production
of the power
when
it iswhen
working
under maximum
ecologicalecological
conditions
is less than
entropy
production
of theplant
power
plant
it is working
under maximum
conditions
is
when
it
is
working
under
maximum
power
conditions.
less than when it is working under maximum power conditions.
between the
the amounts
amounts of
of entropy
entropy production
production for
for two
two regimes:
regimes:
In Figure 13a, we show the ratio between
maximum
ecological
function
and
maximum
power.
In
a
similar
manner,
in
Figure
13b,
we
show
maximum ecological function and maximum power. In a similar manner, in Figure 13b, we show
the
the
ratio
between
the
amounts
of
entropy
production
for
maximum
efficient
power
conditions
ratio between the amounts of entropy production for maximum efficient power conditions and
maximum power conditions.
conditions. In
In addition,
addition, in
in Figure
Figure14,
14,we
weshow
showthe
theratios
ratiosofofentropy
entropyproduction
productionfor
fora
heat
transfer
of
the
Dulong–Petit
type.
In
all
of
our
numerical
calculations
we
have
attempted
to
use
a heat transfer of the Dulong–Petit type. In all of our numerical calculations we have attempted to
for the
involved
parameters
onlyonly
typical
values
reported
in the
literature.
use
for distinct
the distinct
involved
parameters
typical
values
reported
in FTT
the FTT
literature.
(a)
(b)
(c)
Figure
12.Entropy
Entropyproduction
production
under:
(a) maximum
conditions
twoofvalues
of the
Figure 12.
under:
(a) maximum
powerpower
conditions
for two for
values
the parameter
parameter
;
(b)
maximum
ecological
function
conditions
for
two
values
of
the
parameter
; (c)
γ; (b) maximum ecological function conditions for two values of the parameter γ; (c) maximum efficient
maximum
efficientfor
power
conditions
two values
power conditions
two values
of thefor
parameter
γ. of the parameter .
(a)
(b)
Figure 13. Ratio of entropy production between: (a) ecological function conditions and maximum
(c)
Figure 12. Entropy production under: (a) maximum power conditions for two values of the
; (c)
13 of 15
maximum efficient power conditions for two values of the parameter .
parameter
Entropy
2017, 19, 118; (b) maximum ecological function conditions for two values of the parameter
(a)
(b)
Figure 13. Ratio of entropy production between: (a) ecological function conditions and maximum
Figure 13. Ratio of entropy production between: (a) ecological function conditions and maximum
power conditions for two values of the parameter ; (b) ecological function conditions and
power conditions for two values of the parameter γ; (b) ecological function conditions and maximum
maximum
power conditions for two values of the parameter .
Entropy
2017, 19, 118
13 of 15
power conditions for two values of the parameter γ.
Figure 14.
14. Ratio
Ratio of
of entropy
entropy production
production between
between efficient
efficient power
power and
and maximum
maximum power
power and
and ecological
ecological
Figure
function
and
maximum
power
conditions,
respectively,
for
a
heat
transfer
of
the
Dulong–Petit
function and maximum power conditions, respectively, for a heat transfer of the Dulong–Petit type.
type.
4. Conclusions
Conclusions
4.
In this
made
a thermoeconomic
study
of anofirreversible
simplified
thermal
power
In
this work,
work,we
wehave
have
made
a thermoeconomic
study
an irreversible
simplified
thermal
plant model
so-called
NovikovNovikov
engine). engine).
This irreversible
model modifies
the results
power
plant (the
model
(the so-called
This irreversible
model modifies
theobtained
results
by meansby
ofmeans
an endoreversible
one due to
the
inclusion
of the engine’s
internal dissipation
through
obtained
of an endoreversible
one
due
to the inclusion
of the engine’s
internal dissipation
the lumped
R (see Figures
and 8). 7Inand
our8).
study,
westudy,
use different
transfer
the
through
the parameter
lumped parameter
R (see7Figures
In our
we use heat
different
heatlaws:
transfer
Newton’s
law
of
cooling,
the
Stefan–Boltzmann
radiation
law,
the
Dulong–Petit’s
law
and
another
laws: the Newton’s law of cooling, the Stefan–Boltzmann radiation law, the Dulong–Petit’s law and
phenomenological
heat transfer
We also
take
into
account
a costaccount
associated
to the
maintenance
of
another
phenomenological
heatlaw.
transfer
law.
We
also
take into
a cost
associated
to the
the power plant
which
weplant
assume
as proportional
the power output
the plant.
inclusion
maintenance
of the
power
which
we assume astoproportional
to the of
power
outputThe
of the
plant.
of
this
new
cost
has
a
noticeable
impact
on
the
improvement
of
the
optimal
efficiencies
of
the
plant
The inclusion of this new cost has a noticeable impact on the improvement of the optimal efficiencies
model,
nevertheless,
the benefit functions
diminish.
This
effect isThis
onlyeffect
relevant
when
the plant
model
of
the plant
model, nevertheless,
the benefit
functions
diminish.
is only
relevant
when
the
worksmodel
underworks
ME and
EP regimes;
when
the plant
model
maximum
regime,
plant
under
ME andhowever,
EP regimes;
however,
when
theworks
plant at
model
workspower
at maximum
the benefits
diminish
and the
optimaland
efficiency
practically
does not
change.does
We found
that when
power
regime,
the benefits
diminish
the optimal
efficiency
practically
not change.
We
the Novikov
model
the ecological
function,
it remarkably
reduces
the rejected
heatthe
to
found
that when
the maximizes
Novikov model
maximizes
the ecological
function,
it remarkably
reduces
the
environment
in
comparison
to
the
rejected
heat
in
the
case
of
a
power
plant
model
working
rejected heat to the environment in comparison to the rejected heat in the case of a power plant
under the
maximum
regime (see
Figures
9–11).
InFigures
addition,
we analyzed
the we
effect
of internal
model
working
underpower
the maximum
power
regime
(see
9–11).
In addition,
analyzed
the
irreversibilities
on the
reduction of power
and on
total entropy
the model.
In
effect
of internal
irreversibilities
on theoutput
reduction
ofthe
power
output production
and on theoftotal
entropy
summary,
we
have
shown
that
the
maximum
ecological
regime
has
more
advantages
in
the
way
of
production of the model. In summary, we have shown that the maximum ecological regime has
thermoeconomic
the other mentioned
operating
regimes.
more
advantages performance
in the way ofthan
thermoeconomic
performance
than
the other mentioned operating
regimes.
Acknowledgments: The authors acknowledge CONACYT, EDI-IPN, and COFAA-IPN for supporting this
work.
Author Contributions: All three authors contributed equally to the present work. All authors have read and
approved the final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Entropy 2017, 19, 118
14 of 15
Acknowledgments: The authors acknowledge CONACYT, EDI-IPN, and COFAA-IPN for supporting this work.
Author Contributions: All three authors contributed equally to the present work. All authors have read and
approved the final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
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