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)