System Analysis and Design

How does jet power scale?

Analysis of an existing system

• Assume system consists of:
  1. Structural mass
  2. Propelland mass
  3. Tank mass
  • These 3 are arbitrary metrics, and in actuality depend on our goal design
• Assume propellant is used for 2 sequential maneuvers:
  • $$\begin{array}{rl}{M}_{0,1}& =M_S+M_T+M_{P,1}+M_{P,2}+M_{P,\text{remaining}}\\ M_{0,2}& =M_{0,1}-M_{P,1}\\ M_{0,3}& =M_{0,2}-M_{P,2}\end{array}$$
  • Note that here we assume that the same thruster is used for both maneuvers (i.e. $u_1=u_2=u$)
• $$\begin{array}{rl}\Delta V_1& =u\log \left(\frac{M_{0,1}}{M_{0,2}}\right)\\ \Delta V_2& =u\log \left(\frac{M_{0,2}}{M_{0,3}}\right)\\ \Delta V_{\text{total}}& =\Delta V_1+\Delta V_2\\ & =\dots \\ & =u\log \left(\frac{M_{0,1}}{M_{0,3}}\right)\end{array}$$

Optimizing a system

• Construct $\Delta V$ as the function of known parameters:
  • $$\begin{array}{rl}\Delta V& =u\log \left(\frac{M_S+M_T+M_P}{M_S+M_T}\right)\\ \Delta V& =\mathrm{f}\left(u,M_S,M_T,M_P\right)\end{array}$$
  • $M_P$ is the total propellant mass
  • $\Delta V$ is a function of 4 parameters
• Assume $u$ is specified and not being optimized at this point
• Non-dimensionalize:
  • $$\frac{\Delta V}{u}=\log \left(\frac{\frac{M_S}{M_P}+\frac{M_T}{M_P}+1}{\frac{M_S}{M_P}+\frac{M_T}{M_P}}\right)$$
• Now we only have 2 parameters
• $\frac{M_T}{M_P}$ likely directly related
• $\frac{M_S}{M_P}$ might be related:
  • More tank and propellant
  • Assume not related for now
  • As modeled, other items (payload) would be a part of $M_S$
• For the tank:
  • Assume a thin-walled spherical pressure vessel
  • Factor of Safety: $F_S=1$ (for now)
  • $$\begin{array}{rl}\text{Pressure}\times \text{Projected Area}& =\text{Stress}\times \text{Material Area}\\ P(\pi R^2)& ={\sigma }_y(2\pi R)t{F}_S\\ ⟹t& =\frac{PR}{2{\sigma }_y}\end{array}$$
  • Volume:
    • Tank: $V_T=\frac{4}{3}\pi R^3$ ($t\ll R$)
    • Material: $V_M=4\pi R^{2}t=\dots =V_T\left(\frac{3P}{2{\sigma }_y}\right)$
  • Mass:
    • Propellant Density: ${\rho }_P$
    • Material Density: ${\rho }_M$
  • $$\begin{array}{rl}{M}_P& =V_T{\rho }_P\\ M_T& =V_M{\rho }_M\\ \frac{M_T}{M_P}& =\frac{3{\rho }_{M}P}{2{\rho }_P{\sigma }_y}\end{array}$$
  • Generally, would like:
    • $M_T$ small
    • $\frac{{\rho }_M}{\sigma }$ as small as possible (material selection)
    • For ideal gas: lower storage temperature
    • For two-phase: lower storage temperature & low quality (i.e. mostly liquid)

Let’s do it again

• Mass ratio: $MR$
  • Often defined in different ways
  • Sometimes as $>1$
    • $MR=\frac{M_f}{M_0}$
  • Sometimes as $<1$
    • $MR=\frac{M_o}{M_0}$
    • We use this one
• $\Delta V=u\log \left(MR\right)$
  • We know that $MR>0$, because $\Delta V>0$
• Now assume our system:
  1. Structure $M_S$: Fixed and we don’t care about it
  2. Payload $M_L$: What we want to deliver
  3. Propellant $M_P$: Consumable, changes with each maneuver
• Note that this is different from the previous example, so we have a different analysis goal
• Total Initial Mass:
  • $$M_0=M_S+M_P+M_L$$
• Final Mass:
  • $$M_f=M_S+M_P+M_L=M_0-M_P$$
  • All propellant consumed
• Thus:
  • $$MR=\frac{M_0}{M_S+M_L}=\frac{M_0}{M_0-M_P}$$
• Payload Ratio:
  • $$\lambda =\frac{M_L}{M_0-M_L}=\frac{M_L}{M_P+M_S}$$
• Structure Ratio:
  • $$ϵ=\frac{M_S}{M_0-M_L}=\frac{M_S}{M_P+M_S}$$
• Inert Mass Fraction:
  • $$\delta =\frac{M_{\text{in}}}{M_0}$$
• Payload Mass Fraction:
  • $$\lambda =\frac{M_L}{M_0}$$
• Mass Ratio:
  • $$r=\lambda +\delta =\frac{M_f}{M_0}$$
• Thus:
  • $$MR=\frac{1+\lambda }{ϵ+\lambda }=\frac{M_0}{M_f}$$