Convection Correlations: Method

The Correlations, Their Ranges and the Property Rules

Scope

Convection is the one mode of heat transfer that has no closed theory for the cases engineers meet: the coefficient h in Newton's law q″ = h (T_s − T_f) comes from boundary-layer solutions where they exist (the laminar flat plate, the laminar free-convection plate, fully developed laminar tube flow) and from experimental correlations everywhere else. The tool evaluates, for nine geometries, every correlation that the standard handbook (Incropera, DeWitt, Bergman and Lavine, Fundamentals of Heat and Mass Transfer, Chapters 7, 8 and 9) lists for the geometry, prints the range its authors gave, marks the case as inside or outside that range, and recommends the one the handbook prefers. The result is h, the heat flux h ΔT and the heat rate h A ΔT for the entered area. The tool does not solve the energy balance of a heat exchanger (the Heat Exchanger Effectiveness tool does) and does not combine convection with radiation.

Conventions and units

Lengths, the velocity, the density and the specific heat go through the site's unit dictionary. Temperatures are plain numbers, °C in SI mode and °F in USCS mode, because the dictionary's absolute temperature units carry offsets the unit engine cannot apply; the unit switch does not convert them. The viscosity is entered in centipoise in both systems (1 cP = 1 mPa·s = 2.419 lb/(ft·h)); the conductivity in W/(m·K) or Btu/(h·ft·°F) (1 Btu/(h·ft·°F) = 1.7307 W/(m·K)); the expansion coefficient in 1/K or 1/°R. The results without a dictionary type are printed with their unit named: h in W/(m²·K) or Btu/(h·ft²·°F) (1 Btu/(h·ft²·°F) = 5.678 W/(m²·K)), the heat flux in W/m² or Btu/(h·ft²), the heat rate in W or Btu/h. The temperature difference T_s − T_f is positive when the surface heats the fluid; the heat flux and heat rate carry that sign.

Which temperature the properties are taken at

Each correlation was fitted with its properties at a stated temperature, and using another one is a real error, up to 10–20 % for gases with large temperature ratios. The tool follows the handbook's rules: for flow inside a tube the properties are taken at the bulk mean temperature (the entered fluid temperature) except μ_s, taken at the wall; for external forced flow and for free convection at the film temperature T_film = (T_s + T_f)/2; Zukauskas and Whitaker, who fitted their data with free-stream properties, get the free-stream values and Pr_s or μ_s at the surface. With the water or air selection the tool evaluates every property at the right temperature itself; with an entered fluid the same values are used everywhere, and the surface viscosity may be entered for the (μ/μ_s) factors.

Water and air properties

Water: the saturated-liquid table of the NIST Chemistry WebBook (IAPWS-95 density and heat capacity, IAPWS 2008 viscosity, IAPWS 2011 thermal conductivity), one row per degree from the triple point to 100 °C, interpolated linearly; the expansion coefficient is the slope of the table's density, β = −(1/ρ)(dρ/dT), which is zero at the density maximum near 4 °C and negative below it (the tool refuses free convection there). The saturation pressure is a few kPa, but liquid water is nearly incompressible and the values serve at any ordinary pressure. Air: the ideal-gas density ρ = p/(RT) with R = 8.31432/28.9644 kJ/(kg·K), the Sutherland viscosity μ = 1.458 × 10⁻⁶ T^3/2/(T + 110.4) Pa·s and the conductivity k = 2.64638 × 10⁻³ T^3/2/(T + 245.4 × 10^−12/T) W/(m·K) of the U.S. Standard Atmosphere 1976 (equations 51 and 53; sea level 1.7894 × 10⁻⁵ Pa·s and 2.5326 × 10⁻² W/(m·K)), and the heat capacity of the N₂/O₂/Ar mixture from the NIST Shomate fits (1004.7 J/(kg·K) at 300 K; the fits run from 200 to 500 K). These agree with the handbook's Table A.4 within 0.3 % for c_p, 0.5 % for k and 1.7 % for ν at 350 K.

Dimensionless groups

Re = ρ V L_c/μ compares inertia with viscosity and decides the regime; Pr = μ c_p/k = ν/α compares the momentum and thermal diffusivities and decides how the thermal boundary layer sits inside the velocity layer (Pr ≈ 0.7 for gases, 2–10 for water, 10² to 10⁴ for oils, 0.01 for liquid metals); Nu = h L_c/k is the dimensionless coefficient, the ratio of convection to conduction across L_c. In free convection the velocity is not given but set by buoyancy, and Gr = g β |T_s − T_f| L_c³/ν² takes the place of Re²; Ra = Gr Pr is the usual correlating variable, with the transition to turbulence near Ra ≈ 10⁹ on a vertical plate. The characteristic length is the diameter for tubes, cylinders and spheres, the length along the flow for a plate in parallel flow, the height for a vertical plate and the area over the perimeter, A/P, for horizontal plates (a convention that makes one correlation serve several planforms).

Internal flow

Laminar flow (Re < 2,300) that is fully developed in velocity and temperature has an exact Nusselt number independent of Re and Pr: 3.66 for a uniform wall temperature (the first Graetz eigenvalue, 3.6568) and 4.36 for a uniform heat flux (48/11). The velocity profile develops over x_fd,h ≈ 0.05 Re D and the temperature profile over x_fd,t ≈ 0.05 Re Pr D, so a short tube or a high-Pr liquid has a higher average coefficient: Hausen's thermal-entry correlation Nu = 3.66 + 0.0668 Gz/[1 + 0.04 Gz^2/3] with the Graetz number Gz = (D/L) Re Pr applies when the velocity profile is already developed (or Pr ≫ 1), and the Sieder–Tate laminar form Nu = 1.86 Gz^1/3(μ/μ_s)^0.14 to the combined entry length, valid as long as it gives more than 3.66. Turbulent flow: Dittus–Boelter Nu = 0.023 Re^0.8 Prⁿ (n = 0.4 for heating, 0.3 for cooling; Re ≥ 10⁴, 0.6 ≤ Pr ≤ 160, L/D ≥ 10) is the classic form, good to about ±25 %; NACA Report 1020 confirmed the same exponents for air at wall-to-bulk temperature ratios up to 3.5 once the properties were taken at the surface temperature. Sieder–Tate's turbulent form Nu = 0.027 Re^0.8 Pr^1/3(μ/μ_s)^0.14 handles large viscosity variation. Gnielinski's Nu = (f/8)(Re − 1000) Pr/[1 + 12.7 (f/8)^1/2(Pr^2/3 − 1)], with Petukhov's smooth-tube friction factor f = (0.790 ln Re − 1.64)⁻², is accurate to about ±10 % from Re = 3,000 to 5 × 10⁶ and 0.5 ≤ Pr ≤ 2,000 and is the recommended turbulent correlation; Petukhov's own equation, with 1.07 in place of 1 and Re in place of Re − 1000, is its fully turbulent parent. Between Re 2,300 and 3,000 the flow is transitional and no correlation is reliable; the tool applies Gnielinski and says so. For a rough tube Gnielinski's equation may be used with the Colebrook friction factor of the Pipe Flow tool, which raises Nu with f.

Flat plate in parallel flow

Pohlhausen's solution of the energy equation in the Blasius laminar boundary layer gives the local Nu_x = 0.332 Re_x^1/2 Pr^1/3 for Pr ≥ 0.6, within 2 % of the exact value from Pr 0.6 to 100 (the exact coefficient is 0.3321 at Pr = 1 and the Pr^1/3 law is an approximation); since h_x ∝ x^−1/2 the average over the plate is twice the trailing-edge value, Nu_L = 0.664 Re_L^1/2 Pr^1/3. Churchill and Ozoe's form extends it to any Pr (liquid metals). The turbulent boundary layer gives the local Nu_x = 0.0296 Re_x^0.8 Pr^1/3 (0.6 ≤ Pr ≤ 60, Re up to 10⁸); a plate turbulent from the leading edge (tripped) averages to 0.037 Re_L^0.8 Pr^1/3. A smooth plate in a quiet stream is laminar to Re_x,c ≈ 5 × 10⁵ and turbulent beyond, and the mixed average integrates the two: Nu_L = (0.037 Re_L^0.8 − A) Pr^1/3, A = 0.037 Re_x,c^0.8 − 0.664 Re_x,c^1/2 = 871 at 5 × 10⁵. The transition Reynolds number is an input: free-stream turbulence or roughness moves it down to 10⁵ or less.

Cylinder in cross flow and sphere

Hilpert's Nu = C Re^m Pr^1/3 fits the 1933 data for air in five Re bands from 0.4 to 4 × 10⁵ (C, m = 0.989, 0.330; 0.911, 0.385; 0.683, 0.466; 0.193, 0.618; 0.027, 0.805), with film properties; its jumps at the band edges are those of the fit, not of the physics. Zukauskas's Nu = C Re^m Prⁿ(Pr/Pr_s)^1/4 covers 0.7 ≤ Pr ≤ 500 and 1 ≤ Re ≤ 10⁶ with the properties at the free-stream temperature. Churchill and Bernstein's single equation, Nu = 0.3 + 0.62 Re^1/2 Pr^1/3/[1 + (0.4/Pr)^2/3]^1/4 × [1 + (Re/282,000)^5/8]^4/5, holds for all Re Pr ≥ 0.2 and is the recommended one; it is within 20 % of the data over the whole range, which is the state of the art for this geometry (the handbook's Example 7.4 gives 96, 88 and 105 W/(m²·K) from the three correlations for the same cylinder). For a sphere Whitaker's Nu = 2 + (0.4 Re^1/2 + 0.06 Re^2/3) Pr^0.4(μ/μ_s)^1/4 (3.5 ≤ Re ≤ 7.6 × 10⁴, 0.71 ≤ Pr ≤ 380, free-stream properties) is recommended, and Ranz and Marshall's Nu = 2 + 0.6 Re^1/2 Pr^1/3 for drops is shown beside it; both start from the conduction limit Nu = 2 of a sphere in a still fluid.

Free convection

Ostrach (NACA Report 1111, 1953) solved the laminar boundary layer on a vertical isothermal plate exactly: Nu_x = −H′(0)(Gr_x/4)^1/4 with −H′(0) = 0.0812, 0.5046, 0.5671, 0.7165, 1.1694, 2.191 and 3.966 at Pr = 0.01, 0.72, 1, 2, 10, 100 and 1000 (his Table I), and the average over the height is 4/3 of the value at the top. LeFevre's interpolation g(Pr) = 0.75 Pr^1/2/(0.609 + 1.221 Pr^1/2 + 1.238 Pr)^1/4 reproduces the table within 0.1 % (0.7 % at Pr 0.01, where Ostrach's own integration was truncated), and the tool prints Nu_L = (4/3)(Gr_L/4)^1/4 g(Pr) as the theoretical reference for Ra ≤ 10⁹. Churchill and Chu's Nu = {0.825 + 0.387 Ra^1/6/[1 + (0.492/Pr)^9/16]^8/27}² covers the whole range of Ra, laminar and turbulent, and is the recommended correlation; their laminar form Nu = 0.68 + 0.670 Ra^1/4/[1 + (0.492/Pr)^9/16]^4/9 is slightly better below Ra = 10⁹ and sits within 1 % of Ostrach. McAdams's Nu = 0.59 Ra^1/4 (10⁴–10⁹) and 0.10 Ra^1/3 (10⁹–10¹³) are the older forms; the 0.59 lies 14 % above the laminar theory for air, a fit to data that included transition. A vertical cylinder may be treated as a plate when D/L ≥ 35/Gr_L^1/4. Horizontal plates follow McAdams with L = A/P: hot side up or cold side down Nu = 0.54 Ra^1/4 (10⁴–10⁷) and 0.15 Ra^1/3 (10⁷–10¹¹), hot side down or cold side up Nu = 0.27 Ra^1/4 (10⁵–10¹⁰), half the other because the buoyant fluid is trapped against the plate. The horizontal cylinder has Churchill and Chu's Nu = {0.60 + 0.387 Ra^1/6/[1 + (0.559/Pr)^9/16]^8/27}² for Ra ≤ 10¹² and Morgan's five-band C Raⁿ; the sphere Churchill's Nu = 2 + 0.589 Ra^1/4/[1 + (0.469/Pr)^9/16]^4/9 for Ra ≤ 10¹¹ and Pr ≥ 0.7.

How to read the results

The first section prints the properties at the temperature the correlations want, with the Prandtl number from μ c_p/k. The second gives the characteristic length, the area, Re (and the entry lengths of a tube or the transition point on a plate) or Gr and Ra. Table 1 then lists every correlation for the geometry with its Nusselt number, its coefficient, its range and whether the case lies inside it; the bold row is the recommended value carried into the heat-flux section. The spread between the correlations within their ranges, usually 10–25 %, is the honest uncertainty of a convection estimate: when a design hinges on it, a test or a conservative choice is needed. For a tube remember that the printed heat rate assumes a constant wall-to-fluid difference; with a uniform wall temperature the difference falls along the tube and the log-mean value applies, with the outlet temperature from (T_s − T_out)/(T_s − T_in) = exp(−h A/(ṁ c_p)). Figure 2 draws Nu against Re or Ra for each correlation over its own range at the case's Prandtl number, so the position of the case relative to the regime boundaries and the disagreement between the fits are seen at a glance.

Limits

Not covered: tube banks and finned surfaces, non-circular ducts (use the hydraulic diameter in the turbulent correlations, but the laminar values differ), rough tubes (Gnielinski with the Colebrook f), liquid metals in tubes (Pr < 0.5: Skupinski or Seban–Shimazaki forms), boiling and condensation, mixed forced and free convection (Gr/Re² near 1), radiation, enclosures and cavities, inclined plates (replace g by g cos θ for angles up to 60°), surfaces with a uniform heat flux in free convection (the isothermal correlations with the mid-height temperature are within a few percent), and temperature-dependent properties beyond the (μ/μ_s) and (Pr/Pr_s) factors.