Methods
Fourier · Newton · Stefan–Boltzmann · Series resistance (cylindrical & plane) · Lumped capacitance
Standards
ASHRAE Handbook of Fundamentals (2025) · ISO 6946:2017
References
Incropera 8th ed. · Çengel 6th ed. · CRC Handbook · CIBSE Guide C · NIST WebBook
Unit basis
SI input · SI/IP output (display)
Status
Preliminary · reference-informed · engineer verification required
Scope
DENOVA Heat Transfer is a teaching-grade calculator built around the three fundamental modes of heat transfer — conduction, convection, and radiation — plus a set of applications that combine them for real HVAC and refrigeration work (pipe / duct + insulation, composite wall, and transient lumped response). The three modes and the steady-state applications all assume steady state: temperatures are not changing with time and no energy accumulates inside the system — every joule that enters one side leaves the other. The one exception is the transient (lumped-capacitance) application, which deliberately models how a small object's temperature changes over time. Geometric simplifications (1-D plane wall, lumped surface coefficient, grey-body small-surface-in-large-enclosure, cylindrical series resistance) cover the great majority of introductory and many practical analyses.
Conduction · Fourier's law
For a 1-D plane wall of thickness L and area A with constant thermal conductivity k and the two faces held at Th and Tc:
- Heat transfer rate Q = k · A · (Th − Tc) / L
- Linear in temperature difference. The temperature profile through the wall is exactly linear at steady state with constant k.
- Thermal resistance R = L / (k · A)
- Lets us write Q = ΔT / R. Resistances in series add directly — useful for composite walls.
- U-value U = k / L
- Conductance per unit area, the unit used by building codes for envelope performance.
Convection · Newton's law of cooling
For a surface at Ts exposed to a moving fluid of bulk temperature T∞:
- Heat transfer rate Q = h · A · (Ts − T∞)
- The convective coefficient
h rolls fluid properties, flow regime, and geometry into a single number. Realistic values span four orders of magnitude — from ~5 W/m²·K for still air to >10,000 W/m²·K for condensing steam.
- Thermal resistance R = 1 / (h · A)
- Like conduction, lets us write Q = ΔT / R and chain modes in series.
Radiation · Stefan–Boltzmann
For a grey surface of emissivity ε and absolute temperature Ts facing a large enclosure at Tsurr:
- Heat transfer rate Q = ε · σ · A · (Ts⁴ − Tsurr⁴)
- Temperatures must be absolute (kelvin) — the model converts °C or °F automatically. σ = 5.6704 × 10⁻⁸ W/m²·K⁴ is the Stefan–Boltzmann constant.
- Linearized coefficient hr = ε · σ · (Ts² + Tsurr²) · (Ts + Tsurr)
- Lets radiation be treated like convection — combine with a convective coefficient by simple addition at a surface. Only valid as a local approximation; recomputes whenever Ts or Tsurr changes.
- Blackbody emissive power Eb = σ · T4
- The thermal radiation emitted by a perfect emitter at temperature T. A real surface emits
ε · Eb.
Pipe / duct + insulation · series resistance network
A cylindrical composite-and-combined model: hot or cold fluid inside, pipe (or duct) wall, insulation layer, ambient outside, with convection films on both surfaces and optional radiation in parallel at the outer surface. Every layer becomes a resistor in series; the total drives the heat rate. The same math handles liquid pipes, HVAC supply ducts, flue gases, and refrigerant lines — just pick the preset that matches, or dial in your own geometry and boundary conditions. For square or rectangular ducts the cylindrical formulas give a close approximation when you use the equivalent hydraulic diameter Dh = 4A / P.
- Network Q = (T∞,i − T∞,o) / (Rconv,i + Rpipe + Rins + Rconv,o)
- Each interior temperature is recovered by applying
Q · R to the segment leading up to it.
- Cylindrical conduction R = ln(router / rinner) / (2π · k · L)
- The logarithm replaces the plane-wall
L / (k · A) because the heat-flow area grows with radius. For thin walls (router ≈ rinner) this reduces to the plane-wall form.
- Cylindrical convection R = 1 / (h · 2π · r · L)
- Surface area uses the radius at the actual film location — r1 for the inside film, r3 for the outside film.
- Combined outer film ho,eff = ho + hrad
- When outer emissivity
εo > 0, radiation acts in parallel with convection at the outer surface. Because hrad depends on the outer surface temperature (which in turn depends on the total resistance), the solver iterates the outer film coefficient until the temperature stops moving.
- Critical radius rcrit = kins / ho,eff
- For small-diameter pipes or wires, the outer convective resistance falls (more surface area) as you add insulation before the insulation resistance starts to dominate. If
r3 < rcrit, adding insulation can actually increase heat loss. Shown as a dashed amber circle in the diagram when it falls in a visible range.
Composite wall · plane-wall series resistance
A multi-layer plane wall with convective films on each side — the canonical model behind every HVAC envelope U-value and refrigerated-panel sizing calc. Each layer adds its own R = L/k in series, the two films cap the series with 1/h on each side, and the overall U-value is simply 1/Rtotal. The interface temperatures fall out of the network walk-through and tell you where the steepest gradient is (almost always the insulation layer) and whether any surface is at risk of being below the dew-point of the adjacent air.
- Total resistance Rtotal = 1/hi + Σ(Ln/kn) + 1/ho
- Per unit area (m²·K/W). Films and layers are all in series, so resistances simply add. The biggest term dominates the U-value — for a typical insulated wall that's the insulation layer, often by an order of magnitude over everything else.
- Overall U-value U = 1 / Rtotal
- The headline number for envelope calcs. Lower is better. ASHRAE 90.1 prescriptive U-factors for opaque walls are roughly 0.35 W/m²·K (Climate Zone 4–5, steel-frame); for refrigerated walk-ins they're typically 0.2–0.3 W/m²·K depending on temperature class.
- Heat flux & heat flow q″ = U · (Ti − To), Q = q″ · A
- Sign follows Ti − To: positive when inside is warmer (heating-season loss), negative when outside is warmer (cooling-season gain or refrigeration ingress). The KPI cards show magnitudes; the direction indicator shows which way energy moves.
- Interface temperatures Tsi, T12, T23, Tso
- Found by walking the R-network from Ti: Tsi = Ti − q″/hi, then subtract q″·Rn at each layer. The slope of the T(x) line inside each layer scales with that layer's R, which is why the visualisation shows almost all the temperature drop happening across the insulation.
Transient · lumped capacitance
The only non-steady mode in the tool. Models a single small object cooling or heating in a fluid environment when temperature gradients inside the object are negligible — i.e., the object's temperature is essentially uniform at any instant. Useful for thermocouple response time, quench-bath behaviour, small-component thermal response, and any first-pass time-constant estimate. Valid when the Biot number Bi = h·Lc/k ≤ 0.1; beyond that, internal gradients matter and a 1-D transient solver (Heisler charts or finite difference) is needed.
- Time constant τ = ρ · V · c_p / (h · A_s)
- The characteristic time scale of the response. After one τ the temperature has covered 63.2% of the way from T0 to T∞; after 3τ, ~95%; after 5τ, ~99%. ρVcp is the object's thermal capacitance; hAs is its surface conductance. Their ratio gives a time.
- Temperature at time t T(t) = T∞ + (T0 − T∞) · e−t/τ
- An exponential decay toward the asymptote T∞. The driving difference (T0 − T∞) shrinks by a factor of e for every τ that passes. Reversible — if the object is colder than the fluid, T(t) rises toward T∞ from below.
- Characteristic length Lc = V / As
- Volume-to-surface-area ratio. For a sphere of radius r: Lc = r/3. For a plate of half-thickness L: Lc = L. For a long cylinder of radius r: Lc = r/2.
- Biot number Bi = h · Lc / k
- Compares the object's surface conductance (h) to its internal conductance (k/Lc). When Bi ≪ 1, surface convection is the bottleneck and the interior is essentially isothermal — lumped capacitance applies. When Bi ≳ 0.1, internal gradients matter; when Bi ≳ 0.5, the lumped model is genuinely wrong.
- Cumulative energy Q(t) = ρVcp·|T0 − T∞|·(1 − e−t/τ)
- The total energy that has crossed the surface from t = 0 to the queried time. Approaches the full thermal-capacitance change ρVcp·|ΔT0| as t → ∞.
Modelling assumptions
- Steady state — no transient terms. Energy in = energy out at every surface.
- Conduction — 1-D, homogeneous materials, constant
k. Plane wall for the conduction mode; cylindrical (single layer or pipe + insulation) for the pipe mode; stacked plane layers for the composite wall.
- Composite wall — series resistance through plane layers with inside/outside convective films, constant
k per layer. No thermal bridging, framing factors, or 2-D corner/edge effects — for code-compliance U-factors apply the relevant framing correction separately.
- Transient — lumped capacitance: the object's interior is treated as isothermal (valid for Biot ≤ 0.1), with constant properties and a constant film coefficient over the response.
- Convection —
h is taken as a constant. Real h depends on local temperature, flow, and orientation; values shown in presets are mid-range textbook figures.
- Radiation — grey-body assumption (emissivity independent of wavelength), small surface in a large enclosure (so view factor → 1 and the enclosure acts as a blackbody at
Tsurr).
- Sign convention — Q reported as positive in the direction of heat flow. If Tcold > Thot the magnitude is preserved and the direction reversed in the status bar.
Data sources & accuracy notes
Every preset in this calculator is a representative textbook value, not a measurement of a specific commercial product. Values were cross-checked against the references below in May 2026; what follows are the verified ranges and the small adjustments that came out of that audit.
- Thermal conductivity k (47 materials)
-
Pure metals agree across handbooks to within ~2%: silver 428, copper 401, gold 315, aluminum 237, nickel 91, lead 35, titanium 22 W/m·K (CRC / Smithells / Incropera 8th ed. Table A.1). Alloys vary ±10–20% with composition and temper: stainless 304 at 14–17, carbon steel 50–55, brass 70/30 at 110–125, aluminum 6061-T6 at 167. Insulators are nominal values from ASHRAE Handbook of Fundamentals — VIP 0.004–0.008, silica aerogel 0.013–0.020, PUR foam 0.022–0.035, fiberglass batt 0.038–0.046. Real conductivity depends on density, moisture, and mean temperature. Diamond set to 2200 W/m·K for natural type IIa monocrystal (synthetic isotopically-pure can reach 3300). Liquids at 20 °C from NIST Webbook: water 0.60, pure ethylene glycol 0.25 (note: 50/50 EG-water antifreeze is higher, ~0.41), engine oil 0.15. Air at 20 °C is 0.0257, rounded to 0.026.
- Convection coefficient h (23 scenarios)
-
These are order-of-magnitude characteristic values from the standard regime tables (Incropera Table 1.1, Çengel Table 1-5, Engineering Toolbox). Verified ranges: free convection gas 2–25, free convection liquid 50–1000, forced gas 25–250, forced liquid 50–20 000, boiling/condensing 2 500–100 000. Specific values in this app: still air 5, gentle fan 25, strong wind 100, HVAC duct 35, water pipe flow 1 000, fast pipe flow 3 000, pool boiling 5 000, dropwise condensation 50 000. Actual h for a specific situation depends on flow rate, geometry, fluid properties, and surface orientation — calculate via Nusselt-number correlations for design.
- Emissivity ε (29 surfaces)
-
Total hemispherical emissivities at moderate temperatures, from Incropera Table A.11 and the CIBSE Guide Table C3.7. Verified spot-checks: silver polished 0.02, copper polished 0.03, aluminum polished 0.04, aluminum anodized 0.77, galvanized iron 0.28, paint white 0.90, paint matte black 0.95–0.97 (set to 0.97), brick rough 0.93, water 0.96, skin 0.97. Stainless polished refined from 0.15 to 0.17 (textbook range 0.16–0.21). Surface finish drives a 5× swing for metals: polished aluminum is 0.04 but heavily oxidized is 0.20–0.31.
- Pipe / duct scenarios (10 configurations)
-
Plausible textbook-style configurations — geometries are common pipe / duct sizes (¾″ copper, 50 mm sched-40 steel, 400 mm round duct), temperatures are typical industry values (low-pressure steam 150 °C, chilled-water supply 7 °C, HVAC cool supply 13 °C, district heating supply 110 °C, single-wall flue 200 °C). They illustrate the physics, not certify a specific installation. Duct wall thickness is shown at ~5–10 mm for visual clarity even though real residential sheet-metal ducts are 0.6–1.0 mm — wall resistance is negligible compared to insulation either way, so the calculated Q is unaffected.
- Composite wall assemblies (4 configurations)
-
Representative envelope and refrigeration build-ups, layer conductivities from the same ASHRAE / Incropera tables as above: residential wood-frame (12.7 mm drywall · R-13 fiberglass batt · 12 mm OSB, 22 °C → −5 °C winter), commercial steel-stud (16 mm gypsum · 100 mm mineral wool · 90 mm brick veneer), cold-storage panel (steel skin · 100 mm PIR foam · steel skin, −20 °C room → 35 °C ambient), and refrigerator wall (steel · 50 mm PU foam · plastic liner). Film coefficients use ASHRAE surface conductances (~8 W/m²·K still indoor air, ~25 W/m²·K outdoor). These are 1-D centre-of-panel U-factors — they do not include the framing/thermal-bridging correction needed for code-compliance whole-wall U-factors, which you must apply separately.
- Physical constants
-
Stefan–Boltzmann σ = 5.670 374 419 × 10⁻⁸ W/m²·K⁴ (CODATA 2018, exact under the 2019 SI redefinition). Kelvin offset 273.15. Both stored to full precision and used directly in the calculations.
Use this for learning and rough analysis, not for safety-critical or code-compliance design. Real heat-transfer engineering needs measured material data and validated correlations.
References
- Incropera, F., DeWitt, D., Bergman, T., Lavine, A. — Fundamentals of Heat and Mass Transfer, 8th ed., Wiley, 2017. (Property tables A.1–A.11 in appendices; convection-regime values in Ch. 6, 8, 9; cylindrical resistance in Ch. 3.)
- Çengel, Y., Ghajar, A. — Heat and Mass Transfer: Fundamentals and Applications, 6th ed., McGraw-Hill, 2020. (Tables A-1 through A-19 for material properties; Ch. 7–10 for convection correlations.)
- ASHRAE — 2025 ASHRAE Handbook — Fundamentals (SI and I-P editions), American Society of Heating, Refrigerating and Air-Conditioning Engineers. (Ch. 25 "Heat, Air, and Moisture Control in Building Assemblies — Fundamentals"; Ch. 26 "Heat, Air, and Moisture Control — Material Properties"; surface conductances and U-factor calculation conventions used for the composite-wall presets.)
- ISO — ISO 6946:2017, Building components and building elements — Thermal resistance and thermal transmittance — Calculation methods, 3rd ed., International Organization for Standardization (last reviewed and confirmed 2022; corrigendum EN ISO 6946:2017+LC:2021 for European adoption). Series-resistance method for plane multilayer assemblies; this is the standard the composite-wall mode follows for centre-of-panel Rtotal and U.
- Haynes, W. M. (ed.) — CRC Handbook of Chemistry and Physics, 97th ed., CRC Press, 2016. Section 12 "Properties of Solids — Thermal Conductivity of Pure Metals" (values at 300 K used for the metal presets; pure-element k values are essentially invariant across recent editions).
- Gale, W. F. & Totemeier, T. C. (eds.) — Smithells Metals Reference Book, 8th ed., Butterworth-Heinemann, 2004. (Alloy thermal conductivities used to cross-check the metal alloy presets.)
- Mills, A. — Basic Heat and Mass Transfer, 2nd ed., Prentice Hall, 1999. (Emissivity tables, Appendix A.)
- CIBSE — Guide C: Reference Data, Chartered Institution of Building Services Engineers (Section C3, emissivity tables for building materials and construction surfaces).
- NIST — NIST Chemistry WebBook, SRD 69, National Institute of Standards and Technology (webbook.nist.gov/chemistry — fluid thermophysical properties for water, air, refrigerants and glycol mixtures used to set the liquid-property presets).
- Touloukian, Y. S. et al. — Thermophysical Properties of Matter, Vol. 3 (Thermal Conductivity, Nonmetallic Solids), IFI/Plenum.
- CODATA 2018 — recommended values of the fundamental physical constants.