Solving Carnot Heat Engine: Tf = SQRT(TcTh)

[Carlo09]

Homework Statement

A heat pump takes heat from a hot resevoir and dissipates heat to a cold one. Both resevoirs are equal mass and specific heat capacity. Show that as the heat engine does maximal work the final temp of the resevoirs = Tf = SQRT(TcTh)

Homework Equations

Qin = Wout + Qout
mc(Th-Tf) = Wout + mc(Tf -Tc)

Efficiency = Wout/Win = 1- (Qin/Qout)

The Attempt at a Solution

Well I know i somehow need to get TcTh^2 in order to get the solution so I used efficiecy as 1 for maximal work out but I also assumed no work in which means I'm dividing by 0! Or if I say 1= 1-Qin/Qout then Qout = 1-Qin and now I am just confused. Help please :)

 

[hotvette]

Remember from calculus how to find the maximum (or minimim) of a function?

 

[Carlo09]

Remember from calculus how to find the maximum (or minimim) of a function?

Are you meaning differentiate and set to 0 and then solve? Differentiate again and is D^2f(x)/Dx^2 < 0 then it's a maximum?

 

[hotvette]

Are you meaning differentiate and set to 0 and then solve?

Yep, but you should only have to differentiate once. It should be pretty clear whether you have a max or min.

 

[Carlo09]

Yep, but you should only have to differentiate once. It should be pretty clear whether you have a max or min.

Thank you, Which equation do I differentiate?

 

[Carlo09]

Thank you, Which equation do I differentiate?

Ooo right ok so I have dw/dQ_h = (T1-T2)/T1 = 0

therefore T1-T2 = 0 so T1=T2=T and so

dw/dQ_h = (T-T)/T which is 0 which doesn't help me :S