21 - Convection - Introduction

Requirements:

• Gravitational field

• Density change with temperature

Variables: \(h, L, \rho,\mu, k, C_P, \beta, g, \Delta T\)

\(\beta\) is the coefficient of thermal expansion. \[\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T} \right)_P\] For ideal gases, \(\beta=1/T_\infty\)

Dimensions: \(M, L, t, T, Q\)
(where \(Q\) is the dimension for energy).

Grashof number (Gr): \[\text{Gr} = \frac{g \beta \rho^2L^3(T_w-T_{\infty})}{\mu^2}\]

Heat Transfer Correlation: \[\boxed{{\text{Nu} = \phi(\text{Gr}, \text{Pr})}}\]

\[\begin{aligned} \text{Re} &= \frac{Dv\rho}{\mu} = \frac{Dv\rho}{\mu}\frac{Dv}{Dv} = \frac{\rho D^2v^2}{D\mu v} = \frac{\rho D^2v(D/t)}{D^2\mu\frac{v}{D}} \\ &= \frac{\rho D^3(v/t)}{D^2\times\mu\frac{v}{D}} =\frac{ma}{A\,\tau} = \frac{\text{Inertial force}}{\text{Viscous force}} \end{aligned}\]