Non Dimensional Numbers
\[ \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infty) x^3 = \mathrm{Gr}_{x}\mathrm{Pr} \]
Flow over Flat Plate
Local Friction Factor at x(laminar)
Average Friction Factor from 0 to x(laminar)
Local Nusselt number at x(laminar, Isothermal wall)
Average Nusselt number from 0 to x(laminar, Isothermal wall)
Local Friction Factor at x(Turbulent flow)
Local Nusselt Number at x(Turbulent flow, Isothermal wall)
Local Nusselt Number at x(laminar, uniform heat flux)
Local Nusselt Number at x(Turbulent, uniform heat flux)
Cross Flow Over Cylinder
Average Nusselt number(Churchill–Bernstein)
\[ \overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 \]
- There are other correlation for flow over bank of tubes which are not included here but can be found on various literature of heat and mass transfer textbook.
Internal Flow Thru Tube
Friction Factor(Darcy-Weisbach, from Moody Chart)
\[ f_D = 64 / \mathrm{Re}, \text{for laminar flow.} \]\[ {1 \over \sqrt{f_D}}= -2.0 \log_{10} \left(\frac{\epsilon/D}{3.7} + {\frac{2.51}{\mathrm{Re} \sqrt{f_D } } } \right) , \text{for turbulent flow.} \]
Laminar Flow Fully Developed Friction factor and Nusselt Number
Natural Convection
Vertical Plate
\[ \overline{\mathrm{Nu}}_L=\left[0.825+\frac{0.387{\mathrm{Ra}}_L^\frac16}{\left(1+\left(\frac{0.492}{\mathrm{Pr}}\right)^\frac9{16}\right)^\frac8{27}}\right]^2 \]Inclined Plate
\[ {replace\ \ \ g \ \rightarrow \ g\cos (\theta) \ in \ above \ Equation \ of \ vertical \ plate, \ \theta\le60.} \]Horizontal Plate
Upper Surface of Hot Plate or Lower Surface of Cold Plate
\[ \overline{\mathrm{Nu}_L}=0.54{\mathrm{Ra}_L}^\frac14\ \ \ \ (10^4\le{\mathrm{Ra}_L}\le10^7) \]\[ \overline{\mathrm{Nu}_L}=0.15{\mathrm{Ra}_L}^\frac13\ \ \ \ (10^7\le{\mathrm{Ra}_L}\le10^{11}) \]Lower Surface of Hot Plate or Upper Surface of Cold Plate
\[ \overline{\mathrm{Nu}_L}=0.27{\mathrm{Ra}_L}^\frac14\ \ \ \ (10^5\le{\mathrm{Ra}_L}\le10^{10}) \]Horizontal Cylinder
\[ \overline{\mathrm{Nu}}_L=\left[0.6+\frac{0.387{\mathrm{Ra}}_L^\frac16}{\left(1+\left(\frac{0.559}{\mathrm{Pr}}\right)^\frac9{16}\right)^\frac8{27}}\right]^2 \]
