Fourier’s law of heat conduction: For which class of problems does the basic one-dimensional formulation q = -k * A * dT/dx directly give the heat flow with area A and gradient dT/dx?

Introduction / Context:
Fourier’s law is the foundation of conduction analysis. While the vector law applies generally, the familiar textbook expression q = -k * A * dT/dx (with a single spatial gradient) assumes one-dimensional heat flow and constant properties along the direction considered.

Given Data / Assumptions:

• Steady or differential 1-D conduction along x.
• Heat flux normal to area A.
• Thermal conductivity k treated as uniform or known function of T.

Concept / Approach:
In 1-D conduction, temperature varies along only one coordinate. The heat rate is proportional to area, thermal conductivity, and temperature gradient. For 2-D/3-D fields, the full vector form q⃗ = -k ∇T and governing PDEs (Laplace/Poisson) must be solved; direct use of the simple product form with a single gradient is insufficient.

Step-by-Step Explanation:

Verification / Alternative check:
When lateral heat spreading or fins cause 2-D fields, resort to the general form and energy equations; the simple 1-D expression becomes an approximation.

Why Other Options Are Wrong:

• (a) “Irregular surfaces only” is unrelated to dimensionality.
• (b) Non-uniform surface temperatures often lead to multi-D fields.
• (d) Two-dimensional problems require solving PDEs; the 1-D formula is not directly applicable.
• (e) Radiation is governed by Stefan–Boltzmann type laws, not Fourier’s law.

Common Pitfalls:
Applying 1-D conduction formulas to fins or heat-spreading plates without verifying dimensionality.

Final Answer:
One-dimensional conduction cases (through plane walls, long cylinders approximated 1-D)