Question

Which expression is used to determine the heat flux emitted by thermal radiation from a surface? (a) \(-k A \frac{d T}{d x}\) (b) \(-k \operatorname{grad} T\) (c) \(h\left(T_{2}-T_{1}\right)\) (d) \(\varepsilon \sigma T^{4}\) (e) None of them

Short answer

Options: a) -k A (dT/dx) b) -k grad T c) h(T_2-T_1) d) εσT^4 e) None of them Answer: (d) εσT^4

Step-by-step solution

Identifying the Stefan-Boltzmann Law

The Stefan-Boltzmann equation gives the amount of thermal radiation emitted from the surface per unit area per second and is represented by the following formula: \(\phi = \varepsilon\sigma T^{4}\) Here, \(\phi\) represents the heat flux, \(\varepsilon\) is the emissivity of the surface (a dimensionless number between 0 and 1), \(\sigma\) is the Stefan-Boltzmann constant with a value of approximately \(5.67 \times 10^{-8} \; \text{W/m}^2 \text{K}^4\), and \(T\) is the temperature of the surface in Kelvin (K).

Checking the Options

Let's check the given options: (a) \(-k A \frac{d T}{d x}\) : This expression represents the conduction heat flux based on the Fourier's law, which is not the required expression. (b) \(-k \operatorname{grad} T\) : This is another representation of the Fourier's Law for heat conduction, and it is also not the correct expression. (c) \(h\left(T_{2}-T_{1}\right)\) : This expression corresponds to the convective heat transfer, using the convective heat transfer coefficient \(h\), and the temperature difference between two points \(T_1\) and \(T_2\). Thus, this is not the required expression. (d) \(\varepsilon \sigma T^{4}\) : This expression is the Stefan-Boltzmann Law for the heat flux due to thermal radiation, and it matches the correct expression we are looking for. (e) None of them: Since option (d) is the correct expression, we rule out this option.

Conclusion

The correct expression to determine the heat flux emitted by thermal radiation from a surface is: (d) \(\varepsilon \sigma T^{4}\)