Question $2(40$ points$)$

For energy storage, water from a bottom tank needs to be pumped to an upper tank. Then, the potential energy of the water in the upper tank can be used to generate electrical energy. The water is pumped up through a vertical pipe of $0\mathrm{.}05$ m ID$,$ as depicted in the figure. The pump is a positive displacement, which produces a constant flow rate of $\backslash (0\mathrm{.}0004\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash ).$ One possibility to make the system more efficient is to inject compressed air at the base of the riser to reduce the weight of the column and, consequently, reduce the energy required by the pump. Basically, the lower the required pump discharge pressure is$,$ the least energy is comsume by the pump. With this information,

$1.$ Assume that the flow pattern will be bubbly flow for all possible gas injections and estimate the optimal gas flow rate that minimizes the pressure losses in the vertical pipe of the system. Justify why this is the optimal gas injection.

The gas and liquid properties are given in Harvey and are the same form previous problem. You can solve the problem numerically or you can create a table$/$plot that shows the optimal point.