Conduction, Convection, and Radiation Heat Transfer

Heat transfer is energy crossing a system boundary because of a temperature difference. Temperature is a state property; heat is not stored as heat inside an object. Before selecting an equation, draw the control volume and identify heat input, heat output, work, mass flow, internal generation, and stored-energy change. This prevents formulas from being applied to temperatures that do not belong to their boundaries.

At steady state, stored energy does not change with time, but temperatures need not be uniform and heat can still flow continuously. In a transient process, part of the net energy changes internal energy and temperature. A thermal resistance network describes steady linearized paths well; a thermal-capacitance model is additionally needed when storage controls warm-up or cooldown.

Sign conventions should be explicit. One approach treats heat flow into the control volume as positive and writes every boundary term consistently. Another reports positive magnitudes from hot to cold. Either can work, but switching conventions halfway creates apparent energy imbalance. Always verify that predicted heat flows from higher to lower temperature in a passive element.

Fourier's law relates conductive heat flux to the negative temperature gradient. The negative sign states that heat flows toward lower temperature. Thermal conductivity k describes the material's ability to conduct at a specified state. Metals often have high conductivity, insulating solids and gases lower values, and anisotropic composites can conduct differently by direction.

For steady one-dimensional conduction through a plane wall with constant k, heat rate is kA(T_hot - T_cold)/L. Cylinders and spheres use logarithmic or radial geometry because area changes with radius. Temperature-dependent conductivity, internal heat generation, contact resistance, multidimensional spreading, or transient storage require extended models.

Conduction also exists in fluids. At a solid-fluid interface, molecular conduction carries energy through the stationary no-slip layer before fluid motion transports it away. Treating convection as a wholly separate microscopic mechanism can obscure this connection. The convection coefficient packages the combined boundary-layer response rather than replacing Fourier's law at the wall.

Related in this workflow: Conduction Heat Transfer Calculator, Thermal Resistance Calculator.

Newton's law of cooling writes convective heat rate as hA(T_s - T_infinity). The coefficient h is not a basic material property. It depends on fluid properties, velocity, geometry, orientation, surface condition, flow regime, boundary-layer development, and whether motion is forced or buoyancy driven. A value copied from a broad table is usually an order-of-magnitude estimate.

Forced convection uses an external device or pressure difference to move fluid, as with a fan or pump. Natural convection arises when temperature-dependent density differences interact with gravity. Mixed convection occurs when both effects matter. Dimensionless correlations connect h to Reynolds, Prandtl, Grashof, Rayleigh, and Nusselt numbers over documented ranges.

The bulk fluid temperature requires definition. For external flow, a free-stream value may be suitable; for internal flow, the bulk mean temperature changes along the passage. Surface temperature may also vary. Using one average temperature difference can be acceptable for a preliminary estimate, but heat exchangers often require a log-mean difference or a distributed energy balance.

Related in this workflow: Convection Heat Transfer Calculator, Reynolds Number Calculator.

Every surface above absolute zero emits thermal radiation. Net exchange depends on absolute temperature, emissivity, geometry, and irradiation from other surfaces. For a small gray surface facing large isothermal surroundings, a common expression is epsilon sigma A(T_s^4 - T_sur^4). Temperatures must be absolute because the fourth-power law is based on thermodynamic temperature.

Emissivity ranges between zero and one for the gray-surface model and depends on material, finish, oxidation, wavelength, direction, and temperature. A polished metal can radiate very differently from an oxidized or coated surface. Solar absorptivity and thermal emissivity need not be equal when incoming and emitted spectra occupy different wavelength ranges.

Multiple finite surfaces exchange radiation according to view factors and radiosity. Shields, openings, specular reflection, participating gases, flames, and wavelength-selective surfaces require more detailed treatment. Radiation does not require matter between surfaces, but an intervening medium can absorb, emit, or scatter and thereby alter exchange.

Related in this workflow: Radiation Heat Transfer Calculator, Temperature Converter.

Consider heat leaving warm indoor air through a wall to cold outdoor air. Convection carries energy from indoor bulk air to the inner surface, conduction moves it through wall layers, and outdoor convection carries it away. The exterior surface may also exchange radiation with sky and surroundings. Some paths are in series through the wall, while radiation and convection can leave the same surface in parallel.

At a surface energy balance, conductive heat arriving from one side can equal the sum of convective and radiative heat leaving the other at steady state. Because radiation is nonlinear in temperature, the surface temperature may need iteration. A linearized radiation coefficient can combine with convection for a limited temperature range, but its value depends on the operating temperatures and should not be treated as universal.

Contact interfaces add resistance when microscopic roughness leaves imperfect contact and interstitial gaps. Fouling adds deposits, and fins add extended area with a nonuniform temperature. A complete thermal network should include only physically distinct resistances and use a consistent area basis. Double-counting surface films or omitting contact resistance can both distort the result.

Take a 2 m2 wall panel with 0.08 m insulation of conductivity 0.04 W/(m K). Indoor and outdoor convection coefficients are 8 and 25 W/(m2 K), with bulk temperatures 22 degrees C and 2 degrees C. The indoor convection resistance is 1/(8 x 2) = 0.0625 K/W, wall conduction resistance is 0.08/(0.04 x 2) = 1.0 K/W, and outdoor convection resistance is 1/(25 x 2) = 0.020 K/W.

The total series resistance is 1.0825 K/W. Ignoring radiation and thermal bridges for this example, heat loss is (22 - 2)/1.0825 = 18.48 W. The inside surface is approximately 22 - 18.48 x 0.0625 = 20.85 degrees C, and the outside surface is about 2 + 18.48 x 0.020 = 2.37 degrees C.

The small heat rate reflects the strong insulation resistance relative to the two convection films. If radiation to the outdoor surroundings were material, it would add a parallel surface path and change the exterior surface balance. Real wall framing, fasteners, joints, moisture, air leakage, and multidimensional thermal bridges can increase total heat transfer beyond the one-dimensional insulated-area estimate.

Thermal conductivity, viscosity, density, heat capacity, emissivity, and convection correlations depend on temperature and sometimes pressure or composition. Evaluate properties at a justified film, mean, bulk, or local state according to the method. Large temperature ranges may require segmenting or integrating rather than using one value.

Conduction and convection can use temperature differences in kelvin or degrees Celsius because the increment is the same, but radiation requires absolute temperatures in the fourth power. Areas and coefficient bases must match. A cylindrical resistance based on length cannot be mixed with a plane-wall area without transformation. Track W, W/m2, K/W, and m2 K/W carefully because each represents a different quantity.

Report enough significant figures for reproducibility but not more than property and geometry accuracy justify. Convection coefficients are often the least certain input. A sensitivity band for h, emissivity, fouling, or contact resistance can reveal whether refining another parameter would materially improve the estimate.

A frequent error is treating convection coefficient as a fixed fluid property. Another is using Celsius values in the radiation fourth-power expression. A third is adding parallel heat paths as if they were series resistances. Draw the path and identify shared temperature nodes before combining terms. Heat rates add in parallel; temperature drops add along a series path.

Using total area for one layer and exposed area for another can introduce hidden basis errors. In radial systems, area changes with radius. In fins, not all added area remains at base temperature, so fin efficiency matters. In heat exchangers, one overall coefficient must use the same reference area as UA and the resistance terms converted to that basis.

Boundary temperatures are also confused. Ambient air temperature is not automatically the same as mean radiant temperature, and a surface temperature is not the same as fluid bulk temperature. Thermocouple readings can be affected by conduction along wires and radiation. Measurements need an uncertainty and placement review before they are treated as exact boundary conditions.

The equations shown are idealized building blocks. Phase change, reacting flow, participating media, porous materials, moisture migration, anisotropy, turbulence, thermal contact variation, conjugate multidimensional heat transfer, and transient storage can require more advanced models. Correlations should not be extrapolated beyond their geometry and parameter ranges.

Use the calculators to assemble transparent preliminary balances and compare modes, then verify boundary conditions, property sources, coefficient correlations, area bases, and model dimensionality. Safety-critical temperatures, pressure-bound equipment, fire exposure, thermal protection, electronics reliability, and code-regulated systems require applicable standards, validated analysis, testing where appropriate, and qualified engineering review.