A drilling operation involves circulating drilling mud to cool the drill string and maintain safe downhole temperatures. The drill string has a surface temperature of $180\backslash$deg C$,$ and the drilling mud enters the wellbore at $60\backslash$deg C$.$ The convective heat transfer coefficient between the mud and the drill string is $100$ W$/$m$\text{'}.$K$.$ The surface area of the drill string in contact with the mud is $15$ m$?.$ The mud is flowing at a velocity of $1\mathrm{.}5$ m$/$s through a wellbore with a diameter of $0\mathrm{.}2$ m$.$The mud has the following properties:Density p $=1200$ kg$/$m$3$Dynamic viscosity $4=0\mathrm{.}004$ Pa$.$sSpecific heat capacity cp $=2200$ J$/$kg$.$ KThermal

conductivity k $=0\mathrm{.}45$ W$/$m$.$K Answer the following:a$.$ Calculate the rate of heat transfer from the drill string to the mud using Newton's law of cooling.b$.$ Determine the Reynolds number for the mud flow and interpret the flow regime.c$.$ Calculate the Nusselt number for the flow, assuming a characteristic length equal to the wellbore diameter.d$.$ Determine the Prandtl number for the mud.e$.$ Discuss how increasing the flow rate of the mud affects the convective heat transfer rate and flow regime.