# Nonlinear Thermodynamic Analysis and Optimization of a Carnot Engine Cycle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{SH}(source) and T

_{SL}(sink).

_{H}= T

_{SH}− T

_{H}, ΔT

_{L}= T

_{SL}− T

_{L}.

_{H}= 1 corresponds, the heat rate capacity ${\dot{C}}_{SH}$ provides the maximum heat transfer rate to the endoreversible Carnot engine:

_{HSi}is the inlet temperature of the source fluid in the hot-end heat exchanger.

_{L}= T

_{SL}(perfect thermal contact at sink). The efficiency associated to maximum power output arises as:

_{L}/ΔS

_{H}(see Figure 1b).

## 2. Materials and Methods

#### 2.1. Model of Carnot Cycle Engine

_{SH}at the hot-end respectively, T

_{SL}at the cold-end.

#### 2.2. Optimization of the CNCA Machine by Using the Entropic Ratio Method

#### 2.2.1. Optimization General Approach

_{H}≥ 1. Equation (9b) imposes I

_{L}≤ 1.

_{H}= I

_{L}= 1 corresponds to an endoreversible system.

_{H}, that can be either a parameter (constant on a certain domain), or a function to be identified [18], or sought using the Direct Method, for example, by assuming degradation mechanisms [15].

_{lim}is the Carnot cycle efficiency, η

_{C}, given by:

_{II}is not identical to the exergetic efficiency, η

_{ex}. Actually, η

_{ex}is expressed as:

_{ex}is identical to η

_{II}, if T

_{SL}= T

_{0}. This is often close to reality, except certain cases, as for example CHP (Combined Heat and Power) systems.

- entering fluxes, which are generally called EC, energy or material consumptions; to them OF 5 corresponds, namely MIN EC (here ${\dot{Q}}_{H}$);
- an useful flux or Useful Effect, UE (here MAX $\dot{W}$)
- rejects, R, that may be recoverable (RR, recyclable waste), but most often pollutants for the Environment (RP, pollutant rejects); accordingly, OF 6 namely MIN (R).

_{H}, which is a parameter, depending on the problem (Equation (9a)). For example, I

_{H}= f

_{I}(T

_{L}, T

_{H}).

_{L}, T

_{H}. The choice of an objective function (OF) of these variables T

_{L}, T

_{H}(control variables), with C1 as constraint, provides one degree of freedom (DF) leading to the sequential optimization [6]. In this case, K

_{H}, K

_{L}, T

_{SH}, T

_{SL}are parameters (imposed) of the study, i.e., they appear in OF 6 and constraint, C1.

_{SH}, for which the minimum, min I

_{SH}, will be sought. Note that I

_{SH}represents the entropy ratio of the studied system and, therefore, it satisfies the system entropy balance equation:

#### 2.2.2. Particular Results for the Objective Functions MAX $\dot{W}$, MIN I_{SH}

#### Maximization of the Power Output of CNCA Engine

_{SH}= f(T

_{H}, T

_{L}), the Lagrangian function of the optimization can be expressed from Equations (6) and (9a) as follows:

_{H}, T

_{L}corresponding to the general case.

**Remark 1.**

_{SH}= f(T

_{H}, T

_{L}), could be expressed by using FST that is able to provide the entropy generation as a function of the piston speed and temperatures. This improvement makes part of further development of the modeling.

_{I}= I = ct, ${f}_{I,\text{}H}=\frac{\partial {f}_{I}}{\partial {T}_{H}}=0={f}_{I,\text{}L}=\frac{\partial {f}_{I}}{\partial {T}_{L}}$. Then simple analytical calculations provide:

_{H}and K

_{L}.

_{H}and K

_{L}as a constraint C2 that imposes:

_{2}(K

_{H}, K

_{L}):

_{T}, of the temperature difference (T

_{SH}–T

_{SL}), but a decreasing function of the irreversibility, I. The result associated with the endoreversible engine is found.

#### Consideration of System Irreversibility

_{SH}, integrates the irreversibility of the converter by the term, $\sqrt{I}$ (internal irreversibility), and the irreversibility associated with heat transfer, through $\sqrt{{T}_{SH}/{T}_{SL}}$. Both ratios are greater than 1, but yet different.

**Condition of transfer entropies equipartition**

**Remark 2.**

_{L eq}= T

_{SL}(T

_{H eq}= T

_{SH}), which is equivalent to the total reversibility condition (equilibrium thermodynamics).

_{L eq}, T

_{H eq}, T

_{H}, T

_{L}are values of temperatures at transfer entropies equipartition.

**Minimization of the entropy production rate of the system**

_{H}, T

_{L}. In the particular case where I

_{H}= I = constant, we find that the condition minimizing the system entropy production rate is the thermodynamic equilibrium, whatever K

_{H}, K

_{L}are. Hence, ${\dot{S}}_{i\text{}S}$ is an increasing function of T

_{L}and a decreasing one of T

_{H}. One finds a result obtained for the particular case of a thermoelectric system [22].

#### 2.3. Optimization of the CNCA Machine by Using the Entropy Production Rate Method

#### 2.3.1. General Statement of the Optimization

_{H}, T

_{L}. Moreover, it also depends on the characteristic speed of the system, w. This was experimentally confirmed for reverse cycle machines (heat pumps) [21], and it should also apply to engines.

**Entropy production rate of the converter**

- -
- ${\dot{S}}_{C}=\dot{S}=cte$ ($\dot{S}=0$, endoreversible)
- -
- ${\dot{S}}_{C}=S\text{'}\left({T}_{H}-\text{}{T}_{L}\right)$ (linear dependence on temperature difference)
- -
- ${\dot{S}}_{C}=C\cdot ln\frac{{T}_{H}}{{T}_{L}}$ (logarithmic dependence on temperature ratio)

**Heat transfer laws at the source and sink**

_{i}, were supposed constant in Section 2.2, where they were introduced. A step further in the development of a more general model is based on the fact that the literature provides correlations expressed as:

_{i}from Equation (46) could be identified as nonlinear heat transfer laws, namely:

#### 2.3.2. General and Required Condition for the Existence of Optimum Corresponding to MAX $\dot{W}$

_{4}(T

_{H}, T

_{L}, w) is composed as follows:

_{H}, T

_{L}, w) for one constraint, thus the model has two independent variables (2 degrees of freedom).

_{H}, K

_{L}on temperature is negligible, so that f

_{H}(T

_{H}, w) = f

_{H}(w); f

_{L}(T

_{L}, w) = f

_{L}(w). Furthermore, if one supposes identical modes of heat transfer to the hot- and cold-ends, it comes:

_{H}, T

_{L}, the following required condition is derived:

#### 2.3.3. Sequential Optimization of the Engine Power Output

#### Asymptotic Solution for the Case S_{C}(w) Smaller than k_{H}A_{H} and k_{L}A_{L}

_{H}, k

_{L}, A

_{H}, A

_{L}, S

_{C}, T

_{SH}, T

_{SL}, which are parameters. Optimum temperatures, T

_{H}

^{*}, T

_{L}

^{*},are obtained by the asymptotic forms of the following expressions:

#### Optimization of the Physical Geometric Dimensions A_{H}, A_{L}, of the System

_{H}, A

_{L}, are considered as variables, subject to a constraint of finite dimensions given by the A

_{T}parameter. The optimization aims to find the optimal distribution of the heat transfer area between the hot- and cold-ends of the system:

**Remark 3.**

_{H}(w)= k

_{L}(w) = k(w)) and by limiting S

_{C}(w) to the first order form, the expression of the maximum power MAX

_{2}$\left|\dot{W}\right|$ in the limiting case results:

_{T}) and temperatures at the source and sink (T

_{SH}, T

_{SL}).

## 3. Discussion—Partial Conclusion

- (a)
- For a dissipation law, similar to that reported in [18], Equation (51) becomes:$${S}_{C}\left(w\right)={S}_{Co}+{S}_{C}^{\text{'}}\cdot \text{}w$$

**Remark 4.**

_{SH}and T

_{SL}, as well as with the ratio S

_{C}'/k', but decreases with A

_{T}. Small machines have larger optimum speeds.

- (b)
- For a dissipation law as an mth power function of the speed:$${S}_{C}\left(w\right)={S}^{\text{'}}\cdot {w}^{m}$$

_{H}–T

_{L}).

## 4. Extensions—Particular Results

#### 4.1. Optimization Statement on an Entropic Base

_{H}(w) = K

_{L}(w) = K(w)).

_{H}, A

_{L}, of the System” and also the observed trends, in addition to the existence of an optimal characteristic speed providing maximum engine power in the presence of irreversibility.

#### 4.2. Particularization to Calculate the Various Entropy Production Rates

_{H}and K

_{L}parameters. One finds analytically:

_{H}+ K

_{L}= K

_{T}are considered, equipartition of conductances leads to maximum power, expressed by:

_{H}+ K

_{L}= K

_{T}(finite physical dimensions constraint), the total entropy production rate of the endoreversible engine passes through a maximum for equipartition of heat transfer conductances (K

_{H}= K

_{L}= K

_{T}/2).

## 5. Conclusions

_{H}*, T

_{L}*), afterwards the optimal allocation of heat transfer areas (A

_{H}*, A

_{L}*), and finally w*, the engine speed at the optimum operation point. The main findings related to maximizing the engine power extend previous results reported in the literature. Thus, a new generalized form of the nice radical has been proposed in the paper.

_{i}, K

_{i}, A

_{i}) and a characteristic speed of the engine for the first time, to our knowledge. This connection appears highly nonlinear owing to the coupling dimension-speed. The asymptotic behavior near the endoreversibility can provide meaningful analysis results corresponding to realistic tendencies. Primarily, there is an optimum speed associated with the best compromise between transfer and conversion of energy.

- the maximum power supplied by the engine does not correspond systematically to the minimum entropy production rate;
- equipartition of entropy production rate is not associated to the maximum power delivered by the engine (endoreversible or not).

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | heat transfer area | m^{2} |

a | coefficient dependent of the gas nature | - |

b | coefficient related to throttling | - |

c | molecular average speed | m/s |

${c}_{p}$ | mass specific heat at constant pressure | J/kg^{−}K |

${\dot{C}}_{SH}$ | heat rate capacity of the source | W/K |

${\dot{m}}_{SH}$ | mass flow rate of the source fluid | kg/s |

I | entropic ratio | - |

K | heat transfer conductance | W/K |

k | overall heat transfer coefficient | W/m^{2}K |

P_{m, i} | instantaneous mean pressure of the gas | Pa |

ΔP | pressure losses | Pa |

$\dot{Q}$ | heat transfer rate | W |

S | entropy | J/K |

$\dot{S}$ | entropy rate | W/K |

T | temperature | K |

V | volume | m^{3} |

$\dot{W}$ | power output of the engine | W |

w | characteristic speed | m/s |

## Greek symbols

α | intermediate variable |

η_{C} | Carnot cycle efficiency |

η_{ex} | exergetic efficiency |

η_{I} | First Law efficiency |

η_{II} | Second Law efficiency |

## Subscripts

C | related to the converter |

eq | corresponding to equipartition of entropy production between hot-end and cold-end |

f | friction |

H | related to the gas at the hot-end |

L | related to the gas at the cold-end |

SH | related to the source (hot-end) |

SHi | related to the source fluid in the hot-end heat exchanger |

SL | related to the sink (cold-end) |

T | total |

thr | throttling |

0 | related to ambient conditions |

## Superscript

* | related to optimum |

## Acronyms

C1, C2 | Constraints |

CHP | Combined Heat and Power |

CNCA | Carnot–Novikov–Curzon–Ahlborn |

EC | Energy Consumption |

FST | Finite Speed Thermodynamics |

FTT | Finite Time Thermodynamics |

OF | Objective Function |

R | Reject |

UE | Useful Effect |

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**Figure 1.**Carnot–Novikov–Curzon–Ahlborn Machine [3]: (

**a**) scheme; (

**b**) cycle representation in T–s diagram.

**Figure 2.**Energy Triangle connecting the Useful Effect (UE), the Energy Consumption (EC) and the Reject (R).

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**MDPI and ACS Style**

Feidt, M.; Costea, M.; Petrescu, S.; Stanciu, C.
Nonlinear Thermodynamic Analysis and Optimization of a Carnot Engine Cycle. *Entropy* **2016**, *18*, 243.
https://doi.org/10.3390/e18070243

**AMA Style**

Feidt M, Costea M, Petrescu S, Stanciu C.
Nonlinear Thermodynamic Analysis and Optimization of a Carnot Engine Cycle. *Entropy*. 2016; 18(7):243.
https://doi.org/10.3390/e18070243

**Chicago/Turabian Style**

Feidt, Michel, Monica Costea, Stoian Petrescu, and Camelia Stanciu.
2016. "Nonlinear Thermodynamic Analysis and Optimization of a Carnot Engine Cycle" *Entropy* 18, no. 7: 243.
https://doi.org/10.3390/e18070243