Natural convection from a vertical plate

Grashof number: physical meaning

$Gr = \dfrac{g\beta\,\Delta T\,L^3}{\nu^2}$ is the ratio of buoyancy force to viscous force acting on the fluid. In natural convection it plays the role that the Reynolds number plays in forced convection. A higher $Gr$ means stronger buoyancy-driven motion and higher $h$.

(a) Grashof and Rayleigh numbers

$T_f = 60$ °C $= 333.15$ K, so $\beta = 1/333.15 = 3.0\times10^{-3}\ \text{K}^{-1}$.

$$\begin{aligned} Gr_L &= \frac{9.81\times(1/333.15)\times(90-30)\times(0.5)^3}{(18.97\times10^{-6})^2} = 6.14\times10^{8}\\ Ra_L &= Gr_L\,Pr = 6.14\times10^8\times0.7202 = 4.42\times10^{8} \end{aligned}$$

$Ra < 10^9$, so the flow is laminar and the given correlation is valid.

(b) Nusselt number and $h$

$$Nu = 0.59\,Ra^{1/4} = 0.59\,(4.42\times10^8)^{0.25} = 0.59\times145.0 = 85.5$$ $$\bar h = \frac{Nu\,k}{L} = \frac{85.5\times0.02896}{0.5} = 4.95\ \text{W/m}^2\text{K}$$

(c) Heat loss

$$Q = \bar h A\,\Delta T = 4.95\times(0.5\times0.8)\times60 = 119\ \text{W}$$

Answer: $Gr = 6.14\times10^8$, $Ra = 4.42\times10^8$; $Nu = 85.5$; $\bar h = 4.95\ \text{W/m}^2$K; $Q \approx 119$ W.