If the temperature gradient in a rod is uniform, the rate of heat transfer is given by which expression?

• A. q = -k A (dT/dx)
• B. Q = k A ΔT / L
• C. dQ/dt = h A (T_s - T∞)
• D. Q = σ A T^4

Answer: (B)

Conduction through a solid with a uniform temperature gradient follows Fourier’s law. The rate of heat transfer is Q̇ = -k A dT/dx. When the temperature changes linearly from one end to the other over length L, the gradient is constant: dT/dx ≈ ΔT / L. Substituting gives Q̇ = -k A (ΔT / L). Taking the magnitude (heat flowing from hot to cold) yields Q̇ = k A ΔT / L. This is exactly the form that relates heat transfer rate to the temperature difference, cross-sectional area, material conductivity, and length.

The other expressions refer to different forms or mechanisms: a differential form q = -k A dT/dx is the same idea but without collapsing the gradient into ΔT/L and with a sign indicating direction; Q = σ A T^4 is radiative transfer; Q̇ = h A (T_s - T∞) is convection.