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The axes of the turbine efficiency map consist of the speed and pressure ratio in monotonically increasing vectors, speed_vec and pr. Given a speed and a pressure ratio, the speed vector is searched to find the lowest index i of the fitting speed interval. The procedure is repeated for the vector of the pressure ratio and the corresponding index j is found. Now there are four possible efficiency values (the end points of the two fitting intervals) to interpolate between. These four values are the corners of the interpolated field. f(i,j) v f(i,j+1) u f(i+1,j) f(i+1,j+1) f(i’,j’) Figure 21: A point to be interpolated, surrounded by four grid points The variables u and v are the normalized distances (between 0 and 1), according to the following equation: (5.4.2) (5.4.3) where r is the actual pressure ratio, speed is the actual speed and speed_vec and pr_vec are the axes of the interpolation table. The interpolation equation is then given by: With the interpolation method, no new errors are introduced and the accuracy of efficiency model depends solely on the accuracy of the turbine map. The pressure ratio used as the input to the interpolation method is the pressure ratio excluding the turbine diffuser. The diffuser recovers some pressure and the velocity of the gas is decreased. The diffuser increases the efficiency of the turbine, since the turbine with a diffuser can produce the same amount of work but with a lower pressure ratio. In the turbine model, the diffuser is not included, but the efficiency of the turbine is increased by 1% to reflect the effect of the diffuser. The mechanical efficiency of the turbine is taken to 100 %, since the major friction losses in the microturbine come from the two bearings, which are modelled in a separate Loss model, see section 5.10. v = r - pr _ vec[j] pr _ vec[j+ 1] - pr _ vec[j] u = speed - speed_vec[i] speed_vec[i+ 1] - speed_vec[i] 33 (5.4.4) f(i', j')=(1−u)(1−v)⋅ f(i, j)+(1−u)v⋅ f(i, j+1)+u(1−v)⋅ f(i+1, j)+uv⋅ f(i+1, j+1)PDF Image | Modelling of Microturbine Systems
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