At a glance
A route from temperature difference to heat duty
Heat-transfer problems become much easier when you ask the same questions in order: what is the driving temperature difference, what path must heat follow, what resistance does each part of that path add, and is the process steady or changing with time? This guide builds that chain from first principles and then applies it to walls, equipment and heat exchangers.
Steady one-dimensional plane-wall form for constant thermal conductivity.
Surface-to-fluid heat transfer; h depends on the flow and geometry.
Use absolute temperature. Emissivity and geometry matter.
A common language for layered conduction and convection films.
01 // The three modes of heat transfer
The modes are different physical mechanisms, but real engineering systems often use several at the same time.
Conduction
Energy moves through a stationary material because microscopic particles and, in metals, free electrons exchange energy. Thermal conductivity k measures how readily a material conducts heat. High-k materials spread heat effectively; low-k materials insulate.
Convection
Convection combines conduction at a surface with motion in an adjacent fluid. Natural convection is driven by buoyancy; forced convection is driven by fans, pumps or bulk flow. The engineering coefficient h condenses this complex boundary-layer behavior into a usable surface relation.
Radiation
All matter above absolute zero emits electromagnetic radiation. Net radiative transfer depends strongly on absolute temperature, surface emissivity and the geometry between surfaces. Unlike conduction and convection, radiation does not require a material medium.
02 // Heat rate, heat flux and energy balance
Before choosing an equation, separate total thermal power from heat flow per unit area.
| Quantity | Typical symbol | SI unit | Meaning |
|---|---|---|---|
| Heat / thermal energy | Q | J | An amount of transferred energy |
| Heat-transfer rate | q̇ | W = J/s | Thermal power transferred through the whole surface or device |
| Heat flux | q″ | W/m² | Heat-transfer rate divided by area |
| Volumetric heat generation | q‴ | W/m³ | Heat generated inside a volume, such as electrical or nuclear heating |
At steady state with no energy storage, the energy entering a control volume must equal the energy leaving plus any work or other defined sinks. In transient problems, the difference appears as a change in stored internal energy. That single distinction — steady versus transient — determines whether temperatures can be treated as fixed or must be solved as functions of time.
03 // Conduction and thermal resistance
Fourier's law links heat flux to a temperature gradient. For a plane wall with constant k, fixed face temperatures and one-dimensional steady conduction, it reduces to a simple resistance form.
| Form | Relationship | Use |
|---|---|---|
| Plane-wall conduction | q̇ = kA(T₁ − T₂)/L | One homogeneous layer |
| Resistance of one layer | R = L/(kA) | Express the layer as K/W |
| Resistance per unit area | R″ = L/k | Convenient for walls and U-values |
| Series network | q̇ = ΔT / ΣR | Multiple layers and films in one heat-flow path |
Thickness increases resistance linearly, while higher conductivity lowers it. In cylindrical insulation the area changes with radius, so the plane-wall expression no longer applies directly; pipe heat transfer uses the logarithmic cylindrical resistance instead.
04 // Interactive: plane-wall conduction
Change material conductivity, area, thickness and temperature difference to see how each affects steady heat transfer.
Assumes one-dimensional steady conduction, constant k, no contact resistance and uniform temperatures on each face.
05 // Convection and the heat-transfer coefficient h
Newton's law of cooling looks simple; determining h is where fluid mechanics enters the problem.
The engineering relation q̇ = hA(Ts − T∞) describes heat transfer between a surface and the bulk fluid. The coefficient h is not a fixed property of air or water. It changes with velocity, fluid properties, geometry, orientation, surface condition, and whether flow is laminar, transitional or turbulent.
Correlations therefore use dimensionless groups. Reynolds number characterizes the flow regime, Prandtl number compares momentum and thermal diffusivity, and Nusselt number expresses convective heat transfer relative to pure conduction across the fluid layer. A typical workflow is: determine Re and Pr → select a correlation for Nu → calculate h from Nu = hL/k.
Natural convection
Buoyancy creates motion because warmer and cooler fluid have different densities. Orientation and gravity matter, so a hot horizontal surface behaves differently when facing upward versus downward.
Forced convection
A fan, pump or imposed flow raises fluid velocity and usually raises h. This is why forced-air heat sinks, cooling-water loops and high-flow heat exchangers can move much more heat than stagnant surroundings.
Boundary layer
Near a wall the fluid velocity approaches the surface velocity and a thermal gradient develops. Much of convection engineering is really the prediction of this momentum and thermal boundary-layer behavior.
06 // Thermal radiation and emissivity
Radiation becomes increasingly important as absolute temperature rises because emission scales with the fourth power of temperature.
For a small gray surface exchanging radiation with a much larger isothermal surrounding, a common engineering form is q̇ = εσA(Ts⁴ − Tsur⁴). Here ε is emissivity and σ is the Stefan–Boltzmann constant. Temperatures must be in kelvin.
Real enclosures can require view factors, multiple reflections and surface-to-surface radiation networks. A polished metal surface can have very low emissivity, while many paints, oxides and nonmetallic surfaces have much higher emissivity. The same surface treatment can therefore change radiation without materially changing conduction through the object.
07 // Composite walls, films and overall U-value
A useful engineering trick is to convert every series heat-transfer step into a resistance and then add them.
For a flat wall with inside convection, two solid layers and outside convection, the resistance per unit area is:
R″total = 1/hi + L₁/k₁ + L₂/k₂ + 1/ho
The overall coefficient is then U = 1/R″total, and the heat rate becomes q̇ = UAΔT. This same resistance-network idea appears in building envelopes, heat exchangers, insulated vessels and many thermal systems.
08 // Interactive: resistance network & overall U-value
Build a simple two-layer plane wall including inside and outside convection films.
The model uses constant properties, one-dimensional heat flow and no thermal bridges or contact resistances.
09 // Heat exchangers: LMTD, capacity rate and effectiveness
Heat exchangers bring two fluid streams into thermal contact without necessarily mixing them.
LMTD method
When inlet and outlet temperatures are known, the log-mean temperature difference represents the changing driving force along the exchanger. Heat duty is commonly written q̇ = UAFΔTlm, where F corrects for arrangements that differ from ideal parallel or counterflow.
Capacity rate
Each stream has heat-capacity rate C = ṁcp. Ignoring losses and phase change, the heat lost by the hot stream equals the heat gained by the cold stream: q̇ = Ch(Th,in−Th,out) = Cc(Tc,out−Tc,in).
Effectiveness–NTU
When outlet temperatures are unknown, effectiveness ε = q̇/q̇max and NTU = UA/Cmin provide another route. The ε–NTU relation depends on flow arrangement and the capacity-rate ratio Cr = Cmin/Cmax.
10 // Interactive: LMTD and idealized heat duty
Compare parallel-flow and counterflow temperature driving force for the same four terminal temperatures.
Idealized q̇ = UAΔTlm with correction factor F = 1. Real exchanger rating also checks stream energy balance, pressure drop, fouling, property variation and geometry.
11 // Transient heat transfer: when temperature changes with time
Steady-state equations cannot predict how quickly a hot object cools or a cold object warms.
A body stores thermal energy according to its mass and heat capacity. If internal conduction is fast enough that the body remains nearly uniform in temperature, the lumped-capacitance model gives an exponential response:
(T − T∞)/(Ti − T∞) = exp[−hAt/(ρVcp)]
The first screening check is the Biot number, Bi = hLc/k. When Bi is roughly below 0.1, internal temperature gradients are often small enough for lumped analysis. Larger Bi generally requires spatial transient-conduction methods, charts or numerical solutions. The Fourier number, Fo = αt/Lc², compares elapsed time with the material's thermal-diffusion timescale.
12 // Phase change: boiling and condensation
Latent heat can move large amounts of energy with relatively little bulk temperature change.
During boiling or condensation, heat transfer is tied to mass changing phase: q̇ ≈ ṁ hfg for an idealized phase-change duty. In boiling, the heat-transfer coefficient can rise dramatically when nucleate boiling begins, but operation is not unlimited. At sufficiently high surface heat flux, vapor can blanket the surface and sharply reduce cooling; the transition is associated with critical heat flux (CHF). Condensation likewise depends on film behavior, orientation, non-condensable gases and surface condition.
Because these regimes rely strongly on empirical correlations and fluid properties, phase-change equipment should not be designed from a single generic coefficient.
Explore boiling & CHF
Use the EngineerHub boiling visual simulator to see how wall superheat, nucleate boiling and departure toward critical heat flux change the heat-transfer regime.
Condenser heat load
Connect steam condensation duty to cooling-water temperature rise and exchanger effectiveness.
Steam power cycle
See where boiler heat addition and condenser heat rejection sit in the Rankine cycle.
13 // Put the concepts together
A practical sequence for most first-pass heat-transfer problems.
| Step | Question | Typical next move |
|---|---|---|
| 1 | Steady or transient? | Decide whether energy storage must be included. |
| 2 | What modes matter? | Identify conduction, convection, radiation and any phase change. |
| 3 | What is the geometry? | Plane wall, cylinder, sphere, fin, exchanger, enclosure, etc. |
| 4 | What properties are needed? | k, cp, ρ, μ, emissivity, fluid properties — at the right temperature. |
| 5 | Can a resistance network be used? | Combine series/parallel paths and determine U or total resistance. |
| 6 | Are empirical correlations required? | Choose appropriate Re/Pr/Nu, boiling or condensation correlations. |
| 7 | Does the result satisfy energy balance? | Cross-check heat duty on both sides and test limiting cases. |
14 // Background, FAQ, references and limitations
Expand for deeper context, common questions and the technical references behind the guide.
Modern heat-transfer engineering combines conservation of energy with three transport mechanisms. Fourier's law describes conduction, convection correlations link boundary-layer flow to surface heat transfer, and radiation theory handles electromagnetic exchange. These building blocks are then assembled into resistance networks, heat-exchanger methods and transient models.
The key skill is therefore not memorizing isolated equations; it is recognizing which physical resistances and energy-storage terms control the problem, then choosing the simplest model whose assumptions still match the real system.
What are the three modes of heat transfer? Conduction transfers energy through matter, convection couples a surface to moving fluid, and thermal radiation transfers energy by electromagnetic emission and absorption. Most real systems combine them.
What is the difference between heat rate and heat flux? Heat rate is total thermal power in watts; heat flux is heat rate per unit area in W/m².
What is thermal resistance? It is a convenient way to represent how a layer or interface opposes heat flow. Series resistances add, allowing layered walls and convection films to be reduced to an overall U-value.
Is the convection coefficient h a fluid property? No. It depends on fluid properties, but also velocity, geometry, orientation and flow regime. It normally comes from an appropriate correlation, experiment or validated model.
Why must radiation use kelvin? The Stefan–Boltzmann relation uses absolute temperature to the fourth power. Celsius is not an absolute scale.
When is lumped transient analysis valid? A common screening criterion is Bi = hLc/k below about 0.1, indicating relatively small internal temperature gradients.
What is LMTD? It is the logarithmic mean of the temperature difference between hot and cold streams at the two ends of a steady heat exchanger, with the end differences defined according to flow arrangement.
Does increasing insulation always reduce heat transfer? For plane walls, adding low-conductivity insulation increases resistance and lowers heat flow. Cylindrical systems can exhibit a critical-radius effect when added insulation increases external area enough to strengthen convection before resistance dominates.
- ASHRAE Handbook—Fundamentals 2025, Chapter 4: Heat Transfer — engineering treatment of conduction, convection and radiation.
- ASHRAE Handbook—Fundamentals 2025, Chapter 26: Material Properties — thermal-property reference data.
- U.S. DOE Fundamentals Handbook — Heat Transfer and Fluid Flow, Volume 2 — archived training reference for heat-transfer fundamentals.
- NIST/CODATA Fundamental Physical Constants — reference constants including the Stefan–Boltzmann constant.
Material conductivities shown on this page are rounded educational values, not design values for a particular product or temperature.
The three interactive examples deliberately use simplified textbook models. The conduction example assumes a plane wall with constant k. The resistance-network example assumes one-dimensional series heat flow with constant film coefficients and no thermal bridges. The LMTD example uses F = 1 and does not independently enforce stream mass-flow energy balance.
Real engineering analysis may require multidimensional finite-element or CFD models, temperature-dependent properties, contact resistance, fins, radiation view factors, fouling, pressure drop, boiling/condensation correlations, critical-heat-flux margins, manufacturing tolerances and applicable design codes. Use qualified engineering judgement for design decisions.