Question 11.4: The following data are given mPL = 10,000 kg πPL = 0.05 ε = ...
The following data are given
\begin{aligned}m_{P L} &=10,000 kg \\\pi_{ PL } &=0.05 \\\varepsilon &=0.15 \\I_{s p} &=350 s \\g_{o} &=0.00981 km / s ^{2}\end{aligned} (a)
Calculate the payload velocity v_{b o} at burnout, the empty mass of the launch vehicle and the propellant mass for (a) a single stage and (b) a restricted, two-stage vehicle.
(a) From Equation 11.43 we find
v_{b o}=I_{s p} g_{o} \ln \frac{1}{\pi_{P L}(1-\varepsilon)+\varepsilon} (11.43)
v_{b o}=350 \cdot 0.00981 \ln \frac{1}{0.05(1+0.15)+0.15}=5.657 km / sEquation 11.40 yields the gross mass
\pi_{P L}=\frac{m_{P L}}{m_{o}} (11.40)
m_{o}=\frac{10,000}{0.05}=200,000 kgfrom which we obtain the empty mass using Equation 11.34,
\varepsilon=\frac{m_{E}}{m_{E}+m_{p}}=\frac{m_{E}}{m_{o}-m_{P L}} (11.34)
m_{E}=\varepsilon\left(m_{o}-m_{P L}\right)=0.15(200,000-10,000)=28,500 kgThe mass of propellant is
m_{p}=m_{o}-m_{E}-m_{P L}=200,000-28,500-10,000=161,500 kg(b) For a restricted two-stage vehicle, the burnout speed is given by Equation 11.50,
v_{b o_{2-\text { stage }}}=I_{s p} g_{o} \ln \left[\frac{1}{\pi_{P L}^{1 / 2}(1-\varepsilon)+\varepsilon}\right]^{2} (11.50)
v_{b o_{2-\text { stage }}}=350 \cdot 0.00981 \ln \left[\frac{1}{0.05^{1 / 2}(1-0.15)+0.15}\right]^{2}=7.407 km / sThe empty mass of each stage is found using Equations 11.51,
m_{E 1}=\frac{\left(1-\pi_{P L}^{1 / 2}\right) \varepsilon}{\pi_{P L}} m_{P L} \quad m_{E 2}=\frac{\left(1-\pi_{P L}^{1 / 2}\right) \varepsilon}{\pi_{P L}^{\frac{1}{2}}} m_{P L} (11.51)
m_{E 1}=\frac{\left(1-0.05^{1 / 2}\right) \cdot 0.15}{0.05} \cdot 10,000=23,292 kg
m_{E 2}=\frac{\left(1-0.05^{1 / 2}\right) \cdot 0.15}{0.05^{1 / 2}} \cdot 10,000=5208 kg
For the propellant masses, we turn to Equations 11.53
m_{p 1}=\frac{\left(1-\pi_{P L}^{1 / 2}\right)(1-\varepsilon)}{\pi_{P L}} m_{P L} \quad m_{p 2}=\frac{\left(1-\pi_{P L}^{1 / 2}\right)(1-\varepsilon)}{\pi_{P L}^{1 / 2}} m_{P L} (11.53)
m_{p 1}=\frac{\left(1-0.05^{1 / 2}\right) \cdot(1-0.15)}{0.05} \cdot 10,000=131,990 kg
m_{p 2}=\frac{\left(1-0.05^{1 / 2}\right) \cdot(1-0.15)}{0.05^{1 / 2}} \cdot 10,000=29,513 kg
The total empty mass, m_{E}=m_{E 1}+m_{E 2}, and the total propellant mass, m_{p}=m_{p 1}+m_{p 2} are the same as for the single stage rocket. The mass of the second stage, including the payload, is 22.4% of the total vehicle mass.