Applied thermodynamics

BA ELV - 00052_A_A - Rev. 6 03/09/2004

Refining-Petrochemicals-Chemicals-Engineering———

APPLIED CHEMICAL ENGINEERING

APPLIED THERMODYNAMICS

I - LIQUID/VAPOR EQUILIBRIUM OF PURE SUBSTANCES ...................................................... 1

1 - Phase diagram of a pure substance............................................................................................. 12 - Pressure/volume diagram, critical point........................................................................................ 23 - Vaporization and vapor pressure curves...................................................................................... 44 - Enthalpic diagrams of pure substances........................................................................................ 6

II - BEHAVIOR OF GASES........................................................................................................... 12

1 - Perfect gas ................................................................................................................................. 122 - Behavior of real fluids ................................................................................................................. 193 - Law of corresponding states ...................................................................................................... 214 - Law of corresponding states with three parameters .................................................................. 255 - Concept of equation of state ...................................................................................................... 276 - Use of equations of state ........................................................................................................... 29

III - LIQUID-VAPOR EQUILIBRIA OF HYDROCARBON MIXTURES........................................... 30

1 - Equilibrium range for a hydrocarbon mixture.............................................................................. 302 - Flash or liquid/vapor separation of a hydrocarbon mixture ........................................................ 323 - Deviations from ideality - Modern methods for determining equilibrium coefficients.................. 374 - General principles of calculating liquid/vapor equilibria.............................................................. 465 - Isobaric equilibrium diagram of binary mixtures or equilibrium lens........................................... 50

IV - LIQUID/VAPOR EQUILIBRIA OF MIXTURES OF UNIDENTIFIED COMPONENTS ............. 56

1 - Characterization of the volatility of petroleum cuts and crude oils ............................................. 562 - Vaporization curve ..................................................................................................................... 603 - Methods for calculating liquid/vapor equilibria of mixtures of unidentified components ............. 61

V - LIQUID/VAPOR EQUILIBRIA OF NON-IDEAL MIXTURES.................................................... 68

1 - Relationship between liquid phase properties and vaporization behavior, definition of theazeotrope ................................................................................................................................... 68

2 - Examples of non-ideal mixtures and homoazeotropes separation ............................................ 723 - Shift of the azeotropic composition with pressure ...................................................................... 784 - Heteroazeotropy ......................................................................................................................... 81

2004 ENSPM Formation Industrie - IFP Training

00052_A_A © 2004 ENSPM Formation Industrie - IFP Training

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I - LIQUID/VAPOR EQUILIBRIUM OF PURE SUBSTANCES

1 - PHASE DIAGRAM OF A PURE SUBSTANCE

In a pressure/temperature diagram, the domains of existence of a pure substance in its three physicalstates: solid, liquid, and gas or vapor, are bounded by three curves which join at the triple point T.The pressure and temperature at this point allow all three phases to co-exist.

From the triple point, the diagram below shows the characteristic shape of the curves on which twophases co-exist in equilibrium.

TT

Triple point

— LIQUID —

Criticalpoint

GASor

VAPOR

— SOLID —

Pressure

C

Temperature

PTD

TH 0

81 BSublimation

Vaporizatio

n

Fusi

on

- the fusion curve determines the solid/liquid equilibrium conditions

- the sublimation curve corresponds to the solid/vapor equilibrium

- the vaporization curve, often called the vapor pressure curve, characterizes theliquid/vapor equilibrium: this is limited at a point C called the critical point of the puresubstance.

Apart from a few special cases, the triple point of pure substances corresponds to a very low pressure,lower than atmospheric. At the same time, the y-axis coordinated of the critical point C is located in theinterval from 20 to 80 atmospheres for most organic and inorganic compounds.


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The table below gives the triple point pressures and temperatures of a number of pure substances.

TRIPLE POINT

Hydrogen Oxygen Nitrogen ChlorineHydrogen

sulfide AmmoniaCarbondioxide Methane

H2 O2 N2 Cl2 H2S NH3 CO2 CH4

Temperature(°C)

– 259.2 – 218.8 – 210.0 – 101 – 85.7 – 77.8 – 56.6 – 182.5

Pressure (bar) 0.072 0.00152 0.1253 0.014 0.227 6.08 10–4 5.2 0.117

The domain of existence of a pure substance in the fluid state includes all the areas at the right of thefusion and sublimation curves, naturally including those that lie above the critical point. Thesignificance of this particular point can be illustrated by the pressure/volume diagram.

2 - PRESSURE/VOLUME DIAGRAM, CRITICAL POINT

The figure below, for n-pentane, shows the correspondence between the vapor pressure curve of thepressure/temperature diagram and the volume characteristics of the fluids at constant temperatureplotted in the pressure/volume diagram.

Specific volume in cm3/g

Pressurein atm

00

1 0

2 0

3 0

4 0

5 0 100 150 200

Criticalpoint C

VsLs

Temperature in °C

Crit

ical

tem

pera

ture

196

.7°C

00

1 0

2 0

3 0

4 0

1 0

2 0

5 0 0 1000

2

5 5 0 100

Criticalpoint

190°C

190°C150°C

150°C Vs

Saturated vapor

190°C150°C

196.7°C 33.3 atm

4.3 cm3/g33.3 Criticalpressure33.3 atm

— VAPOR —

— LIQUID —

Pressurein atm

205°C

Ls

205 °C

Satu

rate

d liq

uid

D T

H 0

82 B


3

At a given temperature lower than the critical point temperature, the pressure/volume diagram showsthe break in volume existing between the saturated liquid and vapor phases LS and VS.

At 150 °C, for example, under 16 atm pressure, the above diagram shows that the volume per unitmass of n-pentane in the saturated liquid state is about 2 cm3/g (point LS). For the same puresubstance in the saturated vapor state in the same conditions it is 20 cm3/g (point VS).

Segment LS/VS is called the state change plateau, and its length, which materializes the difference involume between the two phases, is found to become shorter as the temperature and pressureincrease. When the pressure and temperature increase, in fact, the volume per unit mass of the liquidphase increases by expansion, while that of the vapor phase decreases under the effect of thepressure. The break in volume is ultimately nullified at the critical point, where the properties of bothphases are identical, thus marking the terminal point of the vapor pressure curve. The criticalconditions common to both phases are defined by the critical volume VC, critical pressure PC andthe critical temperature TC.

Outside the liquid/vapor equilibrium conditions, this diagram also shows the variation in the volume ofthe fluids with pressure. If we observe the isotherms which show the change in volume with pressure,we find that:

- the liquid phase is practically incompressible except when approaching the critical point(vertical isotherm in the P.V diagram),

- the vapor phase increases in volume as the pressure decreases: at low pressure, P.V (Pmultiplied by V) is approximately constant,

- the isotherm corresponding to the critical temperature displays a break point with ahorizontal tangent to the critical point: at this temperature, the two saturated phases VS andLS have the same volume VC,

- the pure substance at a temperature above its critical temperature can no longer exhibittwo-phase equilibrium: it behaves as a compressible fluid up to high pressures, and it isaccordingly said to be a hypercritical or supercritical fluid.

Supercriticalfluid

Liquid

Vapor

Pc

P

Tc 1

Supercriticalfluid

C

D TH

110

B

The critical coordinates, and especially the critical pressure PC and the critical temperature TC,are important data for pure substances, in so far as they are necessary for determining otherthermodynamic properties. Plates A2 give these figures for a large number of pure substances.Additional data can be obtained in the reference book “The Properties of Gases and Liquids”,R.C. Reid, J.M. Prausnitz, and T.K. Sherwood, 3rd Edition, McGraw-Hill Book Company.


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3 - VAPORIZATION OR VAPOR PRESSURE CURVES

The vapor pressure curves help to determine the essential properties of pure substances for theirbehavior with respect to liquid/vapor equilibria: the boiling point and the vapor pressure.

The boiling point of a pure substance, at constant pressure P, is the fixed temperature of thevaporization of a pure substance. Under the same pressure P, the pure substance is found in the liquidstate if the temperature is lower than its boiling point, and in the gas state if it is higher.

The boiling point rises with pressure to reach its highest value at the critical temperature.

Note that, if the vaporization pressure is the normal atmospheric pressure of 1.013 bar, the boilingpoint is said to be the normal boiling point, and is the figure given in the physical constants of thepure substance.

— LIQUID —

pressure

— VAPOR —

C

P

Pressure

1.013 bar

Normalboiling point

Boiling point

curve

Temperature

Vapor

D TH

083

B

The vapor pressure of the liquid pure substance is the equilibrium pressure of the liquid/vapormixture at a given temperature t. Hence it is the pressure in a storage vessel where the pure substanceis present in the liquid state and the gas state simultaneously. It reflects the capacity of the liquid toallow molecules to escape in the gas phase. This property is directly related to the intermolecularbonding forces existing in the liquid phase.

Ps

tEquilibriumpressure or

vapor pressure

VAPOR

LIQUID

CP

t

t

D TH

084

B

Vapor pressureof pure liquid substance

at temperature t

Since the two phases in equilibrium are saturated phases, the vapor pressure is often called thesaturation pressure. It is denoted PS.

The vapor pressure of a pure substance increases with temperature and reaches its maximum value,the critical pressure, at the critical temperature.

Note that, at any temperature lower than the critical temperature, the pure substance is in the vaporstate at any pressure below its vapor pressure, and in the liquid state in the opposite case.


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The data on boiling points, critical conditions and vapor pressures are essential for all calculationspertaining to the behavior of fluids and liquid/vapor equilibria. Some of them are accessible from PlatesA, B and C, which show:

- the physical properties of many pure substances (Plates A1 and A2): overall chemicalformula, melting point, normal boiling point, critical coordinates, density etc.,

- the Cox Chart, which shows the vapor pressure curves of hydrocarbons in the form oflines (Plate B1),

- the coefficients of Antoine’s equation which help to represent the vapor pressure curvesby the equation:

ln PS = A – B

C + T

where:

PS vapor pressure of the pure substance (mmHg),T boiling point (K),A, B, C characteristic constants of each pure substance,

- plate B2 gives the values of the constants A, B and C for a large number of puresubstances: it also indicates the temperature interval Tmin-Tmax in which Antoine’sequation gives the vapor pressure curve with a good degree of accuracy. Plates B5, B6and B7 give a graphic representation of a number of vapor pressure curves obtained byAntoine’s equation,

- data on the vapor pressure curve of water (Plates C),

- coefficients A’, B’, C’ and D’ of Harlacher’s equation.

Harlacher’s equation is a more elaborate form of Antoine’s equation for representing thevapor pressure curve of a pure substance. It has four coefficients and is written as follows:

ln PS = A’ + B’T + C’ ln T +

D’ PS

T2

T is temperature in KelvinPS is pressure in mmHgln is the Napierian logarithm (or natural)

Constants A’, B’, C’ and D’ are given on Plate B8 for a number of pure substances.

Note that the use of Harlacher’s equation requires an iterative calculation. At a given temperature T,successive approximations are used to determine the value of PS, which confirms the equality of thetwo members of Harlacher’s equation.

The following recommendations help to deal with a problem of determining the vapor pressure of apure substance.

- If the vapor pressure is less than 10 mmHg, the above methods are inaccurate.

- If the vapor pressure is lower than 1500 mmHg, Antoine’s equation gives good results.

- If the constants A’, B’, C’ and D’ are available, Harlacher’s equation gives good results inthe interval 10 mmHg < PS < PC.


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4 - ENTHALPIC DIAGRAMS OF PURE SUBSTANCES

a - Quantities of heat involved in heating and cooling a fluid: enthalpies, heat capacities

Starting with the liquid state, for example, suppose one adds heat to a pure substance at constantpressure. The quantity of heat supplied to a unit mass of fluid as a function of temperature measuresits variation in enthalpy. This enthalpy is denoted H and its variation ∆H. They are often expressed inkJ/kg; sometimes in kcal/kg or BTU/lb. Based on a reference state, it gives the heat energy contentper unit mass of the pure substance.

The figure below shows, on pressure/temperature and enthalpy/temperature diagrams, the pathfollowed by a pure substance between a point L in the liquid zone and a point V in the vapor zone.They help to clarify the terminology used and the correspondence between the pressure andtemperature conditions and the enthalpy.

Temperature

VL

tL

tb

tV

P

Pressure

V S

L S

TemperaturetL

tb

tV

Enthalpy

HV

HV S

hL Sh

L LL S

VS

V

D TH

085

BThe initial liquid (point L) is at a pressure P considered at a temperature lower than its boiling point. It issaid to be sub-cooled.

Progressive heating of the liquid to its boiling point is achieved by adding sensible heat because itproduces a variation in temperature. The difference hLs - hL gives this variation in sensible heat.Related to the corresponding temperature difference, it helps to determine the heat capacity c of thepure liquid substance. Thus we have:

c = hLs – hLtb – tL

hLs – hLtbtLc

: in kJ/kg: in °C: in °C: heat capacity in kJ/kg.°C

Over a not too much long temperature interval, the heat capacity c can be considered constant. Theusual values are the following:

- liquid water at low temperature 4.185 kJ/kg.°C- liquid hydrocarbons 2.0 to 3.0 kJ/kg.°C- glycerine 2.43 kJ/kg.°C- mercury 0.138 kJ/kg.°C


7

The saturated liquid (LS) begins to vaporize at the boiling point and its vaporization continues atconstant temperature. The quantity of heat required to obtain the change in state is called the enthalpyof vaporization or latent heat of vaporization (Λ):

enthalpy of vaporizationor Λ = HVs – hLs

heat of vaporization

The enthalpies of vaporization at temperatures sufficiently distant from the critical temperature rangefrom 250 to 420 kJ/kg for hydrocarbons, and about 2000 kJ/kg for water.

At the end of vaporization, the vapor obtained is said to be saturated (VS). Throughout thevaporization, the two phases present are in liquid/vapor equilibrium and are saturated.

The input of sensible heat to the vapor again raises its temperature, removing it from the saturationconditions. This indicates a dry or superheated vapor. HV - HVs measures the correspondingsensible heat.

As for the liquid phase, we can define a mean heat capacity c for heating the vapor over a giventemperature interval. The routine values for some pure substances are:

- 2 kJ/kg.°C for steam- 1 kJ/kg.°C for air- 14.5 kJ/kg.°C for hydrogen

The reverse path is obtained by removing heat. Starting with a dry vapor, it leads successively to thedesuperheat of this vapor down to saturation, and then to its condensation with the restoration oflatent heat, in this case called the heat of condensation, and finally to the sub-cooling of the liquidobtained.

b - Calculation of a heat flow rate

When a fluid undergoes additions or removals of heat accompanied or unaccompanied by changes inphysical state, it is easy to determine the heat flow rates involved if we have its enthalpy before andafter the transformations achieved.

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Conditions Conditions

Input M

enthalpy H1

Physical state L or VTemperature T1Pressure P1

Enthalpy H2

Physical state L or VTemperature T2Pressure P2

HEAT EXCHANGERS

D TH

202

4 A


8

If the enthalpies are mass enthalpies, the heat flow rate Q involved is given by the expression:

Q = M . (H2 – H1)Q heat input in kWM mass flowrate in kg/sH2 and H1 specific enthalpies in kJ/kg

If the transformation concerns a fluid without a change in physical state, i.e. for sensible heat only, it isoften advantageous to calculate the heat flow rate involved by using the heat capacity figure.

21

Input MTemperature T1Pressure P1

Temperature T2Pressure P2

Same physical state in step

Average specified heat of fluid: c

1 2and step

D TH

202

4 B

The sensible heat rate involved is given by:

Q = M . c . ∆T

Q sensible heat input in kWM fluid flowrate in kg/sc specific heat in kJ/kg∆T = T2 – T1 or T1 – T2

change in temperature of fluid

This formula, which is very convenient and commonly used, is limited to heat exchanges withoutchange in phase. In the general case, it is necessary to use the enthalpy and to have diagrams usedto determine the enthalpy of a fluid as a function of the conditions in which it is found.

c - Enthalpy/temperature diagram (H-t diagram)

The previous experiment in the vaporization of a pure substance can be repeated for variouspressures. Starting with a low pressure and by increasing it progressively, we observe:

- an elevation in the vaporization temperature as shown by the vapor pressure curve, up tothe critical temperature

- an increase in the enthalpies of the saturated phases and, in contrast, a decrease in thelatent heat of change of state, which is ultimately nullified when the critical temperature isreached

The heat of vaporization Λ decreases as the pressure increases and becomes zero at the

critical point.

- a steady unbroken change in the enthalpy of the fluid as the pressure becomes higher thanthe critical pressure.


9

By matching the results obtained, we obtain an enthalpy/temperature diagram which is characteristicof the pure substance. The characteristic shape of this diagram is shown in the figure below forn-pentane.

500

0

100

200

300

400

-100 0 100 200 300 400 500

Saturated vapor

critical point

Saturated liquid

1020

50

— n PENTANE — Enthalpy - Temperature diagram

Temperature in °C

Enth

alpy

kca

l/kg

30 40

40 80

150

60

D TH

111

B

This diagram mainly consists of the two enthalpic curves relative to the saturated vapor HVs (t) and thesaturated liquid hLs(t). At a given temperature, the difference between vapor and liquid enthalpy:

HVs - hLs = Λ

represents the heat of vaporization of the pure substance. This decreases progressively as thetemperature rises, and ultimately becomes zero at the critical point, where the enthalpic curves of bothsaturated phases meet.

At a temperature distant from the critical point, the enthalpy of the unsaturated phases is unaffected bythe pressure. This means that, at a given temperature, the enthalpies of the sub-cooled liquid or of thedry vapor are respectively equal to the enthalpies of the saturated liquid and vapor.

As the temperature approaches the critical point, isobars help to account for the influence of thepressure, particularly for the vapor phase.

Above the critical point, the enthalpy of the pure substance in the state of a hypercritical fluid can beobtained as a function of temperature and pressure by prolonging the network of isobaric curves.


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d - Reference conditions of the enthalpy diagram

Enthalpic diagrams are plotted from a reference state chosen arbitrarily, for which the enthalpy iszero. For the diagram given above, the enthalpy reference 0 kcal/kg corresponds to the saturatedliquid state at – 100°C. It was plotted from diagrams of J.B. Maxwell, for which the reference is thesaturated liquid state at – 200°F. Other sources use different references, such as the saturated liquidstate at 0°C. Caution is therefore necessary when simultaneously using diagrams from varioussources.

Plates E1 to E4 show the enthalpy diagrams of a number of hydrocarbons prepared afterJ.B. Maxwell. Others can be obtained from plates E10 to E15.

These diagrams help to determine the enthalpy for a pure substance from the knowledge of itsphysical state, the pressure and the temperature. They offer the possibility of determining theenthalpy of mixtures of known composition by weighting the enthalpies of the pure substances. Othergraphic tools are also available for determining the thermal properties of pure substances. Amongthese, the most commonly used are the Mollier chart and the pressure/enthalpy diagram.

e - Enthalpy/entropy diagram or Mollier chart (H/S diagram)

Mollier charts offer a good representationof the changes in temperature andpressure undergone by a pure gaseoussubstance in compression and expansionprocesses. It is an enthalpy/entropydiagram (H/S) which shows the saturationcurve giving the properties of the puresubstance in the saturated vapor state.Networks of isotherms and isobars help tolocate the superheated vapor and todetermine its enthalpy in given conditionsof temperature and pressure.

saturation line

H

enth

alpy

entropyS

P

isobars

isot

herm

s

t

0.95

0.90

D TH

112

B

Entropy is an important thermodynamic function which is involved in particular in energyconversion processes. It is denoted S and expressed, for example, in kJ/kg.°C.

An isentropic change undergone by a gas during compression or expansion is represented in thisdiagram by a vertical. It is defined as an adiabatic transformation, hence without exchanges of heatwith the exterior, and reversible, i.e. without any degradation in energy. Applied to the expansion ofwater vapor in a steam turbine, it would correspond to a perfectly insulated machine in which thethermal energy of the steam is fully converted into mechanical energy. A real transformation involvingfriction and a degradation of energy in the form of heat would cause an increase in entropy.

Note that Mollier charts, beyond the saturation limit in the two-phase liquid/vapor domain, often have anetwork of curves corresponding to a partial condensation of the pure gaseous substance. These areidentified by a content or mass fraction of vapor in the liquid/vapor mixture.

Plate E5 shows the Mollier chart for propane and plate C7 the Mollier chart for steam and water.


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f - Pressure/enthalpy diagram

Another representation of thermal properties of pure substances is the pressure/enthalpy (P/H)diagram. Plates E6, E16 and E17 are illustrations for propane and for freons 12 and 22. The operatingconditions of refrigerating cycles, which generally operate between two and three different pressures,are extremely well represented by this type of diagram.

Criticalpressure

Satur

ated

liquid

Satu

rate

d va

por

isotherms

t1

Enthalpy H

to

t2

C

t1t

2

Pressure P

D TH

086

B

Changes in state appear in the form of horizontal plateaux of which the length corresponds to the heatof vaporization.

The isotherms help to locate the representative point of the pure substance as a function of thepressure and temperature. These isotherms show a break corresponding to the state change whent < TC (t1 in the diagram). They appear in the form of a continuous curve when t > TC (t0 in thediagram). P/H diagrams often show other properties, such as entropy S and volume per unit mass v (inm3/kg).


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II - BEHAVIOR OF GASES

1 - PERFECT GAS

a - Ideal gas law

The perfect gas represents the limit state of any fluid when the pressure tends towards zero. It is avalid approximation of the behavior of real gases when the pressure is low.

The properties of the perfect gas correspond to those of a fluid in which no intermolecular bondingforce appears. The resulting laws are therefore relatively simple and the behavior of real fluids can beinferred by introducing correction factors due to molecular interactions.

The behavior of the perfect gas, irrespective of the type of gas, is characterized by a law called theideal gas law or Avogadro’s law, which is written:

P . V = n . R . T

P absolute pressure exerted by the gas,V volume occupied by the gas or volume flow rate of gas,n quantity of gas expressed as the number of moles or molar flow rate,T absolute temperature of the gas,R universal gas constant, of which the value depends on the units selected.

The most common units and the corresponding values of R are given in the table below.

P bar atm kg/cm2 PSI a

V m3 or m3/h m3 or m3/h m3 or m3/h Ft3 or Ft3/h

nkmoles orkmoles/h

kmoles orkmoles/h

kmoles orkmoles/h

lbmoles orlbmoles/h

T K K K °R

R 0.08314 0.08205 0.084478 10.73

This law, although approximate for real gases, allows a number of simple calculations. The accuracy,although limited, is often sufficient for an approximate representation of the behavior of real gases. Anumber of applications is given below.


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MOLAR VOLUME OF IDEAL GASES Vm

Vm = R . T

P

Vm molar volume in m3/kmolT temperature in KP pressure in barR universal gas constant = 0.08314

In normal conditions: t = 0°C or T = 273.15 KP = 1 atm = 1.013 bar

the well-known value is obtained: Vm = 22.42 m3/kmol

DENSITY OF AN IDEAL GAS

The density of any gas of molecular weight M (in kg/kmol) is given by the equation as follows:

ρ = molecular weight

molar volume = MVm

Vm in m3/kmol

For an ideal gas:

ρρρρ = M . P R . T

ρ density in kg/m3

M molecular weight in kg/kmolP pressure in barT temperature in KR universal gas constant = 0.08314

The specific gravity of an ideal gas with respect to air (sp.gr. gas) can be obtained from thefollowing equation:

sp.gr.gas = ρgas

ρair =

MgasMair

Air consists essentially of nitrogen (M = 28) and oxygen (M = 32) and has a molecular weightMair = 29 kg/kmol.

Thus we obtain: sp.gr.gas = M29 M in kg/kmol

CONVERSION FROM MASS FLOW RATE TO VOLUME FLOW RATE

In a process operating in steady-state conditions, the mass and volume flow rates of a fluid are relatedby the following equation:

Qm = ρρρρ . Qv

Qm mass flowrate kg/hQv volume flowrate m3/hρ density kg/m3


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If the fluid is an ideal gas: ρ = M . PR . T ; where M is the molecular weight of the gas, and P and T are

the flow conditions, the volume flow rate Qv is then obtained from the mass flow rate Qm by theequation:

Qv = Qm . R . TM . P

Qv in m3/hQm in kg/hT in KP in barM in kg/kmolR = 0.08314

Sometimes, the ratio R/M is used, denoted R and called “gas constant”.

CORRECTION OF VOLUME FLOW RATES

At constant mass flow rate and for a given perfect gas, the above equation shows that if the

temperature and pressure change, the expression Qv . P

T remains constant.

For reference conditions 1 and 2 , therefore:

Qv1 . P1T1

= Qv2. P2

T2or QV2 = QV1 .

T2T1

. P1P2

This expression is often used for the gas volume flow rates in normal or standard conditions.

Normal conditions are defined as follows: t = 0°C (273 K) P = 1 atm = 1.013 bar

which correspond to flow rates expressed in normal m3 per hour (Nm3/h).

Using QVCN to denote the volume flow rate of ideal gas in normal conditions:

QVCN = QV . 273T .

P1.013 or QV = QVCN .

T273 .

1.013P

QV in m3/hT in KP in barQVCN in Nm3/h

In practice, gas volume flow rates are often expressed in conditions slightly different from normalconditions. This has led to the use of standard conditions: P = 1 atm and t = 60°F being theconditions most commonly recognized internationally. For reasons of convenience, it is customary inFrance to use P = 1 atm and t = 15°C, values which are very close to the above, since 60°F = 15.6°C.These different definitions could cause confusion, and it is always preferable to specify the exactconditions in which a gas volume flow rate is given.


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b - Perfect gas: mixtures and partial pressures

If we consider a gas mixture in given conditions exerting a total pressure P, it is often advantageous toknow the participation of each component of the gas mixture to this total pressure.

The share of pressure exerted by a particular component “i” is its partial pressure (PPi). It is definedas the pressure that this component would exert if it alone made up the entire volume flow rateof the mixture.

The figure below shows a flow of a mixture of several pure gaseous substances at temperature T andpressure P. The total volume flow rate is Qv.

xx

xx

xx

xx

x

P

Q vT

x

xx

x

x

xx

x

xx

x

xxx

x

x

D TH

087

A

If ni is the molar flow rate of component “i”, the above definition of partial pressure leads to thefollowing law:

PPi . QV = ni . R . T

This equation can also be written for each one of the components of the mixture.

The summation of these equations for all the components gives:

∑i PPi . QV = ∑

i ni . R . T (1)

At the same time, the ideal gas law can be applied to the entire gas mixture by denoting, as N, the totalmolar flowrate N = ∑

i ni. This gives:

P . QV = N . R . T (2)

By identifying equations (1) and (2), we obtain:

P = ∑∑∑∑i PPi


16

The total pressure P exerted by the gas mixture is equal to the sum of the partial pressures of each ofthe components of the mixture.

The ideal gas law applied to component i and to the gas mixture leads to the two following equations:

PPi . QV = ni . R . T

P . QV = N . R . T

Combining these two equations gives:

PPiP =

niN

niN is the molar fraction yi of component i in the gas mixture or:

PPiP = yi or PPi = P . y i

PPi partial pressure of component iP total pressure (same units as PPi)yi molar fraction of i in vapor phase

The partial pressure of a component of a gas mixture is obtained by multiplying the total pressure by itsmolar fraction in the mixture (Dalton’s law).

These equations, known by the name of Dalton’s law, give the partial pressure of a component of amixture of ideal gases from:

- the total pressure exerted by the mixture which is usually readily accessible bymeasurement

- the molar fraction of the component concerned in the mixture: for ideal gases, the molarfraction yi mol of component i is equal to the volume concentration yi volume:

yi mol = yi volume

If the mass concentration yi mass is known, a conversion is necessary using the followingformula:

yi mol = yi mass . MMi

Mi is the molecular weight of component i,M is the average molecular weight of the gas mixture.


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c - Other properties of ideal gases

While ideal gases all display identical behavior in terms of the properties of pressure, volume, quantityand temperature, this rule does not apply for many of their physical properties, especially theirthermal properties. Tables are given in the literature which list the following for the different puresubstances in the ideal gas state.

MOLAR OR MASS SPECIFIC HEATS AT CONSTANT PRESSURE Cp

The molar specific heat at constant pressure Cp is usually given in joule per mole by Kelvin(J/mol.K) and expresses the quantity of heat to be supplied to a mole of ideal gas to raise itstemperature by 1 K at constant pressure. Sometimes other units as calories or BTU per mol by Kelvinare used.

Its value varies with the temperature, so that Cp is usually given in the form of polynomial functions oftemperature:

Cp = a + bT + cT2 + dT3

T in Kelvina, b, c are given for a large number of pure substances on plate D1

If the temperature of an ideal gas varies from T1 to T2, its corresponding variation in molar enthalpycan be calculated by the equation:

∆H = Cp . (T1 – T2)

where Cp is an average value determined for the temperature interval (T1 – T2).

The molar specific heat at constant volume of the ideal gas Cv is also defined, of which the value isdifferent from Cp. Cv expresses the quantity of heat to be supplied to one mole of ideal gas to raise itstemperature by 1 K at constant volume. The value of Cv is used in the definition of the isentropiccoefficient k of the ideal gas, which determines the behavior of the gas in compression and expansionprocesses.

k = CpCV

k isentropic coefficient of the ideal gasCp molar specific heat at constant pressureCV molar specific heat at constant volume

The value k can be obtained simply for ideal gases using the Mayer equation, which relates Cp andCv to the universal gas constant R. It is written:

Cp – CV = R Cp, CV in kcal/kmol . KR = 1.987 kcal/kmol . K = 8.317 kJ/kmol.K

or:

Cp – CV = RM

Cp, CV in kcal/kg.°CR = 1.987 kcal/kmol.K = 8.317 kJ/kmol.KM molecular weight of gas in kg/kmol


18

The isentropic coefficient k is then obtained by the following equations:

k = Cp

Cp – RCp in kcal/kmol.KR = 1.987 kcal/kmol.K = 8.317 kJ/kmol.K

or:

k = Cp

Cp – RM

Cp in kcal/kmol.KR = 1.987 kcal/kmol.K = 8.317 kJ/kmol.KM in kg/kmol

Note also that the molar specific heat Cpm of a mixture of ideal gases is obtained from the Cpi ofthe components of the mixture by molar weighting:

Cpm = ∑i yi Cpi yi molar fraction of i

STANDARD ENTHALPIES OF FORMATION

The standard enthalpy of formation ( )∆∆∆∆H°T F is defined as the variation in enthalpy which

accompanies the chemical reaction of the formation of a compound from its elements, all taken instandard conditions: same temperature, pressure 1 atmosphere, ideal gas state with some exceptions(e.g. graphite for the element carbon).

The (∆H°T)F help to determine the enthalpy variations during chemical conversions.

They are given on Plates E18 in cal/mol at different temperatures: 0 K, 298 K, 400 K, 600 K, 800 Kand 1000 K for a large number of pure organic and inorganic substances.

STANDARD FREE ENTHALPIES OF FORMATION (∆G°T)F

The standard free enthalpy of formation helps to calculate chemical equilibria. The values of

( )∆G°T F are given on plates E19. As for enthalpy H, the free enthalpy G is expressed in cal/mol. It is

often called the ‘Gibbs Energy’ in English literature.


19

2 - BEHAVIOR OF REAL FLUIDS

a - Definition of the compressibility factor

The differences in behavior between a real fluid and a perfect gas can be identified by observing thechange in the ratio of the volume of real fluid V to that of the perfect gas Vpg as a function of thepressure and temperature of the fluid.

This ratio denoted Z is called the compressibility factor:

Z = V

Vpg

V volume of real fluidVpg volume of ideal gas

The volume of an ideal gas is given by the equation: Vpg = n . R . T

P

Consequently:

Z = P . V

n . R . T

Hence:

P . V = Z . n . R . T

This law can be qualified as the real gas law, and the compressibility factor appears as a correctivefactor to the ideal gas law. If the compressibility factor of the real fluid takes the value 1, its behavior isidentical to that of the ideal gas, and this happens when the fluid is a gas at very low pressure.

If we glance at all the conditions in which a fluid may be found (vapor, saturated or unsaturated liquid,supercritical fluid), we find that, in practice, the compressibility factor deviates quite significantly fromthe value 1, which is characteristic of the ideal gas.

This is obvious for the liquid state in which the real fluid volume is much lower than that of the ideal gas(Z << 1). In the gas state, especially for condensable vapors, wide differences from ideal gas behaviorare also observed. The representation of the volume behavior of a pure substance in the fluid state isclearly illustrated by Amagat’s diagram, which gives isotherms for the variation in Z as a function ofpressure.

b - Amagat’s diagram

The shape of the isotherms plotted on the Amagat diagram shown below can be analyzed on thepressure/temperature diagram, for example by assuming a temperature lower than the criticaltemperature and by increasing the pressure from zero.


20

In these conditions, the value of Z is observed to undergo the following changes.

- At zero or very low pressure, the real gas is similar to the ideal gas and the compressibility factoris approximately equal to 1.

- At higher pressure, the gasapproaches saturation, Z decreasesas the temperature selectedapproaches TC. The compressibilityfactor of the saturated vapor, ZVs,may then be much lower than 1.

- The change of state then leads to asudden dec rease in thecompressibility factor, which takes thevalue ZLs corresponding to thesaturated liquid.

LIQUID

VAPORVS

LSPS

C

Temperature

Pressure

0t

D TH

088

B

- As the pressure then rises in the liquid zone, the compressibility factor increases in so far as thevolume of liquid varies slightly while that of the ideal gas to which it is compared decreases ininverse proportion to the pressure.

Z=PV

RTt = 1.227°C Ideal

gas

t = 77°C

Saturated

vapor

VsZ

Vs

0.75

0.5

0.25

Change ofstate plateau

Criticalcompressibility

factor

Zc

Saturated

liquidL

s A

50 75250 P (atm) 1 0 0

t = 37°C t=tc=32.5°C

Z = 1

ZLs

t = 177°C

7°C

AMAGAT'SDIAGRAM

FORETHANE

t = 2

5°C

t = 7

°C

D TH

113

B


21

Amagat’s diagram given above for ethane C2H6 shows the variation in the compressibility factor atdifferent temperatures.

- the isotherms 7 and 25°C illustrate the above path.

- the critical isotherm plotted for t = tc = 32.5°C shows an inflexion point at the criticalpressure PC. In these conditions, the compressibility factors of the liquid and vapor phasesare equal, and correspond to the critical compressibility factor Zc. It may be observed thatthe value of ZC is 0.285.

- the isotherms plotted at temperatures higher than tC correspond to increasingcompressibility factors which approach the value of 1 at very high temperature. In thehypothetical conditions t = 1227°C, ethane is virtually an ideal gas irrespective of thepressure

A specific Amagat diagram can be plotted for every pure substance. However, all the diagrams, whichare different in terms of temperature and pressure, reveal identical behavior and an approximatelyuniversal value of the critical compressibility factor ZC. Plates A2 give the values of ZC for manypure substances. They are close to 0.27 for hydrocarbons. For other compounds, ZC differs from thisvalue while note deviating from it excessively:

Hydrogen ZC = 0.305Oxygen ZC = 0.288Nitrogen ZC = 0.290Water ZC = 0.229

This similarity in the behavior of different pure substances at the critical point is also found when theyare in relatively identical conditions with respect to critical conditions. This leads to a veryimportant law called the law of corresponding states, which applies not only to compressibilityfactors, but also to many other properties of pure substances.

3 - LAW OF CORRESPONDING STATES

a - Definitions and general diagrams

The law of corresponding states declares that some properties of pure substances are identicalwhen they are in identically reduced conditions. These conditions are defined as follows:

Reduced pressure PR = P

PC

same pressure units for P and PCP pressure of the fluidPC critical pressure of the pure substance

Reduced temperature TR = TTC

T and TC in KelvinT temperature of the fluidPC critical temperature of the pure substance

P and T are the pressure and temperature of the fluid.PC and TC are the critical pressure and the critical temperature of the pure substance.


22

This law applies primarily to the compressibility factor Z. Plate I2 shows the universal diagram usedto determine Z as a function of PR and TR.

The diagram below, taken from “The Properties of Gases and Liquids” (Reid and Sherwood) gives,as an example, the value of the compressibility factor for reduced pressures lower than 1 and reducedtemperatures between 0.6 and 3.

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.300.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.90

0.95

3.002.001.601.401.30

TR

1.05

1.10

1.151.20

1.401.101.00

0.80

0.850.70

0.65

0.60 TR

1.00

Reduced pressure PR

Saturation line

0.00 0.05 0.10

0.800.85

1.00 2.00

0.900.95

0.90 0.75

z

PR

D TH

114

B

Com

pres

sabi

lity fa

ctor

Z =

PV RT

All the isotherms can be seen to converge towards Z = 1 as the reduced pressure tends towards zero.Complementary diagrams help to obtain Z in all conditions up to reduced pressures of 40.

These generalized diagrams actually correspond to an average behavior of the different hydrocarbonsand other compounds. They therefore give a good approximation of the value of the compressibilityfactor.

In using the foregoing correlations for a mixture, it is assumed that PR and TR are replaced bypseudo-reduced coordinates PPR and TPR. These are obtained by molar weighting of the criticalcoordinates of the components, which leads to the calculation of the pseudo-critical pressure of themixture PPC and its pseudo-critical temperature TPC:


23

Pseudo-critical pressure for a mixture PPC = ∑i yi PCi yi molar fraction of i

PCi critical pressure of iP operating pressure

Pseudo-reduced pressure PPR = P

PPC

PPC pseudo-critical pressurePPR pseudo-reduced pressure

Pseudo-critical temperature for a mixture TPC = ∑i yi TCi yi molar fraction of i

TCi critical temperature of iT operating temperature

Pseudo-reduced temperature TPR = T

TPC

TPC pseudo-critical temperatureTPR pseudo-reduced temperature

b - Use of the compressibility factor

The introduction of the compressibility factor in the ideal gas law leads to the real gas law in which theunits used and the value of the constant are identical to those already presented:

P . V = Z . n . R . T

Introducing the volume flow rate Qv gives:

P . Qv = Z . n . R . T

The introduction of Z in the other equations written for the ideal gas leads to the following expressions:

- molar volume of a real gas: Vm = Z R . T

P

- density of a real gas: ρρρρ = M . P

Z . R . T

- relationship between mass flow rate and volume flow rate for a real gas:

Qv = Qmρ

or Qv = Qm . Z . R . T

M . P

- correction of volume flow rates of real gases:

P . QvZ . T = constant hence Qv2 = Qv1 .

Z2Z1

. T2T1

. P1

P2

where Z1 and Z2 are the compressibility factors in conditions 1 and 2 .


24

c - Validity of the law of corresponding states with two parameters

Limited to the generalized diagrams on Plate I2, it appears that the law of corresponding statesremains fairly approximate, and it assigns the same critical compressibility factor ZC = 0.27 to allsubstances, whereas this actually varies from 0.23 for water to 0.3 for hydrogen, while remaining closeto 0.27 for most hydrocarbons.

For other properties to which the law of corresponding states applies, some differences are similarlyobserved in the behavior of the different pure substances. This applies to the vapor pressure curveswhich, plotted in reduced coordinates, should lead to a single generalized curve reflecting thebehavior of any compound at liquid/vapor equilibrium. In fact, the figure below shows that a group ofvapor pressure curves is obtained, showing differences in behavior related to the type and size of themolecules making up the pure substances.

1

0.5

0.1

0.05

0.010.4 0.5 0.6 0.7 0.8 0.9 1

— VAPOR —

— LIQUID —

Reducedsaturationpressure

Ps

Pc

Cm

etha

ne

ammonia

Reduced temperatureT

Tc

Vapor pressure curvesin reduced coordinates

buta

newa

ter

D TH

115

B

In consequence, many improvements to the law of corresponding states have been suggested toobtain greater accuracy in determining the properties of fluids. These commonly consist of introducinga third parameter in addition to TR and PR, to characterize the type of fluid and thus to refine themethod. This is referred to as the law of corresponding states with three parameters.


25

4 - LAW OF CORRESPONDING STATES WITH THREE PARAMETERS

a - Main parameters used

The third parameter is a physicochemical characteristic defined for each pure substance, which can bedetermined for mixtures by weighting. Depending on the choice of this third parameter, manycorrelations have been proposed. The most widely recognized are due to the following:

- J.B. Maxwell, who introduced the molecular weight M of the fluid. The compressibilityfactor Z is thus obtained in a series of charts as a function of TR, PR and M

- Pitzer, who used the acentric factor ω. Related to the volatility characteristics, this factorfrequently appears in data on pure substances. The properties of fluids are then obtained asa function of PR, TR and ω

- Hougen-Watson-Ragatz, who used the value of the critical compressibility factor ZC. Aseries of diagrams gives the properties of the fluids as a function of PR, TR and ZC.

Among these methods, the one that involves the acentric factor ω has undoubtedly found the largestnumber of applications.

b - Definition of the acentric factor

The definition of the acentric factor is based on the differences displayed by hydrocarbons in theirreduced vapor pressure curves.

For simple fluids, like methane, which obey the law of corresponding states with two parameters TRand PR, it is observed that the reduced vapor pressure is very close to 0.1 when the reducedtemperature is 0.7:

for simple fluidsPS

PC = 0.1 if TR =

TTC

= 0.7

The differences in behavior of simple fluids are characterized by the value of the acentric factor definedas follows:

ωωωω = – log10

PS

PC – 1

ω acentric factor

PS vapor pressure at TR = 0.7.

Clearly if PS

PC = 0.1, the acentric factor is zero, and consequently the law with two parameters suffices

to obtain the properties of the pure substances. In practice, as indicated by Plates A2, ratherdiversified values of the acentric factor are observed.


26

It may be advantageous to determine the value of the acentric factor from readily accessibleparameters like TC, PC and Tnb, the normal boiling point of the pure substance. It can be shown that agood approximation of ω can be obtained by the formula:

ω ≅ 37

log10 PCTCTnb

– 1 – 1

Tnb atmospheric boiling point in KPC in atmosphereTC in K

c - Application of the law of corresponding states with three parameters

The calculation of the compressibility factor, for example, using the law of corresponding states withthree parameters, TR, PR and ω, employs the following formula:

Z = Z0 + ωωωω . Z1

Z0 main term obtained as a function of PR and TR, on the chart of Plate I3, or from the table onPlate I3B,

Z1 correction factor given as a function of PR and TR by Plates I3’ or I3’ bisω acentric factor of pure substance, Plates A2.

For mixtures, the reduced pseudo-coordinates PPR and TPR can be calculated, and the acentricfactor is calculated for the mixture ωm:

ωωωωm = ∑∑∑∑i yi ωωωωi

ωi is the acentric factor of component i

yi is the molar fraction of i in the mixture

The law of corresponding states with three parameters also helps to determine the other properties ofreal fluids. The corresponding correlations are usually presented in the form of generalized chartsgiving, as a function of TR, PR and ω, for example, either the correction factors to be applied to theproperties of the fluid considered as an ideal gas (enthalpy, entropy, etc.) or direct values of certainproperties: vapor pressure, fugacity coefficient, etc.

Another method for solving the same type of problem, which is more appropriate for numericalcalculations on a computer, has developed rapidly in the last twenty years: the use of equations ofstate.


27

5 - CONCEPT OF EQUATION OF STATE

Equations of state are intended to represent the behavior of real fluids using a mathematicalexpression f(P,V,T,n) = 0.

The earliest is the Van der Waals equation of state (1872) which modifies the ideal gas law byintroducing two corrective terms:

- the ideal gas pressure is reduced to account for intermolecular bonding forces existing inreal gases. These slow down the molecules, which consequently generate a lower pressure.The corresponding correction factor, called the bonding pressure, is taken as a/V2, wherea is a constant that depends on the type of gas, and V is the volume of gas:

Preal gas = Pperfect gas – a

V2

or: P = Ppg – a

V2

- the ideal gas volume is increased by a factor b which accounts for the actual volume of themolecules of the real gas. b depends on the type of gas and is called the co-volume:

Vreal gas = Videal gas + b

The equation of state of ideal gases is written for one mole of gas:

Ppg . Vpg = R . T

It consequently becomes:

P +

aV2 (V – b) = RT which is Van der Waals equation of state (1872).

From the end of the nineteenth century until recently, many equations of state have appeared for acloser representation of the behavior of real fluids: Clausius (1880), Berthelot (1900), Benedict-Webb-Rubins (1940), Redlich-Kwong, etc. The latter, the Redlich-Kwong equation of state, appearedin 1949 and has a form quite similar to the Van der Waals equation:

P +

a

√ T V (V + b) (V – b) = RT Redlich-Kwong equation of state (1949)

Since its origin, this empirical equation has undergone several hundred changes intended to make itmore efficient. The latest forms (Soave-Redlich-Kwong and Peng-Robinson) are widely used in allcomputer programs for calculating the thermodynamic properties of fluids.

For a pure substance, parameters a and b can be determined from its critical conditions TC and PC:

a = 0.4278 R2 T

2.5C

PCb = 0.0867

RTCPC


28

Since a and b are determined, the equation of state also helps to calculate the critical compressibility

factor ZC = PC VCRTC

.

The value obtained, ZC = 1/3 is too high compared with those obtained by experiment, that usuallyrange between 0.27 and 0.29. This difference shows that the behavior of real fluids is inaccuratelyrepresented by this equation of state in the neighborhood of the critical point.

It is more convenient to use the equation of state by changing the variables.

By introducing the compressibility factor Z = P . VR . T the Redlich-Kwong equation of state is transformed

into a third-degree equation in Z which is often presented as follows:

Z = Z

Z – BP – A2

B . BP

Z + BP

In this equation, A and B are constants that depend on the type of substance and the operatingconditions:

A =

a

R2 T2.5 0.5

= 0.6542

T1.25R P

0.5C

B = b

RT = 0.0867TR PC

Starting with the expression of A and B, the factors A2

B and BP are given by:

A2

B = 4.933

T1.5R

BP = 0.0867 PR

TR

The resolution of this equation implies an iterative calculation which is initiated by assigning an initialvalue to Z in order to calculate the right-hand expression, i.e. to obtain a new value of Z which, ifdifferent from the previous one, is reintroduced into the equation until convergence is obtained.

The equation of state applies to gas mixtures using parameters Am and Bm defined as follows:

Am = ∑∑∑∑i yi Ai Bm = ∑∑∑∑

i yi Bi

where Ai and Bi are parameters A and B of component “i”, and yi is the molar fraction of i.


29

The Redlich-Kwong equation of state has undergone more than 700 changes since 1949. Among thenew forms most widely used are the following:

- the Soave-Redlich-Kwong equation of state (SRK, 1972)

P + a (T, ω)V (V + b) (V – b) = RT

in which the Redlich-Kwong term a

√ T is replaced by a function a (T,ω) depending on the

temperature and the acentric factor.

- the Peng-Robinson equation of state (PR 1976)

P + a (T,ω)

V (V + b) + b (V – b) (V – b) = RT

6 - USE OF EQUATIONS OF STATE

Equations of state not only offer an advantage in their ability to give the volume properties of fluids, butare in fact an excellent basic tool for determining:

- volumes or densities of the vapor and liquid phases: the correct representation of thevolume of the liquid phase is one of the advances achieved by the new equations of theSRK and PR type

- vapor pressures

- liquid/vapor equilibria by means of fugacity coefficients

- enthalpy of real fluids from the enthalpy of the same fluid considered as an ideal gas

- entropy by the same approach

As for the calculation of the thermodynamic properties, such as enthalpy and entropy, the expressionbelow shows the corrective term determined from Redlich-Kwong to be applied to the enthalpy of theideal gas:

H = H0 – RT

3

2 A 2

B ln

1 +

BPZ + 1 – Z

H enthalpy of real fluid,H0 enthalpy of ideal gas,A,B constants of RK equation.

There does not exist today a single equation of state capable of giving excellent results for all theseproperties and for all fluids. Good accuracy can nevertheless be anticipated in the specific validityrange of a proposed equation.


30

III - LIQUID-VAPOR EQUILIBRIA OF HYDROCARBON MIXTURES

1 - EQUILIBRIUM RANGE FOR A HYDROCARBON MIXTURE

An example of a liquid/vapor equilibrium range for a mixture of some hydrocarbons is shown in thepressure/temperature diagram below.

-50 00

5

10

15

20

50 100

D TH

025

B

10 30 50 70

90

100

%molar

C2C3iC4nC4iC5nC5

bubb

lecu

rve

Dew

curv

e

L VLS

VS

Temperature (°C)

Pres

sure

(atm

.)

L + V

— LIQUID — — VAPOR —0

82214241022

Starting with the liquid state on the left-hand side of the diagram and by following the change in themixture at constant pressure and rising temperature by continuous input of heat, the following stepsare observed:

- initial sub-cooled liquid state (point L)

- heating of liquid by input of sensible heat

- initiation of vaporization at the bubble point (or boiling point) of the mixture: the mixture isin the saturated liquid state (point LS)

- progressive vaporization by input of latent heat at rising temperature: the liquid and vaporphases are saturated and in equilibrium, and their composition changes throughoutvaporization

- end of vaporization at the dew-point of the mixture, which is in the saturated vapor state(point VS)

- this is followed by the superheated or dry vapor state at any temperature above the dew-point (point V)

The locus of the bubble points, (corresponding to the bubble point temperature and bubble pointpressure), defines the bubble point curve of the mixture. Similarly, the locus of the dew-points (dew-point temperature and dew-point pressure), characterizes the dew-point curve of the mixture.


31

These two curves appear to be approximately parallel to the vapor pressure curves in the low pressurezone. At higher pressure, they close the liquid/vapor domain by joining the critical point C of themixture.

The figure below, which applies to mixtures of ethane and heptane, shows complete ranges ofliquid/vapor equilibria. These are inscribed between the two vapor pressure curves of the puresubstances. They approach one or the other as a function of composition.

C2

C3

0 4 0 8 0 120 160 200 240 280

C

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

100

9 0

LIQUID-VAPOR EQUILIBRIUM DOMAINSOF MIXTURES OF ETHANE AND n-HEPTANE

Mixture 1 containing 96.8 mol % of ethane

n-heptane

C4

etha

ne

C

C1

Temperature °C

Atm

- p

ress

ure

Mixture 2 containing 77.1 mol % of ethaneMixture 3 containing 58.7 mol % of ethaneMixture 4 containing 26.5 mol % of ethane

D TH

116

B

It may be observed in a given range that the critical point C of the mixture located at the junction ofthe bubble point and dew-point curves of the mixture does not occupy a particular position.

It is generally not found at the maximum pressure point of the liquid/vapor equilibrium range(cricondenbar) nor at the maximum temperature (cricondentherm).

Note that the liquid/vapor equilibrium range of mixtures extends much further in pressure than that ofthe pure substances. It is limited by the bell curve which joins the critical points of the differentmixtures.


32

2 - FLASH OR LIQUID/VAPOR SEPARATION OF A HYDROCARBON MIXTURE

a - Liquid/vapor separation

A liquid/vapor separation or flash of a hydrocarbon mixture is achieved by carrying out a partialvaporization and then separating the liquid and vapor phases in a separator drum.

The vapor phase leaves at the top of the drum, and the liquid phase at the bottom.

P

t

Feed

composition zi Saturatedvapor

Saturatedliquidcomposition xi

composition yi

D TH

089

B

A V

L

The temperature and pressure maintained in the drum determine the vaporized percentage of themixture, and hence, considering the feed flow rate A, the vapor and liquid flow rates V and Lrespectively.

The liquid and vapor phases in contact at the same temperature and the same pressure are said to bein liquid/vapor equilibrium.

- the vapor phase is a saturated vapor at its dew-point- the liquid phase is a saturated liquid at its bubble point.

Both phases have different compositions:

- the molar fractions in the vapor phase are denoted yi- the molar fractions in the liquid phase are denoted xi

The relationship between the compositions of the two phases can be identified in terms of the vaporpressure of the liquid phase.


33

b - Vapor pressure or bubble point pressure of the liquid phase, Raoult’s law

The operating pressure P of the above vessel is the vapor pressure which is also called the bubblepoint pressure of the liquid mixture:

P = PSm

P operating pressure of drum

PSm vapor pressure or bubble point pressure of liquid mixture

This vapor pressure PSm is related to:

- the type and molar fraction of the components in the liquid phase- the operating temperature of the drum

For hydrocarbon mixtures of the same chemical family and at low pressure, it can be estimated by asimple calculation resulting from the application of the approximate Raoult’s law.

This postulates that the vapor pressure of the liquid is obtained from the specific vapor pressures ofthe components weighted by the molar fractions:

PSm = ∑i

PSi . xi RAOULT’S LAW

PSi is the vapor pressure of component i attemperature of drum

xi is the molar fraction of i in the liquidphase

Liquid mixtures which obey this law are said to be ideal solutions and this can be considered to applyto mixtures of hydrocarbons belonging to the same chemical family and at low pressure.

Varying deviations from the behavior of ideal solutions appear when:

- the mixture contains aromatics or hydrocarbons of very different sizes- polar compounds are present: CO2, H2S, water etc.

Using Raoult’s law, it is possible to determine the vapor phase composition and, using a liquid/vaporequilibrium coefficient, identify the differences in composition between the two phases.

c - Vapor composition and equilibrium coefficient

Raoult’s law helps to identify the contribution of each component present in the liquid phase to thevapor pressure of the liquid, i.e. to the operating pressure P.

For substance i, for example, this contribution is called the partial vapor pressure of i, andcorresponds to the partial pressure PPi of this component in the gas phase:

partial vapor pressure of i = PSi . xi = PPi


34

Dalton’s law, which is applicable assuming that the vapor phase is an ideal gas, serves to use yi, themolar fraction of i in the gas phase:

yi = PPiP hence yi =

PSi . xiP

This equation helps to determine the composition of the vapor phase from the calculation of the partialvapor pressures of each component of the liquid phase.

It is also advantageous to use the ratio yi/xi of the molar fractions of component i in the two phases,which characterizes the behavior of this component in the liquid/vapor separation.

This ratio is called Ki, the equilibrium coefficient (or K value) of component i.

The above equation helps to write:

Ki = yixi

= P

Si

P

Ki equilibrium coefficient of iyi molar fraction of i in the vapor phasexi molar fraction of i in the liquid phase

PSi vapor pressure of i at operating temperature

P total operating pressure

Ki characterizes the volatility of i in the liquid/vapor separation conditions. For ideal liquid phases andideal gas vapor phases, the above expression shows that Ki only depends on:

- the type of component i

- the temperature ( )through PSi

- the pressure P

This factor is extremely important in all calculations of liquid/vapor separation (flash, distillation,absorption, stripping etc.) because the practical approach to these problems always first requires thedetermination of the values of the equilibrium coefficients of the components concerned.

Many methods have therefore been developed for determining the values of these equilibriumcoefficients as accurately as possible. The simplest and also the most limited are graphic methods.

d - Simple graphic methods for determining equilibrium coefficients

Determining the equilibrium coefficients is only a simple problem for ideal solutions or mixtures whichdisplay behavior relatively similar to that of ideal solutions.

This applies to hydrocarbon mixtures of similar size and chemical family and at low pressure.

The methods applicable in this case include:

- Raoult’s law already presented Ki = P

Si

P valid at low pressure and for similar hydrocarbons

with similar volatilities,

- the Scheibel and Jenny nomograph (Plate J1), very practical to use and self-correcting inso far as it inherently limits the number and type of components: 18 components from C1 toC14

- the Maxwell “fugacity function” of which the diagrams (Plates J2) give the product P . Kias a function of temperature and pressure for the first eight paraffinic hydrocarbons.


35

Faced with a problem of liquid/vapor separation, access to the equilibrium coefficient helps to calculatethe flash conditions, and especially the compositions of the liquid and vapor phases. Yet this access isrelatively complex when deviations in behavior from ideal solutions appear.

The equilibrium coefficients also help to measure and compare volatilities of different componentsinvolved in liquid/vapor separations.

e - Volatility and relative volatility

Irrespective of the complexity of the method used to determine an equilibrium coefficient, its valueappears as an indicator of its volatility in the flash conditions:

Ki > 1 indicates a light or volatile componentKi < 1 corresponds to a non-volatile or heavy substance

In the main industrial separation processes based on differences in volatility, it is often advantageousto characterize the volatility difference existing between two components to be separated and thus togauge the difficulty of the process.

For ideal mixtures, the volatility difference between two substances i and j can be characterized fromthe ratio of their equilibrium coefficients.

This leads to the definition of a relative volatility of component i with respect to component j ααααij.

αij = KiKj

αij is the relative volatility of i with respect to j

If i is more volatile than j: Ki > Kj and αij > 1. If not, αij < 1.

This parameter is highly advantageous in distillation studies of hydrocarbon mixtures, because itremains approximately constant for all the operating conditions of a column:

αij = KiKj

=

PSi

P

PSj

P

= P

Si

PSj

By using Raoult’s expression, αij appears to be given by the ratio of the vapor pressures of i and j. It is

possible to conceive that, in a column, despite the variations in temperature, the ratio PSi /P

Sj remains

relatively constant. An average value can be selected corresponding, for example, to an intermediatetemperature in the column. This offers a qualitative indication of the volatility difference existingbetween the substances to be separated.


36

Applied to a number of common refinery and petrochemical separations, this method for determiningthe relative volatility gives the following results.

Component 1 Component 2 Operatingpressure αααα1-2 Industrial column

Propane Isobutane 17 bar 2.44 Depropanizern-butane Isopentane 11 bar 2.07 Debutanizer

Isopentane n-pentane 2.5 bar 1.23 DeisopentanizerMetaxylene Orthoxylene 1 bar 1.15 Aromatics

Note that the volatility difference between two substances is very sensitive to the pressure, and that itincreases when the pressure decreases for hydrocarbon mixtures. A separation of such mixtures bydistillation is hence always easier at lower pressure.

The table below shows the change in relative volatility of isopentane with respect to n-pentane atdifferent pressures. In each case, the temperature selected corresponds to the average of the boilingpoints of the two components.

Pressure in atm Averagetemperature Kic5 Knc5 ααααic5/nc5

0.2 – 10 1.17 0.76 1.541 32 1.13 0.85 1.33

2.5 62 1.07 0.87 1.235 91 1.07 0.92 1.16

10 120 1.04 0.93 1.09


37

3 - DEVIATIONS FROM IDEALITY - MODERN METHODS FOR DETERMININGEQUILIBRIUM COEFFICIENTS

a - Behavior of non-ideal solutions - Activity coefficient

Applied to a binary mixture a/b (a light component, b heavy component), Raoult’s law can be showngraphically by an isothermal diagram giving the vapor pressure of liquid mixtures a/b at constanttemperature as a function of their composition expressed in molar fractions xa and xb.

Raoult's law

Vaporpressure

PS

1

xa

0

PS

PS

a

m

b

b a

xm

Constant temperature

D TH

090

B

PSm = PSa . xa + PSb . xb

PSm = PSa . xa + PSb . (1 – xa)

PSm = (PSa – PSb) . xa + PSb

At a given temperature, this equation is that of a line which is based on the vapor pressures of the puresubstances:

if xa = 1 PSm = PSaif xa = 0 PSm = PSb

The line thus obtained helps to determine the vapor pressure of liquid mixtures a/b graphically as afunction of their composition xa.

Starting with this diagram, the demonstration of non-ideal behavior of a real solution is reflected bythe appearance of a deviation in the evolution of the vapor pressure.

As indicated in the figures below, positive or negative deviations are observed with respect to the vaporpressure values indicated by the Raoult’s law.


38

P

T

Vapor pressure o f mixtures

PSb

(heavy)

Positivedeviation

xa0

PSa

(light)

1

P

T

PSb

(heavy)

Negativedeviation

xa0

PSa

(light)

1

P

b a b a

P

Raoult's law

Raoult's law

D TH

091

B

These deviations indicate the extent of the molecular interactions in the liquid phase.

Raoult’s law reflects the forces of attraction existing in the liquid phase between the hydrocarbonmolecules of the same size and the same chemical family. The appearance of stronger forces ofattraction leads to greater cohesion of the liquid phase and a lower vapor pressure than the oneindicated by Raoult (negative deviation).

On the contrary, a higher vapor pressure than the Raoult figure indicates weaker forces of attractionbetween the molecules of the liquid phase (positive deviation).

The consideration of a non-ideal behavior corresponding to the deviations from Raoult’s law uses acorrection factor for the partial vapor pressures of each component. This factor, denoted γi, is called

the activity coefficient of component i. It depends on the temperature, the type and theconcentrations of the other components.

The vapor pressure of the liquid mixture is written then:

PSm = ∑∑∑∑i

PSi . xi . γγγγi

PSi vapor pressure of i componentxi molar fraction of i in the liquid phaseγi activity coefficient of i

PSm vapor pressure of the mixture


39

For a non-ideal solution in equilibrium with a perfect gas/gas phase, we can write:

yi = PPiP =

PSi . xi . γi

P

or

Ki = yixi

= PSi . γi

P

Ki equilibrium coefficientPSi vapor pressure of iγi activity coefficient of i

P total pressure

The determination of the equilibrium coefficient Ki therefore requires the knowledge of the activitycoefficient γi, whose value depends on the composition of the liquid phase. Ki then depends on i, T, P

and all the xi.

Depending on the type of mixture, different methods are proposed:

- for mixtures of non-polar compounds (e.g. hydrocarbons), the deviations are smaller andthe activity coefficients can be determined by the theory of regular solutions due toScatchard and Hildebrand. The calculation is then predictive. The recent equations ofstate also help to characterize the behavior of the liquid phase in this case

- for mixtures containing polar compounds, the deviations are much wider and the activitycoefficients can only be calculated from a minimum of experimental data.

The application of the different existing models is relatively complex. The most commonlyused are the following ones:

Models for determining the activity coefficients of non-ideal mixtures:

Wilson (1963) NRTL (1968)UNIQUAC (1975) UNIFAC (1975)

Note that the development of these methods designed to calculate the behavior of liquid mixtures ofpolar compounds is relatively recent.

Moreover, in addition to the deviations encountered in the liquid phase are those relative to the vaporphase. Consideration of these differences from ideal solutions and from ideal gases involves a specificthermodynamic quantity, the fugacity.


40

b - Concept of fugacity

A situation of liquid/vapor equilibriumillustrated by the figure opposite ischaracterized by the co-existence of twophases, separated by an interface, at thesame temperature and pressure.

Considering the movements of themolecules, equilibrium means thatmolecular transfers from one phase toanother occur in equal numbers.Intuitively, moreover, these transfersappear to be the result of two drivingproperties specific to the fluids: the vaporpressure of the liquid causes transfer inthe liquid/vapor direction, and the vaporphase pressure is the driving force oftransfer from vapor to liquid.

P

T

L

V

D TH

092

A

This concept of what occurs is perfectly justified for ideal solutions and ideal gases, but not for realfluids. For these, the equilibrium conditions require a definition of a corrected pressure which is reallythe effective pressure governing the liquid/vapor equilibria. Apart from ideal gases, this effectivepressure is different from the real pressure, and is called the fugacity (f). It is naturally expressed inpressure units and the condition of liquid/vapor equilibrium of the two saturated phases VS and LS iswritten:

fVS = fLS

Liquid/vapor equilibriumfugacity of saturated vapor = fugacity of saturated liquid

For a mixture at liquid/vapor equilibrium, the equilibrium condition is that the fugacities in the liquid andvapor phases must be equal for each component i:

fVi = f

Li for each component “i” of the mixture

Fugacity of i component in the vapor phase = fugacity of i component in the liquid phase

Based on this equality, it is possible to determine the equilibrium coefficient Ki of each component byexpressing the fugacities.

If the vapor phase can be considered as an ideal gas, and if the liquid phase is an ideal solution, wehave:

fugacity of i in vapor phase fVi = P . y i = PPi partial pressure of i in vapor phase

fugacity of i in liquid phase fLi = P

Si . x i partial vapor pressure of i

at liquid/vapor equilibrium fVi = f

Li for each component

thus: P . y i = PSi . x i


41

This returns us to the expression already obtained from the laws of Raoult and Dalton, which gives theideal equilibrium coefficient:

Ki = yixi

= P

Si

PKi ideal equilibrium coefficient

For the more general case of a real gas vapor phase and a non-ideal solution liquid phase, thefugacities of the components in each of the phases are expressed in a more complex manner.

c - Fugacity of component i in vapor phase

The fugacity of i in the vapor phase is obtained by correcting the foregoing expression and by

introducing a correction factor called ϕVi , the fugacity coefficient of component i in the vapor

phase. This gives:

fVi = P . yi . ϕ

Vi

ϕVi is the fugacity coefficient of component i. It depends on the temperature, pressure and

composition of the mixture. Its value naturally tends towards 1 as the pressure tends towardszero:

ϕVi → 1 when P → 0

A number of methods are used to determine the fugacity coefficient of i in the vapor phase. Theyinclude the following ones:

- the law of corresponding states which usually gives ϕVi as a function of TR and PR in

generalized graphic form, and possibly a third parameter.

- equations of state: by way of example, the expression below gives the value of ϕVi

obtained from the Redlich-Kwong equation of state:

ln ϕVi = (Z – 1)

BiB – ln (Z – BP) –

A2

B

2 AiA –

BiB ln

1 +

BPZ

A,B parameters of the mixture,Ai,Bi parameters of component i,Z compressibility factor of the vapor mixture.


42

d - Fugacity of component i in the liquid phase

Two main forms are available for expressing fLi , the fugacity of component i in the liquid phase.

- the first and the most classic consists in expressing the activity coefficient γi of

component i in the liquid phase.

At low pressure, we have the following expression already presented:

fLi = P

Si . xi . γi

If the pressure is higher, the vapor pressure PSi must be replaced by f

*Li , the fugacity of pure

component i in the liquid state in the conditions of liquid/vapor equilibrium. f*Li appears as

the corrected vapor pressure. In consequence:

fLi = f

*Li . xi . γi

The fugacity of pure component i in the saturated liquid state is obtained by:

f*Li = P

Si . ϕ*S

i . e

V*iL ( )P – PSi

RT

PSi is the vapor pressure of i at equilibrium temperature,

ϕ*Si is the fugacity coefficient of pure component i in the vapor state and at the

equilibrium temperature under pressure PSi

P is the equilibrium pressureV*iL molar volume of pure substance i in the saturated liquid state at the

equilibrium temperature

- the second uses the same formalism as for the vapor phase. A fugacity coefficient of

component i in the liquid phase ϕϕϕϕLi is accordingly defined.

Since the total pressure is P, and the molar fractions in the liquid phase are xi, we have:

fLi = P . xi . ϕ

Li

fLi fugacity of i in the liquid phase

P total pressure

ϕLi fugacity coefficient of pure i in the liquid phase


43

Since ϕVi , ϕL

i depends on the temperature, pressure and composition of the liquid phase, it

can be calculated from an equation of state capable of representing the liquid phase. The

expression giving the fugacity coefficient is of the same type as that which gives ϕVi , but the

compressibility factor used is that of the liquid phase. For hydrocarbon mixtures, the

calculation of ϕLi is obtained for example:

• by the SRK equation of state• by the Peng-Robinson equation of state

Using the explanation thus obtained for fugacities fVi and f

Li , it is possible to clarify overall

the contents of the different thermodynamic methods available for calculating theliquid/vapor equilibria.

e - Methods for determining equilibrium coefficients

Apart from a number of graphic methods already presented, the determination of the equilibriumcoefficients Ki requires thermodynamic calculations that are relatively complex, and are also capable ofpredicting the properties of the phases in liquid/vapor equilibrium. The methods used are oftenqualified as thermodynamic models. For the user of chemical engineering software, the problemoften faced is to select, from a range of thermodynamic models, the one which is appropriate to themixtures treated and the operating conditions. It is therefore important to know the general propertiesand fields of application of the main models used. Plate J4, ‘Choice of a thermodynamic model’,indicates the main possibilities available today in this area. Among these, the following can bedistinguished.

- MODELS OF CHAO AND SEADER (1961) AND OF GRAYSON AND STREED (1963)

The definition of the fugacities fVi = P . yi . ϕϕϕϕ

Vi

and fLi = f

*Li . xi . γγγγi

helps to write at equilibrium f Li = f

Vi , or:

Ki = yixi

= f

*LiP .

γi

ϕvi

The equilibrium coefficient in this correlation is calculated from three distinct parameters:

ϕϕϕϕvi fugacity coefficient of component i in gas phase, determined using the Redlich-Kwong

equation of state and the expression previously presented

γγγγi activity coefficient, calculated from the Scatchard-Hildebrand theory of regular solutions

f *Li fugacity of pure component i in the liquid state

RT . ln . γi = V*Li . (δi – δm)2


44

V*Li molar volume of pure liquid component i at 25°C

δi solubility parameter of component i defined by the expression below: it is calculated at 25°C

because its value varies little with temperature:

δi =

∆H

V25 – RT

V*L25

12

∆HV25 enthalpy of vaporization at 25°C

V*L25 liquid molar volume at 25°C

δm average solubility parameter calculated by volumetric weighting of δi

δm = ∑i

xvi . δi xvi volume fraction of i

f *LiP is obtained by the law of corresponding states with three parameters:

log f*LiP = log

f

*LiP

0

+ ωi log

f

*LiP

1

log

f

*LiP

0 = A0 +

A1TR

+ A2 TR + A3 T2R + (As + A6 TR + A7 T

2R) PR + (A8 + A9 TR) P

2R – log PR

log

f

*LiP

1 = – 4.23893 + 8.65808 TR –

1.22060TR

– 3.15224 T3R – 0.025 (PR – 0.6)

A0 to A9 are universal constants except for methane and hydrogen, for which special constants aredefined.

To summarize, the data on each component needed to make these correlations are limited to:

- critical temperature Tci- critical pressure Pci- acentric factor ωi

- enthalpy of vaporization at 25°C HiV25

- liquid molar volume at 25°C Vi*L25

This model’s few data requirements have helped to use it to deal with liquid/vapor equilibria ofmixtures of unidentified components: petroleum cuts, crude oils.


45

This is because these can be divided into pseudo-components characterized by their specific gravity at15°C, their tmav and their molecular weight. Using these data alone, correlations usually based on theproperties of pure substances of similar volatility help to assign values for the five parameters Tc, Pc,

ω, ∆HV25, v

*L25.

This makes it possible to calculate the equilibrium coefficients of these pseudo-components, and thusoffers predictive models capable of calculating the liquid/vapor equilibria of mixtures of identifiedhydrocarbons and petroleum cuts.

The range of validity of the Chao and Seader model was limited to 500°F (260°C) and 2000 psia(140 bar). It was extended by Grayson and Streed in 1963 to 900°F (482°C) and 8000 psi (560 bar).

The accuracy of this method, has been evaluated by different authors at a relative mean error of 8% inthe equilibrium coefficients. But it deteriorates considerably if the reduced temperature of certaincomponents exceeds the value of 1.3 or if the mixture contains H2S and CO2.

- THE SOAVE-REDLICH-KWONG MODEL (SRK) (1972)

In this recent model, the fugacities of component i in the two phases are expressed as follows:

f Vi = P . yi . ϕϕϕϕ

Vi

f Li = P . xi . ϕϕϕϕ

Li

At equilibrium: f Li = f

Vi

hence: Ki = yixi

= ϕL

i

ϕVi

In this case, both fugacity coefficients are calculated from the same equation of state, which isthe Redlich-Kwong equation improved by Soave. This equation introduces interaction parameters kijbetween the components of the mixture, which are adjusted to experimental data. The parameters kijoffer a better representation of non-ideality.

For hydrocarbon mixtures belonging to the same chemical family, the kij can be taken as zero, but thisdoes not apply in the presence of hydrocarbons of different families, H2S, CO2 etc.

Even in the absence of these coefficients kij, this new method is considered superior to the method ofChao and Seader. It helps to deal with pseudo-components, but it remains limited to non-polarcompounds.

It also offers the advantage of dealing better with liquid/vapor equilibria at high pressure.


46

- THE PENG-ROBINSON MODEL (1976)

Similar to the SRK model, the Peng-Robinson model is derived from the Redlich-Kwong equation ofstate. It offers greater accuracy in predicting the densities of the liquid phase.

- OTHER MODELS

Plate J4 shows other models capable of determining the equilibrium coefficients of hydrocarbons froman equation of state. The Benedict-Webb-Rubin model, which has eight parameters, is ideal forcalculating liquid/vapor equilibria at low temperature (t < – 30°C).

The “sour water” model applies to mixtures of H2S, NH3, CO2 and H2O. It applies to process waterstrippers, and is essentially based on experimental data.

Models that represent the liquid phase (activity coefficients γi) are indispensable if the problem is

concerned with mixtures of polar compounds. The models already presented (Wilson, NRTL,UNIQUAC etc.) are difficult to use, and demand the availability of binary experimental data or a databank. The Wilson model is restricted to the representation of liquid/vapor equilibria. The NRTL andUNIQUAC models can also deal with liquid/liquid equilibria.

Note also that, in the absence of experimental data, use can be made of the semi-predictive UNIFACmodel, which is based on set contributions.

4 - GENERAL PRINCIPLES OF CALCULATING LIQUID/VAPOR EQUILIBRIA

In the general case of the calculation of aflash: bubble point, dew-point, partiallyvaporized mixture, the following data areavailable:

- feed flow rate A,

- feed composition Zi (n components),

- model for determining equilibriumcoefficients Ki as a function oftemperature t, pressure P and, ifapplicable, vapor and liquid phasecompositions yi and xi respectively.

P

tA

V, yi

L, xi

zi

D TH

093

A


47

EQUATIONS GOVERNING THE SYSTEM

Material balance equationsOverall balance A = L + V 1 equationBalance on one component i A . zi = L . xi + V . yi n equations

Normation equation ∑i xi = 1

of or 1 equation

compositions ∑i yi = 1

Liquid/vapor equilibrium equations yi = Ki . xi n equations ————————— 2 n + 2 equations

UNKNOWN VARIABLES

Concentrations xi and yi 2 unknown variablesFlow rates V and L 2 unknown variablesPressure P and temperature T 2 unknown variables

————————— 2 n + 4 unknown variables

The system is hence defined if two parameters are fixed. The most common cases are describedbelow.

Different kinds of problem to be solved:

- bubble point temperature at a given pressure- dew point temperature at a given pressure- flash at a given pressure and temperature

a - Calculation of the bubble point of a mixture at a given pressure

Data:P imposed

bubble point V = 0)

In this case: xi = zi ∀i (the feed is liquid)

hence: yi = Ki . zi with ∑i yi = 1

The system to be resolved is thus reduced to:

∑i Ki . zi = 1


48

The trial and error method is implemented.

The following resolution flow chart applies when Ki only depends on the pressure and temperature(ideal mixtures).

Hypothesis on t Calculation of Ki Calculation of ΣKi • zi t = t bubble= 1

≠ 1

D T

H 2

061

A

To account for the influence of vapor and liquid phase compositions on Ki, it is necessary to add aninternal redefinition loop of the Ki after calculating the compositions.

Hypothesis on t Calculation of Ki Calculation of ΣKi • ziΣyi = 1Σyi = Ct

Σyi 1

Σyi varies

t = t bubble

D TH

206

1 B

b - Calculation of the dew-point of a mixture at a given pressure

Data:P imposed

dew-point L = 0

In this case: yi = zi (the feed is vapor)

hence: zi = Ki . xi or ziKi

= xi with ∑i xi = 1

The system is reduced to:

∑i

ziKi

= 1

The trial and error calculation is represented on the following flow chart:

Hypothesis on t Calculation of Ki Calculation of Σ zi = 1

≠ 1

t = t dew

D TH

206

1 C

Ki


49

As above, a complementary loop redefining the Ki after calculating the compositions is necessary if Kiis a function of xi and yi.

c - Calculation of a flash

Data:flash pressure P

flash temperature t

The flow rates and compositions of two phases are calculated:

A . zi = V . yi + L . xi

yi = Ki . xi

xi = Azi

L + V . Ki

and ∑i xi = 1 = ∑

i

A . ziL + V . Ki

If Ki only depend on P and t, their value can be determined and the following trial and error methodimplemented:

Hypothesis on L V = A - L Calculation of xi= 1

≠ 1

Calculation of Σxi

D TH

206

1 D

L solution

If the Ki depend on P, t, xi and yi, an additional loop is necessary:

Hypothesis on L V = A - L Calculation of Ki= 1

L solution

1

Calculation of xiand yi

D TH

206

1 E

Σxi = Ct

Σxi varies


50

5 - ISOBARIC EQUILIBRIUM DIAGRAM OF BINARY MIXTURES OR EQUILIBRIUMLENS

a - Construction of the equilibrium lens

The equilibrium lens of a binary mixture is an isobaric diagram which gives the equilibriumtemperatures (bubble point and dew-point) of a binary mixture as a function of its composition.

This diagram is easily obtained by graphic construction from the liquid/vapor equilibrium domainspresented below for different mixtures of two components a and b, where a is the volatile componentand b is the heavy component.

Temperature (°C)

Pressure (atm)

2 Mixture number

Bubble point curve

Dew point curve

Mixtures 90% a1

10% b70% a

230% b50% a

50% b30% a

470% b10% a

590% b

P

bp a Bubblepoint ofmixture

Dewpoint ofmixture

3 3

bp b

VP c

urve

of s

ubst

ance

a

1

12

32

3 4

45

3

5

D TH

026

B

VP c

urve

of s

ubsta

nce

b

An horizontal plotted for the pressure selected P gives the equilibrium temperatures of the differentmixtures, and particularly the extreme equilibrium temperatures which correspond to the boiling pointsof the two pure substances. Plotted on a composition/temperature graph, these temperatures revealtwo sets of points defining two curves which make up the equilibrium diagram of mixture a/b underpressure P.


51

The upper curve connects the dew-points and the lower curve connects the bubble points.

— LIQUID —

— VAPOR —

P

0 1 0 3 0 5 0 7 0 9 0 100

5 4 3 2 1Mixtures

Boiling pointof b

Boiling pointof a

Temperature Temperature

ab

D TH

094

B

This diagram is very useful, particularly in distillation, because, at constant pressure, it helps todetermine the relationships between the compositions of the saturated phases (liquid or vapor)and their temperature.

If the equilibrium coefficients Ka and Kb of the two components are available at pressure P and inthe temperature interval between the boiling points tbp.a and tbp.b, the lens can be generated asfollows:

- choice of a temperature t tbp.a < t < tbp.b

- determination of Ka = yaxa

and Kb = ybxb

at t and P

- determination of liquid composition at its bubble point at temperature t:

Ka xa + Kb xb = 1 (bubble point)with xa + xb = 1hence Ka xa + Kb (1 – xa) = 1

xa = 1 – Kb

Ka – Kb

- determination of vapor composition in equilibrium with the liquid:

ya = Ka xa

This calculation can be repeated for different temperatures until a sufficient number of points has beenobtained to plot the two curves.

The use of this diagram can be illustrated by the study of a flash.


52

b - Flash of a binary mixture

The representation of a flash is shown below.

z, x, y

x

L + V

LIQUID

VAPOR

t (°C)

P

0 1

t

t (°C)

x

LVA

P

t

y

x

z

V

L

A

z y

ab

t (°C)

P

0 1

t

t (°C)

LVA

P

t

y

x

z

V

L

A

z y

ab

D TH

095

B

Boilingpoint of b

Boilingpoint of a

Compositions z, y and x are fractions or molar percentages of volatile component a.

Point A representing the feed of composition z is located in the liquid/vapor domain at temperature t.The two phases in liquid/vapor equilibrium at temperature t are at their bubble and dew-pointsrespectively. This helps to position points L and V and to determine the compositions x and y ofthese two phases. The difference between the values of y and x illustrates the separation made by anequilibrium stage and materializes the selectivity of the operation.

The vapor flow rate V and liquid flow rate L can be obtained from the compositions. The materialbalance of the flash drum is written:

Overall balance A = L + V

Balance on “a” Az = Lx + Vy

Or Az = (A – V) x + y

Hence V = A . z – xy – x and L = A – V

The application of this formula is known by the name of the “lever rule” or “rule of inverse segments”.Geometrically, in fact, (z - x) represents segment AL and (y - x) represents segment LV:

V = A . ALLV L = A .

AVLV


53

To facilitate the analysis of the operation of distillation columns, it is also advantageous to representthe molar enthalpy diagram of binary mixtures. The standard representation corresponding to anideal mixture is shown below.

By using ha and hb to denote themolar enthalpies of a and b in thestate of pure liquid substances attemperature t, and Ha and Hb themolar enthalpies of the samecomponents in the vapor state, themolar enthalpies HV and hL of theliquid and vapor phases inequilibrium can be obtained by theequations:

HV = Ha y + Hb (1 – y)hL = ha x + hb (1 – x)

This calculation, which assumesideal behavior of the mixture, can berepeated for different compositionsof the saturated liquid and vaporphases. The result is an upper curvegiving HV as a function of y and alower curve giving hL as a function ofx.

This diagram shows Λa and Λbwhich are the molar heats ofvaporization of substances a and bunder pressure P. It may beobserved that, for an ideal mixture,the molar enthalpy curves of the twophases approach two parallels and,in consequence, the molar heats ofvaporization vary only slightly withthe composition.

Temperature

P

t

L + V

LVA

A

V

Ha

Hv

a

b

L

hb

ha

Hb

0

h L

x, y, z 1

molarenthalpy

ab

x z yD

TH 0

96 B

Boilingpoint of b

Boilingpoint of a

The oblique line joining the two phases in equilibrium is usually represented by an arrow, and is calledthe tie line.

The relative constancy of molar heats of vaporization is important in distillation columns. Indeed, ontheir trays we observe a partial condensation of the vapor flow and partial vaporization of the liquidflow. These phenomena concern mixtures of different compositions but with similar molar heatvaporization.

Consequently, molar flowrates concerned by vaporization and condensation are roughly equal whichleads, finally, to a certain constancy of the molar flowrates of saturated liquid and vapor in the industrialdistillation columns.


54

In general, the shape of the equilibrium lens shows the selectivity of the phase equilibria, and thediagrams corresponding to two components of similar volatilities or, on the contrary, very differentvolatilities are shown below.

t

P

t

t

P

t

α high compared to 1α close to 1

x z y x z y

D TH

097

B

For the same ideal hydrocarbon mixture, the influence of pressure in differences in volatility leads tothe lenses below. The equilibrium temperatures are obviously shifted with the pressure, and asimultaneous drop in selectivity of liquid/vapor equilibria which accompanies an increase in pressure issimultaneously observed.

t (°C)

y3

y2

y1

1x0

P1

P2

P3

mol % of component ain mixture

P1 < P 2 < P 3

t1

t2

t3

b a D TH

098

B


55

c - y/x diagram

To illustrate the selectivity of liquid/vapor equilibria, it is also advantageous to represent the vaporphase composition y at a given pressure as a function of the liquid phase composition x. The diagramobtained is a square diagram. Since the values of y and x are conventionally related to the volatilecomponent, y is greater than x and the corresponding curve is located above the diagonal of thesquare which corresponds to equation y = x.

The construction of this diagram from the lens is shown below. The y/x diagram is the basis of anumber of methods of graphic calculation of distillation columns for binary mixtures. It is usedparticularly in the MacCabe and Thiele method.

Temperature

1000x 1 x 2

Compositionof liquid

Pure heavycomponent

Compositionof vapor

100

y 2

y 1

Equilibriumlens

Pure lightcomponent

y - x Diagram

t 2

t 1

P

P

x 1y 1 x 2 y 2

t

y :

x :

y = x

D TH

099

B


56

IV - LIQUID/VAPOR EQUILIBRIA OF MIXTURES OF UNIDENTIFIED COMPONENTS

Mixtures of unidentified components are petroleum cuts, petroleum products, crude oils or pyrolysis cuts.Their components are so numerous and similar in terms of volatility that it is impossible to identify them bystandard analytical methods like chromatography. These products are also sometimes qualified as ‘complexcontinuums’.

1 - CHARACTERIZATION OF THE VOLATILITY OF PETROLEUM CUTS AND CRUDEOILS

Chromatographic analysis helps to identify the lightest components up to the C7 to C10 hydrocarbons.As a rule, analyses of crude oils stop at C5. Two standard methods are available beyond this level tocharacterize the volatility of a cut: TBP distillation, a rigorous, lengthy and costly analysis, and ASTMdistillation, a rapid routine quality control analysis.

In recent years, simulated distillation, which helps to determine the “distribution of the boilingranges of petroleum fractions by gas chromatography” appears capable of providing a morerigorous approach in determining the composition of petroleum cuts.

a - TBP (True Boiling Point) distillation

This is a batch distillation which exploits a high separating power obtained by using a large numberof contact stages and a high reflux rate.

Standard ASTM D 2892 describes a TBP distillation method called 15/5 because the distillationcolumn has 15 theoretical contact stages and the operating reflux ratio is maintained at 5.

The analysis consists in distilling about 10 liters of product or crude oil in this column and recording thecorrespondences between the quantities collected (mass and volume) and the temperatures observedat the top of the column, which is illustrated below.

The operation is performed first at atmospheric pressure and then under vacuum to prevent thermaldegradation of the hydrocarbons.

The temperatures observed under vacuum are corrected and adjusted to atmospheric pressure.


57

Solenoid

Water on condenser

Vent

Valve

Condenser

Thermocouple

Heating circuitcompensation

FlaskHeatingcontrol

Thermocouple

Packed column

Vent

To vacuumpump

D AN

A 09

5 B

- 15 à 18 theoretical contact stages- reflux ratio = 5

T.B.P. DISTILLATION principle

Due to the separating power employed, the components of the product analyzed are relatively wellfractionated in this test and proceed towards the top of the column by order of volatility. The toptemperature therefore reflects the boiling points of the components which reach the top of the columnsuccessively.

One of the test results is the TBP distillation curve, which gives the correspondence between volumeor mass collected and boiling point. It thus indicates the composition of a cut or a crude oil in the senseof the volatility of its components.


58

The curves below show the shape of the TBP curves for a complex continuum and for a binary mixture.

Temperature °C

% distilled

Temperature °C

m % distilled

1 atm.

Temperature °C

ab0 1000 100 D

TH 2

021

B

TBP distillation curve

tbpb

tbpa

TBP curve Equilibrium lens TBP curvefor a complex cut under P = 1 atm of mixture m

for mixture a – b

b - ASTM distillation

ASTM distillation is a batch distillation without reflux using about one liquid/vapor equilibrium stage.

For light products, Standard ASTM D 86 is used, which corresponds to atmospheric distillation. Forheavy products, vacuum distillation is performed to prevent thermal degradation according to StandardASTM D 1160.

In terms of significance, the ASTM method resembles the simple progressive vaporization SPV,which is a theoretical batch distillation strictly comprising one theoretical stage.

The SPV analysis shows that:

- its initial point is logically the bubble point of the product- its final point is the boiling point of the heaviest component of the mixture

In actual fact, the following differences are observed with ASTM:

- the ASTM IP is related to the bubble point of the product, but is lower than it because ofthe initial rectification that occurs following the generation of a reflux by the cold columnduring temperature conditioning

- the ASTM FP is much lower than the boiling point of the heaviest components, becausedistillation is generally incomplete, and a non-distillable residue usually remains in the testconditions

Note that the ASTM IP is affected by the overall composition of a cut, whereas the ASTM FP mainlydepends on the heaviest components of the mixture.


59

c - Simulated distillation

The simulated distillation of a petroleum cut gives correspondence between the percentage distilledand the boiling point by a gas chromatography analysis.

The chromatographic column actually separates the hydrocarbons of the sample analyzed according totheir volatility. This column is calibrated by the injection of a mixture of n-paraffins with knownboiling points, which help to make a temperature graduation of the chromatogram obtained.Measuring the areas between the peaks of the n-paraffins makes it possible to determine therelationship between the mass percentages and the boiling points.

C22

C24C23

C25

C26

C30

C27

C28

C29

C32

C34

C36

C38

C40

C42

1 µl cut 370-535°C

Signal intensity

Elution timecorresponding to

boiling temperatures

Example of result 370-535°C cut

D TH

202

9 B

Two ASTM methods using this principle exist:

- D 2887 available for petroleum cuts with boiling points lower than 538°C- D 3710 available for lighter cuts (< 250°C)


60

2 - VAPORIZATION CURVE

The vaporization curve, which is also called the “Flash Curve”, gives the percentage vaporized as afunction of temperature for a given cut and at a given pressure.

Widely used in simplified design methods for draw-off columns, the flash curve is characterized by itstwo extreme points:

- the initial point or point 0% vaporized, which is the bubble point of the mixture- the final point or point 100% vaporized is the dew-point of the mixture

FC at P

FC atm

P t(°C)0% 100%

1 atm

tbubble

tbubble

tdew

tdew

t

t

0 100% vaporized D AN

A 20

39 B

The atmospheric flash curve corresponds to the normal pressure of 1 atm.

Compared with the ASTM and TBP curves, this curve is flatter as shown by the figures below for acomplex cut and for a binary mixture.

% distilledor vaporised

Temperature °C

m % distilledor vaporised

Temperature °C

b a

TBPASTM

FC

TBPASTM

FC

0 1000 100

— Complex cut — — Binary mixture — D TH

202

2 A

The flash curve naturally shifts with the pressure. It is important because, prior to the existence of therecent thermodynamic models, it offered a relatively simple means of dealing with problems ofliquid/vapor equilibria in complex continuums.

Indeed, determination of crude oils and atmospheric residues flash curves is necessary in shortcutcalculations of atmospheric distillation and vacuum distillation.


61

3 - METHODS FOR CALCULATING LIQUID/VAPOR EQUILIBRIA OF MIXTURES OFUNIDENTIFIED COMPONENTS

a - Breakdown into pseudo-components

The modern approach of these calculations makes use of the breakdown of the mixture analyzed intopseudo-components whose physical properties can be defined sufficiently as inputs in computerizedmethods for calculating liquid/vapor equilibria.

This method requires the availability of a TBP distillation curve of the mixture. The fine breakdowngives the characteristics of small range cuts (2 or 3°C width or 2%). Then the breakdown policy oftenconsists in defining the pseudo-components corresponding to a constant temperature interval ∆∆∆∆t(e.g. 10, 20 and 30°C). Their characteristics are obtained by weighting the properties of the smallrange cuts.

The figure below locates the pseudo-component i of which the following can be defined:

- mass concentration in the mixture- the weighted average boiling point which can be treated as tmav for narrow cuts

% weight distilled

Temperature °C

∆t

% weight

pseudocomponent i

Definedcomponents

t mavi

D T

H 2

023

A

TBP

distill

ation

curv

e

At the same time, if a curve giving the specific gravities of the small range cuts is available, it is

possible to determine the d154 of pseudo-component i.

Pseudo component i tmavi boiling temperature

zi mass concentration

d154i specific gravity at 15°C


62

Using these basic data, the values of the different parameters needed for the thermodynamic modelsmust be generated. Taking the case of the Chao and Seader model, the calculation of the equilibriumcoefficient Ki of pseudo-component i requires:

TCi critical temperaturePCi critical pressureωi acentric factor

V25 liquid molar volume at 25 °C∆H25 enthalpy of vaporization at 25 °C (or solubility parameter δi)

The basic data required to characterize each pseudo-component are:

- weighted average boiling point treated as tmav- specific gravity

d1515 or Spgr 60

60 = 1.00096 d154

or °API = A = 141.5

Spgr6060

– 131.5

- and possibly the molecular weight M, which can be determined as follows:

M = 5.805 x 10–5 t2.3776mav

spgr0.9371tmav in K

Critical coordinates can be obtained by:

- correlations using the Winn chart:

TC = exp (4.2009 tmav0.08615 Spgr0.04614)

1.8 tmav

TC in K

PC = 6.1483 . 1012 Spgr2.4853

tmav2.3177PC in Pa

- Cavett’s correlation, which uses a set of universal constants a0 to a6 and b0 to b7:

TC = a0 + a1 tmav + a2 t 2mav + a3 t

3mav + a4 A tmav + a5 A t

2mav + a6 A2 t

2mav

PC = b0 + b1 tmav + b2 t 2mav + b3 t

3mav + b4 A tmav + b5 A t

2mav + b6 A2 t

2mav

- Lee-Kesler method (1975)

TC = 189.8 + 450.6 spgr + tmav (0.4244 + 0.1174 spgr) + (14410 – 100688 spgr)

tmav

ln PC = 5.68925 – 0.0566spgr – 10–3 tmav

0.436392 +

4.12164spgr +

0.213426spgr2

+ 10–7 t2mav

4.75794 +

11.819spgr +

1.53015spgr2 – 10–10 t

3mav

2.45055 +

9.901spgr2


63

The acentric factor ω of which the definition is:

ω = – log10

PS

PC TR = 0.7

– 1

may be placed in the following form in order to obtain ω from TC, PC and tmav:

ω = 37 .

log10 PCTC

tmav – 1

PC in atmTC and tmav in K

ω can also be determined by the Lee-Kesler equation (1975) with:

PSbR reduced saturation pressure (at normal boiling point PS = 1 atm)TbR reduced normal boiling point = tmav/TC

ω = ln P

SbR – 5.92714 + 6.09648/TbR + 1.28862 ln TbR – 0.169347 T

6bR

15.2518 – 15.6875/TbR – 13.4721 ln TbR + 0.43577 T6bR

The liquid molar volume at 25°C is given by:

liquid molar volume V25 = Molecular weightDensity at 25°C =

Mρ25

ρ25 = 0.98907 Spgr 6060

M obtained by the chart of the characterization factor or by the formula:

M = 42.965 [ ]exp (2.097 . 10–4 tmav) – 7.78712 spgr + 2.0847610–3 . tmav . spgr ( )t1.26007mav . spgr4.98308

This formula can be applied to pseudo-component with a density at 15°C less than 0.97 and a boilingpoint less than 840K.

Other equations established by Lee Kesler (1975) are given in literature.

The enthalpy of vaporization at 25°C ∆∆∆∆HV25 is calculated using Kistiakowski’s formula which gives

∆HV for the normal boiling point:

∆HVtmav

= 7.58 + 4.571 ln (tmav)

tmav in °R

Watson’s formula then gives the variation in ∆HV with temperature:

∆HV25

∆HVtmav

=

TC – 537

TC – Tmav

0.38TC in °R


64

The solubility parameter δi can then be calculated by the equation:

δi =

∆H

V25 – RT

V25

12

The work done for each pseudo-component transforms a product with an infinite number of unidentifiedcomponents into a perfectly defined mixture of pseudo-components which can be treated like puresubstances by chemical engineering software.

Yet the results thus obtained can only approach reality and require very close attention on the part ofthe user.

Before the advent of these modern methods, the treatment of the liquid/vapor equilibria of the cutsessentially made use of the vaporization curves. Many statistical correlations were developed in thepast to obtain them, and several of them are still equally valid today.

b - Statistical correlations

Statistical correlations are generally presented in graphic form. Some of them are digitized. They helpin particular to:

- switch from TBP to ASTM and vice versa- switch from TBP or ASTM to the atmospheric flash or vacuum flash curve

Simultaneously, other correlations help to correct the flash curves as a function of pressure, and even,in a liquid/vapor equilibrium, to determine the properties of the saturated phases.

We shall now describe some of these correlations.

PASSAGE FROM TBP TO ASTM

• Edmister correlation

This correlation is very widely used, particularly in the following cases:

- conversion of the TBP portions of the crude oil corresponding to the desired distillation cutsinto ASTM, in order to characterize the products.

- determination of the TBP of a cut when the ASTM is available.

- use of computer distillation calculations which give the compositions of the cuts in the formof compositions of pseudo-components, each of them corresponding to a boiling point tmav.Using these data, it is therefore possible to define a TBP, which can then be transformedinto ASTM if necessary. This use entailed the digitization of the correlation.

Plate J9 from ring-binder 0 shows the curves used.

The principle of the method consists of the following:

- first, positioning the 50% point of the required curve from the 50% point of the known curve(bottom right-hand part of the diagram)

- secondly, correlating the temperature differences 0 to 10°C, 10 to 30°C, 30 to 50°C etc. ofthe known curve with the corresponding differences in the desired curve

- thirdly, constructing the curve from the 50% point by adding or deducting the calculateddifferences


65

• Riazi method

This method published by API, allows to transform a ASTM D86 curve into a TBP curve with thefollowing relation:

TTBP = a TbASTM

a and b are parameters linked to % distilled volume.

% distilled volume a b

0 0.9177 1.0019

10 0.5564 1.0900

30 0.7617 1.0425

50 0.9013 1.0176

70 0.8821 1.00226

90 0.9552 1.0110

95 0.8177 1.0355

On the contrary, ASTM D86 curve can be obtained from TBP curve:

TASTM = a’ Tb’TBP

With following a’ and b’ values.

% distilled volumeor vaporized volume a’ b’

0 1.08947 0.99810

10 1.71243 0.91743

30 1.29838 0.95923

50 1.10755 0.98270

70 1.13047 0.97790

90 1.04643 0.98912

95 1.21455 0.96572


66

PASSAGE FROM TBP OR ASTM TO THE ATMOSPHERIC VAPORIZATION CURVE

Two main methods are available for this transformation:

- the J.B. Maxwell method is presented on Plate J10- the Edmister method is presented on Plate J11

Maxwell’s correlation is based on the different slopes of the distillation curves and the flash curve. Itconsists successively of the following:

- defining a DRL line (distillation curve reference line) from the 10 and 70% points of the TBP

TBPreal → DRL

- plotting an FRL line (flash curve reference line) from the correlations giving the following asa function of the 10 to 70 slope of the DRL:

• the slope of the FRL• the ∆t 50 (DRL - FRL), i.e. t50FRL

DRL → FRL

- from the differences read for each percent distilled between DRL and TBP (∆tD),determining the differences ∆tF between FRL and atmospheric flash curve: the graphiccorrelation shows that, except for the low percentages distilled, ∆tF is one-third of ∆tD,

∆tD → ∆tF

- finally plotting the atmospheric flash curve.

FRL → FCATM

Edmister’s correlation uses a similar approach to the one for passage from TBP to ASTM.


67

SHIFT OF THE FLASH CURVE WITH PRESSURE

The figure below shows the characteristic shape of the liquid/vapor domain of a petroleum cut shownon a Cox Chart. The ‘iso percentage vaporized’ curves are lines approximately parallel to the vaporpressure lines of the hydrocarbons, but they converge at high pressure at a point F. This point only hasgeometric significance, and is called the focal point. It is obviously different from the critical point C ofthe mixture.

5004003002001501005 01 00,1

0,2

0,40,60,81

2

468

1 0

2 0

4 06 08 0

100

Critical pointsof paraffins

0% bubble point curve

305 0

70

90 100%

dew

poi

nt c

urve

Pressureatm

F

C

Temperature °C

10

D TH

117

B

Two main methods are used to shift the vaporization curve.

- the first consists in converting the atmospheric curve by the 50% point on the CoxChart. By plotting a line from the 50% part of the flash curve, interpolated in the network ofvapor pressure lines, one can determine the variation in temperature undergone by thispoint as a function of pressure. The same variation is then applied to all the points of thecurve. This method is valid for a slight variation in pressure

- the second is based on the positioning of the focal point which, once in place, helps toplot the network of lines corresponding to the different percentages vaporized, and thus toobtain the vaporization curve at any pressure.

To position the focal point, J.B. Maxwell proposed interpolating a line in the network of vaporpressure lines of the Cox Chart. This lines passes through the 40% point of theatmospheric vaporization curve (temperature 40 °C, pressure 1 atm) and is prolongedover a temperature interval of 83 °C beyond the intersection with the locus of the criticalpoints of the paraffins. The terminal point obtained is the desired focal point.


68

V - LIQUID/VAPOR EQUILIBRIA OF NON-IDEAL MIXTURES

1 - RELATIONSHIP BETWEEN LIQUID PHASE PROPERTIES AND VAPORIZATIONBEHAVIOR, DEFINITION OF THE AZEOTROPY

a - Pure substances

The behavior of a pure substance in liquid/vapor equilibria is determined primarily by the properties ofits liquid phase, of which the cohesion is ensured by intermolecular forces. These intermolecularforces are mainly due to the electron movements within the molecules which give rise toelectromagnetic fields.

The molecules consequently undergo forces of mutual attraction of which the intensity variesaccording to the type of the molecule and the conditions in which the molecules are found.

These forces act at short distance and decrease very rapidly as the molecules separate from eachother. They have little effect on the behavior of gases, but are decisive for liquids, whosevaporization behavior they influence.

Strong forces of attraction reflect good cohesion of the liquid phase. In terms of liquid/vaporequilibrium, this gives rise to:

- a low vapor pressure: the molecules are ‘retained’ in the liquid phase- a high boiling point necessary to overcome these forces during vaporization

By way of example, this makes it possible to explain the following:

- the progression of boiling points and hence of vapor pressures of normal paraffinichydrocarbons with the number of carbon atoms, because the forces of attraction are greateras the chain length increases:

tboiling (°C) tboiling (°C) tboiling (°C)

C1 – 161.5 C12 217 C24 403C4 – 0.5 C16 287 C28 440C8 + 126 C20 350 C34 492

- the differences in boiling point and vapor pressure between n-paraffins and iso-paraffins:

nC4 – 0.5°C nC5 + 36°CiC4 – 11.8°C iC5 + 28°C

Branched chain hydrocarbons generally have lower boiling points and higher vaporpressures, because the cohesion forces of the liquid phase are weaker.

This is due to the branches of the chains, which hinder the approach of the moleculestowards each other and thus limit the effect of the forces of attraction by a distance effect.


69

- the abnormally high boiling point of water:

H H

O

+ +

—

The water molecule is in fact a polar molecule. The shift of theelectrons in the molecule from the hydrogen atoms to the oxygenatoms gives rise to a genuine electric dipole.

This polarity brings about bonds in the liquid phase between the hydrogen atoms of onemolecule and the oxygen atoms of the neighboring molecules.

These intermolecular bonds, called hydrogen bonds, confer an extraordinary cohesion toliquid water, which therefore has a very low vapor pressure and very high boiling point withrespect to the size of the molecule.

b - Binary mixtures

The vaporization behavior of binary mixtures is conditioned by the extent of the interaction forces whichappear between the mixed molecules in the liquid phase. Between two neighboring hydrocarbons ofthe same chemical family, the forces are observed to evolve regularly as a function of the molecularproportions between those corresponding to one of the pure substances and those corresponding tothe other. This exhibits the ideal behavior characterized by Raoult’s law.

If the forces of attraction in the liquid mixture phase are weak, or even in the case of repulsion betweenthe molecules, a higher vapor pressure is observed than the level anticipated by Raoult’s law.Simultaneously, the equilibrium lens is deformed because the mixtures with higher vapor pressureshave lower boiling points.

Figures A, B and C show the relationship between the isothermal diagram giving the vapor pressureof mixtures of two components a/b (a light/b heavy) and the isobaric equilibrium lens.

t

mol. % light

Figure AIdeal mixture

Pt

heavy C. lig h t C .

t

Ps heavy C.

Figure BPositive deviation

Pt

PP t

Figure CAzeotropy

Pt


P

Raoult law

Azeotrope


tb light C.t

b light C.tb light C.

tb heavy C.

tb heavy C.

tb heavy C.

Ps heavy C.Ps heavy C.

Ps light C.Ps light C.Ps light C.

D TH

100

B


70

Figure A relates to an ideal mixture where Raoult’s law is respected and also displays thecorresponding equilibrium lens.

Figure B corresponds to a mixture displaying positive non-ideality: the vapor pressure observed ishigher than that predicted by Raoult’s law. This means a downward deformation of the equilibrium lensbecause the mixtures with high vapor pressure have a lower boiling point. Note that ‘light’ rich mixtureshave a boiling point approximately equal to that of the ‘light’ component itself. Simultaneously, thecurve for saturated vapors is also shifted downward, causing a proximity of the two curves, which isdetrimental to the selectivity of liquid/vapor equilibria in the zone of high ‘light’ concentrations.

Figure C is characteristic of azeotropy, which occurs when the curve giving the vapor pressure of themixtures passes through an extreme (minimum or maximum). The mixture which has this maximum orminimum vapor pressure consequently has a minimum or maximum boiling point in the diagram ofequilibrium temperatures.

For this maximum or minimum boiling point, the liquid/vapor equilibrium imposes the presence of avapor in equilibrium at the same temperature. Hence, at this point, this requires the junction of theboiling point curves of the liquids and the dew-point curves of the vapors. The isobaric equilibriumdiagram is thus divided into two lenses. Figure C is said to be concerned with a minimum azeotrope.At the same time, since the example presented implies the presence of a one-phase mixture in theliquid phase (no liquid/liquid separation), the azeotrope is called a homoazeotrope.

We know that with a constant temperature, the vapor pressure of a non-ideal a-b mixture is given by:

Ps = Psa xa γa + P

sb xb γb

γa and γb are the activity coefficients a and b components.

Thus Ps = Psa xa γa + P

sb γb (1 – xa)

= xa ( )Psa γa – P

sb γb + P

sb γb

If the curve Ps = f(x) at constant temperature present an extremum, then:

d Ps

d x T = 0 thus P

sa γa = P

sb γb

In this conditions, if the equilibrium coefficients are expressed by:

Ka = yaxa

= P

sa γa

P and Kb = ybxb

= P

sa γb

P

thus αa-b = KaKb

= P

sa γa

Psb γb

= 1

There is no more selectivity of liquid-vapor equilibrium for the special composition of this mixture.

The vapor phase occurs during the vaporization’s step is the same as the liquid phase.


71

The figures below show the two typical cases of homoazeotropy.

heavy C. light C.

Pressure

heavy C. light C.

Pressure

P P

% mol. light

T T

m1 m2

heavy C. light C. heavy C. light C.m1 m2

TemperatureTemperature

bp heavy

tb min

Ps heavy

Ps max

Ps light

Ps heavy

Ps light

Ps min

D TH

101

B

bp heavy

bp lightbp light

bp max

m1: homoazeotrope at minimum boiling point

m2: homoazeotrope at maximum boiling point

The presence of azeotropes generally complicates problems of separation by distillation, because theazeotropes behave as pure substances:

- fixed boiling point

- vapor generated with composition identical to the liquid, hence no liquid/vaporselectivity


72

2 - EXAMPLES OF NON-IDEAL MIXTURES AND HOMOAZEOTROPES SEPARATION

a - Separation of ideal mixtures

A number of examples helps to position the different problems connected with non-ideality and todemonstrate the resulting difficulties in the feasibility of separations by distillation.

For an ideal binary mixture, the shape of the equilibrium lens guarantees constant selectivity ofliquid/vapor equilibrium. In fact, it can be considered that the ideal equilibrium lens reflects a constantvalue for the relative volatility of the light component with respect to the heavy component. In so far asthe volatility difference between the two components to be separated is sufficiently large, a distillationcolumn helps to obtain each of the two components in the desired purity.

Temperature

tb

light C.

tb

heavy C.

Light C.Heavy C.

P

light C.

heavy C.

t b heavy C.

tb light C.

P

α ≅ constant

composition

D TH

102

BWe obtain under pressure supposed constant:

- at the top of the column, the volatile component a and the temperature of the head of thecolumn is established at teba

- at the bottom of the column the heavy component b is established with its boiling point tebb

For non-ideal mixtures, the shape of the lens indicates wide variations in volatility, which often restrictthe possibilities of separation by distillation, as shown by the examples below.


73

b - Non-ideal mixtures: n-heptane-benzene and n-hexane-benzene

This mixture is characteristic of non-ideality which appears in aromatic/non-aromatic mixtures.

n-heptane-benzene mixture

The figures hereunder show the isothermal (at 80°C) and isobaric (at 1 atm) diagram of n-heptane-benzene mixture.

80

90

0 0,2 0,4 0,6 0,8 1

Bz 80,10°C

z,x,y benzene(molar fraction)

Tem

pera

ture

°C

100

Equilibrium diagram ofbenzene-n-heptane mixture

at P = 1 atm

D TH

047

D

P = 1 atm

Bz + nC7

nC70,75

1

P sbenzene

P

x

xbenzene

xheptane

0 0.5

0.51 0

0,5

Vapor tension ofheptane-benzenemixtures at 80°C

T = 80°C

Raoult

P s(atm)

P sheptane

nC7 98,42°C

Except for areas with high heptane concentrations, this mixture displays a highly positive deviationfrom Raoult’s law without an apparition of azeotropic phenomenon.

The strongest vapor pressures observed lead to lower boiling point in particular for mixtures with highbenzene content.

The isobaric equilibrium diagram shows a bad selectivity of liquid-vapor equilibriums, for highbenzene concentration (> 0.9). The values near liquid concentrations x and vapor concentrations y atthe equilibrium displays a relative volatility of benzene in comparison with n-heptane close to 1 in thisarea.

According to this particular behaviour, benzene-n-heptane mixture which is treated in a classicaldistillation column can be split into a nC7 residue which can be pure and a distillate rich in benzene butobviously with a small quantity of nC7.

For low benzene concentrations, it displays a highly positive deviation from Raoult’s law, whichreflects, in the zone of high hexane contents, a vapor pressure nearly as high as that of pure hexane.

This gives a very flat equilibrium curve corresponding to nearly constant equilibrium temperatures inthe neighbourhood of hexane.


74

From the distillation standpoint, this diagram shows that the volatility difference between these twocompounds varies widely according to the composition:

- αhexane/benzene is much larger than 1 for hexane contents lower than 60 vol %

- αhexane/benzene is very close to 1 in the zone above 80% hexane

This indicates that separation could help to obtain pure benzene at the bottom, but that, on thecontrary, at the top, hexane may not be purified and may exit with 10 to 20% benzene.

For mixtures with high content of nC6, the boiling points are very close to nC6 boiling point. Thesemixtures which almost present an azeotropic character are characteristic of a bad selectivity in areaswith a high n-hexane content.

This example also shows the risks of error induced by the assumptions of constant relative volatility forhydrocarbons.

6 5

0 0.2 0.4 0.6 0.8 1

n-hexane

benzene

+

benzene

n-hexane

benzene

Benzene80.1 °C

Tem

pera

ture

(°C)

8 0

7 5

7 0

68.74°Cn-Hexane

P = 1 atm

z,x,y Benzene (molar fraction)

Equilibrium diagramBenzene - n-Hexane

D TH

118

B


75

c - Benzene/cyclohexane azeotropic mixture

This is a rare case of azeotropy in a hydrocarbon mixture. It is due to the fact that benzene(tnb = 80.1°C) and cyclohexane (tnb = 80.7°C) have very similar volatilities. Their vapor pressurecurves are extremely close to each other and, at a given temperature, indicate nearly identical vaporpressure for both pure substances.

A weak non-ideality (positive here)suffices in this case to confer a highervapor pressure on benzene/cyclohexanemixtures than the vapor pressure of thepure substances, thus causing azeotropy.

The opposite diagram shows the shape ofthe vapor pressure curve for cyclohexane-benzene mixtures around 80°C.

PT

PsCH Ps

Bz

Psmax

BzAzeotropeCH D TH

204

5 B

The equilibrium diagram plotted below at 1 atm pressure shows that the azeotropic mixture contains55 mol % benzene and has a boiling point of 77.4°C.

Benzene (molar fraction)

0 0.2 0.4 0.6 0.8 1

7 5

77.4

8 0

Cyclohexane80.7°C

P = 1 atm

Benzene80.1°C

Tem

pera

ture

(°C)

8 5

z,x,y

Equilibrium diagramBenzene - Cyclohexane

0.55

D TH

119

B


76

cyclohexane

benzene

(Z BZ < 0.5)

cyclohexane

azeotrope

D TH

103

B

Depending on the feed composition, adistillation column helps to obtain one ofthe two pure substances at the bottomand the azeotrope at the top. The figureopposite corresponds to a feed containingless than 50% benzene.

d - Ethanol/hexane azeotropic mixture

Ethanol and hexane display very high positive non-ideality in mixtures in the liquid state shown bya very high maximum vapor pressure on the below drawn at a temperature diagram of 30°C. Thisgives rise to a vapor pressure extreme which leads to a minimum homoazeotrope at atmosphericpressure.

P

x

200

100

PSethanol

0 0,5 0,77 1xhexane

Vapor pressure ofhexane-ethanolmixtures at 30°C

PS hexane

P Smaximum

T = 30°C

247

P SmmHg

D TH

046

B


77

The figure below shows the diagram obtained:

- boiling point of ethanol 78.3°C- boiling point of hexane 68.7°C- boiling point of azeotrope 59.6°C, corresponding to a mixture containing 67.5 mol % hexane

7 5

7 0

6 5

6 059.6

5 5

78.3°C8 0

Tem

pera

ture

°C

P = 1 atm

68.7°C

67.5

mol. % hexane

EQUILIBRIUM CURVES OFHEXANE - ETHANOL MIXTURES

ETHANOL HEXANE D TH

048

B

In terms of separation, such a mixture cannot be separated by simple distillation:

- a feed of composition Z < 0.675 hexane gives the azeotrope at the top and highconcentration ethanol at the bottom

- a feed of composition Z > 0.675 hexane gives the azeotrope in the distillate and hexane inthe residue

AZEOTROPE

ETHANOL

AZEOTROPE

HEXANE

D PC

D 03

4 C

Z < 0.675 Z > 0.675


78

e - Chloroform/acetone azeotropic mixture

Mixtures with high negative non-ideality giving rise to a maximum azeotrope are relatively few innumber. Dilute acidic solutions and the acetone/chloroform mixture shown below are examples.

In this case, distillation produces one of the two pure substances at the top and the azeotrope at thebottom.

Temperature °C

6 4

6 2

6 0

5 8

5 6

H

61.2

0 0.2 0.4 0.6 0.8 1.0

Acetone - chloroformmaximum azeotrope

P = 1 atm

Molar concentration of acetone

56.5

64.5°C

0.345

D TH

120

B

3 - SHIFT OF THE AZEOTROPIC COMPOSITION WITH PRESSURE

Pressure alters the equilibrium temperatures, and these affect the cohesion forces in the liquid phase.

The azeotropic diagrams not only undergo a shift in the temperatures as the pressure is changed, buta change in the azeotropic composition is also observed.

For an ethanol/n-hexane mixture, the azeotropic composition shifts towards higher hexaneconcentrations as the pressure is decreased.

P (mmHg) Boiling point azeotrope Composition of azeotrope(mol % hexane)

760 59.6°C 67.5247 30°C 7756.4 0°C 82


79

For the ethanol/water mixture, it is interesting to see the azeotrope present at atmospheric pressure(89.5 mol % ethanol, tnb = 78.15°C) disappear when going to an absolute pressure of 100 mmHg.

The two diagrams given below show the possibility of a method of separation by distillation ofwater/ethanol mixtures.

100

9 0

8 0

7 0

6 0

5 0

4 0

3 0

Tem

pera

ture

°C

0 .895

Water - Ethanol equilibriumat 760 and 100 mm Hg

P = 760 mm Hg

P = 100 mm Hg

0 0.2 0.4 0.6 0.8 1Molar fraction of ethanol

78.3°C

51.2°C

35°C

78.15°C

D TH

121

B


80

Starting with a feed of composition Z < 89.5 mol % ethanol, the distillation scheme uses a first columnoperating at 100 mmHg. This produces pure water at the bottom and a distillate of compositionxD > 0.895 at the top. This is then separated in an atmospheric column giving the azeotrope at the top,recycled to the first column, and pure ethanol at the bottom. The flow chart is shown below.

WATERETHANOL FEED

Z < 0.895ethanol

WATER ETHANOL

100mmHg

1atm.

XD > 0.895 Azeotrope

D TH

204

6 B


81

4 - HETEROAZEOTROPY

Heteroazeotropy adds the presence of partial or total immiscibility in the liquid phase to themechanism of azeotropy.

a - Heteroazeotropy with partial immiscibility

WATER ISOBUTANOL DIAGRAM

• Solubility diagram - Water-isobutanol diagram

Partial immiscibility is reflected in the temperature/composition binary diagram by the presence of aliquid/liquid equilibrium curve which divides the diagram into a two-phase zone inside the curve and aone-phase zone outside.

In the two-phase zone, the mixture is separated into two liquid phases which are said to be inliquid/liquid equilibrium.

The solubility diagram helps to indicate the conditions of this equilibrium. The figure below gives theexample of the water/isobutanol mixture.

0

0.02

0 .2 0.4

0.53

0 .80.6

P 2 M P 1

5 0

3 0

0

100

150Critical temperature

Tem

pera

ture

°C

of dissolution

Domain with2 liquid phases

Solubility diagramWater - isobutanol

1 liquid phase

D TH

122

B

Molar fractionof isobutanol


82

At a given temperature, a liquid mixture represented by a point located in the two-phase domain isseparated into two liquid phases whose compositions are given, for the temperature concerned, by theintersections with the equilibrium curve.

At 30°C for example, a mixture Mcontaining 20 mol % isobutanol separatesinto a phase P1 containing 53%isobutanol and a phase P2 containing 2%.

30°C

M(20% iB)

P1 : 53% iB

P2 : 2% iB

D TH

204

4 A

The compositions of the liquid phases in equilibrium are modified with the temperature. Note that theequilibrium curves move closer together, so that the difference in composition between the two liquidphases in equilibrium is reduced as the temperature rises. The two curves ultimately join at a maximumtemperature above which two liquid phases can no longer exist. This temperature is called the criticaldissolution temperature (CDT).

Depending on the choice of a pressure, the liquid/vapor equilibrium temperatures may be lower thanthe CDT, giving rise to the presence of a heteroazeotrope.

• Isotherm diagram

The figure below shows water-isobutanol isotherm diagram which present a partial immiscibility inliquid phase.

The mixture displays a high positive non-ideality. The immiscibility area is represented by a constantvapor pressure because liquid phases compositions at the equilibrium are fixed at a constanttemperature.

Zoneof immiscibility

Pswater

Psdiphasicmixture

PsiB

IsobutanolWater

T

D TH

204

7 B

P P


83

• Heteroazeotropic diagram

At atmospheric pressure, the diagram above shows that the water/isobutanol mixture has an azeotrope(34 mol % isobutanol, 89.5°C). This is located in the two-phase liquid zone, showing that theheteroazeotropy corresponds to a three-phase equilibrium involving:

- a liquid phase with an overall composition of 34% butanol, but separated into two liquidphases in equilibrium (2 and 40%)

- a vapor phase containing 34% isobutanol

Note that the three-phase equilibrium imposes a constant value on the vapor pressure of the mixture,because all the liquids of which the average composition lies inside the miscibility curves have thesame boiling point.

0

Tem

pera

ture

en

°C Liquid - VaporEquilibrium diagramWater - isobutanol

P = 1 atm

100°C

2 Liquid phases

8 5

89.59 0

9 5

100

105

107.1°C

0.340.4 0.6 0.8 10.2

x or y molar fraction of isobutanol0.035 D TH

124

B


84

• Separation of a binary heteroazeotropic mixture

The separation of heteroazeotropic mixtures by distillation generally requires the use of twocolumns.

Starting with a one-phase liquid feed rich in isobutanol (Z > 40%), a first column C1 helps to separatethe isobutanol with the desired purity at the bottom and the heteroazeotrope at the top.

After condensation, the heteroazeotrope settles into two liquid phases of which the compositions aregiven by the miscibility curves.

The isobutanol-rich phase is sent as reflux to the first column, and the water-rich phase is treated in thesecond column C2 producing:

- the azeotrope at the head, recycled to the condensation system- water at the bottom in the desired purity

The flow chart of the process is shown below.

iB rich phase

Water rich

phase

C2

C1

D TH

105

B

WATER +ISOBUTANOL

ZiB > 40%

ISOBUTANOL

WATER

• Other examples of binary heteroazeotropic mixtures

Water/furfural and water/MEK diagrams are similar to the water/isobutanol diagram. The partialimmiscibility curve in the liquid phase is however wider.

Separation of the mixtures by distillation is carried out as described above.


85

WATER/FURFURAL DIAGRAMS

0

3 5

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100%1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

100

110

120

130

140

97,9

Tem

pera

ture

°C

162°C

100°C

— Vapor phase —

1 liq

uid

phas

e

1 liq

uid

phas

e

WATER - FURFUROL1 ATM

D TH

125

B

WATER FURFUROL

Dew point curve

Bubble point curve

Miscibility curve

— 2 Liquid —phases

Furfurolweight percentage


86

WATER/METHYL-ETHYL-KETONE (MEK) DIAGRAM

Temperature °C

mol % of methyl ethyl ketone

P = 1 atm

P = 6.8 atm

P = 17 atm

P = 34 atm

P = 3.4 atm

6 0

8 0

100

120

140

160

180

200

220

240

260

D TH

126

B

0 10 20 30 40 50 60 70 80 90 100

WATER-METHYL ETHYL KETONE

MECWATER

This diagram also shows the shift in the composition of the heteroazeotrope with pressure.

At 34 atm, the equilibrium curves leave the solubility diagram and give rise to a homoazeotrope.


87

b - Heteroazeotrope with total immiscibility

Water/hydrocarbon mixtures provide the most common examples of heteroazeotropy with totalimmiscibility. The corresponding solubilities are in fact very low at moderate temperature, and thewater/hydrocarbon liquid mixtures can always be considered as two-phase mixtures.

This gives rise to special behavior which can be approached from the particularly simple vaporpressure law applicable to these mixtures.

• Vapor pressure of a water/hydrocarbon mixture

The figure below shows a water/hydrocarbon mixture in liquid/vapor equilibrium.

P

t

GAS SATURATEDWITH WATER

LIQUIDHYDROCARBONS

WATER D TH

106

B

MIXTURE WATER+

HYDROCARBONS

The equilibrium conditions are characterized as follows.

- an equilibrium pressure P or vapor pressure PSm of the liquid mixture equal to the sum of

the vapor pressures of water PSwater and of the liquid hydrocarbons P

SHC.

P = PSm = P

Swater + P

SHC

This behavior is explained by the fact that, due to the total immiscibility, no intermolecularbonding force exists between the water and the hydrocarbons in the liquid phase. Each ofthe two components behaves as if it were alone in liquid/vapor equilibrium, and thusexerts its own vapor pressure.

The first consequence of this behavior is that the addition of liquid water to hydrocarbonsgives rise to an overall mixture whose vapor pressure and hence volatility are increased.This means lower vaporization temperatures.


88

- a vapor phase composition which depends directly on the respective values of the vaporpressures of water and the hydrocarbons. Dalton’s law, applicable at low pressure, helps todetermine the water content of the vapor phase ywater by relating the partial pressure of

water or PSwater to the total pressure P.

This gives: ywater = P

SwaterP

PSwater depends on the temperature of the drum (see vapor pressure curve of water)

P is the total pressure

The gas resulting from this liquid/vapor equilibrium is said to be saturated with water.

Obviously we similarly have:

yHC = P

SHCP = 1 – ywater

The behavior of mixtures of water and hydrocarbons is therefore reflected by very simple laws.

• Isobaric equilibrium diagram of a water-benzene mixture

In terms of liquid/vapor equilibria, it is advantageous to apply these properties to construct the isobaricequilibrium diagram corresponding to a binary water/hydrocarbon mixture. The example given belowfor the water/benzene mixture is representative of binary water/hydrocarbon mixtures, with theunderstanding that the diagram obtained is deformed according to the specific volatility of thehydrocarbon combined with the water.

The liquid-vapor equilibrium diagram is drawn at atmospheric pressure.

It shows the following results:

- the benzene boiling point: 80.1°C

- the water boiling point: 100°C

- the fixed boiling point for all two-phases liquid benzene-water: 69.4°C

- the presence of an heteroazeotrope at 70.5% mol of benzene

- a supposed total immiscibility zone in liquid phase


89

Applied to the water/benzene binary mixture, the two above equations give the equilibrium diagramunder atmospheric pressure shown below.

100

9 0

8 0

7 0

2 0

Tem

pera

ture

°C

0 4 0 6 0 8 0 100

8 0 6 0 4 0 2 0 0100mol % water

mol % benzene

Liquid (water) ++ vapor (water+ Benzene)

Equilibrium diagramliquid - vapor

WATER - BENZENE

P = 1 atm

Vapor(water + Benzene)

2 liquids(water and benzene)

Liquid(Benzene)

+Vapor

(Water + Benzene)

tbp benzene80.1°C

69,4°C

70.5

29.5

D TH

127

B

tbp water

In fact, all the water/benzene mixtures have the same boiling point at atmospheric pressure. Thismust be such that the following equation is satisfied:

PSm = P

Sbenzene + P

Swater = 1 atm

This temperature can be determinedby successive approximations, leadingto an equality satisfied at atemperature of 69°C.

At this temperature, we have:

PSbenzene = 0.705 atm

and PSwater = 0.295 atm

hence: PSm = 1 atm

1

0.705

0.295

69.4 80.1 100

BZWater

Pressure

Temperature °C

D TH

205

2 B


90

Consequently, all water/benzene liquid mixtures can be seen to liberate a vapor of the same molarcomposition:

ybenzene = P

Sbenzene

PSm

= 0.705

1 = 0.705 ywater = P

Swater

PSm

= 0.295

1 = 0.295

This particular composition, for which contact occurs between the vapor and liquid equilibrium curves,thus corresponds to the water/benzene heteroazeotrope.

By applying the same method, this mechanism of heteroazeotropy can be identified for all binarymixtures of water and hydrocarbons. This results in diagrams having a similar shape to that of thewater/benzene mixture. The heteroazeotropic composition approaches the hydrocarbon for mixturesof water and light hydrocarbons (e.g. water/propane), or, on the contrary, approaches water in thereverse case (e.g. water/C25).

If we now examine the problem of the condensation of a gaseous mixture of water and benzeneunder atmospheric pressure, two different developments can be observed depending on thecomposition of the mixture.

Benzene-rich mixture

ybenzene = 0.8ywater = 0.2

0.2

0.4

0.6

0.8

Pressureatm.

6 0 6 9 8 0 100 110 120

Benzene

ps wate

r

ps be

nzen

e

Water

pp water

Temperature °C

pp b

enze

ne

93.5

Condensationwater + benzene

1.0

07 3

condensationof benzene

0.705

0.295D

TH 1

07 B

The above figure shows the water/benzene gaseous mixture in the pressure/temperature diagram.

Starting with a temperature of 110°C, for example, and by cooling the mixture, the representativepoints of benzene and water shift towards lower temperatures while remaining at a partial pressure of0.8 atm for benzene and 0.2 atm for water.

At a temperature of about 73°C, the representative point of benzene reaches its vapor pressure curve,whereas the point of water remains in the gas domain.


91

Consequently, benzene alone begins to condense at this point. The mixture has reached the dew-point appearing in the binary equilibrium diagram shown earlier.

The process then continues with a decrease in the temperature as the benzene is condensed. Thisleads to a reduction in the partial pressure of benzene, which is now its vapor pressure, and,consequently, since the total pressure is constant, to an increase in the partial pressure of water vapor.

At 69°C, the representative point of water reaches its equilibrium curve and the condensation of bothcomponents of the mixture terminates in heteroazeotropic conditions: constant composition andtemperature.

Water-rich mixture ywater = 0.8ybenzene = 0.2

0.2

0.4

0.6

0.8

Pressureatm.

6 0 6 9 8 0 100 110 120

Water

Benzene

pp benzene

condensationof water

Temperature °C

pp w

ater

93.5

condensationwater + benzene

1.0

0

0.705

0.295

D TH

108

B

ps wate

r

ps be

nzen

e

For this mixture, contrary to the previous case, the representative point of water reaches its vaporpressure curve first at the dew-point of the mixture: 93.5°C. Water is then condensed alone up to69°C, a temperature at which the partial pressure of benzene, which is the complement at 1 atm of thevapor pressure of water, reaches its equilibrium pressure. Both components continue to condense inheteroazeotropic conditions.

By repeating this analysis for different compositions, the dew-point curves appearing in theheteroazeotropic diagram can be generated point by point.


92

• Applications: hydrocarbon drying column

A rather common application of this type of behavior in liquid/vapor equilibrium corresponds tohydrocarbon drying process by azeotropic distillation.

The wet hydrocarbon (containing dissolved water) is fed to a distillation column. At the bottom the dryhydrocarbon normally exits, while a composition close to the heteroazeotrope exits at the top.

Water

D T

H 1

09 B

WETHYDROCARBON

DRY HYDROCARBON

tbp azeo

tbp HC

P

The heteroazeotrope separates into two liquid phases after condensation. The hydrocarbon phase isrefluxed to the column and the aqueous phase is removed.

In all cases, regardless the hydrocarbons and water volatility, the water is removed at the top of thecolumn.

Figures above show azeotropic diagrams with water and light or heavy hydrocarbons.

tbpwater

tbpHC

P

Water Light HC

tebwater

tbpHCP

Water Heavy HC D TH

204

8 B

t t t t

Water - light HC Water - heavy HC

tbpazeotbpazeo

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Applied thermodynamics