External forced convection

So far the convection coefficient has simply been given. From this lesson on, we find it. Northgate needs \(h\) for several external flows: wind across the flat roof of the boiler hall, wind across a section of steam pipe that runs outdoors on a pipe rack between buildings, and flue gas flowing across the tube bank of the planned economizer. Each is an external flow, in which the fluid flows over a body and the boundary layer on the body grows without being confined by walls. The tools are boundary layer theory, dimensionless groups and empirical correlations, and the skill is in choosing the right correlation and evaluating it with the right properties.

Velocity and thermal boundary layers

When fluid flows over a surface, the layer in contact with it is brought to rest, and a velocity boundary layer forms in which the velocity rises from zero at the wall to the free-stream value \(u_\infty\). If the surface is at a different temperature from the fluid, a thermal boundary layer also forms, across which the temperature changes from \(T_s\) to \(T_\infty\). At the wall itself, heat passes by conduction through the stationary fluid, so the local heat flux is \(q_s'' = -k_f\,\partial T/\partial y\) at \(y = 0\), and the convection coefficient is

Anything that steepens the temperature gradient at the wall, such as higher velocity, a thinner boundary layer or turbulent mixing, raises \(h\). On a flat plate the boundary layer starts thin at the leading edge and thickens downstream, so the local coefficient is highest at the leading edge and falls with distance.

The flow in the boundary layer starts laminar, with smooth, orderly streamlines. At some distance it becomes unstable and passes through a transition region to turbulent flow, in which eddies mix fluid across the layer. Turbulent mixing greatly increases momentum and heat transfer near the wall, so the local coefficient jumps at transition and then falls slowly. On a smooth flat plate, transition is usually taken to occur at a critical Reynolds number of about \(5 \times 10^5\), although roughness, free-stream turbulence and vibration can trigger it earlier.

Dimensionless groups

Correlations for \(h\) are written in terms of three dimensionless groups. The Reynolds number compares inertia with viscous forces and decides whether the flow is laminar or turbulent. The Prandtl number, a fluid property, compares momentum diffusivity with thermal diffusivity and so relates the thicknesses of the velocity and thermal boundary layers: about 0.7 for gases, 1.5 to 7 for water depending on temperature, and hundreds or thousands for oils. The Nusselt number is the dimensionless convection coefficient, the ratio of convection to conduction across the fluid layer.

Dimensional analysis shows that for forced convection in a given geometry, \(\text{Nu} = f(\text{Re}, \text{Pr})\). Experiments and theory fix the function. Properties are normally evaluated at the film temperature, \(T_f = (T_s + T_\infty)/2\), which accounts approximately for their variation across the boundary layer. The Nusselt number uses the fluid's conductivity, not the solid's, which distinguishes it from the Biot number of Lesson 5.

The flat plate in parallel flow

For laminar flow over an isothermal flat plate, boundary layer theory gives exact results. The average coefficient over a laminar plate of length \(L\) is \(\overline{\text{Nu}}_L = 0.664\,\text{Re}_L^{1/2}\text{Pr}^{1/3}\), valid for Pr above about 0.6. For a fully turbulent plate, \(\overline{\text{Nu}}_L = 0.037\,\text{Re}_L^{4/5}\text{Pr}^{1/3}\). Real plates longer than the transition length have a laminar leading section followed by turbulent flow; combining the two with transition at \(5\times10^5\) gives the mixed-boundary-layer correlation

When \(\text{Re}_L\) is much larger than the critical value, the laminar section is a small fraction of the plate and the fully turbulent correlation is adequate. For a uniform surface heat flux instead of a uniform temperature, the local Nusselt number is about 36 percent higher in laminar flow and about 4 percent higher in turbulent flow.

# External forced convection: a flat roof, a steam pipe in the wind and a tube bank in flue gas
import math
# Air at atmospheric pressure (approximate table values): T (K), nu (m2/s), k (W/m.K), Pr
AIR = [(200, 7.59e-6, 0.0181, 0.737), (250, 11.44e-6, 0.0223, 0.720), (300, 15.89e-6, 0.0263, 0.707),
       (350, 20.92e-6, 0.0300, 0.700), (400, 26.41e-6, 0.0338, 0.690), (450, 32.39e-6, 0.0373, 0.686),
       (500, 38.79e-6, 0.0407, 0.684), (600, 52.69e-6, 0.0469, 0.685), (700, 68.10e-6, 0.0524, 0.695)]
def air(T):
    for a, b in zip(AIR, AIR[1:]):
        if a[0] <= T <= b[0]:
            f = (T - a[0]) / (b[0] - a[0])
            return tuple(x + f * (y - x) for x, y in zip(a[1:], b[1:]))
    raise ValueError("temperature outside table")

# 1. Flat plate: wind blowing along the 30 m flat roof of the boiler hall
U, L, Tf = 6.0, 30.0, 260.0
nu, k, Pr = air(Tf)
Re = U * L / nu
x_c = 5e5 * nu / U
Nu_lam = 0.664 * Re**0.5 * Pr**(1/3)
Nu_mix = (0.037 * Re**0.8 - 871) * Pr**(1/3)
print(f"roof: Re_L = {Re:.2e}, transition at x = {x_c:.2f} m, h(all laminar, wrong) = {Nu_lam*k/L:.2f}, h(mixed) = {Nu_mix*k/L:.2f} W/m2.K")

# 2. Cylinder in cross flow (Churchill-Bernstein): a bare DN200 pipe outdoors on a rack
def churchill_bernstein(Re, Pr):
    return 0.3 + 0.62 * Re**0.5 * Pr**(1/3) / (1 + (0.4/Pr)**(2/3))**0.25 * (1 + (Re/282000)**(5/8))**(4/5)
D, Ts = 0.2191, 175.0
for U, Tinf in ((1.0, -30.0), (8.0, -30.0), (8.0, 20.0)):
    Tf = (Ts + Tinf) / 2 + 273.15
    nu, k, Pr = air(Tf)
    Re = U * D / nu
    h = churchill_bernstein(Re, Pr) * k / D
    print(f"pipe, wind {U:3.0f} m/s, air {Tinf:4.0f} C: Re = {Re:8.0f}, h = {h:5.1f} W/m2.K, convection loss {h*math.pi*D*(Ts-Tinf):5.0f} W/m")

# 3. Bank of bare staggered tubes in flue gas (Zukauskas), properties of air used for flue gas
def zukauskas_staggered(Re, Pr, ST, SL):
    if Re < 1e3:   C, m = 0.71, 0.5
    elif Re < 2e5: C, m = (0.35 * (ST/SL)**0.2, 0.60) if ST/SL < 2 else (0.40, 0.60)
    else:          C, m = 0.031 * (ST/SL)**0.2, 0.8
    return C * Re**m * Pr**0.36
D, ST, SL, V, Tg = 0.0508, 0.100, 0.090, 8.0, 500.0     # tube OD, pitches (m), face velocity (m/s), gas (K)
SD = math.hypot(SL, ST/2)
Vmax = V * ST / (ST - D) if SD > (ST + D) / 2 else V * ST / (2 * (SD - D))
nu, k, Pr = air(Tg)
Re = Vmax * D / nu
h = zukauskas_staggered(Re, Pr, ST, SL) * k / D
print(f"tube bank: Vmax = {Vmax:.1f} m/s, Re_D,max = {Re:.0f}, h = {h:.1f} W/m2.K (20 or more rows)")
for N, Cr in ((2, 0.76), (5, 0.92), (10, 0.97)):
    print(f"  with {N:2d} rows the average h is about {Cr*h:.1f} W/m2.K")

roof: Re_L = 1.46e+07, transition at x = 1.03 m, h(all laminar, wrong) = 1.75, h(mixed) = 13.14 W/m2.K
pipe, wind   1 m/s, air  -30 C: Re =    10697, h =   7.5 W/m2.K, convection loss  1058 W/m
pipe, wind   8 m/s, air  -30 C: Re =    85576, h =  26.2 W/m2.K, convection loss  3691 W/m
pipe, wind   8 m/s, air   20 C: Re =    75593, h =  25.6 W/m2.K, convection loss  2729 W/m
tube bank: Vmax = 16.3 m/s, Re_D,max = 21295, h = 98.7 W/m2.K (20 or more rows)
  with  2 rows the average h is about 75.0 W/m2.K
  with  5 rows the average h is about 90.8 W/m2.K
  with 10 rows the average h is about 95.8 W/m2.K

Cylinders and spheres in cross flow

Flow across a cylinder is more complex. The boundary layer forms at the front stagnation point and grows around the cylinder, but the pressure rises on the rear half, and the boundary layer separates, leaving a wake of recirculating eddies. The local coefficient is highest at the front, falls toward separation and rises again in the wake. Correlations therefore average over the whole surface. The Churchill-Bernstein correlation covers the whole range of Re·Pr above about 0.2:

Simpler power-law forms, \(\overline{\text{Nu}}_D = C\,\text{Re}_D^m\text{Pr}^{1/3}\) with \(C\) and \(m\) tabulated by Reynolds number range (the Hilpert correlation), are also widely used. For spheres, the Whitaker correlation \(\overline{\text{Nu}}_D = 2 + (0.4\,\text{Re}_D^{1/2} + 0.06\,\text{Re}_D^{2/3})\text{Pr}^{0.4}(\mu/\mu_s)^{1/4}\) applies; the constant 2 is the pure conduction limit for a sphere in a stagnant fluid. Typical uncertainties of all these correlations are 15 to 25 percent, a reminder that convection coefficients are estimates, not exact values.

Tube banks

Heat exchangers such as economizers, air heaters and air-cooled condensers use banks of tubes across which the outside fluid flows. Tubes are arranged in line or staggered, with transverse pitch \(S_T\) and longitudinal pitch \(S_L\). The flow accelerates through the gaps between tubes, so the Reynolds number is based on the maximum velocity, which in a staggered bank occurs either in the transverse gap or in the diagonal gap, whichever is smaller. The Zukauskas correlation, \(\overline{\text{Nu}}_D = C\,\text{Re}_{D,\max}^m\,\text{Pr}^{0.36}(\text{Pr}/\text{Pr}_s)^{1/4}\), with \(C\) and \(m\) depending on Reynolds number and arrangement, applies to banks of 20 or more rows. For fewer rows, a correction factor reduces the average, because the first rows see less turbulence than the inner rows.

US design guides take wind and temperature data from ASHRAE climatic design tables for each weather station; the Minneapolis-St. Paul design conditions combine low temperatures with significant winter winds. In US units, the 8 m/s wind is about 18 mph and the coefficient of 26.2 W/m²·K is about 4.6 Btu/h·ft²·°F; the bare pipe loses about 3,800 Btu/h per foot by convection on such a night.