Exercises: Life Support and Human Spaceflight

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, and the per-crew-day planning figures from the chapter (oxygen $0.84\ \text{kg}$, potable water $2.5\ \text{kg}$, food $1.8\ \text{kg}$, so a supplied total of $5.14\ \text{kg}$; carbon dioxide out $\approx 1.0\ \text{kg}$). Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold first. For "implement it" problems, do not run the code: hand-trace it and record the result in an # Expected output: comment, exactly as the chapter does.

Part A — Warm-ups: tally and convert (⭐)

28.1 † How much oxygen must be supplied for a crew of $3$ on a $30$-day mission (open-loop)?

28.2 How much potable water does a crew of $6$ drink over one year ($365$ days)? Give the answer in tonnes.

28.3 † What is the total supplied-consumables mass per crew member for a $500$-day mission with no recycling?

28.4 A cabin is held at $100\ \text{kPa}$ total pressure with a $24\%$ oxygen fraction. What is the partial pressure of oxygen $p_{\text{O}_2}$? Is that comfortable, hypoxic, or a fire-risk concern?

28.5 † How much carbon dioxide does a crew of $4$ produce in one week, and why does this quantity, rather than the cabin volume, set how hard the scrubber must work?

28.6 A centrifuge habitat spins at $\omega = 0.40\ \text{rad/s}$ with the crew at radius $r = 40\ \text{m}$. What artificial gravity does the floor provide, in $\text{m/s}^2$ and in units of $g$?

Part B — Loop closure and chemistry (⭐⭐)

28.7 † Balance the Sabatier reaction $\text{CO}_2 + \text{H}_2 \rightarrow \text{CH}_4 + \text{H}_2\text{O}$ (fill in the coefficients). State how many molecules of hydrogen it needs per molecule of carbon dioxide, and use that number to explain in one sentence why a Sabatier-based system recovers only about half the crew's oxygen.

28.8 Oxygen is regenerated by electrolysis, $2\,\text{H}_2\text{O} \rightarrow 2\,\text{H}_2 + \text{O}_2$. Using molar masses (O$_2$ = $32\ \text{g/mol}$, H$_2$O = $18\ \text{g/mol}$), how much water must be split to make one crew member's daily $0.84\ \text{kg}$ of oxygen? How much hydrogen is produced?

28.9 † A water recovery system is upgraded from $90\%$ to $98\%$ recovery. For a crew of $4$ on a $500$-day mission, how much makeup water mass does the upgrade save? (Use potable water $2.5\ \text{kg}$ per crew-day.)

28.10 The Bosch reaction $\text{CO}_2 + 2\,\text{H}_2 \rightarrow \text{C} + 2\,\text{H}_2\text{O}$ can close the oxygen loop nearly completely where Sabatier reaches only $\sim 50\%$. Explain, in terms of where the hydrogen ends up in each reaction, why Bosch closes the loop better — and name the operational penalty that has kept it off flown spacecraft.

Part C — Implement it in Python (⭐⭐)

Write each function, hand-trace it for the given input, and record the result in an # Expected output: comment. Do not run it.

28.11 † Write consumables(crew, days, rate=5.14) returning the open-loop supplied mass in kg. Trace consumables(4, 180).

28.12 Write breakeven(hw_mass, crew, open_rate=5.14, closed_rate=2.47) returning the mission day at which closed-loop mass undercuts open-loop for a fixed crew. Trace breakeven(1500, 4).

28.13 † Write ag_rpm(r, a=9.81) returning the spin rate in rpm needed to produce acceleration a at radius r. (Recall $\omega = \sqrt{a/r}$ and $\text{rpm} = \omega\cdot 60/2\pi$.) Trace ag_rpm(50).

Part D — Find the error (⭐⭐)

28.14 † A systems engineer writes: "Our Sabatier reactor consumes all of the crew's exhaled CO$_2$, so it recovers $100\%$ of their oxygen." Identify the flaw and give the correct approximate recovery, with the reason.

28.15 A design proposes a $1\,g$ centrifuge with the crew at radius $r = 5\ \text{m}$, and claims it will be "perfectly comfortable." Compute the required spin rate, the Coriolis acceleration on a hand moving at $1\ \text{m/s}$, and the gravity gradient over a $1.8\ \text{m}$ standing height. Is the claim credible?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

28.16 † (Sizing.) A crew of $6$ flies a $400$-day mission. (a) Find the open-loop supplied consumables mass. (b) Find the closed-loop resupply mass assuming food is not recycled ($1.8\ \text{kg}$ per crew-day), water is recovered at $90\%$, and oxygen at $50\%$, on top of a fixed recycler hardware mass of $2{,}500\ \text{kg}$. (c) Which architecture is lighter, and by how much? (d) Which single commodity now dominates the closed-loop mass, and what does that imply for future systems?

28.17 (Mission project.) Write your mission's crew note. If your track (A/B/C/D) is uncrewed, state that ECLSS is not applicable and write one sentence naming what human-rating would add to your MDR budgets. If you are sketching a crewed variant, size its consumables for your crew and duration and record the open- or closed-loop decision.

28.18 † (Storm shelter.) You want a shelter wall of areal density $20\ \text{g/cm}^2$ over $8\ \text{m}^2$ of surface, built from the mission's water. (a) How much water mass does that require? (b) If the mission stores well over that mass of water, argue why the shelter's shielding is "nearly free" in launch mass. (c) Food can also serve as shielding — name one drawback of relying on it over the course of a long mission.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

28.19 † Why can't you just shield the whole spacecraft against galactic cosmic rays as thoroughly as a storm shelter? Frame your answer with a mass argument (how shielding mass scales) and one physics point about GCR.

28.20 (Back of the envelope.) A $1\,g$ rotating habitat at a comfortable $2\ \text{rpm}$ needs a radius of about $224\ \text{m}$. Estimate the circumference of the ring, and compare it to the ISS (whose main truss is about $109\ \text{m}$ long). What does the comparison say about building artificial-gravity stations today?

28.21 † Why can't you just fill the cabin with pure oxygen at one atmosphere to simplify the atmosphere system (no nitrogen to carry, no partial-pressure bookkeeping)? Give the two reasons the chapter raises, and the historical event that settled the question.

Part G — Interleaved & synthesis (Chapters 1, 3, 24, 25) (⭐⭐ / ⭐⭐⭐)

28.22 † (Ch. 25.) An oxygen-generation assembly draws a continuous $1.0\ \text{kW}$. How many kilowatt-hours does it consume over a $180$-day mission? Explain in one sentence why this figure feeds directly into the power-budget and solar-array sizing of Chapter 25.

28.23 (Ch. 24.) The crew and their electronics dissipate $2.0\ \text{kW}$ of heat into the cabin. In vacuum this can only leave by radiation. Using $P = \varepsilon\sigma A T^4$ with $\varepsilon = 0.85$, $\sigma = 5.67\times10^{-8}\ \text{W/m}^2\text{K}^4$, and a radiator temperature $T = 300\ \text{K}$, estimate the radiator area needed. Why does this tie the life-support heat load directly to the thermal-control chapter?

28.24 † (Ch. 3.) A Mars mission would need about $18.5\ \text{t}$ of open-loop consumables, but only about $11\ \text{t}$ with recycling. Explain, invoking the rocket equation, why "saving $7.5\ \text{t}$ of consumables" actually saves much more than $7.5\ \text{t}$ of launched mass.

28.25 (Ch. 1.) Chapter 1 listed radiation and microgravity among the reasons space is hard. Name one countermeasure this chapter gives for each, and state in one sentence why neither fully solves the problem.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: correct set-up of the mass or rotation model, explicit units, a stated Tier for any assumed figure, and a sanity check on the final number.