Gas dynamics and jet propulsion – presentationof problemsanswers

Isentropic Process-Nozzle,Diffuser– Unit 1
1

2

Throat area A* correspond to Mach Number 1.
Thrust Force F* correspond to Mach Number 1.
3

Gas Dynamics and Jet Propulsion – Unit 1
Problem: The stagnation enthalpy fluxes entering and leaving a control
volume containing a turbine stage are 25 and 10 kJ/s. Heat loss is 2.5 kJ/s. Find
the power developed by the turbine.
4

Problem: Determine the mach number of an aircraft at which the velocity
temperature of air at the entry of engine equals its static temperature.
5

Problem: Determine the velocity of air corresponding to a velocity
temperature of 1 0C . (γ = 1.4; Cp = 1005 J/kgK)
Given: Velocity temperature Tc = 1 0C = 274 K (=1+273)
(γ = 1.4; R = 287 J/kgK)
6

An air jet at 400 K has sonic velocity. Find (a) Velocity of sound at 400 K, (b)
Velocity of sound at stagnation state, (c ) Maximum velocity of jet, (d)
stagnation enthalpy and (e) Crocco number.
Given: Velocity of jet, c = 400 m/s; Static temperature of the jet, T = 400 K
R = 287 J/kgK and γ = 1.4 for air
‘a = 400.9 m/s; a0 = 439.2 m/s; c max = 982 m/s; h0 = 482.2 kJ/kg; Cr = 0.4
7

Problem: A jet of gas at 593 K (γ = 1.4; R = 469 J/kgK) has a mach number of
1.2. Find for static and stagnation conditions (a) Velocity of sound, (b)
enthalpy, and (c ) maximum attainable velocity.
Given: Mach Number M = 1.2; Static Temperature T = 593 K
γ = 1.4; R = 469 J/kgK)
8

Problem: Air flows from a reservoir at 550 kPa and 70 0C. Assuming isentropic
flow calculate the velocity, temperature, pressure, and density at a section
where mach number is 0.6.
Given:
M= 0.6 T0 = 343 K p0 = 550 kPa
Isentropic table provides the following Data:
M M* T//T0 p/p0 A/A* F/F*
0.6 **** ***
T = p =
9

Problem: Air at a pressure of 3 bar and temperature 500 K flows with a
velocity of 200 m/s in a 30 cm diameter duct. Find (a) mass flow rate, (b)
stagnation temperature, (c ) mach number, (d) stagnation pressure for
compressible flow and (e) stagnation pressure for incompressible flow.
Given: p = 3 bar; T = 500 K; c = 200 m/s;
10

Problem: An air stream at 1 bar, and 400 K flowing with a velocity of 400 m/s
is brought to rest isentropically. Determine the stagnation pressure and
temperature.
Given:
T =400 K p = 1 bar c = 400 m/s
Isentropic table provides the following Data:
**** **** ***
T0 = p0 =
11

Problem: An aircraft flies at 800 kmph at an altitude of 10000 m. the air is
reversibly compressed in an inlet diffuser. If the mach number at exit of the
diffuser is 0.36, find (a) entry mach number, (b) velocity, pressure, and
temperature of air at diffuser exit.
Given: z = 10000 m; c = 800 kmph = 800/3.6 = 222.22 m/s; M2=0.36
From table, at z = 10000 m
T1 = ; p1 = ;
M1 **** ***
T0 = p0 =
0.36 **** ***
T2 = ; p2 = ;
12

Problem: Air enters a straight axisymmetric duct at 27 0C, 3.45 bar and 150
m/s. It leaves at 40C, 2.058 bar and 1260 m/s. Under adiabatic flow conditions
for an inlet cross sectional area of 500 cm2, determine the stagnation
temperature, maximum velocity, mass flow rate and exit area.
Given: T1 = 270C = 300 K; p1 = 3.45 bar; c1 = 150 m/s; A1 = 500 cm2
T2 = 40C = 277 K; p2 = 2.058 bar; c2= 260 m/s
M1
T0 = p0 = A*=
M2 **** *** ***
A2 = cm2
T1 = 300 K; p1 = 3.45 bar A1 = 500 cm2
13

Problem:A conical air diffuser has an intake area of 0.11 m2 and an exit area of
0.44 m2. Air enters the diffuser with a static pressure of 0.18 MPa, static
temperature 37 oC , and velocity of 267 m/s. Calculate(a) mass flow rate of air
through the diffuser, (b) mach number, pressure and temperature of air
leaving the diffuser, and (c ) net thrust acting upon the diffuser due to
diffusion. (γ=1.4, Cp = 1005J/kgK) .
14

Problem: Air (γ=1.4, Cp = 1005 J/kgK) at p1 = 3x105 N/m2, and temperature 500 K
flows with a velocity of 200 m/s in a 30 cm diameter duct. Find (a) mass flow
rate, (b) stagnation temperature, (c ) mach number, and stagnation pressure
assuming the flow is compressible and incompressible respectively.
16

Problem: A conical diffuser has entry and exit diameters of 15 cm and 30 cm
respectively. The pressure, temperature, and velocity of air at entry are 0.69
bar, 340 K, and 180 m/s respectively. Determine (a) the exit temperature, (b) exit
velocity, and (c ) force exerted on the diffuser. Assume isentropic flow with
γ=1.4, Cp = 1005J/kgK) .
17

Problem: Air flowing in a 30 cm diameter duct has a velocity of 300 m/s, pressure
1.0 bar and temperature 290 K. Assume γ=1.4, R= 287 J/kgK and determine (a)
mass flow ate, (b) stagnation temperature and pressure, (c ) velocity of sound in
dynamic and stagnation conditions, and (d ) stagnation pressure assuming constant
density.
19

Problem: An air plane travels at M=2 at an elevation where the temperature is
233 K. Find the speed of the air plane in kmph. Assume γ=1.4.
21

Problem: A jet fighter is flying at M=2.5. It is observed directly overhead at a
height of 10 km. How much distance it would cover before sonic boom is
heard on the ground.
22

Problem: Air is discharged from a reservoir at P0 = 6.91 bar, t0 = 235 0C through a
nozzle to an exit pressure of 0.98 bar. If the flow rate is 3600 kg/hr, determine for
isentropic flow (a) area, pressure, and velocity at throat, (b) area and mach number
at exit and (c ) maximum possible velocity.
23

Problem: In an isentropic flow through a diffuser the inlet area is 0.15 m2. At the
inlet, velocity = 240 m/s, static temperature = 300 K, static pressure = 0.7 bar. Air leaves
the diffuser with a velocity of 120 m/s. Calculate (a) the mass flow rate, (b) stagnation
temperature and pressure, (c ) exit area, and (d) entropy change across the diffuser.
24

25
1. Write the continuity equations.
2. Write the momentum equations.
3. Write the energy equations.
4. Write the relation for compressibility ratio in terms of Mach
number.
5. Write the dynamic equation.
6. Write the relations for M*, Cr, T0/T, T0/T*, .
7. Draw the h-s diagram for isentropic and adiabatic diffuser
process.
8. Write the equation for diffuser efficiency for small pressure
rise and large pressure rise in a diffuser.
9. Draw the h-s diagram for isentropic and adiabatic nozzle
process.
10. Write the equation for nozzle efficiency.
11. What is the use of isentropic table?

26
12. What is the basic difference between compressible and
incompressible fluid flow?
13. What is the meaning of stagnation state? Give example.
14. Distinguish between static and stagnation pressure.
15. What is the use of mach number?
16. Give four reference velocities used in non-dimensional form.
17. What are the different regions of compressible flow?
18. What happens to mach number value if a an airplane goes to higher
altitudes maintaining the same speed?
19. Define mach angle, mach wedge, and mach cone.
20. What happens to air properties if air flowing through a nozzle is
heated?
21. State Fliegner’s formula.
22. What is the use of impulse function?
23. What is the condition for choked flow to occur in a nozzle?
24. Draw the variation of P/P0 along the length of convergent divergent
nozzle acting as diffuser, nozzle and venturi.
25. Air from reservoir is discharged through a nozzle. Show the variation
of pressure and variation of temperature along the nozzle.

END of Unit 1
27

Fanno Flow , Rayleigh Flow– Unit 2

Problem: Air enters a combustion chamber with a certain mach number and
sufficient heat is added to obtain a stagnation temperature ratio of 3 and a final
mach number of 0.8. Determine the mach number at entry and the percentage loss
in static pressure. Take γ=1.4, Cp = 1005 J/kgK for air.
32

Problem: the friction factor for a 50 mm dia steel pipe is 0.005. At the inlet to the
pipe the velocity is 70 m/s, temperature is 80 0C, and the pressure is 10 bar. Find the
temperature, pressure, and mach number at exit if the pipe is 25 m long. Also
determine the maximum possible length.
33

Problem: A circular duct passes 8.25kg/s of air at an exit mach number of 0.5. the
entry pressure and temperature are 3.45 bar and 38 0C respectively and the
coefficient of friction is 0.005. If the mach number at entry is 0.15, determine (a)
the duct diameter, (b) length of the duct, (c ) pressure and temperature at the
exit, and (d) stagnation pressure loss.
34

Problem: A combustion chamber in a gas turbine plant receives air at 350 K, 0.55
bar and 75 m/s. the air fuel ratio is 29 and calorific value of the fuel is 41.87 MJ/kg.
Assume γ=1.4, R= 287J/kgK for the gas determine (initial and final mach
numbers, (b) final pressure, temperature and velocity of the gas, (c ) percentage
stagnation pressure loss in the combustion chamber and (d) maximum stagnation
temperature attainable.
36

Problem: A long pipe of 0.0254 m diameter has a mean coefficient of friction
0.003. Air enters the pipe at a mach number of 2.5, stagnation temperature of
310 K and static pressure 0.507 bar. Determine for a section at which the
mach number reaches 1.2 (a) static pressure and temperature, (b) stagnation
pressure and temperature, ( c ) velocity of air, (d) distance of this section from
inlet and (e) mass flow rate of air.
38

Problem: The mach number at the exit of a combustion chamber is 0.9. the
ratio of stagnation temperatures at exit and entry is 3.74. If the pressure and
temperature of the gas at exit are 2.5 bar and 1273 K respectively, determine
(a) mach number, pressure, and temperature of the gas at entry, (b) heat
supplied per kg of the gas and (c ) maximum heat that can be supplied.
40

Problem: Air enters a constant area duct at M1 = 3, P1 = 1 atm, and T1 300 K.
Inside the duct heat added per unit mass is q = 3x105 J/kg. Calculate the flow
properties mach number (M2) , stagnation pressure (P02)and temperature
(T02), and static pressure (P2) temperature (T2) and density (ρ2) at the exit.
42

Problem: Air at an inlet temperature of 600C flows with subsonic velocity through
an insulated pipe having inside diameter of 50 mm and a length of 5 m. The
pressure at the exit of the pipe is 101 kPa and the flow is choked at the end of the
pipe. If the friction factor 4f = 0.05, determine the inlet mach number, mass flow
rate and exit temperature.
43

1. Derive Fanno line.
2. Derive Rayleigh line.
3. Write the general gas dynamic equation.
4. What are the uses of Fanno table?
5. What are the uses of Rayleigh table?
6. Differentiate between Rayleigh and Fanno flows.
7. Show that M = 1 at s max in a Rayleigh flow.
8. Show that M= 1/√γ at hmax in a Rayleigh flow.
9. Define Fanno flow.
10. What are the assumptions in Fanno flow?
11. Draw the Fanno line in h-s diagram.
12. Give examples for Fanno flow.
13. How does the pressure and density vary along the length of pipe in a Fanno flow when the
upstream condition is supersonic?
14. How does the pressure and density vary along the length of pipe in a Fanno flow when the
upstream condition is subsonic?
15. Define choking in Fanno flow.
16. How friction causes choking in Fanno flow?
17. What is the effect of increasing the length after reaching critical condition in a Fanno flow.
18. Distinguish between Isothermal and Fanno flow.
19. Define Rayleigh flow or Diabatic flow.
20. What are the assumptions in Rayleigh flow?
21. Give examples for Rayleigh flow.
22. Heat addition to a gas may cool the gas. Explain with T-s diagram. 45

END of Unit 2
46

Normal Shock, Oblique Shock– Unit 3

Problem: When a convergent divergent nozzle is operated at off design condition, a
normal shock occurs at a section where the cross sectional area is 18.75 cm2 in the
diverging position. At inlet to the nozzle, stagnation state is given as 0.21 MPa and 36
0C. The throat area is 12.5 cm2 and exit area is 25 cm2. Estimate the exit mach
number, exit pressure and loss in the stagnation pressure for the flow through nozzle.
49

Problem: A convergent divergent nozzle is designed to expand air from a reservoir in
which the pressure is 800 kPa and temperature is 40 0C to give a mach number at exit of
2.5. The throat area is 25 cm2. Find (a) mass flow rate, (b) exit area and (c ) when normal
shock appears at a section where the area is 40 cm2, determine the pressure and
temperature at exit.
51
2
throat
1 M=1

Problem: A bow shock occurs in front of a pitot tube when it is used in a
supersonic flow field. It measures 16 kPa and 70 kPa for static pressure
upstream of the shock and the pressure at the mouth of the tube respectively.
Estimate the mach number of the supersonic flow. If the stagnation
temperature is 300 0C. Calculate the static temperature and total pressure
upstream and down stream of the pitot tube.
53

Problem: The ratio of the exit to entry area in a subsonic diffuser is 4.0. the mach
number of a jet of air approaching the diffuser at P0 = 1.013 bar, T = 290 K is 2.2.
There is a standing normal shock wave just outside the diffuser entry. The flow in
the diffuser is isentropic. Determine at the exit of the diffuser (a) mach
number, (b) temperature, and (c ) pressure. What is the stagnation pressure loss
between the initial and final state of the flow.
54

Problem: Air at M = 2.5 enters a convergent duct with a area ratio of A2/A1= 0.5.
Normal shock occurs at a test section where At/A1 = 0.6. For this condition find exit
mach number and pressure ratio across the duct.
56

Problem: A gas (γ = 1.3) at P1 = 345 mbar, T1 = 350 K and M1 = 1.5 is to be
isentropically expanded to 138 mbar. Determine (a) the deflection angle, (b)
final mach number and temperature of the gas.
57

Problem: A jet of air at mach number 2.5 is deflected inwards at the corner of
a curved wall. The wave angle at the corner is 60 0. Determine the deflection
angle on the wall, pressure and temperature ratios and final mach number.
58

An oblique shock wave at angle of 330 occurs at the leading edge of a
symmetrical wedge. Air has mach number of 3.2 upstream temperature of
300 K and upstream pressure of 11 bar. Determine (a) Downstream
pressure, (b) Downstream temperature, (c ) wedge angle, and (d) downstream
mach number.
59

Problem: An explosion in air (γ=1.4) creates a spherical shockwave
propagating radially into still air at standard conditions. At that instant the
pressure just inside the shock is 13.789 bar(abs). Estimate (a) shock speed c
ab, (b) air velocity c just inside the shock.
Given: Px=1.01325 bar; Tx= 150C = 288 K; Py=13.789 bar
** ** 13.6087 **** ****
60

61
1. Show the shock process in Fanno and Rayleigh line.
2. Why shock is impossible in a subsonic flow?
3. Why rarefaction shocks are not possible?
4. Write the equation for strength of shock.
5. Write the change in entropy across a shock.
6. How would you measure irreversibility of a shock?
7. Is shock an irreversible process?Why?
8. Write the Prandtl Meyer relation.
9. Write the Rankine Hugoniot equation.
10. Differentiate between normal and oblique shocks
11. Define a simple wave.
12. What is a wave motion?
13. Define shock wave.
14. What are the effects of a shock wave?
15. What is a normal shock wave?
16. How do the properties change across a normal shock?
17. Is the flow process across a normal shock equilibrium one?
18. What are the applications of a moving shock?
19. Are the shock waves created by supersonic aircraft desirable?
20. What is a shock polar?
21. Define oblique shock.
22. Differentiate between normal shock and oblique shock.
23. Show Rankine Hugoniot curve.

END of Unit 3
62

Performance of Turbo Jet Components– Unit 4
S.No. Component Process Detail Process reference Efficiency
1 Diffuser Increase pressure –
transformation
[ i-1 ]
Adiabatic
[ i-1’ ]
Isentropic
2 Compressor Increase pressure –
work transfer
[ 1-2 ] [ 1-2’ ]
Isentropic
3 Combustor Heat added-
Constant pressure
[ 2-3 ] [ 2-3’ ]
Rayleigh
4 Turbine Increase velocity –
work transfer
[ 3-4 ] [3-4’ ]
Isentropic
5 Nozzle Increase velocity –
transformation
[ 4-e ] [ 4-e’ ]
Isentropic
Components of Turbo Jet:

64

65

66

Problem: The flight speed of a turbojet is 600 kmph at 10000 m altitude. The
density of air at that altitude is 0.17 kg/m3.. The drag for the plane is 6.8 kN.
The propulsive efficiency of the jet is 60%. Calculate the specific fuel
consumption (SFC), air-fuel ratio (AF), and jet velocity (cj). Assume calorific
value of the fuel as 45 MJ/kg and overall efficiency of the turbo jet as 18%.
67

Problem: The diameter of the propeller of an aircraft is 2.5 m. It flies at a
speed of 500 kmph at an altitude of 8000 m. For a flight to jet speed ratio of
0.75, determine (a) mass flow rate through the propeller, (b) thrust produced,
(c ) specific thrust, (d) specific impulse, and (e) thrust power.
68

Problem: A turbo jet propels an aircraft at a speed of 900 kmph while taking 3000 kg
of air per minute. The isentropic enthalpy drop in the nozzle is 200 kJ/kg and nozzle
efficiency is 90%. The air fuel ratio is 85 and the combustion efficiency is 95%. The
calorific value of the fuel is 42 MJ/kg. Calculate (a) thrust power, (b) power output of
the engine, (c ) thermal efficiency, (d) propulsive efficiency, (e) Specific Thrust and (f)
Specific Impulse.
69
Continued..

70

71
Problem: A turbo jet aircraft flies with a velocity of 300 m/s at an altitude where air
pressure is 0.35 bar and temperature – 400C. The compressor has a pressure ratio of 10
and temperature of gases at the turbine inlet is 1100 0C. Air enters the compressor at the
rate of 50 kg/s. Estimate (a) the temperature and pressure of gases at turbine exit, (b)
temperature and Velocity of gases at the nozzle exit, (c ) Flight to Jet speed Ratio and (d)
propulsive efficiency of the cycle.

Problem: A jet propelled plane has 2 jets, 250 mm diameter and net power at
turbine is 3 MW. Fuel consumption per kWhr is 0.42 kg with CV = 49 MJ/kg.
When flying at a speed of 300 m/s in an atmosphere having density of 0.168
kg/m3. the air fuel ratio is 53. Calculate (a) absolute velocity of jet, (b)
resistance of the plane, (c ) Thrust Power, (d) Engine Power Output, and (e)
Propulsive Efficiency, overall efficiency and thermal efficiency.
73

75
1. Draw the h-s diagram for diffuser process.
2. Write the equation for diffuser efficiency for small pressure rise and large
pressure rise in a diffuser.
3. Write the equation for isentropic compression in a compressor.
4. Draw the h-s diagram for compression process in a compressor.
5. Write the equation for compressor efficiency of the compressor.
6. Draw the h-s diagram for combustion process in a combustion chamber.
7. Write the equation for heat transferred per kg of gases in a combustion
chamber.
8. Write the equation for combustion efficiency of the combustion chamber.
9. Draw the h-s diagram for turbine process.
10. Write the equation for isentropic efficiency of a turbine.
11. Draw the h-s diagram for nozzle process.
12. Write the equation for nozzle efficiency.
13. Write the equation for exit velocity of gases in a nozzle.

76
14. Write the relation for thrust force considering the mass flow rate of fuel.
15. Write the relation for thrust force neglecting the mass flow rate of fuel.
16. Write the equation for thrust power or propulsive power of a turbo jet.
17. Write the equation for power output of engine PE.
18. Write the equation for propulsive efficiency of a turbo jet.
19. Write the equation for propulsive efficiency in terms of flight to jet speed
ratio.
20. What is the condition for maximum thrust power?
21. What is the propulsive efficiency for maximum thrust condition?
22. Write the relations for mechanical efficiency, combustion efficiency and
thermal efficiency.
23. Write the relation for overall efficiency? Express the relation also in terms of
mechanical efficiency, combustion efficiency and thermal efficiency.
24. Define TSFC.
25. Define specific thrust and specific impulse.
26. Define effective jet velocity.

END of Unit 4
77

Problem5.1: If a rocket has to develop a specific impulse of 250 s, calculate the effective
jet velocity. If the rocket to jet speed ratio is 0.75 and propellant flow rate is 100
kg/min, calculate the thrust power. If the overall efficiency is 20%, determine the required
heating value for the propellant.
81

Problem: Calculate the thrust specific impulse, propulsive efficiency, thermal
efficiency and overall efficiency of a rocket engine from the following data: Effective
jet velocity = 1250 m/s; Flight to jet speed ratio = 0.8; Oxidizer flow rate =3.5 kg/s;
Fuel flow rate = 1kg/s; Heat of reaction of exhaust gases=2500 kJ/kg.
82

Problem: The effective jet velocity from a rocket is 2700 m/s. the forward flight
velocity is 1300 m/s. Propellant consumption is 78.6 kg/s. Calculate (a) thrust; (b)
thrust power, (c )specific impulse, (d) engine power output and (e ) propulsive
efficiency.
83

Problem: The specific impulse of a rocket is 125 s. and the propellant flow
rate is 44 kg/s. The nozzle throat area is 18 cm2 and the pressure in the
combustor is 25 bar. Determine (a) thrust coefficient, (b) propellant flow
coefficient, (c ) specific propellant consumption, and (d) characteristic velocity.
84

Problem: A rocket has following data: Propellant flow rate = 5 kg/s; Nozzle exit
diameter = 10 cm; Nozzle exit pressure = 1.02 bar; Ambient pressure = 1.013 bar;
Thrust chamber pressure = 20 bar; Thrust = 7 kN; Determine (a) the effective jet
velocity, (b) actual jet velocity, (c ) specific impulse, and (d) specific propellant
consumption. Recalculate the values of thrust and specific impulse for an altitude
where the ambient pressure is 10 m bar.
85

Problem: The following data refers to a rocket: Propellant flow rate = 193 kg/s;
thrust chamber pressure = 27 bar and temperature = 3000 K; Nozzle exit
diameter = 600 mm; Nozzle exit pressure = 1.1 bar; Ambient pressure = 1.013
bar; Thrust produced = 380 kN; Find the effective jet velocity, specific impulse
and specific propellant consumption. Also the rocket speed is 250 kmph and heat
of reaction of propellant gases is 6500 kJ/kg. Find the propulsive , thermal and
overall efficiencies.
86

Problem: A rocket has an initial mass of 3600 kg at take off. The effective jet
velocity for the duration of powered flight 80 sec remains constant at 2070 m/s.
The maximum velocity of the rocket during the flight is twice the effective jet
velocity. Calculate the propellant flow rate, thrust developed, and the altitude
gains during powered and coasting flights.
88

Problem: A rocket engine has the following data. Combustion chamber pressure is 38 bar.
Combustion chamber temperature is 3500 K. Oxidizer flow rate is 41.67 kg/s. Mixture ratio
is 5. Properties of exhaust gases are Cp/Cv = 1.3 and R = 0.287 kJ/kgK. The expansion takes
place to the ambient pressure of 0.0582 bar. Calculate the nozzle throat area, thrust, thrust
coefficient, exit velocity of the exhaust and maximum possible exhaust velocity .
89

90

1. Write the equation for velocity of rocket.
2. Define orbital velocity.
3. Define escape velocity.
4. Define characteristic velocity.
5. Define thrust coefficient.
6. Write the relation for coasting vertical flight gain.
7. Write the relation for vertical power flight gain.
8. Define exit pressure ratio.
9. Write the equation for thrust of a rocket at an altitude.
10.Write the equation for thrust of a rocket at sea level.
11.Write the equation for mass flow rate propellant.
12.Define mass ratio MR.
13.Define propellant mass fraction.
14.Define flight to jet speed ratio of a rocket.
15.Write the equation for propulsive efficiency of a rocket.
16.Write the equation for propulsive efficiency of a rocket in terms of flight to jet speed
ratio.
17.Write the equation for thrust power or propulsive power of a rocket.
18.Write the equation for power output of engine PE.
19.Write the relations for thermal efficiency and overall efficiency.
20.Define specific impulse.
21.Define specific propellant consumption.
91

END of Unit 5
92

Gas dynamics and jet propulsion – presentationof problemsanswers

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (20)

Similar to Gas dynamics and jet propulsion – presentationof problemsanswers

Similar to Gas dynamics and jet propulsion – presentationof problemsanswers (20)

More from Vaidyanathan Ramakrishnan

More from Vaidyanathan Ramakrishnan (6)

Gas dynamics and jet propulsion – presentationof problemsanswers