$(20$ points$)$ Two$-$thirds of the energy content of fuel entering a $1,000$ MW rated nuclear power plant is removed by condensing cooling water that is withdrawn from a local river $($there are no stack losses, as is the case for a fossil fuel$-$fired plant$).$ The river has an upstream flow of $100\frac{m^3}{s}$ and temperature of $20C.$

a$.(10)$ If the cooling water is only allowed to rise in temperature by $10C,$ what flow rate from the river would be required?

b$.(10)$ During the summer months the flow from the river is only $28\frac{m^3}{s}.$ If the power plant is only allowed to discharge water at a rate of $10\frac{m^3}{s}$ and the temperature of the river after the plant discharge mixes with the river is $27C,$ what is the temperature of the cooling water before it mixes with the river? Ignore all heat losses to the environment.