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Heat transfer: preserve the circuit when paths become parallel

A thermal-resistance diagram encodes which heat rate is common and which temperature difference is common. A bridge changes that diagram: series formulas cannot simply be averaged across parallel areas.

Course topic: Heat and Mass Transfer

Study route and exercise: University Reading List editorial project. How we curate · Sources checked:

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Scope and responsibility

Steady one-dimensional conduction with convection films, an interface resistance and a two-column bridge approximation. Lienhard is a free engineering spine; Bergman offers broad heat/mass coverage; Nellis and Klein emphasize worked modeling and computation. Results describe an invented ideal wall, not tested building performance.

Choose your starting point

I need a free engineering course

Choose Lienhard after elementary thermodynamics and fluid mechanics.

I need a broad heat-and-mass reference

Choose Bergman and collaborators for mechanisms and engineering problems.

I want model construction and computation

Use Nellis and Klein, checking equations independently of software.

Compare the three books

A Heat Transfer Textbook

John H. Lienhard V; John H. Lienhard IV · Edition: 6 · 2024

Free course spine

Choose it when
You know energy balances, units and elementary fluid mechanics.
Use another route when
You expect free access to remove engineering prerequisites.
Selected reading
Thermal resistance and overall heat-transfer coefficients, then steady conduction.
Limits and trade-offs
The free personal/nonprofit instructional grant is not an unrestricted reuse licence; the course still requires a defensible physical model.

Try before choosing

Put both films, both slabs and the contact in one per-area resistance sum.

Check your work: All five terms have m² K/W; films are 1/h and slabs L/k.

Official free reading ↗

Book record and access options →

Fundamentals of Heat and Mass Transfer

Theodore L. Bergman; Adrienne S. Lavine; Frank P. Incropera; David P. DeWitt · Edition: 8 · 2018

Broad engineering reference

Choose it when
You want conduction to connect with convection, radiation and mass transfer.
Use another route when
You need only a compact wall-conduction treatment.
Selected reading
Eighth-edition one-dimensional steady conduction and convection-boundary resistance models.
Limits and trade-offs
Breadth and problem variety help later; distinguish total K/W from per-area m² K/W before using a chart.

Try before choosing

Compute the temperature jump across a contact with R″c=0.05 and q″=50.

Check your work: The jump is 2.5 K; there are two different surface temperatures at the contact.

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Introduction to Engineering Heat Transfer

G. F. Nellis; S. A. Klein · 2021

Modeling and computation route

Choose it when
You want detailed derivation followed by numerical implementation.
Use another route when
You want software to choose the physical network automatically.
Selected reading
One-Dimensional Steady-State Conduction, with introductory modeling and later extended surfaces as needed.
Limits and trade-offs
Computational tools can solve the chosen equations but do not validate an incorrect series/parallel topology; licences are separate.

Try before choosing

For two area fractions at common ΔT, decide what to average.

Check your work: Average area-weighted conductances 1/R″, not R″ itself.

Book record and access options →

Check your preparation

Can I keep heat rate and heat flux separate?

Q has units W; q″=Q/A has W/m². For a uniform column q″=ΔT/R″.

Is temperature continuous across this contact?

Not in the ideal resistance model: ΔTc=q″R″c. Energy conservation gives equal flux through it.

What changes for parallel columns?

The ambient temperature difference is common, but each column has its own heat flux; total heat rate is the sum over areas.

A six-step study plan

These are your own study marks, not an assessment. They are not sent to this site.

  1. Declare the idealizations

    Selected reading: Energy balances.

    What to do: List steady state, constant properties, no generation and omitted lateral spreading/radiation.

    Check your work: Each column has one constant heat flux.

  2. Build the series network

    Selected reading: Resistance and overall coefficient.

    What to do: Tabulate each per-area resistance with units.

    Check your work: The baseline total is 0.60 m² K/W.

  3. Recover heat rate

    Selected reading: One-dimensional conduction.

    What to do: Calculate q″ and Q separately.

    Check your work: 50 W/m² and 100 W for 2 m².

  4. Recover interface temperatures

    Selected reading: Contact resistance.

    What to do: Calculate every temperature drop from the hot air to cold air.

    Check your work: Drops sum to 30 K and the contact has a 2.5 K jump.

  5. Introduce the bridge

    Selected reading: Parallel networks and the follow-up.

    What to do: Calculate both column conductances, then their area-weighted sum.

    Check your work: The bridge is 0.42 m² K/W and total Q=760/7 W.

  6. Audit a tempting shortcut

    Selected reading: Your network diagrams.

    What to do: Compute what averaging resistances would predict and explain the failure.

    Check your work: Equal temperature difference, not equal flux, is the parallel-path condition.

Worked exercise

A wall with two films and a contact jump

A=2 m² separates air at 30°C and 0°C. Film coefficients are hin=10 and hout=20 W/(m² K). Layers have L1=0.10 m,k1=0.50 W/(m K), L2=0.05 m,k2=0.25 W/(m K), with R″c=0.05 m² K/W between them. Find heat flux, rate and surface temperatures.

Assumptions

Steady one-dimensional flow from hot to cold, constant properties, no generation, radiation or lateral spreading. Contact has negligible thickness but finite resistance. Celsius differences equal kelvin differences.

  1. R″=1/10+0.10/0.50+0.05+0.05/0.25+1/20=0.60 m² K/W. Thus q″=30/0.60=50 W/m² and Q=Aq″=100 W.
  2. Starting at hot air, the inside film drop is 5 K, so hot wall surface is 25°C. Layer 1 drops 10 K to 15°C.
  3. The contact drops 2.5 K to 12.5°C; layer 2 drops 10 K to outer wall 2.5°C; the outer film drops 2.5 K to the 0°C air.

Answer and interpretation

q″=50 W/m², Q=100 W; ordered solid/contact temperatures are 25,15,12.5,2.5°C.

Test the result

Drops 5+10+2.5+10+2.5=30 K. The same flux crosses each series element; forcing the two contact temperatures equal would discard its stated resistance.

Change an assumption

A bridge adds a parallel path

Keep all conditions, but 20% of the area has layer-2 conductivity 2.5 W/(m K). Treat the ordinary and bridge areas as independent one-dimensional columns, each with the same films and contact. Find total heat rate.

  1. Ordinary area 1.6 m² retains R″=0.60 and q″=50. Bridge area 0.4 m² has R″b=0.10+0.20+0.05+0.05/2.5+0.05=0.42 m² K/W.
  2. Bridge flux is 30/0.42=500/7 W/m². Add rates: Q=1.6·50+0.4·500/7=760/7 W.
  3. The effective conductance per total area is 0.8/0.60+0.2/0.42=38/21 W/(m² K). Averaging resistances instead gives 0.564 and an incorrect 60/0.564≈106.383 W.

Answer and interpretation

Q=760/7≈108.571 W in the independent-column approximation.

Test the result

The result exceeds the no-bridge 100 W and lies below an all-bridge wall. Real lateral spreading would require a different model; it is explicitly omitted here.

Common mistakes

Mixing total and per-area resistance

A missing area changes units and the predicted rate.

One contact temperature

A finite contact resistance permits a jump.

Averaging a series resistance over parallel areas

Parallel paths share ΔT; add their conductances weighted by area.

Before moving on

  • Construct a dimensionally consistent thermal network.
  • Recover the contact-side temperatures.
  • Model a parallel bridge and explain its approximation.

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