Solved Problems And Sheet.doc 646s26

Subject: Power Plant Technology Chapter: Steam Turbines Lecturer:Dr. S .H .Faisal

i

e b

e ae

re

i

ri

ai

fi=ri sin βi

f=fi-fe

fe=re sin βe

Q1//In an impulse stage the mean diameter of the blade ring is 800 mm and the speed of rotation is 3000 rpm. The steam issues from the nozzles with a velocity of 300 m/s and the nozzle angle is 20°. The blades are symmetric and the blade velocity coefficient (k) is 0.86 . What is the power developed in the stage when the axial thrust on the blades is 140 N.? Solution:

ωi=ri cos βi

ωe=re cos βe

ω= ωi + ωe DN  * 0.8 * 3000   125.6 m / s 60 60 ai  300m / s ,  i  20 o From the velocity triangular: ai sin  i 300 * sin 20 o tan  i    0.6565 ai cos  i  b 300 cos 20 o  125.6 Since the blades are symmetric, then:  i   o  33.3o Also from the velocity triangular: ai sin  i  ri sin  i b

-

102.61  ri sin 33.3  ri  187 m / s  ro  0.86 *187  161m / s

The axial thrust is given by:

 [ ri sin  i  re sin  e ] Faxial  m  [187 sin 33.3(1  0.86)] m  *14.3654  140 m   9.7456kg / s m the power developed  [ ri cos  i  re cos  e ] Power  m  355.84kW

Q2//The nozzles of the impulse stage of a turbine receive steam at 15 bar and 300°C and discharge it at l0bar. The nozzle efficiency is 95% and the nozzle angle is 20°. The blade speed is that required for maximum blade efficiency, and the inlet angle of the blades is that required for entry of the steam without shock. The blade exit angle is 5° less than the inlet angle. The blade friction factor is 0.9. Calculate for a steam flow of 1350 kg/h, (a) the axial thrust. (b) The diagram power, and (e) the diagram efficiency. Problems sheet 19

i

e b

e ae

re

ωe=re cos βe

i

ri

ai

fi=ri sin βi

f=fi-fe

fe=re sin βe

Solution:

ωi=ri cos βi ω= ωi + ωe

h1  kJ / kg 

The state "2s" is in the superheated region T1s  250 , h 2s  2943.1 kJ/kg ai 

2 nozz ( h1  h2 s ) 

2 * 0.95 *1000 * (3038.9  2943.1)  426.63m / s

For maximum efficiency: cos  i cos 20 o b    0.4699 ai 2 2  b  200.45

tan  i 

ai sin  i 145.916   0.7279 ai cos  i  b 200.45   i  36   e  36  5  31o

ri 

ai sin  i  248.25m / s sin  i

re  ri * k  223.42m / s

  [ri cos  i  re cos  e ]  392.35m / s f  [ ri sin  i  re sin  e ]  30.85m / s

1350 * 30.85  11.57 N 3600 1350   m * 392.35  147.13N 3600

f  Faxial  m Fdriving

Problems sheet

20

   147.13 * 200.45 * 10 3 Power developed  bm  29.492kW

blade 

29492  0.864  86.4% 1 1350 2 * * (426.63) 2 3600

******Unsolved Problems****** Q1// the velocity of steam at inlet to simple impulse turbine is 1000 m/s and the nozzle single is 20o. The blade speed is 400 m/s and the blades are symmetrical. Determine the blade angle, tangential force, diagrame power, axial thrust, and the diagram efficiency for frictionless blades. If the relative velocity at exit is reduced by friction to 80% of that at inlet, then recalculates the diagram power, axial thrust, and the diagram efficiency. Take mass flow rate 1 kg/s .Answer (32.5o, 1065.27 N, 426.27 kW, 0 N, 85.25 %,* 383.643 kW, 67.89 N, 76.7%) Q2// a single row impulse turbine receive 3 kg/s of steam with velocity 425 m/s . The blade speed ratio is 0.4 and the output power is 171.58 kW. If the frictional losses in the moving blade amount to 14.92 kW, determine the diagram efficiency and the blade velocity coefficient. The nozzles angle is 16o . Answer: (63.32%; 0.927) Q3//a simple impulse turbine (De Laval turbine) is supplied with steam at 15 bar, 400 oC-superheat. The steam expands in the nozzles, which have an efficiency of 90%, to pressure of 1bar. Assuming the nozzle angle is 20o, blade velocity coefficient is 0.8, and symmetrical blades; determine for ideally maximum blade efficiency condition: 1.The blade speed. 2. The blade angle. 3.The power output for m  9000 kg / hr . 4.Set the summation of nozzle losses, useful output work, blade frictional losses, and leaving kinetic energy; which must be equal to the isentropic enthalpy drop. answ(493.902 m/s; 36.05o;1097.75kW;nozzle losses:61.46 kJ/kg;energy output:439.1 kJ/kg ;friction losses:67.18 kJ/kg; residual kinetic energy:46.24 kJ/kg) Q4//The rotational-speed of an impulse turbine wheel is 3000 r.p.m. The nozzles are inclined at 20o and their efficiency is 0.89. The isentropic heat drop for the stage is 159 kJ/kg. If the blade speed ratio is 0.4, the blade speed coefficient is 0.82, and the blade efficiency is 0.76. Find for steam flow rate of 15 kg/s: 1. The mean blade ring diameter. 2. The power developed by the stage. Answer (1.355m, 1613.2 kW,

).

Q5// The nozzles of the impulse stage of turbine receive steam at 15 bar, 300 oC and discharge it at 10 bar. The nozzle efficiency is 95 % and the nozzle angle is 20o. The blade speed is that required for maximum power and the inlet angle of the blades is that required for entry of steam without shock .the blade exit angle is 5o less than the inlet angle. The blade velocity coefficient is 0.9. Calculate for steam flow rate 1350 kg/hr: 1. The diagram power. 2. The diagram efficiency. Answer (30.3 kW , 86.3 % ) Q6// the steam from the nozzle of single wheel impulse turbine discharges with velocity of 600 m/s at angle of 20o . The blade wheel rotates at 3000 rev/min and the mean blade radius is 590mm. the axial velocity of steam at exit from the blade ( f e ) is 164 m/s and the blades are symmetrical, calculate: 1. The blades angles. 2. The diagram power. 3. The diagram efficiency . 4. The blade velocity coefficient. Answer (28o ,126.3 kW , 70% ,0.799)

Problems sheet

21

Q7//for DeLaval turbine of symmetrical and frictionless blades degree ,derive an expression for blade efficiency showing the optimum operation of blade speed ratio (b/a i) and the corresponding maximum efficiency power. Answer:b  4

b b (cos  i  ) ai ai

d (b ) 0 by making d ( b ) we find ai

Where (

b ) is the blade speed ratio ai

b ai

 optimum

The velocity diagram will be:

cos  i 2  b2 ; maxb  cos  i ; power= 2m 2

b b max

b ae

mri re

ai

b ai b ai

optimum

Q.8//in a single stage simple impulse turbine the steam flows ata rate of 5kg/s. It has rotor of 1.2m diameter running at 3000 rpm. Nozzle angle is 18˚, blade speed ratio is 0.4 and velocity coefficient is 0.9, outlet angle of blade is 3˚ less than inlet angle. Determine blade angles and power developed. Q.9// In a certain stage of an impulse turbine, the nozzle angle 20˚ with the plane of the wheel. The mean diameter of the blade ring is 2.8 metres. Its develop 55kw at 2400 rpm. Four nozzles, each of 10 mm diameters expand steam isentropically from 15 bar and 250˚ C to 0.5 bar. The axial thrust is 3.5 N. Calculate blade angle at entrance and exit and power lost in blade friction.

Problems sheet

22