Solving Carnot Heat Engine: Tf = SQRT(TcTh)

Click For Summary

Homework Help Overview

The discussion revolves around a heat pump and its relationship with a Carnot heat engine, specifically focusing on deriving the final temperature of the reservoirs as Tf = SQRT(TcTh). The problem involves concepts from thermodynamics and calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to manipulate equations related to heat transfer and efficiency but expresses confusion regarding the assumptions made, particularly about work and efficiency. Some participants suggest using calculus to find maxima or minima, questioning how to apply differentiation effectively in this context.

Discussion Status

The discussion is ongoing, with participants exploring different mathematical approaches to the problem. There is a focus on differentiating equations to find maximum work output, but no consensus has been reached on the specific equation to differentiate or the implications of the results.

Contextual Notes

Participants are grappling with assumptions about work input and output, as well as the implications of setting efficiency to 1 for maximal work. There is an acknowledgment of potential confusion regarding the mathematical steps involved in the differentiation process.

Carlo09
Messages
13
Reaction score
0

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 :)
 
Physics news on Phys.org
Remember from calculus how to find the maximum (or minimim) of a function?
 
hotvette said:
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?
 
Carlo09 said:
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.
 
hotvette said:
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 said:
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
 

Similar threads

Calculating the waste heat of Carnot engine
  • · Replies 4 ·
Replies
4
Views
2K
How Much Water Can a 1KW Carnot Engine Freeze in 5 Minutes?
  • · Replies 3 ·
Replies
3
Views
2K
Efficiency of Stirling Cycle - Is it the same as the Carnot Engine?
  • · Replies 14 ·
Replies
14
Views
2K
Temperature in a Carnot heat engine
  • · Replies 2 ·
Replies
2
Views
3K
Engine working between multiple temperature baths worse than Carnot
  • · Replies 1 ·
Replies
1
Views
1K
Carnot engine efficiency problem
  • · Replies 3 ·
Replies
3
Views
4K
Efficiency and Temperature in Heat Engine Cycles: Approaching Parts B, C, and D
  • · Replies 1 ·
Replies
1
Views
1K
Max Work From Heat Engine Reservoirs
  • · Replies 11 ·
Replies
11
Views
3K
Work done by a Carnot Engine. Melting ice
  • · Replies 2 ·
Replies
2
Views
3K
What is the power output of the heat engine?
  • · Replies 1 ·
Replies
1
Views
8K