Forced and Natural Convection
Newton's law of cooling, Nusselt, Prandtl, Reynolds and Grashof numbers, thermal boundary layers, standard correlations for plates and pipes, and natural convection.
Convection is heat transfer between a surface and a moving fluid. Its strength is the heat transfer coefficient h, found from dimensionless correlations.
Newton's law of cooling
Q = h A (T_s − T_∞)
Typical h (W/m²·K): free convection of air 5–25; forced air 25–250; forced water 100–20,000; boiling and condensation 2,500–100,000.
Dimensionless numbers
| Number | Definition | Meaning |
|---|---|---|
| Nusselt Nu | hL/k (k of the fluid) | Convection / conduction across the layer |
| Prandtl Pr | ν/α = μc_p/k | Momentum / thermal diffusivity |
| Reynolds Re | VL/ν | Inertia / viscous (forced convection) |
| Grashof Gr | gβΔT L³/ν² | Buoyancy / viscous (natural convection) |
| Rayleigh Ra | Gr · Pr | Natural convection strength |
| Stanton St | Nu/(Re Pr) = h/(ρVc_p) | Used in analogies |
Pr is about 0.7 for air, 7 for water and much less than 1 for liquid metals. The ratio of the velocity to thermal boundary layer thickness is about Pr^(1/3).
Forced convection correlations
- Flat plate, laminar (Re < 5 × 10⁵): local Nu_x = 0.332 Re_x^(1/2) Pr^(1/3); average Nu = 0.664 Re_L^(1/2) Pr^(1/3).
- Pipe, laminar, fully developed: Nu = 3.66 (constant wall temperature) or 4.36 (constant heat flux), independent of Re.
- Pipe, turbulent (Dittus–Boelter): Nu = 0.023 Re^0.8 Pr^n, n = 0.4 for heating the fluid, 0.3 for cooling.
Reynolds analogy (Pr = 1): St = C_f/2. The Chilton–Colburn analogy extends it: St Pr^(2/3) = C_f/2.
Natural convection
Fluid moves because heated fluid is lighter. Correlations take the form Nu = C Ra^n: n ≈ 1/4 for laminar and 1/3 for turbulent flow (Ra above about 10⁹ on a vertical plate).
Worked example
Water flows in a tube of 20 mm diameter with Re = 20,000 and Pr = 5, being heated; k = 0.6 W/m·K.
- Nu = 0.023 × 20,000^0.8 × 5^0.4 = 0.023 × 2759 × 1.904 = 120.8.
- h = Nu k/D = 120.8 × 0.6/0.02 = 3624 W/m²·K.
Practice
A surface at 80 °C loses heat to air at 20 °C with h = 15 W/m²·K over 2 m². What is the heat loss in W?
Show answer and explanation
Answer: 1800Q = hAΔT = 15 × 2 × 60 = 1800 W.
Fully developed laminar flow in a tube of 10 mm diameter has a constant wall temperature; k = 0.6 W/m·K. What is h in W/m²·K?
Show answer and explanation
Answer: 219.6Nu = 3.66, so h = 3.66 × 0.6/0.01 = 219.6 W/m²·K.
Which dimensionless number decides heat transfer in natural (free) convection, together with Pr?
Show answer and explanation
Answer: BIn natural convection the flow is driven by buoyancy, measured by Gr (or Ra = Gr Pr).
For air (Pr ≈ 0.7) flowing over a plate, compared with the velocity boundary layer, the thermal boundary layer is:
Show answer and explanation
Answer: Aδ/δ_t ≈ Pr^(1/3); for Pr < 1 the thermal layer is thicker. For water (Pr ≈ 7) it is thinner.