# Chapter 7 : Rocket propulsion¶

## Example 7.1 page : 21¶

In :
import math

#Input data
mp = 12. 				#flow rate in kg/s
Ae = 335.*10**-4 				#exit area in m**2
Ce = 2000. 				#exhaust velocity in m/s
h = 10. 				#altitude in km
Pe = 1.*10**5 				#exhaust pressure in Pa
P0 = 1.*10**5 				#p0 = atomspheric pressure in Pa at h = 0.
P10 = 0.25*10**5 				#atmospheric pressure in Pa umath.sing gas tables

#Calculations
Fs = mp*Ce*10**-3 				#thrust of motor at sea level math.since pe = p0 in kN
F10 = ((mp*Ce) + Ae*(Pe-P10))*10**-3 				#thrust of motor at altitude of 10km in kN

#Output
print 'A)thrust of motor at sea level is %3i kN upwards  \
\nB)thrust of motor at an altitude 10km is %3.4f kN'%(Fs,F10)

A)thrust of motor at sea level is  24 kN upwards
B)thrust of motor at an altitude 10km is 26.5125 kN


## Example 7.2 page : 22¶

In :
import math

#Input data
P0 = 38*10**5 				#combustion chamber pressure in Pa
T0 = 3500 				#combustion chamber temperature in K
ma = 41.67 				#oxidizer flow rate in kg/s
MR = 5 				#mixture ratio
k = 1.3 				#adiabatic consmath.tant
R = 287 				#gas consmath.tant in J/kg-K
Pamb = 0.0582*10**5 				#ambient pressure in Pa
Pe = Pamb 				#exhaust pressure at sea level in Pa

#Calculation
mf = ma/MR 				#mass flow of fuel in kg/s
mp = mf+ma 				#propellant mass flow in kg/s
Cp = (k*R)/(k-1) 				#specific heat at consmath.tant pressure in J/kg-k
p = P0/Pe 				#ratio of pressures at combustion chamber and exhaust
Me = ((((p**((k-1)/k))-1)*2)/(k-1))**0.5 				#Mach number
t = 1/(1+(((k-1)/2)*Me**2)) 				#ratio of exhaust temperature to combustion temperature
Te = t*T0 				#exhaust temperature in Kelvin
a = (1/Me)*(((2/(k+1))+(((k-1)/(k+1))*Me**2))**((k+1)/(2*(k-1)))) 				#ratio of areas at exit section and throat section of the nozzle
Ce = (k*R*Te)**0.5*Me 				#exit velocity in the exhaust in m/s
Cj = Ce 				#average effective jet velocity in m/s, math.since Pe = Pamb
P1 = P0*(2/(k+1))**(k/(k-1)) 				#pressure at throat section in Pa
T1 = T0*(2/(k+1)) 				#temperature at throat section in K
d1 = P1/(R*T1) 				#density of fuel at throat section in kg/m**3
C1 = (k*R*T1)**0.5 				#velocity at throat section in m/s
A1 = (mp/(d1*C1))*10**4 				#nozzle throat area in cm**2
Ae = a*A1 				#exit area in cm**2
F = (mp*Ce)*10**-3 				#thrust in kN
Cmax1 = (2*Cp*T0)**0.5 				#maximum possible velocity in m/s
Cf = (F*10**3)/(P0*A1*10**-4) 				#thrust coefficient, F in kN and A1 in m**2
Cch1 = Cj/Cf 				#characteristic velocity in m/s

#Output
print 'A)nozzle throat area is %3.2f cm**2  \
\nB)thrust is %3.1f kN  \
\nC)thrust coefficient is %3.2f  \
\nD)characteristic velocity is %3i m/s  \
\nE)exit velocity in exhaust is %3i m/s \
\nF)maximum possible exhaust velocity is %3i m/s'%(A1,F,Cf,Cch1,Ce,Cmax1)

A)nozzle throat area is 197.65 cm**2
B)thrust is 130.0 kN
C)thrust coefficient is 1.73
D)characteristic velocity is 1502 m/s
E)exit velocity in exhaust is 2599 m/s
F)maximum possible exhaust velocity is 2950 m/s


## Example 7.3 page : 23¶

In :
import math

#Input data
a = 3. 				#exit area to throat area ratio
T0 = 2973. 				#combustion chamber temperature in K
P0 = 20.*10**5 				#combustion chamber pressure in Pa
k = 1.3 				#adiabatic consmath.tant
R = 248. 				#gas consmath.tant in J/kg-K
Pamb = 1.*10**5 				#ambient pressure in Pa
Me = 2.52 				#mach number for k = 1.3 and a = 3 umath.sing gas tables
g = 9.81 				#acceleration due to gravity in m/s**2

#Calculation
p = 1/((1+(((k-1)/2)*Me**2))**(k/(k-1))) 				#ratio of pressures at exhaust and combustion chamber
Pe = p*P0 				#exhaust pressure in Pa
t = 1/(1+(((k-1)/2)*Me**2)) 				#ratio of exhaust temperature to combustion temperature
Te = t*T0 				#exhaust temperature in Kelvin
Ce = (k*R*Te)**0.5*Me 				#exit velocity in the exhaust in m/s
M = (Pe*Ce)/(R*Te) 				#propellant mass flow per unit area of exit in kg/m**2-s
Fa = ((M*Ce)+(Pe-Pamb))*10**-3 				#thrust per unit area of exit in N/m**2
Is = (Fa*10**3)/(M*g) 				#specific impulse in sec

#Output
print 'A)thrust per unit area of exit is %3.2f kN/m**2  \
\nB)specific impulse is %3.2f sec'%(Fa,Is)

A)thrust per unit area of exit is 918.93 kN/m**2
B)specific impulse is 181.98 sec


## Example 7.4 page : 24¶

In :
import math

#Input data
mp = 5. 				#propellent flow rate in kg/s (mismath.sing data)
de = 0.10 				#nozzle exit diameter in m
Pe = 1.02*10**5 				#nozzle exit pressure in Pa
Pamb = 1.013*10**5 				#ambient pressure in Pa
P0 = 20. 				#thrust chamber pressure in Pa
F = 7000. 				#thrust in N
u = 1000. 				#rocket speed in m/s
g = 9.81 				#acceleration due to gravity in m/s**2

#Calculation
Cj = F/mp 				#effective jet velocity in m/s
Ca = Cj-u 				#absolute jet velocity in m/s
wp = mp*g 				#weight flow rate of propellent in N/s
Is = F/(wp) 				#specific impulse in sec
SPC = 1/Is 				#specific propellent consumption in sec**-1

#Output
print 'A)effective jet velocity is %3i m/s  \
\nB)specific impulse is %3.2f sec  \
\nC)specific propellent consumption is %3.3f s**-1  \
\nD)absolute jet velocity is %3i m/s'%(Cj,Is,SPC,Ca)

A)effective jet velocity is 1400 m/s
B)specific impulse is 142.71 sec
C)specific propellent consumption is 0.007 s**-1
D)absolute jet velocity is 400 m/s


## Example 7.5 page: 24¶

In :
import math

#Input data
Cj = 2700. 				#average effective jet velocity in m/s
u = 1350. 				#forward flight velocity in m/s
mp = 78.6 				#propellant mass flow in kg/s

#Calculation
s = u/Cj 				#effective jet speed ratio
np = (2*s)/(1+s**2) 				#propulsive efficiency
F = Cj*mp*10**-3 				#thrust in kN
Pt = F*u*10**-3 				#Thrust power in MW, F in N

#Output
print 'A)thrust is %3.2f kN  \
\nB)Thrust power is %3.3f MW  \
\nC)propulsive efficiency is %3.1f'%(F,Pt,np)

A)thrust is 212.22 kN
B)Thrust power is 286.497 MW
C)propulsive efficiency is 0.8


## Example 7.6 page : 24¶

In :
import math
from scipy.integrate import quad

#Input data
mi = 15000. 				#mass of the rocket in kg
mp = 125. 				#propellant mass flow in kg/s
Cj = 2000. 				#velocity of gases coming out in m/s
t = 70. 				#time interval in sec
t0 = 0. 				#lower limit in integration in sec
t1 = 70. 				#upper limit in integration in sec
g = 9.81 				#acceleration due to gravity in m/s**2

#Calculation
u = (-Cj*(math.log(1-((mp*t)/mi))))-(g*t) 				#velocity attained in 70 sec in m/s

def f0(t):
return ((-2000*(math.log(1-((125*t)/15000))))-(g*t))

h1 = ( quad(f0,t0,t1))

h2 = (u**2/(2*g))*10**-3 				#Distance reached after fuel last i.e. after 70 sec due to kinetic energy by umath.sing KE = PE in km
h = h1+h2 				#maximum height the rocket will reach in km

#Output
print 'A)velocity attained in %i sec is %3.2f m/s \
\nB)maximum height the rocket will reach is %3.3f km'%(t,u,h)

A)velocity attained in 70 sec is 1064.24 m/s
B)maximum height the rocket will reach is 28476.353 km


## Example 7.7 page : 25¶

In :
import math

#Input data
A1 = 18.*10**-4 				#throat area in m**2
P0 = 25.*10**5 				#combustion chamber pressure in Pa
Is = 127.42 				#specific impulse in sec
wp = 44.145 				#weight flow rate of propellent in N/s
g = 9.81 				#acceleration due to kravity in m/s**2

#Calculation
F = Is*wp 				#thrust in N
mp = wp/g 				#propellant mass flow in kg/s
Cj = F/mp 				#average effective jet velocity in m/s
Cf = F/(P0*A1) 				#thrust coefficient
Cw = wp/(P0*A1)/10**-3 				#propellent weight flow coefficent *10**-3
SPC = (wp/F)/10**-3 				#specific propellent consumption in sec**-1 *10**-3
Cch1 = Cj/Cf 				#characteristic velocity in m/s

#Output
print 'A)thrust coefficient is %3.2f  \
\nB)propellent weight flow coefficent is %3.2f*10**-3  \
\nC)specific propellent consumption is %3.2f*10**-3 s**-1  \
\nD)characteristic velocity is %3.0f m/s'%(Cf,Cw,SPC,Cch1)

A)thrust coefficient is 1.25
B)propellent weight flow coefficent is 9.81*10**-3
C)specific propellent consumption is 7.85*10**-3 s**-1
D)characteristic velocity is 1000 m/s


## Example 7.8 page : 26¶

In :
import math

#Input data
m1 = 200. 				#internal mass in kg
m2 = 130. 				#mass after rocket operation in kg
m3 = 110. 				#payload,non-propulsive structure, etc in kg
tp = 3. 				#rocket operation duration in sec
Is = 240. 				#specific impulse in sec
g = 9.81 				#acceleration due to kravity in m/s**2

#Calculation
MR = m2/m1 				#mass ratio
Mp = m1-m2 				#mass of propellant in kg
mp = Mp/tp 				#propellent flow rate in kg/s
wp = mp*g 				#weight flow rate of propellent in N/s
IMF = (m2-m3)/(m1-m3) 				#initial mass fraction
PMF = 1-IMF 				#propellant mass fraction
F = Is*wp 				#thrust in N
TWRi = F/(m1*g) 				#initial thrust to weight ratio
TWRf = F/(m2*g) 				#final thrust to weight ratio
av = F/m2 				#Maximum accelaration of the vechicle in m/s**2
Cj = Is*g 				#effective exhaust velocity in m/s
It = Is*Mp*g*10**-3 				#total impulse in kN-s, units of the answer given in the book is wrong
IWR = (It*10**3)/((m1-m3)*g) 				#impulse to weighr ratio, It in N-s

#Output
print 'A)mass ratio is %3.2f  \
\nB)propellent mass fraction is %3.3f  \
\nC)propellent flow rate is %3.1f kg/s \
\nD)thrust is %3.1f N \
\nE)thrust to weight ratio is %3.2f intial) and %3.2f final) \
\nF)accelaration of the vechicle is %3.2f m/s**2 \
\nG)effective exhaust velocity is %3.1f m/s \
\nH)total impulse is %3.3f kN-s \
\nI)impulse to weighr ratio is %3.2f'%(MR,PMF,mp,F,TWRi,TWRf,av,Cj,It,IWR)

A)mass ratio is 0.65
B)propellent mass fraction is 0.778
C)propellent flow rate is 23.3 kg/s
D)thrust is 54936.0 N
E)thrust to weight ratio is 28.00 intial) and 43.08 final)
F)accelaration of the vechicle is 422.58 m/s**2
G)effective exhaust velocity is 2354.4 m/s
H)total impulse is 164.808 kN-s
I)impulse to weighr ratio is 186.67


## Example 7.9 page : 27¶

In :
import math

#Input data
u = 2800. 				#rocket speed in m/s
Cj = 1400. 				#effective exhaust velocity in m/s
mp = 5. 				#propellent flow rate in kg/s
q = 6500. 				#heat of propellent per kg of propellant mixture in kJ/kg

#Calculation
s = u/Cj 				#effective jet speed ratio
np = (2*s)/(1+s**2) 				#propulsive efficiency
F = Cj*mp*10**-3 				#thrust in kN
Pt = F*10**3*u*10**-6 				#Thrust power in MW, F in N
Pe = Pt/np 				#engine outputin MW
nth = Pe*10**3/(mp*q) 				#thermal efficiency, Pe in kW
no = np*nth 				#overall efficiency

#Output
print 'A)propulsive efficiency is %3.1f  \
\nB)propulsive power is %3.1f MW \
\nC)engine outut is %3.1f MW \
\nD)thermal efficiency is %3.4f  \
\nE)overall efficiency is %3.3f'%(np,Pt,Pe,nth,no)

A)propulsive efficiency is 0.8
B)propulsive power is 19.6 MW
C)engine outut is 24.5 MW
D)thermal efficiency is 0.7538
E)overall efficiency is 0.603


## Example 7.10 page : 27¶

In :
import math

#Input data
Cj = 1250. 				#effective exhaust velocity in m/s
s = 0.8 				#effective jet speed ratio i.e. flight to jet speed ratio
ma = 3.5 				#oxidizer flow rate in kg/s
mf = 1. 				#fuel flow rate in kg/s
g = 9.81 				#acceleration due to gravity in m/s**2
q = 2500.*10**3 				#heat of propellent per kg of propellant mixture in J/kg

#Calculation
u = s*Cj 				#flight velocity in m/s
mp = ma+mf 				#propellant mass flow in kg/s
F = Cj*mp*10**-3 				#thrust in kN
wp = mp*g 				#weight flow rate of propellent in N/s
Is = (F*10**3)/(wp) 				#specific impulse in sec,F in N
np = (2*s)/(1+s**2) 				#propulsive efficiency
nth = 0.5*mp*((Cj**2+u**2)/(mp*q)) 				#thermal efficiency
no = np*nth 				#overall efficiency

#Output
print 'A)thrust is %3.3f kN \
\nB)specific impulse is %3.2f sec \
\nC)propulsive efficiency is %3.4f  \
\nD)thermal efficiency is %3.4f  \
\nE)overall efficiency is %3.1f'%(F,Is,np,nth,no)

A)thrust is 5.625 kN
B)specific impulse is 127.42 sec
C)propulsive efficiency is 0.9756
D)thermal efficiency is 0.5125
E)overall efficiency is 0.5


## Example 7.11 page : 28¶

In :
import math

#Input data
mp = 193. 				#propellent flow rate in kg/s
P1 = 27.*10**5 				#pressure at throat section in Pa
T1 = 3000. 				#temperature at throat section in K
de = 0.6 				#nozzle exit diameter in m
Pe = 1.1*10**5 				#exhaust pressure in Pa
Pamb = 1.013*10**5 				#ambient pressure in Pa
F = 380*10**3 				#thrust of motor in N
u = 694.44 				#flight velocity in m/s
g = 9.81 				#acceleration due to gravity in m/s**2
q = 6500*10**3 				#heat of propellent per kg of propellant mixture in J/kg

#Calculation
Ae = (math.pi*0.6**2)/4 				#exit area in m**2
Cj = F/mp 				#average effective jet velocity in m/s
Ce = (F-((Pe-Pamb)*Ae))/mp 				#exhaust velocity in m/s, wrong answer in textbook
wp = mp*g 				#weight flow rate of propellent in N/s
Is = (F)/(wp) 				#specific impulse in sec
SPC = (wp/F)/10**-3 				#specific propellent consumption in sec**-1 *10**-3
Pt = F*u*10**-6 				#Thrust power in MW
Pl = (0.5*mp*((Cj-u)**2))*10**-6 				#Power loss in exhaust in MW
Pe = Pt+Pl 				#engine output in MW
np = Pt/Pe 				#propulsive efficiency
nth = Pe*10**3/(mp*q*10**-3) 				#thermal efficiency and Pe,q in kW
no = np*nth 				#overall efficiency

#Output
print 'A)effective jet velocity is %3.4f m/s \
\nB)Actual jet velocity is %3.4f m/s \
\nC)specific impulse is %3.1f sec \
\nD)specific propellent consumption is %3.4f*10**-3 sec**-1 \
\nE)propulsive efficiency is %3.5f  \
\nD)thermal efficiency is %3.3f  \
\nE)overall efficiency is %3.5f'%(Cj,Ce,Is,SPC,np,nth,no)

A)effective jet velocity is 1968.9119 m/s
B)Actual jet velocity is 1956.1665 m/s
C)specific impulse is 200.7 sec
D)specific propellent consumption is 4.9824*10**-3 sec**-1
E)propulsive efficiency is 0.62736
D)thermal efficiency is 0.335
E)overall efficiency is 0.21035


## Example 7.12 page : 29¶

In :
import math

#Input data
m1 = 3600. 				#internal mass in kg
Cj = 2070. 				#average effective jet velocity in m/s
tp = 80. 				#rocket operation duration in sec
g = 9.81 				#acceleration due to gravity in m/s**2

#Calculation
up = 2*Cj 				#flight velocity in m/s
MR = 1/math.exp((up+(g*tp))/Cj) 				#mass ratio
m2 = MR*m1 				#mass after rocket operation in kg
PMF = 1-MR 				#propellant mass fraction
Mp = m1-m2 				#mass of propellant in kg
mp = Mp/tp 				#propellent flow rate in kg/s
F = Cj*mp*10**-3 				#thrust in kN
Zp = (((1+((1-(1/PMF))*math.log(1/MR)))*Cj*tp)-(0.5*g*tp**2))*10**-3 				#powered altitude gain in km
Zc = ((0.5*up**2)/g)*10**-3 				#coasting altitude gain in km
Z = Zp+Zc 				#maximum altitude in km

#Output
print 'A)flow rate of propellent is %3.2f kg/s \
\nB)thrust developed is %3.3f kN \
\nC)altitude gains during powered and coasting flights are %3.3f km and %3.3f km respectively'%(mp,F,Zp,Zc)

A)flow rate of propellent is 40.83 kg/s
B)thrust developed is 84.521 kN
C)altitude gains during powered and coasting flights are 93.987 km and 873.578 km respectively


## Example 7.13 page : 29¶

In :
import math

#Input data
s = 0.2105 				#effective jet speed ratio
Is = 203.88 				#specific impulse in sec
tp = 8 				#rocket operation duration i.e. burn out time in sec
g = 9.81 				#acceleration due to kravity in m/s**2

#Calculation
Cj = g*Is 				#average effective jet velocity in m/s
up = s*Cj 				#maximum flight speed in m/s
MR = 1/math.exp((up+(g*tp))/Cj) 				#mass ratio
PMF = 1-MR 				#propellant mass fraction
Zp = (((1+((1-(1/PMF))*math.log(1/MR)))*Cj*tp)-(0.5*g*tp**2))*10**-3 				#powered altitude gain in km
Zc = ((0.5*up**2)/g)*10**-3 				#coasting altitude gain in km
Z = Zp+Zc 				#maximum altitude in km

#Output
print 'A)effective jet velocity is %3i m/s \
\nB)mass ratio and propellent mass fraction are %3.2f and %3.2f respectively \
\nC)maximum flight speed is %3.2f m/s \
\nD))altitude gains during powered and coasting flights are %3.3f km and %3.3f km respectively'%(Cj,MR,PMF,up,Zp,Zc)

A)effective jet velocity is 2000 m/s
B)mass ratio and propellent mass fraction are 0.78 and 0.22 respectively
C)maximum flight speed is 421.01 m/s
D))altitude gains during powered and coasting flights are 1.601 km and 9.034 km respectively


## Example 7.14 page : 30¶

In :
import math

#Input data
R0 = 6341.6*10**3 				#radius of earth at mean sea-level in m
g = 9.809 				#acceleration due to gravity in m/s**2
Z1 = 0 				#altitude at sea-level in m
Z2 = 300*10**3 				#altitude above sea-level in m

#Calculation
uorb1 = R0*math.sqrt(g/(R0+Z1)) 				#orbit velocity of a rocket at mean sea level in m/s
uesc1 = math.sqrt(2)*uorb1 				#escape velocity of a rocket at mean sea level in m/s
uorb2 = R0*math.sqrt(g/(R0+Z2)) 				#orbit velocity of a rocket at an altitude of 300 km in m/s
uesc2 = math.sqrt(2)*uorb2 				#escape velocity of a rocket at an altitude of 300 km in m/s

#Output
print 'A)orbit and escape velocities of a rocket at mean sea level are %3i m/s and %3i m/s \
\nB)orbit and escape velocities of a rocket at an altitude of 300 km are %3.1f m/s and %3.2f m/s'%(uorb1,uesc1,uorb2,uesc2 )

A)orbit and escape velocities of a rocket at mean sea level are 7886 m/s and 11153 m/s
B)orbit and escape velocities of a rocket at an altitude of 300 km are 7706.8 m/s and 10899.08 m/s