The following data are given mPL = 10,000 kg πPL = 0.05 ε = 0.15 ISP = 350s go = 0.00981 km/s² (a) Calculate the payload velocity vbo 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.

Question

The following data are given

[latex]\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}[/latex] (a)

Calculate the payload velocity [latex]v_{b o}[/latex] 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.

Accepted Answer

(a) From Equation 11.43 we find

[latex]v_{b o}=I_{s p} g_{o} \ln \frac{1}{\pi_{P L}(1-\varepsilon)+\varepsilon}[/latex] (11.43)

[latex]v_{b o}=350 \cdot 0.00981 \ln \frac{1}{0.05(1+0.15)+0.15}=5.657 km / s[/latex]

Equation 11.40 yields the gross mass

[latex]\pi_{P L}=\frac{m_{P L}}{m_{o}}[/latex] (11.40)

[latex]m_{o}=\frac{10,000}{0.05}=200,000 kg[/latex]

from which we obtain the empty mass using Equation 11.34,

[latex]\varepsilon=\frac{m_{E}}{m_{E}+m_{p}}=\frac{m_{E}}{m_{o}-m_{P L}}[/latex] (11.34)

[latex]m_{E}=\varepsilon\left(m_{o}-m_{P L} ight)=0.15(200,000-10,000)=28,500 kg[/latex]

The mass of propellant is

[latex]m_{p}=m_{o}-m_{E}-m_{P L}=200,000-28,500-10,000=161,500 kg[/latex]

(b) For a restricted two-stage vehicle, the burnout speed is given by Equation 11.50,

[latex]v_{b o_{2- ext { stage }}}=I_{s p} g_{o} \ln \left[\frac{1}{\pi_{P L}^{1 / 2}(1-\varepsilon)+\varepsilon} ight]^{2}[/latex] (11.50)

[latex]v_{b o_{2- ext { stage }}}=350 \cdot 0.00981 \ln \left[\frac{1}{0.05^{1 / 2}(1-0.15)+0.15} ight]^{2}=7.407 km / s[/latex]

The empty mass of each stage is found using Equations 11.51,

[latex]m_{E 1}=\frac{\left(1-\pi_{P L}^{1 / 2} ight) \varepsilon}{\pi_{P L}} m_{P L} \quad m_{E 2}=\frac{\left(1-\pi_{P L}^{1 / 2} ight) \varepsilon}{\pi_{P L}^{\frac{1}{2}}} m_{P L}[/latex] (11.51)

[latex]m_{E 1}=\frac{\left(1-0.05^{1 / 2} ight) \cdot 0.15}{0.05} \cdot 10,000=23,292 kg[/latex]
[latex]m_{E 2}=\frac{\left(1-0.05^{1 / 2} ight) \cdot 0.15}{0.05^{1 / 2}} \cdot 10,000=5208 kg[/latex]

For the propellant masses, we turn to Equations 11.53

[latex]m_{p 1}=\frac{\left(1-\pi_{P L}^{1 / 2} ight)(1-\varepsilon)}{\pi_{P L}} m_{P L} \quad m_{p 2}=\frac{\left(1-\pi_{P L}^{1 / 2} ight)(1-\varepsilon)}{\pi_{P L}^{1 / 2}} m_{P L}[/latex] (11.53)

[latex]m_{p 1}=\frac{\left(1-0.05^{1 / 2} ight) \cdot(1-0.15)}{0.05} \cdot 10,000=131,990 kg[/latex]
[latex]m_{p 2}=\frac{\left(1-0.05^{1 / 2} ight) \cdot(1-0.15)}{0.05^{1 / 2}} \cdot 10,000=29,513 kg[/latex]

The total empty mass, [latex]m_{E}=m_{E 1}+m_{E 2}[/latex], and the total propellant mass, [latex]m_{p}=m_{p 1}+m_{p 2}[/latex] 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.

Question 11.4: The following data are given mPL = 10,000 kg πPL = 0.05 ε = ...

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