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: 05.03.2007

Handling time: 120 Minutes

ISE Bachelor Course Designated scores:

Permitted utilities: Exercise 1 ( 24 points) Exercise 2 ( 20 points) Exercise 3 ( 16 points) Exercise 4 ( 18 points) Exercise 5 ( 22 points)

Table of formulas (provided), pocket calculator

Σ 100 points

Department of Mechanical Engineering Turbomachinery Exercise 1 ( 24 points)

The blade of a single stage radial compressor has to be designed. The inflow of the blade is swirl free. Following data are known: Mass flow rate: = 1,3 kg/s m& Pressure ratio of stage: p3/p1 = 1,6 State of the air at inlet: Τ1 = 288 K p1 = 1 bar Material properties of air: κ = 1,4 = const. R = 287 J/(kg K) Static polytropic efficiency (stage): η = 0,8 Static polytropic efficiency (impeller): η’’ = 0,85 Flow coefficient plane 1: ϕ1 = 0,28

Flow coefficient plane 2: ϕ2 = 0,25 Diameter ratio: D1/D2 = 0,5 Ratio of relative velocities: w2/w1 = 0,73.

Assumption: Polytropic change of state without heat transfer. Differences in geodetic height can be neglected.

1.1 Determine the optimal number of revolution n for the specific revolution of σyM = 0,12 and the optimal blade diameter D2 by means of the Cordier-diagram.

1.2 Draw the non-dimensional velocity diagrams and calculate the relative flow angles β1 und β2.

1.3 Calculate the thermodynamic state (p2, T2) in the plane 2.

1.4 Calculate the blade widths (b1 and b2) at the inlet and the outlet of the blade (see fig. 1.1).

Fig. 1.1

Department of Mechanical Engineering Turbomachinery

Fig.1.2: Cordier-diagram

Abteilung Maschinenbau Strömungsmaschinen Exercise 2 ( 20 Points)

The following design data from an adiabatic compressor stage (impeller + guide vane) are known: Temperature behind impeller T2 = 300 K Pressure ratio of guide vane p3/p2 = 1.05

Polytropic ratio of impeller 1"

"

−nn = 3,0

Specific gas constant R = 287 J/(kg K) Isentropic exponent κ = 1,4 Speccific heat at const. pressure cp = 1004,15 J/(kg K) Enthalpie difference of guide vane = 6,55 kJ/kg 'hΔ

Enthalpie difference of impeller = 15,05 kJ/kg ''hΔ Absolute velocity 1cr = 3cr

2.1 Calculate the polytropic efficiency of the impeller qh

y−Δ

= ''

''''η and the guide vane

qhy−Δ

= ''η

'

2.2 Determine the specific flow work (y, y’, y’’), the dissipation (j, j’, j’’) and the increase of entropy (Δs, Δs’, Δs’’) in the impeller, the guide vane and in the stage. Give also the spec. work (aEA).

2.3 Draw a sketch of an temperature entropy diagram (T-s diagram) and mark the areas that represent the quantities y’, y’’, j’, j’’, Δh’ and Δh’’ .

Abteilung Maschinenbau Strömungsmaschinen Exercise 3 ( 16 Points)

Given are the data of the first stage of an axial flow turbine. This stage satisfies the repeating condition and has a swirl free inflow. The following data of the stage are also known:

Pressure ratio p0/p2 = 2,5

Flow coefficient ϕ 1 = 0,4 ϕ 2 = 0,6 Enthalpie coefficient Ψ h = -3 Static inlet temperature T0 = 1173 K static Pressure at inlet p0 = 5 bar static polytropic efficiency η T = 0,9 The working fluid can be estimated to be an ideal fluid with the following fluid parameters: Specific gas constant R = 289 J/(kg K) Specfic heat capacity cp = 1,05 kJ/(kg K) Isentropic exponent κ = 1,38 The change in state in this turbine stage is adiabatic.

3.1 Calculate the degree of reaction ρh.

3.2 Draw (in scale) the dimensionless velocity triangles of the stage and show also the parameters ρh, Ψ h and ϕ 1, ϕ 2 in the triangles.

3.3 Give the isentropic static efficiency (η T)s of this turbine stage.

3.4 Calculate the termodynamic state at impeller inlet (plane 1; T1 = ?, p1 = ?) with the assumption of an equal polytropic ratio ν in impeller and guide vane ( ν = ν’ = ν’’ ). Draw this change in state qualitatively in a h,s – diagram.

Department of Mechanical Engineering Turbomachinery Exercise 4 ( 18 points)

Figure 4.1 shows two sections of a rotor blade of an axial flow low pressure steam turbine. The blade is part of a repeating stage, in which an adiabatic change of state takes place.

4.1 Complete the velocity diagrams for both sections at inlet and outlet of the blade for the condition of a swirl free outflow. Give attention to the correct length of the vector of the circumferential speed and label the velocity vectors using the letters u, c and w.

4.2 Transfer the velocity diagrams of the two sections into the turbine typical non-dimensional plot and mark the characteristics ϕ, ψh und ρh as distance. Scale: u/u = 3cm.

4.3 What are the names of the two stage types presented in both sections?

4.4 Specify the ratio of the two associated diameters ( ) for a constant specific work along the blade radius. Deduce an equation for the ratio and calculate the amount of .

ΙΙΙ DD /DD /

DD /ΙΙΙ

ΙΙΙ

Schnitt I

Schnitt II

I I

II II

Figure 4.1: Blade of a low pressure steam turbine

Department of Mechanical Engineering Turbomachinery Exercise 5 (22 points)

For an adiabatic axial compressor stage the following design data are known:

Repeating condition: 31 ccrr

=

Swirl free through flow to the impeller: cu1 = 0 m/s

Constant flow coefficient: ϕD = constant

For the same through flow conditions and with the same number of revolutions the blades of

the impeller and the stator should be investigated for the following design conditions:

3c2b1a

Dht

Dht

Dht

=ψ

=ψ

=ψ

∞

∞

∞

,

,

,

)))

10.1 Show the non-dimensional velocity triangles for the cases a) to c).

(Assumption for on infinite number of blades: flow follows perfect the blade shape).

10.2 Draw the blades for the impeller and for the stator in plane cascades for the cases a) to

c) in a qualitative correct manner into the given sheet and give the correct names for

the different cases.

10.3 Specify the kinematic degrees of reaction ρh for the cases a) to c).

10.4 Draw the slope of the theoretical characteristics for the three cases from ϕmin = 0 to

ϕmax = 1,5ϕD into the given diagram ( diagramht −ϕψ ∞ , ) .

ϕ

ψht 8

0

1

2

3

Department of Mechanical Engineering Turbomachinery

To exercise 10 Impeller Stator

a)

b)

c)