Introduction & Context

The one-dimensional steady-state heat-flow calculation through a plane wall is the simplest—and most frequently used—analogy for transport phenomena in process engineering. It couples three core ideas that re-appear in momentum, heat and mass transfer:

  • A driving force (temperature difference, pressure difference, concentration difference).
  • A geometric resistance (thickness ÷ area).
  • A material property that converts the gradient into a flux (thermal conductivity, viscosity, diffusivity).

Because the same algebraic skeleton is reused for Fourier’s law, Ohm’s law, Fick’s law and Darcy’s law, mastering the wall analogy gives engineers a mental template for scaling equipment, estimating losses, checking insulation thickness, or scoping energy balances in reactors, heat exchangers, dryers, building envelopes and pipelines.

Methodology & Formulas

  1. Convert thickness to base units
    \( L = \dfrac{t_{\text{mm}}}{1000} \)
  2. Compute cross-sectional area
    \( A = H \cdot W \)
  3. Define the driving force
    \( \Delta T = T_{\text{in}} - T_{\text{out}} \)
  4. Calculate thermal resistance
    \( R_{\text{wall}} = \dfrac{L}{k \cdot A} \)
  5. Obtain total heat-flow rate (Fourier’s law)
    \( q = \dfrac{\Delta T}{R_{\text{wall}}} \)
  6. Convert to heat flux
    \( q'' = \dfrac{q}{A} \)
Regime Condition Implication
Steady state \( \partial T/\partial t = 0 \) Storage term zero; inflow = outflow
1-D conduction \( \nabla^2 T = \partial^2 T/\partial x^2 \) Lateral edge losses neglected
Constant properties \( k \neq f(T) \) Linear temperature profile