A vertical plate of height \(L\) at a temperature of \(T_{\mathrm{s}}\) is immersed in a fluid at a temperature of \(T_{\mathrm{a}}\). Local density variation due to temperature causes a flow, which causes an enhancement in the heat transfer rate.

Note that the heat flux depends now on the following variables:

\[q=f(\Delta T, \beta, g, ho, \mu, k, L)\]

where \(\beta\) is the volumetric coefficient of thermal expansion.

(a) What is the expression for \(\beta\) based on the ideal-gas law? What are its units?

(b) Identify the dimensionless groups needed to correlate the data on heat transfer in such systems using the method of matrix algebra.

(c) Suggest expressions to correlate data on natural convection heat transfer.