Question

Using the conversion factors between W and Btu/h, \(\mathrm{m}\) and \(\mathrm{ft}\), and \(\mathrm{K}\) and \(\mathrm{R}\), express the Stefan-Boltzmann constant $\sigma=5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\( in the English unit \)\mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot \mathrm{R}^{4}$.

Short answer

Question: Convert the Stefan-Boltzmann constant from SI (W, m, K) units to English (Btu/h, ft, R) units. Answer: Approximately, 1.714 × 10^-9 Btu/h*ft^2*R^4.

Step-by-step solution

Convert W to Btu/h

Multiply the given Stefan-Boltzmann constant by the W to Btu/h conversion factor: \(\sigma = 5.67 \times 10^{-8} \frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{K}^4} \cdot \frac{3.412142\, \mathrm{Btu/h}}{1\, \mathrm{W}}\)

Simplify and cancel W

Now we can simplify the expression by canceling out watts (W): \(\sigma = (5.67 \times 10^{-8}) \times (3.412142) \frac{\mathrm{Btu/h}}{\mathrm{m}^2 \cdot \mathrm{K}^4}\)

Convert m to ft

Multiply the expression by the m to ft conversion factor: \(\sigma = (5.67 \times 10^{-8}) \times (3.412142) \cdot \left( \frac{1\, \mathrm{m}}{3.28084\, \mathrm{ft}} \right)^2 \cdot \frac{\mathrm{Btu/h}}{\mathrm{ft}^2 \cdot \mathrm{K}^4}\)

Simplify and cancel m

Now we can simplify the expression by canceling out meters (m): \(\sigma = (5.67 \times 10^{-8}) \times (3.412142) \cdot \left( \frac{1}{3.28084} \right)^2 \frac{\mathrm{Btu/h}}{\mathrm{ft}^2 \cdot \mathrm{K}^4}\)

Convert K to R

Multiply the expression by the K to R conversion factor: \(\sigma = (5.67 \times 10^{-8}) \times (3.412142) \cdot \left( \frac{1}{3.28084} \right)^2 \cdot \left( \frac{1\,\mathrm{K}}{1.8\,\mathrm{R}} \right)^4 \frac{\mathrm{Btu/h}}{\mathrm{ft}^2 \cdot \mathrm{R}^4}\)

Simplify and cancel K

Now we can simplify the expression by canceling out Kelvin (K): \(\sigma = (5.67 \times 10^{-8}) \times (3.412142) \cdot \left( \frac{1}{3.28084} \right)^2 \cdot \left( \frac{1}{1.8} \right)^4 \frac{\mathrm{Btu/h}}{\mathrm{ft}^2 \cdot \mathrm{R}^4}\)

Calculate the final expression

Now we can calculate the final expression: \(\sigma = (5.67 \times 10^{-8}) \times (3.412142) \times \left( \frac{1}{3.28084} \right)^2 \times \left( \frac{1}{1.8} \right)^4 \frac{\mathrm{Btu/h}}{\mathrm{ft}^2 \cdot \mathrm{R}^4} \approx 1.714 \times 10^{-9} \, \frac{\mathrm{Btu/h}}{\mathrm{ft}^2 \cdot \mathrm{R}^4}\) So, the Stefan-Boltzmann constant in Btu/h, ft, and R is \(\sigma \approx 1.714 \times 10^{-9} \, \frac{\mathrm{Btu/h}}{\mathrm{ft}^2 \cdot \mathrm{R}^4}\).