Question

Fourier's law of heat conduction relates

• A. Heat flux to temperature gradient
• B. Heat flux to pressure gradient
• C. Heat flux to velocity gradient
• D. Heat flux to density

Hint:

Remember the core transport analogies:
• Fourier's Law: Heat Flux \(\propto\) Temperature Gradient
• Fick's Law: Mass Flux \(\propto\) Concentration Gradient
• Ohm's Law: Current Density \(\propto\) Voltage Gradient
• Newton's Law: Shear Stress \(\propto\) Velocity Gradient

Answer 1

The Correct Option is A

Concept: Fourier's Law of Heat Conduction is a phenomenological governing principle that describes the rate at which thermal energy moves through a medium via conduction. In its vector notation form, it states that the local heat flux density is directly proportional to the negative spatial gradient of temperature. Mathematical Formulation:
In a single dimension (\(x\)), Fourier's law is expressed mathematically as: \[ q_x'' = -k \frac{dT}{dx} \] where:
• \(q_x''\) represents the heat flux (\(\text{W/m}^2\)), which is the heat transfer rate per unit surface area normal to the direction of flow (\(q_x'' = Q/A\)).
• \(k\) is the intrinsic thermal conductivity of the material medium (\(\text{W/m}\cdot\text{K}\)).
• \(\frac{dT}{dx}\) represents the temperature gradient (\(\text{K/m}\) or \(^\circ\text{C/m}\)), showing how temperature changes along the spatial coordinate path.
• The negative sign explicitly enforces compliance with the Second Law of Thermodynamics, confirming that thermal energy flows spontaneously from regions of high temperature to low temperature. Let us review the alternate physical transport mechanisms described in options (2), (3), and (4):
• Option (2) relates to Darcy's Law or Hagen-Poiseuille flow, where fluid volumetric flow rate/flux is driven by a pressure gradient (\(\frac{dP}{dx}\)).
• Option (3) relates to Newton's Law of Viscosity, which defines shear stress (\(\tau\)) as proportional to a velocity gradient (\(\frac{du}{dy}\)).
• Option (4) is incorrect because mass density is a state property and does not act as a direct linear driving force for heat flux. Thus, Fourier's law explicitly couples heat flux directly to the temperature gradient, matching Option (1).