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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 into it’s pure components? Ex. Mining, Desalination, Material Purification, Recycling.
Pure Component 2
Pure Component 1
Mixture 12
inW outQ
dNi,sys
dt= Ni,in ! Ni,out
dEdt
= ! Qout + Win + H12 ! H1 ! H2
dSdt
= ! Qout
T0+ S12 ! S1 ! S2 + Sirr
Win = (( H1 + H2 ) ! H12 ) ! To(( S1 + S2 ) ! S12 ) + To Sirr
Win = ! N12 ( hmix ! T0 smix ) + T0 Sirr
Balance Eq’ns for Mass, Energy & Entropy
Sirr
Win = ! N12 g
omix + T0 Sirr
wmin = Wmin N12
= ! g#mix
Win = ! N12 ( hmix ! T0 smix ) + T0 Sirr
Minimum Work of Separation
Gibbs Free Energy of Mixing* Δgo
mix = Δhomix –T0 Δso
mix.
Δgomix ≈ –T0 Δsmix = –T0 (s12 –x1s1 – x2s2)
For non-interacting molecules entropy can dominate often resulting in a negative Gibbs Free Energy and hence spontaneous mixing. I.e. Δgo
mix < 0
* at standard conditions
S = k ln Ω
Boltzmann’s entropy equation
! =n!
r!(n r)!How many ways can “r” atoms be positioned in a lattice with “n” locations?
wmin = T0Δsmix = k T0 (ln Ω12)
Ex. 4 atoms in 8 locations
! 12 =n!
r!(n r)!=8!4!4!
=701
wmin = ! T0R(x ln x + (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
Multi-component System
! =n!
n1!n2 !.....nj !
wmin = ! T0R xii=1
j
ln xi
“Separation”
wmin = ! T0R xii=1
n
ln xi
))xln(NxlnN(RTW )N(min
i !+!= 1210
))1ln(ln)1(( 210)1(
min1 xNxNRTW N !+!!=!
wmin, 1 = T0R(ln 1x1
)
“Extraction”
Separation Examples
• From the atmosphere • From the Ocean • Solutions
– Polymer – Water based – Liquid metals (activity coef)
The minimum work to separate O2 from the atmosphere
ex,O2o = T0R(ln
1xO2) ! 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
Energykg(target)
=kg (processed)kg (target)
i Energykg (processed)
~ 1g
i Energykg (processed)
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)
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].
Exergy of a Mixture
eoxx, mixture = xi! eox, i + RT0 xi! ln xi
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)
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