Prof. Dr.-Ing. F.-K. Benra. ISE Bachelor Course

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1 University Duisburg-Essen Campus Duisburg Faculty of Engineering Science Department of Mechanical Engineering Name Matr.- Nr. Examination: Fluid Machines Examiner: Prof. Dr.-Ing. F.-K. Benra Date of examination: Handling time: 120 Minutes ISE Bachelor Course Designated scores: Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 ( 18 points) ( 18 points) ( 24 points) ( 24 points) ( 16 points) Σ 100 points Permitted utilities: Table of formulas (provided), pocket calculator

2 Exercise 1 (18 points) The potential energy of a water reservoir which is placed on a hill should be deployed to make electric power. To design a water turbine for a power station the following data are known: Rotational speed of turbine: n = 1000 min -1 Available head: H = 85 m Water flow rate at design point: V & = 3,1 m 3 /s 1.1 Find out which type of turbine should be used according to the Cordier-diagram. Justify your choice. 1.2 Indicate the amounts of the following non-dimensional machine parameters of the chosen turbine: - specific diameter δ ym - through flow coefficient ϕ M - non-dimensional pressure parameter ψ ym - specific speed σ ym 1.3 Calculate from the determined non-dimensional machine parameters the outer diameter of the impeller. 1.4 Sketch a meridional view of the stator and of the impeller of the chosen turbine und give numbers to the different planes.

3 Name Matr.- Nr. To exercise 1 Figure 1: Cordier-diagram

4 Exercise 2 (18 points) 2.1 Show the isentropic and the polytropic changes of state in a qualitative h,s-diagram for an adiabatic expansion process of a turbine. The pressure rate is the same for both changes in state. 2.2 Outline the energy fractions Δh, Δh s, the kinetic energy at inlet and at outlet of the turbine as well as the change of total enthalpy Δh t in the h,s-diagram. 2.3 Mark in a T,s-diagram the change in state of an adiabatic expansion process of a turbine and show the energy fractions Δh, y and j. 2.4 Show the isentropic and the polytropic efficiency of an adiabatic expansion process of a turbine by marking the appropriate parameters in the h,s-diagram. 2.5 Constitute if the polytropic or the isentropic efficiency of an adiabatic expansion process between the same pressure rate of a turbine is higher by using the T,s-diagram.

5 Exercise 3 (24 points) Oxygen is delivered to a higher pressure level in a plant by a multistage turbo compressor. The complete change in state from inlet to outlet of the machine can be regarded as a polytropic and adiabatic process. In the concerning pressure and temperature range the fluid Oxygen can be considered as an ideal gas which has constant specific thermal capacities. The following data are known: Specific thermal heat capacity of O 2 : c p = 0,917 kj/(kgk) Specific gas constant of O 2 : R = 259,8 J/(kgK) Adiabatic change in state: q = 0 Polytropic efficiency of compressor: η pol = 0,82 Static temperature at inlet: T E = 298 K Static pressure at inlet: p E = 1 bar r r Absolute velocities at inlet and outlet: c E =c A Compressor pressure ratio: π C,EA = Determine the polytropic exponent of the complete change in state between inlet and outlet of the compressor. 3.2 Indicate the static temperature and the static pressure at the outlet of the machine. Assume the polytropic exponent to be n = 1,4 if you did not find out in part Calculate the specific work a EA, the difference of specific enthalpy Δh EA, the specific flow work y EA and the specific dissipation j EA of the machine. 3.4 Indicate the change in state between inlet and outlet of the machine in a T,s-diagram and allocate the energy fractions a EA, Δh EA, y EA and j EA. 3.5 Calculate the static temperature T A,s under the assumption of an isentropic change in state with the same pressure ratio and indicate the static isentropic efficiency η s.

6 Exercise 4 (24 points) A stage of a radial compressor consists of an impeller and a vaned diffuser. The outflow out of the impeller is in radial direction (β = 90 deg). The stage is subject to be a repeating stage r r ( c 1=c 3 ). The fluid which is to be compressed can be regarded as an ideal gas which has constant specific thermal capacities. The change in state is adiabatic (q = 0). The following data are known: Constant through flow coefficient: ϕ 1 = ϕ 2 = ϕ 3 = 0,4 Impeller diameter ratio: d 1 /d 2 = 0,55 Impeller speed at outer diameter: u 2 = 245 m/s Swirl angle at impeller inlet: α 1 = 60 deg 4.1 Outline a meridional view of the compressor stage and indicate the appropriate planes. 4.2 Draw in scale the non-dimensional velocity triangles of the compressor stage. 4.3 Sketch the circular cascades of the impeller and of the diffuser vanes (view to an axis normal cut). 4.4 Calculate the specific work a, the kinematic degree of reaction ρ h, the enthalpy parameter Ψ h, the enthalpy differences of the impeller Δh and of the stator Δh of the stage. 4.5 Depict a complete h,s-diagram of the stage. Identify all energy values. Take the partitioning of the change in state into account which should be exact in a qualitative manner.

7 Exercise 5 (16 points) The given figure 5.1 shows the non-dimensional characteristic (Ψ y = f (ϕ 2 )) of a turbomachine stage which has a swirl free oncoming flow. The stage behaves like a repeating stage. 5.1 Is the turbomachine which produces the given characteristic a turbine or is it a compressor? Please justify your choice. 5.2 Insert the theoretical characteristic Ψ ht = f (ϕ 2 ) which evolves under the assumption of an infinite number of blades in both blade rows. For ϕ 2 = 1,5 the total enthalpy coefficient is Ψ ht = Give a number for the impeller outlet flow angle (β 2 > 90 deg or β 2 = 90 deg or β 2 < 90 deg). 5.4 Calculate the total polytropic efficiency for the design point of the machine. The design point is designated as A. Use this amount of efficiency and draw a curve of the efficiency for the complete operating range of the machine in a qualitative manner into the η t, ϕ 2 -diagram. 5.5 Calculate the specific losses for the design point A of the machine. The impeller speed at the outer diameter is u 2 = 250 m/s. 5.6 Draw the non-dimensional velocity triangles of the turbomachine stage in a qualitative manner.

8 Name Matr.- Nr. Figure 5.1

Prof. Dr.-Ing. F.-K. Benra. ISE batchelor course

Prof. Dr.-Ing. F.-K. Benra. ISE batchelor course University Duisburg-Essen Campus Duisburg Faculty of engineering Science Department of Mechanical Engineering Examination: Fluid Machines Examiner: Prof. Dr.-Ing. F.-K. Benra Date of examination: 06.03.2006