Understanding Exergy: Confusion about Internal Energy Change in an Environment

[Red_CCF]

Hi

In the textbook I was reading, they were trying to derive the formula for exergy and I got confused. They defined an environment (with constant pressure P0 and constant temperature T0) containing closed system as the overall system that is closed. The closed system undergoes a process and they used first law to write energy balance of the overall system. From that they determined that the change in total internal energy of the environment deltaU as T0deltaS - P0deltaV (first TdS equation). This is where I got confused since I don't understand how, if the temperature and pressure remained constant, how can there by an internal energy change of the environment.

Thank you

 

[Zeppos10]

Reading your question, I picked up 'Moran and Shapiro' (2/e, 1998) and on p267 I see a similar, rather confusing, formulation (they call it availability instead of exergy). Are you reading the same text, or is it copied from this one ?

 

[Red_CCF]

Reading your question, I picked up 'Moran and Shapiro' (2/e, 1998) and on p267 I see a similar, rather confusing, formulation (they call it availability instead of exergy). Are you reading the same text, or is it copied from this one ?

Yes I'm using Moran and Shapiro 6th edition, Fundamental of Engineering Thermodynamics, but they changed the name to exergy for this edition. It's their proof on the formula that I'm confused about

 

[Red_CCF]

Anyone?

 

[shushu97]

The environment is assumed to be infinite so the temperature and pressure remain constant.
At the same time there must be conservation of energy.

 

[shushu97]

The environment is assumed to be infinite so the temperature and pressure remain constant.
At the same time there must be conservation of energy.

 

[mbeychok]

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It covers the following subjects:

-- Potential energy
-- Kinetic energy
-- Energy from heat
-- Chemical energy
-- Electrostatic energy
-- Electric energy
-- Equivalence of energy and mass
-- Energy in quantum mechanics

​

 

[Red_CCF]

The environment is assumed to be infinite so the temperature and pressure remain constant.
At the same time there must be conservation of energy.

Hi

I understand that the temperature and pressure must be constant, but I don't understand how the internal energy of the environment can change given that it is a function of temperature and pressure which is constant.

 

[shushu97]

The energy of the combined system (system and environment) must be conserved.
So any change in the energy of the system must be absorbed by the environment.
Imagine considering the vicinity of the system for the conservation of energy and then moving far away in the environment where properties are not affected.

 

[shushu97]

The energy of the combined system (system and environment) must be conserved.
So any change in the energy of the system must be absorbed by the environment.
Imagine considering the vicinity of the system for the conservation of energy and then moving far away in the environment where properties are not affected.

 

[Zeppos10]

The energy of the combined system (system and environment) must be conserved. So any change in the energy of the system must be absorbed by the environment. .

This statement is incorrect because there is also the work Wc given up by the system but not absorbed by the environment.

There is a certain danger in trying to explain what the textbook is trying to say, but I will try anyway.
The first usefull interpretation is to assume that Wc is identical to what other textbooks call shaft work or 'usefull work'. So availability / exergy is the max usefull work the system can do in the given environment. The latter part of this definition is essential and explains why T0, and p0 must show up: however it would be less confusing if T and p of the environment are indicated by Te and pe.)
Moran and Shapiro are possibly the only ones to introduce symbols for extensive environmental quantitites (environmental internal energy, environmental volume and environmental entropy). Ofcourse they have to eliminate them because these are fictional quatities. Most of the derivation = elimination of these.

The last suggestion that is worthwhile considering: what is the difference between the Gibbs free energy G and availability/exergy A ? The only difference is the G counts from a standard state, A counts from the environmental state: these 2 reference however are commonly taken to be equal: (298K,1 atm).

 

[shushu97]

Second law of thermodynamics (Kelvin-Planck statement): "It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of energy by work to its surroundings while receiving energy by heat transfer from a single thermal reservoir".

 

[Zeppos10]

Second law of thermodynamics (Kelvin-Planck statement): "It is impossible for any system to operate in and deliver a net amount of energy by work to its surroundings while receiving energy by heat transfer from a single thermal reservoir".

Moran and Shapiro (chapter on exergy) do not discuss a thermodynamic cycle.
However, it would be interesting to know where they would put a second thermal reservoir: from the other hand if you have 2 infinite heat reservoirs, you have infinite capacity to do work.

 

[Red_CCF]

This statement is incorrect because there is also the work Wc given up by the system but not absorbed by the environment.

There is a certain danger in trying to explain what the textbook is trying to say, but I will try anyway.
The first usefull interpretation is to assume that Wc is identical to what other textbooks call shaft work or 'usefull work'. So availability / exergy is the max usefull work the system can do in the given environment. The latter part of this definition is essential and explains why T0, and p0 must show up: however it would be less confusing if T and p of the environment are indicated by Te and pe.)
Moran and Shapiro are possibly the only ones to introduce symbols for extensive environmental quantitites (environmental internal energy, environmental volume and environmental entropy). Ofcourse they have to eliminate them because these are fictional quatities. Most of the derivation = elimination of these.

The last suggestion that is worthwhile considering: what is the difference between the Gibbs free energy G and availability/exergy A ? The only difference is the G counts from a standard state, A counts from the environmental state: these 2 reference however are commonly taken to be equal: (298K,1 atm).

Hi

I guess the only question that I have about the derivation is that the book claims that internal energy for a closed system can change without a change in pressure and temperature which I thought was not possible unless the environment was in saturation, but I'm not sure now since the first T-dS equation (TdS = dU+pdV) is still valid mathematically if I just insert constant T and p, so can internal energy change for constant T and p and how?

 

[Zeppos10]

Hi

the book claims that internal energy for a closed system can change without a change in pressure and temperature which I thought was not possible unless the environment was in saturation,

I do not know what you mean by "unless the environment was in saturation", but
A case where U changes under 'constant T and p' is a mix of 2 mmol H2 and 1 mmol O2 at environmental p and T: if the reation is allowed to proceed, heat will bereleased to the environment (taken as isothermal) and work is done (taken as isobaric) by the environment on the system if the reaction to water proceeds to completion. However to do so the actual T and p in the systems does not remain exactly equal to the T and p of the environment.
(note that he process is irreversible and this might be a cause of confusion.)

 

[Red_CCF]

I do not know what you mean by "unless the environment was in saturation", but
A case where U changes under 'constant T and p' is a mix of 2 mmol H2 and 1 mmol O2 at environmental p and T: if the reation is allowed to proceed, heat will bereleased to the environment (taken as isothermal) and work is done (taken as isobaric) by the environment on the system if the reaction to water proceeds to completion. However to do so the actual T and p in the systems does not remain exactly equal to the T and p of the environment.
(note that he process is irreversible and this might be a cause of confusion.)

Hi, thanks for the response

In the Moran and Shapiro book they state that specific internal energy was a function of temperature and pressure in chapter 3 (so I guess they did not consider chemical reactions) so that's why I'm confused as if the T and p remain constant the function would give the same values.

What I meant by the saturation thing is that my professor stated that the only time when internal energy changes for constant T and p is when the system is in its saturation region (i.e. as liquid water changes to water vapour, the flat line on the T-v phase diagram) as T and p are dependent and another property is required to determine internal energy.

Also, with regards to your example, if heat is released and work is done on the system, how do we know that the final internal energy did change (i.e. they're not equal)?

Something I find interesting is that on the fifth edition of the same book, the authors state that the intensive properties of the environment doesn't change but the extensive does. Another thing is that they never explicitly said the combined system (environment and closed system inside) was closed, only that its volume didn't change, so I'm not sure if they allow for mass change of the combined system or not.

Thanks very much for all your help

 

[Zeppos10]

It is usefull to distinguish serveral aspects here:
1. Moran&Shapiro Ch3 is about pure substances, where in Ch 7 it is not.
2. I assume now that 'saturated' means liquid or solid in equilibrium with its vapor, leaving 'environment' (see above) alone.
3. According to the phase rule (F=C-P+2), a 1 component system with two phases has only 1 degree of freedom: if you choose p, you cannot choose T, vice versa.
4. Does the relation dU=TdS-pdV hold for this system ?? : I never saw a text where it is claimed that it does not hold for this case (C=1,P=2) , although the relation itself is used to state that U is a function of two wariables, and constant T and p does not change that, if considered from a mathematical point of view. This causes some conceptual pain, meaning: there is an inconsistency here I think.
5. Consider 1 mole of water {H2O(l)==H2O(g)} at 99C in contact with an environment of 101C, 1 atm. Note there is not (but almost) thermal equilibrium, there is not (but almost) mechanical equilibrium) but there is internal equilibrium between the two phases. The system will absorb heat Q = DH= +44 kJ (evaporation) and the system will expand doing extenal work w=-pdV=~ -2.3 kJ. U=H-pV= ~+41.7. Note that I used the external work done, ie I used external pressure.

Point 4 is the crux I think: I hope this to be useful.

 

[Red_CCF]

It is usefull to distinguish serveral aspects here:
1. Moran&Shapiro Ch3 is about pure substances, where in Ch 7 it is not.
2. I assume now that 'saturated' means liquid or solid in equilibrium with its vapor, leaving 'environment' (see above) alone.
3. According to the phase rule (F=C-P+2), a 1 component system with two phases has only 1 degree of freedom: if you choose p, you cannot choose T, vice versa.
4. Does the relation dU=TdS-pdV hold for this system ?? : I never saw a text where it is claimed that it does not hold for this case (C=1,P=2) , although the relation itself is used to state that U is a function of two wariables, and constant T and p does not change that, if considered from a mathematical point of view. This causes some conceptual pain, meaning: there is an inconsistency here I think.
5. Consider 1 mole of water {H2O(l)==H2O(g)} at 99C in contact with an environment of 101C, 1 atm. Note there is not (but almost) thermal equilibrium, there is not (but almost) mechanical equilibrium) but there is internal equilibrium between the two phases. The system will absorb heat Q = DH= +44 kJ (evaporation) and the system will expand doing extenal work w=-pdV=~ -2.3 kJ. U=H-pV= ~+41.7. Note that I used the external work done, ie I used external pressure.

Point 4 is the crux I think: I hope this to be useful.

Hi

So what would the internal energy be a function of in a system with multiple substances other than T and p? I always thought that we just superposition the independent internal energies of each substance and that it would still be a function of T and p?

Sorry for the confusion, I meant saturated liquid/vapor as liquid/vapor at saturation/boiling temperature (the temperature where the flat line occurs on the phase diagram).

I'm not sure if this is correct but it would solve all of my confusions. My only confusion with the proof was that I thought internal energy didn't change due to the constant T and p.

I went through their wording very carefully and found that the book states that the overall system (env. + closed system inside) as a constant volume system while I initially thought it to be closed. Since they didn't state the overall system as closed like the others, it means that it's a control volume and mass can flow in and out as the closed system changes volume, which would explain why internal energy can change (and why they used total internal energy rather than specific internal energy) given constant T and p. Is this reasoning correct?

Thanks