thermodynamics of separation what is the minimum work to separate a mixture into it’s pure...
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Pure Component 2
Pure Component 1
Mixture 12
inW&outQ&
Thermodynamics of separation
What is the minimum work to separate a mixture intoit’s pure components? Ex. Mining, Desalination, MaterialPurification, Recycling.
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Pure Component 2
Pure Component 1
Mixture 12
inW&outQ&
dNi,sys
dt= &Ni,in − &Ni,out
dE
dt=−&Qout + &Win + &H12 − &H1 − &H2
&Win =(( &H1 + &H2 )− &H12 )−To(( &S1 + &S2 )−&S12 ) +To
&Sirr
&Win =−&N12 (Δhmix −T0Δsmix) +T0
&Sirr
Balance Eq’ns for Mass, Energy & Entropy
Sirr
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&Win =−&N12Δgo
mix +T0&Sirr
wmin =
&Wmin
&N12
=−Δg∗mix
&Win =−&N12 (Δhmix −T0Δsmix) +T0
&Sirr
Minimum Work of Separation
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Gibbs Free Energy of Mixing*
Δgomix = Δho
mix –T0 Δsomix.
Δgomix–T0 Δsmix =
–T0 (s12 –x1s1 – x2s2)
For non-interacting molecules entropy can dominateoften resulting in a negative Gibbs Free Energy and hence spontaneous mixing. I.e. Δgo
mix < 0
* at standard conditions
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S = k ln
Boltzmann’s entropy equation
=n!
r!(n − r)!
How many ways can “r”atoms be positioned ina lattice with “n” locations?
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wmin = T0Δsmix = k T0 (ln 12)
Ex. 4 atoms in 8 locations
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wmin =−T0R(xlnx+ (1−x)ln(1−x))
Using Stirling’s Approximation
Where x is mol fraction r/n, and R = k Navo
ln N! = N ln N - N
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Multi-component System
=n!
n1!n2 !.....n j !
wmin =−T0R xii=1
j
∑ lnxi
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“Separation”
wmin =−T0R xii=1
n
∑ lnxi
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))xln(NxlnN(RTW )N(min
i −+−= 1210
))1ln(ln)1(( 210)1(
min1 xNxNRTW N −+−−=−
wmin, 1 =T0R(ln1x1
)
“Extraction”
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Separation Examples
• From the atmosphere
• From the Ocean
• Solutions– Polymer– Water based– Liquid metals (activity coef)
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The minimum work to separate O2 from the atmosphere
ex,O2
o =T0R(ln1
xO2
) ≅−298(K )×8.314(J / molK )ln(0.212) =3.84(kJ / mol)
In wet air you get 3.97 kJ/mol : compare with Szargut
Table from the EngineeringToolBox.com
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Energy
kg(target)=
kg(processed)kg(target)
gEnergy
kg(processed)
~1gg
Energykg(processed)
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energy requirements for mining and milling, possible future trends
Chapman and Roberts p 113 & 116
underground ~ 1000/g (MJ/t metal)
open pit ~ 400/g (MJ/t metal)
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Sherwood plot showing the relationship between the concentration of a target material in a feed stream and the market value of (or cost to remove) the target material [Grübler 1998].
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Exergy of a Mixture
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CRUST at To, po
Ore value at mine
Pure ore (e.g. Fe2O3)
Pure metal Metal alloy
Mixing in product
Mixing in waste stream
Further mixing and corrosion
Exergy
Purification Stages
Recycle to pure metal
Theoretical Exergy Values for a metal extracted from the earth’s crust shown at various stages of a product life cycle (not to scale)