Optics 016 · How hot can sunlight make a tube

Solar Concentrator & Étendue

Why can’t any lens or mirror concentrate sunlight onto an arbitrarily small spot? Because no passive optical system can shrink étendue, the beam's area multiplied by its spread of directions. Squeeze the light onto a smaller spot and it must arrive from a wider range of angles, and the angles eventually run out. The highest possible concentration is set by the refractive index and the acceptance angle, with different limits for trough-shaped and dish-shaped concentrators.

Aim a parabolic trough or a still CPC at the sun, shrink the absorber, and see why concentration stops at a limit set by the sun’s size.

Aperture width over absorber size $\text{—}$

Share of the sunlight reaching the absorber $\text{—}$

Concentration delivered, against the limit for this light $\text{—}$

Absorber temperature with no heat taken away $\text{—}$

Average reflections before reaching the absorber $\text{—}$

Focal length and tube diameter, or CPC height and absorber width $\text{—}$

Share of all the absorber could take in that the sunlight fills $\text{—}$

Limit for a dish that concentrates in both directions $\text{—}$

Rays traced for this record $\text{—}$

Optics 016

Why can’t any mirror concentrate sunlight onto an arbitrarily small spot?

Compound parabolic concentratorSwap the tracking trough for a CPC that sits still.Half-angle the absorber accepts

\theta_{\mathrm{a}}

0.75^{\circ},\ d = 70\,\mathrm{mm}

Rim angle of the trough

\phi

68.4^{\circ}

Angular radius of the light: the sun, widened by mirror errors

\delta

0.27^{\circ}

Sun’s angle from the collector’s axis

\varepsilon

0^{\circ}

A smaller absorber raises the concentration only until it starts to miss the sun’s image; no design passes the limit set by the sun’s size.

The Lab traces sunlight in the cross-section of a long collector, so every ray stays in one plane and the limits are the two-dimensional ones; a dish, which concentrates in both directions, appears only as a readout. In the cross-section the sunlight is spread evenly across the sun’s angular radius of 0.266°, from the IAU 2015 nominal solar radius and the astronomical unit, and the sun’s effective temperature is 5772 K; a real disc seen this way is brightest in the middle, and the real sun is also darker towards its edge. The slider for the light’s angular radius widens this disc to stand in for mirror and tracking errors. Mirrors reflect perfectly, the absorber is black, and there is no glass envelope, air or heat loss, so the temperature shown is the stagnation temperature of a black absorber that radiates away exactly what it absorbs (Rabl 1975). The default trough is the LUZ LS-2 module tested at Sandia: a 5 m aperture, 1.84 m focal length and a 70 mm tube (Dudley et al. 1994). The CPC’s walls follow Rabl’s construction and reproduce the height and reflector length of his Table II. Sunlight is traced forward with 4000 rays; the profile and the map along the absorber are traced backward from 48 cells, which can misplace a few per cent of the light when the sun sits exactly on a CPC’s acceptance edge.

Physics tutorial

Concentrate sunlight and find where it stops

BackgroundA solar thermal plant heats a fluid with concentrated sunlight. A curved mirror gathers the light falling on a wide opening, the aperture, onto a much smaller absorber, a black tube or plate. The concentration says how many times brighter the light on the absorber is than sunlight falling straight on a plate.

Why it mattersThe hotter the absorber gets, the better a heat engine runs on it, so designers want to squeeze the light onto as small an absorber as they can. The sun is not a point, though: seen from Earth its disc is 0.53° across, and that width decides how far the light can be squeezed.

Start with the essentials

Focus question: The LS-2 trough gathers sunlight across 5 m onto a 70 mm tube, 22.7 times concentrated. Why can no mirror of any shape push sunlight in a trough past 215 times, and what does a thinner tube actually buy?
One-sentence intuition: Light entering an aperture of width $W$ from within $\pm\delta$ of the sun’s centre has an étendue, its width times its spread of directions, of $2W\sin\delta$ . Mirrors can trade width for spread but can never shrink the product. An absorber of size $A$ takes in light from at most every direction above it, an étendue of $2A$ , so $C=W/A\le1/\sin\delta$ , 215 for the sun. A tube smaller than the sun’s image simply misses the light from the sun’s edges. A dish at its own limit makes every point of the absorber see the sun in every direction, and the absorber reaches the sun’s temperature; it cannot get hotter.

Core mathematical model

The étendue limit

\begin{gathered}2W\sin\delta\le2A\\C=\frac{W}{A}\le\frac{1}{\sin\delta}\approx215\\C_{\mathrm{dish}}\le\frac{1}{\sin^{2}\delta}\approx46\,200\end{gathered}

Sunlight through the aperture $W$ spreads over $\pm\delta$ ; the absorber $A$ can accept at most every direction on its side, $\pm90^{\circ}$ . Mirrors keep the light’s étendue fixed and the absorber can take in no more than its own, which gives the limit $1/\sin\delta$ in two dimensions. A dish, focusing in both directions, gets $1/\sin^{2}\delta$ (Rabl 1975, Eqs. II-14 and II-15).

The sun’s size

\begin{gathered}\sin\delta=\frac{R_{\odot}}{1\,\mathrm{au}},\qquad\delta=0.266^{\circ}\\E_{\odot}=\sigma T_{\odot}^{4}\sin^{2}\delta=1361\,\mathrm{W/m^{2}}\end{gathered}

The IAU’s 2015 nominal solar radius, 695 700 km, seen from 1 au gives the sun’s angular radius $\delta$ . A black surface facing a sun of effective temperature $T_{\odot}=5772\,\mathrm{K}$ receives the irradiance $E_{\odot}$ , which matches the IAU’s nominal $1361\,\mathrm{W/m^{2}}$ : the disc’s size and its brightness agree.

A trough with a tube

\begin{gathered}d=2r\sin\delta,\qquad W=2r\sin\phi\\C=\frac{W}{\pi d}=\frac{\sin\phi}{\pi\sin\delta}\end{gathered}

The mirror’s rim, at distance $r$ from the focus and rim angle $\phi$ , makes the largest image of the sun, so the tube must be at least $d$ across to catch it. A tube absorbs, and loses heat, over its whole circumference $\pi d$ , so the concentration is aperture over circumference. It peaks at $\phi=90^{\circ}$ at $1/(\pi\sin\delta)=68.4$ (Rabl 1975, Eq. VI-1).

A compound parabolic concentrator

\begin{gathered}C=\frac{W}{2a'}=\frac{1}{\sin\theta_{\mathrm{a}}}\\L=\left(\frac{W}{2}+a'\right)\cot\theta_{\mathrm{a}}\end{gathered}

Each wall is a piece of a parabola whose focus is the opposite end of the flat absorber $2a'$ and whose axis is tilted by the acceptance half-angle $\theta_{\mathrm{a}}$ . Every ray entering within $\pm\theta_{\mathrm{a}}$ reaches the absorber and every other ray goes back out, so the CPC reaches the limit for that acceptance. Its height $L$ and wall length match Rabl’s Table II.

How hot the absorber gets

T=T_{\odot}\left(C\sin^{2}\delta\right)^{1/4}

A black absorber that loses heat only by radiating warms until it radiates $\sigma T^{4}$ as fast as it absorbs $C E_{\odot}$ . This stagnation temperature is 394 K for a bare plate facing the sun, 860 K for the LS-2 and 1507 K at the trough limit. At the dish limit, $C\sin^{2}\delta=1$ , it reaches the sun’s own 5772 K (Rabl 1975, Eq. II-23).

Common difficulties

A bigger mirror does not make a hotter spot

Typical misconceptionA large enough mirror could make its focus hotter than the sun.

Better mental modelThe sun’s angular size caps the concentration, whatever the mirror’s size. At the dish limit the absorber sees the sun in every direction and reaches 5772 K, the sun’s temperature. If it got hotter, heat would be flowing on its own from the cooler sun to the hotter absorber, which the second law forbids.

Aperture over absorber is not the concentration

Typical misconceptionDividing the aperture by the absorber size gives the concentration.

Better mental modelOnly when all the light lands. The 16.9 mm tube has a ratio of 94.2 but catches 87 % of the light, so it delivers 81.8.

A tube cannot reach the limit

Typical misconceptionA perfect parabolic trough reaches the étendue limit.

Better mental modelA trough focusing onto a tube reaches at most $1/\pi$ of it, 68.4 against 215, because each point of the tube sees the sun in only a narrow band of directions. Reaching the limit takes a non-imaging design such as a CPC, which at $\pm30^{\circ}$ gives exactly 2.00, the limit for that acceptance.

Run the experiment

01
Watch the light reach the tube
Keep the default preset, LS-2 trough on target: all the light caught.
What to observe: Every drawn ray lands on the 70 mm tube. Aperture over absorber, $5\,\mathrm{m}/(\pi\cdot70\,\mathrm{mm})$ , is 22.7, and all the sunlight arrives, so 22.7 is also the concentration delivered, against the 215 the sun allows. With no heat taken away the tube would reach 860 K.
02
Let the tracking slip
Drag the sun sideways, or load Tracking slips: light misses the tube.
What to observe: Nothing is lost until the sun is about 0.5° off the axis. At 0.75°, 15 % of the light passes beside the tube, drawn in grey, and the concentration falls to 19.4. At 1° only 42 % arrives. The acceptance curve in the data row shows the drop.
03
Shrink the tube
Load Thinner tube: hotter, but light is lost, or lower the acceptance half-angle.
What to observe: A 16.9 mm tube catches only 87 % of the light but concentrates what it catches 81.8 times, so it would reach 1184 K; aperture over absorber says 94.2. Shrinking it to 14.1 mm loses more light and gains nothing: in the sweep chart the delivered curve flattens near 82, 38 % of the limit, while aperture over absorber keeps rising.
04
Open the rim to 90°
Load The best a trough can do.
What to observe: A 90° rim with a 23.3 mm tube, just wide enough for the sun’s image from the rim, catches all the light and delivers 68.4, which is $215/\pi$ . In the map of directions each point of the tube sees the sun in only a narrow band; the band fills 32 % of the map, so the trough uses only that share of what the tube could take in.
05
Stop tracking
Load A still CPC catches the sun 20° off.
What to observe: The CPC accepting $\pm30^{\circ}$ catches all the light from a sun 20° off its axis, after 0.72 reflections on average, and delivers 1.88; with the sun on its axis it delivers 2.00, the limit for light spread over $\pm30^{\circ}$ . Half a degree past 30° it catches nothing. It gives up most of the 215 so that it can sit still while the sun moves, and reaches only 461 K.
06
Blur the sun
Return to the default and raise the angular radius of the light to 1°.
What to observe: Light spread over 1°, as from a mirror with a slight figure error, sets a limit of 57.3 instead of 215. The 70 mm tube now catches 93 % and delivers 21.2. At 3° the limit is 19.1 and the tube catches a third of the light.

Solar Concentrator & Étendue

Concentrate sunlight and find where it stops

Core mathematical model

The étendue limit

The sun’s size

A trough with a tube

A compound parabolic concentrator

How hot the absorber gets

Common difficulties

A bigger mirror does not make a hotter spot

Aperture over absorber is not the concentration

A tube cannot reach the limit

Run the experiment

Watch the light reach the tube

Let the tracking slip

Shrink the tube

Open the rim to 90°

Stop tracking

Blur the sun

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