# Thermodynamic Analysis of Closed Steady or Cyclic Systems

## Abstract

**:**

## 1. Introduction

#### 1.1. Model

**Figure 1.**A closed reversible or irreversible system, A, bounded by constant thermal resistances linked to isothermal reservoirs.

#### 1.2. Aims

## 2. Concept Development

#### 2.1. One Heat Source and One Heat Sink with Reversible Heat Transfer

**Figure 2.**A closed reversible system, A, bounded by two isothermal reservoirs, without intermediate thermal resistances.

- a heat engine (which we denote as HE1) accepting heat transfer from a high temperature heat source at ${T}_{1}$ and rejecting heat at ${T}_{0}$ (if irreversible, the entropy transfer rate out at ${T}_{0}$ is greater than the entropy transfer rate in at ${T}_{1}$),
- a heat engine (HE2) accepting heat transfer at ${T}_{0}$ and rejecting heat to a low temperature heat sink at ${T}_{1}$ (if irreversible, the entropy transfer rate out at ${T}_{1}$ is greater than the entropy transfer rate in at ${T}_{0}$),
- a heat pump (which we denote as a reverse heat engine, Type 1, RHE1) rejecting heat to a high temperature sink at ${T}_{1}$ and accepting heat transfer at ${T}_{0}$ (if irreversible, the entropy transfer rate out at ${T}_{1}$ is greater than the entropy transfer rate in at ${T}_{0}$) or,
- a refrigerator (RHE2) accepting heat transfer from a low temperature source at ${T}_{1}$ and rejecting heat at ${T}_{0}$ (if irreversible, the entropy transfer rate out at ${T}_{0}$ is greater than the entropy transfer rate in at ${T}_{1}$).

#### 2.2. One Heat Source and One Heat Sink with Thermal Resistances

**Figure 3.**A closed reversible system, A, bounded by two constant thermal resistances linked to isothermal reservoirs.

#### 2.3. Heat Input Rate and Thermal Efficiency of a Heat Engine at Maximum Power

#### 2.4. Exergy Analysis Principles to Be Applied

**Figure 4.**The exergy transfer rate, ${\dot{X}}_{i,j}$, corresponding to a heat transfer rate ${\dot{Q}}_{i}$ at ${T}_{i}$, relative to an isothermal reservoir at temperature ${T}_{j}$ (virtual reversible heat engines, HE${}_{\mathrm{rev}}$, are employed).

**2P**, “…all the products of a generic equipment have the same unit exergetic cost” is key to addressing furcations (or forks) in the flow of exergy through an overall system or plant. Lozano and Valero published a comprehensive paper on the theory of exergetic cost in 1993 [21]. In this publication, the corresponding proposition,

**P4b**, had the wording “…if a unit has a product composed of several flows, then the same unit exergetic cost will be assigned to all of them”.

#### 2.5. Exergy Analysis of a Reversible System Linked without Thermal Resistances to Its Heat Source and Sink

#### 2.6. Exergy Analysis of a Reversible System Linked to Its Heat Source and Sink by Thermal Resistances

**Figure 5.**Rational efficiency versus the principal heat transfer rate, ${\dot{Q}}_{1}$, for heat engines (${T}_{1}>{T}_{0}$ and ${T}_{1}<{T}_{0}$).

**Figure 6.**Rational efficiency versus the principal heat transfer rate, ${\dot{Q}}_{1}$, for reverse heat engines (${T}_{1}>{T}_{0}$ and ${T}_{1}<{T}_{0}$).

#### 2.7. The Temperature versus Entropy Transfer Rate Diagram

**Figure 7.**A diagram of temperature versus entropy transfer rates for a reversible heat engine linked by thermal resistances to two isothermal reservoirs, one of which is at the environmental temperature, ${T}_{0}$.

**Figure 8.**A diagram of the exergy transfer rates for the reversible heat engine linked by thermal resistances to two isothermal reservoirs, one of which is at the environmental temperature, ${T}_{0}$.

#### 2.8. An Irreversible System Linked to Its Heat Source and Sink by Thermal Resistances

#### 2.9. Lost Work Rate and the $T-\dot{S}$ Diagram

**Figure 10.**$T-\dot{S}$ Diagrams for an irreversible Reversed Heat Engine 1 (RHE1) and an irreversible RHE2.

- A reverse heat engine can operate between thermal reservoirs at ${T}_{\mathrm{L}}$ and ${T}_{\mathrm{H}}={T}_{0}$, but with only an infinitesimal heat transfer rate at ${T}_{\mathrm{L}}$ (regarded as a highly irreversible refrigerator), or at ${T}_{\mathrm{L}}$ again when ${T}_{\mathrm{L}}={T}_{0}$ (regarded as a highly irreversible heat pump), Figure 10.
- There can be a net rate of work input accompanied by a corresponding net rate of heat rejection at ${T}_{0}$, e.g., a brake or a churn; Figure 11, work rate dissipation.
- There can be internal heat transfer through the system in the direction from the hotter surface to the colder one with zero net work output (a thermal resistance); Figure 11, heat transfer rate.

**Figure 11.**Diagrams of temperature versus entropy transfer rates for irreversible work rate dissipation, as in a brake, and heat transfer rate in a thermal resistance.

## 3. The Environmental Temperature and Multiple Isothermal Reservoirs

#### 3.1. One Heat Source and Two Environmental Heat Sinks

**Figure 12.**A diagram of temperature versus entropy transfer rates for a reversible heat engine linked by thermal resistances to one heat source and two “environmental” heat sinks.

#### 3.2. Generalization for Multiple Thermal Reservoirs

#### 3.3. The Nature of the Environmental Reference Temperature

## 4. Methodology Suggestions

**Figure 13.**Example diagrams of temperature versus entropy transfer rates for closed, steady or cyclic systems.

## 5. The Irreversible Carnot System of Finite Thermal Resistance

**Figure 14.**Temperature versus specific entropy diagrams for irreversible Carnot cycles undergone by a two phase working fluid. For a normal heat engine the sense is clockwise, while for an reverse heat engine the sense is anticlockwise.

**Figure 15.**Summary of the equivalence of an irreversible Carnot system having a given rational efficiency relative to ${T}_{0}$ and a Curzon Ahlborn system having a certain resistance

## 6. Overall and Subsystem Rational Efficiencies

## 7. Discussion

## 8. Conclusions

## Conflicts of Interest

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McGovern, J. Thermodynamic Analysis of Closed Steady or Cyclic Systems. *Entropy* **2015**, *17*, 6712-6742.
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McGovern, Jim. 2015. "Thermodynamic Analysis of Closed Steady or Cyclic Systems" *Entropy* 17, no. 10: 6712-6742.
https://doi.org/10.3390/e17106712