15.2 Heat Transfer Mechanisms

Heat Transfer Mechanisms

Heat transfer describes how thermal energy moves from regions of higher temperature to regions of lower temperature. It occurs through three distinct mechanisms: conduction, convection, and radiation. Each mechanism operates by different physical principles, but all three drive systems toward thermal equilibrium.

Mechanisms of Heat Transfer

Conduction transfers thermal energy through direct contact between particles. When faster-moving (hotter) particles collide with slower-moving (cooler) neighbors, kinetic energy passes between them. This process is driven by temperature gradients and occurs in solids, liquids, and gases.

• Conduction is fastest in materials with closely packed atoms and free electrons, like metals (copper, silver)
• Conduction is slowest in materials with loosely packed molecules, like air or fiberglass, which is why these materials work well as insulators

Convection transfers heat through the bulk movement of a fluid (liquid or gas). The fluid itself carries thermal energy from one place to another, combining conduction at the particle level with large-scale fluid motion.

• Natural convection is driven by buoyancy forces. When a fluid is heated, it becomes less dense and rises, while cooler, denser fluid sinks to replace it. Hot air rising above a radiator is a classic example.
• Forced convection is driven by an external mechanism like a fan, pump, or wind. A convection oven uses a fan to circulate hot air, increasing the rate of heat transfer compared to natural convection alone.
• Large-scale examples include ocean currents and atmospheric wind patterns.

Radiation transfers heat through electromagnetic waves and requires no medium at all. Every object above absolute zero (0 K) emits thermal radiation.

• The amount of radiation emitted depends on the object's temperature and its surface properties (specifically, its emissivity)
• Radiation is the dominant heat transfer mechanism in space, since there's no matter to support conduction or convection. This is how the Sun's energy reaches Earth.

Heat Equation for Conduction

Fourier's law of heat conduction quantifies how quickly thermal energy flows through a material. For one-dimensional steady-state conduction, the heat flux (energy per unit area per unit time) is:

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

The total rate of heat transfer through a cross-sectional area $A$ is:

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

• $Q$: rate of heat transfer (W)
• $k$: thermal conductivity of the material (W/m·K), a property that describes how easily heat flows through it
• $A$: cross-sectional area perpendicular to the heat flow (m²)
• $\frac{dT}{dx}$: temperature gradient (K/m)

The negative sign indicates that heat flows in the direction of decreasing temperature, from hot to cold.

Thermal resistance provides a useful way to analyze conduction problems, especially for layered materials. It's analogous to electrical resistance in circuits:

$R = \frac{L}{kA}$

• $L$: material thickness (m)
• $k$: thermal conductivity (W/m·K)
• $A$: cross-sectional area (m²)

For composite materials (like a wall made of drywall, insulation, and brick), you calculate total heat flow in three steps:

1. Calculate each layer's thermal resistance: $R_i = \frac{L_i}{k_i A_i}$
2. Sum the resistances in series: $R_{total} = R_1 + R_2 + \cdots + R_n$
3. Find the heat flow using the overall temperature difference: $Q = \frac{\Delta T}{R_{total}}$

This works just like adding resistors in series in a circuit, where $\Delta T$ plays the role of voltage and $Q$ plays the role of current.

Factors in Convective Transfer

Newton's law of cooling gives the rate of convective heat transfer between a surface and a surrounding fluid:

$Q = hA(T_s - T_f)$

• $h$: convective heat transfer coefficient (W/m²·K)
• $A$: surface area exposed to the fluid (m²)
• $T_s$: surface temperature (K)
• $T_f$: fluid temperature (K)

The value of $h$ is not a simple material property. It depends on the fluid, the flow conditions, and the geometry. That's what makes convection problems more complex than conduction.

Fluid Properties Affecting Convection

Several fluid properties influence how effectively convection transfers heat:

• Density affects buoyancy forces. Larger density differences between hot and cold fluid drive stronger natural convection.
• Viscosity influences how easily the fluid flows. Low-viscosity fluids like water convect more readily than high-viscosity fluids like honey.
• Thermal conductivity of the fluid affects how quickly heat transfers at the fluid-surface interface.
• Specific heat capacity determines how much energy the fluid can carry per unit mass. Water's high specific heat (4,186 J/kg·K) makes it an excellent coolant.

Flow Characteristics

The nature of the flow significantly impacts convection efficiency:

• Laminar flow consists of smooth, parallel layers with predictable paths. It occurs at low velocities and transfers heat relatively slowly.
• Turbulent flow involves chaotic mixing, which breaks up the thermal boundary layer and greatly increases heat transfer rates. It occurs at higher velocities.
• The boundary layer is the thin region of fluid near the surface where velocity and temperature change rapidly. A thinner boundary layer means faster heat transfer.

Dimensionless Numbers

Engineers use dimensionless numbers to characterize convective systems without needing to solve the full fluid dynamics equations:

• Reynolds number (Re): ratio of inertial forces to viscous forces. It predicts whether flow will be laminar or turbulent. For flow in a pipe, turbulence typically begins around $Re \approx 4000$.
• Nusselt number (Nu): ratio of convective to conductive heat transfer across the boundary. A higher Nu means convection is more effective relative to conduction alone.
• Prandtl number (Pr): ratio of momentum diffusivity to thermal diffusivity. It relates the thickness of the velocity boundary layer to the thermal boundary layer.

Stefan-Boltzmann Law for Radiation

The Stefan-Boltzmann law describes the total radiant energy emitted per unit area by a surface at temperature $T$:

$E = \epsilon \sigma T^4$

• $E$: radiant power emitted per unit area (W/m²)
• $\epsilon$: emissivity of the surface, ranging from 0 to 1. A perfect blackbody has $\epsilon = 1$; a shiny metallic surface might have $\epsilon \approx 0.05$.
• $\sigma$: Stefan-Boltzmann constant, $5.67 \times 10^{-8}$ W/m²·K⁴
• $T$: absolute temperature in Kelvin

Notice the $T^4$ dependence. This means radiation increases very steeply with temperature. Doubling the absolute temperature of an object increases its radiated power by a factor of 16.

Net radiative heat transfer between an object and its surroundings is:

$Q = \epsilon \sigma A(T_1^4 - T_2^4)$

• $A$: surface area (m²)
• $T_1$, $T_2$: absolute temperatures of the two surfaces (K)

When $T_1 > T_2$, the object emits more radiation than it absorbs, so $Q$ is positive (net energy loss from the hotter surface).

Additional Radiation Concepts

• View factor: the fraction of radiation leaving one surface that actually strikes another surface. It depends on the geometry and relative orientation of the surfaces (e.g., parallel plates vs. concentric spheres).
• Kirchhoff's law of thermal radiation: at thermal equilibrium, a surface's emissivity equals its absorptivity. Good emitters are also good absorbers, and poor emitters (like polished metal) are poor absorbers but good reflectors.

Applications

• Solar panels absorb radiation from the Sun and convert it to usable energy
• Thermal imaging cameras detect infrared radiation emitted by objects to map temperature differences
• Radiant floor heating systems warm a room primarily through radiation rather than convection