Question

Consider a rocket engine in which the combustion chamber pressure and temperature are 30 atm and 3756 K, respectively. The area of the rocket nozzle exit is 15 m2 and is designed so that the exit pressure exactly equals ambient pressure at a standard altitude of 25 km. For the gas mixture, assume that γ = 1.18 and the molecular weight is 20. At a standard altitude of 25 km. Calculate the:

a. specific impulse
b. exit velocity
c. mass flow
d. thrust
e. throat area.

Answer 1

Answer:

• 379.48 s ( specific impulse )
• 3678.75 m/s ( exit velocity )
• 263.21 kg/s ( mass flow )
• 979855.3 N ( thrust )
• 0.168 $m^{2}$ ( throat area )

Explanation:

A) Specific impulse

standard altitude = 25 km

take free steam pressure ( $p_{0}$ ) = 2527.3 N/$m^{2}$

Exit pressure = ambient pressure. therefore exit pressure ( $p_{e}$ ) = 2527.3 N/$m^{2}$

Applying specific impulse equation

$I_{sp}$ = $\frac{1}{g_{0} }$ [ $\frac{2Y T_{0} }{Y - 1} (\frac{R}{M} )[ 1 - (\frac{P_{e} }{p_{0} })^{\frac{y-1}{1} } ]]^{\frac{1}{2} }$     equation 1

given

$g_{0}$ = 9.81 m/$s^{2}$ , M = 20 kg/kmol, R = 8314.47 j/kmol.k

y = 1.18, $T_{0}$ = 3756 k,  $p_{e }$ = 2527.3 N/$m^{2}$, $p_{0}$ = 30 atm

substitute the given data into equation 1

$I_{sp}$ ( specific impulse ) = 379.48 s

B) exit velocity

Relating the equation for isentropic process, pressure and temperature at the combustion chamber , the exit temperature (Te) can be calculated using this formula :

$\frac{p_{e} }{p_{0} }$ = $(\frac{T_{e} }{T_{0} })^{\frac{y}{y-1} }$

SUBSTITUTE the value of the above parameters

make (Te) subject of the equation Te = 1273.11 k

next calculate for constant specific pressure ( Cp )

Cp = $\frac{y}{y-1}( \frac{R}{M})$

substitute the value of the above parameters into the equation

Cp = 2725.3 J/kg.k

Finally with Te and Cp known calculate the exit velocity

Ve = $\sqrt{2Cp(To - Te)}$

Cp = 2725.3 J/kg.k

To = 3756 k

Te = 1273.11 k

substitute the given values into the equation

Ve = 3678.75 m/s ( exit velocity )

C ) Mass flow

firstly calculate the density of the gas at exit ( De )

De = $\frac{Pe}{(\frac{R}{M}) } Te$

substituting the values of the parameters into the equation

De ( density of gas at exit ) = 0.00477 kg/m$^{3}$

finally calculate the Mass flow at the exit

$m_{e}$ = De*Ve*Ae

De = 0.00477 kg/$m^{3}$

Ve = 3678.75 m/s

Ae = 15 $m^{2}$

therefore $m_{e}$ = 263.21 kg/s

D ) thrust

calculate the weight flow rate rate ( w ) firstly

w = mg ( mass flow * speed of gravity )

= 263.21 * 9.81 = 2582.1 N/s

finally calculate thrust by applying this equation for thrust

T ( thrust ) = specific impulse * weight flow rate

= 379.48 * 2582. 1  = 979855.3 N

E ) throat area

calculate the throat area using this equation

m = $\frac{PoA}{\sqrt{To} } \sqrt{\frac{y}{(\frac{R}{M} }) } (\frac{2}{y+1} )^{\frac{(Y+1)}{(y-1)} }$

substituting the values of : m , M , R, To, y, and Po into the equation and making A subject of the equation

A = $\frac{263.21 kg/s}{1568.1\sqrt{N.kg /m^{2} \sqrt{m} } }$    = 0.168 $m^{2}$ ( throat area )