$2.$ A pump delivers $\backslash (6\mathrm{.}19\backslash$times $10^\{-3\}\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash )$ sugar solution in a tank having a density of $\backslash (1200\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ and a viscosity of $31$ centipoises through a $50$ m long, straight, horizontal sanitary pipe of $35\mathrm{.}61$ mm inside diameter. The end of the pipe is at atmospheric pressure. Total friction loss in the pipe is $\backslash (868\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kg$\}\backslash ).$

a$)$ Draw the diagram for the flow system;

b$)$ Determine the Reynolds number and flow regime $($laminar or turbulent flow$)$;

c$)$ Calculate the gauge pressure at the discharge of the pump $($i$.$e$.$ at the entrance of the $50$ m pipe$)$;

d$)$ If the pipeline is NOT horizontal, and the flow in a certain part is upward, calculate the kW power needed for the pump with an efficiency of $\backslash (68\backslash \%\backslash )($The water exit point is $10\mathrm{.}7$ m above the entrance point, and the additional friction loss of pipe fittings is $\backslash (17\mathrm{.}8\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kg$\}\backslash ).)$