MECH3610 · Advanced Thermofluids
External Forced Convection
This chapter applies the empirical correlations for forced flow over external surfaces — flat plates (laminar, turbulent and mixed), cylinders and spheres in cross-flow — to predict the convection coefficient and heat rate. Selecting the correct correlation from the Reynolds number and evaluating properties at the film temperature is the examinable skill, and it feeds the Forced-Convection lab, the 30% mid-term and the property-lookup drills the open-book exam rewards.
What this chapter covers
- 01Flat-plate laminar local Nu_x = 0.332 Re_x^{1/2} Pr^{1/3} and average Nu_L = 0.664 Re_L^{1/2} Pr^{1/3}
- 02Turbulent local Nu_x = 0.0296 Re_x^{0.8} Pr^{1/3} and mixed average Nu_L = (0.037 Re_L^{0.8} - 871) Pr^{1/3}
- 03Uniform-heat-flux variants (0.453 laminar, 0.0308 turbulent local coefficients)
- 04Deciding laminar vs turbulent vs mixed from Re_x,c ~ 5e5, and local vs average h
- 05Cylinder in cross-flow via the Churchill-Bernstein correlation; non-circular sections via tabulated C, m
- 06Sphere in cross-flow via the Whitaker correlation with the viscosity-ratio correction
- 07Tube banks: maximum velocity, array-average Nu with C1, C2, m from tables
- 08Evaluating all properties at the film temperature T_f and reading the right table row quickly
Average convection coefficient and heat rate for air over a flat plate
- +1Film temperature T_f = (80 + 20)/2 = 50 C ~ 323 K; evaluate air properties there. With those values in hand we can size the flow.
- +1Reynolds number Re_L = U*L/nu = 10*0.5/18.2e-6 = 2.75e5. This is below the transition value ~5e5, so the boundary layer is laminar over the whole plate.
- +1Use the laminar average-plate correlation Nu_L = 0.664 Re_L^{1/2} Pr^{1/3} = 0.664 * (2.75e5)^{0.5} * (0.70)^{1/3} = 0.664 * 524 * 0.890 = 310.
- +1Average coefficient h = Nu_L*k/L = 310*0.028/0.5 = 17.3 W/m2-K. Per metre of width the area is L*1 = 0.5 m2, so Q = h*A*(T_s - T-inf) = 17.3*0.5*(80 - 20) = 520 W per metre of width.
Key terms
- Laminar flat-plate correlation
- For a plate at uniform temperature, local Nu_x = 0.332 Re_x^{1/2} Pr^{1/3} and average Nu_L = 0.664 Re_L^{1/2} Pr^{1/3}, valid for Pr > 0.6 and Re below the transition ~5e5.
- Mixed boundary-layer correlation
- When the plate runs from laminar into turbulent (transition at Re_x,c ~ 5e5), the average is Nu_L = (0.037 Re_L^{0.8} - 871) Pr^{1/3}; the -871 term removes the over-count of the leading laminar stretch.
- Churchill-Bernstein correlation
- A single correlation for a cylinder in cross-flow spanning a wide Re_D range, giving the array-average Nu_D from Re_D and Pr; used with film-temperature properties.
- Whitaker (sphere) correlation
- Nu_D = 2 + (0.4 Re_D^{1/2} + 0.06 Re_D^{2/3}) Pr^{0.4} (mu-inf/mu_s)^{1/4} for a sphere in cross-flow, notable for the viscosity-ratio correction and the conduction floor Nu_D = 2.
- Local vs average coefficient
- The local h_x applies at a point x; the average h-bar over a length L (h-bar = (1/L) integral of h_x) gives the total rate Q = h-bar*A_s*(T_s - T-inf). Correlations come in both local and averaged forms.
- Critical Reynolds number
- Re_x,c ~ 5e5 for a smooth flat plate marks the laminar-to-turbulent transition; it decides which correlation applies and whether a mixed-layer treatment is needed.
External Forced Convection FAQ
How do I decide which flat-plate correlation to use?
Compute Re_L. If it is below ~5e5 the plate is laminar throughout, so use the 0.664 average. If it is above, part of the plate is turbulent: use the mixed correlation (0.037 Re^0.8 - 871) Pr^{1/3}, unless the boundary layer is tripped turbulent from the leading edge, in which case use the all-turbulent 0.037 Re^0.8 Pr^{1/3}.
Why do I keep evaluating properties at the film temperature?
Because air and other fluids have temperature-dependent viscosity, conductivity and Prandtl number, and the film temperature T_f = (T_s + T-inf)/2 is the representative average across the boundary layer. In an open-book exam the challenge is finding the right T_f row in the property table quickly, so practise that lookup.
What is special about the sphere correlation?
The Whitaker sphere correlation keeps a floor of Nu_D = 2 (the pure-conduction limit for a sphere in a still infinite medium) and carries a viscosity-ratio correction (mu-inf/mu_s)^{1/4} evaluated with the surface viscosity — a detail that is easy to drop under time pressure.
How is this examined in MECH3610?
As flat-plate and cross-flow calculations in the Week-3 material, the Forced-Convection lab and the open-book mid-term, where marks go to the regime decision, the correct correlation and property evaluation at T_f. Confirm the coverage and permitted tables on Moodle.
Exam move
Build a one-page correlation map: geometry (plate / cylinder / sphere) by regime (laminar / turbulent / mixed) with the matching Nu formula and its validity range. Practise the workflow every time — film temperature, look up properties, compute Re, decide the regime, pick local or average, apply the correlation, then h = Nu*k/L and Q = h*A*deltaT. Rehearse the property-table lookup at T_f until it is fast, because the open-book exam rewards speed there. Keep the mixed-layer -871 term and the sphere's Nu = 2 floor in your notes so you do not forget them. Confirm the exam format on Moodle.
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