MECH3610 · Advanced Thermofluids
Extended Surfaces & Fins
Fins extend a surface to add convective area and boost heat rejection, and this chapter derives the fin equation and its four standard tip solutions. Choosing the correct tip boundary condition (convective, adiabatic, prescribed-tip or infinite) is a recurring exam decision, and fin efficiency, effectiveness and arrays are examined via the Week-2 tutorials, the Forced-Convection lab and the 30% open-book mid-term.
What this chapter covers
- 01The fin equation d2theta/dx2 - m^2 theta = 0 with excess temperature theta = T - T-inf and m^2 = hP/(kA_c)
- 02General solution theta = C1 e^{mx} + C2 e^{-mx} and the fin heat-rate scale M = sqrt(h*P*k*A_c)*theta_b
- 03Four tip boundary conditions: convective, adiabatic (Q = M tanh mL), prescribed-tip, and infinite fin (Q = M)
- 04The corrected length L_c = L + A_c/P (pin fin L + D/4, rectangular fin L + t/2) to fake a convective tip
- 05Infinite-fin validity: tanh(mL) ~ 1 needs mL >= 2.65
- 06Fin efficiency eta_fin = Q_fin/Q_max and Q_fin = eta_fin*h*A_fin*(T_b - T-inf), read from efficiency charts
- 07Fin effectiveness eps_fin = Q_fin/Q_no-fin; worthwhile when eps_fin >~ 2, best with high k, low h, slender fins
- 08Fin arrays: A_t = N*A_f + A_b and Q_t = h*A_t[1 - (N*A_f/A_t)(1 - eta_f)]*theta_b
Pin fin with an adiabatic tip: heat rate and tip temperature, with a validity check
- +1Cross-section A_c = pi*D^2/4 = pi*(0.005)^2/4 = 1.963e-5 m2 and perimeter P = pi*D = 0.01571 m. Then m = sqrt(h*P/(k*A_c)) = sqrt(40*0.01571/(180*1.963e-5)) = sqrt(177.8) = 13.33 /m, so mL = 13.33*0.050 = 0.667.
- +1The heat-rate scale is M = sqrt(h*P*k*A_c)*theta_b. Inside the root: 40*0.01571*180*1.963e-5 = 2.221e-3, whose square root is 0.04712; with theta_b = 100 - 25 = 75 K, M = 0.04712*75 = 3.53 W.
- +1For an adiabatic tip Q = M*tanh(mL) = 3.53*tanh(0.667) = 3.53*0.583 = 2.06 W.
- +1The adiabatic-tip profile gives the tip excess theta_L/theta_b = 1/cosh(mL) = 1/cosh(0.667) = 1/1.231 = 0.813, so T_L = 25 + 0.813*75 = 85.9 C. Because mL = 0.667 is far below 2.65, the infinite-fin approximation would have been wrong here (it would have predicted the tip cooled to ambient) - the short fin is the right model.
Key terms
- Fin equation
- The 1-D energy balance on a constant-area fin, d2theta/dx2 - m^2 theta = 0, in the excess temperature theta = T - T-inf. Its solutions give the temperature along the fin for each tip condition.
- Fin parameter (m)
- m = sqrt(h*P/(k*A_c)) [1/m], where P is the perimeter and A_c the cross-section. Large m (high h, thin low-k fin) means the excess temperature decays quickly along the fin.
- Heat-rate scale (M)
- M = sqrt(h*P*k*A_c)*theta_b [W], the natural scale of fin heat rate; the actual rate is M times a tip-dependent factor (tanh mL for an adiabatic tip, 1 for an infinite fin).
- Fin efficiency (eta_fin)
- eta_fin = Q_fin/Q_max, the actual fin heat rate divided by the rate if the whole fin were at the base temperature. Read from standard efficiency charts vs a profile parameter; then Q_fin = eta_fin*h*A_fin*theta_b.
- Fin effectiveness (eps_fin)
- eps_fin = Q_fin/Q_no-fin, the fin heat rate divided by the bare-base rate. A fin is only worthwhile when eps_fin is roughly 2 or more; effectiveness rises with high k, low h and a slender (high P/A_c) shape.
- Corrected length (L_c)
- L_c = L + A_c/P adds a small length increment so an adiabatic-tip formula accounts for tip convection: L + D/4 for a pin fin, L + t/2 for a straight rectangular fin.
Extended Surfaces & Fins FAQ
How do I choose the right tip boundary condition?
First compute mL. If mL >= 2.65 the fin behaves as effectively infinite and Q = M. If not, use the adiabatic-tip solution Q = M*tanh(mL) — and if the problem stresses tip convection, add it via the corrected length L_c = L + A_c/P. Use the prescribed-tip formula only when the tip temperature is actually given.
What is the difference between fin efficiency and fin effectiveness?
Efficiency compares the fin to an ideal fin at uniform base temperature (eta = Q_fin/Q_max, always <= 1). Effectiveness compares the finned surface to the same area with no fin at all (eps = Q_fin/Q_no-fin, and you want it well above 1). A fin can be efficient yet ineffective if the base area it covers was already a good radiator.
When are fins actually worth adding?
When effectiveness eps_fin >~ 2, which favours a high-conductivity material, a low convection coefficient (so fins help most in gases and natural convection, not in vigorous liquid flow) and a slender, high-perimeter shape. That is why air-cooled engines and heat sinks use many thin aluminium fins.
How is this examined in MECH3610?
Fins appear in the Week-2 tutorials (turbine-blade and finned-cylinder problems), feed the Forced-Convection lab, and recur on the open-book mid-term where the key marks are for selecting the correct tip model and reading the efficiency chart. Confirm the mid-term coverage on Moodle.
Exam move
Learn the fin workflow as a decision tree: compute m and mL, test mL against 2.65 to pick infinite vs finite, then apply the matching tip formula for theta(x) and Q. Practise both the base-heat-rate route (Q = M times a tip factor) and the efficiency route (Q = eta_fin*h*A_fin*theta_b) so you can handle whichever the question hands you, and rehearse reading a fin-efficiency chart for an annular fin. For arrays, keep the area bookkeeping tidy: total area A_t = N*A_f + A_b. Keep efficiency and effectiveness cleanly separated in your head. Confirm the mid-term format on Moodle.
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