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# Radiative Heat Transfer Relations ## Nomenclature | SYMBOL | DESCRIPTION | UNITS | | :------ | :------------------------------------------- | :--------------------------------------------------------- | | A...

# Radiative Heat Transfer Relations ## Nomenclature | SYMBOL | DESCRIPTION | UNITS | | :------ | :------------------------------------------- | :--------------------------------------------------------- | | A | Area | $\left[\mathrm{m}^{2}\right]$ | | F | Shape factor | $--$ | | G | Irradiation | $\left[\mathrm{W} / \mathrm{m}^{2}\right]$ | | J | Radiosity | $\left[\mathrm{W} / \mathrm{m}^{2}\right]$ | | Q | Heat transfer rate | $[\mathrm{W}]$ | | T | Temperature | $[\mathrm{K}]$ or $\left[{ }^{\circ} \mathrm{C}\right]$ | | $\varepsilon$ | Emissivity | $--$ | | $\rho$ | Reflectivity | $--$ | | $\sigma$ | Stefan-Boltzmann constant, $5.67 \times 10^{-8}$ | $\left[\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right]$ | ## Equations ### Blackbody * $E_{b}=\sigma T^{4}$ ### Gray body * $E=\varepsilon \sigma T^{4}$ * $0 \leq \varepsilon \leq 1$ * $\varepsilon=\alpha$ ### Radiosity * $J=\varepsilon E_{b}+\rho G$ * $J=\varepsilon E_{b}+\rho G=\varepsilon E_{b}+(1-\varepsilon) G$ ### Space resistance * $R_{i}=\frac{1-\varepsilon_{i}}{A_{i} \varepsilon_{i}}$ ### Surface resistance * $R_{i j}=\frac{1}{A_{i} F_{i j}}$ ### Heat exchange between two black surfaces * $Q_{12}=A_{1} F_{12}\left(E_{b 1}-E_{b 2}\right)$ * $Q_{12}=A_{1} F_{12} \sigma\left(T_{1}^{4}-T_{2}^{4}\right)$ ### Heat exchange between two gray surfaces * $Q_{12}=\frac{E_{b 1}-E_{b 2}}{R_{1}+R_{12}+R_{2}}$ * $Q_{12}=\frac{E_{b 1}-E_{b 2}}{\frac{1-\varepsilon_{1}}{A_{1} \varepsilon_{1}}+\frac{1}{A_{1} F_{12}}+\frac{1-\varepsilon_{2}}{A_{2} \varepsilon_{2}}}$ * $Q_{12}=\frac{A_{1} \sigma\left(T_{1}^{4}-T_{2}^{4}\right)}{\frac{1-\varepsilon_{1}}{\varepsilon_{1}}+\frac{1}{F_{12}}+\frac{A_{1}}{A_{2}} \frac{1-\varepsilon_{2}}{\varepsilon_{2}}}$ ### Two large parallel plates * $Q_{12}=\frac{A \sigma\left(T_{1}^{4}-T_{2}^{4}\right)}{\frac{1}{\varepsilon_{1}}+\frac{1}{\varepsilon_{2}}-1}$ ### Two long concentric cylinders * $Q_{12}=\frac{A_{1} \sigma\left(T_{1}^{4}-T_{2}^{4}\right)}{\frac{1}{\varepsilon_{1}}+\frac{A_{1}}{A_{2}}\left(\frac{1}{\varepsilon_{2}}-1\right)}$ ### Sphere inside enclosure * $Q_{12}=\frac{A_{1} \sigma\left(T_{1}^{4}-T_{2}^{4}\right)}{\frac{1}{\varepsilon_{1}}+\frac{A_{1}}{A_{2}}\left(\frac{1}{\varepsilon_{2}}-1\right)}$ ### Radiation shield * $Q_{13}=Q_{32}$ * $Q_{12}=\frac{A \sigma\left(T_{1}^{4}-T_{2}^{4}\right)}{\left(\frac{1}{\varepsilon_{1}}+\frac{1}{\varepsilon_{2}}-1\right)+\left(\frac{1}{\varepsilon_{31}}+\frac{1}{\varepsilon_{32}}-1\right)}$ ### Simplified form for radiation shield * $\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}$ * $Q_{12, \text { with shield }}=\frac{1}{n+1} Q_{12, \text { without shield }}$