05 - 5 E fficiency of H eat Engines at M a x i m u m Power...

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5 Efficiency of Heat Engines at Maximum Power 5.1 maximum power output The thermal efficiency of a Carnot cycle operating between high temperature (TH) and low temperature (T,) reservoirs is given by Efficiency of an internally reversible heat engine when producing This cycle is extremely idealised. It requires an ideal, reversible heat engine (internally reversible) but, in addition, the heat transfer from the reservoirs is also reversible (externally reversible). To achieve external reversibility it is necessary that the tempera- ture difference between the reservoirs and the engine is infinitesimal, which means that the heat exchanger surface area must be very large or the time to transfer heat must be long. The former is limited by size and cost factors whilst the latter will limit the actual power output achieved for the engine. It is possible to evaluate the maximum power oufpuf achievable from an internally reversible (endoreversible) heat engine receiving heat irreversibly from two reservoirs at TH and T,. This will now be done, based on Bejan (1988). Assume that the engine is a steady-flow one (e.g. like a steam turbine or closed cycle gas turbine): a similar analysis is possible for an intermittent device (e.g. like a Stirling engine). A schematic of such an engine is shown in Fig 5.1. The reservoir at transfers heat to the engine across a resistance and it is received by the engine at temperature In a similar manner, the engine rejects energy at T2 but the cold reservoir is at It can be assumed that the engine itself is reversible and acts as a Carnot cycle device with This thermal efficiency is less than the maximum achievable value given by eqn (5.1) because T,/T, > T,/T,. The value can only approach that of eqn (5.1) if the temperature drops between the reservoirs and the engine approach zero.
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86 Eficiency of heat engines at maximum power Fig. 5.1 Internally reversible heat engine operating between reservoirs at TH and T, The heat transfer from the hot reservoir can be defined as QH = uHAH(TH - TI) where UH = heat transfer coefficient of hot reservoir (e.g. kW/m2 K) and AH = area of heat transfer surface of hot reservoir (e.g. m2) = rate of heat transfer (e.g. kW). (5.3) The heat transfer to the cold reservoir is similarly By the first law Qc = UcAc(T2 - Tc) (5.4) w=QH- QC (5.5) Now, the heat engine is internally reversible and hence the entropy entering and leaving it must be equal i.e. This means that It is possible to manipulate these equations to give W in terms of TH, Tc, UHAH, and the ratio T2/Tl = z. From eqn (5.4) and, from eqn (5.6),
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Eficiency of an internally reversible heat engine 87 Hence Rearranging gives and hence w t l+- ( 2:;) (5.10) (5.11) Thus the rate of work output is a function of the ratio of temperatures of the hot and cold reservoirs, the ratio of temperatures across the engine and the thermal resistances. The optimum temperature ratio across the engine (t) to give maximum power output is obtained when -0 aw at -- Differentiating eqn (5.1 1) with respect to t gives 1 (1 - t) (t - TJT,) - - f1 - 7. t2 1+- Hence aw/& = o when t = 00 or = Tc/TH.
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05 - 5 E fficiency of H eat Engines at M a x i m u m Power...

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