Chapter 24 — Exercises: Thermal Control

These problems build the reflex every thermal engineer needs: reach for the radiative balance, put units on everything, and sanity-check the number against a body you can picture. Work in SI (kelvin for temperature; convert to $^\circ\text{C}$ only to report), and keep the constants $S = 1361\ \text{W/m}^2$ and $\sigma = 5.670\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$ close at hand.

Difficulty tiers. ⭐ builds a single idea with given numbers; ⭐⭐ combines two or three steps and asks for a sanity check; ⭐⭐⭐ is open-ended, design-flavored, or spans several chapters.

Solutions policy. Problems marked † and all odd-numbered problems have full worked solutions in appendices/answers-to-selected.md. Try them completely before looking — a thermal answer you did not struggle for will not survive a real design review.

Part A — Calculate it (real numbers)

A1. ⭐ † Compute the solar flux at Venus ($0.723\ \text{AU}$) and at Saturn ($9.537\ \text{AU}$). By what factor does the flux at Venus exceed that at Saturn?

A2. ⭐ A grey isothermal sphere ($\alpha = \varepsilon$) sits at Jupiter's distance ($5.203\ \text{AU}$). Find its equilibrium temperature. (Hint: it scales from the $278\ \text{K}$ 1-AU value as $d^{-1/2}$.)

A3. ⭐⭐ † A flat plate faces the Sun at $1\ \text{AU}$, absorbing on one face and radiating only from that same face. It is coated with white paint, $\alpha = 0.20$, $\varepsilon = 0.90$. Find its equilibrium temperature in $^\circ\text{C}$. Then repeat for bare polished aluminum ($\alpha = 0.15$, $\varepsilon = 0.05$) and comment on which you would put on a sun-facing surface you want to keep cool.

A4. ⭐⭐ A spacecraft must reject $750\ \text{W}$ of waste heat from a radiator running at $T = 310\ \text{K}$ with $\varepsilon = 0.85$, pointed at deep space (no absorbed sunlight). What radiator area is needed? If the allowable radiator temperature were raised to $350\ \text{K}$, what area would it need instead — and by what factor did the area shrink?

A5. ⭐⭐ † A battery box is held at $278\ \text{K}$ ($+5\ ^\circ\text{C}$) in eclipse. It presents $0.15\ \text{m}^2$ to deep space through MLI of effective emittance $\varepsilon^{*} = 0.04$. What heater power is required? If the same box lost its blanket and radiated as bare anodized aluminum ($\varepsilon = 0.76$), what heater power would it then demand?

A6. ⭐⭐⭐ A cubical electronics box, $0.3\ \text{m}$ on a side, floats far from any planet at $1\ \text{AU}$, dissipating $40\ \text{W}$ internally. One face (area $0.09\ \text{m}^2$) faces the Sun; all six faces radiate. The whole box is painted white ($\alpha = 0.20$, $\varepsilon = 0.90$) and is isothermal. Find its equilibrium temperature. (Set absorbed = emitted, with $A_{\text{sun}} = 0.09\ \text{m}^2$ and $A_{\text{rad}} = 6 \times 0.09 = 0.54\ \text{m}^2$, and don't forget the internal power.)

A7. ⭐⭐ † A downward-facing (nadir) panel of area $0.5\ \text{m}^2$ in low Earth orbit sees Earth filling its view. It absorbs albedo (with $\alpha = 0.20$) at a peak flux of $400\ \text{W/m}^2$ and Earth infrared (which it absorbs with its IR emissivity $\varepsilon = 0.88$) at $237\ \text{W/m}^2$. Compute the total power the panel absorbs from Earth, then find the equilibrium temperature at which its own emission (from that one face) balances the load. What does the result tell you about mounting a radiator on the nadir face?

Part B — Simulate it (write Python; do not run it)

Write clean Python and give the answer as a hand-computed # Expected output: comment. Never execute it.

B1. ⭐ † Write a function t_eq(alpha, eps, a_sun, a_rad, flux=1361.0, q_int=0.0) returning the radiative equilibrium temperature in kelvin. Call it for the grey sphere (1, 1, 1, 4) and the black 1-sided plate (1, 1, 1, 1), printing each in $^\circ\text{C}$.

B2. ⭐⭐ Extend thermal.py with radiator_area(q_reject, t_rad, eps=0.85, alpha=0.0, flux=0.0) that returns the area (m²) to reject q_reject watts at t_rad kelvin, accounting for absorbed sunlight alpha*flux. Use it to reproduce the two radiator sizes of Section 24.4 (shaded, and white in sunlight).

B3. ⭐⭐ † Write a loop that tabulates the solar flux and grey-sphere equilibrium temperature for Mercury, Earth, Mars, and Neptune (distances $0.387$, $1.000$, $1.524$, $30.07\ \text{AU}$). Print each temperature in kelvin and $^\circ\text{C}$.

B4. ⭐⭐⭐ Write a simple time-stepping simulation of the eclipse cooldown of Section 24.6 that, unlike the linear estimate, updates the radiated power each step as $\varepsilon\sigma A T^4$. Step a $35$-minute eclipse in $60\ \text{s}$ increments for the component in that example ($m = 10\ \text{kg}$, $c_p = 900$, $A = 0.3\ \text{m}^2$, $\varepsilon = 0.8$, $T_0 = 290\ \text{K}$, $Q_{\text{int}} = 5\ \text{W}$) and report the final temperature. Explain why it is warmer than the linear estimate's $268.7\ \text{K}$.

Part C — Design it (Mission Design Checkpoint work)

C1. ⭐⭐ † For your chosen mission track, write the three-part thermal note described in this chapter's Mission Design Checkpoint: (a) your solar flux from the inverse-square law; (b) the hot-case and cold-case equilibrium temperature of a representative radiator surface ($\alpha = 0.2$, $\varepsilon = 0.85$, area your choice) with a stated internal power; (c) the resulting heater power and radiator area. State every assumption.

C2. ⭐⭐⭐ A Track-C Mars orbiter and a Track-A GEO comsat carry the same electronics box dissipating $100\ \text{W}$. Size a sun-shaded radiator for each (choose reasonable radiator temperatures) and explain physically why the Mars orbiter's radiator, despite the far weaker sunlight at Mars, is not proportionally smaller. What actually sets the radiator size?

C3. ⭐⭐⭐ Your spacecraft's power budget (previewing Chapter 25) is tight, and the thermal team is asking for $25\ \text{W}$ of survival heaters in eclipse. Propose two design changes that would reduce that heater demand without adding heater power, and estimate the effect of one of them quantitatively. (Think about what sets the heat loss.)

Part D — Why can't you just… (conceptual)

D1. ⭐ † Why can't you just cool an overheating spacecraft component by "running a fan over it," the way you would cool a laptop? What is physically absent?

D2. ⭐ Why can't you just paint the whole spacecraft black so it "radiates heat away efficiently"? What does black paint do to the absorbed side of the balance?

D3. ⭐⭐ † Why can't a radiator that works perfectly on the shaded side of a spacecraft simply be moved to a sunlit side "since space is cold everywhere"? Quantify the penalty for a black radiator ($\alpha = 0.95$) at $300\ \text{K}$ forced to face the Sun at $1\ \text{AU}$.

D4. ⭐⭐ Why can't you reach cryogenic temperatures on an infrared telescope with insulation alone — why is a sunshield plus, ultimately, a cryocooler needed to get to a few kelvin?

D5. ⭐ † Gold foil reflects most of the sunlight that hits it ($\alpha = 0.25$), far more than black paint ($\alpha = 0.95$). Why, then, does a gold-foiled surface run hotter in sunlight than a black one? What is gold foil actually good for on a spacecraft, if not keeping things cool?

Part E — Back of the envelope (order of magnitude)

E1. ⭐ † Estimate, to one significant figure, the equilibrium temperature of a fast-spinning grey asteroid at $2.5\ \text{AU}$. Is your answer above or below the freezing point of water?

E2. ⭐⭐ A $1\ \text{m}^2$ black surface faces the Sun at Earth's distance and can radiate from that one face only. Without a calculator, argue that its temperature is "a bit over $100\ ^\circ\text{C}$" using only the fact that a blackbody at $\sim 394\ \text{K}$ radiates about $1361\ \text{W/m}^2$.

E3. ⭐⭐ † A cryogenic tank of liquid oxygen ($h_{fg} \approx 213\ \text{kJ/kg}$) suffers a $10\ \text{W}$ heat leak. Estimate the boiloff in kg/day and comment on whether a week-long loiter is feasible for a tank holding $50\ \text{kg}$.

E4. ⭐⭐ Using the ideal MLI relation $\varepsilon^{*} \approx 1/[(N+1)(2/\varepsilon - 1)]$ with a per-layer emissivity $\varepsilon = 0.03$, show that even a modest $N = 10$ shields predicts an ideal effective emittance far below the $0.01$–$0.05$ that flown blankets actually achieve. What does that gap tell you about what really limits MLI performance — the physics of the layers, or the workmanship of the installation?

Part F — Find the error

F1. ⭐⭐ † An engineer computes the equilibrium temperature of a sunlit spherical satellite and writes: "The sphere radiates from $4\pi r^2$ and absorbs on $4\pi r^2$, so $T = (S/\sigma)^{1/4} = 394\ \text{K}$." Diagnose the mistake and give the correct temperature.

F2. ⭐⭐ A student sizes a radiator for the hot case using the spacecraft's launch-day white paint, $\alpha = 0.20$. A reviewer rejects the analysis. Why? What value of $\alpha$ should the hot case use, and which direction does the error push the predicted temperature?

F3. ⭐⭐⭐ † A design memo claims: "Our MLI blanket has 20 layers, so from $q = \sigma(T_h^4 - T_c^4)/[(N+1)(2/\varepsilon - 1)]$ its effective emittance is $0.0012$; we will size the survival heaters on that basis." Identify the physical reason this is optimistic and state a defensible $\varepsilon^{*}$ to design with, explaining which way the heater sizing errs if the ideal value is used.

Part G — Interleaved (mixing earlier chapters)

G1. ⭐⭐ † (with Ch. 7) Contrast the heat load on a re-entry heat shield with the heat load on an orbiting radiator: name the source, the rough magnitude, and the timescale of each, and state why one problem is solved with ablative material and the other with paint and radiators.

G2. ⭐⭐ (with Ch. 23) A radiator panel is added to a spacecraft whose mass budget is already at its limit. Frame the radiator as an instance of theme #4 (mass is the enemy): what is being traded, and why can the mass not simply be "found" elsewhere?

G3. ⭐⭐⭐ † (with Ch. 3 and Ch. 22) A cryogenic upper stage loses liquid hydrogen to boiloff at $0.97\ \text{kg/day}$ (Section 24.5) while it loiters in orbit awaiting its burn. If the stage carries $120\ \text{kg}$ of usable LH2 for a $v_e = 4{,}400\ \text{m/s}$ engine and a $2{,}000\ \text{kg}$ dry-plus-payload mass, estimate how much delta-v the stage loses per day of loiter. (Use the rocket equation; treat the boiloff as a slow reduction in propellant.)

G4. ⭐⭐ (with Ch. 9) Explain why a sun-synchronous "dawn–dusk" orbit is thermally special: how does keeping the orbit plane near the terminator change the eclipse fraction and the solar load, and why is that attractive for both thermal and power design?

G5. ⭐⭐ † (previewing Ch. 25) Your survival heaters draw $25\ \text{W}$ during each $35$-minute eclipse of a $90$-minute orbit. (a) How much energy (in watt-hours) do the heaters alone demand from the battery each eclipse? (b) What average power, spread over the whole orbit, does that represent? (c) In one sentence, explain how this thermal load enlarges the battery you will size in Chapter 25 — the direct coupling between the thermal and power subsystems.

A note on checking your work

Every temperature you compute should pass three tests before you trust it: (1) units — did you work in kelvin and only convert to $^\circ\text{C}$ at the end? (2) sign of sanity — is a sunlit low-emissivity surface hotter than a shaded high-emissivity one, as it must be? (3) the sphere benchmark — does your answer sit sensibly relative to the $278\ \text{K}$ grey-sphere-at-1-AU anchor, scaled for distance and $\alpha/\varepsilon$? An answer that fails any of these is telling you to look again before you build.