This cheat sheet covers the main convection correlations used in undergraduate heat transfer and thermal engineering. Students need it because convection problems often require choosing the correct dimensionless numbers, flow regime, geometry, and boundary condition before using a formula. It is designed as a quick reference for estimating heat-transfer coefficients for internal flow, external flow, and natural convection.
The core idea is that convection heat transfer is usually calculated from q = h A (Ts - Tinf), where h comes from a Nusselt number correlation. Reynolds number identifies laminar or turbulent forced flow, while Grashof and Rayleigh numbers describe buoyancy-driven natural convection. Prandtl number links momentum and thermal diffusion, and fluid properties are usually evaluated at the film temperature unless a correlation states otherwise.
Key Facts
- Newton's law of cooling is q = h A (Ts - Tinf), where h is the convection heat-transfer coefficient.
- The Nusselt number is Nu = h Lc / k, so the heat-transfer coefficient is h = Nu k / Lc.
- The Reynolds number is Re = rho V Lc / mu = V Lc / nu, and it compares inertial forces to viscous forces.
- The Prandtl number is Pr = nu / alpha = cp mu / k, and it compares momentum diffusivity to thermal diffusivity.
- For internal pipe flow, laminar flow is usually ReD < 2300, transitional flow is 2300 < ReD < 10000, and turbulent flow is ReD > 10000.
- For turbulent flow in smooth circular tubes, the Dittus-Boelter correlation is NuD = 0.023 ReD^0.8 Pr^n, with n = 0.4 for heating and n = 0.3 for cooling.
- For natural convection, the Rayleigh number is RaL = GrL Pr = g beta (Ts - Tinf) Lc^3 Pr / nu^2.
- For external flow over a flat plate, the local laminar boundary layer often transitions near Rex = 5 x 10^5, depending on surface roughness and free-stream turbulence.
Vocabulary
- Convection
- Heat transfer between a surface and a moving fluid caused by both molecular diffusion and bulk fluid motion.
- Heat-transfer coefficient
- The proportionality constant h in q = h A (Ts - Tinf) that measures how effectively convection transfers heat.
- Nusselt number
- A dimensionless number Nu = h Lc / k that compares convection heat transfer to pure conduction through a fluid layer.
- Reynolds number
- A dimensionless number Re = rho V Lc / mu that indicates whether forced flow is likely laminar, transitional, or turbulent.
- Prandtl number
- A dimensionless number Pr = nu / alpha that compares the rate of momentum diffusion to the rate of thermal diffusion.
- Rayleigh number
- A dimensionless number Ra = Gr Pr that indicates the strength of buoyancy-driven natural convection.
Common Mistakes to Avoid
- Using the wrong characteristic length is a major error because Lc changes with geometry, such as tube diameter for pipe flow or plate length for flat-plate flow.
- Mixing average and local Nusselt numbers gives the wrong heat-transfer coefficient because local values apply at one position while average values apply over a surface.
- Applying a turbulent correlation to laminar flow is wrong because the velocity and thermal boundary layers behave differently in each regime.
- Evaluating properties at an arbitrary temperature can cause large error because viscosity, density, and thermal conductivity may change significantly with temperature.
- Forgetting unit consistency in Re, Nu, and h calculations is wrong because dimensionless groups require compatible base units before substitution.
Practice Questions
- 1 Air flows over a flat plate with V = 8 m/s, L = 0.50 m, and nu = 1.6 x 10^-5 m^2/s. Calculate ReL and decide whether the flow is likely laminar if transition occurs near ReL = 5 x 10^5.
- 2 Water flows through a tube with D = 0.020 m, V = 1.5 m/s, rho = 998 kg/m^3, and mu = 0.0010 Pa s. Calculate ReD and classify the flow as laminar, transitional, or turbulent.
- 3 A correlation gives Nu = 85 for air over a surface with k = 0.026 W/(m K) and Lc = 0.10 m. Calculate the convection coefficient h.
- 4 Explain why a heat-transfer correlation for forced convection over a flat plate should not be used for natural convection from a vertical wall, even if both surfaces are the same size.