Free and Mixed Convection Correlations
Natural Convection
Semi-Infinite Vertical Plate
Turbulence Transition: Uniform wall temperature, Rayleigh number is defined as \mathrm{Ra}_x=\mathrm{Pr}\mathrm{Gr}_x=\frac {g\beta x^3(T_s-T_{\infty})}{\nu\alpha}
\mathrm{Ra}_x\approx10^9
Turbulence Transition, Benjan (1993): Uniform wall temperature, more accurate transition for 10^{-3}<\mathrm{Pr}<10^3, Grashof number is defined as \mathrm{Gr}_x=\frac {g\beta l^3(T_s-T_{\infty})}{\nu}
\mathrm{Gr}_x\approx10^9
Turbulence Transition: Uniform heat flux, modified Rayleigh number defined as \mathrm{Ra}_x^*=\mathrm{Gr}_x^*\mathrm{Pr}=\frac {g\beta q_s''x^4}{k\nu\alpha}
\mathrm{Ra}_{x,\mathrm{cr}}^*\approx10^{13}\tag{10.6.5}
Profiles
Analytical: Uniform wall temperature, Grashof number defined as \mathrm{Gr}_x=\frac {g\beta x^3(T_s-T_{\infty})}{\nu}
\frac {\delta}x=3.93\mathrm{Pr}^{-1/2}\mathrm{Gr}_x^{-1/4}\left(\mathrm{Pr}+\frac {20}{21}\right)^{1/4}\tag{10.5.23}
Analytical: Uniform heat flux, modified Grashof number defined as \mathrm{Gr}_x^*=\frac {g\beta q_s''x^4}{k\nu^2}
\frac {\delta}x=\left[\frac {360(\mathrm{Pr}+0.8)}{\mathrm{Pr}^2\mathrm{Gr}_x^*}\right]^{1/5}\tag{10.5.34}
Analytical: Uniform heat flux, modified Grashof number defined as \mathrm{Gr}_x^*=\frac {g\beta q_s''x^4}{k\nu^2}
T_s-T_{\infty}=\frac {1.622q_s''x}k\left(\frac {\mathrm{Pr}+0.8}{\mathrm{Pr}^2\mathrm{Gr}_x^*}\right)^{1/5}\tag{10.5.37}
Heat Transfer Coefficient
Laminar Flow
Analytical: Calculate \mathrm{Nu}_l using any applicable correlation.
\langle\mathrm{Nu}_l\rangle_l=\frac 43\mathrm{Nu}_l\tag{10.4.16}
LeFevre and Ede (1965): Uniform wall temperature
\begin{aligned} \mathrm{Nu}_{x,\mathrm{UWT}} & =\phi(\mathrm{Pr})\mathrm{Gr}_x^{1/4} \\ \phi(\mathrm{Pr}) & =\frac {0.75\mathrm{Pr}^{1/4}}{\left[4\left(0.609+1.221\mathrm{Pr}^{1/2}+1.238\mathrm{Pr}\right)\right]^{1/4}} \end{aligned}\tag{10.4.14}
Analytical: Uniform wall temperature, Grashof number defined as \mathrm{Gr}_x=\frac {g\beta x^3(T_s-T_{\infty})}{\nu}
\mathrm{Nu}_{x,\mathrm{UWT}}=0.508\mathrm{Pr}^{1/2}\left(\mathrm{Pr}+\frac {20}{21}\right)^{-1/4}\mathrm{Gr}_x^{1/4}\tag{10.5.25}
Analytical: Uniform heat flux, modified Grashof number defined as \mathrm{Gr}_x^*=\frac {g\beta q_s''x^4}{k\nu^2}
\mathrm{Nu}_{x,\mathrm{UHF}}=0.62\left(\frac {\mathrm{Pr}^2\mathrm{Gr}_x^*}{\mathrm{Pr}+0.8}\right)^{1/5}\tag{10.5.35}
McAdams (1954): Uniform wall temperature, for \mathrm{Pr}\approx1, for 10^4<\mathrm{Ra}_l<10^9, Rayleigh number defined as \mathrm{Ra}_x=\mathrm{Pr}\mathrm{Gr}_x=\frac {g\beta x^3(T_s-T_{\infty})}{\nu\alpha}, calculate fluid properties at T_{\mathrm{film}}=\frac 12(T_s+T_{\infty})
\langle\mathrm{Nu}_{l,\mathrm{UWT}}\rangle_l=0.59\mathrm{Ra}_l^{1/4}\tag{10.6.1}
Churchill and Chu (1975): Uniform wall temperature, valid for all \mathrm{Ra}_l and \mathrm{Pr}, fluid properties calculated at T_{\mathrm{film}}=\frac 12(T_s+T_{\infty})
\langle\mathrm{Nu}_{l,\mathrm{UWT}}\rangle_l=\left[0.825+\frac {0.387\mathrm{Ra}_l^{1/6}}{\left[1+(0.492/\mathrm{Pr})^{9/16}\right]^{8/27}}\right]^2\tag{10.6.3}
Churchill and Chu (1975): Uniform wall temperature, for \mathrm{Ra}_l<10^9, calculate fluid properties at T_{\mathrm{film}}=\frac 12(T_s+T_{\infty})
\langle\mathrm{Nu}_{l,\mathrm{UWT}}\rangle_l=0.68+\frac {0.67\mathrm{Ra}_l^{1/4}}{\left[1+(0.492/\mathrm{Pr})^{9/16}\right]^{4/9}}\tag{10.6.4}
Vliet and Liu (1969): Uniform heat flux, water only, modified Rayleigh number is defined as \mathrm{Ra}_x^*=\mathrm{Gr}_x^*\mathrm{Pr}=\frac {g\beta q_s''x^4}{k\nu\alpha}, average Nusselt number defined as \langle\mathrm{Nu}_l\rangle_l=\frac {q_s''l}{k\left(\langle T_s\rangle_l-T_{\infty}\right)}
\begin{aligned} \mathrm{Nu}_{x,\mathrm{UHF}} & =0.60\left(\mathrm{Ra}_x^*\right)^{1/5}\qquad\qquad10^5<\mathrm{Ra}_x^*<10^{13} \\ \langle\mathrm{Nu}_{l,\mathrm{UHF}}\rangle_l & =1.25\mathrm{Nu}_l\qquad\qquad\qquad10^5<\mathrm{Ra}_x^*<10^{11} \end{aligned}\tag{10.6.7}
Vliet and Ross (1975): Uniform heat flux, air only, modified Rayleigh number is defined as \mathrm{Ra}_x^*=\mathrm{Gr}_x^*\mathrm{Pr}=\frac {g\beta q_s''x^4}{k\nu\alpha}, average Nusselt number defined as \langle\mathrm{Nu}_l\rangle_l=\frac {q_s''l}{k\left(\langle T_s\rangle_l-T_{\infty}\right)}
\langle\mathrm{Nu}_{l,\mathrm{UHF}}\rangle_l=0.55\left(\mathrm{Ra}_l^*\right)^{1/5}\tag{10.6.12}
Turbulent Flow
Oosthuizen and Naylor (1999): Smooth surface, \frac 17 power law velocity profile, valid for \mathrm{Pr}=0.7, Grashof number defined as \mathrm{Gr}_x=\frac {g\beta x^3(T_s-T_{\infty})}{\nu}
\begin{aligned} \mathrm{Nu}_{x} & =0.0185\mathrm{Gr}_x^{0.4} \\ \langle\mathrm{Nu}_{l}\rangle_l & =0.0154\mathrm{Gr}_l^{0.4} \end{aligned}\tag{10.5.44}
Churchill and Chu (1975): Uniform wall temperature, valid for all \mathrm{Ra}_l and \mathrm{Pr}, calculate fluid properties at T_{\mathrm{film}}=\frac 12(T_s+T_{\infty})
\langle\mathrm{Nu}_{l,\mathrm{UWT}}\rangle_l=\left[0.825+\frac {0.387\mathrm{Ra}_l^{1/6}}{\left[1+(0.492/\mathrm{Pr})^{9/16}\right]^{8/27}}\right]^2\tag{10.6.3}
McAdams (1954): Uniform wall temperature, valid for \mathrm{Pr}\approx1 and 10^9<\mathrm{Ra}_l<10^{13}, Rayleigh number defined as \mathrm{Ra}_x=\mathrm{Pr}\mathrm{Gr}_x=\frac {g\beta x^3(T_s-T_{\infty})}{\nu\alpha}, calculate fluid properties at T_{\mathrm{film}}=\frac 12(T_s+T_{\infty})
\langle\mathrm{Nu}_l\rangle_l=0.1\mathrm{Ra}_l^{1/3}\tag{10.6.2}
Vliet and Liu (1969): Uniform heat flux, water only, modified Rayleigh number is defined as \mathrm{Ra}_x^*=\mathrm{Gr}_x^*\mathrm{Pr}=\frac {g\beta q_s''x^4}{k\nu\alpha}, average Nusselt number defined as \langle\mathrm{Nu}_l\rangle_l=\frac {q_s''l}{k\left(\langle T_s\rangle_l-T_{\infty}\right)}
\begin{aligned} \mathrm{Nu}_{x,\mathrm{UHF}} & =0.568\left(\mathrm{Ra}_x^*\right)^{0.22}\qquad\qquad10^{13}<\mathrm{Ra}_x^*<10^{16} \\ \langle\mathrm{Nu}_{l,\mathrm{UHF}}\rangle_l & =1.136\mathrm{Nu}_l\qquad\qquad\qquad\,\,2\times10^{13}<\mathrm{Ra}_x^*<10^{16} \end{aligned}\tag{10.6.9}
Vliet and Ross (1975): Uniform heat flux, air only, modified Rayleigh number is defined as \mathrm{Ra}_x^*=\mathrm{Gr}_x^*\mathrm{Pr}=\frac {g\beta q_s''x^4}{k\nu\alpha}, average Nusselt number defined as \langle\mathrm{Nu}_l\rangle_l=\frac {q_s''l}{k\left(\langle T_s\rangle_l-T_{\infty}\right)}
\langle\mathrm{Nu}_{l,\mathrm{UHF}}\rangle_l=0.17\left(\mathrm{Ra}_l^*\right)^{1/4}\tag{10.6.13}
Vertical Cylinder
Heat Transfer Coefficient
Laminar Flow
LeFevre and Ede (1965), Sparrow and Gregg (1956): Uniform wall temperature, valid for \mathrm{Pr}\leq1 and \frac Dl>35\mathrm{Gr}_l^{-1/4} or if \mathrm{Pr}\gtrsim1 and \frac Dl>(\mathrm{Gr}_l\mathrm{Pr})^{-1/4}
\begin{aligned} \mathrm{Nu}_{x,\mathrm{UWT}} & =\phi(\mathrm{Pr})\mathrm{Gr}_x^{1/4} \\ \phi(\mathrm{Pr}) & =\frac {0.75\mathrm{Pr}^{1/4}}{\left[4\left(0.609+1.221\mathrm{Pr}^{1/2}+1.238\mathrm{Pr}\right)\right]^{1/4}} \end{aligned}\tag{10.4.14}
LeFevre and Ede (1965): Uniform wall temperature, valid for \mathrm{Pr}\leq1 and \frac Dl\leq35\mathrm{Gr}_l^{-1/4} or if \mathrm{Pr}\gtrsim1 and \frac Dl\leq(\mathrm{Gr}_l\mathrm{Pr})^{-1/4}
\begin{aligned} \mathrm{Nu}_{x,\mathrm{UWT}} & =\left[\frac {7\mathrm{Ra}_x\mathrm{Pr}}{5(20+21\mathrm{Pr})}\right]^{1/4}+\frac {4x(272+345\mathrm{Pr})}{35D(64+63\mathrm{Pr})} \\ \langle\mathrm{Nu}_{l,\mathrm{UWT}}\rangle_l & =\frac 43\left[\frac {7\mathrm{Ra}_l\mathrm{Pr}}{5(20+21\mathrm{Pr})}\right]^{1/4}+\frac {4l(272+345\mathrm{Pr})}{35D(64+63\mathrm{Pr})} \end{aligned}\tag{10.4.22}
Turbulent Flow
Horizontal Flat Plate
Heat Transfer Coefficient
Laminar Flow
McAdams (1954): Uniform wall temperature, cooled upward-facing surfaces and heated downward-facing surfaces, valid for 10^5\lesssim\mathrm{Ra}_{l_c}\lesssim10^{10}
\langle\mathrm{Nu}_{l_c}\rangle=0.27\mathrm{Ra}_{l_c}^{1/4}\tag{10.7.2}
where l_c=\frac Ap is the total plate surface area normalized by its perimeter.
McAdams (1954): Uniform wall temperature, cooled upward-facing surfaces and heated downward-facing surfaces
\begin{aligned} \langle\mathrm{Nu}_{l_c}\rangle & =0.54\mathrm{Ra}_{l_c}^{1/4}\qquad\qquad\qquad10^5\lesssim\mathrm{Ra}_{l_c}\lesssim10^7 \\ \langle\mathrm{Nu}_{l_c}\rangle & =0.15\mathrm{Ra}_{l_c}^{1/3}\qquad\qquad\qquad10^7\lesssim\mathrm{Ra}_{l_c}\lesssim10^{11} \end{aligned}\tag{10.7.3}
Fujii and Imura (1972): Uniform heat flux, cooled upward-facing surfaces and heated downward-facing surfaces
\begin{aligned} \langle\mathrm{Nu}_{l_c}\rangle & =0.16\mathrm{Ra}_{l_c}^{1/3}\qquad\qquad\qquad\mathrm{Ra}_{l_c}<2\times10^8 \\ \langle\mathrm{Nu}_{l_c}\rangle & =0.13\mathrm{Ra}_{l_c}^{1/3}\qquad\qquad\qquad\mathrm{Ra}_{l_c}>5\times10^8 \end{aligned}\tag{10.7.5}
Inclined Flat Plate
Turbulence Transition, Lloyd and Sparrow (1970): Uniform wall temperature
| \phi | \mathrm{Ra}_x |
|---|---|
| 0^{\circ} | 8.7\times10^8 |
| 20^{\circ} | 2.5\times10^8 |
| 45^{\circ} | 1.7\times10^7 |
| 60^{\circ} | 7.7\times10^5 |
Turbulence Transition, Vliet (1969): Uniform heat flux
| \phi | \mathrm{Ra}_x^* |
|---|---|
| 0^{\circ} | 5\times10^{12}-10^{14} |
| 30^{\circ} | 3\times10^{10}-10^{12} |
| 60^{\circ} | 6\times10^7-10^9 |
Heat Transfer Coefficient
Laminar Flow
All correlations for natural convection on vertical flat surfaces are applicable provided that g is replaced with g\cos\phi, where \phi is the inclination angle from the vertical, for all of these models. See Sec. 1.1.2.1 for relevant correlations.
Intermittent Flow
Jaluria (2003): Uniform heat flux, upward-facing inclined surface, valid for 10^5<\mathrm{Ra}_l\cos\phi<10^{11} where 15^{\circ}<\phi<75^{\circ} and \mathrm{Gr}_l>\mathrm{Gr}_{\mathrm{cr}}, average Nusselt number definition based on \left|\langle T_s\rangle_l-T_{\infty}\right|
\langle\mathrm{Nu}_l\rangle_l=0.14\left[\mathrm{Ra}_l^{1/3}-\mathrm{Ra}_{\mathrm{cr}}^{1/3}\right]+0.56\left(\mathrm{Ra}_{\mathrm{cr}}\cos\phi\right)^{1/4}\tag{10.8.1}
The critical Rayleigh number is \mathrm{Ra}_{\mathrm{cr}}=\mathrm{Gr}_{\mathrm{cr}}\mathrm{Pr} where \mathrm{Gr}_{\mathrm{cr}} is the critical Grashof number. From Fujii and Imura (1972)
\mathrm{Gr}_{\mathrm{cr}}=\begin{cases} 5\times10^9 & \qquad\qquad\phi=15^{\circ} \\ 2\times10^9 & \qquad\qquad\phi=30^{\circ} \\ 10^8 & \qquad\qquad\phi=60^{\circ} \\ 10^6 & \qquad\qquad\phi=70^{\circ} \end{cases}\tag{10.8.4}
Turbulent Flow
All correlations for natural convection on vertical flat surfaces are applicable provided that g is replaced with g\cos\phi, where \phi is the inclination angle from the vertical, for all of these models. See Sec. 1.1.2.2 for relevant correlations.
Submerged Bodies
Turbulence Transition: Rayleigh number defined as \mathrm{Ra}_D=\frac {g\beta\left|T_s-T_{\infty}\right|D^3}{\nu\alpha}
\mathrm{Ra}_D\gtrsim10^9
Heat Transfer Coefficient
Laminar Flow
General Geometries
Heat Transfer Coefficient
Laminar Flow
Yovanovich (1987): Reference Fig. 1 for submerged body geometries, valid for \mathrm{Ra}_{l_c}<10^8, characteristic length l_c=\sqrt A where A is the total surface area.
The coefficient G_{l_c} is a geometric parameter and \langle\mathrm{Nu}_{l_c}\rangle_{\mathrm{Ra}_{l_c}\to0} is average Nusselt number under pure conduction.
\langle\mathrm{Nu}_{l_c}\rangle=\langle\mathrm{Nu}_{l_c}\rangle_{\mathrm{Ra}_{l_c}\to0}+\frac {0.67G_{l_c}\mathrm{Ra}_{l_c}^{1/4}}{\left[1+\left(0.492/\mathrm{Pr}\right)^{9/16}\right]^{4/9}}\tag{10.9.2}
See Table 1 for G_{l_c} and \langle\mathrm{Nu}_{l_c}\rangle_{\mathrm{Ra}_{l_c}\to0}.
| Body Shape | G_{l_c} | \langle\mathrm{Nu}_{l_c}\rangle_{\mathrm{Ra}_{l_c}\to0} |
|---|---|---|
| Sphere | 1.023 | 3.545 |
| Bi-sphere | 0.928 | 3.475 |
| Cube 1 | 0.951 | 3.338 |
| Cube 2 | 0.990 | 3.338 |
| Cube 3 | 1.014 | 3.338 |
| Vertical Cylinder | 0.967 | 3.444 |
| Horizontal Cylinder | 1.019 | 3.444 |
| Cylinder at 45^{\circ} | 1.004 | 3.444 |
| Prolate Spheroid (C/B=1.93) | 1.012 | 3.566 |
| Prolate Spheroid (C/B=0.5) | 0.973 | 3.529 |
| Oblate Spheroid (C/B=0.1) | 0.768 | 3.342 |
Churchill and Chu (1975): Long horizontal cylinders, valid for all \mathrm{Pr} and 10^{-5}\leq\mathrm{Re}_D\leq10^{12}
\langle\mathrm{Nu}_D\rangle=\frac {\langle h\rangle D}k=\left\{0.6+\frac {0.387\mathrm{Ra}_D^{1/6}}{\left[1+\left(0.559/\mathrm{Pr}\right)^{9/16}\right]^{8/27}}\right\}^2\tag{10.9.5}