The paper develops generalizing entropic approaches of irreversible closed cycles. The mathematical models of the irreversible engines (basic, with internal regeneration of the heat, cogeneration units) and of the refrigeration cycles were applied to four possible operating irreversible trigeneration cycles. The models involve the reference entropy, the number of internal irreversibility, the thermal conductance inventory, the proper temperatures of external heat reservoirs unifying the first law of thermodynamics and the linear heat transfer law, the mean log temperature differences, and four possible operational constraints, i.e., constant heat input, constant power, constant energy efficiency and constant reference entropy. The reference entropy is always the entropy variation rate of the working fluid during the reversible heat input process. The number of internal irreversibility allows the evaluation of the heat output via the ratio of overall internal irreversible entropy generation and the reference entropy. The operational constraints allow the replacement of the reference entropy function of the finite physical dimensions parameters, i.e., mean log temperature differences, thermal conductance inventory, and the proper external heat reservoir temperatures. The paper presents initially the number of internal irreversibility and the energy efficiency equations for engine and refrigeration cycles. At the limit, i.e., endoreversibility, we can re-obtain the endoreversible energy efficiency equation. The second part develops the influences between the imposed operational constraint and the finite physical dimensions parameters for the basic irreversible cycle. The third part is applying the mathematical models to four possible standalone trigeneration cycles. It was assumed that there are the required consumers of the all useful heat delivered by the trigeneration system. The design of trigeneration system must know the ratio of refrigeration rate to power, e.g., engine shaft power or useful power delivered directly to power consumers. The final discussions and conclusions emphasize the novelties and the complexity of interconnected irreversible trigeneration systems design/optimization.
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