# On reversible, endoreversible, and irreversible heat device cycles versus
Carnot cycle: A pedagogical approach to account for losses

Gonzalez-Ayala J[1], Angulo-Brown F[2], Calvo Hernández[1] A and Velasco S[1]
[1] Departamento de Física aplicada, Universidad de Salamanca, 37008
Salamanca, Spain
[2] Departamento de Física, Escuela Superior de Física y Matemáticas,
Instituto Politécnico Nacional, Edif. No. 9, U.P. Zacatenco, 07738, México D.
F., México

###### Abstract

In this work we analyze the deviations of reversible cycles (for both heat
engines and refrigerators) from the corresponding Carnot cycle operating
between the same extreme temperatures and deviations of irreversible cycles
from its corresponding reversible realization while putting emphasis on the
corresponding losses. The endoreversible models fit in the proposed framework.
Two suitable loss factors, which don’t need the explicit calculation of
entropy variations, are introduced. The behavior of these factors and their
interplay allow for a clear and pedagogical visualization of where external
and internal irreversibilities are located and their intensities in terms of
the main variables describing the cycle. The analysis could be used as a
starting point for more advanced studies on modeling and optimization of real
devices and installations.

###### pacs:

05.70.Ln, 05.77.-a; 84.60.Bk

This is the preprint of the published verison available at
10.1088/0143-0807/37/4/045103

##  1 Introduction

It is known from the maximum work theorem [1, 2, 3] that for all processes,
leading a thermodynamical system from an initial to a final state, the
delivery of work is maximum for a reversible process. Moreover, from Carnot
theorem and its equivalency with second law of thermodynamics it is known that
the efficiency of an energy converter operating between two external
reservoirs is given by the Carnot efficiency, for heat engines, and the Carnot
coefficient of performance (COP) for refrigerators. We stress that
reversibility along any thermodynamic process requires a succession of
infinite equilibrium states, thus implying infinite time of realization. This
is not the case of heat engines in real life, where we use heat engines
working with a non-zero net power output operating at finite times. In this
way, it has been long known that the actual performance of any heat device is
accompanied by losses in regard to a reversible realization, meanwhile the
reversible realization will display a decrement in its performance with
respect to the Carnot realization.

Below, two statements over a generic non-isothermal energy converter will be
used here to account for two classes of losses. We consider both heat engines
and refrigerators devices and the focus is on the losses of each device for
the specific job it was designed: to produce work for a heat engine and to
extract cooling heat for a refrigerator. To get this and from the perspective
of an introductory thermodynamics, we introduce in each case two suitable loss
factors, which don’t need the explicit calculation of entropy variations, a
subtle and sometimes difficult task. The endoreversible systems composed by an
inner reversible subsystem coupled with two external reservoirs by an
irreversible interaction are included in a natural way in the proposed
framework.

These factors, closely related with the second-law efficiency [4, 5], allow
for a clear visualization of the behavior of losses, their nature and
intensity in terms of the main variables describing the cycle as well as of
the resulting parametric power versus efficiency curves. Thus, it can guide a
first approach to the design and optimization of real heat devices, an issue
of primary interest due to the energy shortage [6]. Besides, the results can
be instructive in order to stressing how the imperfections of any heat
converter, mainly irreversibilities coming from finite-time and finite-size,
provoke that the heat input in a heat engine is not fully available as work
output, a situation we face also every day in real life.

The paper is organized as follow: In Sect. 2 we define the two loss factors on
thermodynamics basis. In order to show how the loss factors work and their
influence on the main energetic magnitudes we present two representative and
pedagogical cycles: a Brayton-like cycle in Sect. 3 and a Carnot-like
irreversible heat engine in Sect. 4. In Sect. 5 the analysis is extended to
refrigeration cycles in order to show its unified validity with heat devices.
Finally, some specific remarks on the loss factors are presented in Sect. 6.

##  2 Heat engine cycles.

Figure 1: Schematic diagram of a Carnot cycle (a), a generic heat engine (b)
and a direct heat leak (c) between two external reservoirs at temperatures
$T_{h}$ and $T_{c}$ (see text). Work output decreases from the maximum
possible value $W_{C}$ in (a) to its minimum $W=0$ value in (c).

Usually the second law of Thermodynamics for closed systems is expressed as
the increasing entropy principle $\Delta S\geq 0$ [1, 2, 3, 4, 5], where the
equality applies to reversible processes. When applied to a heat engine
performing a cyclic process it states that the best conversion of an amount of
heat input $Q_{h}\geq 0$ in work output $W\leq 0$ is trough a Carnot heat
engine. For such a conversion, see Fig. 1a, the efficiency is
$\eta_{C}=1-\tau$, where $\tau\equiv T_{c}/T_{h}\leq 1$ is the ratio of the
temperature of the cold thermal bath $T_{c}$ to the temperature of the hot
thermal bath $T_{h}$; the work output $|W|_{max}=|Q_{h}|\eta_{C}\equiv|W_{C}|$
is thus the maximum possible work delivered from a fixed amount of heat
$Q_{h}$. Obviously, the variation of the overall entropy is $\Delta
S=-\frac{|Q_{h}|}{T_{h}}+\frac{|Q_{c}|}{T_{c}}=0$ since the inherent
reversibility of the Carnot cycle and no lost work is generated
$|W_{lost}|=T_{c}\Delta S=0$. On the opposite side we can imagine a situation
where the same heat $Q_{h}$ is entirely dissipated in the cold thermal bath
without the realization of work output (see Fig. 1c). In these circumstances
$\Delta
S=-\frac{|Q_{h}|}{T_{h}}+\frac{|Q_{h}|}{T_{c}}=\frac{|Q_{h}|}{T_{c}}\eta_{c}\equiv\Delta
S_{max}$ is the maximum possible value of the entropy variation and, as a
consequence, $|W_{lost}|=T_{c}\Delta S_{max}$ amounts the maximum possible
value that becomes unavailable to do work. In between these two limit
situations we consider a generic heat engine $M$ working between the same
external heat reservoirs $T_{h}$ and $T_{c}$ and with the same heat input
$Q_{h}$ (see Fig. 1b). Now, according with the first law for any cyclic
process ($\Delta U=0$)
$|W|=|Q_{h}|-|Q_{c}|=|Q_{h}|(1-\frac{|Q_{c}|}{|Q_{h}|})\equiv|Q_{h}|\eta$,
where $\eta$ is the first-law conversion efficiency (i.e. the fraction of heat
input converted in work output) and $\Delta
S=-\frac{|Q_{h}|}{T_{h}}+\frac{|Q_{c}|}{T_{c}}=\frac{|Q_{h}|}{T_{c}}(\eta_{c}-\eta)$.
In order to account for deviations in work output from the Carnot performance
we propose a total loss factor $d$ measuring the lost work given by the heat
engine $M$ with respect to the maximum possible work output performed by the
Carnot cycle, $W_{C}$:

$\displaystyle
d=\frac{|W|_{lost}}{|W|_{max}}=\frac{|W|_{max}-|W|}{|W|_{max}}\equiv\frac{|W|_{C}-|W|}{|W|_{C}}=$
$\displaystyle=\frac{\eta_{C}-\eta}{\eta_{C}}\leq 1$ (1)

where the last equality is due to the assumption of the same heat input.

From this definition we can divide the total losses distinguishing the
reversible realization of $M$ from the corresponding Carnot cycle, and the
real irreversible performance from the reversible case, respectively, by using
the following relation:

$d=\frac{\eta_{C}-\eta}{\eta_{C}}\equiv\frac{\eta_{C}-\eta_{rev}}{\eta_{C}}+\frac{\eta_{rev}-\eta}{\eta_{rev}}\left(\frac{\eta_{rev}}{\eta_{C}}\right)\leq
1$ (2)

or

$d=d_{e}+\left(\frac{\eta_{rev}}{\eta_{C}}\right)d_{i}=d_{e}+d_{i}-d_{e}d_{i}$
(3)

where

$d_{e}=1-\frac{\eta_{rev}}{\eta_{C}}\leq 1$ (4)

and

$d_{i}=1-\frac{\eta}{\eta_{rev}}\leq 1$ (5)

The factor $d_{e}$ accounts for differences in work output between the Carnot
and the idealized (reversible) cycles while the second one, $d_{i}$, is a
measure of the difference between the idealized and actual performances. Note
in Eq. 3 that $d=d_{e}+d_{i}-d_{e}d_{i}$ includes a coupling term $d_{e}d_{i}$
as a notable feature.

Obviously, if $M$ is a reversible Carnot heat engine working between the same
extreme temperatures and with the same heat input $d_{e}=0$. For any different
reversible cycle $\eta_{rev}\leq\eta_{C}$ and $d_{e}\leq 1$. Since $d_{e}$
involves only the consideration of reversible processes, this loss factor
could be interpreted and measured in geometric terms: for the same heat input
in the Carnot and in the idealized heat engines,
$d_{e}=\frac{\eta_{C}-\eta_{rev}}{\eta_{C}}=\frac{|W_{C}|-|W_{rev}|}{|W_{C}|}$
accounts for the (normalized) difference between the areas enclosed by the two
cycles. In this geometrical context the losses are related to a comparison of
areas which do not generate a real dissipation; this will be shown
specifically in the next section (for an example, see Fig. 3a). Alternatively,
as it will be seen in section 4, $d_{e}$ can be interpreted in terms of the
so-called endoreversibility [6] since it represents an internally reversible
cycle coupled to external heat baths. In this case the losses accounted for
$d_{e}$ should be attributed to the coupling of the inner part with the
external heat baths and the corresponding entropy-producing irreversibilities
variation. The quantitative equivalence of both interpretations has been
pointed out in [7], where the role of the dissipation of an endoreversible
engine in the context of finite-time thermodynamics (FTT) has been analyzed.

On the other hand, $d_{i}$ measures the differences in work output between the
reversible and irreversible performances of $M$ and its origin is due to
internal irreversibilities. We can say that $d_{i}$ is operational in nature
since it accounts for internal irreversibilities (as frictions, viscosities
and so on) in a real engine with respect to its idealized, reversible
counterpart

##  3 Brayton-like heat engine

In order to show how the loss factors work and their influence on the power
output and efficeincy of a heat engine we elect as a first representative
example a simple irreversible Brayton heat engine, which is well known both in
classical equilibrium thermodynamics and in more advanced disciplines devoted
to optimization and control of energy conversion processes. The Brayton cycle
is the usual model of a continuous-combustion gas turbine engine where
compressed air enters in a combustion chamber and it is mixed with fuel. The
mixture is burned out and the resulting gas at high temperature and high
pressure is expanded across the turbine, where work is done and the gas
returns to approximately the atmospheric pressure. Then, gas is exhausted to
the ambient.

Figure 2: Brayton-like cycle in the $p$-$V$ and $T$-$S$ diagrams.
Irreversibilities are introduced in the adiabatic processes $1\rightarrow 2$
and $3\rightarrow 4$ (see text).

Main internal irreversibilities in both recuperative and non-recuperative
Brayton heat engine includes effects coming from fluid friction of air in the
turbine and compressor. Volume increases in the heat addition and drops in the
heat rejection processes [8, 9, 10]. Here and for simplicity we avoid the
effects of the regenerator and the pressure drops while the irreversibilities
in the compressor and turbine are accounted by the corresponding isentropic
efficiencies $\eta_{c}$ and $\eta_{t}$, which quantify the degree to which
adiabatic branches deviate from being isentropic. We assume a mass flow rate
$\dot{m}$ as an ideal gas with constant heat capacities and running an ideal
air-standard Brayton cycle $1\rightarrow 2\rightarrow 3\rightarrow 4$ (see
Fig. 2). Then $\eta_{c}=\frac{T_{2s}-T_{1}}{T_{2}-T_{1}}\leq 1$ and
$\eta_{t}=\frac{T_{3}-T_{4}}{T_{3}-T_{4s}}\leq 1$. Now
$|\dot{Q}_{in}|=\dot{m}c_{p}(T_{3}-T_{2})$ and
$|\dot{Q}_{out}|=\dot{m}c_{p}(T_{4}-T_{1})$. These magnitudes can be easily
worked out in terms of the main characteristics of the cycle as
$a=r_{p}^{\frac{\gamma-1}{\gamma}}$
($r_{p}=\frac{P_{2s}}{P_{1}}=\frac{P_{3}}{P_{4}s}$), $\gamma=c_{p}/c_{v}$,
$\tau=\frac{T_{1}}{T_{3}}$, and the isentropic efficiencies $\eta_{c}$ and
$\eta_{t}$. The final results read as [8, 9, 10]:

Figure 3: Behavior of $\overline{W}(a,\tau,\eta_{c},\eta_{t})$,
$\eta(a,\tau,\eta_{c},\eta_{t})$, $d_{e}(a,\tau)$,
$d_{i}(a,\tau,\eta_{c},\eta_{t})$ and $d(a,\tau,\eta_{c},\eta_{t})$ for a
Brayton cycle; see Eqs. (7)-(11). In (a) it is shown the behavior of $d_{e}$,
$\eta$ and $\overline{W}$ for the reversible performance
($\eta_{c}=\eta_{t}=1$; that is, $d_{i}=0$). In (b) and (c) the evolution of
$d_{i}$, $d_{e}d_{i}$, and $d$ is also plotted for values
$\eta_{c}=\eta_{t}=0.9$ and $\eta_{c}=\eta_{t}=0.8$, respectively. In (d) the
parabolic and loop behavior of the work-efficiency parametric curves is
depicted. In all cases $\tau=0.2$ ($\eta_{C}=0.8$).

$\displaystyle|\dot{Q}_{in}|(a,\tau,\eta_{c})=\dot{m}c_{p}(T_{3}-T_{2})=$
$\displaystyle\dot{m}c_{p}T_{3}\left[1-\tau-\frac{\tau}{\eta_{c}}(a-1)\right]$
(6)
$\dot{W}(a,\tau,\eta_{c},\eta_{t})=\dot{m}c_{p}T_{3}\left[\eta_{t}-\frac{\tau
a}{\eta_{c}}\right]\left[1-\frac{1}{a}\right]$ (7)
$\eta(a,\tau,\eta_{c},\eta_{t})=\frac{\left(\eta_{t}-\frac{\tau
a}{\eta_{c}}\right)\left(1-\frac{1}{a}\right)}{1-\tau-\frac{\tau}{\eta_{c}}(a-1)}$
(8)

Note that when $\eta_{c}=\eta_{t}=1$, i.e., compressor and turbine internal
irreversibilities are absent, Eqs. (6)-(8) reduce to the well-known equations
of a reversible Brayton cycle. In particular, the reversible efficiency is

$\eta_{rev}(a,\tau)=\eta(a,\tau,\eta_{c}=1,\eta_{t}=1)=1-\frac{1}{a}$ (9)

From Eqs. (8) and (9), the loss factors in Eqs. (4) and (5) are given,
respectively, by:

$d_{e}(a,\tau)=1-\frac{\eta_{rev}(a,\tau)}{1-\tau}=\frac{1-a\tau}{a(1-\tau)}$
(10) $\displaystyle
d_{i}(a,\tau,\eta_{c},\eta_{t})=1-\frac{\eta(a,\tau,\eta_{c},\eta_{t})}{\eta_{rev}(a,\tau)}=$
$\displaystyle=\frac{\eta_{c}(1-\tau)+\tau-\eta_{c}\eta_{t}}{\eta_{c}(1-\tau)+\tau-a\tau}$
(11)

In Fig. 3a we plot the behavior of normalized work output,
$\overline{W}(a,\tau)\equiv\dot{W}(a,\tau)/\dot{m}c_{p}T_{3}$, the efficiency,
and the external loss factor $d_{e}$ for the idealized version of the Brayton
heat engine $\eta_{c}=\eta_{t}=1$ ($d_{i}\equiv 0$). In Figs. 3(b)-3(c) we
show the evolution of the normalized power output
$\overline{{W}}(a,\tau,\eta_{c},\eta_{t})\equiv\dot{W}(a,\tau,\eta_{c},\eta_{t})/\dot{m}c_{p}T_{3}$,
the efficiency, the operating loss factor $d_{i}$, the coupling term
$d_{e}d_{i}$, and the total loss factor $d$ for values of
$\eta_{c}=\eta_{t}=0.9$ and $0.8$, which account for increasing
irreversibilities.

In the reversible performance, Fig. 3a, we observe how the efficiency
monotonically increases from zero (at $a=1$) to the maximum possible Carnot
value $\eta_{C}=1-\tau=0.8$ (at $a=1/\tau=5;\tau=0.2$) while $d_{e}$, which
measures departures from the Carnot efficiency, shows a monotonically
decreasing behavior from $1$ to $0$. The work output shows a maximum value at
$a=1/\sqrt{\tau}$ and two null values at the extreme allowed values of $a$:
$a=1$ and $a=1/\tau$. A peculiar value is the efficiency at maximum work which
take the values
$\eta_{rev}(a=\frac{1}{\sqrt{\tau}},\tau)=1-\sqrt{\tau}\equiv\eta_{CA}$, i.e.,
the Curzon-Ahlborn efficiency [11, 12]. The parametric plot work-efficiency is
an open parabolic-like curve (see Fig. 3d), characteristic of the so-called
endoreversible cycles (internally reversible but with thermal couplings to
external heat baths) and also of the most idealized cycles considered in
classical equilibrium thermodynamics (Otto, Diesel, Atkinson). However, only
those with two adiabatic branches alternating with two others of the same
nature (Brayton, Otto) show the CA-efficiency at maximum work while those with
two adiabatic branches alternating with two others of the different nature
(Diesel, Atkinson) show an efficiency at maximum power depending on the
specific physical properties of the working fluid, though nearly to the CA-
value [13, 14].

The evolution with irreversibilities is shown in Figs. 3b and 3c. Opposite to
$d_{e}$, the internal loss factor $d_{i}$ shows a monotonically increase with
the pressure ratio $a$ since, as expected, higher pressure ratios induces
higher internal losses in the compressor and turbine. The total loss factor
$d=d_{e}+d_{i}-d_{e}d_{i}$ (see Eq. 3) thus shows an inverted parabolic-like
shape modulated by the coupling term with a well defined minimum value at the
same $a$-value as the maximum of the efficiency (and closer to maximum power)
and with two maximum values ($d=1$) when efficiency (and power) are null.
Indeed, as the frictional fluid losses increase (lower values of $\eta_{t}$
and $\eta_{c}$) $d_{i}$ (and $d_{e}d_{i}$) increases in intensity; as a direct
consequence we observe lower values of the efficiency accomplished with lower
values of power output. The evolution of the parametric plot power-efficiency
is now loop-like (see Fig. 3d) with close but non-coincident maximum power and
maximum efficiency points. Such behavior has been identified as a sign of
actual heat engine models [10, 14, 15]. As internal irreversibilities increase
the loop-like behaviors are narrower and with smaller maximum power and
maximum efficiency values.

##  4 Irreversible Carnot-like heat engine

Figure 4: Scheme of an irreversible Carnot-like heat engine (left) and its T-S
diagram (right).

As a second and representative example, we elect an irreversible Carnot-like
heat engine. This model, as an extension of the original Curzon-Ahlborn
endoreversible heat engine and related simplified versions [16, 17], has been
widely used in finite-time thermodynamics [6]. Following [18] we assume a
working system running a Carnot-like cycle where the isothermal heat addition
and heat rejection processes are at temperatures $T^{\prime}_{h}$ and
$T^{\prime}_{c}$, respectively, see Fig. 4. The inner part is coupled to
external heat baths at temperatures $T_{h}$ and $T_{c}$ by appropriate heat
transfer laws, $Q_{h}$ and $Q_{c}$, and heat leak $Q_{i}$ between the external
reservoirs is allowed. The simplest mathematical version corresponds to the
assumption that all heat transfer processes follow a linear law with the
temperature difference and constant thermal conductances $\sigma$. In this
case the heat fluxes can be mathematically described by

${|\dot{Q}_{h}|}=\sigma_{\rm h}\left(T_{\rm h}-T^{\prime}_{\rm
h}\right)\equiv\sigma_{\rm h}T_{h}\left(1-\frac{1}{a_{h}}\right)$ (12)
${|\dot{Q}_{c}|}=\sigma_{\rm c}\left(T^{\prime}_{\rm c}-T_{\rm
c}\right)\equiv\sigma_{\rm c}T_{c}\left(a_{c}-1\right)$ (13)
${|\dot{Q}_{i}|}=\sigma_{\rm i}\left(T_{\rm h}-T_{\rm
c}\right)\equiv\sigma_{\rm i}T_{h}\left(1-\tau\right)$ (14)

where $a_{h}=T_{h}/T^{\prime}_{h}\geq 1$, $a_{c}=T^{\prime}_{c}/T_{c}\geq 1$.

The reversible or irreversible nature of the inner cycle is accounted by the
Clausius relation

$I\frac{|\dot{Q}_{c}|}{T^{\prime}_{c}}-\frac{|\dot{Q}_{h}|}{T^{\prime}_{h}}=0.$
(15)

If the inner cycle is reversible $I=1$ and if irreversible $I<1$. The two
internal temperatures $T^{\prime}_{h}$ and $T^{\prime}_{c}$ (i.e., $a_{h}$ and
$a_{c}$) are not independent variables. Substituting Eqs. (12) and (13) in
(15) we obtain:

$a_{c}=\frac{I}{I-\sigma_{hc}(a_{h}-1)}$ (16)

where $\sigma_{hc}=\sigma_{h}/\sigma_{c}$. From the above equations, the power
$\dot{|W|}=|\dot{Q}_{h}|-|\dot{Q}_{c}|$ and the efficiency
$\eta=\dot{|W|}/(|\dot{Q}_{h}|+|\dot{Q}_{i}|)$ read as:

$\displaystyle\dot{|W|}(a_{h},I,\sigma_{h},\sigma_{c},\tau)=$
$\displaystyle\sigma_{h}T_{h}\frac{I\left(a_{h}-1\right)-\sigma_{hc}\left(a_{h}-1\right)^{2}-\tau\left(a_{h}^{2}-a_{h}\right)}{a_{h}\left(I+\sigma_{hc}\right)-\sigma_{hc}a_{h}^{2}}$
(17) $\displaystyle\eta(a_{h},I,\sigma_{hc},\sigma_{ih},\tau)=$
$\displaystyle\left[1-\frac{a_{h}\tau}{I-\sigma_{hc}\left(a_{h}-1\right)}\right]\left[\frac{\left(a_{h}-1\right)}{\left(a_{h}-1\right)+\sigma_{ih}\left(1-\tau\right)a_{h}}\right]$
(18)

where $\sigma_{ih}=\sigma_{i}/\sigma_{h}$. If $I=1$ and $\sigma_{i}=0$ the
model recovers the original Curzon-Ahlborn endoreversible model (as an
internally reversible Carnot cycle externally coupled to external heat baths
by linear heat transfer laws)[3, 11].

The consideration of values $I<1$ and $\sigma_{i}\neq 0$ thus give additional
information on the heat leak and internal dissipations. As in the previous
section, the two loss factors are defined as:

$d_{e}(a_{h},\sigma_{hc},\tau)=1-\frac{\eta(a_{h},I=1,\sigma_{hc},\sigma_{ih}=0,\tau)}{1-\tau}$
(19)
$d_{i}(a_{h},I,\sigma_{hc},\sigma_{ih},\tau)=1-\frac{\eta(a_{h},I,\sigma_{hc},\sigma_{ih},\tau)}{\eta(a_{h},I=1,\sigma_{hc},\sigma_{ih}=0,\tau)}$
(20)

Figure 5: Evolution of the behavior of $\eta$ and normalized power
$\overline{W}=\dot{W}/\sigma_{h}T_{h}$ (Eqs. 17 and 18) and $d_{e}$, $d_{i}$,
$d_{e}d_{i}$ and $d$ (see Eqs. 19 and 20) for the irreversible Carnot-like
heat engine for the labeled parameter $I$ and $\sigma_{ih}$. In (d) the
corresponding parametric plots of the $\overline{W}-\eta$ curves are shown. In
all cases $\sigma_{hc}=1$ and $\tau=0.2$.

In the endoreversible limit ($I=1$, $\sigma_{ih}=0$) the monotonic decrease of
the efficiency $\eta$ is accomplished by the corresponding monotonic increase
of the external loss factor $d_{e}$, see Fig. 5a. The influence of $I$ shows
the same behavior for the internal loss factor $d_{i}$, see Fig. 5b. As the
influence of the heat leak ($\sigma_{ih}\neq 0$) is taken into account we
observe how $d_{i}$, $d_{e}d_{i}$ play a central role yielding in $d$ an
inverted parabolic behavior, see Fig. 5c. As in above section for the Brayton
cycle, the minimum of $d$ is coincident with the maximum efficiency and closer
to the maximum power. The resulting parametric power-efficiency curves are
also shown in Fig. 5d. Obviously, particular differences in $d_{e}$, $d_{i}$,
$d_{e}d_{i}$ and $d$ between this model and the Brayton model arise due to the
different nature of the external couplings (isobaric heat transfers in a
Brayton heat engine or isothermal heat transfers in an endoreversible Carnot)
and of internal dissipation (non adiabatic compressor and turbine performances
in a Brayton cycle or heat leak and internal dissipation in the irreversible
Carnot-like engine).

##  5 Refrigeration cycles

Figure 6: Schematic diagram of a Carnot refrigeration cycle (a), a generic
refrigerator (b) and of a direct dissipative process of work input in heat (c)
between two external reservoirs at temperatures $T_{h}$ and $T_{c}$ (see
text). Cooling load decreases from (a) to (c).

The analysis for heat engine cycles in Sect. II can be extended for
refrigerator cycles. In this case our interest is the heat $Q_{c}\geq 0$
extracted for the working fluid from a thermal reservoir at $T_{c}$ (the cold
space). The second law states that the maximum coefficient of performance
$\epsilon=Q_{c}/W$ is obtained when the working fluid runs a Carnot
refrigeration cycle by the input of a work $W\geq 0$, see Fig. 6a. Then
$\epsilon_{C}=\frac{\tau}{1-\tau}$ and in this case the overall entropy
production is null, $\Delta S=\frac{|Q_{h}|}{T_{h}}-\frac{|Q_{c}|}{T_{c}}=0$,
and $Q_{c}\geq 0$ is the maximum possible value given by
$Q_{c}=W\epsilon_{C}=Q_{c,max}\equiv Q_{c,C}$. In the opposite side we have a
situation, see Fig 6c, where with the same work input $W$ the cycle works
between the same external temperatures without any heat extracted from the
thermal bath at $T_{c}$. All input work is dissipated to the hot reservoir. In
such a situation the entropy variation is the maximum possible $\Delta
S=\frac{|Q_{h}|}{T_{h}}=\frac{W}{T_{h}}=\Delta S_{max}$, the COP is null and
no heat is extracted from the low temperature reservoir, $Q_{c}=0$. In between
these two extreme cases we consider a generic refrigerator F working between
the same temperatures and with the same energy requirement $W$, see Fig. 6b.
Now $Q_{c}=W\epsilon$, $\Delta
S=\frac{Q_{c}+W}{T_{h}}-\frac{Q_{c}}{T_{c}}=\frac{W}{T_{h}}\left(1-\frac{\epsilon}{\epsilon_{C}}\right)$
and the difference of heat absorbed in comparison with the most favorable case
is $Q_{c,lost}=W(\epsilon_{C}-\epsilon)$. In order to account for losses we
focus on the ratio between the lost refrigeration load to the maximum possible
refrigeration load for the same work requirement:

$d=\frac{Q_{c,lost}}{Q_{c,max}}=\frac{Q_{c,max}-Q_{c}}{Q_{c,max}}\equiv\frac{Q_{c,C}-Q_{c}}{Q_{c,C}}=1-\frac{\epsilon}{\epsilon_{C}}$
(21)

where the last equality follows from the assumption of equal work input in
both cycles. Following the same steps that for heat engines we divide
deviations from the Carnot performance of any actual refrigerator in two
contributions as:

$d=\frac{\epsilon_{C}-\epsilon}{\epsilon_{C}}\equiv\frac{\epsilon_{C}-\epsilon_{rev}}{\epsilon_{C}}+\left(\frac{\epsilon_{rev}-\epsilon}{\epsilon_{rev}}\right)\frac{\epsilon_{rev}}{\epsilon_{C}}\leq
1$ (22)

or

$d=d_{e}+\left(\frac{\epsilon_{rev}}{\epsilon_{C}}\right)d_{i}=d_{e}+d_{i}-d_{e}d_{i}$
(23)

with

$d_{e}=1-\frac{\epsilon_{rev}}{\epsilon_{C}}\leq 1$ (24)

and

$d_{i}=1-\frac{\epsilon}{\epsilon_{rev}}\leq 1$ (25)

The application of above definition to a particular cycle needs the
specification of the reversible and irreversible models which are compared
with the Carnot refrigeration between the same external reservoirs. Here we
don’t present any particular result but the evaluation of the loss factors for
the inverted Brayton and Carnot-like cycles presented above could be an
instructive exercise for students and interested readers. However, special
care should be taken when a heat engine cycle is reversed [19]. Only the
Carnot cycle automatically yields a refrigerator (heat pump) cycle in all
circumstances. For other well-known cycles (as Otto, Diesel, Brayton,
Atkinson) the inversion can yield refrigerator cycles only under restricted
conditions for the involved temperatures [19].

##  6 Some remarks and conclusions

Notice that the form for counting losses expressed in Eq. (3) for heat engines
and Eq. (23) for refrigerators is the same. In any case, when one of the loss
factor is zero the total loss factor is due to the other one, as would be
expected. However, an interesting feature of this expression is the negative
crossing term in Eq. (3) or Eq. (23) when both kind of losses are present. In
this case the simple addition of losses give an over-counting which must be
decreased by the interplay accounted by the negative term. The interplay
between the external and internal loss factors to account for the so called
external and internal irreversibilities [20] could be of some utility to
explain observed reversible landmarks for efficiencies stemming from finite-
time thermodynamics [21] as the maximum work efficiency of some reversible
cycle. From the analyzed examples we see that loop-like behavior of the power-
efficiency in heat engines curves can be attributed in general to the internal
irreversibilities while the open, parabolic-like behavior comes from the
external losses. Accordingly, the interplay between both $d_{e}$ and $d_{i}$
being non zero accounts for the participation of both quantities in the
appearing of such behavior.

Recently, Wu [20] published an instructive paper on the three factors causing
the thermal efficiency of a heat engine to be less than unity. The first
factor is the nonzero reference point due to the unreachable absolute zero
temperature and, as a consequence, is induced by unavailability rather than
irreversibility. The second factor is the external irreversibility coming from
coupling of the heat engine to external heat sources; and the third one is
coming from the inherent internal irreversibility. From a quantitative point
of view Wu quantifies the second-law overall efficiency of the heat engine
$\eta_{II}=\eta_{I}/\eta_{C}$ [4, 5] as
$\eta_{II}=\eta_{int,ir}\eta_{ext,ir}$, where
$\eta_{ext,ir}=\eta_{rev}/\eta_{C}$ and $\eta_{int,ir}=\eta_{I}/\eta_{rev}$.
In these expressions $\eta_{rev}$ is the efficiency of ideal cycles as those
which are internally reversible but they may involve irreversibilities that
are external to the system, $\eta_{C}$ is the Carnot efficiency of the Carnot
cycle working between the same extreme temperatures, and $\eta_{I}$ is the
first-law thermal efficiency. However, one must be careful when considering
the second factor, since an external coupling with two reservoirs is a
characteristic of a endoreversible model rather than that of a reversible
system. In other words, it must be specified if the irreversibility is real
(intrinsic to the model) or coming from a geometrical nature (no real
dissipation). Note that in our analysis the total loss factor is
$d=1-\eta_{II}$ while $d_{e}=1-\eta_{ext,ir}$ and $d_{i}=1-\eta_{int,ir}$. For
refrigerator cycles, not considered by Wu [20], the similarity can be extended
straightforwardly: $d=1-\epsilon_{II}$ where $\epsilon_{II}$ is the second law
COP defined [4, 5] as the ratio of the firs-law COP $\epsilon_{I}$ to the
Carnot COP $\epsilon_{C}$, $\epsilon_{II}=\epsilon_{I}/\epsilon_{C}$.

In summary, the simple and pedagogical analysis presented here allows for a
clear visualization of two classes of losses in the specific job of a heat
device when measured from the corresponding Carnot performance. The first one
measures the losses due to external coupling of the idealized, reversible
operation with external heat baths while the second one measures losses of the
actual performance coming from internal dissipations. The analysis does not
need explicit evaluation of the entropy generation to link the thermodynamic
non-ideality of the design to the physical characteristics of the system, a
subtle and sometimes difficult task. The results could guide students and
readers to a more detailed knowledge on how external and internal losses,
together with their interplay, affect the performance of real heat devices.
Hence, it is a starting point for more advance studies on modeling and
optimization of real devices and installations, an unavoidable task in a
sustainable use of the natural resources.

### Acknowledgment

Angulo-Brown F an Gonzalez-Ayala J acknowledge CONACYT-MÉXICO and COFFA-SIP-
IPN-MÉXICO; Velasco S and Calvo Hernández A acknowledge Ministerio de Economía
y Competitividad of Spain under Grant ENE2013-40644R.

### References

## References

  * [1] Zemansky M W and Dittman R H 1997 Heat and Thermodynamics 7th edn (New York: McGraw-Hill)
  * [2] Cengel Y A and Boles A M 2002 Thermodynamics: An Engineering Approach 5th edn (Boston: McGraw-Hill)
  * [3] H. B. Callen, 1985 Thermodynamics and an introduction to Thermostatistics, 2nd Ed. (New York: J. Wiley and Sons ) 1985.
  * [4] A. Bejan, Advance Engineering Thermodynamics, 3rd edn (Hoboken: John Wiley and Sons, 2006) pp 109-115
  * [5] T. V. Marcella, Entropy production and the second law of thermodynamics: an introduction to the second law analysis , Am. J. Phys. 60(10), 888-895 (1992)
  * [6] Durmayaz A, Sogut O S, Sahin B and Yavuz H 2004 Optimization of thermal systems based on finite-time thermodynamics and thermoeconomics Progr. Energy Combust. Sci. 30 175–217
  * [7] Gonzalez-Ayala J 2014 Algunos aspectos relacionados con la segunda ley de la termodinámica: relaciones matemáticas, restricciones físicas y procesos de optimización. PhD Thesis ESFM-IPN-México
  * [8] Van Wylen G, Sonntag R E and C Borgnakke C 1994 Fundamentals of Classical Thermodynamics 4th edn (New York: J. Wiley and Sons )
  * [9] Calvo H A, Medina A, and Roco J M M 1995 Power and efficiency in a regenerative gas turbine, J. Phys D: Appl. Phys. 28 (10) 2020-2023
  * [10] Gordon J M and Huleihil M 1992 General performance characteristics of real heat engines, J. Appl. Phys. 72(3), 829-837
  * [11] Curzon F L and Alborn B 1975 Efficiency of a Carnot engine att maximum power output , Am. J. Phys.43, 22-24
  * [12] Calvo H A, Roco J M M, Medina A, Velasco S, Guzmán-Vargas L 2015 The maximum power efficiency $1-\sqrt{\tau}$: Research, education, and bibliometric relevance, The European Physical Journal Special Topics 224 (5), 809-823
  * [13] Leff H S 1987 Thermal efficiency at maximum work output: new results for old heat engines , Am. J. Phys. 55(7), 602-610
  * [14] Gordon J M 1991 Generalized power versus efficiency characteristics of heat engines: the thermoelectric generator as an instructive illustration, Am. J. Phys. 59(6), 551-555 (1991)
  * [15] Qin X, Chen L, Sun F and Wu C 2003 The universal power and efficiency characteristics for irreversible reciprocating heat engines cycles, Eur. J. Phys. 24, 359-366
  * [16] Agrawal D C 2009 A simplified version of the Curzon-Ahlborn engine, Eur. J. Phys. 30, 1173-1179
  * [17] Pescetti D 2014 On the optimization of endoreversible processes, Eur. J. Phys. 35, 025014
  * [18] Chen J 1994 The maximum power output and maximum efficiency of an irreversible Carnot heat engine, J. Phys. D: Appl. Phys. 27,1144-1149
  * [19] Ferreira da Silva M F 2012 Some considerations about thermodynamic cycles, Eur. J. Phys. 33, 13-42
  * [20] Wu J 2015 Three factors causing the thermal efficiency of a heat engine to be less than unity and their relevance to daily life, Eur. J. Phys. 36, 015008
  * [21] Gonzalez-Ayala J, Arias-Hernandez L A and Angulo-Brown F 2013 Connection between maximum-work and maximum-power thermal cycles, Physical Review E 88 (5), 052142