Step 1: Understanding the Problem:
The question asks to identify the proportional relationship of heat flux during conduction heat transfer as defined by Fourier's Law of Heat Conduction.
Conduction is the transfer of thermal energy between microscopic particles through collisions and direct contact.
Step 2: Key Formula or Approach:
Fourier's Law of Heat Conduction is mathematically expressed as:
\[ q = -k A \frac{dT}{dx} \]
Where:
\( q \) is the rate of heat flow in Watts (\( \text{W} \))
\( k \) is the thermal conductivity of the material in \( \text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1} \)
\( A \) is the surface area perpendicular to the direction of heat flow in \( \text{m}^2 \)
\( \frac{dT}{dx} \) is the temperature gradient in \( \text{K}\cdot\text{m}^{-1} \)
The heat flux (\( q'' \)), which is defined as the rate of heat transfer per unit area (\( q'' = \frac{q}{A} \)), is given by:
\[ q'' = -k \frac{dT}{dx} \]
Step 3: Detailed Explanation:
The physical significance of the parameters in Fourier's law is detailed below:
• Proportionality to Gradient: The expression clearly shows that heat flux (\( q'' \)) is directly proportional to the temperature gradient (\( \frac{dT}{dx} \)).
A gradient represents the rate of change of temperature with respect to distance, which is more specific than just a simple temperature difference.
• Significance of the Negative Sign: The negative sign in the equation is of paramount physical importance.
According to the Second Law of Thermodynamics, heat spontaneously flows from a region of higher temperature to a region of lower temperature.
This means that as distance (\( x \)) increases, the temperature (\( T \)) decreases, making the temperature gradient (\( \frac{dT}{dx} \)) a negative quantity.
To ensure that the calculated heat flow vector in the direction of decreasing temperature is positive, the negative sign is introduced.
Therefore, heat flux is proportional to the negative temperature gradient.
• Analysis of other options:
- Temperature difference alone (Option A) is incorrect because it neglects the thickness (distance) of the material.
- Fluid velocity (Option C) is a parameter relevant to convective heat transfer (Newton's Law of Cooling), not conduction.
- Surface area only (Option D) is incorrect because heat flux is independent of the total surface area by definition.
Step 4: Final Answer:
Hence, Fourier's Law states that heat flux is proportional to the negative temperature gradient.