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}
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}
\begin{aligned} \frac {\delta}x & =\left[\frac {360(\mathrm{Pr}+0.8)}{\mathrm{Pr}^2\mathrm{Gr}_x^*}\right]^{1/5} \\ \mathrm{Nu}_{x,\mathrm{UHF}} & =0.62\left(\frac {\mathrm{Pr}^2\mathrm{Gr}_x^*}{\mathrm{Pr}+0.8}\right)^{1/5} \\ 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} \end{aligned}
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 Liu (1969): 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, if \mathrm{Pr}=0.7 and using 1/7 approximation for turbulent boundary layer, 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, for \mathrm{Pr}\approx1, for 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 Liu (1969): 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, if \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 tempearture, if \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}