Fins and Extended Surfaces

Fins and Extended Surfaces

Fins and Extended Surfaces

A fin is a piece of conducting material attached to a surface so that more area is exposed to the fluid. Heat enters the fin at its base, flows along it by conduction and leaves its surface by convection; because the fin is cooler than its base the extra surface does not work at full strength, and the measure of how much of it works is the fin efficiency. The theory below is the classical one-dimensional analysis of Harper and Brown (1922) and Gardner (1945), in the form Incropera and DeWitt tabulate it (Section 3.6, Tables 3.4 and 3.5). The tool evaluates the closed forms exactly, draws the temperature along the fin and places the fin on the efficiency chart of its family.

Assumptions

Steady state; the temperature varies only along the fin (one-dimensional conduction), which requires the Biot number h(A_c/P)/k across the half thickness to be small, below about 0.1; uniform conductivity k; a uniform convection coefficient h over the surface, the same value at the tip when the tip convects; a uniform fluid temperature T_∞; a base held at T_b; no radiation and no contact resistance at the base. With θ = T − T_∞ the energy balance on a slice of the fin gives

d/dx (k A_c dθ/dx) = h P θ,

where A_c(x) is the section and P(x) the perimeter. The fin parameter at the base section is m = √(hP/(kA_c)) and M = √(hPkA_c) θ_b is the heat rate an infinitely long fin of that section would carry.

Fins of uniform section (straight rectangular fin, cylindrical pin)

With A_c and P constant the equation is θ″ = m²θ, whose solutions are hyperbolic functions. The base condition θ(0) = θ_b is common to all; the tip condition selects the solution (Incropera Table 3.4):

• Convection from the tip (case A), −k θ′(L) = h θ(L): θ/θ_b = [cosh m(L − x) + (h/mk) sinh m(L − x)] / [cosh mL + (h/mk) sinh mL] and q_f = M [sinh mL + (h/mk) cosh mL] / [cosh mL + (h/mk) sinh mL]. The ratio h/mk is small for a thin metal fin and the solution is then close to case B.
• Adiabatic tip (case B), θ′(L) = 0: θ/θ_b = cosh m(L − x)/cosh mL and q_f = M tanh mL, so that η_f = tanh(mL)/(mL) exactly. This is the form Harper and Brown derived (their eq. 5) for the cooling fins of air-cooled engines.
• Prescribed tip temperature (case C), θ(L) = θ_L: θ/θ_b = [(θ_L/θ_b) sinh mx + sinh m(L − x)]/sinh mL and q_f = M (cosh mL − θ_L/θ_b)/sinh mL. A tip held hotter than cosh mL would allow draws heat into the base and the heat rate is negative.
• Infinitely long fin (case D): θ/θ_b = e^−mx and q_f = M. It has no finite surface, so no efficiency; its effectiveness is √(kP/(hA_c)). A real fin is "infinite" once mL exceeds about 2.65, where tanh mL = 0.99.

The tool uses the exact perimeter of the rectangular section, P = 2(w + t), so a narrow fin's edges are counted; Incropera's Table 3.5 writes m = √(2h/kt) for a wide fin (w ≫ t), which is the same thing when the edges are negligible.

The Harper–Brown corrected length

Harper and Brown noticed that the heat leaving a convecting tip can be accounted for by lengthening an adiabatic fin so that its added surface equals the tip area: L_c = L + A_c/P, which is L + t/2 for a wide straight fin and L + D/4 for a pin. Then η_f ≈ tanh(mL_c)/(mL_c) and q_f ≈ M tanh(mL_c). Their report compares this with an exact two-dimensional solution of a deliberately exaggerated case (a thick fin of poor conductivity) and finds the difference "not quite 3 parts in 500"; for an aluminium fin of ordinary proportions it is below 0.1 %. Incropera gives the condition ht/k ≤ 0.0625 for the approximation to hold within the usual accuracy. The tool prints the exact case-A value and the corrected-length value side by side; for the default aluminium fin they agree to six figures.

Tapered straight fins and the conical pin

When the section tapers the equation has variable coefficients and its solutions are modified Bessel functions. Measuring x′ = L − x from the tip:

• Triangular profile, thickness t x′/L: d/dx′(x′ θ′) = m²L θ with m = √(2h/kt), so θ/θ_b = I₀(2m√(Lx′))/I₀(2mL) and η_f = I₁(2mL)/(mL I₀(2mL)).
• Parabolic profile, half thickness (t/2)(x′/L)²: d/dx′(x′² θ′) = m²L² θ, an Euler equation with the power solution θ/θ_b = (x′/L)ⁿ, n = −½ + √(¼ + (mL)²), and η_f = 2/[1 + √(1 + 4(mL)²)]. At mL = 1 this is the golden ratio 0.618.
• Conical pin, diameter D x′/L with m = √(4h/kD): d/dx′(x′² θ′) = m²L x′ θ, so θ/θ_b = √(L/x′) I₁(2m√(Lx′))/I₁(2mL), finite at the tip (mL/I₁(2mL)), and η_f = (2/mL) I₂(2mL)/I₁(2mL).

These solutions neglect the slant of the surface and the fin's edges in the differential equation (P = 2w or πD per unit length), which is what makes them closed forms; Gardner and Incropera then refer the efficiency to the true slant surface A_f of Table 3.5 (2w√(L² + (t/2)²) for the triangle, w[C₁L + (L²/t) ln(t/L + C₁)] for the parabola, (πD/2)√(L² + (D/2)²) for the cone) and write q_f = η_f h A_f θ_b. The tool follows that convention and also prints the heat rate of the one-dimensional equation itself; the two differ by the slant factor √(1 + (t/2L)²), a hundredth of a percent for a slender fin. The tapered profiles have no tip area, so the tip-condition choice does not apply to them; their natural condition is regularity at the sharp tip.

The annular fin

A flat ring of thickness t from r₁ to r₂ on a tube: d/dr(r θ′) = m² r θ with m = √(2h/kt), the modified Bessel equation of order zero, so θ = C₁ I₀(mr) + C₂ K₀(mr). The constants follow from θ(r₁) = θ_b and the tip condition at r₂, which the tool imposes exactly for all four conditions (the convecting tip gives a second linear equation in C₁, C₂). The heat rate is the conduction through the base ring, q_f = −k (2πr₁t) θ′(r₁) = −k A_c m [C₁ I₁(mr₁) − C₂ K₁(mr₁)]. With an adiabatic tip the result is Incropera's eq. 3.91, and the efficiency is C₂[K₁(mr₁) I₁(mr₂) − I₁(mr₁) K₁(mr₂)]/[I₀(mr₁) K₁(mr₂) + K₀(mr₁) I₁(mr₂)] with C₂ = (2r₁/m)/(r₂² − r₁²). Incropera handles a convecting tip by evaluating this at the corrected radius r_2c = r₂ + t/2 (his Figure 3.19 is drawn that way); the tool prints that value beside the exact one, and for the Example 3.10 fins they differ by 0.03 %.

The modified Bessel functions are computed in the tool's library: I_n by its ascending series, whose terms are all positive, and K_n by the integral K_n(x) = ∫₀^∞ e^{−x cosh u} cosh(nu) du with the trapezoidal rule, which converges exponentially for this integrand. Both agree with scipy to twelve figures over the range a fin can produce; mr₂ above 600 overflows and is refused.

Efficiency, effectiveness and resistance

The fin efficiency η_f = q_f/(h A_f θ_b) compares the fin with one of the same surface held entirely at the base temperature; it falls from 1 for a very short or very conductive fin towards 1/(mL) for a long one. The fin effectiveness ε_f = q_f/(h A_c θ_b) compares the fin with the bare patch of base it covers; a fin is justified when ε_f is at least about 2, and for the infinite fin ε_f = √(kP/(hA_c)), which says fins pay when the conductivity is high, the fin is thin (large P/A_c) and the convection coefficient is low (gases rather than liquids, free rather than forced convection). The fin resistance R_f = θ_b/q_f = 1/(η_f h A_f) lets the fin be placed in a thermal circuit.

Fin arrays

For N identical fins on a base whose unfinned (prime) area between them is A_b, the total surface is A_t = N A_f + A_b and the overall surface efficiency is η_o = 1 − (N A_f/A_t)(1 − η_f), the prime surface counting at full strength; the total heat rate is q_t = η_o h A_t θ_b = N q_f + h A_b θ_b, and its ratio to the heat the bare base would give, h (A_b + N A_c) θ_b, is the gain of finning. A contact resistance at the fin roots (attached rather than integral fins) is not included.

The efficiency chart

Gardner plotted the efficiency of each profile against a single abscissa, L_c^3/2(h/(kA_p))^1/2, where A_p is the profile area (L_ct, Lt/2, Lt/3 for the rectangular, triangular and parabolic straight fins). Written that way the curves compare fins of the same material volume: at a given abscissa the parabolic fin is the most efficient and the rectangular the least, which is Schmidt's observation that the concave parabola is the profile of least material for a given duty. For the uniform fin the abscissa is mL_c/√2, for the triangle mL, for the parabola mL√(3/2); the tool evaluates it with the fin's own m so that the design point lies on its curve. The annular curves are drawn for r_2c/r₁ = 1 (the straight fin), 2, 3 and 5 and for the fin's own ratio, the efficiency falling as the ratio grows because more of the surface lies far from the base. Pin fins are charted against mL directly.

Units and the Biot check

Lengths and the base area go through the site's unit system (mm or in, cm² or in²); k, h and the temperatures are plain numbers in the units named on the form (W/(m·K), W/(m²·K) and °C in SI mode; Btu/(h·ft·°F), Btu/(h·ft²·°F) and °F in USCS mode, converted inside the tool with 1 Btu/(h·ft·°F) = 1.7307 W/(m·K) and 1 Btu/(h·ft²·°F) = 5.6783 W/(m²·K)). The fin parameter is reported per metre or per foot accordingly and the heat rates in W or Btu/h. The Biot number across the half thickness is printed with every result; when it exceeds 0.1 the temperature varies across the section as well as along the fin, the one-dimensional heat rate is an upper bound, and the fin should be made thinner or of a better conductor.

Limits

Not covered: temperature-dependent k and h, variable h along the fin (boundary-layer development, fin-to-fin interference in dense arrays), radiation, contact resistance at the base, two-dimensional conduction in thick fins (Bi above 0.1), fins of other profiles (hyperbolic, trapezoidal, parabolic spines beyond the comparison curve), longitudinal fins on cylinders of small radius, and the optimum-profile and optimum-spacing problems. For a heat exchanger whose finned surface this tool rates, the overall efficiency feeds the UA of the Heat Exchanger Effectiveness tool.