# Problems

When solving problems, three appendixes (see end of book) may be helpful:

Appendix l. Conversion Factors and Constants Appendix 2. Properties of the Earth's Standard Atmosphere Appendix 3. Summary of Key Equations

1. Prove that the value of the reaction thrust F equals twice the total dynamic pressure across the area A for an incompressible fluid as shown below.

Flow rate w

Force F

Flow rate w

### Force F

2. The following data are given for a certain rocket unit: thrust, 8896 N; propellant consumption, 3.867 kg/sec; velocity of vehicle, 400 m/sec; energy content of propellant, 6.911 MJ/kg. Assume 100% combustion efficiency.

Determine (a) the effective velocity; (b) the kinetic jet energy rate per unit flow of propellant; (c) the internal efficiency; (d) the propulsive efficiency; (e) the overall efficiency; (f) the specific impulse; (g) the specific propellant consumption. Answers: (a) 2300 m/sec; (b) 2.645 MJ-sec/kg; (c) 38.3%; (d) 33.7%; (e) 13.3%; (f) 234.7 sec; (g) 0.00426 sec"1.

3. A certain rocket has an effective exhaust velocity of 7000 ft/sec; it consumes 280 lbm/sec of propellant mass, each of which liberates 2400 Btu/lbm. The unit operates for 65 sec. Construct a set of curves plotting the propulsive, internal, and overall efficiencies versus the velocity ratio u/c (0 < u/c < 1.0). The rated flight velocity equals 5000 ft/sec. Calculate (a) the specific impulse; (b) the total impulse; (c) the mass of propellants required; (d) the volume that the propellants occupy if their average specific gravity is 0.925.

Answers: (a) 217.5 sec; (b) 3,960,000 lbf-sec; (c) 18,200 lbm; (d) 315 ft3.

4. For the rocket in Problem 2, calculate the specific power, assuming a propulsion system dry mass of 80 kg and a duration of 3 min.

5. For the values given in Table 2-1 for the various propulsion systems, calculate the total impulse for a fixed propellant mass of 2000 kg.

6. A jet of fluid hits a stationary flat plate in the manner shown below.

(a) If there is 50 kg of fluid flowing per minute at an abolute velocity of 200 m/sec, what will be the force on the plate?

(b) What will this force be when the plate moves in the direction of flow at u = 50 km/h?

7. Plot the variation of the thrust and specific impulse against altitude, using the atmospheric pressure information given in Appendix 2, and the data for the Minuteman first-stage rocket thrust chamber in Table 11-3. Assume that p2 = 8.66 psia.

8. Derive an equation relating the mass ratio 1VR and the propellant mass fraction. Answer: f = 1 â€” 1VR.

SYMBOLS (English engineering units are given in parentheses)

A, nozzle throat area, m2 (ft2)

A2 exist area of nozzle, m2 (ft2)

c effective velocity, m/sec (ft/sec)

c* characteristic velocity, m/sec (ft/sec)

go standard sea level acceleration of gravity, 9.80665 m/sec2

7, specific impulse, sec

I, impulse or total impulse, N-sec (lbf-sec)

J conversion factor or mechanical equivalent of heat, 4.184 J/cal or 1055

J/Btu or 778 ft-lbf/Btu.

m mass, kg (slugs) (1 slug = mass of 32.174 lb of weight at sea level)

m mass flow rate, kg/sec (lbm/sec)

my final mass (after rocket propellant is ejected), kg (lbm or slugs)

mp propellant mass, kg (lbm or slugs)

w0 initial mass (before rocket propellant is ejected), kg (lbm or slugs)

ambient or atmospheric pressure, Pa (lbf/ft2)

p2 rocket gas pressure at nozzle exit, Pa (lbf/ft2)

pl chamber pressure, Pa (lbf/ft2)

Ps specific power, J/sec-kg (ft-lbf/sec-lbm)

Qr heat of reaction per unit propellant, J/kg (Btu/lbm)

t time, sec u vehicle velocity, m/sec (ft/sec)

v2 gas velocity leaving the rocket, m/sec (ft/sec)

w weight flow rate, N/sec (lbf/sec)

w0 initial weight, N or kg-m/sec2 (lbf)