# Heat transfer: preserve the circuit when paths become parallel — Study worksheet [Study guides](https://unireadinglist.com/guides/heat-transfer/) [Course topic: Heat and Mass Transfer](https://unireadinglist.com/subjects/engineering/#course-heat-transfer) Sources checked: 2026-11-04 Study route and exercise: University Reading List editorial project. 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 · 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. Your notes: ________________________________ [Official free reading](https://ahtt.mit.edu/) [Book record and access options](https://unireadinglist.com/subjects/engineering/a-heat-transfer-textbook/) ### Fundamentals of Heat and Mass Transfer Theodore L. Bergman; Adrienne S. Lavine; Frank P. Incropera; David P. DeWitt · 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. Your notes: ________________________________ ISBN: 9781119353881 [Book record and access options](https://unireadinglist.com/subjects/engineering/fundamentals-heat-mass-transfer/) ### 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. Your notes: ________________________________ ISBN: 9781107179530 [Book record and access options](https://unireadinglist.com/subjects/engineering/introduction-to-engineering-heat-transfer-nellis-klein/) ## Check your preparation ### Can I keep heat rate and heat flux separate? Your notes: ________________________________ ### Is temperature continuous across this contact? Your notes: ________________________________ ### What changes for parallel columns? Your notes: ________________________________ ## A six-step study plan - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ ## 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. Your notes: ________________________________ ## 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. Your notes: ________________________________ ## Before moving on - Construct a dimensionally consistent thermal network. - Recover the contact-side temperatures. - Model a parallel bridge and explain its approximation. ## Answer and interpretation ### 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 wall with two films and a contact jump 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. 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 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. 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. ## Sources and edition checks - [Authors: sixth-edition heat-transfer text](https://ahtt.mit.edu/) The sixth-edition author text covers thermal resistance and expects elementary thermodynamics and fluid mechanics. - [Wiley: exact eighth-edition product](https://www.wiley.com/en-us/fundamentals-of-heat-and-mass-transfer-8th-edition-p-9781119353881) The exact selected eighth-edition product is the 2018 ISBN; other formats have different release fields. - [Cambridge: selected heat-transfer hardcover](https://assets.cambridge.org/97811071/79530/copyright/9781107179530_copyright_info.pdf) The selected hardcover copyright page says first published 2021; its modeling route is not a guarantee of software entitlement. 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