Thermodynamics Thermodynamics

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Lecture

Lecture 4. 4.

Thermodynamics Thermodynamics

[Ch. 2]

• Energy, Entropy, and Availability Balances

• Phase Equilibria

- Fugacities and activity coefficients - K-values

• Nonideal Thermodynamic Property Models

- P-v-T equation-of-state models - Activity coefficient models

• Selecting an Appropriate Model

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Thermodynamic Properties Thermodynamic Properties

• Importance of thermodynamic properties and p y p p equations in separation operations

– Energy requirements (heat and work) Energy requirements (heat and work) – Phase equilibria : Separation limit

– Equipment sizing Equipment sizing

• Property estimation

– Specific volume, enthalpy, entropy, availability, fugacity, activity, etc.

– Used for design calculations

 Separator size and layout

A ili Pi i l

 Auxiliary components : Piping, pumps, valves, etc.

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Energy

Energy, Entropy and Availability , Entropy and Availability Balances

Balances Balances Balances

s in T Q ,

Heat transfer in and out

s out T

Q , O f d

s

Q ,in

… …

s

Q ,out

One or more feed streams flowing into the system are separated into two or more

d t t th t fl

v b s h P T z

n, _i, , , , , ,

:

Streams in product streams that flow

out of the system.

Separation process (system) :

:

: : :

n Molar flow rate

b h P T (system)

: :

Streams out LW

S_irr,

 ^z Mole fraction

T Temperature P Pressure

v b s h P T z

n, _i, , , , , ,

(Surroundings)

P Pressure

h Molar enthalpy s Molar entropy

… …

Shaft work in and out

out

W )s in (

W )s 0 (

T ^b Molar availability

v Specific volume

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Energy

Energy Balance Balance

• Continuous and steady-state flow system Continuous and steady state flow system

• Kinetic, potential, and surface energy changes are neglected

neglected

• First law of thermodynamics (conservation of energy)

(stream enthalpy flows + heat transfer + shaft work)leaving system

( t th l fl + h t t f + h ft k)

- (stream enthalpy flows + heat transfer + shaft work)entering system

= 0

t f i t

( nh Q W  

_s

)  ( nh Q W  

_s

)  0

 

out of in to

system system

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Entropy

Entropy Balance Balance

• The first law provides no information on energy efficiency

• Second law of thermodynamics

(stream entropy flows + entropy flows by heat transfer)leaving system

- (stream entropy flows + entropy flows by heat transfer )entering system

   

= production of entropy by the process

out of in to

system system

irr

s s

Q Q

ns ns S

T T 

   

   

   

   

 

system system

- Production of entropy

- Irreversible increase in the entropy of the universe

- Quantitative measure of the thermodynamic inefficiency of a process

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Availability (

Availability (Exergy Exergy) ) Balance Balance

• The entropy balance contains no terms related to shaft work

Th t i diffi lt t l t ith ti

• The entropy is difficult to relate with power consumption

• Availability (exergy) : Available energy for complete conversion to shaft work

s T h

b  

₀

to shaft work

• Stream availability function :

a measure of the maximum amount of stream energy that can be (Entropy balance)  T₀ - (Energy balance)

a measure of the maximum amount of stream energy that can be

converted into shaft work if the stream is taken to the reference state

0 0

1 1

       

       

       

       

   

 ^nb ^Q ^T _T ^W

^s

 ^nb ^Q ^T _T ^W

^s

^LW

in to out of

system

   T

^s

  

system

   T

^s

  

(stream availability flows + availability of heat + shaft work)entering systementering system

- (stream availability flows + availability of heat + shaft work)leaving system

= loss of availability (lost work)

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Lost Work, Minimum Work,

Lost Work, Minimum Work, and and Second Law Efficiency

Second Law Efficiency Second Law Efficiency Second Law Efficiency

• Lost work, LW   T S

₀ _irr

- The greater its value, the greater is the energy inefficiency

- Its magnitude depends on the extent of process irreversibilities

• Minimum work of separation W

- Reversible process : LW = 0

• Minimum work of separation, W

_min

– Minimum shaft work required to conduct the separation – Equivalent to the difference Equivalent to the difference

in the heat transfer and shaft work

_min

out of in to

system system

W   nb   nb

system system

) separation of

work (minimum

W

• The second-law efficiency

) separation of

work actual

t (equivalen

) separation of

work (minimum

min



 

W LW

 W

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Phase

Phase Equilibria Equilibria

• The phase equilibria of the given system provide possible equilibrium compositions (separation limit)

equilibrium compositions (separation limit)

• Equilibrium : Gibbs free energy for all phases is a minimum

( )

G G T P N N( , , ₁, ₂,...,N_C) G  G T P N N N

i i

dG   SdT  VdP    dN

components

( ) ( )

  

  

 

 

^p ^p

system i i

dG dN

(at constant T & P)







^N _i ^p

i

dN

⁽¹⁾ ⁽ ⁾

,

 

 

 

system i i

phases components P T

(conservation of moles of each species, no reaction)



p i

i

dN

2

) ( )

2 ( )

1

( _i

....

_i ^N

i

 

   

( p , )

The chemical potential of a particular species in a multicomponent system is identical in all phases at physical equilibrium.

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Fugacities

Fugacities and Activity Coefficients and Activity Coefficients

• Chemical potential

Units of energy

• More convenient quantities

– Units of energy

– Not easy to understand physical meaning

q

– Fugacity : pseudo-pressure

– Equality of chemical potentials  equality of fugacities )

/

exp( RT

C

f_i 



_i

– Fugacity coefficient

 Ratio of fugacity and pressure

P f_i

i  /

 

_i  f y P_i / _i

 Reference : ideal gas – Activity

 Ratio of fugacities

/ _i0 i

i f f

a 

At equilibrium,

(1) (2) (N)

i i i

f  f    f

 Ratio of fugacities

 Reference : ideal solution

– Activity coefficients



_i  a_i / x_i  f_i / x f_i _i⁰

T

⁽¹⁾

 T

⁽²⁾

   T

⁽^N⁾

(1) (2) (N)

i i i

a  a    a

 Ratio of activity and composition

 Departure from ideal solution behavior

i i i fi i fi

 T  T    T

(1) (2) (N)

P  P    P

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K

K- -Values Values

• Phase equilibrium ratio : ratio of mole fractions of a species present in two phases at equilibrium

• K-value (vapor-liquid equilibrium ratio; K-factor)

i i

/

i

K  y x

: for the vapor-liquid case

• Distribution coefficient (liquid-liquid equilibrium ratio) : for the liquid-liquid case

(1) (2)

Di i

/

i

K  x x

• Relative volatility Relative volatility : for the for the vapor liquid vapor-liquid case case

ij

K

i

/ K

j

 

• Relative selectivity : for the liquid liquid case

ij

K

Di

/ K

Dj

 

• Relative selectivity : for the liquid-liquid case

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Phase Equilibrium Calculations (VLE) Phase Equilibrium Calculations (VLE)

iV iL

f  f

For vapor-liquid equilibrium

• Ideal gas + Ideal solution

t

iV iL

f f

For vapor liquid equilibrium

sat i i

i

P x P

y 

P P x

K y

sat i i

i

 



• Phi-Phi approach : equation-of-state form of K-value

P x P

y

_i _iL _i

iV



 

iV iL

K

i



 

o o

 f  

• Gamma-Phi approach : activity coefficient form of K-value

o iV

y P

i iL

x f

i iL

  

^iL ^iL ^{iL iL}

i

iV iV

K f

P

  

 

 

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Nonideal

Nonideal Thermodynamic Thermodynamic Property Models

Property Models Property Models Property Models

• No universal equations are available for computing, for

nonideal mixtures, values of thermodynamic properties such as density, enthalpy, entropy, fugacities, and activity

ffi i t f ti f T P d h iti

coefficients as functions of T, P, and phase composition.

⇒ (1) P-v-T equation-of-state models ( )

• P-v-T equation-of-state models

(2) Activity coefficient or free-energy models q

Nonideality is due to (1) the volume occupied by the molecules and (2) intermolecular forces among the molecules

e.g. the van der Waals equation

RT a

2

RT a P  v b  v



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Useful Equations of State Useful Equations of State

• Mixing rules

C



C _0.5



^C

1 1

( )

C C

i j i j

i j

a y y a a

 

 

  

 

 

1

C

i i i

b y b



 

(14)

Models for

Models for Activity Coefficients Activity Coefficients

) ,....,

, ,

(

₁ ₂ _C

i

 T x x x

 

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Notes on

Notes on Using Using Phase Equilibrium Phase Equilibrium Models

Models Models Models

• Low pressure VLE

Gamma Phi approach recommended – Gamma-Phi approach recommended

– Poynting correction (modified Raoult’s law) required for medium pressure – Cannot be applied when T or P condition exceeds critical T, P

• High pressure VLE

– Phi-Phi approach recommended

– Special care should be taken for polar components (alcohols waterSpecial care should be taken for polar components (alcohols, water, acids, amines, etc.)

• Check binary interaction parameters matrix

If parameters exist use them – If parameters exist, use them – If parameters do not exist,

 Try to obtain by regression of experimental data

U t ib ti th d ( UNIFAC)

 Use group contribution method (e.g. UNIFAC)

• Special applications  Specialized models required

– Polymer solutionPolymer solution – Electrolyte solution

– Biomolecular applications

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Selecting an Appropriate Model Selecting an Appropriate Model

(LG): light gases (HC): hydrocarbons

(PC): polar organic compounds (A): aqueous solutions (PC): polar organic compounds (A): aqueous solutions (E): electrolytes

• If the mixture is (A) with no (PC)

- If (E) are present  modified NRTL equation - If (E) are not present  a special model

• If the mixture contains (HC), covering a wide boiling rage

 The corresponding-states method of Lee-Kesler-Plöcker

If h b ili f i f (HC) i id

• If the boiling range of a mixture of (HC) is not wide

- For all T and P  the P-R equation

- For all P and noncryogenic T  the S-R-K equation

• If the mixture contains (PC)

- For all T, but not P in the critical region  the Benedict-Webb- Rubin-Starling method

• If the mixture contains (PC)

- If (LG) are present  the PSRK method

- If (LG) are not present  a suitable liquid-phase  method

Thermodynamics Thermodynamics

Lecture

Lecture 4. 4.