24-Bld-A4 Building Environmental Control Systems · May 2017

Question 3 of 14: Flood-Lamp Illuminance Cone

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams May 2017 — 07-Bld-A4, Building Environmental Control Systems (Building Engineering). 3-hour closed-book exam: Section 1 (four 20-mark essay/calculation questions) + Section 2 (ten 2-mark multiple-choice questions).

Reference texts: ASHRAE Handbook — Fundamentals (2021 ed.); ASHRAE Standard 62.1, Ventilation for Acceptable Indoor Air Quality (2022 ed.); McQuiston, Parker & Spitler, Heating, Ventilating, and Air Conditioning: Analysis and Design (6th ed.); National Building Code of Canada 2020 (NBCC); National Energy Code of Canada for Buildings 2020 (NECB).

Question 1.3: Flood-Lamp Illuminance Cone (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Lamp power 45 W; luminous efficacy 15 lm/W; full beam angle 45° (half-angle 22.5°); mounting height 4 m above the floor, aimed straight down; assume uniform luminous intensity within the beam cone. Display diameter to be "just within the cone": 1.8 m.

Find. The beam-spot diameter D and peak (on-axis) illuminance E at each 1 m interval from the lamp down to the floor, and the height above the floor at which a 1.8 m-diameter circular display exactly fills the beam cone.

Fig. 1.3 — Illuminance cone: lamp 4 m above the floor, half-angle 22.5°. Numeric spot diameter and peak illuminance at each level are collected in the Final Results table.

Approach. Convert lamp wattage and efficacy to total luminous flux, spread it uniformly over the beam's solid angle to get luminous intensity, then apply the inverse-square law and simple beam-cone geometry at each 1 m interval.

Total luminous flux and beam intensity. $\Phi = P\times\text{efficacy} = 45\times15 = 675$ lm. The beam solid angle (half-angle $\theta=22.5^{\circ}$) is $\Omega = 2\pi(1-\cos\theta) = 2\pi(1-\cos 22.5^{\circ}) = 0.4783$ sr, assuming uniform intensity across the cone: $$\boxed{I = \Phi/\Omega = 675/0.4783 = 1411\ \text{cd}}$$
Spot diameter and illuminance at each 1 m interval. At distance $d$ (m) from the lamp along the beam axis, the spot radius is $r=d\tan\theta$ (so $D=2d\tan\theta$) and the on-axis illuminance follows the inverse-square law, $E=I/d^2$ (normal incidence, since the beam points straight down): $$D(d)=2d\tan 22.5^{\circ}=0.828\,d\ \text{m},\qquad E(d)=1411/d^2\ \text{lux}$$ Evaluating at $d=1,2,3,4$ m (heights above floor 3, 2, 1, 0 m) gives the table below.
Height for the 1.8 m display "just within the cone." The display's diameter must equal the cone's diameter at that distance: $1.8 = 2d\tan 22.5^{\circ}$, so $$d = \dfrac{1.8}{2\tan 22.5^{\circ}} = 2.173\ \text{m from the lamp}$$ $$\boxed{\text{height above floor} = 4 - 2.173 = 1.83\ \text{m}}$$

Final Results — Question 1.3
Distance from lamp, d (m)	Height above floor (m)	Spot diameter, D (m)	Peak illuminance, E (lux)
1	3	0.83	1411.3
2	2	1.66	352.8
3	1	2.49	156.8
4	0 (floor)	3.31	88.2
1.8 m display height above floor = 1.83 m

Topic: Point-source photometry — luminous flux, solid angle, intensity, illuminance.

Key relations: $\Phi=P\times\text{efficacy}$; solid angle of a cone $\Omega=2\pi(1-\cos\theta)$; $I=\Phi/\Omega$; inverse-square law $E=I/d^2$ (at normal incidence).

Why this works: A flood lamp's beam angle defines the solid angle over which its flux is concentrated; treating the intensity as uniform within that cone (a standard first-order lighting-design idealization) lets total flux be converted directly to a candela rating, after which every other level's illuminance and spot size follow from pure geometry and the inverse-square law — no photometric distribution curve is needed for this level of design.

Common pitfall: Using the FULL beam angle instead of the HALF-angle in the tangent and solid-angle formulas, or forgetting the cos(incidence angle) factor for surfaces not perpendicular to the beam axis (not needed here since the lamp points straight down onto a horizontal floor/display).

Source: ASHRAE/IES Handbook, Lighting Fundamentals; Building Engineering course notes on photometry (illuminance, luminous intensity, and beam geometry).

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