Convection

When a fluid absorbs heat, its molecules gain kinetic energy and, in most fluids, move further apart. This thermal expansion decreases the fluid's density (mass per unit volume). The key relation is ρ = ρ₀[1 − β(T − T₀)], where β is the thermal expansion coefficient, ρ₀ is the reference density, and T₀ is the reference temperature. For water around room temperature, β ≈ 2×10⁻⁴ K⁻¹; for air at 20°C, β ≈ 3.4×10⁻³ K⁻¹. Even small temperature differences produce measurable density contrasts.

Archimedes' principle tells us that an object (or parcel of fluid) immersed in a denser surrounding fluid experiences an upward buoyant force. A warm, less dense parcel of fluid surrounded by cooler, denser fluid therefore feels a net upward force: F_buoy − F_gravity = (ρ_cool − ρ_warm)Vg, where V is the parcel's volume and g is gravitational acceleration. This net force accelerates the warm parcel upward. Simultaneously, the cooler, denser fluid above descends to replace it. The result is a circulating loop — a convection current.

In a fluid heated from below (such as water in a pot or the lower atmosphere), the entire layer becomes unstable: warm fluid continuously rises from the bottom, cools at the top by losing energy to the surroundings, and then sinks back down. This forms organised convection cells. In laboratory conditions these appear as regular hexagonal Bénard cells. The onset of convection is predicted by the dimensionless Rayleigh number Ra = (gβΔTL³)/(νκ), where ΔT is the temperature difference across the layer of height L, ν is the kinematic viscosity, and κ is the thermal diffusivity. Convection begins when Ra exceeds approximately 1100 for a layer heated from below.

The rate at which heat is transferred by convection from a solid surface to a fluid is described by Newton's Law of Cooling: Q̇ = hA(T_s − T_f), where h is the convective heat transfer coefficient (W m⁻² K⁻¹), A is the surface area in contact with the fluid, T_s is the surface temperature, and T_f is the fluid temperature far from the surface. The value of h is not a material constant — it depends on fluid properties, flow speed, and geometry. For free convection in air, h ≈ 5–25 W m⁻² K⁻¹; for forced convection in water, h can reach 500–10 000 W m⁻² K⁻¹. This is why stirring hot soup makes it cool so much faster.

In forced convection, an external agent moves the fluid. The relevant dimensionless number is the Reynolds number Re = ρvL/μ, which compares inertial forces to viscous forces; high Re means turbulent, efficient heat transfer. The Nusselt number Nu = hL/k relates the actual convective heat transfer to what conduction alone would achieve; Nu > 1 always in convection. Empirical correlations link Nu to Re and the Prandtl number Pr = ν/κ. For natural convection, Nu is correlated with Ra instead. These dimensionless groups allow engineers to scale results from lab experiments to full-size industrial systems.

• \dot{Q} — Rate of heat transfer (W (watts)): The thermal energy transferred per second from the surface to the fluid.
• h — Convective heat transfer coefficient (W m⁻² K⁻¹): Measures how effectively the particular fluid and flow geometry transfer heat. Higher h means faster convective cooling.
• A — Surface area (m²): The area of the solid surface in contact with the moving fluid. Larger area means more heat transferred.
• T_s — Surface temperature (K or °C): The temperature at the surface of the solid object losing (or gaining) heat.
• T_f — Fluid temperature (K or °C): The bulk temperature of the fluid far from the surface, also called the free-stream temperature.

Problem. A flat metal plate of area 0.50 m² is maintained at 80°C. It is cooled by natural convection in air at 20°C. The convective heat transfer coefficient for this setup is h = 12 W m⁻² K⁻¹. (a) Calculate the rate of heat loss from the plate. (b) How much energy is lost in 5 minutes?

Answer. The plate loses heat at 360 W, and transfers 108 kJ of thermal energy in 5 minutes.

• Conduction — Conduction is the competing heat transfer mechanism through stationary matter; comparing it with convection clarifies when each process dominates.
• Thermal Radiation — Radiation is the third mode of heat transfer and, unlike convection, requires no medium — understanding all three modes gives a complete picture of thermal physics.
• Buoyancy and Archimedes' Principle — Buoyancy is the fundamental mechanism driving natural convection — the upward force on less dense warm fluid creates the circulation currents.
• Thermal Expansion — Thermal expansion reduces fluid density when heated, which is the first step in establishing the density difference that drives natural convection.