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azimuth angleψ . ψ V =Vr0 ψ,θ 0r M Abidat, M K Hamidou, M Hachemi And M Hamel Or ψ AρV=AρV1.− (01b) of the velocity at the ψ ψ ψ,θ 0 0 0 360 The free vortex law is used to compute the tangential component V (02) At each cross section, the flow is assumed to be tangent to the centroid line defined by equation 04 and the absolute flow angle αψ is given by: ψ tan(α )= dr (03) ψ rdψ ψ ,θ ψ A relationship between the centroid radius rψ and the azimuth angle ψ is obtained by an analytical method based on Bezier polynomials, as described by Forest [10] and Casey [11]. This method was chosen for its simplicity and flexibility. The form of the curve r = r(ψ ) can be modified simply by varying the position of the points describing it. In this case, a third order polynomial is used to define the centroid curve (figure 01), while the intermediate points P1 and P2 are determined by requirements of slope and curvature at end points. →→→→→→→ 3223 OP(u)=ψi+r j=(1−u)OP+3u(1−u)OP+3u (1−u)OP+uOP (04) ψ0123 The velocity Vψ at each cross section centroid is obtained by equation 05. V V= ψ,θ (05) ψ tan(α ) ψ ψ Combining equation 05 with the energy equation for an adiabatic flow and the equations 01b and 03 gives the cross section area at each azimuth angle. A = qm,ψ (06) ψ ψ ψ,θ ρV The selection of the cross section and the vaneless nozzle shapes shown in figure 02 has been made according to the following considerations: • The distance between the bearing housing and the back of the rotor is limited so that possible rotor vibrations, which can be damaging to the machine, can be avoided. • The need to have a straight vaneless duct upstream of the rotor so that the flow is directed properly at the meridional direction. • The simplicity of the casing geometry so that it can be easily manufactured. The method described above has been used to design the mixed flow turbine volute shown in figure 03. The whole turbine with three different rotors has been tested in a cold rig test at Imperial College by Abidat et al [9].PDF Image | ANALYSIS OF RADIAL AND MIXED FLOW TURBINE VOLUTES
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