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oddjobmj
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Homework Statement
Pumped Energy Storage.
A water reservoir has surface area A and depth D. The water flows down pipes and through turbines to generate electric power. The bottom of the reservoir is at height H above the turbines. The depth D of water in the reservoir decreases at a rate δ. (a) Calculate the total gravitational potential energy U. (b) Calculate the available power P = |dU/dt|, i.e., available for conversion to electric power.
DATA: A = 8 ×105 m2; D = 15 m; H = 108 m; δ = 0.5 m per hour; density = 1.0 ×103 kg/m3.
Homework Equations
U=mgh
Power=Energy/time
The Attempt at a Solution
To find the power I want to integrate mgh with respect to h from the initial 108 m to the total 123 m.
U=[itex]\frac{3465gm}{2}[/itex]
The mass of the tank is A*D*Density=1.2*107 kg and g=9.8
U=2.03742*1011 J
As far as power goes can I not simply divide the total potential by the total time it takes to empty the reservoir? If I do that I get 1.89*106 J/s.
Alternatively if I integrate the function for U divided by t from t=0->108000 seconds (30 hours)
[itex]\U/t[/itex]=[itex]\int2.03742*10^{11}/t[/itex]=2.03742*1011Log[108000]=2.36135*1012 J/s
Those are wildly different answers and neither of them in combination with the first is correct. Both parts have to be correct for me to check my answers.
Where am I messing up? Thank you!