MECH3610 · Advanced Thermofluids
Free (Natural) Convection
When there is no imposed flow, buoyancy from density differences drives the motion, and this chapter introduces the Grashof and Rayleigh numbers and the correlations that give Nu = f(Ra) for plates, cylinders and spheres. Recognising a free-convection situation, computing Ra correctly (including beta = 1/T_f for a gas) and picking the correct Ra-range correlation are the examinable skills for the 30% open-book mid-term and final.
What this chapter covers
- 01Grashof number Gr_L = g*beta*(T_s - T-inf)*L^3/nu^2 as the buoyancy-to-viscous ratio
- 02Volumetric thermal-expansion coefficient beta, equal to 1/T_f for an ideal gas (T_f in kelvin)
- 03Rayleigh number Ra_L = Gr_L*Pr = g*beta*(T_s - T-inf)*L^3/(nu*alpha)
- 04Vertical-plate correlations: Nu_L = 0.59 Ra^{1/4} (10^4-10^9), 0.10 Ra^{1/3} (10^9-10^13), and Churchill-Chu
- 05Horizontal-plate correlations by orientation (hot-up vs hot-down) with characteristic length L_c = A_s/P
- 06Horizontal-cylinder and sphere correlations (Churchill-Chu type)
- 07Mixed convection: Nu^n ~ Nu_forced^n +/- Nu_free^n (plus assisting, minus opposing)
- 08Deciding free vs forced dominance and selecting the Ra-range-appropriate correlation
Free convection from a heated vertical plate: Rayleigh number and h
- +1Film temperature T_f = (70 + 20)/2 = 45 C = 318 K, so for a gas beta = 1/T_f = 1/318 = 3.14e-3 /K. Evaluate air properties at T_f.
- +1Rayleigh number Ra_L = g*beta*(T_s - T-inf)*L^3/(nu*alpha) = 9.81*3.14e-3*(70 - 20)*(0.3)^3/(17.8e-6*25.2e-6) = 0.04165/4.486e-10 = 9.29e7.
- +1Ra_L = 9.29e7 lies in the 10^4-10^9 band, so use the vertical-plate correlation Nu_L = 0.59 Ra_L^{1/4} = 0.59 * (9.29e7)^{0.25} = 0.59 * 98.2 = 57.9.
- +1Average coefficient h = Nu_L*k/L = 57.9*0.0276/0.3 = 5.33 W/m2-K. (For scale, the plate then sheds about 5.33*0.3*50 = 80 W per metre of width.)
Key terms
- Grashof number (Gr)
- Gr_L = g*beta*(T_s - T-inf)*L^3/nu^2, the ratio of buoyancy to viscous forces in free convection; it plays the role Reynolds number plays in forced convection.
- Rayleigh number (Ra)
- Ra_L = Gr_L*Pr = g*beta*(T_s - T-inf)*L^3/(nu*alpha). Its magnitude sets both the flow regime and which free-convection correlation applies.
- Thermal-expansion coefficient (beta)
- beta [1/K] measures how much a fluid expands per degree; for an ideal gas beta = 1/T_f with T_f in kelvin. It drives the buoyancy force in Gr and Ra.
- Vertical-plate correlation
- Nu_L = 0.59 Ra_L^{1/4} for laminar free convection (Ra in 10^4-10^9) and 0.10 Ra_L^{1/3} for turbulent (10^9-10^13); the Churchill-Chu form covers all Ra.
- Mixed convection
- The regime where forced and free convection are comparable; the combined Nusselt number follows Nu^n ~ Nu_forced^n +/- Nu_free^n, with + when buoyancy assists the forced flow and - when it opposes.
- Plate characteristic length
- For a horizontal plate the length in Ra is L_c = A_s/P (surface area over perimeter), not a simple side, and the correlation depends on whether the hot face is up or down.
Free (Natural) Convection FAQ
How do I know it is free rather than forced convection?
Free convection dominates when there is no imposed flow (still air or liquid) and the motion is driven purely by buoyancy. A telltale is a small convection coefficient — single digits to a few tens of W/m2-K for air — versus the hundreds or thousands typical of forced flow. When both act, compare Gr to Re^2; if Gr/Re^2 ~ 1 the regime is mixed.
What is beta and why is it 1/T_f for a gas?
beta is the volumetric thermal-expansion coefficient, the fractional volume change per degree. For an ideal gas density is inversely proportional to absolute temperature, which gives beta = 1/T exactly; evaluated at the film temperature that is beta = 1/T_f, with T_f in kelvin.
Which Rayleigh-number correlation do I use?
Compute Ra first, then match it to the correlation's validity band: for a vertical plate, 0.59 Ra^{1/4} for 10^4-10^9 and 0.10 Ra^{1/3} for 10^9-10^13, or the all-range Churchill-Chu form. Horizontal plates, cylinders and spheres each have their own correlations, so identify the geometry and orientation too.
How is free convection examined in MECH3610?
As mid-term and final calculations for heated plates, pipes and enclosures where the marks are for recognising the free-convection regime, computing Ra with the right beta, and choosing the correct correlation band. Because the exam is open-book, the equation sheet carries these relations. Confirm the coverage on Moodle.
Exam move
Treat free convection as the buoyancy analogue of forced convection: Ra plays the role Re did, and you still finish with Nu, h = Nu*k/L and Q. Drill the Ra computation, being careful with beta = 1/T_f (kelvin) for gases and with the horizontal-plate characteristic length A_s/P. Keep a small table of correlations by geometry and Ra band so you can select quickly in the open-book exam. Build the instinct that free-convection h is small; a large value means you misidentified the regime. Rehearse the mixed-convection combining rule for problems where a fan or wind is present. Confirm the exam format on Moodle.
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