Finite–time exergy with a finite heat reservoir and generalized radiative heat transfer law
Shaojun Xia, Lingen Chen*, Fengrui Sun
Postgraduate School, Naval University of Engineering, Wuhan 430033, P.R. China, Fax: 0086–27–83638709; Tel: 0086–27–83615046, *E–mail: lgchenna@yahoo.com; lingenchen@hotmail.com
Recibido el 4 de enero de 2010
Aceptado el 13 de abril de 2010
Abstract
The problem of the maximum work that can be extracted from a system consisting of one finite heat reservoir and one subsystem with the generalized radiative heat transfer law [q ∞ Δ (Tn)] is investigated in this paper. Finite–time exergy is derived for a fixed duration and a given initial state of the subsystem by applying optimal control theory. The optimal subsystem temperature configuration for the finite–time exergy consists of three segments, including the initial and final instantaneous adiabatic branches and the intermediate heat transfer branch. Analyses for special examples show that the optimal configuration of the heat transfer branch with Newton's heat transfer law [q ∞ Δ (T)] is that the temperatures of the reservoir and the subsystem change exponentially with time and the temperature ratio between them is a constant; The optimal configuration of the heat transfer branch with the linear phenomenological heat transfer law [q ∞ Δ (T–1)] is such that the temperatures of the reservoir and the subsystem change linearly and non–linearly with time, respectively, and the difference in reciprocal temperature between them is a constant. The optimal configuration of the heat transfer branch with the radiative heat transfer law [q ∞ Δ (T4)] is significantly different from those with the former two different heat transfer laws. Numerical examples are given, effects of changes in the reservoir's heat capacity on the optimized results are analyzed, and the results for the cases with some special heat transfer laws are also compared with each other. The results show that heat transfer laws have significant effects on the finite–time exergy and the corresponding optimal thermodynamic process. The finite–time exergy tends to the classical thermodynamic exergy and the average power tends to zero when the process duration tends to infinitely large. Some modifications are also made to the results from recent literatures.
Keywords: Finite time thermodynamics; finite–time exergy; finite heat reservoir; generalized radiative heat transfer law; optimal control.
Resumen
En este trabajo se investiga el problema del maximo trabajo que es posible extraer del sistema consistente en un recipiente térmico finito y un subsistema con la ley generalizada de transferencia de calor por radiación [q ∞ Δ (Tn)]. Se obtiene la exergía de tiempo finito para una duración fija y un estado inicial del subsistema dado aplicando la teoría de control óptimo. La configuración (óptima de temperatura del subsistema para la exergía de tiempo finito consiste en tres segmentos: la rama instantánea adiabática inicial y final, y la rama de transferencia de calor intermedia. El análisis de ejemplos especiales muestra que la configuración óptima de la rama de transferencia de calor con la ley de Newton de transferencia térmica [q ∞ Δ (T)] es aquella en la que la temperatura del recipiente y del subsistema cambian exponencialmente con el tiempo y la razón de temperaturas es constante. La configuración óptima de la rama de transferencia térmica con la ley lineal fenomenológica [q ∞ Δ (T–1)] es aquella en la que las temperaturas del recipiente y del subsistema cambian lineal y no linealmente con el tiempo respectivamente y la diferencia en la temperatura recíproca entre ellos es constante. La configuración óptima para la rama de transferencia térmica con la ley radiativa de transferencia de calor [q ∞ Δ (T4)] es significativamente diferente de las que emplean las dos leyes anteriores. Se dan ejemplos numéricos, se analizan los efectos de los cambios en la capacidad calorífica del recipiente en los resultados optimizados, y los resultados para los casos con alguna ley especial de transferencia térmica se comparan unos con otros. Los resultados muestran que las leyes de transferencia térmica tienen efectos significativos en la exergía de tiempos finitos y en el proceso termodinámico óptimo correspondiente. La exergía de tiempos finitos tiende a la de la termodinámica clásica y la potencia promedio tiende a cero cuando la duración del proceso tiende a ser infinitamente largo. También se hacen algunas modificaciones a resultados recientemente publicados.
Descriptores: Termodinámica a tiempos finitos; exergía a tiempos finitos; recipiente térmico finito; ley generalizada de transferencia de calor por radiación; control óptimo.
PACS: 44.10.+i;44.40.+a
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Acknowledgements
This paper is supported by The National Natural Science Foundation of P. R. China (Project No. 10905093), Program for New Century Excellent Talents in University of P.R. China (Project No. NCET–04–1006) and The Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (Project No. 200136).
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