Newton's Law of Cooling

Convective heat transfer is governed by Newton's Law of Cooling, which relates heat transfer to the temperature difference between a surface and surrounding fluid:

Q = h * A * (Ts - Tf)

Where Q is heat transfer rate (W), h is the convection coefficient (W/m2-K), A is surface area (m2), Ts is surface temperature, and Tf is fluid temperature. The challenge lies in determining h, which depends on fluid properties, flow conditions, and geometry.

Natural vs Forced Convection

Natural (Free) Convection: Fluid motion is driven solely by buoyancy forces arising from density differences due to temperature gradients. Hot fluid rises, cool fluid sinks, creating circulation patterns. Characterized by the Grashof and Rayleigh numbers. Typical h values: 5-25 W/m2-K for air, 100-1000 W/m2-K for water.

Forced Convection: Fluid motion is driven by external means such as fans, pumps, or wind. The flow velocity is imposed rather than arising from buoyancy. Characterized by the Reynolds number. Generally produces much higher h values: 25-500 W/m2-K for air, 500-20,000 W/m2-K for water.

When both mechanisms are significant (mixed convection), the Richardson number Ri = Gr/Re2 determines which dominates. For Ri << 1, forced convection dominates; for Ri >> 1, natural convection dominates.

Dimensionless Numbers & Correlations

Nusselt Number (Nu): Ratio of convective to conductive heat transfer. Nu = hL/k, where L is characteristic length and k is fluid thermal conductivity. Higher Nu indicates more effective convection.

Nu = hL/k (definition used to find h from correlations)

Reynolds Number (Re): Ratio of inertial to viscous forces, determines flow regime (laminar/turbulent).

Re = VL/nu = rho*V*L/mu

Transition typically occurs: Re ~ 5x10^5 for flat plates, Re ~ 2300 for pipe flow.

Prandtl Number (Pr): Ratio of momentum diffusivity to thermal diffusivity. Pr = nu/alpha = mu*Cp/k. For air Pr ~ 0.7, for water Pr ~ 1-10, for oils Pr ~ 100-10000.

Grashof Number (Gr): Ratio of buoyancy to viscous forces in natural convection.

Gr = g*beta*dT*L^3/nu^2

Rayleigh Number (Ra): Product of Grashof and Prandtl numbers, characterizes natural convection.

Ra = Gr * Pr

Common Nusselt Correlations

Forced Convection - Flat Plate:

• Laminar (Re < 5x10^5): Nu = 0.664 * Re^0.5 * Pr^(1/3)
• Turbulent (Re > 5x10^5): Nu = 0.037 * Re^0.8 * Pr^(1/3)

Forced Convection - Internal Pipe Flow:

• Laminar (Re < 2300): Nu = 3.66 (constant wall temp) or Nu = 4.36 (constant heat flux)
• Turbulent: Dittus-Boelter: Nu = 0.023 * Re^0.8 * Pr^n (n=0.4 heating, 0.3 cooling)
• Turbulent (better accuracy): Gnielinski: Nu = (f/8)(Re-1000)Pr / [1 + 12.7(f/8)^0.5(Pr^(2/3)-1)]

Natural Convection - Vertical Plate:

• Laminar (Ra < 10^9): Nu = 0.59 * Ra^0.25
• Turbulent (Ra > 10^9): Nu = 0.1 * Ra^(1/3)

External Flow Over Cylinder:

• Churchill-Bernstein (all Re): Complex correlation accounting for full range of flow conditions

Thermal Boundary Layer

Heat transfer occurs through a thin thermal boundary layer adjacent to the surface. In this layer, temperature transitions from surface temperature to bulk fluid temperature. The boundary layer thickness depends on flow velocity and fluid properties. Turbulent flow has a thinner thermal boundary layer and higher heat transfer coefficients than laminar flow.