University Duisburg-Essen Campus Duisburg Faculty of Engineering Science Examination: Fluid Machines Examiner: Prof. Dr.-Ing. F.-K. Benra Date of examination: 07.08.2006 Handling time: 120 Minutes ISE Bachelor Course Designated scores: Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 ( 18 points) ( 20 points) ( 22 points) ( 24 points) ( 16 points) Σ 100 points Permitted utilities: Table of formulas (provided), pocket calculator
Exercise 1 (18 points) For an exhaust turbo-supercharger the impeller of a single stage centripetal turbine should be designed for following data: mass flow rate: m& = 0,95 kg/s pressure ratio: p 0 /p 2 = 1,82 inlet pressure: p 0 = 1,82 bar inlet temperature: T 0 = 710 K b 1 Fluid: combustion gas with D1 R = 288 J/(kg K) κ =1,35 = const. Di2 Dm2 Da2 The single stage is a repeating stage with swirlfree outflow and with adiabatic change in state. For the design following data are known: Static polytropic efficiency of the stage: η T = 0,82 Static polytropic efficiency of the stator: η T = 0,84 Kinematic degree of reaction: ρ h = 0,5 Flow coefficient: ϕ 1 = ϕ 2 = 0,72 Diameter ratio: D m2 /D 1 = 0,46 The change in state in stator and rotor is approximately polytropic. 1.1 Calculate the rotational speed of the machine for an optimum specific speed of σ ym = 0,15. 1.2 Calculate the impeller diameter D 1. Use the enclosed Cordier-diagram. 1.3 Calculate the pressure and the temperature at impeller inlet (p 1, T 1 ) and at impeller outlet (p 2, T 2 ). 1.4 Calculate the width b 1 of the impeller, as well as the inner diameter D i2 and the outer diameter D a2 of the impeller.
Exercise 2 (20 points) A multi-stage, radial turbo compressor compresses oxygen to a higher pressure level. The change in state from inlet to outlet of the machine is approximately polytropic and adiabatic. For the mentioned pressure and temperature range, the oxygen can be described as an ideal gas with constant specific heat. Following data are known: Specific heat of O 2 : c p = 917 J/kgK Specific gas constant of O 2 : R = 259,8 J/kgK Polytropic efficiency η = 0,8 Inlet temperature T E = 295 K Inlet pressure: p Ε = 1 bar Velocities in absolute frame of reference: c r E = c r A Pressure ratio: p A /p E = 6,5 2.1 Calculate the mean polytropic exponent for the change in state from inlet to outlet of the machine. 2.2 Calculate the temperature and the pressure at outlet of the machine. 2.3 Calculate the specific work a EA, the specific change in enthalpy Δh EA, the specific flow work y EA and the specific dissipation j EA of the machine. 2.4 Draw the polytropic change in state of the machine from inlet to outlet in a T,sdiagram. Mark the flow work y EA, the change in enthalpy Δh EA and the dissipation j EA. 2.5 Calculate the temperature T A,S at machine outlet for the case of an isentropic change in state. Calculate the isentropic efficiency. η S
Exercise 3 ( 22 points) A radial blower for the delivery of carbon dioxide is equipped with an impeller with radial ending blades (ß 2 = 90 o ). The flow in front of the impeller is swirl free. The blower stage satisfies the repetition condition. In addition the following data are known: Measured volume flow: V & = 10 m /s Density of the gas: ρ = 1,98 kg/m 3 Outer impeller diameter: D 2 = 1 m Inner impeller diameter: D 1 = 0,6 m Meridional velocity: c m1 = c m2 = c m3 = 30 m/s Mechanical efficiency: η mech = 0,98 Rotational speed: n = 1470 min -1 For the change in state inside the machine the gas can be assumed as incompressible. The machine can be assumed as adiabatic. 3.1 Draw a sketch of a meridional section of the machine. Indicate the flow planes 1, 2, 3. 3.2 Draw a sketch of the impeller in a cutting plane normal to the axis of rotation. Show the blades in this plane as a circular flat cascade. Show qualitatively the right shape of the blades and indicate the direction of rotation of the impeller. 3.3 Draw the non-dimensional velocity triangles of this blower stage. (scale u 2 /u 2 = 1 =ˆ 10 cm) 3.4 Calculate the kinematic degree of reaction ρ h and the enthalpy coefficient ψ h of the stage.
Exercise 4 ( 24 points) For an axial turbine stage the following data at midspan are known: Adiabatic change in state: q = 0 Constant flow coefficient: ϕ 0 = ϕ 1 = ϕ 2 = 0,417 Inflow velocity: c 0 = 125 m/s Swirl free impeller outflow: α 2 = 90 deg Kinematic degree of reaction: ρ h = 0 Diameter of the mean streamline: D m0 = D m1 = D m2 = 0,662 m Rotational speed: n = 144 s -1 Flow conditions upstream and downstream of the stage are the same ( repeating stage) 4.1 Draw the non-dimensional velocity diagrams to scale and show the quantities of ϕ, ψ h and ρ h in the velocity diagrams (scale: u/u = 5 cm). 4.2 What is the name of this kind of turbine stage? 4.3 Show the thermodynamic process of the turbine stage in a qualitatively correct h,sdiagram and show the enthalpy differences Δh, Δh, Δh, Δh t, Δh s and Δh s. 4.4 Draw a qualitatively correct meridional section of the stage and the trend of the static pressure from the entrance to the outlet of the stage in a diagram. 4.5 Draw circumferential sections of the stator and of the rotor blade rows with a minimum of two blades for each row. 4.6 Determine the enthalpy coefficient ψ h and the specific work a of this turbine stage.
Exercise 5 ( 16 points) In the enclosed diagram, the non-dimensional head characteristic ψ yt = f (ϕ 2 ) of a turbomachine stage is given. The following additional data are known: Adiabatic change in state: q = 0 Swirl free impeller inflow: α 1 = 90 deg Constant flow coefficient: ϕ 1 = ϕ 2 = ϕ 3 = 0,4 Ratio of impeller diameters: d 1 /d 2 = 0,6 Circumferential speed at impeller exit: u 2 = 250 m/s Flow conditions upstream and downstream of the stage are the same ( repeating stage) 5.1 Is the machine a turbine or a compressor? 5.2 Draw into the given diagram the theoretical head characteristic ht = f (ϕ2) (after streamline theory) of the stage. For ϕ 2 = 1,5 the amount of ψht is zero. 5.3 What kind of blades has the impeller (forward swept, radial ending, backward swept)? 5.4 Calculate the total polytropic efficiency η pol,t,a for the design point A of the stage. Draw into the given diagram [η pol,t = f (ϕ 2 )] the efficiency curve of the stage in a qualitative manner, outgoing from the efficiency η pol,t,a at the design point. 5.5 Calculate the specific losses j for the design point of the stage. ψ 5.6 Draw the non-dimensional velocity triangles of the stage for the design point (scale: u 2 / u 2 = 10cm).
Exercise 5: Non-dimensional characteristic Name: Matr.-Nr: