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Thermophysical propertiescalculation / prediction overview

Supplement of the course:Computer-aided process simulation

Presented by B. Behzadi

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Equipment & process design (general or detailed) in the chemicalprocess industries is based on the (thermophysical) properties of theprocessed/utility material involved. Due to the large number of speciesand phases that are potentially encountered, general methods havebeen developed to calculate/predict the properties of pure componentsand their mixtures at different process conditions.

Properties mostly encountered by chemical engineers:General properties:§ Molecular weight (MW)§ Critical properties: pressure (Pc), temperature (Tc), volume (Vc),

compressibility factor (Zc)§ Acentric factor (w)§ Normal boiling point (Tbp)§ Normal freezing point (Tf)§ Vapor pressure (Psat)§ Density (r)

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Transport (& other related) properties:Momentum§ Viscosity (m,h)§ Surface tension (s)

Mass§ Binary diffusion constants (Dij)§ Overall mass transfer coefficient (ki)

Heat§ Heat capacity (CP)§ Enthalpy (H)§ Entropy (S)§ (Latent) heat of vaporization (DHvap)§ (Latent) heat of fusion (DHfus)§ Thermal conductivity (k)

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Other:§ Heat (enthalpy) of mixing (DHmix)§ Volume change of mixing (DVmix)§ Solubility or Henry’s constants (Hij)

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Calculation / Prediction methods:1. Empirical methods; polynomials, correlations … etc.§ No specific theoretical background for correlation, only based on

fitting to experimental data§ Validity depends on database§ Very accurate over specified T (&P) ranges; extrapolation not

allowed, care must be taken especially when logarithmic ortrigonometric equations are included

§ Generally not applicable to unknown species

Due to their high accuracy, valid empirical methods are usually thepreferred choice if they fall within the specified ranges, especially forpure components

Example: saturated liquid viscosity

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h = ATB

Ln(h) = A+B/TLn(h) = A+B/T+CT+DT2

Other properties: ideal gas or liquid heat capacity, ideal gasviscosity, ideal gas or liquid thermal conductivity, surface tension

2. Semi-empirical methods (correlations)

§ Based on theory§ Include additional modifications to improve real data representation§ Include ‘fitted’ constants§ Best performance in specified T & P ranges; usually some

extrapolation possible§ Usually cover wider ranges of T & P compared to empirical methods§ Possible extension to components in the same classification

(homologous series, etc.); less fitting necessary

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Example: gas viscosityKinetic gas theory

Method of Chung et al.

theory)Enskog-(Chapman

nsinteractioularintermolecincludes)(

gasideal1

)( 2

e

sh

kTf

MWTconst

v

v

v

=W

=WW

´=

factornassociatioempiricalmomentdipolereduced

factoracentricdiametercollision

energyninteractio059035.02756.01

785.40

809.0

2593.12593.1

r

2

32

31

k

kFV

MWTF

V

TkTTk

rC

vC

C

rC

C

mwse

mw

h

se

e

++-=

W´

=

=

=®=

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3. Group contribution methods§ Predictive (± 30% error)§ Usually semi-empirical§ Commonly applied when component falls in a class for which the

previous two methods are inapplicable, or when the necessaryexperimental property is not available (e.g. Tc, Pc for complexmolecules)

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Phase-equilibrium calculations

Basis: equality of chemical potentials of each species in all phasesmi

I = miII = mi

III … @ mixture T, P, compositiondmi = RTdLnfi → fi

I = fiII = fi

III = … iso-fugacity condition

Commonly encountered cases : VLE, LLE, VLLE, LSE, VLSE, LLSE

Equation of state models : Mostly Helmholtz free energy basis;e.g. :

EOS)Robinson-(Peng)()(

ln)21()21(ln

22

,

bVbbVVa

bVRTP

VAP

VbVRT

bVbV

baA

nT

-++-

-=®

÷øö

çèæ

¶¶

-=

÷øö

çèæ -

-úû

ùêë

é

++-+

=

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Gas phase only; Ideal gas equation, virial series,Benedict-Webb-Rubin-Starling (1940)

Cubic EOS : for simple (nearly spherical) normal moleculesMostly modifications of the van der Waals equation (1873):

P=RT/(V-b)-a/V2

Redlich-Kwong (RK- 1949) Soave (SRK-1972)Peng-Robinson (PR-1976) Patel-Teja (PT- 1982)Stryjec-Vera (PRSV-1986) Yu-Lu (YL-1987)Trebble-Bishnoi (1987) Schwartzentruber-Renon (1989)

Applicable to all phases, especially vapor & liquid, relatively simple, smallnumber of parameters, good prediction of normal fluid mixtures usingsimple mixing rules (for model parameters) and combining rules (formodel constants) + binary interaction coefficients

Suitable constants/modifications must be used for associating, polar,complex fluids

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Equations of state for chain molecules: perturbation theories

Perturbed hard chain theory (PHCT- 1975)based on free volume effects (meets gas limit)good results for polymer solutions, petroleum fractions

Simplified version (SPHCT)

A version including effects of anisotropic multipolar forces (dipoles,quadropoles): Perturbed anisotropic chain theory (PACT); suitablefor complex fluids

Equations of state for associating fluids : perturbation theories

Some sort of pseudo-chemical equilibria assumedIncludes association sites derived from Wertheim'sthermodynamic perturbation theory (TPT1)

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detailed molecular theories sometimes including manyadjustable parameterswell suited for complex fluidsgood prediction properties especially at high pressurescomplex solution of equilibrium conditions

Associated perturbed anisotropic chain theory (APACT- 1986)

Z=1+Zrep+Zattr+Zassoc

Statistical associating fluid theory (SAFT- 1988)

A=Aideal+Aseg(hs+disp)+Achain+Aassoc

most commonly used : Huang-Radosz (1990)also: simplified SAFT (SSAFT), hard-sphere (HS-SAFT), Lennard-Jones(LJ-SAFT),square-well (SW-SAFT), and a generalized variable range

potential (SAFT-VR)

Cubic plus association equations of state (CPA)

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Activity coefficient models: Gibbs free energy basislngi=(d(GE/RT)/dni)nj

more accurate liquid phase fugacities using simpler models (in terms ofnumber of parameters) compared to EOS's

must be used with EOS when vapor phase is present

Margules (2 suffix, 3suffix); molecules of nearly the same size, non-polarNRTL (non-random two-liquid theory); strongly non-ideal systems,

applicable to immiscible systemsFlory-Huggins ; polymersWilson’s; practical two parameter, not applicable for immiscibility,

applicable to associating/polar solvents

UNIQUAC; derived for Guggenheim’s quasi-chemical theory

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gE=gE(combinatorial)+gE(residual)

Simple (2parameter), widely applicable (VLE, LLE, polar, associating)

UNIFAC (universal functional activity coefficient); group contributionmethod on the basis of UNIQUAC

EOS mixing rules from excess Gibbs energy models:

Assumptions:1- excess Gibbs energy: EOS = activity coefficient model2- b parameter in cubic EOS = infinite pressure V3- excess volume = 0

Huron-Vidal (and modifications MHV1, MHV2) mixing rulesEOS amix,bmix=f(aii,bii,activity model (NRTL,…))

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Wong-Sandler mixing rules, replace GE by AE due to theoretical reasoning

No independent interaction parameters necessary (predictive for mixtures)

Determination of the number of phases (stabilityanalysis)

Basis: minimization of the Gibbs free energy of the systemMost common method: Michelsen’s method (and extensions)

sy' trialallphases,all0))(()(:conditionStabilityconditionsonminimizatiNaroundGofexpansionseriesTaylor

GandGphasesnewofnumbersmoleand-Nintophasesplit

phasestudiedaofenergyGibbsinitial

0

0

III

00

³-=

®+-+=D

=

å

å

ii

ii

III

iii

yyyF

GGGG

nG

mm

ee

m

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Different implementation techniques, different minimization algorithms

mi is written in terms of fugacity/activity depending on thermodynamicmodel (EOS/activity coefficient)

Note : number of EOS roots DOES NOT give number of phases present

stability analysis also gives aninitial guess for compositions tobe used in phase equilibriumcalculations (VERY important)

Multi-component phase envelope

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Robust phase equilibrium calculations

T-P flash calculation basics (VLE); the direct substitution method

Substitution is performed until convergence is reachedSimilar implementation for multiphase systems

ii

ii

i

iiii

i

iii

i

iii

i

i

iiii

yxyx

VKKzVFyx

VKzKyVK

zxKxyLV

yxyx

,new)(Kmodelamic thermodynVVnew0F(V)ofsolution),(orKforguessinitial

conditionfugacity-iso theand

0))1(1()1()(satisfymustVand,

))1(1())1(1(1

0)(11

i

i

Þ+®=+

=-+-=

-+=-+=®==+

=-®==

å

å å å

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Possible problems:1- convergence at high pressures & near critical points2- Non-condensables with activity coefficient models (especially in oil &gas systems with C1 & C2, CO2, H2S, N2 etc

Solutions:(1)better initial guesses/phase recognition conditions !)فوت کوزه گری(redefinition of functions/variables (logarithmic definition, etc.)Acceleration techniques: better modification of Ki’s

Ki(m+1) = Ki

(m) [fiI/fi

II]g g=f(g) g i= LnfiII – Lnfi

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(2)Use of empirical equations for extrapolation of pure-componentfugacities

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Electrolyte systems:Feature: Long-range ionic forces presentEffects: vapor pressure decrease, salting out (/in) of solvents (useful insolvent extraction), present in (or enhancement) of chemical reactions

Additional necessary equilibrium condition: electroneutrality

n+n+ + n-n- ... = 0Frequently used models:Debye-Huckel model (1924): assumes point charges, applicable at very

low molalities (0.1M-1M max), or in combination with other effects

Pitzer (and extensions): virial type, exact, many parameters necessary

Mean-spherical approximation (MSA): takes into account ionic size(usually concentration dependent), accurate up to medium/highmolalities (6M and above)

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Specific modelsSour-gas processing (amine treatment) :§ Non-proprietary models: take into account equilibrium reactions

among present species, activity coefficients of neutral / ioniccomponents, electroneutrality, material balance,gas-phase non-ideality, simultaneous solution model for multi-variable system ofequations

Kent-Eisenberg model (1976): approximate, good for initial guess,activity coefficients = 1

Deshmukh-Mather (1982): NRTL + EOS, wide database used forParameter adjustment (renewed recently)

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Hydrate formation prediction

Combination of water + (primarily) methane ice-like crystalsgas molecules in ice cavities

Models based on probability of presence of gas molecule in cavity

mH2O=m0H2O + RT∑ni Ln(1 -∑yki) (Van der Waals – Platteeuw model)

(1959)number of cavities probability

Other models:Parrish-Prausnitz (1972), Ng-Robinson (1976)

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Petroleum fractionsCharacterization (distillation curves)

Light components measured separately to be estimated if not availableTrue boiling point (TBP- usually all results are converted to TBP

distillation curve)ASTM-D86 (atmospheric)ASTM-D1160 (heavy fractions)ASTM-D2887 (gas chromatography)Equilibrium flash vaporization (EFV)The curves are converted to mol% basis for pseudo-component

identification

0

100

200

300

400

500

600

0 20 40 60 80 100

vol %

T

0100200300400500600700800900

1000

100 200 300 400 500 600

TM

W0

0.5

1

1.5

2

2.5

0 200 400 600

T

sp. g

r.

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Important initial parameters (overall or per fraction):Watson K-factor : K = (Mean average boiling point)^(1/3) / (sp.gr.60/60)

~ 12from distillation curve calculated from

distillation curve

API = 141.5/(sp.gr.60/60) - 131.5ü sp.gr. or MW curve critical P & T , acentric factor, etc

pseudo-component definition

ü Pseudo-components thermophysical properties calculation

petroleum-specific correlations normal calculations(API methods, etc) (EOS, empirical, etc)

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Related references§ The properties of gases and liquids; Reid,Prausnitz,Poling§ Design institute for physical properties (DIPPR); AIChE§ Data prediction manual, manual for predicting chemical process design

data; Danner, Daubert§ API technical databook – petroleum refining, vols I-III (AOI)§ Perry’s chemical engineers’ handbook§ Physical properties’ prediction, Yaws§ M. Michelsen, Fluid Phase Equilibria 4 (1980) pp. 1-10 & 9 (1982) pp1-

40 and more recent modifications (stability analysis and equilibriumcalculation methods)

§ R.A. Heidemann, Fluid Phase Equilibria 14 (1983) pp. 55-78 and otherrecent modifications (equilibrium calculation methods, especially high P)