Convection: Forced & Free (Nusselt Correlations)

Convection is a core GATE ME topic (~2 marks). The dimensionless-number framework (Nu, Re, Pr, Gr) and the difference between forced and free convection are the reliably tested ideas.

Newton's law & dimensionless numbers

Newton's law of cooling: Q = h·A·(T_s − T∞), where h = convective heat-transfer coefficient — the quantity all correlations aim to find.

Key dimensionless groups:

• Nusselt Nu = hL/k — dimensionless h (ratio of convective to conductive transfer across the fluid layer).
• Prandtl Pr = μc_p/k = ν/α — ratio of momentum to thermal diffusivity (Pr ≈ 0.7 for air, ~7 for water, tiny for liquid metals).
• Reynolds Re = ρVL/μ — flow regime (forced convection).
• Grashof Gr = gβΔT·L³/ν² — buoyancy/viscous ratio (free convection). Rayleigh Ra = Gr·Pr.

Forced vs free convection

• Forced convection (fan/pump-driven): Nu = f(Re, Pr); e.g. Dittus–Boelter for turbulent pipe flow Nu = 0.023·Re^0.8·Pr^n (n = 0.4 heating, 0.3 cooling).
• Free/natural convection (buoyancy-driven): Nu = f(Gr, Pr) = f(Ra); slower, lower h.
  The convective boundary layer sets h — thinner layer (higher velocity/turbulence) → higher h.

Exam Tricks & Tips

• 🎯 Nu = hL/k is dimensionless h — every correlation ends by giving Nu, from which h = Nu·k/L.
• 🎯 Forced convection: Nu = f(Re, Pr); free convection: Nu = f(Gr, Pr) — the single most-tested distinction.
• 🎯 Prandtl Pr = ν/α links velocity and thermal boundary layers: Pr > 1 (oils) → thermal layer thinner; Pr < 1 (metals) → thicker.
• 🎯 Dittus–Boelter Nu = 0.023 Re^0.8 Pr^n (n = 0.4 heating, 0.3 cooling) — the go-to turbulent-pipe correlation.
• 🎯 Grashof plays the role of Reynolds in free convection — buoyancy drives the flow instead of an external pump.
• ❌ Common mistake: using a forced-convection (Re-based) correlation for a still-air natural-convection problem — with no forced flow you must use Gr/Ra-based correlations.

Expected exam pattern

A 1-mark Nu/Pr/h relationship or "forced vs free" MCQ, and a 2-mark h-from-correlation or Newton's-cooling problem (find Q or surface temperature). The Nu = f(Re,Pr) vs f(Gr,Pr) split and Pr meaning are frequent points.

Quick recap

Q = hA(Ts − T∞). Nu = hL/k (dimensionless h), Pr = ν/α, Re (forced), Gr (free), Ra = Gr·Pr. Forced: Nu = f(Re, Pr), e.g. Dittus–Boelter 0.023Re^0.8Pr^n. Free: Nu = f(Gr, Pr). Thinner boundary layer → higher h.

Cover the answer, recall, then check. 11 cards on convection for GATE ME.

Q1. State Newton's law of cooling.
A1. Q = h·A·(T_s − T∞), where h = convective heat-transfer coefficient.

Q2. Define the Nusselt number.
A2. Nu = hL/k — dimensionless convective coefficient (convection/conduction across the fluid). Then h = Nu·k/L.

Q3. Define the Prandtl number.
A3. Pr = μc_p/k = ν/α — ratio of momentum to thermal diffusivity. ≈0.7 for air, ~7 for water.

Q4. On what does forced-convection Nusselt depend?
A4. Nu = f(Re, Pr) — Reynolds (flow) and Prandtl (fluid) numbers.

Q5. On what does free-convection Nusselt depend?
A5. Nu = f(Gr, Pr) = f(Ra) — Grashof (buoyancy) and Prandtl numbers.

Q6. Define the Grashof number.
A6. Gr = gβΔT·L³/ν² — the buoyancy-to-viscous force ratio driving natural convection.

Q7. State the Dittus–Boelter correlation.
A7. Nu = 0.023·Re^0.8·Pr^n for turbulent pipe flow, n = 0.4 (heating) or 0.3 (cooling).

Q8. Define the Rayleigh number.
A8. Ra = Gr·Pr — governs the onset and strength of natural convection.

Q9. What does the Prandtl number tell you about boundary layers?
A9. Pr > 1 → thermal boundary layer thinner than velocity layer (oils); Pr < 1 → thicker (liquid metals); Pr ≈ 1 → comparable (gases).

Q10. Forced vs natural convection — key difference?
A10. Forced uses external flow (fan/pump), correlations in Re; natural is buoyancy-driven, correlations in Gr — natural gives lower h.

Q11. How does the boundary layer affect h?
A11. A thinner boundary layer (higher velocity/turbulence) gives a steeper temperature gradient at the wall and thus a higher h.

Convection is heat carried by moving fluid — the h in Q = hAΔT. Unlike conduction, h is not a material property; it depends on flow, geometry and fluid, and we find it from dimensionless Nusselt-number correlations. GATE tests the dimensionless groups and how to pick and use a correlation.

Core concept: the convective coefficient h is packaged in the Nusselt number Nu = hL/k, which experiments correlate against Reynolds (forced) or Grashof (free) and Prandtl numbers — recognise the situation, pick the correlation, extract h.

Deep explanation

Beginner — Newton's law of cooling and Nusselt

Convection: Q = hA(T_s − T_∞). The Nusselt number Nu = hL/k is the dimensionless temperature gradient at the wall — the ratio of convective to (fluid) conductive heat transfer. Nu = 1 means pure conduction; higher Nu means stronger convection.

Intermediate — the governing dimensionless numbers

• Reynolds Re = ρVL/μ — forced-convection flow character.
• Prandtl Pr = μc_p/k = ν/α — ratio of momentum to thermal diffusivity (how thick the velocity vs thermal boundary layer is). Pr ≈ 0.7 for air, ~7 for water, ≪1 for liquid metals.
• Grashof Gr = gβΔTL³/ν² — buoyancy to viscous forces, driving free (natural) convection.
• Rayleigh Ra = Gr·Pr governs natural convection onset.

Advanced — choosing correlations

• Forced convection: Nu = f(Re, Pr). Turbulent pipe flow (Dittus–Boelter): Nu = 0.023 Re^0.8 Pr^n (n = 0.4 heating, 0.3 cooling). Laminar fully-developed pipe with constant wall temperature: Nu = 3.66 (constant!). Flat plate laminar: Nu = 0.664 Re^0.5 Pr^(1/3).
• Free convection: Nu = f(Gr, Pr) = C(Ra)^n, exponent ~1/4 (laminar) to 1/3 (turbulent).
• Which mode? If Gr/Re² ≫ 1, buoyancy dominates (free); if ≪ 1, forced dominates; ~1 is mixed. Compute h from Nu = hL/k once the correlation gives Nu.

Worked example

Water (k = 0.6 W/mK, Pr = 6, ρ = 1000, μ = 0.001) flows at 2 m/s through a 25 mm pipe. Estimate h using Dittus–Boelter (heating).
Re = ρVD/μ = 1000 × 2 × 0.025/0.001 = 50,000 (turbulent).
Nu = 0.023 Re^0.8 Pr^0.4 = 0.023 × (50,000)^0.8 × 6^0.4 = 0.023 × 5,215 × 2.05 = 245.9.
h = Nu·k/D = 245.9 × 0.6/0.025 = 5,902 W/m²K.

GATE relevance

Convection correlations (Dittus–Boelter, Nu = 3.66 for laminar pipe), the meaning of Re, Pr, Gr, and computing h from Nu = hL/k are core Heat Transfer questions. Distinguishing forced from free convection via Gr/Re² and interpreting the Prandtl number are common concept items.

Exam tricks & shortcuts

• Get h from Nu = hL/k after the correlation gives Nu — never forget the k/L factor.
• Fully-developed laminar pipe flow: Nu = 3.66 (constant T) or 4.36 (constant flux) — a fixed number, no calculation.
• Prandtl links boundary layers: Pr = ν/α; air ~0.7, water ~7, liquid metals ≪1.
• Mnemonic: "Nu is h-L-over-k — that's how h finds its way."

Key formulas

• Nusselt: Nu = hL/k; Reynolds Re = ρVL/μ; Prandtl Pr = μc_p/k; Grashof Gr = gβΔTL³/ν².
• Forced (flat plate, laminar): Nu = 0.664 Re^0.5 Pr^(1/3).
• Forced (pipe, turbulent, Dittus–Boelter): Nu = 0.023 Re^0.8 Prⁿ (n = 0.4 heating, 0.3 cooling).
• Free convection: Nu = C(Gr·Pr)ⁿ.

Nusselt number gives the convection coefficient; forced convection scales with Re, free convection with Grashof.

Worked Example

Problem: Air flows over a flat plate in laminar forced convection. The average Nusselt number follows Nu = 0.664·Re^0.5·Pr^(1/3). For Re = 1 × 10⁵, Pr = 0.7, plate length L = 0.5 m, and air conductivity k = 0.026 W/m·K, find the average convective heat transfer coefficient.

Solution:
Evaluate each factor of the correlation:
Re^0.5 = (1 × 10⁵)^0.5 = 316.2.
Pr^(1/3) = (0.7)^(1/3) = 0.888.

Average Nusselt number:
Nu = 0.664 × 316.2 × 0.888 = 186.4.

The Nusselt number is the dimensionless heat transfer coefficient, Nu = hL/k, so:
h = Nu·k/L = (186.4 × 0.026)/0.5 = 4.847/0.5 ≈ 9.69 W/m²·K.

Answer: Nu ≈ 186 and the average heat transfer coefficient h ≈ 9.7 W/m²·K.