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From Cengel (1998) we get the following expression for isentropic compression where the subscript is denotes the variable at isentropic conditions: (5.1.8) 11 The equation for the specific work can now be calculated using the inlet and outlet pressure and κ−1 T pκ 2,is = 2 Tp input temperatures as follows: κ−1 κ p κ −wis =( )RT1 2 −1 κ−1 p1 (5.1.9) This is only valid for ideal processes, processes that are adiabatic and reversible. To compensate for actual conditions, we use the thermodynamic variable, isentropic efficiency, defined as the ratio of the enthalpy change at isentropic conditions and the actual enthalpy change: ηis =h2,is −h1 =cp(T2,is −T1)=T2,is −T1 (5.1.10) h2 −h1 cp(T2 −T1) T2 −T1 The value of the isentropic efficiency can be determined through experiments, where the temperatures are measured. Results from tests show that the efficiency does not however remain constant for all pressure ratios. For increasing pressure ratios, there is a decrease in isentropic efficiency. A physical explanation is that the increase in temperature due to friction in one stage of the compression results in more work being required in the next stage, Cohen, (1996). Apart from friction, there is another factor that contributes to a lower efficiency. If another speed is used, different from the rotational speed the compressor is optimally designed for, this can cause a difference in alignment with the gas flow and the impeller vanes. If the flow enters in an angle different from the angle of the impeller vanes, there will be small turbulent eddies right behind the vanes and the efficiency of the compressor will decrease. The actual specific work required can be formulated as: κ pκ−1 1 −w=( )RT1( 2)κ −1 κ−1 p1 ηis The actual power needed to run the compressor is then simply: Pcompressor = m& ⋅ w There can also be mechanical losses in the compressor. The compressor is mounted on a single (5.1.11) shaft and the relation between torque and power can be written as: P ⋅η =τ ⋅ω compressor mec compressor (5.1.13) (5.1.12) where ω is the angular velocity of the shaft, τ is the torque the compressor consumes and ηmec is the mechanical efficiency of the compressor. 5.2 The compressor model in Modelica Based on the equations derived above a model of the compressor was developed. The model is taken from Perez (2001) and then modified. There are some properties that cannot be calculated analytically, e.g. the isentropic efficiency and the mass flow through the compressor. These must instead come from empirical data, which is given in the form of a compressor map. In the map, the mass flow and efficiency are given as functions of other known variables, usually the speed and the pressure ratio. The compressor map 24PDF Image | Modelling of Microturbine Systems
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