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?

Question

Solve this multiple-choice question and choose the correct option.

Curioustab · Accepted Answer

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: 1) Assume 1-D slab: T = T(x); then q = -k * A * dT/dx.2) Integrate over thickness L with boundary temperatures T1 and T2 to get Q = k * A * (T1 - T2) / L (for constant k).3) For long cylinders/spheres with small radial gradients across thickness, approximate 1-D forms are used with appropriate areas. 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)