Turbomachinery assignment

MECH557 HW# 3

Due Date: 11/21/13

Problem 1 Consider a compressor with a pressure ratio of 8. The inlet total temperature is 300oK and the outlet total temperature is 586.4oK. Calculate the isentropic efficiency and the polytropic efficiency.

Problem 2

Problem 3 A small axial fan (D ~ 7.5 cm) for computer cooling applications has a blade leading-edge angle  of 71.5o. The incoming flow is axial.

(a) Estimate the maximum ideal static efficiency of the fan. (b) Estimate the blade trailing-edge angle  for maximum ideal static efficiency. (c) If a stator is employed, what is the maximum ideal static efficiency of the

corresponding stage? (d) Calculate the pressure coefficient across the rotor, i.e.

  2/2

2

U ppC atm

rotorp  



(e) Calculate the pressure coefficient across the stator, i.e.

  2/2 23

U ppC

statorp  



(f) If an ideal diffuser is placed behind the fan stage to recover 100% of the dynamic pressure associated with the axial velocity exiting the fan stage, calculate the pressure coefficient across the diffuser, i.e.

  2/2 34

U ppC

diffuserp  



where station 4 is the diffuser exit plane.

Problem 4 Your manager asks you if it is worthwhile to add a diffuser to an axial fan. The fan flow rate is 0.083 m3/s, with Dh = 4 cm and Dt = 7.5 cm. The fan static pressure rise (gage) is 60 Pa. The constraint is that the diffuser-length to fan-diameter ratio should not exceed 2. Use the performance correlations of Sovran & Klomp (1967) for conical diffusers.

(a) How much pressure can you gain with the added diffuser (in Pa)? What is the corresponding diffuser exit diameter?

(b) Any problem(s) with using the Sovran & Klomp correlations for this application?

Problem 5 The figure shows a control volume with periodic side walls surrounding one blade in a two-dimensional cascade of airfoils that turn the flow from inlet angle  to outlet angle . The inlet velocity relative to the blade is W1, and the density of the fluid is  and may be assumed constant for this particular flow. If there are no losses in stagnation pressure, show that the axial velocity component stays constant and the sums of the forces exerted by the fluid on the blade in the z- and –directions are (per unit blade height)

Problem 6 By applying the Bernoulli equation to the relative flow in an axial compressor rotor, show that the static pressure rise across the rotor is

p p C C U C C2 1 1 2

2 2

2 1 1 2

2        ( )

Problem 7 For a repeating stage, the reaction ratio R is defined as the ratio of the static pressure rise across the rotor to the total pressure rise across the rotor. Show that, in the absence of friction, R can be expressed as

R   1 2

1 1 2  (tan tan )

Where is the flow coefficient  defined as = Cx/U. Show that the reaction R can also be expressed in terms of the rotor relative flow angles as

 21 tantan2 1  R

For a repeating stage, C1 = C3, and hence for an ideal flow

13

12

0103

12

0102

12

pp pp

PP pp

PP ppR

 

  

  



which is the ratio of the static pressure rise across rotor to the stage static pressure rise. Here, station 3 is the stator exit. Problem 8 Using the typical cascade performance data shown in the attached figure, show how, for a stagger angle of 40o, the turning angle and pressure coefficient depends on the inlet flow angle. Plot (1-2) and ( ) / ( / )p p W2 1 1

2 2  as a function of 1 for solidities of 1.0 and 1.5, assuming no stagnation pressure loss. Hint: express ( ) / ( / )p p W2 1 1

2 2  in terms of the inlet flow angle 1 and the outlet flow angle 2.

Problem 9 A compressor cascade has a space/chord ratio of unity and blade inlet and outlet metal angles of 50o and 20o, respectively. If the blade camber line is a circular arc (i.e. maximum camber at 50% chord) and the cascade is designed to operate at Howell’s nominal condition, determine the fluid deflection, incidence and ideal lift coefficient at the design point. At an off-design condition where the incidence angle i=3.8o, estimate the fluid deflection  and the drag coefficient. Problem 10 A single-stage axial compressor uses 65(12)10 blading with a solidity at the mean diameter of 1.0 (see Fig. 7.26 given in Problem 4). It is designed for a reaction R=50% and a flow coefficient =0.5.

(a) Show that the relationship between the inlet and outlet flow angles is  2

1 12 

tan ( tan ) . (b) From the test data given in Fig. 7.26, plot 2 versus 1 at the design points

(midpoint between the stalling limits) for different stagger angles. On the same figure, plot the corresponding stagger angle  versus 1.

(c) From the figure developed in parts (a) and (b), choose a suitable stagger angle, and rotor and stator flow angles.

(d) Determine the ideal stage work characteristic ( versus ) between stalling limits. Recall that for ideal flow,

 21 tantan1  