Heat Conduction Calculator: Multi-Layer Walls & Cylinders

Steady-state conduction through a plane wall: Q = kAΔT/L, where k is the material's thermal conductivity, A the cross-sectional area, ΔT the temperature difference, L the wall thickness. Rewrite as Q = ΔT/R with R = L/(kA), and you can stack thermal resistance in series the same way as electrical resistors. Multi-layer wall? Add the R-values. Drywall plus insulation plus sheathing plus outside air film: R_total = ΣR_i, and Q follows from ΔT and R_total.

The trap is forgetting that a 2×6 stud wall isn't all insulation. Studs every 16 inches act as thermal bridges that turn a nominal R-19 batt into a real R-14 wall. Code U-values measure the real number, not the nominal. This page covers steady-state conduction through plane and cylindrical walls, series resistance networks, and selection criteria for insulation materials. If you're sizing pipe insulation, designing building envelopes, or analyzing heat exchangers, the resistance-addition principle is the same; material selection and code requirements differ by application.

Heat moves through a wall by every mode you can name, and the conduction calculator is the middle act. At the inside surface, warm room air transfers heat to the drywall by free convection (h ≈ 8 W/m²K) and radiation from every other surface in the room. Conduction takes over inside the wall. At the outside surface, wind-driven forced convection (h ≈ 25 W/m²K in light wind) and longwave radiation to the sky carry heat away. Sky temperature on a clear winter night can sit 20°C below ambient air, which is why frost forms on horizontal surfaces before the air drops below freezing.

Phase change shows up at the dew point. Water vapor condensing on a cold sheathing surface releases 2257 kJ/kg of latent heat right where you don't want it, soaking the insulation and dropping its k by a factor of 5 once moisture content exceeds 5% by volume. That's the physical reason building science obsesses about vapor barriers and dew-point location inside the wall stack.

Heat flows through building walls in series: inside air film, drywall, insulation, sheathing, siding, outside air film. Each layer adds its own thermal resistance, and they sum directly: R_total = R₁ + R₂ + R₃ + ... For conduction through a uniform layer R = L/(k·A), where L is thickness, k is thermal conductivity, and A is area.

That ignores framing. Studs at 16" o.c. occupy about 25% of the wall area, and wood (R-1.25/inch) carries far less R than insulation. The area-weighted average drops to R-14 to R-15. That's the reason continuous exterior insulation is often required to meet code.

Cylindrical resistance

Pipes carry heat radially through surfaces of growing area, so the resistance formula changes: R_cyl = ln(r₂/r₁)/(2πkL). The logarithm falls out of integrating Fourier's law in cylindrical coordinates. Example: 2" NPS pipe (r₁ = 30.5 mm), 2" mineral-wool insulation (r₂ = 81.5 mm), k = 0.04 W/mK gives R = ln(81.5/30.5)/(2π × 0.04 × 1) = 3.9 K·m/W per meter of pipe length.

Parallel paths and the critical radius

Studs and cavity insulation form parallel thermal paths. Use the area-weighted U: U_total = f_stud × U_stud + f_cavity × U_cavity. ISO 6946 calls these out as "upper and lower limits" and averages them. For very small pipes, adding insulation can paradoxically increase heat loss until you exceed the critical radius r_cr = k/h. With k = 0.04 W/mK and h = 10 W/m²K, r_cr = 4 mm; for any pipe larger than 8 mm bare diameter, more insulation always means less heat loss. The concept matters for instrument tubing and electrical wires, rarely for building plumbing.

All R-value math here assumes steady state: temperatures aren't changing with time. That's valid when boundary conditions are constant long enough for the temperature profile to stabilize. The characteristic time τ = L²/α_diff (where α_diff = k/(ρ·c_p) is thermal diffusivity) tells you how quickly a material responds.

For daily heating/cooling cycles, massive walls (concrete, masonry, CMU with grout fill) may never reach steady state; transient analysis is needed and thermal mass becomes a useful design feature. For process equipment with constant operating temperature, steady state is usually valid after startup.

Biot number Bi = hL/k decides which regime fits. Bi < 0.1 means lumped capacitance applies and the part is roughly isothermal at any instant. Bi > 1 means internal gradients dominate and you need 1D or 2D transient solutions. ASHRAE provides response factors and conduction transfer functions for hour-by-hour building load calculations precisely because daily cycles sit in the awkward middle range.

Thermal conductivity k spans orders of magnitude. Selection depends on temperature range, moisture exposure, fire rating, and cost. Copper at 400 W/mK is what you want in a heat sink; mineral wool at 0.035 is what you want in a wall. Same SI unit, four orders of magnitude apart.

For building insulation, foam boards have the best R/inch but may need a 15-minute thermal barrier (½" gypsum) per IBC fire code. For pipe insulation, the material must handle operating temperature and form around pipes. Always check maximum service temperature before selecting.

Beyond k

Thermal diffusivity α_diff = k/(ρ·c_p) decides transient response. Specific heat c_p sets the energy stored per kg per °C: water is 4186 J/kgK, dry concrete around 880, dry wood around 1700, EPS foam around 1500 (but the foam is so light per m³ that volumetric heat capacity ρc_p stays low). Surface emissivity ε governs radiation: bright aluminum foil ε ≈ 0.03, painted gypsum ε ≈ 0.9, oxidized galvanized steel ε ≈ 0.28. Drop ε on a roof from 0.9 to 0.3 with a cool-roof coating and you cut summer attic load by 20 to 40%.

Climate-zone targets (IECC 2021)

ASHRAE 90.1 and IECC 2021 specify maximum U-factors as assembly values that include framing fraction. To hit U-0.064 in a 2×6 wall you generally need R-20 cavity insulation plus R-5 continuous exterior insulation, not just R-16 in the stud bays.

Cold bridges (Eurocode language: thermal bridges) are the gap between calculated R and measured R. A 2×6 stud at 16" o.c. occupies 25% of the wall area at R-1.25/inch versus R-3.5/inch for the cavity insulation. Steel studs are worse: k = 50 W/mK, three orders of magnitude higher than mineral wool. ISO 6946 gives correction factors; ASHRAE 90.1 Appendix A tabulates assembly U-factors that already include framing.

Edge effects show up at window heads, foundation tops, parapet roofs, and balcony slabs. A concrete balcony slab punching through an exterior wall is a textbook two-dimensional thermal bridge that can carry 10 to 20% of the whole-wall heat loss in a single linear meter. Linear thermal transmittance ψ (W/mK per linear meter) is added on top of the area-based U:

Surface resistance accounts for the boundary layer of still air clinging to inside and outside surfaces. ISO 6946 fixes inside R_si = 0.13 m²K/W (h ≈ 7.7 W/m²K) and outside R_se = 0.04 m²K/W (h ≈ 25 W/m²K). These are seasonal averages; on a still winter night with strong sky radiation, the effective outside h drops and the outside surface can run colder than air, driving condensation on single-pane windows and uninsulated metal cladding.

Limitations and Assumptions

Problem: An exterior wall has three layers: 12 mm gypsum board (k = 0.17 W/mK), 90 mm fiberglass batt (k = 0.04 W/mK), and 100 mm brick veneer (k = 0.7 W/mK). Inside air sits at 21°C with h_i = 8 W/m²K. Outside air is -5°C with h_o = 25 W/m²K. Find the U-value, the heat flux per square meter, and the temperature at each interface. Compare to IECC 2021 climate zone targets.

The fiberglass layer carries 86% of the resistance and 86% of the temperature drop, exactly as it should. Note where the dew point lands. Inside air at 21°C and 40% RH has a dew point of about 7°C, which is reached partway through the fiberglass batt. That's why this assembly needs an interior vapor retarder in cold climates: condensation inside the insulation kills its k.

Code check: U = 0.38 W/m²K is roughly U-0.067 in IP units, just above the IECC 2021 Zone 5/6 target of U-0.064. Adding 50 mm of XPS continuous insulation outside the brick would push R_total to about 4.4 m²K/W and U to 0.23 W/m²K (≈ U-0.040), comfortably under any U.S. residential climate zone requirement.