B Supplementary Material 2

B.1 Analytical Solutions for Direct Flux from
Power Cosine Sources

Direct flux originating from a source with flux distribution \( \Gamma _{src} \)

\{begin}{align*} F_i & = \int _{\Omega _{HS}} \Gamma _{src}(\mathbf {\Theta }) (\mathbf {\Theta } \cdot \mathbf {n_{i}}) d\bm {\omega }_\mathbf {\Theta }, \quad \text {with} \quad d\bm {\omega }_\mathbf {\Theta } = sin\theta d\theta d\varphi \ . \{end}{align*}

Direct flux originating from a source with power cosine distribution

\{begin}{align*} F_i & = \int _{\Omega _{HS}} \left [\cos (\theta )^n \mathbf {\Theta }(\varphi ,\theta ) \cdot \mathbf {n_i}(\varphi ,\theta ) \right ]sin\theta d\theta d\varphi \ . \{end}{align*}

Direct flux originating from a source with power cosine distribution on a horizontal surface, i.e., \( \mathbf {n_i}(\varphi ,0) \)

\{begin}{align*} F_{horizontal} & = \int _0^{\pi /2}\int _0^{2\pi } \left [\cos (\theta )^n \underbrace {\mathbf {\Theta }(\varphi ,\theta ) \cdot \mathbf {n_i}(\varphi ,0)}_{\cos (\theta )} \right ]sin\theta d\theta d\varphi \\ & = \int _0^{\pi /2}\int _0^{2\pi } \left [\cos (\theta )^n cos(\theta ) \right ]sin\theta d\theta d\varphi \\ & = \int _0^{\pi /2}\int _0^{2\pi } \left [ \cos (\theta )^{n+1}\right ] sin \theta d\theta d\varphi = \frac {2}{n+2} \pi \ . \{end}{align*}

E.g., for \( n=1 \) (i.e., a diffuse source)

\{begin}{align*} F_{horizontal} & = \int _0^{\pi /2}\int _0^{2\pi } \left [ \cos (\theta )^{2}\right ] sin \theta d\theta d\varphi \\ & = 2\pi \int _0^{\pi /2} \left [\cos (\theta )^{2}\right ] sin(\theta )d\theta \\ & = 2\pi \left [ -\frac {1}{3} cos(\pi /2)^3 +\frac {1}{3} cos(0)^3\right ] = 2\pi \left [\frac {1}{3}\right ] = \frac {2}{3} \pi \ . \{end}{align*}

Direct flux originating from a source with power cosine distribution on a vertical surface, i.e., \( \mathbf {n_i}(\varphi ,\pi /2) \)

\{begin}{align*} F_{vertical} & = \int _0^{\pi /2}\int _0^{\pi } \left [\cos (\theta )^n \underbrace {\mathbf {\Theta }(\varphi ,\theta ) \cdot \mathbf {n_i}(\varphi ,\pi /2)}_{\sin (\theta )\cos (\varphi - \pi /2)} \right ]sin\theta d\theta d\varphi \\ & = \int _0^{\pi /2}\int _0^{\pi } \left [ cos(\theta )^n (sin(\theta )cos(\varphi - \pi /2))\right ] sin(\theta )d\theta d\varphi \\ & = \int _0^{\pi /2}\int _0^{\pi } \left [ cos(\theta )^n (sin(\theta )sin(\varphi ))\right ] sin(\theta )d\theta d\varphi \\ & = \int _0^{\pi /2} \left [ cos(\theta )^n sin(\theta )(-cos(\pi )) -cos(\theta )^n sin(\theta )(-cos(0)) \right ] sin(\theta )d\theta \\ & = \int _0^{\pi /2} \left [ cos(\theta )^n sin(\theta )(1) -cos(\theta )^n sin(\theta )(-1) \right ] sin(\theta )d\theta \\ & =2 \int _0^{\pi /2} \left [ cos(\theta )^n sin(\theta ) \right ] sin(\theta )d\theta \ . \{end}{align*}

E.g., for \( n=1 \) (i.e., a diffuse source)

\{begin}{align*} F_{vertical} & =2 \int _0^{\pi /2} \left [ cos(\theta )^n sin(\theta ) \right ] sin(\theta )d\theta \\ & =2 \left [ \frac {1}{3} \right ] = \frac {2}{3} \ . \{end}{align*}

or other exponents

\{begin}{align*} F_{vertical} & =2 \int _0^{\pi /2} \left [ cos(\theta )^n sin(\theta ) \right ] sin(\theta )d\theta \\ & =2 \left [ \frac {\pi }{16} \right ] \quad (n=2) \\ & =2 \left [ \frac {2}{15} \right ] \quad (n=3) \\ & = \dots \\ & =2 \left [ \frac {21\pi }{2048} \right ] \quad (n=10) \ . \{end}{align*}