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Qwestrum Engineering360 · Chemical Engineering · Heat Transfer

Conduction Through Walls

Steady conduction is solved with Fourier’s law recast as a thermal-resistance network: the heat rate equals the overall temperature difference divided by the sum of series resistances L/(kA) for walls or ln(r₂/r₁)/(2πkL) for pipes.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • ThermalresistanceRth=L(kA)forplanewallThermal resistance R_{th} = \frac{L}{(kA)} for plane wall
  • Steady 1-D conduction — no internal generation unless stated
  • Contact resistance adds extra R at interfaces

Topic details

Introduction

This McCabe-Smith / Kern topic treats the wall or pipe as an electrical analogue where temperature difference is voltage, heat rate is current and L/(kA) is resistance. Composite walls add resistances in series, parallel paths add in parallel, and interface imperfections appear as an extra contact resistance. The method scales directly to lagged pipes and furnace walls.

Key relations & formulas

q=−kAdTdxq = -k A \frac{dT}{dx}
(Fourier law)
q=(T1−T2)/Σ(LikiA)q = (T_{1} - T_{2}) / Σ(\frac{L_{i}}{k_{i}} A)
(composite plane wall, series resistance)
q=2πL(T1−T2)ln(r2r1)q = 2\pi L\frac{(T_{1} - T_{2})}{ln}(\frac{r_{2}}{r_{1}})
(cylindrical radial conduction)

Notation and sign conventions

Relation 1 —
q=−kAdTdxq = -k A \frac{dT}{dx}
q=−kAdTdxq = -k A \frac{dT}{dx}
(Fourier law)
Write this relation with symbols exactly as in Process Heat Transfer — Kern before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
q=q =
q=(T1−T2)/Σ(LikiA)q = (T_{1} - T_{2}) / Σ(\frac{L_{i}}{k_{i}} A)
(composite plane wall, series resistance)
Write this relation with symbols exactly as in Process Heat Transfer — Kern before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
q=2πLq = 2\pi L
q=2πL(T1−T2)ln(r2r1)q = 2\pi L\frac{(T_{1} - T_{2})}{ln}(\frac{r_{2}}{r_{1}})
(cylindrical radial conduction)
Write this relation with symbols exactly as in Process Heat Transfer — Kern before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Concept in depth

Fourier’s law states that heat flux is proportional to the negative temperature gradient, with conductivity k as the proportionality. In one-dimensional steady state with no generation, the heat rate is constant through every layer, so the same q passes through each resistance and the temperature drop across a layer is proportional to its resistance. For cylinders the area grows with radius, which is why the resistance carries a logarithm rather than a simple L/(kA) form — and why adding insulation to a small pipe can, below the critical radius, actually increase heat loss.

Assumptions and validity limits

State assumptions explicitly before using any relation for conduction through walls — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Heat Transfer (Chemical) viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Heat Transfer (Chemical) papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to conduction through walls.
4. Use equation 1:
q=−kAdTdxq = -k A \frac{dT}{dx}
.
5. Use equation 2:
q=q =
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Conduction Through Walls appears in heat exchangers and reactors. In Indian chemical curricula this topic is tested because it connects theory to heat exchange in process equipment.
GATE and semester exams often combine conduction through walls with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use conduction through walls?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Students use the plane-wall formula for pipes (ignoring the logarithm), forget to include contact or convective film resistances in the series sum, and mix up which temperature difference (overall versus per-layer) belongs with which resistance. Assuming steady state when generation is present is another slip.

Quick revision checklist

Before attempting conduction through walls problems, confirm you can:
1.
ThermalresistanceRth=L(kA)forplanewallThermal resistance R_{th} = \frac{L}{(kA)} for plane wall

2. Steady 1-D conduction — no internal generation unless stated
3. Contact resistance adds extra R at interfaces
Revise the solved examples in Process Heat Transfer — Kern and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Two-layer composite wall

Problem

A wall has a 0.2 m brick layer (k = 0.7 W/m·K) and 0.05 m insulation (k = 0.04 W/m·K), area 1 m², with faces at 200 °C and 30 °C. Find the heat rate.

Solution

R = 0.2/0.7 + 0.05/0.04 = 0.286 + 1.25 = 1.536 K/W. q = ΔT/R = (200−30)/1.536 = 110.7 W. The insulation dominates the resistance despite being thinner.

Conceptual check — Conduction Through Walls

Problem

In a Heat Transfer (Chemical) semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of conduction through walls." What should a complete answer include?

Exams & GATE

Kern Ch. 2 — draw thermal circuit analogy like electrical resistors.

📖 Standard books (India)

  • Process Heat Transfer — Kern

    Read: Syllabus unit

    Heat exchangers and process heat transfer

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