Natural convection

In much of a building, and in much of a plant, nothing drives the air but temperature itself. Warm air beside a hot surface is less dense than the air around it, so it rises, and cooler air flows in to replace it. This is natural, or free, convection. It heats the patient rooms of Riverside General Hospital through wall-mounted convector panels, it carries heat away from bare pipes in Northgate's mechanical rooms, it governs heat loss from the tops of storage tanks, and it moves air inside the gap of every double-glazed window. Its coefficients are small, typically 2 to 10 W/m²·K in air, which is why radiation often matters as much. This lesson develops the physics and the main correlations, and shows how to tell whether a flow is natural, forced or mixed.

Buoyancy-driven flow

In natural convection the fluid velocity is not given; it is produced by the heat transfer itself. Near a heated vertical plate, a boundary layer forms in which the velocity is zero at the wall, rises to a maximum and falls back to zero at the edge of the layer, while the temperature falls from \(T_s\) to \(T_\infty\). The driving force is buoyancy, proportional to the density difference, which for small temperature differences is \(\rho\beta(T_s - T_\infty)\), where \(\beta\) is the volumetric thermal expansion coefficient. For an ideal gas, \(\beta = 1/T\), with \(T\) in kelvin, evaluated at the film temperature.

The relevant dimensionless group, playing the role that the Reynolds number plays in forced convection, is the Grashof number, the ratio of buoyancy to viscous forces. Combined with the Prandtl number, it gives the Rayleigh number, which governs heat transfer:

On a vertical plate the boundary layer is laminar for Rayleigh numbers below about \(10^9\) and becomes turbulent above that. Because \(\text{Ra}\) grows with \(L^3\), large surfaces reach turbulence: a 0.6 m panel in a room has \(\text{Ra} \approx 7\times10^8\), just laminar, while a 3 m wall would be turbulent.

Correlations for external natural convection

Natural convection correlations take the form \(\overline{\text{Nu}}_L = C\,\text{Ra}_L^n\), with \(n = 1/4\) for laminar and \(1/3\) for turbulent flow, or more general forms that cover both. The most widely used for a vertical plate is the Churchill-Chu correlation, valid for all Rayleigh numbers:

It also applies to vertical cylinders whose diameter is large compared with the boundary layer thickness. For a long horizontal cylinder, Churchill and Chu give a similar form based on the diameter, valid for \(\text{Ra}_D\) up to about \(10^{12}\):

Horizontal plates behave differently on their two faces. On the upper face of a hot plate, or the lower face of a cold one, buoyant plumes rise freely from the surface and heat transfer is relatively strong: \(\overline{\text{Nu}}_L = 0.54\,\text{Ra}_L^{1/4}\) for \(10^4\) to \(10^7\) and \(0.15\,\text{Ra}_L^{1/3}\) above that. On the lower face of a hot plate, warm fluid is trapped beneath and must creep out to the edges, so heat transfer is weak: \(\overline{\text{Nu}}_L = 0.52\,\text{Ra}_L^{1/5}\). For these correlations the characteristic length is the plate area divided by its perimeter. Spheres have their own correlation, approaching \(\text{Nu} = 2\) as the Rayleigh number falls.

# Natural convection: a hospital convector panel, bare pipe in the mechanical room, a tank top and a window
import math
g = 9.81
AIR = [(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)]
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:]))
def Ra(Ts, Tinf, L):
    Tf = (Ts + Tinf) / 2 + 273.15
    nu, k, Pr = air(Tf)
    alpha = nu / Pr
    return g * (1 / Tf) * abs(Ts - Tinf) * L**3 / (nu * alpha), k, Pr, nu

def vertical_plate(Ra, Pr):                 # Churchill and Chu, all Rayleigh numbers
    return (0.825 + 0.387 * Ra**(1/6) / (1 + (0.492/Pr)**(9/16))**(8/27))**2
def horizontal_cylinder(Ra, Pr):            # Churchill and Chu, Ra up to about 1e12
    return (0.60 + 0.387 * Ra**(1/6) / (1 + (0.559/Pr)**(9/16))**(8/27))**2
sigma = 5.670e-8

# 1. A wall-mounted panel at 70 C, 0.6 m high, 1.2 m long, in a 21 C patient room
H, W, Ts, Tr = 0.6, 1.2, 70.0, 21.0
R, k, Pr, _ = Ra(Ts, Tr, H)
h = vertical_plate(R, Pr) * k / H
q_c = h * H * W * (Ts - Tr)
q_r = 0.9 * sigma * H * W * ((Ts+273.15)**4 - (Tr+273.15)**4)
print(f"panel front face: Ra = {R:.2e} ({'laminar' if R < 1e9 else 'turbulent'}), h = {h:.2f} W/m2.K, "
      f"convection {q_c:.0f} W, radiation {q_r:.0f} W")

# 2. A bare horizontal DN50 hot-water pipe (OD 60.3 mm) at 90 C in a 25 C mechanical room
D, Ts, Tr = 0.0603, 90.0, 25.0
R, k, Pr, _ = Ra(Ts, Tr, D)
h = horizontal_cylinder(R, Pr) * k / D
h_r = 0.85 * sigma * ((Ts+273.15)**4 - (Tr+273.15)**4) / (Ts - Tr)
print(f"bare pipe: Ra_D = {R:.2e}, h_conv = {h:.2f}, h_rad = {h_r:.2f} W/m2.K, loss {(h+h_r)*math.pi*D*(Ts-Tr):.0f} W/m")

# 3. The flat top of a 3 m diameter hot-water storage tank, 60 C surface in 20 C air (hot face up)
Dt, Ts, Tr = 3.0, 60.0, 20.0
Lc = (math.pi * Dt**2 / 4) / (math.pi * Dt)                 # area / perimeter
R, k, Pr, _ = Ra(Ts, Tr, Lc)
Nu_up = 0.54 * R**0.25 if R < 1e7 else 0.15 * R**(1/3)
Nu_dn = 0.52 * R**0.2                                       # hot face down (for comparison)
print(f"tank top: Ra = {R:.2e}, h(hot face up) = {Nu_up*k/Lc:.2f}, h(hot face down) = {Nu_dn*k/Lc:.2f} W/m2.K")

# 4. Air gap in a double-glazed control-room window, 1.2 m tall, panes at 15 C and -10 C
Hg, T1, T2 = 1.2, 15.0, -10.0
for Lg in (0.006, 0.013, 0.025, 0.050):
    R, k, Pr, _ = Ra(T1, T2, Lg)
    Nu_e = max(1.0, 0.42 * R**0.25 * Pr**0.012 * (Hg/Lg)**-0.3) if R > 1e3 else 1.0
    print(f"window gap {Lg*1000:4.0f} mm: Ra_L = {R:8.0f}, Nu = {Nu_e:.2f}, effective k = {Nu_e*k:.4f} W/m.K, "
          f"flux across the gap {Nu_e*k*(T1-T2)/Lg:5.1f} W/m2")

# 5. Is the panel's flow mixed? Compare buoyancy with a 0.3 m/s draft from a supply diffuser
R, k, Pr, nu = Ra(70.0, 21.0, 0.6)
Gr = R / Pr; Re = 0.3 * 0.6 / nu
print(f"Gr/Re^2 = {Gr/Re**2:.1f} ({'natural' if Gr/Re**2 > 10 else 'mixed' if Gr/Re**2 > 0.1 else 'forced'} convection dominates)")

panel front face: Ra = 7.27e+08 (laminar), h = 5.13 W/m2.K, convection 181 W, radiation 234 W
bare pipe: Ra_D = 8.25e+05, h_conv = 6.52, h_rad = 7.04 W/m2.K, loss 167 W/m
tank top: Ra = 1.26e+09, h(hot face up) = 5.89, h(hot face down) = 1.25 W/m2.K
window gap    6 mm: Ra_L =      728, Nu = 1.00, effective k = 0.0244 W/m.K, flux across the gap 101.5 W/m2
window gap   13 mm: Ra_L =     7404, Nu = 1.00, effective k = 0.0244 W/m.K, flux across the gap  46.8 W/m2
window gap   25 mm: Ra_L =    52659, Nu = 1.98, effective k = 0.0483 W/m.K, flux across the gap  48.3 W/m2
window gap   50 mm: Ra_L =   421273, Nu = 4.11, effective k = 0.1000 W/m.K, flux across the gap  50.0 W/m2
Gr/Re^2 = 10.1 (natural convection dominates)

Enclosures

Natural convection inside a closed space, such as the gap between two panes of glass, a wall cavity or the air space in a solar collector, is described by an effective conductivity: \(k_{\text{eff}} = k\,\overline{\text{Nu}}\), so that the heat flux is \(k_{\text{eff}}(T_1 - T_2)/L\), where \(L\) is the gap width. For a vertical enclosure heated on one side and cooled on the other, there is no flow at all to speak of below a Rayleigh number of about 1000; the air conducts as if it were still, and \(\text{Nu} = 1\). Above that, a circulating cell forms and the Nusselt number rises. Correlations depend on the aspect ratio \(H/L\); one widely used form for tall enclosures is \(\overline{\text{Nu}}_L = 0.42\,\text{Ra}_L^{1/4}\text{Pr}^{0.012}(H/L)^{-0.3}\).

Mixed convection

When a forced flow is weak and temperature differences are large, buoyancy and forced flow act together. The ratio \(\text{Gr}_L/\text{Re}_L^2\) measures their relative strength: if it is much less than 1, forced convection dominates; if it is much greater than 1, natural convection dominates; near 1, both matter and the flow is mixed. A common approximation combines the two Nusselt numbers as \(\text{Nu}^3 = \text{Nu}_F^3 \pm \text{Nu}_N^3\), with the plus sign when the flows assist each other and the minus sign when they oppose. For the convector panel, a 0.3 m/s draft from a nearby supply diffuser gives \(\text{Gr}/\text{Re}^2 \approx 10\), so natural convection still dominates, but a stronger draft would change the panel's output, which is why the hospital's air distribution design keeps supply jets away from the emitters.

Hydronic emitters sold in Canada are rated by manufacturers under standard test methods, and Canadian designers often derate catalogue outputs to the lower supply temperatures used in modern district and condensing-boiler systems. Fenestration performance is rated under CSA standards and the National Building Code, and window U-values are reported in W/m²·K; in Winnipeg's climate, triple glazing has become common in new institutional buildings.