Discrete Hamiltonian Analysis of Endoreversible Thermal Cascades

  • S. Sieniutycz
  • R. S. Berry
Chapter

Abstract

Endoreversible multistage processes which yield mechanical work are optimized by a relatively little-known discrete maximum principle of Pontryagin’s type. A discrete optimization approach extends the classical method, well known for continuous systems in which a Hamiltonian is maximized with respect to controls. Equations of dynamics which follow from energy balance and transfer equations are difference constraints for optimizing work. Irreversibilites caused by the energy transport are essential. Variation of efficiency is analyzed in terms of the heat flux. Enhanced bounds for the work released from an engine system or added to a heat-pump system are evaluated. Lagrangians of work functionals, canonical equations, and structure of the Hamiltonian function are all discrete characteristics which reach their continuous conterparts in the limit of an infinite number of stages. For a finite-time passage of a resource fluid between two given temperatures, optimality of an irreversible process manifests itself as a connection between the process duration and an optimal intensity expressed in terms of the Hamiltonian. Extremal performance functions that describe extremal work are found in terms of final states, process duration, and number of stages. A discrete extension of classical thermal exergy to systems with a finite number of stages and a finite holdup time of a resource fluid is one of the main results. This extended exergy, that has an irreversible component, simplifies to the classical thermal exergy in the limit of infinite duration and an infinite number of stages. The extended exergy exhibits a hysteretic property as a decrease of maximum work received from a multistage engine system and an increase of minimum work added to a heat-pump system, two properties which are particularly important in high-rate regimes.

Keywords

Heat Flux Heat Pump Legendre Transformation Adjoint Variable Engine Mode 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I. I. Novikov: The efficiency of atomic power stations (a review), J. Nuclear Energy II 7, 125–128, (1958). English translation from Atom Energy 3, 409-412, (1957).Google Scholar
  2. [2]
    F. L. Curzon and B. Ahlborn: Efficiency of Carnot engine at maximum power output, Amer. J. Phys. 43, 22–24, (1975).ADSCrossRefGoogle Scholar
  3. [3]
    B. Andresen, P. Salamon, and R. S. Berry: Thermodynamics in finite time: extremals for imperfect heat engines, J. Chem. Phys. 66, 1571–1577, (1977).ADSCrossRefGoogle Scholar
  4. [4]
    A. De Vos: Endoreversible thermodynamics of solar energy conversion, Oxford University Press, Oxford, pp. 29–51, 1992.Google Scholar
  5. [5]
    L. Chen and Z. Yan: The effect of heat-transfer law on performance of a two-heat-source endoreversible cycle, J. Chem. Phys. 90, 3740–3743, (1989).ADSCrossRefGoogle Scholar
  6. [6]
    J. M. Gordon: Maximum power point characteristics of heat engines as a general thermodynamic problem, Amer. J. Phys. 57, 1136–1142, (1989).ADSCrossRefGoogle Scholar
  7. [7]
    S. Sieniutycz: The constancy of Hamiltonian in a discrete optimal process, Reports of Inst. of Chem. Engng. Warsaw Tech. Univ. vol. 2, pp. 399–429, 1973.Google Scholar
  8. [8]
    S. Sieniutycz: Optimization in process engineering, 1st ed., Wydawnictwa Naukowo Techniczne, Warsaw, 1978.Google Scholar
  9. [9]
    Z. Szwast: Discrete algorithms of maximum principle with constant Hamiltonian and their selected applications in chemical engineering, PhD Thesis, Institute of Chemical Engineering at the Warsaw University of Technology, Warsaw, 1979.Google Scholar
  10. [10]
    Z. Szwast: Enhanced version of a discrete algorithm for optimization with a constant Hamiltonian, Inz. Chem. Proc. 3, 529–545, (1988).Google Scholar
  11. [11]
    S. Sieniutycz and Z. Szwast: Practice in Optimization: Process Problems, Wydawnictwa Naukowo Techniczne, Warsaw, 1982.Google Scholar
  12. [12]
    S. Sieniutycz and Z. Szwast: A discrete algorithm for optimization with a constant Hamiltonian and its application to chemical engineering, Internat. J. Chem. Engng. 23, 155–166, (1983).Google Scholar
  13. [13]
    L. T. Fan and C. S. Wang: The Discrete Maximum Principle, a Study of Multistage System Optimization, Wiley, New York, 1964.Google Scholar
  14. [14]
    V. G. Boltyanski: Optimal Control of Discrete Systems, Nauka, Moscow, 1973.Google Scholar
  15. [15]
    L. T. Fan: The Continuous Maximum Principle, a Study of Complex System Optimization, Wiley, New York, 1966.Google Scholar
  16. [16]
    W. Findeisen, J. Szymanowski, and A. Wierzbicki: Theory and Computational Methods of Optimization, Panstwowe Wydawnictwa Naukowe, Warsaw, 1980.Google Scholar
  17. [17]
    R. E. Bellman: Adaptive Control Processes: a Guided Tour, Princeton University Press, Princeton, 1961.MATHGoogle Scholar
  18. [18]
    R. Aris: Discrete Dynamic Programming, Blaisdell, New York, 1964.MATHGoogle Scholar
  19. [19]
    S. Sieniutycz: Endoreversible modeling and optimization of multistage thermal machines by dynamic programming, in: Advances in Recent Finite-Time Thermodynamics (ed. C. Wu), Nova Science, New York, 1999.Google Scholar
  20. [20]
    V. G. Boltyanski: Mathematical Methods of Optimal Control, Nauka, Moscow, 1969.Google Scholar
  21. [21]
    S. Sieniutycz: Carnot problem of maximum work from a finite resource interacting with environment in a finite time, Physica A 264, 234–263, (1999).CrossRefGoogle Scholar
  22. [22]
    P. Salamon, P. A. Nitzan, B. Andresen, and R. S. Berry: Thermodynamics in finite time IV: Minimum entropy production and the optimization of heat engines, Phys. Rev. A 21, 2115–2129, (1980).MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    B. Andresen and J. Gordon: Optimal heating and cooling strategies for heat exchanger design, J. Appl. Phys. 71, 76–79, (1992).ADSCrossRefGoogle Scholar
  24. [24]
    R. S. Berry, V. A. Kazakov, S. Sieniutycz, Z. Szwast, and A. M. Tsirlin: Thermodynamic Optimization of Finite Time Processes, Wiley, Chichester, 2000.Google Scholar
  25. [25]
    B. Andresen, M. H. Rubin, and R. S. Berry: Availability for finite time processes. General theory and model, J. Phys. Chem. 87, 2704–2713, (1983).CrossRefGoogle Scholar
  26. [26]
    V. A. Mironova, A. M. Tsirlin, V. A. Kazakov, and R. S. Berry: Finitetime thermodynamics. Exergy and optimization of time-constrained processes, J. Appl. Phys. 76, 629–636, (1994).ADSCrossRefGoogle Scholar
  27. [27]
    R. S. Berry: Exergy and optimization of time-constrained processes, Periodica Polytech. Ser. Phys. Nucl. Sci. 2, 5–14, (1994).Google Scholar
  28. [28]
    S. Sieniutycz: Hamilton-Jacobi-Bellman theory of dissipative thermal availability, Phys. Rev. E 56, 5051–5064, (1997).MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    A. De Vos, J. Chen and B. Andresen: Analysis of combined systems of two endoreversible engines, Open Sys. Inform. Dynam. 4, 3–14, (1997).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • S. Sieniutycz
  • R. S. Berry

There are no affiliations available

Personalised recommendations