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From Finite Time to Finite Physical Dimensions Thermodynamics: The Carnot Engine and Onsager’s Relations Revisited

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Published/Copyright: March 7, 2018
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Journal of Non-Equilibrium Thermodynamics
From the journal Journal of Non-Equilibrium Thermodynamics Volume 43 Issue 2

Abstract

Many works have been devoted to finite time thermodynamics since the Curzon and Ahlborn [1] contribution, which is generally considered as its origin. Nevertheless, previous works in this domain have been revealed [2], [3], and recently, results of the attempt to correlate Finite Time Thermodynamics with Linear Irreversible Thermodynamics according to Onsager’s theory were reported [4].

The aim of the present paper is to extend and improve the approach relative to thermodynamic optimization of generic objective functions of a Carnot engine with linear response regime presented in [4]. The case study of the Carnot engine is revisited within the steady state hypothesis, when non-adiabaticity of the system is considered, and heat loss is accounted for by an overall heat leak between the engine heat reservoirs.

The optimization is focused on the main objective functions connected to engineering conditions, namely maximum efficiency or power output, except the one relative to entropy that is more fundamental.

Results given in reference [4] relative to the maximum power output and minimum entropy production as objective function are reconsidered and clarified, and the change from finite time to finite physical dimension was shown to be done by the heat flow rate at the source.

Our modeling has led to new results of the Carnot engine optimization and proved that the primary interest for an engineer is mainly connected to what we called Finite Physical Dimensions Optimal Thermodynamics.

Keywords: Carnot engine; steady state modeling; linear irreversible thermodynamics; objective functions; optimization; Finite Time and Finite Physical Dimensions Thermodynamics

A.1 Existence of a low value limit for q2 when the power output is optimized

For σS≥0, eq. (7) imposes q2≤1, which is the first upper bound.

For W˙≥0, eq. (14) imposes for the engine a new condition. Hence, by replacing eq. (7) in (14) one gets

(A.1)W˙=JHηC−TCS1LHH·JH2+1q2−11LHHJH−LHHXH2.

By developing the calculation, an inequality between JH=Q˙H and q2 finally results. Thus, W˙≥0 implies

(A.2)JHηCTCS≥1LHH·JH2+1q2−11LHHJH−LHHXH2.

By considering eq. (16) one successively gets

(A.3)JHTHS−TCSTHSTCSLHH−JH2≥1q2−1JH−LHHXH2,
(A.4)JHXHLHH−JH2≥1q2−1JH−LHHXH2,
(A.5)−JHJH−LHHXHJH−LHHXH2≥1q2−1,
(A.6)1−JHJH−LHHXH≥1q2,
(A.7)LHHXHLHHXH−JH≥1q2,
(A.8)q2≥1−JHLHHXH=qlim2.

It results from the above demonstration that, for a given Q˙H(=JH), q2∈[qlim2,1].

Regarding the value of JH, one can conclude that:

  1. JH = 0 corresponds to strong coupling and thus to the equilibrium thermodynamics limit;

  2. for small values of JH, one retrieves the case developed in Section 4 (relaxed tight-coupling);

  3. for JH values close to LHHXH ones, qlim2 tends to zero, but this limit seems unrealistic.

Looking to the definition of q2 given by

(A.9)q2=LHC2LHHLCC,

one can see that the limit concerns the phenomenological coefficient such as:

(A.10)LHC≥LHHXH−JHLCCXH.

This condition corresponds to a physical limitation relative to irreversibilities and coupling related to JH – available heat rate at the source (with JH≤LHHXH).

A second way consists in supposing the coupling parameter imposed. Thus, the heat rate at the source JH is limited by

(A.11)LHHXH≥JH≥1−q2LHHXH.

Inequality (A.11) shows that the allowed variation domain for JH decreases with q2, and it is limited by JH=LHHXH, when q2 = 0.

References

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Received: 2017-9-12
Revised: 2018-2-4
Accepted: 2018-2-13
Published Online: 2018-3-7
Published in Print: 2018-4-25

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Research Articles
  4. GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems
  5. Ergodicity, Maximum Entropy Production, and Steepest Entropy Ascent in the Proofs of Onsager’s Reciprocal Relations
  6. Entropy Generation Minimization in Dimethyl Ether Synthesis: A Case Study
  7. Systematic Constraint Selection Strategy for Rate-Controlled Constrained-Equilibrium Modeling of Complex Nonequilibrium Chemical Kinetics
  8. General Properties for an Agrawal Thermal Engine
  9. Novikov Engine with Fluctuating Heat Bath Temperature
  10. From Finite Time to Finite Physical Dimensions Thermodynamics: The Carnot Engine and Onsager’s Relations Revisited
  11. Non-Equilibrium Dislocation Dynamics in Semiconductor Crystals and Superlattices
  12. A Thermodynamical Theory with Internal Variables Describing Thermal Effects in Viscous Fluids
  13. A Thermodynamically Consistent Approach to Phase-Separating Viscous Fluids
https://doi.org/10.1515/jnet-2017-0047
Keywords for this article
Carnot engine; steady state modeling; linear irreversible thermodynamics; objective functions; optimization; Finite Time and Finite Physical Dimensions Thermodynamics

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Research Articles
  4. GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems
  5. Ergodicity, Maximum Entropy Production, and Steepest Entropy Ascent in the Proofs of Onsager’s Reciprocal Relations
  6. Entropy Generation Minimization in Dimethyl Ether Synthesis: A Case Study
  7. Systematic Constraint Selection Strategy for Rate-Controlled Constrained-Equilibrium Modeling of Complex Nonequilibrium Chemical Kinetics
  8. General Properties for an Agrawal Thermal Engine
  9. Novikov Engine with Fluctuating Heat Bath Temperature
  10. From Finite Time to Finite Physical Dimensions Thermodynamics: The Carnot Engine and Onsager’s Relations Revisited
  11. Non-Equilibrium Dislocation Dynamics in Semiconductor Crystals and Superlattices
  12. A Thermodynamical Theory with Internal Variables Describing Thermal Effects in Viscous Fluids
  13. A Thermodynamically Consistent Approach to Phase-Separating Viscous Fluids
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