Advanced Thermodynamics - The Second Law of Thermodynamics · 2018-03-01 · The Second Law of Thermodynamics Outline Second Law Equilibrium Principle Two Axioms 1 Second Law Closed

Outline Second Law Equilibrium Principle Two Axioms

Advanced ThermodynamicsThe Second Law of Thermodynamics


1 Second LawClosed SystemOpen Systems

2 Equilibrium

3 Principle

4 Two Axioms

Dr. Ayat Gharehghani

Assistant Professor

School of Mechanical Engineering

Iran University of Science & Technology, Tehran, IRAN

Winter, 2018


Closed System

System & One Heat Reservoir

Since Count Rumford & Joules to Carnot and then up to day,many experiments have been carried out; In each experiment,the apparatus (a closed system) absorbed work and rejectedheat to the ambient, i.e., the only heat (or temperature)reservoir with which it could communicate. It was never theother way around; All the attempts to construct a heat enginethat would operate cyclically as a closed system while inpossible contact with a single heat reservoir have failed.


Closed System

Cycle & Two Heat ReservoirReversible cycle was introduced by Sadi Carnot in 1824 & isfamous as Carnot cycle under influence of Emile Clapeyron:

(−Q2/Q1)rev = T2/T1 ⇒Q1

T1+

Q2

T2≤ 0

Image adapted from: Bejan, A., Advance Engineering Thermodynamics, 3rd Ed.,John Wiley, 2006.;


Closed System

Cycle & Two Heat Reservoir-Contd.

One should note that, in principle, it could be demonstratedthat when in contact with two reservoirs, the engine would notreject heat to, or absorb heat from both reservoirs! This wouldbe discussed in the classroom.


Closed System

Cycle & Arbitrary Heat Reservoir

We saw for cycle in contact with one & two temperaturereservoirs that:

Q1

T1≤ 0

Q1

T1+

Q2

T2≤ 0

Now, we are going to use the mathematical inductiontechnique to prove that the above correlation holds forarbitrary number of heat reservoirs!


Closed System

Cycle & Arbitrary Heat Reservoir-Contd.

So, we will assume that for a system be in contact with n heatreservoirs we have:

Q1

T1+

Q2

T2+ · · ·+ Qn

Tn≤ 0

and will try to prove that for a system in contact with n + 1 heatreservoirs:

n+1∑i=1

Qi

Ti≤ 0


Closed System

Cycle & Arbitrary Heat Reservoir

Image adapted from the same reference;

We put the Reservoir Tn+1 in contact with a reversible cycle to bring it to itsoriginal state. For the composite system A + Tn + C, we have:

∑ni=1

QiTi

+Qn,C

Tn≤ 0

Qn,C/Tn+Qn+1,C/Tn+1=0−−−−−−−−−−−−−−−→

∑ni=1

QiTi

− Qn+1,CTn+1

≤ 0Qn+1+Qn+1,C=0−−−−−−−−−−→

∑ni=1

QiTi

+Qn+1,C

Tn+1≤ 0


Closed System

Entropy Again...!Let’s consider a continuous variation of system boundarytemperature as the cycle is executed in contact with an infinitesequence of temperature reservoirs, each contributing heatinteraction of δQ or Qdt. We will have:∮

δQT≤ 0 Where

∮δQrev

T=0

If the net change in δQrev/T is zero at the end of a reversiblecycle, δQrev/T represents change in a thermodynamic property:

dS =δQrev

T

This property was named entropy by Clausius in 1865. Prior tothat, it was called thermodynamic function by Rankine!


Closed System

Process & Arbitrary No. of Heat ReservoirsConsider a change between states 1 & 2 in an arbitrary processwhich is reversible (it is a part of a reversible cycle); we have:

dS =δQrev

TIntegrate−−−−−→ S2 − S1 =

∫ 2

1

δQrev

T

This process can also be a part of the cycle 1→ 2→ 1. Theprocess 2→ 1 takes place along a reversible path. We have:∮

δQT≤ 0⇒

∫ 2

1

δQT

+

∫ 1

2

δQrev

T≤ 0∫ 2

1

δQT

Entropy Transfer

≤ S2 − S1Entropy Change


Closed System

Aftermaths

The yielded equation (2nd Law of thermodynamics for process)states that the entropy transfer (nonproperty) never exceeds theentropy change. The strength of that inequality is measured orstated with a new definition, i.e., entropy generation or entropyproduction, which is never negative.

Sgen = S2 − S1 −∫ 2

1

δQT≥ 0


Closed System

Example

Two bodies of water with masses m1 and m2 and temperaturesT1 and T2 are instantaneous heat reservoirs for a reversible heatengine. Please find the final equilibrium temperature of the twobodies of water, T∞ and the total work delivered by the engine.

Take the two bodies of water and the engine sandwiched inbetween as the aggregate system.

1st Law : −Wi−f = Uf −Ui2nd Law : Sgen,i−f = Sf − Si = 0


Closed System

Example-Contd.

Uf −Ui = (Uf −Ui)m1 + (Uf −Ui)engine + (Uf −Ui)m2

= m1c(T∞ − T1) + 0 + m2c(T∞ − T2)

Sf − Si = (Sf − Si)m1 + (Sf − Si)engine + (Sf − Si)m2

= m1c lnT∞T1

+ 0 + m2c lnT∞T2

= 0

Remark: du = cdT & ds = cdTT


Closed System

Example-Contd.

From the second law, we have:

T∞ = Tα1 T1−α

2where−−−→ α = m1/(m1 + m2)

Combining the above equation with the first law we wouldhave:

Wi−f = m1cT1

[1−

(T2

T1

)1−α]+ m2cT2

[1−

(T1

T2

)α]If α→ 0, the results may be reduced to:

T∞ = T2 & Wi−f = m1c(

T1 − T2 − T2 lnT1

T2

)


Open Systems

An Assessment on the Concepts

Image depicted by the Author;

Image adapted from: Bejan, A., Advance Engineering Thermodynamics, 3rd Ed.,John Wiley, 2006.;


Open Systems

Back to Second Law

Sclosed,t = Sopen,t +∆SinSclosed,(t+∆t) = Sopen,(t+∆t) +∆Sout

(∆S)in, out = (s∆M)in, out = (sm)in, out∆t

Sgen=S2−S1−∫ 2

1δQT ≥0

−−−−−−−−−−−−−→from t to t+∆t

∆Sgen = Sopen,(t+∆t) − Sopen,t −Qt

Ti+ (ms)out∆t− (ms)in∆t ≥ 0

We invoke the limit ∆t→ 0, drop the term “open”, andconsider any number of spots for heat and mass transfer...


Open Systems

Back to Second Law-Contd.

Sgen︸︷︷︸Entropy

GenerationRate

=dSdt︸︷︷︸

Rate ofEntropyAccum.

Inside theControlVolume

−∑

i

Qi

Ti︸︷︷︸EntropyTransfer

Rate(via heattransfer)

+∑out

ms−∑

in

ms︸︷︷︸Net Entropy

Flow Rate Outof the

Control Volume(via mass flow)

≥ 0

Remark: The fundamental distinction between heat transfer &work transfer is made by 2nd


Open Systems

Other FormsUsing terminology of control volume we can write:

Sgen =

∫V

∂ρs∂t

dV +

∫A

1T

q · ndA +

∫Aρsv · ndA ≥ 0

For point size control volume, the surface integrals aretransformed using divergence theorem and mass continuityequation is invoked, i.e.,

s′′′gen = ρDsDt

+ ∇ · qT≥ 0

& for volumetric rate of entropy generation in W/(K ·m3):

Sgen =

∫V

s′′′gendV

law: heat transfer is the energyinteraction accompanied by entropy transfer, whereas worktransfer is the energy interaction that takes place in the absenceof entropy transfer.


Open Systems

Another Example from A. Bejan

An evacuated & rigid bottle is surrounded by the atmosphere(T0, p0). The bottle neck valve opens & the air flows graduallyin. The thin conductive wall of bottle lets trapped air & theatmosphere eventually reach thermal equilibrium. The trappedair & atmosphere reach mechanical equilibrium at the end sincethe valve remains open.

At the beginning, the system is evacuated, while at the end it isfilled with air at atmospheric conditions:

M1 = 0U1 = 0S1 = 0

−→{

T2 = T0p2 = p0


Open Systems

To assess the reversibility of the filling process, the amount ofentropy generated along the entire process should becalculated:

Sgen =

∫ 2

1Sgendt←− Sgen =

dSdt− ms0 −

QT0≥ 0

We also have:dMdt

= m & Q1−2 0

Sgen = S2 − S1 − s0(M2 −M1) +p0VT0

=p0VT0

> 0

The filling process is irreversible, since the entropy generationterm is definitely positive.

Another Example from A. Bejan-Contd.

= -p V

So, we perform the integral and have (Note that :M1=0, S1=0)


Open Systems

Example Concluding Remark

Sgen is directly proportional to the work p0V done by theatmosphere on the mass that ultimately resides in the bottle!p0V work is lost. The final equation can be rewritten as:

Wlost = T0Sgen


Are We on a Wrong Path?!!1 A closed thermodynamic system is in equilibrium if it

undergoes no further changes in the absence of“interactions” with the environment.

2 The equal sign in the analytical statement of the 2nd

3 So, the inequality sign describes the others!!!!4 Entropy definition refers to a succession of equilibrium

states. Temperature describes quantitatively therelationship of thermal equilibrium between a system &environment (0th law).

On what basis do we use equilibrium concepts to expressdeparture from reversible processes?

This is called Gouy-Stodola theorem. The quality called“entropy generation” is proportional to the work that wasavailable but was not delivered to the user.

law isassociated with reversible processes (successions ofequilibrium states).


Local Thermodynamic Equilibrium

+ This assumption means that if we consider anyinfinitesimally small subcompartment of mass ∆m andvolume ∆V, and instantly isolate this sample, we willobserve no changes in the state of this sample as we followits evolution in time.

+ At equilibrium, the properties that describe the intensivestate of the sample, are related to each other in the samemanner as the properites of the finite-size batch of thematerial in equilibrium.


Entropy Maximum & Energy Minimum

For a closed system with adiabaticrigid boundary & not penetratedwith a rotating shaft or energizedpower cable:

1st Law : dU = 02nd Law : dS ≥ 0

Principle of Entropy Increaseor

Entropy Maximum Principle

During an arbitrary changeof state, the energy of anisolated system is fixed,whereas the entropyincreases, or, at its best,remains constant.

This is the principle of entropy increase or, alternatively,entropy maximum principle. Since in the course of each change theentropy cannot decrease, at the end of the series of possible changes, the entropyreaches its greatest algebraic value.


Is It Possible?Experiences show that an isolated system can behave in one ofthe following two ways:

+ Its state remians unchanged regardless of the time,considered as stable equilibrium.

+ The state changes by chance or triggered by a“disturbance” that is sufficiently weak to qualify aszero-energy interaction.

Joule’s free-expansion experiment


Facts about Joule’s free-expansion experiment+ The equilibrium temperatures before and after expansion

are practically the same⇒ The internal energy ofpermanent gases are functions of temperature only.

+ State of the system, initially in constrained equilibrium,changed with removing of internal constraint, and settledin a state of equlibrium without any internal constraints,which is a stable equilibrium.

+ The state changes by chance or triggered by a“disturbance” that is sufficiently weak to qualify aszero-energy interaction

+ This observation, the modification of internal geometry ofan isolated system, is the essence of constructal law, thatguides the time evolution of the configurations of the flowsystem.



+ These observations can be generalized for systems whose internal constitution depends initially on an arbitrary number of internal constraints.

+ The geometry of the isolated system at equilibrium is described by a number of deformation parameters, Xk

+ For example, the geometry of the gas–vacuum system designed by Joule is described by only two deformation parameters, the volume occupied by the gas, Vg, and the volume occupied by the vacuum, Vv; if V is the total volume of the isolated system, the (Vg, Vv) set changes from (V/2, V/2) to (V, 0) as the internal constraint is removed.

+ The Xk set of deformation parameters of the general system is shown attached to the vertical axis in Fig. (next page) ,The Xk values change as the internal constitution of the system changes with each removal of an internal constraint

+ The drawing is meant to suggest that the bottom plane corresponds to the configuration in which the system is free of internal constraints; in other words, the bottom plane is the locus of stable equilibrium states.

+ It follows that the attainment of maximum entropy in the limit of zero internal constraints is represented by the curve drawn in the constant-U plane. To say that after the removal of all possible internal constraints an isolated system reaches its entropy maximum is to make two analytical statements about the vicinity of a stable equilibrium state in an isolated system:

Facts-Contd.; Deformation Parameters: XkOn the U-cte plane: (dS)U = 0 (d2S)U < 0On the S-cte plane: (dU)S = 0 (d2U)S > 0

Image adapted from: Bejan, A., Advance Engineering Thermodynamics, 3rd Ed., John Wiley, 2006.;

The energy of the universe is constant;

The entropy of the universe stives to attain a maximum value;


An Example

Image depicted by the author;

The initial (A-state) internal energy and entropy of the system:UA − U0 =

4∑i=1

mc(Ti − T0) = 6mcT0

SA − S0 =

4∑i=1

mc lnTi

T0= 3.18mc



An Example-Contd.

Image depicted by the author;

There is more than 1 sequence in which the partitions can be removed;Removal of each particle leads to increase in the entropy of 4m system;The unconstrained stable equilibrium is unique, with the entropygreater than any preceding system.The final value of entropy is a supremum not maximum, and the finalvalue of internal energy is an infimum not minimum.


Constantin Caratheodory

Constantin Caratheodory-German Mathematician; 1873-1950


Introduction

The essence of the efforts to mathematize and generalizethermodynamics:

2 In the immediate neighborhood of every state of thesystem, there are other states that can’t be reached from thefirst by an adiabatic process.

Axiom II is used to prove the existence of “Reversible &Adiabatic Surfaces”, the property “Entropy”, the property

“Thermodynamic Temperature.


Assessment on Axiom 1

The change in energy is equal to the negative of adiabatic worktransfer:

−W1−2,adiabatic = U2 −U1

Obviously, the heat transfer is defined as the differencebetween the adiabatic & actual work transfers!

δQ− δW = dU⇒ δQ = δW − δWadiabatic

From the same statement:

δQ = dU −∑

i

YidXi

1 The work is the same in all adiabatic processes that take asystem from a given initial state to a given final state.


Assessment on Axiom 2



Reversible & Adiabatic Surfaces


−W1→2 = U2 −U1 but can state 2 land anywhere on the verticalline V = V2 of the U − V plane?


Remark

The states with volume V2 that are adiabatically accessible fromstate 1, can not be located under state 2rev in the U − V plane⇒All the states of volume V2 that are situated under state 2rev arenot accessible adiabatically from state 1.The unique point 2rev on the V = V2 line divides it intoadiabatically accessible and adiabatically in-accessible parts.


Remarks-Contd.



Remarks Generalization



Entropy

Considering the general case with arbitrary number ofdeformation coordinates (Xi), leads to existence ofconstant-empirical entropy hypersurfaces σ(U,Xi) = σ0 in thehyperspace (U,Xi). This existence of constant-σ surfaces can beused to prove the existence of thermodynamic properties.


Derivation

σ varies monotonically with θ; So, the state of A + B system canbe specified in terms of either (θ,VA,VB) or (σ,VA,VB). Anystate-dependent quantity such as U can be regarded as afunction of σ, VA, VB. Through an infinitesimal change weobtain:

dU =

(∂U∂σ

)VA,VB

dσ +

(∂U∂VA

)σ,VB

dVA +

(∂U∂VB

)σ,VA

dVB

We had from the first axiom:

δQ=dU−∑

i

YidXi

−−−−−−−−−−−−−→ dU = δQrev − pAdVA − pBdVB


Derivation-Contd.Subtract these equations to have:

0 =

(∂U∂σ

)VA,VB

dσ − δQrev +

[(∂U∂VA

)σ,VB

+ pA

]dVA

+

[(∂U∂VB

)σ,VA

+ pB

]dVB

Since σ, VA & VB can be varied independently, one will have:

δQrev =

(∂U∂σ

)VA,VB

dσ

−pA =

(∂U∂VA

)σ,VB

− pB =

(∂U∂VB

)σ,VA


We are Just Beginning...!

The 1st equation can be written as:

δQrev = λdσ

where

λ =

(∂U∂σ

)VA,VB

Is λ similar to other thermodynamic properties, i.e., two-foldderiven? Can we implement parameter separation?


Deduction

For the closed systems A and B, we have:{δQrev,A = λAdσA

δQrev,B = λBdσB

Now, the question is that, does the empirical entropy follow thegeneral lines produced by clausius entropy?

σ?= σ(σA, σB)


Deduction-Contd.

In fact, the heat transferred to A + B is the sum of heat transferinteractions experienced by A and B individually;

δQrev = δQrev,A + δQrev,B

λdσ = λAdσA + λBdσB

dσ =λA

λdσA +

λB

λdσB

Whenever σA & σB are held constant, the σ is constant. Also,

λA

λ= f1(σA, σB)

λB

λ= f2(σA, σB)


Deduction-Contd.

In fact, the heat transferred to A + B is the sum of heat transferinteractions experienced by A and B individually;

δQrev = δQrev,A + δQrev,B

λdσ = λAdσA + λBdσB

dσ =λA

λdσA +

λB

λdσB

Whenever σA & σB are held constant, the σ is constant. Also,

λA

λ= f1(σA, σB)

λB

λ= f2(σA, σB)


Deduction-Contd.

It is obvious that following λA & λB, λ is also independent ofVA & VB. So,

λ(θ, σA, σB), λA(θ, σA), λB(θ, σB)

Bringing the previous frame into account

∂

∂θ

(λA

λ

)= 0 &

∂

∂θ

(λB

λ

)= 0

Which is the same as:

∂

∂θ(lnλ) =

∂

∂θ(lnλA) =

∂

∂θ(lnλB)


Deduction-Contd.

σA & σB can be varied independently of θ; So,

∂

∂θ(lnλ) = ϕ1(θ)

⇒ λ = f (σ) exp [ϕ1(θ)dθ]

⇒ λ = f (σ)ϕ(θ)


Other Forms for the Axioms

Keenan & Shapiro imagined a world in which people weremore familiar with thermal effects & their quantitative analysisthan with mechanics. Their new axioms are

I’ The heat transfer is the same in all zero-work processesthat take a system from a given initial state to a given finalstate.

II’ In the immediate neighborhood of every state of thesystem, there are other states that can not be reached fromthe first via a zero work process.

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Advanced Thermodynamics - The Second Law of Thermodynamics · 2018-03-01 · The Second Law of Thermodynamics Outline Second Law Equilibrium Principle Two Axioms 1 Second Law Closed