Convection - Engineering Notes

01 // PHYSICAL MECHANISM

Conduction vs. Convection

Both require a material medium.
Convection requires fluid motion.
Conduction in a fluid is the limiting case of convection (quiescent fluid).

Newton's Law of Cooling

\dot{q}_{\text{conv}} = h(T_s - T_\infty)

h = Convection heat transfer coefficient [W/m²·K]

No-Slip Condition

A fluid in direct contact with a solid "sticks" to the surface due to viscous effects, and there is no slip.

"Heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, since the fluid layer is motionless."

\dot{q}_{\text{conv}} = \dot{q}_{\text{cond}} = -k_{\text{fluid}} \left. \frac{\partial T}{\partial y} \right|_{y=0}

02 // CLASSIFICATION OF FLUID FLOWS

Viscous vs. Inviscid

Viscous: Frictional effects are significant.

Inviscid: Viscous forces are negligibly small compared to inertial/pressure forces.

Internal vs. External

External: Unbounded fluid over a surface (plate, wire, pipe exterior).

Internal: Fluid completely bounded by solid surfaces (pipe flow).

Compressible vs. Incompressible

Incompressible: Density remains nearly constant (liquids, low-speed gas).

Compressible: Density changes significantly (high-speed gas, Ma > 0.3).

Laminar vs. Turbulent

Laminar: Highly ordered fluid motion, smooth layers.

Turbulent: Highly disordered, velocity fluctuations.

Natural vs. Forced

Forced: Fluid forced by external means (pump, fan).

Natural: Fluid motion due to buoyancy effects (density differences).

Steady vs. Unsteady

Steady: No change at a point with time.

Unsteady: Properties change with time.

03 // VELOCITY BOUNDARY LAYER

The region of the flow above the plate bounded by δ in which the effects of the viscous shearing forces caused by fluid viscosity are felt.

Boundary Layer Thickness

\delta \text{ defined where } u = 0.99V

Wall Shear Stress

\tau_w = \mu \left. \frac{\partial u}{\partial y} \right|_{y=0}

Friction Coefficient

C_f = \frac{\tau_w}{\frac{1}{2}\rho V^2}

[DIAGRAM: Velocity Profile & Boundary Layer Growth]

04 // THERMAL BOUNDARY LAYER

The flow region over the surface in which the temperature variation in the direction normal to the surface is significant.

Thermal Boundary Layer Thickness

\delta_t \text{ defined where } \frac{T - T_s}{T_\infty - T_s} = 0.99

Prandtl Number

Pr = \frac{\text{Molecular diffusivity of momentum}}{\text{Molecular diffusivity of heat}} = \frac{\nu}{\alpha} = \frac{\mu c_p}{k}

[DIAGRAM: Temperature Profile & Thermal Boundary Layer]

Liquid Metals

Pr ≪ 1

Heat diffuses faster

Gases

Pr ≈ 1

Similar rates

Oils

Pr ≫ 1

Momentum diffuses faster

05 // LAMINAR & TURBULENT FLOWS

Reynolds Number

Ratio of inertial forces to viscous forces.

Re = \frac{\text{Inertial forces}}{\text{Viscous forces}} = \frac{V L_c}{\nu} = \frac{\rho V L_c}{\mu}

Large ReInertial forces dominate (Turbulent)

Small ReViscous forces dominate (Laminar)

Critical Reynolds Number

The value of Re at which the flow becomes turbulent.

Flat Plate

Re_{cr} \approx 5 \times 10^5

Pipe Flow

Re_{cr} \approx 2300

06 // DIMENSIONLESS NUMBERS

Nu

Nusselt Number

Nu = \frac{h L_c}{k}

Dimensionless convection heat transfer coefficient. Ratio of convection to conduction across the fluid layer.

Re

Reynolds Number

Re = \frac{V L_c}{\nu}

Ratio of inertial forces to viscous forces. Determines flow regime (Laminar vs Turbulent).

Pr

Prandtl Number

Pr = \frac{\nu}{\alpha}

Ratio of momentum diffusivity to thermal diffusivity. Relates velocity and thermal boundary layers.