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Irreversible thermodynamics, a.k.a. Non-equilibrium thermodynamics(an introduction)
Ron ZevenhovenÅbo Akademi University
Thermal and Flow Engineering Laboratory / Värme- och strömningstekniktel. 3223 ; [email protected]
Process EngineeringThermodynamicscourse # 424304.0 v. 2015
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5.1 Equilibrium, classicalthermodynamics vs.irreverisible (non-equilibrium)thermodynamics
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Irreversible thermodynamics /1
Classical thermodynamics deals with driving forces for chemical reactions and with equilibrium, with equilibriumdescriptions following from the 1st and 2nd laws of thermodynamics for closed systems.
Open systems with heat, mass and/or electricitytransported over system boundaries are muchmore important in engineering applications.
Change and movement is more common thanequilibrium state (also in nature as a result of the constant energy influx from the sun).
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Equilibrium (HK65, chapter 4) /1
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Equilibrium (HK65, chapter 4) /2
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Irreversible thermodynamics /2
Irreversible thermodynamics addresses non-equilibrium situations, assuming reversibilityon a small scale (i.e. local equilibrium) and linear transport processes.
A starting point was the work of Thomson (later Lord Kelvin) on thermo-electricity, i.e. interacting transport of heat and electric charge in the 1850’s
The main goal is to describeinteracting transport processes, taking into accountentropy production and the 2nd Law of thermodynamics
An Estonian-German physicist Thomas Seebeck (1770-1831) twisted two wires of different metals together and heated the point at where they were joined. He produced a small current of electricity. This is called thermo-electricity and is known in physics as the "Seebeck Effect". P
ic: h
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Transport processes (linear) /1
Entropy production and Fourier’s law:
force driving coeff. transportflux :general in
: :
: :
ii
yxymmI
MQ
XLJdx
dvJNewton
dx
dV
A
iJOhm
dx
dcD
A
nJFick
dx
dT
A
QJFourier
2
22
222
ba
:lossexergy
:T and T betweenfer Heat trans
dx
dT
T
T
volume
xEΔ
T
TAJTSTxEΔ
T
TAJ
T
TQ S
dt
dS
T
TdQ
TT
TTdQdS
o
Qo
geno
Qgengen
ba
bagen
A m2
(voltage V, specif. conductance σ = 1/ρ, where spec. resistance ρ = R∙A/Δx)
dx
R = electricalresistance, Ω
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Transport processes (linear) /2
Similar for electric current (or mass diffusion or fluid flow):
This gives the general description Ji = L·Xi, and
Cross-effects and interactions can be described too:
2
:loss exergy *
dx
dV
T
T
volume
xEΔ
T
ViTSTxEΔ
T
Vi
dt
dSS
dt
dSTViQ
oo
geno
gen
TVΔ
X,iJ:Ohm;TTΔ
X,JJ:Fourier
JX;XJS
iiiQi
iiiigen
and and
for force driving the is where
relations) reciprocal s(Onsager' L Lwhere
and ;
jiij
22212122121111
XLXLJXLXLJ
V =voltage
flows heat
or mass no *
dt
dSSgen
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Transport processes (linear) /3
For example, simultaneous heat and mass transfer
This gives the general description
The reciprocal relations are also known as the 4th Law of Thermodynamics.
for entropy production with two coupled flows
D tcoefficien diffusionmass ~ L , tyconductivi heat ~ L
and t;coefficien usionthermodiff theis LL where
and
MMQQ
QMMQ
λ
XLXLJ;XLXLJMMMQMQMMQMQQQQ
j
jiji XLJ
LLLand,L,LS
XLXX)LL(XLXJXJS
XLXLJ;XLXLJ
gen
gen
thatimplies
and
Note:L12
may be< 0 !
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Irreversible thermodynamics /3
Irreversible or non-equilibrium thermodynamicsdescribes transport processes in systems not in global equilibrium.
The 2nd Law is reformulated in terms of entropyproduction, Ṡgen, assuming local equilibrium.
The approach is very powerful for the analysis of simultaneous transfer of heat and mass, or mass and electric charge, etc, as for example found in membrane separationsor electrolyte systems, or systems where gravityis important.
iiigengen
o
lost
gen
XJSSTW
S
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Driving forces; entropy production For heat exchange the entropy production rate is the
product of the thermodynamic driving force X = Δ(1/T) and the resulting flow J = Q.
For more general systems, for example an isolatedsystem separated into sections by a membranepermeable only to one species (e.g. species ”1”):
iii
gen
gen
XJ
T
µn
T
pV
TQS
T
µ
T
µ
dt
dn
T
p
T
p
dt
dV
TTdt
dQ
dt
dS
111
2
2
1
11
2
2
1
11
21
1
1
11
V1, n1,p1,T1, µ1
V2, n2,p2,T2, µ2
.
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An example
Source:K07
zT
yT
xT
T
andvolume
S
Here
gen
/
/
/
:
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5.2 Maximum entropy production
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Maximum entropy production For combined transport processes, the final outcome is
not only determined by balances for conservedproperties such as mass, energy, charge and momentum
For simulantaneous transport the increased degreeof freedom results in interactions
The final outcome is thenmore strongly governed by the 2nd law of thermodynamicsand, unless the deviations from equilibrium are small, the process tends to maximiseentropy production Maximum entropy
analysis is also used asa statistics method P
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Max. entropy production example/1
An electric heating element distributes heat Q intoQ1+ Q2 while heating up two different streams. Input energy Q results in temperature Th for the heating element. The system is well insulated.
For both streams the energy balance equation gives Qi= ṁi·ΔTi·cpi = Ui·Ai·ΔTlm,i, and Q = Q1 + Q2 is fixed. (Assume A1 = A2, or even U1·A1 = U2·A2).
How will the input heat energy Q be distributed?
T=ThHeat Q
Flow ṁ1, T1, cp1
Flow ṁ2, T2, cp2
Q1
Q2
Flow ṁ2, T4, cp2
Flow ṁ1, T3, cp1
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Max. entropy production example/2
For one of the streams the entropy generation is
The result follows from max {Ṡgen1(Q1) + Ṡgen2 (Q2)}
T=ThHeat Q
Flow ṁ1, T1, cp1
Flow ṁ2, T2, cp2
Q1
Q2
Flow ṁ2, T4, cp2
Flow ṁ1, T3, cp1
stream other the for similar also while
;cm
QTT
)TT
ln(cmT
dTcmdT
T
cmdSS
,p
,p
T
T,p
T
T
,pT
T,gen
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Max. entropy production example/3
Thus,
which can be solved for Q1, giving Q2= Q - Q1, finally giving temperatures T3 and T4.
T=ThHeat Q
Flow ṁ1, T1, cp1
Flow ṁ2, T2, cp2
Q1
Q2
Flow ṁ2, T4, cp2
Flow ṁ1, T3, cp1
)QQ(Tcm
cm
QTcm
cm
,dQ
Sd
dQ
Sd
,p
,p
,p
,p
,gen,gen
gives which
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5.3 Thermo-electricity
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Entropy generation /1
Consider a slice of a material that conduct heat and electricity; the heat flux Ф” (W/m2) and electriccurrent density I” (A/m2).
The entropy production rate per unit volume (area·dx) ds’’’/dt for the heat flow is given by
V= electric potentialT = temperature
Pic, source: B01
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Entropy generation /2
Pic, source: B01
see alsonext slide
See alsoslides 7,8
T
dV
dx
''idtvolume
ds
dt
'''ds
Then for the total volume A·dx:
For the electric current density (Ohm’s law):with σ = 1/ρ, with specific electrical resistance ρ
The entropy production as a result of Ohmic losses, per unit volume:
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Entropy generation /3
Then for the total volume:
The processes can be coupled using:
with
Pic, source: B01
)R( resistance with Ω
dxdV
Tσ
TσI
dxTdxρIdxT
dRareaIdxareaT
dRIvolumeT
Qddt
'''ds
:Note
''''
''
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Entropy generation /4
Pic, source: B012
22
2112
211
222
21
T
dT
L
LL
T
dTLΦ
T
dT
L
LT
dV
0I
R
dVT
dVL
I
0dT
22
According to Ohm’s law, if dT = 0:
with resistance R = ρ·dx/A, for thickness dx, area A
Fourier’s law follows from I = 0: with Φ = G∙dT, heat conductance G = λ·A/dx.
A third material property is needed to describe the cross phenomena: the Seebeck coefficient, θ, defined as:
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Entropy generation /5
This gives for current I: Eliminating the voltage gradient dV/dx from the
expressions gives
which with conductance G = λ·area/dx can also be written as
combining Fourier’s law and the Seebeck effect.
Pic, source: B01
thermo-electricity
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The Seebeck effect
For combined heat flow and electric current, with I = 0:
This means that the voltage difference ΔV (”thermo-current”) for the system in the Fig. can be can be calculated as:
Δ
Δ
Reference temperature
Thermometry , i.e. a thermocouple Pic, source: B01
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The Peltier effect /1
Consider again combined heat flow and electric current, but now with dT = 0. The Onsager expressions nowgive: and
Combining dT = 0 with gives
which implies that an electric current involves a heat flow as well, which enters and leaves the material without causing heating or cooling.
Pic, source: B01
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The Peltier effect /2
If, however, two different materials a and b are used (as in a thermocouple), a heating or cooling effectis obtained at the contact point of materials a and b.
The netto heat flow, Фab is determined by the Seebeck coefficients for the two materials, θa and θb:
The propertyis referred to as the Peltier-coefficient for the material set a-b.
Pic, source: B01
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5.4 Power from osmosis
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The mixing of sea water with fresh water gives a (mixing) exergyeffect that can be exploited
Installing a membranesystem at 120-150 m below the fresh water intake allows for a significant extra hydropower effect
Pic, source: KB08
Work from a saline power plant /1
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Osmosis leads to transport of water across the membranefrom the fresh water to the salt water side solution
This gives a hydrostaticpressure difference
Pic, source: KB08
Work from a saline power plant /2
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The fresh water will go through the membrane, against a pressure difference; the pressure on the sea water side is not high enough to prevent fresh water movement.
Pic, source: KB08
Work from a saline power plant /3
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In reality, only part of the potential energy is recovered; the unit won’t be as low as 163 m below the fresh water intake.
Entropy production in the membrane (combined heat and mass transfer) is important
Work from a saline power plant /4
Pic, source: KB08
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Sources B01: Bart, G.C.J. Advanced thermodynamics (in Dutch) course compendium
Delft Univ. of Technol., Delft (2001) Chapter 11 B97: Bejan, A. Advanced engineering thermodynamics John Wiley & Sons
(1997) Chapter 12 FFK88: Förland, K.S., Förland, T., Kjelstrup, S. Irreversible thermodynamics.
Tapir Akademisk Förlag, Trondheim (1988) HK65: Hatsopoulos, G.N., Keenan, J.H. Principles of general thermodynamics.
R.E. Krieger Publ. Co. (1965) Chapter 4 H84: Hoogendoorn, C.J. Advanced thermodynamics (in Dutch) course
compendium Delft Univ. of Technol., Delft (1984) K07: Koper, G.J.M. An introduction to chemical thermodynamics, VSSD, Delft
(2007) Chapter 13 KB08: Kjelstrup, S., Bedeaux, D. Non-equilibrium thermo-dynamics of
heterogeneous systems, World Scientific (2008) KBJG10: Kjelstrup, S., Bedeaux, D., Johanessen, E., Gross, J. ”Non-equilibrium
thermodynamics for engineers”, World Scientific (2010) SAKS04: de Swaan Arons, J., van der Kooi, H., Sankaranarayanan, K. Efficiency
and sustainability in the efficiency and chemical industries. Marcel Dekker, New York (NY) 2004