Extended surfaces and fins

The flue gas leaving Northgate's hot-water boilers is still at about 165 °C, and Kofi wants an economizer to recover some of that heat into the district return water. The difficulty is the gas side. A gas has a low convection coefficient, perhaps 50 to 100 W/m²·K, while water inside a tube has a coefficient of several thousand. With bare tubes, the gas-side resistance would dominate and the economizer would be enormous. The standard remedy is to add fins to the gas side, increasing its area many times over. This lesson develops the theory of extended surfaces and applies it to the economizer tubes and, as a second example, to a thermowell in the steam main, where the same physics produces a measurement error.

When fins help

Newton's law, \(q = hA(T_s - T_\infty)\), offers three ways to increase heat transfer: raise \(h\), raise the temperature difference, or raise the area. When the first two are fixed by the process, extending the surface is often the cheapest option. Fins appear on car radiators, motorcycle cylinders, electronic heat sinks, air-conditioning coils, transformer tanks and the finned tubes of economizers and air-cooled condensers. They are effective when the convection coefficient on their side is low, as with gases, and nearly useless when it is high, as with boiling liquids, because the fin's own conduction resistance then limits it.

The fin equation

Consider a fin of constant cross-sectional area \(A_c\) and perimeter \(P\), attached to a wall at temperature \(T_b\) and surrounded by fluid at \(T_\infty\). If the fin is thin, its temperature varies only along its length \(x\). An energy balance on a slice of thickness \(dx\), with conduction in and out along the fin and convection from its surface, gives

The general solution is a combination of \(e^{mx}\) and \(e^{-mx}\), or equivalently of \(\cosh mx\) and \(\sinh mx\). The parameter \(m\), in 1/m, measures how quickly the fin temperature decays toward the fluid temperature: a high \(h\) or a low \(k\) makes the decay fast. The two boundary conditions are the base temperature, \(\theta(0) = \theta_b\), and a condition at the tip. Four standard tip conditions give four solutions.

In practice the adiabatic-tip solution is used with a corrected length \(L_c = L + t/2\) for a rectangular fin of thickness \(t\) (or \(L + D/4\) for a pin fin), which adds the tip area to the side area and reproduces the convective-tip result closely. A fin with \(mL\) greater than about 2.5 behaves as if it were infinitely long: \(\tanh 2.5 = 0.987\), so making it longer adds almost nothing.

Fin efficiency and effectiveness

Two measures describe how well a fin performs. The fin efficiency compares the actual heat rate with the ideal rate if the whole fin were at the base temperature:

The fin effectiveness compares the heat rate with the fin to the heat rate from the base area it covers without the fin: \(\varepsilon_f = q_f/(hA_{c,b}\theta_b)\). A fin is worth adding only if its effectiveness is well above 1; designers usually look for 2 or more. For an array of fins on a base, the overall surface efficiency combines finned and unfinned area:

where \(N\) is the number of fins, \(A_f\) the area of one fin and \(A_t\) the total area, fins plus exposed base. The overall surface efficiency enters heat exchanger calculations directly: the gas-side resistance of a finned tube is \(1/(\eta_o h A_t)\).

Annular fins, the discs wrapped around economizer tubes, have a cross-section that grows with radius, so their efficiency is somewhat lower than that of a straight fin with the same height. The exact solution uses modified Bessel functions and is usually read from charts; the program below instead solves the annular fin equation numerically, which gives the same answer and shows how such charts are made.

# Fins: a finned economizer tube and a thermowell in the steam main
import math

def straight_fin(h, k, t, L):
    """Straight rectangular fin per 1 m of width, corrected length for tip convection."""
    m = math.sqrt(2 * h / (k * t)); Lc = L + t / 2
    return m, math.tanh(m * Lc) / (m * Lc)

def annular_fin_eff(h, k, t, r1, r2, n=400):
    """Annular fin efficiency by finite differences on (1/r) d/dr (r dT/dr) = m^2 (T - Tinf), adiabatic corrected tip."""
    r2c = r2 + t / 2; m2 = 2 * h / (k * t); dr = (r2c - r1) / n
    r = [r1 + i * dr for i in range(n + 1)]
    a, b, c, d = [0.0]*(n+1), [0.0]*(n+1), [0.0]*(n+1), [0.0]*(n+1)
    b[0], d[0] = 1.0, 1.0                                  # theta = 1 at the base
    for i in range(1, n):
        rm, rp = r[i] - dr/2, r[i] + dr/2
        a[i], c[i] = rm / dr**2, rp / dr**2
        b[i] = -(rm + rp) / dr**2 - m2 * r[i]
    a[n], b[n] = 2 * (r[n] - dr/2) / dr**2, -2 * (r[n] - dr/2) / dr**2 - m2 * r[n]   # adiabatic tip (ghost node)
    for i in range(1, n + 1):                              # Thomas algorithm
        w = a[i] / b[i-1]; b[i] -= w * c[i-1]; d[i] -= w * d[i-1]
    th = [0.0]*(n+1); th[n] = d[n] / b[n]
    for i in range(n - 1, -1, -1): th[i] = (d[i] - c[i] * th[i+1]) / b[i]
    q = sum(((th[i] + th[i+1]) / 2) * 2 * math.pi * ((r[i] + r[i+1]) / 2) * dr for i in range(n)) * 2 * h
    q_max = h * 2 * math.pi * (r2c**2 - r1**2)
    return q / q_max

# Economizer tube in the flue gas of a hot-water boiler
D, t, H, s = 0.0508, 0.0012, 0.015, 0.004        # tube OD, fin thickness, fin height, fin pitch (m)
k, h, dT = 45.0, 60.0, 120.0                     # carbon steel; gas-side coefficient W/m2.K; gas minus tube wall (K)
r1, r2 = D/2, D/2 + H
eta_ann = annular_fin_eff(h, k, t, r1, r2)
m, eta_str = straight_fin(h, k, t, H)
print(f"m = {m:.1f} 1/m, straight-fin estimate eta = {eta_str:.3f}, annular fin (numerical) eta = {eta_ann:.3f}")

N = 1 / s                                         # fins per 1 m of tube
A_fin = N * 2 * math.pi * ((r2 + t/2)**2 - r1**2)
A_base = math.pi * D * (1 - N * t)
A_tot = A_fin + A_base
eta_o = 1 - A_fin / A_tot * (1 - eta_ann)
q_bare = h * math.pi * D * dT
q_fin = eta_o * h * A_tot * dT
print(f"per 1 m of tube: fin area {A_fin:.3f} m2, base area {A_base:.4f} m2, overall surface efficiency {eta_o:.3f}")
print(f"heat rate bare {q_bare:.0f} W/m, finned {q_fin:.0f} W/m, ratio {q_fin/q_bare:.1f}")
eps_f = eta_ann * 2*math.pi*((r2+t/2)**2 - r1**2) / (2*math.pi*r1*t)
print(f"single-fin effectiveness = {eps_f:.1f}")

# The same fins in aluminum, and in steel on the water side (h = 3000) where fins would not pay
print(f"aluminum fin (k = 200): eta = {annular_fin_eff(h, 200.0, t, r1, r2):.3f}")
print(f"steel fin on the water side (h = 3000): eta = {annular_fin_eff(3000.0, k, t, r1, r2):.3f}")

# Thermowell in the 1 MPa steam main treated as a hollow pin fin with an insulated tip
Do, Di, kw, hw = 0.012, 0.007, 16.0, 150.0       # stainless steel well, steam-side h
P, Ac = math.pi * Do, math.pi / 4 * (Do**2 - Di**2)
mw = math.sqrt(hw * P / (kw * Ac))
T_steam, T_base = 250.0, 230.0                   # well base cooled by a poorly insulated nozzle
for L in (0.040, 0.060, 0.120):
    T_tip = T_steam - (T_steam - T_base) / math.cosh(mw * L)
    print(f"thermowell L = {L*1000:3.0f} mm: mL = {mw*L:.2f}, tip reads {T_tip:.1f} C, error {T_steam - T_tip:.2f} K")

m = 47.1 1/m, straight-fin estimate eta = 0.852, annular fin (numerical) eta = 0.818
per 1 m of tube: fin area 1.627 m2, base area 0.1117 m2, overall surface efficiency 0.830
heat rate bare 1149 W/m, finned 10391 W/m, ratio 9.0
single-fin effectiveness = 27.8
aluminum fin (k = 200): eta = 0.951
steel fin on the water side (h = 3000): eta = 0.156
thermowell L =  40 mm: mL = 2.75, tip reads 247.5 C, error 2.54 K
thermowell L =  60 mm: mL = 4.13, tip reads 249.4 C, error 0.64 K
thermowell L = 120 mm: mL = 8.26, tip reads 250.0 C, error 0.01 K

The program also shows why fins are not used on the water side of the same tube. With a coefficient of 3000 W/m²·K, \(mL_c\) rises to about 5 and the fin efficiency falls to 16 percent: most of the fin sits at nearly the water temperature and does almost nothing. The water side already has a small resistance, so the right design puts all the extended surface on the gas side.

Fins in electronics and equipment

Plant equipment uses fins in many smaller ways. The variable-frequency drives on Northgate's distribution pumps dissipate several kilowatts each through aluminum heat sinks with straight plate fins, cooled by fans; their design balances fin spacing against the fan's pressure drop, because fins packed too closely choke the airflow and the boundary layers between them merge. Transformers carry finned radiators. Air-cooled condensers on the chillers use thin aluminum plate fins bonded to copper tubes, often with a wavy or louvred pattern that breaks up the boundary layer and raises h. In each case, the governing parameters are the same: \(m\), \(L_c\), the fin efficiency, and the trade-off between added area and added flow resistance.

A thermowell as a fin

Temperature sensors in pipes are mounted in thermowells, closed tubes that protrude into the flow so the sensor can be removed without draining the line. A thermowell is a fin in reverse: heat flows from the fluid into the well and along it toward a base that may be cooler than the fluid, especially if the nozzle where the well is mounted is poorly insulated. The tip, where the sensor sits, therefore reads below the true fluid temperature. With an adiabatic tip, the error is

US practice rates thermowells mechanically by ASME PTC 19.3 TW, which checks the well's natural frequency against vortex-shedding frequencies and its stresses at the design flow. In US units, the economizer's 10.4 kW per meter is about 10,800 Btu/h per foot of tube, and the gas-side coefficient of 60 W/m²·K is about 10.6 Btu/h·ft²·°F, typical of finned-tube economizers in boiler stacks.