A water distribution system is to be designed for a city with a population of $100000$ and a requirement of $450\frac{L}{?}$ person$/$day$.$ The pipe network is presented in the figure. Determine the flow rate distribution using the Cross Method. The desired accuracy is $Q{10}^{-4}\frac{m^3}{s}.$ All pipes are cast iron with an equivalent roughness height of $0\mathrm{.}15mm.$ Use a kinematic viscosity of $1\mathrm{.}12{10}^{-6}\frac{m^2}{s}$ Ignore local losses. The solution should include:

a$)$ Nodes, pipes, and circuits identification.

b$)$ Initial distribution of flows. Continuity at the nodes should be satisfied.

c$)$ Equations used. Description of the iteration.

d$)$ Final flow distribution, indicate flow rates and their direction in the pipe network.

f$)$ Friction coefficients and flow regime.

g$)$ Energy losses due to friction for all pipes.