Question

Using the conversion factors between W and Btu/h, 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

Answer: The Stefan-Boltzmann constant in English units is 1.724 x 10⁻⁹ Btu/h·ft²·R⁴.

Step-by-step solution

Identify the conversion factors

The main conversion factors we need are: 1. Power: W to Btu/h 2. Length: m to ft 3. Temperature: K to R For power (W to Btu/h): 1 W = 3.412142 Btu/h For length (m to ft): 1 m = 3.28084 ft For temperature (K to R): 1 K = 1.8 R Now we can continue with the conversion.

Convert power

We start by converting W to Btu/h: 1 W = 3.412142 Btu/h So, \(\sigma = 5.67 \times 10^{-8} \frac{\text{W}}{\text{m}^{2} \cdot \text{K}^{4}} \cdot \frac{3.412142 \, \text{Btu/h}}{1 \, \text{W}}\)

Convert length and temperature

Convert m² to ft² and K⁴ to R⁴ by applying the conversion factors: Length: We need to square the conversion factor from m to ft: \((1 \, \text{m} \cdot 3.28084 \, \text{ft})^2 = 10.7639 \, \text{ft}^2\) Temperature: We need to raise the conversion factor from K to R to the power of 4: \((1 \, \text{K} \cdot 1.8 \, \text{R})^4 = 10.4976 \, \text{R}^4\) Now we can plug these conversion factors in: \(\sigma = 5.67 \times 10^{-8} \frac{\text{Btu/h}}{\text{ft}^{2} \cdot \text{R}^{4}} \cdot \frac{10.7639 \, \text{ft}^2}{1 \, \text{m}^2} \cdot \frac{10.4976 \, \text{R}^4}{1 \, \text{K}^4}\)

Calculate the final result

Multiply the Stefan-Boltzmann constant by the conversion factors: \(\sigma = 5.67 \times 10^{-8} \cdot 3.412142 \cdot 10.7639 \cdot 10.4976 \, \frac{\text{Btu}}{\text{h} \cdot \text{ft}^{2} \cdot \text{R}^4}\) \(\sigma = 1.724 \times 10^{-9} \, \frac{\text{Btu}}{\text{h} \cdot \text{ft}^{2} \cdot \text{R}^{4}}\) So, the Stefan-Boltzmann constant in English units is: \(\sigma =1.724 \times 10^{-9} \, \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot \mathrm{R}^{4}\)

Key concepts

Thermodynamics

The field of thermodynamics is the study of heat, temperature, and their relation to energy and work. Within this discipline, a fundamental concept is the understanding of how thermal energy is transferred between systems. One aspect to consider is the Stefan-Boltzmann law, which tells us that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of the black body's temperature, expressed as \( \sigma T^4 \). The Stefan-Boltzmann constant \( \sigma \) is a critical value in this law, facilitating the calculation of radiative heat transfer from an object’s temperature.

The exercise provided involves the application of this law in different unit systems, a task common in the world of thermodynamics where measurements may be taken or required in various units depending on the geographical location or industry standards. Understanding how to convert between these units is essential for accurate communication and interpretation of scientific data.

Heat Transfer

In terms of heat transfer, the Stefan-Boltzmann law's primary role is in quantifying radiative transfer—one of the three modes of heat transfer, alongside conduction and convection. This type of heat transfer is significant in many applications, ranging from astrophysical calculations to engineering problems involving thermal systems.

Importance of Accurate Conversion

For example, when engineers design systems such as spacecraft, solar panels, or thermal insulations, they need to take into account the amount of radiative heat transfer that can occur, which is calculated using the Stefan-Boltzmann constant. The challenge, as shown in the exercise, is ensuring that when the constant is used in formulas, it's expressed in the correct unit system to match the other parameters in the calculation. Failing to convert the units properly could lead to significant errors, impacting the effectiveness or safety of the system being designed.

Unit Conversion

The unit conversion process is integral in both academic and professional sciences. In practice, we might encounter the need to convert units from the International System of Units (SI) to the British Imperial System or the United States Customary Units (USCS). When dealing with thermodynamic equations, as seen in our exercise example, precision in unit conversion is imperative to obtain accurate results.

Understanding Conversion Factors

To successfully engage in unit conversion, one must comprehend the conversion factors for each unit. As seen in the exercise, knowing that 1 W is equivalent to 3.412142 Btu/h or 1 m equals 3.28084 ft makes it possible to perform a series of conversions that transforms the Stefan-Boltzmann constant to the required unit system. Converting correctly represents a real-world implication when different industries work with varying measurement systems and need to communicate findings or specifications unambiguously.