thermodynamics of separation what is the minimum work to separate a mixture into it’s pure...

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.

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

&Win =−&N12Δgo

mix +T0&Sirr

wmin =

&Wmin

&N12

=−Δg∗mix

&Win =−&N12 (Δhmix −T0Δsmix) +T0

&Sirr

Minimum Work of Separation

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

S = k ln

Boltzmann’s entropy equation

=n!

r!(n − r)!

How many ways can “r”atoms be positioned ina lattice with “n” locations?

wmin = T0Δsmix = k T0 (ln 12)

Ex. 4 atoms in 8 locations

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

Multi-component System

=n!

n1!n2 !.....n j !

wmin =−T0R xii=1

j

∑ lnxi

“Separation”

wmin =−T0R xii=1

n

∑ lnxi

))xln(NxlnN(RTW )N(min

i −+−= 1210

))1ln(ln)1(( 210)1(

min1 xNxNRTW N −+−−=−

wmin, 1 =T0R(ln1x1

)

“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,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

Energy

kg(target)=

kg(processed)kg(target)

gEnergy

kg(processed)

~1gg

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

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)