# Chapter 6 - Radial Flow Gas and Steam Turbines¶

## Ex 6.1 Page 266¶

In [1]:
from __future__ import division
from math import atan, pi, tan, degrees, cos
#input data
P00=3#The pressure at which air is received in bar
T00=373#The temperature at which air is received in K
rt=0.5#The rotor tip diameter of turbine in m
rh=0.3#The rotor exit diameter of the turbine in m
b=0.03#The rotor blade width at entry in m
b11=60#The air angle at rotor entry in degree
a11=25#The air angle at nozzle exit in degree
Ps=2#The stage pressure ratio
nn=0.97#The nozzle efficiency
N=7200#The speed of the turbine rotation in rpm
R=287#The universal gas constant in J/kg.K
Cp=1005#The specific heat of air at constant pressure in J/kg.K
r=1.4#The ratio of specific heats of air

#calculations
U1=(3.14*rt*N)/60#Peripheral velocity of impeller at inlet in m/s
Cr=U1/(1/tan(pi/180*a11)-1/tan(pi/180*b11))#The radial velocity at inlet in m/s
ps1=Cr/U1#Flow coefficient
DR=((1-(ps1*1/tan(pi/180*b11)))/2)#Degree of reaction
nts=((sl*U1**2)/(Cp*T00*(1-((1/Ps)**((r-1)/r)))))#Stage efficiency of the turbine
C2=Cr#Absolute velocity at the exit in m/s
U2=(3.1415*rh*N)/60#Peripheral velocity of impeller at exit in m/s
b22=degrees(atan(C2/U2))#The air angle at rotor exit in degree
dT=DR*U1*Cr*1/tan(pi/180*a11)/Cp#Total actual change in temperature in a stage turbine in K
dT0=(U1*Cr*1/tan(pi/180*a11))/Cp#The total change in temperature in turbine in K
T02=T00-dT0#The exit absolute temperature in K
T2=T02-((C2**2)/(2*Cp))#The actual exit temperature in K
T1=dT+T2#The actual inlet temperature in K
Cx1=Cr*1/tan(pi/180*a11)#Inlet absolute velocity of air in tangential direction in m/s
C1=Cx1/cos(pi/180*a11)#Absolute velocity at the inlet in m/s
dT1=(C1**2/2)/(Cp*nn)#The absolute change in temperature at the first stage in K
dP1=(1-(dT1/T00))**(r/(r-1))#The absolute pressure ratio in first stage
P1=dP1*P00#The inlet pressure in bar
d1=(P1*10**5)/(R*T1)#The inlet density in kg/m**3
A1=3.1415*rt*b#The inlet area of the turbine in m**2
m=d1*A1*Cr#The mass flow rate of air at inlet in kg/s
P2=P00/Ps#The exit pressure in bar
d2=(P2*10**5)/(R*T2)#The exit density of air in kg/m**3
bh=(m/(d2*3.1415*rh*Cr))#Rotor width at the exit in m
W=m*U1*Cx1*10**-3#The power developed by the turbine in kW

#output
print '(a)\n    (1)The flow coefficient is %3.3f\n    (2)The loading coefficient is %3.3f\n(b)\n    (1)The degree of reaction is %0.2f %% \n    (2)The stage efficiency of the turbine is %0.2f %% \n(c)\n    (1)The air angle at the rotor exit is %3.2f degree\n    (2)The width at the rotor exit is %0.2f cm\n(d)\n    (1)The mass flow rate is %3.2f kg/s\n    (2)The power developed is %3.2f kW'%(ps1,sl,DR*100,nts*100,b22,bh*100,m,W)
# answer in the textbook is not correct.

(a)
(1)The flow coefficient is 0.638
(b)
(1)The degree of reaction is 31.58 %
(2)The stage efficiency of the turbine is 72.12 %
(c)
(1)The air angle at the rotor exit is 46.75 degree
(2)The width at the rotor exit is 6.31 cm
(d)
(1)The mass flow rate is 11.78 kg/s
(2)The power developed is 572.01 kW


## Ex 6.2 Page 270¶

In [2]:
from math import sin
#input data
P0=4#Overall stage pressure ratio
T00=557#Temperature at entry in K
P3=1#Diffuser exit pressure in bar
m=6.5#Mass flow rate of air in kg/s
ps1=0.3#Flow coefficient
N=18000#Speed of the turbine in rpm
Dt=0.42#Rotor tip diameter in m
D2m=0.21#Mean diameter at rotor exit in m
R=287#The universal gas constant in J/kg.K
Cp=1.005#The specific heat of air at constant pressure in kJ/kg.K
r=1.4#The ratio of specific heats of air

#calculations
U1=(3.1415*Dt*N)/60#Peripheral velocity of impeller at inlet in m/s
Cr1=ps1*U1#The radial velocity at inlet in m/s
a11=degrees(atan(Cr1/U1))#The nozzle exit air angle in degree
W=m*U1**2*10**-3#Power developed by turbine in kW
dT=(1/P0)**((r-1)/r)#The total isentropic temperature ratio in entire process
T3s=dT*T00#The final isentropic temperature at exit in K
dh2=W/m#The absolute enthalpy change in the first two stages in kJ/kg
ns=dh2/(Cp*(T00-T3s))#The stage efficiency of the turbine
T02=T00-(W/(m*Cp))#The absolute temperature at the entry of second stage in K
T03=T02#The absolute temperature at exit of second stage in K
dH=Cp*(T02-T3s)#The total enthalpy loss in kJ/kg
dHn=dH/2#The enthalpy loss in the nozzle in kJ/kg
C1=Cr1/sin(pi/180*a11)#Absolute velocity at the inlet in m/s
dH0=((C1**2)/(2000*Cp))+(dHn)#The isentropic absolute enthalpy loss in nozzle in kJ/kg
dT0=dH0/Cp#The isentropic absolute temperature loss in nozzle in K
T1s=T00-dT0#The isentropic temperature at the entry in K
P1=P0*(T1s/T00)**(r/(r-1))#The pressure at the entry of turbine in bar
T1=T00-((C1**2)/(2000*Cp))#The temperature at the entry of turbine in K
d1=(P1*10**5)/(R*T1)#The density of the air at inlet in kg/m**3
b1=m/(d1*Cr1*3.141*Dt)#The width of the rotor at inlet in m
C2=Cr1#The avsolute velocity at the second stage entry in m/s
T2=T02-((C2**2)/(2000*Cp))#The temperature at the second stage entry in K
P23=(T2/T03)**(r/(r-1))#The pressure ratio at the second stage
P2=P23*P3#The pressure at the second stage in bar
d2=(P2*10**5)/(R*T2)#The density of the air at second stage in kg/m**3
C2=Cr1#The absolute velocity at the second stage in m/s
A2=m/(d2*C2)#The area of cross section at the second stage in m**2
h2=(A2/(3.14*D2m))#The rotor blade height at the exit in m
M1=C1/(r*R*T1)**(1/2)#The mach number at the nozzle
U2=(3.14*D2m*N)/60#The Peripheral velocity of impeller at exit in m/s
M2r=(((C2**2)+(U2**2))**(1/2))/(r*R*T2)**(1/2)#The mach number at the rotor exit
Ln=(dHn*10**3)/((C1**2)/2)#The nozzle loss coefficient
Lr=(dHn*10**3)/(((((C2**2)+(U2**2))**(1/2))**2)/2)#The rotor loss coefficient

#output
print '(a)The nozzle exit air angle is %3.2f degree\n(b)The power developed is %3.1f kW\n(c)The stage efficiency is %0.2f %%\n(d)The rotor width at the entry is %0.2f cm\n(e)The rotor blade height at the exit is %0.2f cm\n(f)\n    (1)The mach number at the nozzle exit is %3.4f\n    (2)The mach number at the rotor exit is %3.2f\n(g)\n    (1)The nozzle loss coefficient is %3.4f\n    (2)The rotor loss coefficient is %3.3f'%(a11,W,ns*100,b1*100,h2*100,M1,M2r,Ln,Lr)

(a)The nozzle exit air angle is 16.70 degree
(b)The power developed is 1018.4 kW
(c)The stage efficiency is 85.58 %
(d)The rotor width at the entry is 2.76 cm
(e)The rotor blade height at the exit is 9.99 cm
(f)
(1)The mach number at the nozzle exit is 0.9489
(2)The mach number at the rotor exit is 0.58
(g)
(1)The nozzle loss coefficient is 0.1546
(2)The rotor loss coefficient is 0.496


## Ex 6.3 Page 274¶

In [3]:
from math import acos
#input data
ntt=0.9#Total-to-total efficiency
P00=300#The pressure at entry to the nozzle in kPa
T00=1150#The temperature at entry to the nozzle in K
T1=1013#The static temperature at the outlet of the nozzle in K
P03=100#The pressure at the outlet of the diffuser in kPa
R=284.5#The universal gas constant in J/kg.K
Cp=1.147#The specific heat of air at constant pressure in kJ/kg.K
r=1.33#The ratio of specific heats of given gas

#calculations
U1=(ntt*Cp*1000*T00*(1-((P03/P00)**((r-1)/r))))**(1/2)#The impeller tip speed in m/s
T01=T00#The absolute temperature at the entry in K
C1=(2000*Cp*(T01-T1))**(1/2)#The absolute velocity at the inletof turbine in m/s
a11=acos(pi/180*U1/C1)#The flow angle at the nozzle oulet in degree
M1=C1/(r*R*T1)**(1/2)#The mach number at the nozzle outlet

#output
print '(a)The impeller tip speed is %3.1f m/s\n(b)The flow angle at the nozzle oulet is %3.2f degrees\n(c)The mach number at the nozzle outlet is %3.2f'%(U1,a11,M1)
# answer in the textbook is not correct fot part(b)

(a)The impeller tip speed is 532.2 m/s
(b)The flow angle at the nozzle oulet is 1.55 degrees
(c)The mach number at the nozzle outlet is 0.91


## Ex 6.4 Page 275¶

In [4]:
#input data
D1=0.09#Rotor inlet tip diameter in m
D2t=0.062#Rotor outlet tip diameter in m
D2h=0.025#Rotor outlet hub diameter in m
d2=1.8#Density of exhaust gases at impeller exit in kg/m**3
C2s=0.447#Ratio of absolute velocity and isentropic velocity at exit
U1Cs=0.707#Ratio of impeller tip velocity and isentropic velocity

#calculations
U1=(3.1415*D1*N)/60#The impeller tip speed in m/s
Cs=U1/U1Cs#Isentropic velocity in m/s
C2=C2s*Cs#Absolute velocity at the exit in m/s
A2=(3.141/4)*((D2t**2)-(D2h**2))#Area at the exit in m**2
Q2=A2*C2#Volume flow rate at the impeller exit in m**3/s
M=d2*Q2#Mass flow rate in kg/s
W=M*U1**2#Power developed in W

#output
print '(a)Volume flow rate at the impeller exit is %3.3f m**3/s\n(b)Power developed is %0.3f kW'%(Q2,W/1000)

(a)Volume flow rate at the impeller exit is 0.226 m**3/s
(b)Power developed is 8.127 kW


## Ex 6.5 Page 276¶

In [5]:
#input data
P00=3.5#Total-to-static pressure ratio
P2=1#Exit pressure in bar
T00=923#Inlet total temperature in K
D=0.45#Rotor diameter ratio
N=16000#Speed from nozzle in rpm
a11=20#Nozzle exit angle in degree
nn=0.95#Nozzle efficiency
b1=0.05#Rotor width at inlet in m
R=287#The universal gas constant in J/kg.K
Cp=1005#The specific heat of air at constant pressure in J/kg.K
r=1.4#The ratio of specific heats of air

#Calculations
T2s=T00*(1/P00)**((r-1)/r)#Isentropic temperature at the exit in K
Cs=(2*Cp*(T00-T2s))**(1/2)#The isentropic velocity in m/s
U1=U1Cs*Cs#The impeller tip speed in m/s
D1=(U1*60)/(3.14*N)#Rotor inlet diameter in m
D2=D*D1#Rotor outlet diameter in m
Cr2=U1*tan(pi/180*a11)#The relative velocity at the exit in m/s
U2=(3.1415*D2*N)/60#Peripheral velocity of impeller at exit in m/s
b22=degrees(atan(Cr2/U2))#The air angle at rotor exit in degree
T02=T00-((U1**2)/(Cp))#The absolute temperature at the exit in K
T2=T02-((Cr2**2)/(2*Cp))#The temperature at the exit of turbine in K
T1=T2+((U1**2)/(2*Cp))#The temperature at the entry of turbine in K
T1s=T00-((T00-T1)/nn)#Isentropic temperature at the entry in K
P1=P00*(T1s/T00)**(r/(r-1))#The pressure at the entry stage in bar
d1=(P1*10**5)/(R*T1)#The density of the air  at the inlet in kg/m**3
A1=3.1415*D1*b1#The area at the inlet in m**2
Cr1=Cr2#The relative velocity at the entry in m/s
m=d1*A1*Cr1#The mass flow rate for a 90degree IFR turbine Degree of Reaction is 0.5 in kg/s
W=(m*U1**2)*10**-6#Power developed in MW
d2=(P2*10**5)/(R*T2)#The density of the air at the exit in kg/m**3
b2=m/(d2*3.141*D2*Cr2)#Rotor width at the exit in m
D2h=D2-b2#Hub diameter at the exit in m
D2t=D2+b2#Tip diameter at the exit in m
nts=(W*10**6)/(m*Cp*(T00-T2s))#Total-to-static efficiency
C1=U1/cos(pi/180*a11)#Absolute velocity at the entry in m/s
Ln=(Cp*(T1-T1s))/((C1**2)/2)#Nozzle enthalpy loss coefficient
W2=((U2**2)+(Cr2**2))**(1/2)#Resultant relative velocity at the exit in m/s
T2s=T1*(P2/P1)**((r-1)/r)#Isentropic temperature at the exit in K
Lr=(Cp*(T2-T2s))/((W2**2)/2)#Rotor enthalpy loss coefficient

#output
print '(a)\n    (1)Rotor inlet diameter is %3.2f m\n    (2)Rotor outlet diameter is %3.3f m\n(b)The air angle at rotor exit is %3.2f degree\n(c)The mass flow rate for a 90degree IFR turbine Degree of Reaction is 0.5 is %3.2f kg/s\n(d)Power developed is %3.3f MW\n(e)\n    (1)Hub diameter at the exit is %3.4f m\n    (2)Tip diameter at the exit is %3.4f m\n(f)Total-to-static efficiency is %3.4f\n(g)Nozzle enthalpy loss coefficient is %3.4f\n(h)Rotor enthalpy loss coefficient is %3.4f'%(D1,D2,b22,m,W,D2h,D2t,nts,Ln,Lr)

(a)
(1)Rotor inlet diameter is 0.59 m
(2)Rotor outlet diameter is 0.265 m
(b)The air angle at rotor exit is 38.95 degree
(c)The mass flow rate for a 90degree IFR turbine Degree of Reaction is 0.5 is 14.21 kg/s
(d)Power developed is 3.456 MW
(e)
(1)Hub diameter at the exit is 0.0834 m
(2)Tip diameter at the exit is 0.4466 m
(f)Total-to-static efficiency is 0.8712
(g)Nozzle enthalpy loss coefficient is 0.0526
(h)Rotor enthalpy loss coefficient is 0.3396


## Ex 6.6 Page 280¶

In [6]:
#input data
P00=700#Total-to-static pressure ratio
T00=1145#Inlet total temperature in K
P1=527#The pressure at the entry stage in bar
T1=1029#The temperature at the entry of turbine in K
P2=385#The pressure at the second stage in bar
T2=915#The temperature at the second stage entry in K
T02=925#The absolute temperature at the exit in K
D2mD1=0.49#The ratio of rotor exit mean diameter to rotor inlet diameter
R1=8.314#The gas constant of given gas in kJ/kg.K
r=1.67#The ratio of specific heats of the gas
m=39.94#Molecular weight of a gas

#calculations
R=R1/m#The universal gas constant in kJ/kg.K
Cp=(r*R)/(r-1)#The specific heat of air at constant pressure in kJ/kg.K
T2ss=T00*(P2/P00)**((r-1)/r)#Isentropic stage temperature at the exit in K
nts=(T00-T02)/(T00-T2ss)#Total-to-static efficiency of the turbine
U1=(Cp*1000*(T00-T02))**(1/2)#The impeller tip speed in m/s
D1=(U1*60)/(3.1415*N)#Rotor inlet diameter in m
D2m=D1*D2mD1#Rotor exit mean diameter in m
C1=(2*Cp*(T00-T1))**(1/2)#Absolute velocity at the entry in m/s
T1s=T00*(P1/P00)**((r-1)/r)#Isentropic temperature at the entry in K
Ln=(Cp*(T1-T1s))/((C1**2)/2)#Nozzle enthalpy loss coefficient
C2=(2*Cp*1000*(T02-T2))**(1/2)#The temperature at the exit of turbine in K
U2=(3.14*D2m*N)/(60)#Peripheral velocity of impeller at exit in m/s
W2=((C2**2)+(U2**2))**(1/2)#Resultant relative velocity at the exit in m/s
T2s=T1*(P2/P1)**((r-1)/r)#stage temperature at the exit in K
Lr=(Cp*1000*(T2-T2s))/((W2**2)/2)#Rotor enthalpy loss coefficient
ntt=1/((1/nts)-((C2**2)/(2*U1**2)))#Total-to-total efficiency

#output
print '(a)Total-to-static efficiency of the turbine is %0.1f %%\n(b)\n    (1)Rotor inlet diameter is %3.3f m\n    (2)Rotor exit mean diameter is %3.3f m\n(c)\n    (1)Nozzle enthalpy loss coefficient is %3.4f\n    (2)Rotor enthalpy loss coefficient is %3.4f\n(d)Total-to-total efficiency is %0.2f %%'%(nts*100,D1,D2m,Ln,Lr,ntt*100)

(a)Total-to-static efficiency of the turbine is 90.1 %
(b)
(1)Rotor inlet diameter is 0.269 m
(2)Rotor exit mean diameter is 0.132 m
(c)
(1)Nozzle enthalpy loss coefficient is 0.0625
(2)Rotor enthalpy loss coefficient is 0.2138
(d)Total-to-total efficiency is 93.95 %