The hottest a sunlit receiver can get is where the sunlight it absorbs equals the heat it radiates, T = (efficiency × concentration × 1,000 W/m² ÷ 5.67 × 10⁻⁸)^(1/4), which for an ideal black receiver is 91°C with no concentration, 580°C at 30x, 879°C at 100x and 1,776°C at 1,000x. No mirror can concentrate beyond about 46,000x, because the sun is half a degree wide, and a real receiver delivers useful power only far below these stagnation temperatures.
Part of Concentrating Optics
The concentration ratio is the mirror area divided by the receiver area; the receiver temperature follows from balancing absorbed sunlight against radiated heat.
The Energy Balance Sets A Ceiling
A receiver of area A_r takes in η × C × G × A_r, where G is the beam power per square metre (1,000 W/m² here, the parent page’s peak figure), C is the concentration ratio and η is the optical efficiency of the mirror (reflectance, coverage, accuracy). It radiates σT⁴ × A_r, where σ = 5.67 × 10⁻⁸ W/m²·K⁴ (Stefan-Boltzmann constant, standard physical value). Taking absorptivity equal to emissivity (a black receiver), the limit is reached when the two are equal:
T = (η × C × G ÷ σ)^(1/4)
This ignores convection and conduction, so it is an upper bound on temperature.
Worked Example: The Table
Compute (η = 1; and η = 0.8 for a mirror that reflects and covers 80% of the beam):
| C | T at η = 1 | T at η = 0.8 |
|---|---|---|
| 1 | 364 K = 91°C | 345 K = 71°C |
| 10 | 648 K = 375°C | 613 K = 340°C |
| 30 | 853 K = 580°C | 807 K = 533°C |
| 100 | 1,152 K = 879°C | 1,090 K = 817°C |
| 300 | 1,517 K = 1,243°C | 1,434 K = 1,161°C |
| 1,000 | 2,049 K = 1,776°C | 1,938 K = 1,665°C |
One line in detail, C = 100, η = 1: 100 × 1,000 ÷ 5.67 × 10⁻⁸ = 1.76 × 10¹² K⁴; the fourth root is 1,152 K; subtracting 273 gives 879°C.
The parent page’s table (flat plate 80-100°C; 30-80x 250-400°C; 100-300x 400-800°C; 500-1,000x 800-1,500°C) lies below these limits in every row, as it should. The gap between the limit and the quoted value is the convection and conduction loss, which this model leaves out.
How Far Can Concentration Go?
The sun’s disc has an angular radius θ of about 0.267° (an angular diameter of about 0.53°, a standard astronomical value). Conservation of the beam’s angular spread (the etendue argument of non-imaging optics, a textbook result, recalled) caps the concentration:
- Line focus (trough): C_max = 1 ÷ sin θ = 1 ÷ 0.00465 = 215.
- Point focus (dish): C_max = 1 ÷ sin²θ = 1 ÷ (0.00465)² = 46,200, about 46,000.
At C = 46,000 and η = 1, the formula gives T = 5,337 K, just below the sun’s surface temperature of about 5,772 K, which is a consistency check: no receiver can get hotter than the source. The parent page’s 30-80x for a trough is far below 215, and a salvaged tiled dish is far below 46,000, because slope errors and tile size (not the sun’s width) set the spot (Mirror Tile Parabola).
Stagnation Is Not Delivered Power
A receiver that is delivering heat to a load must be cooler than its stagnation temperature. The fraction of absorbed power lost by radiation alone is σT⁴ ÷ (η × C × G). Inputs: C = 100, η = 0.8, so absorbed flux = 80,000 W/m².
- T = 500°C (773 K): σT⁴ = 5.67 × 10⁻⁸ × 773⁴ = 20,260 W/m²; loss fraction 0.25, so delivered fraction 0.75.
- T = 700°C (973 K): σT⁴ = 50,850 W/m²; delivered fraction 0.36.
- T = 800°C (1,073 K): σT⁴ = 75,200 W/m²; delivered fraction 0.06.
At 100x, then, the receiver is a good heat source at 500°C and a poor one at 800°C. To work at higher temperature you raise C. This is the arithmetic behind the rule that process temperature demands concentration. Real receivers lose more by convection, so use these fractions as optimistic.
Method: Estimate For Your Dish
Step 1: Measure the spot with a card at the focus (Accuracy & Focus Testing) and compute C = mirror area ÷ spot area.
Step 2: Choose η from a power measurement, or 0.6 as a first guess.
Step 3: Compute the stagnation limit and the loss fraction at your working temperature.
Step 4: If the loss fraction exceeds about a third, increase C, shrink the spot, or insulate the receiver.
Common Mistakes
- Reading the stagnation temperature as the working temperature.
- Counting concentration from mirror area alone. Only the energy inside the receiver counts.
- Forgetting that the table is an upper bound. Convection and conduction reduce every figure.
Safety
The numbers describe a surface hot enough to ignite and melt, in light that blinds
A receiver at several hundred degrees ignites wood and burns on a touch; a concentrator that produces the temperatures above also focuses enough light to blind permanently and painlessly. Keep the focus covered when idle and never look at it.
What’s Next
- Learn next: Accuracy & Focus Testing — measuring C for a real dish
- Combine with: The Greenhouse Effect in Collectors — the same balance at low concentration