Understand Kelvin-Plank Statement for 2nd Law of Thermodynamics

[integrate]

Thermodynamics 2nd Law Please Help! :(

I am going crazy trying to understand the Kelvin-Plank statement for the 2nd Law of Thermodynamics. It states:

"It is impossible for any system to operate in a thermodynamic cycle and transfer energy by work to the surrounding while receiving heat from a single thermal reservoir."

Let's define:

T_H : temperature of the hot reservoir
Q_H : heat transfer input from the hot reservoir
T_C : temperature of the cold reservoir
Q_C : heat transfer out from the cold reservoir
W_net : work output

A perpetual motion machine: A system (engine) undergoing a cycle by exchanging heat transfer with a single reservoir (Q_H) and all of it is converted into work (W_net). This means that all of the input was converted into the output (100% efficiency). The engine does not violate the principle of energy conservation (1st Law). Hence, Q_H = W_net.

Can someone explain how the 2nd law is violated with detail or through contradiction?

Does this have anything to do with reversible or irreversible processes? Is there an assumption I need to make? I understand that for a cycle to be reversible, heat needs to be rejected/extracted in order to bring the entropy change equal to 0. Reversible means that the state of the system and surroundings remain constant (i.e. have the same initial and final properties). In an irreversible process, the system, the surroundings or both do not remain constant. For example, the creation of friction (heat) to the surroundings increases the entropy of the surroundings.

Furthermore, the Kelvin-Plank statement says,

W_Cycle \leq 0 (for single reservoir)

In detail it says,

W_Cycle < 0 (Internal irreversibilities present) (for single reservoir)
W_Cycle = 0 (No irreversibilities present) (for single reservoir)

What do these inequalities mean? I don't understand why.

I have read all the other threads involving thermodynamics and they did not help much because they were not specifically related to my question. Please help! :(

 

[Mapes]

Can someone explain how the 2nd law is violated with detail or through contradiction?

Work is energy transfer without entropy transfer. Since this hypothetical devices inputs energy in the form of heat and outputs energy in the form of work, the Second Law is violated because total entropy has been reduced.

Does this have anything to do with reversible or irreversible processes?

The limitation holds even if assuming perfectly efficient, reversible processes (e.g., the Carnot cycle assumes reversibility and still cannot achieve 100% efficiency of heat->work transfer).

 

[integrate]

Work is energy transfer without entropy transfer. Since this hypothetical devices inputs energy in the form of heat and outputs energy in the form of work, the Second Law is violated because total entropy has been reduced.

How has the total entropy been reduced? What do you mean by reduced? less than 0, which is impossible? Heat was added to the system and and since you said work is energy transfer without entropy transfer, therefore the entropy of the system should be greater than 0. Isn't entropy a function of heat transfer and temperature of the system?

 

[Mapes]

Right, the total entropy of the universe is decreased, which is a violation of the Second Law, which is why perpetual motion machines are impossible. The entropy doesn't leave when the energy is transferred out as work, and it can't stay in the machine because machines (in thermodynamic analyses) operate cyclically by definition; thus, they must return to their initial state. So there's a contradiction. Make sense?

 

[rcgldr]

"It is impossible for any system to operate in a thermodynamic cycle and transfer energy by work to the surrounding while receiving heat from a single thermal reservoir."
T_H : temperature of the hot reservoir
T_C : temperature of the cold reservoir

Didn't you just violate the premise by having two reservoirs, as opposed to a "single reservoir"? I think the idea here is that the system would eventually reach the same temperature as the reservoir, thus ending the thermodynamic cycle.

On a side note, in the case of an thermalcouple in space, the "cold" reservoir dumps it's heat into space, so does that make space the "cold" reservoir, even though it's void of mass? Or is this simply not a closed system since it dumps heat into empty space which is unbounded?

 

[integrate]

The entropy doesn't leave when the energy is transferred out as work, and it can't stay in the machine because machines (in thermodynamic analyses) operate cyclically by definition; thus, they must return to their initial state.

So what you are saying is that as a process (engine) completes a cycle it comes back to its original state/properties (by definition of the word "cycle") and if the entropy never leaves the system, then the system never went back to its original state and hence never completed the cycle. The only way the entropy can leave the system and allow the process (engine) to complete the cycle is through a heat transfer to another reservoir (T_C). Right?

But on the other hand it is possible to have heat transfer from a single reservoir and produce work (W_net), it is just that it would NOT be a "thermodynamic cycle". Right?

 

[Mapes]

Exactly. A classic example is a container of alcohol, whose relatively high vapor pressure pushes up a weight and vents the system, allowing the weight to fall again. The system, though isothermal, appears to produce work from nowhere. But the working fluid is continually lost, so the system is not a true machine. (A similar example is the "drinking bird" toy.) It's just a temporary process that exploits the difference in alcohol vapor pressure inside and outside the container.

 

[integrate]

Thank you very much Mapes. I really appreciate your kind help. :)

 

[rcgldr]

A similar example is the "drinking bird" toy.

The bird "dips" it's "beak" into a small cup of water, then uses the evaporating water as the cold reservoir, and the water gets used up. The bird will work as long as head is colder than the base, so as an alternative, it can be changed to a "sun" bird by placing a heat radiating black body under the bird with a light (or sun) to provide a mild heat source, although the red color versions will fade in sunlight.

http://en.wikipedia.org/wiki/Drinking_bird

Urban legend? Apparently the old birds used ethanol, but in the USA there was no liquor tax on them. Urban legend has it that people would buy the birds instead of everclear just to get the ethanol.

 

[Mapes]

I meant "similar" in the sense that only one heat reservoir is required (taking the system boundaries to include the device and the liquid water), that the working fluid is exhausted during operation, that the process was only pseudocyclic, and that the device can't be considered a perpetual motion machine. Come on, obviously some of the details of operation are going to be different.

 

[des51]

Oh jeff, you are a stickler for the details aren't you. Mapes' explanation was very helpful.

 

[rcgldr]

Sorry, my response was harsh. I re-phrased it.

 

[Mapes]

Hey, no offense taken. I always enjoy reading your posts, Jeff.