3 (a) The flow entering an inward-flow turbine may be assumed to be frictionless and two-...

 

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3 (a) The flow entering an inward-flow turbine may be assumed to be frictionless and two- dimensional. The flow can be approximated by the superposition of a free vortex of strength I and a sink of strength m. Water leaves the guide passages at a radius of 1.5 m. The velocity is 30 m/s and makes an angle of 60° to the radial direction. The water then enters the runner blade at a radius of 1.25 m. The density of water is 1000 kg/m³. (i) Determine the strength I' of the free vortex and strength m of the sink? (ii) If the pressure at the exit of the guide passage is 200 kPa (gauge), what is the gauge pressure at the inlet of the runner blade? (6 marks) (iii) Based on the inlet pressure at the runner blade, what is the likely problem that the turbine will encounter at the exit of the runner blade? Explain your answer and suggest ways to overcome the problem encountered. (2 marks) (Note that following expressions may be used: v₁ = (avz at Vr (b) Consider steady flow of an incompressible Newtonian liquid between two horizontal infinitely long, concentric cylinders as shown in Figure 2. The outer cylinder is stationary, and the inner cylinder moves in the axial direction with a velocity U. The flow is caused entirely by the motion of the inner cylinder and the pressure gradient in the axial direction is zero. 2πU (r-r In 4 + Vz 772 Figure 2 (1) Derive an expression for the axial velocity V₂ (r). (ii) State the boundary conditions at the inner and outer radii? (iii) Show that the volume flow rate Q can be expressed as Əv₂) ap дz U === 3 2ær Fixed wall = vo (4 marks) 2πr + 1/{7 m (² (5 marks) In cylindrical coordinates, the Navier-Stokes equation for the axial velocity V₂ (r) is: əvz ve dvz əvz + ər r 20 = + Pg₂ + μ[ ²1 (70V²₂) +- a - rər 101, 07, 2 მ02 дz ər (5 marks) (3 marks) •+22] 3 (a) The flow entering an inward-flow turbine may be assumed to be frictionless and two- dimensional. The flow can be approximated by the superposition of a free vortex of strength I and a sink of strength m. Water leaves the guide passages at a radius of 1.5 m. The velocity is 30 m/s and makes an angle of 60° to the radial direction. The water then enters the runner blade at a radius of 1.25 m. The density of water is 1000 kg/m³. (i) Determine the strength I' of the free vortex and strength m of the sink? (ii) If the pressure at the exit of the guide passage is 200 kPa (gauge), what is the gauge pressure at the inlet of the runner blade? (6 marks) (iii) Based on the inlet pressure at the runner blade, what is the likely problem that the turbine will encounter at the exit of the runner blade? Explain your answer and suggest ways to overcome the problem encountered. (2 marks) (Note that following expressions may be used: v₁ = (avz at Vr (b) Consider steady flow of an incompressible Newtonian liquid between two horizontal infinitely long, concentric cylinders as shown in Figure 2. The outer cylinder is stationary, and the inner cylinder moves in the axial direction with a velocity U. The flow is caused entirely by the motion of the inner cylinder and the pressure gradient in the axial direction is zero. 2πU (r-r In 4 + Vz 772 Figure 2 (1) Derive an expression for the axial velocity V₂ (r). (ii) State the boundary conditions at the inner and outer radii? (iii) Show that the volume flow rate Q can be expressed as Əv₂) ap дz U === 3 2ær Fixed wall = vo (4 marks) 2πr + 1/{7 m (² (5 marks) In cylindrical coordinates, the Navier-Stokes equation for the axial velocity V₂ (r) is: əvz ve dvz əvz + ər r 20 = + Pg₂ + μ[ ²1 (70V²₂) +- a - rər 101, 07, 2 მ02 дz ər (5 marks) (3 marks) •+22]