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Given a known pressure ratio and speed we want to know the corresponding mass flow. In the original map this might be a problem since the curves for the lower speeds are not monotonic, i.e. there is no unique solution to the problem. With the use of elliptic curves we get monotonic yet accurate functions. An obvious drawback with the method is the large manual part of the function fitting. It makes the process less automatic, when new compressors are introduced, but on the other hand it gives a model that works for the whole operating range. The errors introduced come firstly from the curve fitting procedure, and secondly, new errors arise when the curves are parameterized with speed. The error of the mass flow model is around 2.9 %, around the normal operating point of 70 000 rpm, see the result section, and is in general somewhat higher for lower speeds. Especially when the gas turbine operates near the surge line where the model curve is almost flat; a small error in the pressure ratio gives a larger error in the mass flow. However the gas turbine operates only in the lower regions during the start up or stop procedure and then the functionality of the model is emphasized over absolute accuracy. The surge points and choke points for each speed are gathered and parameterized with speed. During simulation these polynomial functions are evaluated with the current speed to get an estimate of how close the current operating point is to the surge and choke line. Compressor map and model curves, at various speeds High speed Surge line Choke line Low speed Corrected mass flow, mdotcorr = mdot sqrt(T1) / p1 Figure 17: Compressor map (solid blue) and fitted ellipsoid curves (dashed red) Summary of the development of the mass flow model of the compressor: 1. The value of the parameter a is taken from the map, i.e. the mass flow value at choking conditions. 2. The parameter z is set to an arbitrarily value, e.g. 5. 3. One data point (x, y) is taken from the map, where the ellipsoid curve and data will be identical matched. Using this point, equation (5.2.4) is solved for the parameter b. 4. The curve is plotted and compared to data from the map. By visual inspection, the parameters a and z are modified to ensure a better fit inside the operating range. 28 Pressure ratio, p2/p1PDF Image | Modelling of Microturbine Systems
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