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66 &&& η=W =QH −QL [5.2.5] && QH QH In practice, α and β are functions of physical heat exchanger size and performance. If heat exchanger performance is approximately constant then physical heat exchanger size scales with α and β. In addition, as heat exchanger size increases it follows that related plant equipment must also scale up to accommodate larger fluid flow rates and power outputs. α and β are then representative of total power cycle size, defined here as UA, the total overall heat transfer coefficient: UA = α + β [5.2.6] α, β, TH, and TL are arbitrarily defined parameters. Equations [5.2.1] through [5.2.5] make up a system of 5 equations with 6 unknown variables. Consider that TH and TL are fixed boundary conditions that the cycle must operate within. Consider also that the power output is a salable product. Finite-time thermodynamics then suggests that the optimal power cycle design would extract the maximum possible power from the thermal resources. This observation is used as the impetus to derive the efficiency at which maximum power is produced in terms of the thermal resource temperatures. Equations [5.2.1] through [5.2.5] are used to obtain an expression for cycle power output in terms of thermal resource and power cycle temperature ratios, τ=TL/TH and θ=T2/T1 [Yilmaz, 2006]: T (θ−τ)(1−θ) W= H & [5.2.7] The maximum power point is found by setting the derivative of work with respect to θ to zero: θ& =τ=TL [5.2.8] dW =0 TH dθ The maximum power, and the corresponding efficiency are identified by substituting equation [5.2.8] into equation [5.2.7] and equation [5.2.8] into equation [5.2.4], respectively: θ⎛1+1⎞ ⎜α β⎟ ⎝⎠PDF Image | Organic Rankine Cycle Solar-Thermal Powerplants
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