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5. For each speed there is a different set of parameters. The values of the parameters are now parameterized with speed in Matlab to polynomial functions to achieve a continuous model. 6. With the complete model, the mass flow is now continuously given by speed and pressure ratio. Modelling the isentropic efficiency was harder. As can be seen in the figure below, there is no ellipsoid form to use. For a given rotational speed, the maximum efficiency is well-known, i.e. the top value of the curve. Knowing the maximum efficiency, we can approximate the efficiency map with parabolic degradation curves. The amount of curvature or degradation is denoted d and is fitted based on numerous data to get parameterization for all speeds. The following equation is taken from Gustafsson (1998). η = η m a x − d ( m& − m& m a x e f f ) 2 (5.2.6) Compressor efficiency from map and model at various speeds 70 000 rpm 66 500 rpm 63 000 rpm Corrected mass flow, mdotcorr=mdot sqrt(T1)/p1 Figure 18: Isentropic efficiency from map (solid blue) and model (dashed red) as a function of speed and mass flow As can be seen in the figure above, the curves are not exactly symmetrical. Near choking conditions (to the right), the performance is decreased rapidly. Another difficult part in this case is that the curvature changes a factor of 20 from the lowest to the highest speed. It is hard to create a continuous function that gives the correct value at every speed. Instead a simple interpolation function was used. At steady state the microturbine is very close to optimum, where the model is very accurate. At the extreme ends of the operating range, as e.g. near surge and choking, the accuracy decreases, due to errors both in the modelled mass flow and modelled efficiency. So why did the interpolation method not work for the compressor mass flow or efficiency? The first obvious drawback is that for lower speeds the map does not have a unique solution for a given pressure and speed. As we did for the continuous ellipsoid curves we can manually change 29 Isentropic efficiencyPDF Image | Modelling of Microturbine Systems
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