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the pressure curves such that we have a monotonic map. A more important drawback is that the derivative of the mass flow would not be continuous with a bilinear interpolation. This can lead to numerical problems in the simulation. For some parameters like mass flow and pressure it is essential to have the derivative continuous, in order to do physically correct modelling. There are other interpolation methods, e.g. using cubic splines to generate continuous maps. The drawback here is then that we need to calculate 16 spline coefficients for each grid point in the interpolation map. In our case with 176 grid points and four interpolation tables we would need to calculate 11264 coefficients. Doing good modelling with interpolation methods is not an easy subject and is beyond the scope of this thesis. The bilinear interpolation method was tested for the compressor mass flow and efficiency, but could not prove to be reliable in the whole operating range, even after the compressor map had been manually altered to provide unique solutions for lower speeds. In most cases the simulation was halted or got stuck due to chattering (discontinuity sticking). What happens is that a variable, e.g. speed is changed causing the interpolation index of the axis of the interpolation table to also change. The interpolated output data, e.g. the mass flow, causes the speed to change back and then the index is changed back as well and so on and so forth. This might be partly caused by the fact that the data is not rectangular. With this means that each speed has a different pressure ratio interval, e.g. for 20 000 rpm the pressure ratio ranges from 1.18 to 1.02, whereas for 70 000 rpm the pressure ratio ranges from 5.2 to 4.8. This means that e.g. index 1 (the first vertical column) represents different pressure ratios depending on speed. The turbine data, however, is in rectangular form. All the speeds have the same range of pressure ratio and e.g. the index 1 (the first vertical column) always represent data at a pressure ratio of 1.25. The reasons behind this are simple. The compressor compresses the air and can for a certain speed only create a certain maximum pressure ratio. The higher the speed the higher is the pressure ratio created. The turbine expands the gas and work for all different pressure ratios and speeds, since it uses the pressure ratio created by the compressor. Another problem is when the numerical solver of the n:th order expects to find equations that are n times differentiable and the equations are only differentiable of a lower order. Often the solver succeeds despite this, but there are always reasons to be careful. The mechanical efficiency of the compressor is taken to 100 %, since the major friction losses in the microturbine come from the two bearings, which are modelled in a separate submodel, see section 5.10. 5.3 The turbine equations Before the exhaust gas enters the turbine, it passes through a nozzle where the velocity is increased and the pressure decreases. The high-speed gas exerts a force on the turbine blades and the geometry of the blade causes the turbine to rotate, thus producing mechanical work. After the turbine rotor, the gas passes through the diffuser of the turbine, where again the velocity is decreased and the pressure increased but not as much as after the compressor. The thermodynamic equations are in large parts the same as for the compressor. Instead of repeating all of the derivations, there will be references to the corresponding compressor section. We have the same energy balance as we used for the compressor. Also the same approximations can be made about the turbine. The only difference is that work is now done by the gas on the surroundings, i.e. on the turbine. The equation for the isentropic work of the turbine is therefore exactly as equation (5.1.9) but with the sign changed. From equation (5.1.8) we get the relation for isentropic expansion and by including the isentropic efficiency from equation (5.1.10), the equation for the produced actual work can be rewritten as: 30PDF Image | Modelling of Microturbine Systems
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