Heliostat Aiming Geometry

What is modelled

A flat mirror sends the sun to a fixed point if its normal bisects the unit vector to the sun and the unit vector to the receiver. The effective area shrinks with the cosine of the incidence angle. A small tracking error spreads the reflected spot, and the part of the spot outside the receiver is spilled.

Equations

• Declination δ = 23.45° sin(360°(284+n)/365); elevation sin E = sinφ sinδ + cosφ cosδ cos h.
• Mirror normal n = (s + t)/|s + t|; efficiency cosθ = s·n = √((1 + s·t)/2).
• A mirror aim error of σ moves the reflected ray by 2σ; spot spread σimg = 2 d σ /√2 per axis estimate; intercepted fraction 1 - e-r²/(2σ²).
• Day length from cos h0 = -tanφ tanδ.

Checks run in the page

• Equal incidence and reflection angles, and the half-angle identity.
• Sunrise by bisection against the closed form.
• Spillage from 200000 random aim errors against the analytic fraction.
• Maximum declination and a daily integral by two quadrature rules.

Honest simplifications

A flat mirror, a point receiver direction, a circular receiver facing the mirror, no atmospheric attenuation, no blocking or shading between mirrors and the sun treated as a point. Not a plant design tool.

• Press Winter morning and watch the cosine efficiency and day length fall.
• Press Far mirror: the spot spreads and the receiver spills more light.
• Press Poor tracking and enlarge the receiver to win the light back.