3.2 Heat transfer mechanisms

Heat transfer mechanisms describe how energy moves from one place to another due to temperature differences. The three main types are conduction, convection, and radiation, each governed by distinct physics and quantified by different equations.

These mechanisms matter for designing any thermal system. Factors like temperature gradients, surface area, and material properties determine how quickly heat flows, which affects everything from building insulation to engine cooling to spacecraft design.

Heat Transfer Mechanisms

Mechanisms of heat transfer

Conduction transfers heat through direct contact between particles. More energetic (hotter) particles collide with less energetic (cooler) neighbors, passing kinetic energy along. It occurs in solids, liquids, and gases, but solids conduct best because their particles are tightly packed. Metals like copper and aluminum are especially good conductors.

The driving force behind conduction is a temperature gradient within the material. Heat always flows from higher-temperature regions toward lower-temperature regions.

Convection transfers heat through the bulk movement of a fluid (liquid or gas). It combines two effects: conduction within the fluid and the physical motion of the fluid carrying energy with it.

• Natural convection happens when buoyancy drives the flow. Hot fluid is less dense, so it rises while cooler fluid sinks, creating circulation (think of hot air rising above a radiator).
• Forced convection uses an external device like a fan or pump to move the fluid, which typically produces much higher heat transfer rates.

Radiation transmits heat through electromagnetic waves, primarily in the infrared spectrum. Unlike conduction and convection, radiation requires no medium at all. This is how the Sun heats the Earth across the vacuum of space. The rate of radiative heat transfer depends on temperature, surface properties (emissivity and absorptivity), and the geometry between emitting and absorbing surfaces.

Factors affecting heat transfer

Temperature gradient is the difference in temperature between two points in a system. A steeper gradient produces a higher rate of heat transfer. For example, hot coffee cools much faster in a cold room (large temperature difference) than in a warm room (small temperature difference).

Surface area determines how much area is available for heat exchange. Larger surface areas allow more heat to flow. This is why car radiators use thin fins to maximize surface area, and why heat exchangers in power plants have extensive tube networks.

Material properties play several roles:

• Thermal conductivity ($k$) measures how well a material conducts heat. Copper ($k \approx 400$ W/m·K) transfers heat far more readily than wood ($k \approx 0.15$ W/m·K).
• Emissivity ($\varepsilon$) describes how effectively a surface emits thermal radiation compared to a perfect blackbody. Values range from 0 to 1. Black paint and oxidized metals have high emissivity (~0.9), while polished metals are low (~0.1).
• Absorptivity ($\alpha$) quantifies how well a surface absorbs incoming thermal radiation, also ranging from 0 to 1. Dark-colored objects absorb more radiation than light-colored ones, which is why a black car gets hotter in sunlight than a white one.

Conduction and Convection Heat Transfer

Fourier's law in conduction

Fourier's law is the fundamental equation for conduction. It quantifies how much heat flows through a material based on the temperature gradient, the material's conductivity, and the cross-sectional area:

$q = -kA\frac{dT}{dx}$

• $q$: heat transfer rate (W)
• $k$: thermal conductivity of the material (W/m·K)
• $A$: cross-sectional area perpendicular to heat flow (m²)
• $\frac{dT}{dx}$: temperature gradient (K/m)

The negative sign accounts for the fact that heat flows in the direction of decreasing temperature. If temperature decreases as $x$ increases, $\frac{dT}{dx}$ is negative, and the two negatives make $q$ positive in the direction of heat flow.

A few key concepts build on this law:

• Steady-state conduction means the temperature distribution doesn't change with time. The heat flowing into any section equals the heat flowing out. Insulated building walls in winter approximate this once temperatures stabilize.
• One-dimensional conduction simplifies the analysis to heat transfer along a single direction, such as straight through a flat wall.
• Thermal resistance is analogous to electrical resistance. For a plane wall: $R = \frac{L}{kA}$, where $L$ is the wall thickness (m). Just as voltage drives current through electrical resistance, temperature difference drives heat flow through thermal resistance. You can even add thermal resistances in series for multi-layer walls, just like resistors in a circuit.

Convective heat transfer analysis

Newton's law of cooling is the governing equation for convection:

$q = hA(T_s - T_\infty)$

• $q$: convective heat transfer rate (W)
• $h$: convective heat transfer coefficient (W/m²·K)
• $A$: surface area exposed to the fluid (m²)
• $T_s$: surface temperature (K)
• $T_\infty$: fluid temperature far from the surface (K)

The convective heat transfer coefficient $h$ is not a simple material property. It depends on several factors:

• Fluid properties: density, viscosity, specific heat, and thermal conductivity
• Flow characteristics: velocity and whether the flow is laminar (smooth, orderly) or turbulent (chaotic, with mixing). Turbulent flow generally gives higher $h$ values because the mixing brings fresh fluid into contact with the surface more effectively.
• Surface geometry and roughness: flat plates, cylinders, and spheres each produce different flow patterns and therefore different $h$ values.

The Nusselt number ($Nu$) is a dimensionless parameter that compares convective to conductive heat transfer in a fluid:

$Nu = \frac{hL}{k}$

Here $L$ is a characteristic length (m) and $k$ is the fluid's thermal conductivity (W/m·K). A Nusselt number of 1 would mean heat crosses the fluid by conduction alone. Higher values indicate that convection is significantly enhancing the heat transfer beyond what conduction alone would provide.

Thermal Radiation

Principles of thermal radiation

Blackbody radiation provides the theoretical upper limit for thermal emission. A perfect blackbody absorbs all incident radiation and emits the maximum possible energy at any given temperature. No real surface does this, but it serves as the reference standard.

The spectral distribution of blackbody emission follows Planck's law:

$E_{b\lambda} = \frac{C_1}{\lambda^5\left[\exp\left(\frac{C_2}{\lambda T}\right)-1\right]}$

• $C_1 = 3.742 \times 10^{-16}$ W·m² and $C_2 = 1.439 \times 10^{-2}$ m·K are radiation constants
• $\lambda$: wavelength (μm)
• $T$: absolute temperature (K)

This equation tells you how much energy a blackbody emits at each wavelength. As temperature increases, the peak of the spectrum shifts to shorter wavelengths (this is Wien's displacement law in action, and it's why hotter objects glow from red to white).

To get the total emissive power across all wavelengths, you use the Stefan-Boltzmann law:

$E_b = \sigma T^4$

• $E_b$: total blackbody emissive power (W/m²)
• $\sigma = 5.67 \times 10^{-8}$ W/m²·K⁴

The $T^4$ dependence is important: doubling the absolute temperature increases radiation by a factor of 16. This makes radiation dominant at high temperatures.

Emissivity ($\varepsilon$) is the ratio of a real surface's emissive power to that of a blackbody at the same temperature. A perfect blackbody has $\varepsilon = 1$. Typical values: polished metals ~0.05–0.1, oxidized metals ~0.5–0.8, non-metallic surfaces ~0.85–0.95. For a real surface, the emitted power is:

$E = \varepsilon \sigma T^4$

Kirchhoff's law states that for a surface in thermal equilibrium, emissivity equals absorptivity at the same temperature and wavelength: $\varepsilon = \alpha$. This means a good emitter is also a good absorber, and a poor emitter is a poor absorber (and therefore a good reflector).

Absorptivity ($\alpha$) is the fraction of incident radiation a surface absorbs, ranging from 0 (perfectly reflective) to 1 (perfectly absorptive). Surfaces with high absorptivity heat up more in radiative environments. This is why solar collectors use dark, high-absorptivity coatings, and why reflective surfaces are used as heat shields.