Question

Which expression is used to determine the heat flux for conduction? (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

Answer: (b) \(-k \operatorname{grad} T\)

Step-by-step solution

Review options given

Here are the options provided for heat flux in conduction: (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 Let us analyze each option to determine the correct expression.

Analyzing option (a) - \(-k A \frac{dT}{dx}\)

The formula in option (a) uses the following variables: - \(k\): thermal conductivity - \(A\): area - \(\frac{dT}{dx}\): temperature gradient This is actually the formula for heat transfer due to conduction. However, we're looking for the formula for heat flux, which is the heat transfer per unit area. We need to modify this formula to find the heat flux.

Analyzing option (b) - \(-k \operatorname{grad} T\)

The formula in option (b) uses the following variables: - \(k\): thermal conductivity - \(\operatorname{grad} T\): temperature gradient This is the general form for determining heat flux due to conduction, which applies to one, two, or three dimensions. The negative sign indicates that heat flows in the direction of decreasing temperature, which maintains the second law of thermodynamics. This expression is the correct one for heat flux due to conduction.

Reviewing other options

Now that we've found the correct expression for heat flux due to conduction, let's quickly review the other options to understand what they represent. (c) \(h\left(T_{2}-T_{1}\right)\) - This formula represents heat transfer due to convection, where \(h\) is the heat transfer coefficient, and \((T_{2}-T_{1})\) is the difference in temperature between two locations. (d) \(\varepsilon \sigma T^{4}\) - This formula represents radiative heat transfer, where \(\varepsilon\) is the emissivity of the surface, \(\sigma\) is the Stefan-Boltzmann constant, and \(T\) is the absolute temperature of the object. (e) None of them - This option is incorrect since we have already identified the correct expression for heat flux due to conduction. In conclusion, the correct expression for determining heat flux due to conduction is option: - (b) \(-k \operatorname{grad} T\)