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.
Understanding Convection Correlations Reference
A convection correlation is not a universal law. It is a compact summary of measurements, theory, or both for a particular flow situation. Before selecting one, identify the geometry and the direction of motion.
Air moving across a plate behaves differently from water flowing through a tube. Flow over a cylinder differs from flow over a sphere. The characteristic length must match the correlation.
It may be tube diameter, plate length measured from the leading edge, hydraulic diameter, or object height. A wrong length changes the dimensionless groups and can produce a very believable but incorrect result.
The reason these groups work is the boundary layer. Fluid touching a solid surface has nearly zero velocity relative to that surface. Its temperature begins to change near the wall too.
Farther from the wall, the fluid still has its bulk velocity and temperature. The thickness and mixing within these layers control heat transfer. Laminar flow has smooth layers with limited mixing.
Turbulent flow continually moves warmer and cooler fluid between the wall region and the main stream. This usually raises the heat transfer coefficient, though it raises pressure loss and pumping power too. Surface roughness can trigger turbulence earlier, so an ideal smooth-surface correlation may not represent a real pipe or heat exchanger.
Internal-flow problems need extra care near an inlet. Velocity and temperature profiles need distance to develop after fluid enters a tube. A correlation for fully developed flow should not be used too close to the entrance.
Thermal development can continue after the velocity profile is fully developed, especially for fluids with large Prandtl number. Oil is a common example because heat diffuses through it slowly. Students should check whether the wall has nearly constant temperature or nearly constant heat flux.
These are different physical conditions and often use different correlations. They should also distinguish bulk mean fluid temperature from wall temperature. The bulk mean temperature represents the energy carried by the flowing fluid, not simply the temperature at the tube center.
Natural convection has no fan or pump setting the velocity. Motion develops because warmer fluid becomes less dense and rises while cooler fluid moves downward. Object orientation matters greatly.
A hot upward-facing horizontal surface can produce stronger motion than a hot downward-facing one. A tall vertical wall gives buoyancy more distance to build a moving boundary layer. Natural convection is found around radiators, laptop casings, power electronics, hot drink containers, building walls, and pipes in still rooms.
Its coefficient is often much smaller than that of forced convection, so a small fan can make a large thermal difference. For reliable calculations, use fluid properties at the stated reference temperature, check every correlation range, keep units consistent, and report the result as an estimate rather than an exact value.