## 1. Introduction

A valuable tool for validating and improving knowledge of nature is using models. A scientific model is an abstract, conceptual, graphic, or visual representation of phenomena, systems, or processes to analyze, describe, explain, simulate, explore, control, and predict these phenomena or processes. A model allows determining a final result from appropriate data. The creation of models is essential for all scientific activity. Moreover, a given physics theory is a model for studying the behavior of a complete system. The model is applied in all areas of physics, reducing the observed behavior to more basic fundamental facts, and helps to explain and predict the behavior of physical systems under different circumstances.

In classical equilibrium thermodynamics, the simplest model of an engine that converts heat into work is the Carnot cycle. The behavior of a heat engine working between two heat reservoirs, modeled as this cycle, is expressed by the relation between the efficiency

A more realistic cycle than the Carnot cycle is a modified cycle taking into account the processes time of heat transfer between the system and its surroundings, in which the working temperatures are different of those its reservoirs [1], obtaining the efficiency

Two operating regimens of a heat engine with the same type of parameters have been established: maximum power output regimen as in [1] and maximum effective power regimen, taking into account the entropy production through a function called ecological function, which represents the relationship between power output

An alternative to analyze the Curzon–Ahlborn cycle, taking into account some effects that are nonideal to the adiabatic processes through the time of these processes, is the model proposed in [5] and in [7]. It allows to find the efficiency of a cycle as a function of the compression ratio,

In the present paper, the performance of a non-endoreversible heat engine modeled as a Curzon–Ahlborn cycle is analyzed. The procedure in [5] is combined with the procedure in [16], arriving to linear approaches of the efficiency as a function of a parameter that contains the compression ratio in both regimens maximum power output and maximum ecological function. From the limit values of the non-endoreversibility parameter and the compression ratio, the known expressions of the efficiency found in the literature of finite-time thermodynamics are recovered. Also, an analysis of the Stirling and Ericsson cycles is made, when the existence of a finite time for the heat transfer for isothermal processes is assumed, and assuming they are not endoreversible cycles, through the non-endoreversibility parameter that represents internal irreversibilities of them. Some results in [22] are used, and from the expressions obtained for the power output function and ecological function, the methodology to obtain a linear approximation of efficiency including an adequate parameter is shown, similar to those used in case of the Curzon–Ahlborn cycle. Variable changes are made right, like those used in [5] and in [23,24]. In order to make the present paper self-contained, a review of results for instantaneous adiabatic case is presented. All quantities have been taken in the International System of Measurement.

## 2. Linear approximation of efficiency: endoreversible Curzon–Ahlborn cycle

In a previous published chapter by InTech [25], we devoted to analyze the Curzon and Ahlborn cycle under the following conditions: without internal irreversibilities and non instantaneous adiabats. We have shown some results in case of the Newton heat transfer law (Newton cooling law) and the Dulong and Petit heat transfer law, namely, heat transfer law like

### 2.1. Known results and basic assumptions

Since the pioneer paper [1], the so-called finite time thermodynamics has been development. They proposed a model of thermal engine shown in Figure 1, which has the mentioned Curzon–Ahlborn–Novikov–Chambadal efficiency, as a function of the cold reservoir temperature

In this cycle,

Equation (1) has been obtained at maximum power output regimen and recovered later by some procedures [5,10,26,27] among others. Moreover, in [4] was advanced an optimization criterion of merit for the Curzon and Ahlborn cycle, taking into account the entropy production, the ecological criterion, by maximization of the ecological function,

where *P* is the power output,

By contrast, following the procedure in [5], the form of the ecological function and its efficiency was found using the Newton heat transfer law and ideal gas as working substance in [12] and using the Dulong–Petit heat transfer law for ideal gas as working substance in [28]. Hence, as the upper limit of the efficiency of any heat engine is the Carnot efficiency, the temperatures of the reservoir equal those of the heat engine. Thus, the definition of efficiency of an engine working in cycles leads to the Carnot efficiency, fulfilling

With

Thus, the problem of finding the efficiency of a heat engine modeled as a Curzon–Ahlborn cycle, maximizing power output or maximizing ecological function, becomes the problem of finding a function

Similar results are obtained with a nonlinear heat transfer, like the Dulong and Petit heat transfer. Assuming the same thermal conductance

By contrast, assuming the heat flows

where *R* is the general constant of gases. The parameter

being the times for the isothermal processes,

and the times for the adiabatic processes have been assumed to be

with

The maximization conditions

where

that is *z* is given by a power series in

Upon susbtituting Equation (16) in Equation (6) and because the terms in Equation (16) are positive, an upper bound for the efficiency is obtained when

The equivalent of Equation (16) for the ecological function with this procedure was obtained in [12] by substituting Equation (9) in Equation (3), and the entropy production,

Upon maximizing the function *z* and *u* is obtained:

The equation that *z* obeys at the maximum of the ecological function is obtained as follows:

We find, upon taking the implicit successive derivatives of Equation (20) with respect to

Furthermore, using Equation (21), we can write the efficiency as a power series in

When

which is the maximum possible one for this operating regimen. From Equations (16) and (21) a linear approximation for the efficiency

and the corresponding linear approximation of ecological efficiency is as follows:

As it is known in real compressors, the percent of volume in the total displacement of a piston into a cylinder is called the dead space ratio, and it is defined as

### 2.2. Nonlinear heat transfer law

The ecological efficiency has been calculated using Dulong and Petit’s heat transfer law in [30], maximizing ecological function for instantaneous adiabats. When the time for all the processes of the Curzon and Ahlborn cycle is taken into account, efficiencies in both regimens, maximum power output and maximum ecological function, can be obtained, following the procedure employed. Suppose an ideal gas as a working substance in a cylinder with a piston that exchanges heat with the reservoirs, and using a heat transfer law of the form

where

Equation (27) implies that the times along the isothermal processes in Figure 1 are, respectively,

The corresponding heat exchanged

where

where

with the same used parameters. By means of

and the resulting expression for the implicit function *k*,

(33) |

With reasonable approximations, only for the exponents in Equation (33), the following can be obtained:

Equation (34) allows to the explicit expression for the function

Taking now

The numerical results for *z* obtained from Equation (34) can be expressed as a power series in the parameter

One can find

Now, the ecological function for Curzon and Ahlborn engine takes the form

We find the function

and from the second one,

Substituting now Equation (40) for

The analytical solution of Equation (42) is not feasible when the exponents of

For the case

Any other root has no physical meaning because efficiencies must always be positive. Adequate comparison between fitted numerical values of

At last taking,

## 3. The non-endoreversible Curzon and Ahlborn cycle

By contrast, in finite time, thermodynamics is usually considered an endoreversible Curzon–Ahlborn cycle, but in nature, there is no endoreversible engine. Thus, some authors have analyzed the non-endoreversible Curzon and Ahlborn cycle. Particularly in [16] has been analyzed the effect of thermal resistances, heat leakage, and internal irreversibility by a non-endoreversibility parameter, advanced in [14],

where

Following the procedure in [16], have been found expressions to measure possible reductions of undesired effects in heat engines operation [17], and has been pointed out that *I*_{S} is not dependent of

Moreover, in [31] has been applied variational calculus showing that the saving function in [17] and modified ecological criteria are equivalent. In this section, internal irreversibilities are taken into account to obtain Equation (4), replacing

### 3.1. Curzon and Ahlborn cycle with instantaneous adiabats

Suppose a thermal engine working like a Curzon and Ahlborn cycle, in which an internal heat by internal processes of working fluid appears, assuming ideal gas as working fluid. The Clausius inequality with the parameter of non-endoreversibility becomes

The changes of entropy are

The volumes in the states of change of process in the cycle are

Assume the exchange of heat as Equation (8) with

with the changes

Also, the variation of entropy in the cycle can be written as

and Equation (18) is modified as

Now, the following is obtained from the conditions

and a physically possible solution for *Z*_{I} is found, which leads to the efficiency

when the change *u* and

For the suitable values of parameter

### 3.2. Curzon–Ahlborn cycle with noninstantaneous adiabats

In order to include the compression ratio in the analysis of Curzon and Ahlborn cycle, it is necessary to suppose finite time for the adiabatic processes. Hence, as it is known, with ideal gas as working fluid and using the Newton heat transfer law, the following can be written:

and because

Then again, internal energy

The integration of Equation (62) leads to the time of the adiabatic expansion in the cycle,

and taking into account the form that acquires the yielded heat

and the total time of the non-endoreversible cycle is as follows:

So that a new expression for power output in the cycle using the changes of variables in Equation (54) and Equation (56) is found, namely,

with

In order to maximize power output, Equation (66), the conditions

and in case of maximizing ecological function, the following is obtained:

When

Parameters

and in case of maximum ecological function,

The linear approximation of efficiency, at maximum power output and at maximum ecological function, can now be derived from Equation (71) or Equation (72), respectively, as

It is important to note that compression ratio has no arbitrary values, as discussed in Section 2.1. Thus, for

## 4. Stirling and Ericsson cycles

As it is known, the thermal engines can be endothermic or exothermic. Among the first engines, the best known are Otto and Diesel, and among the second two engines, very interesting and similar to the theoretical Carnot engine are Stirling and Ericsson engines [32,33]. In particular, a Stirling engine is a closed-cycle regenerative engine initially used for various applications, and until the middle of last century, they were manufactured on a large scale. However, the development of internal combustion engines from the mid-nineteenth century and the improvement in the refining of fossil fuels influenced the abandonment of the Stirling and Ericsson engines in the race for industrialization, gradually since the early twentieth century. Reference [34] is an interesting paper devoted to Stirling engine.

In the classical equilibrium thermodynamics, Stirling and Ericsson cycles have an efficiency that goes to the Carnot efficiency, as it is shown in some textbooks. These three cycles have the common characteristics, including two isothermal processes. The objection to the classical point of view is that reservoirs coupled to the engine modeled by any of these cycles do not have the same temperature as the working fluid because this working fluid is not in direct thermal contact with the reservoir. Thus, an alternative study of these cycles is using finite time thermodynamics. Thus, since the end of the previous century, and on recent times, the characteristics of Stirling and Ericsson engines have resulted in renewed interest in the study and design of such engines, and in the analysis of its theoretical idealized cycle, as it is shown in many papers, [22,35-37] among others. Nevertheless, the discussion on these engines and its theoretical model has not been exhausted.

In this section, an analysis of the Stirling and Ericsson cycles from the viewpoint of finite time thermodynamics is made. The existence of finite time for heat transfer in isothermal processes is proposed, but the cycles are analyzed assuming they are not endoreversible cycles, through the factor that represents their internal irreversibilities [14], so that the proposed heat engine model is closer to a real engine. Some results in reference [22] are used, and a methodology to obtain a linear approximation of efficiency, including adequate parameters, is shown. Variable changes are made right, like those used in [5] and in [23,25]. This section is a summary of obtained results in [38].

### 4.1. Stirling cycle

Now, as it is known, Stirling cycle consists of two isochoric processes and two isothermal processes. At finite time, the difference between the temperatures of reservoirs and the corresponding operating temperatures is considered, as shown in Figure 3. To construct expressions for power output and ecological function for this cycle, some initial assumptions are necessary. First, the heat transfer is supposed as Newton’s cooling law for two bodies in thermal contact with temperatures

On the other hand, it is assumed that the internal processes of the system cause irreversibilities that can be represented by the factor

Power output is defined as

With ideal gas as working substance for an isothermal process, the equation of state leads to

An assumption for the cycle is that heating and cooling at constant volume is performed as

where it is not difficult to show that it meets

By contrast, from the equilibrium conditions, it can be assumed

and the heating and cooling, respectively, from the first law of thermodynamics are

and the time for each isochoric processes is given as

The time for the isothermal processes can be found from Equation (77) as

The negative sign in

Since its definition and taking into account Equation (76), the power output of cycle is written as

Now, with the change of variables used in the previous section in Equations (54) and (56), and taking into account the ratio of temperatures of the heat reservoirs, used in Equations (1) and (2), with the parameter

The optimization conditions

and from the second one, a solution physically adequate

(88) |

Thus, that the efficiency at maximum power output can be written as

For known values of parameters

The analysis for ecological function is similar to power output, and also leads to similar results. The shape of function

which leads to the ecological function as

Where

As in the case of power output, in order to find the efficiency at maximum ecological function, there are two conditions, namely,

(94) |

The efficiency for the Stirling cycle at maximum ecological function can be written now as

and

The existence of a finite heat transfer in the isothermal processes is affected with the assumption of a non-endoreversible cycle with ideal gas as working substance. Power output and ecological function have also an issue that shows direct dependence on the temperature of the working substance. Expressions obtained with the changes of variables have the virtue of leading directly to the shape of the efficiency through

### 4.2. Ericsson cycle

The Ericsson cycle consisting of two isobaric processes and two isothermal processes is shown in Figure 4. Now, it follows a similar procedure as in the Stirling cycle case. Thus, the hypothesis on constant heating and cooling, now at constant pressure, is expressed as

It is true that

The equilibrium condition now is

and the time for a constant pressure process is given as

The time for the isothermal processes can also be obtained from Equation (77) and can be written as

and the total time of cycle is now

so the power output of cycle from its definition and taking into account Equation (76) remains

With the change of variables used in the previous section, now the expression for the power output of the non-endoreversible Ericsson cycle is

which is essentially found for the Stirling cycle, with factor

(105) |

Thus, at maximum power output regimen, the efficiency of non-endoreversible Ericsson cycle is

The analysis for the case of ecological function is similar to the case of power output and also leads to similar results. The shape of the function

from which the ecological function for the Ericsson cycle can be written as

where the parameter

(109) |

The efficiency for Ericsson cycle at maximum ecological function can be written now as

## 5. Concluding remarks

The developed methodology leads directly to appropriate expressions of the objective functions simplifying the optimization process. This methodology shows the consequences of assuming non-endoreversible cyle in the process of isothermal heat transfer through the factor