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10 3. TURBOCHARGERS, WITH FOCUS ON THE TURBINE where ω ̄ is the angular velocity and T ̄Shaft is the shaft torque. In the computations, the shaft torque is obtained by integrating the element forces due to the shear and pressure forces times the radial coordinate over the turbine wheel: T ̄Shaft = r ̄×(f ̄·nˆ)dS (3.2) S Another important quantity is the turbine efficiency, ηis. Commonly one defines the efficiency as the ratio of shaft power to the maximal isentropic power of the driving gas flow: η= PS (3.3) is γ−1 m ̇ · cp · T01 · (1 − p2 γ ) p01 where m ̇ is the mass flow, T01 and p01 is the total gas temperature and pressure before the turbine and p2 is the static pressure downstream of the turbine. cp is the specific heat and γ is the specific heat ratio. This definition works well for non-pulsatile flow and for pulsatile flow at moderate frequencies. For higher frequencies, the non-constant phase shift between the mass flow, pressure and shaft torque makes it impossible to define an isentropic efficiency in a proper way. The fundamental parameters can either be predicted by using analytic or semi-analytic models or by testing the turbocharger under controlled condi- tions. The fundamental parameters are often non-dimensionalized, see Table 1, to take account for the inlet conditions and are often reported in com- pressor and turbine maps. Since the diameter, D for a particular turbine is constant, it can be dropped. The turbines are also tested with the actual work- ing gas, and hence, the gas constant R and γ can also be dropped. By doing so, the quasi-non dimensional parameters are obtained. The performance can also be reported as the referred parameters, which have the same units as the fundamental parameters. In the referred parameters, the inlet pressure and temperatute are referred to reference values. The performance of a turbine is tested in gas stand where the turbine is driven by gas from an external compressor. The measurement can be performed both with cold and hot gas. In the later case, fuel is injected and burnt in a combustion chamber upstream of the turbine. Since the rotational speed is very high, it is very hard to directly measure the shaft power output of the turbine. Therefore, the compressor is throttled to control the load on the turbine, and by measuring the shaft speed, the inlet and outlet pressure, temperature and the mass flow through the turbine and the compressor, respectively, the efficiency of the turbine can be computed. This efficiency is the mechanical efficiency, since the bearing losses are included. The mechanical efficiency can be divided intoPDF Image | Numerical computations of the unsteady flow in a radial turbine
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