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7/30/2019 SeparThermo

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1Lecture 3: Separations Basics

Thermodynamics of Separations

This lecture primarily focuses on the thermodynamics of separations. Well cover:

Phase Stability and the Gibbs phase rule.

A simple separation based on a vapor-liquid phase diagram.

The lever rule.

Equilibrium ratios (K-values, distribution coefficients, etc.).

The activity, and activity coefficients.

Measures of separations.


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Conditions for Phase Stability

U TS PV inii

G U TS PV

G TS PV inii TS PV

G inii

dU TdSPdV idnii

Integrate dU

From the definition of G

Substituting U into our expressionfor G

Gives the Gibbs Free Energy in terms

of chemical potentials and concentrations.


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The Gibbs-Duhem Equation

G i nii

dG d i nii

dG nidii

idnii

Starting with our expressions forthe Gibbs free energy

We consider how it changes for

an infinitesimal process

We see that G changes because the

amount of components changes or

because the chemical potential of the

components changes

GUTS PV

dG SdT VdP i dnii

dG dU d TS d PV

Since these two expressions for dG are equivalent we can equate them to find:

SdT VdP nidii

0The Gibbs-Duhem Equationwhich relates T, P and at

equilibrium in a single phase

system

G changes because T changes

or P changes or because the

composition changes

Consider a single phase in equilibrium:


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Gibbs Phase Rule

Each phase of a system in internal equilibrium is governed by its own Gibbs-Duhem equation:

SdT VdP nidii 0

Each phase is described by C+2 intensive variables:

T, P and the C chemical potentials.

Since the Gibbs-Duhem expression relates these C+2 variables,

only C+1 of them are independent.

If phases are in equilibrium with each other, then we have only one T and one P for

all the phases, so we still have C+2 variables. However, we now have relationships

between the C+2 variables since we have a Gibbs-Duhem expression for each phase.

f C 2

One can independently vary f intensive variables for a system

of C components and still keep phases in equilibrium.

Gibbs Phase Rule:Note: F is the number of degrees

of freedom and P is the number of

equilibrium phases.


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Vapor-Liquid Equilibrium

T

0

XB

1

V

L

T

0 XB 1

V

L

Lever Rule:

fL XB XB

V

XBL XB

V

fV XB

L XBXB

L XBV

Note that if we start with a

liquid of composition XB(L) andheat to the bubble point we

start to form a vapor with composition

XB(V), which is rich in the volatile

component. This vapor could be

collected, and cooled to the bubble

point line to create a liquid

concentrated in the volatile species.

Note that this is not really a practical

process because _______________?

XB(L)XB(V)

Dew point line

Bubble point line

If we start with a liquid of the same

composition, but heat above thebubble point as shown we can perform

a separation; however the resulting

products are less pure.

Note that the liquid could be

collected, and heated to the dew point

line to create a liquid even more

concentrated in the non-volatile species.

XB


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Equilibrium Ratios

T

0 XB 1

V

L

This type of separation is possible because

there is an equilibrium ratio different from one. That is,

the K-value for the component of interest issubstantially different from one.

A K-value is the ratio of the amount of component i in

one phase to another.

For liquid-liquid equilibrium it is often

called a distribution coefficient:

Ki xi

1

xi2

Note that the partial pressure of component i in the

vapor is used rather than the mole fraction, which

implies that we are assuming an ideal gas.

For liquid-vapor equilibrium it is

called a vapor-liquid equilibrium ratio:

Ki yixi


7/127Lecture 3: Separations Basics

Difficult Cases

T

0 XB 1

V

L

Note that the vapor-liquid equilibrium shown on the right

will be a difficult system to separate because the difference in

concentrations of the vapor and liquid phases is small.

T

0 XB 1

V

L

Note that the azeotropic vapor-liquid equilibrium shown on the right

will be a difficult system to separate because the difference in

concentrations of the vapor and liquid phases is small.

Azeotrope composition

Why are these phase diagrams like this?



Ideal Mixtures

If theliquid phase is ideal, then the separation would be

difficult because there would be no tendency to vaporize one

component versus the other. In other words, in order to use

the above type of separation process the mixture shouldnot

be ideal.

dG dH TdS

GM HM TSM

We know that:

And for mixing:

By definition for

an ideal mixture:GM TSM T Rxi lnxi

i

Since there is no enthalpic effect,

the Gibbs free energy of mixing

is only due to the randomness

created by mixing. This is

the same for the any ideal mixture.



Activity

The activity indicates how different the partial pressures of a vapor in equilibrium with a condensed

phase are from the mole fractions of the condensed phase.

By definition for

an ideal mixture:GM T Rxi lnxi

i

For a non-ideal

mixture:GM T Rxi lnai

i

ai

Pi

Pi0

ai fi

fi0 Use fugacities

if gas doesnt

behave ideally.

GM T Rxi lnii

T Rxi lnxii

The activity coefficient is defined by:

ai ixi

Activity

Gas constant

Activity coefficient

An activity coefficient equal to one indicates that the

mixture is behaving ideally for that component.



Activity

Note that there is more of the component indicated by the small

green circles in the vapor phase above its pure form than above

the mixture. Its activity is _____. The activity of the other componentis______?

The enthalpic contribution to non-ideality changes the amount of

one species in the vapor phase relative to the amount that you would

expect based on the composition of the liquid phase in equilibrium

with the vapor.



Measures of Separation

Split Fraction: SFi ni(1)

niF The Split Fraction is the ratio of the amount of

component in a product stream to the feed stream.

Split Ratio: SRi ni(1)

ni2

The Split Ratio is the ratio of the amount of

component in two product streams.

Separation Power: SPi,j

Ci1

Ci2

Cj1

Cj2

The separation power is the a ratio of split ratios.

A Separation Power near 1 indicates a poor separation.

Concentration

Number of moles of i



Next Lecture

Next lecture will show how to apply some of our thermodynamics to

flowing systems, and show how data of mixture compositions are sometimes

tabulated graphically. Specifically, we cover:

Energy and entropy balances in flowing systems

Availability and lost work

Gibbs Phase Rule for flowing systems and system specification Graphical determination of vapor-liquid equilibrium of hydrocarbon systems

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