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Thermodynamic modeling with equations of state: present challenges with established methods Øivind Wilhelmsen, Ailo Aasen, Geir Skaugen, Peder Aursand, Anders Austegard, Eskil Aursand, Magnus Aa. Gjennestad, Halvor Lund, Gaute Linga, and Morten Hammer Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b00317 • Publication Date (Web): 10 Mar 2017 Downloaded from http://pubs.acs.org on March 14, 2017

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Industrial & Engineering Chemistry Research

Thermodynamic modeling with equations of state: present challenges with established methods Øivind Wilhelmsen, Austegard,

‡

∗,†,‡

Ailo Aasen,

Eskil Aursand,

‡

†, ‡

Geir Skaugen,

‡

Magnus Aa. Gjennestad,

Linga,

‡

and Morten Hammer

Peder Aursand,

‡

Halvor Lund,

‡ ‡

Anders Gaute

‡

†Norwegian University of Science and Technology, Department of Energy and Process Engineering, NO-7491 Trondheim, Norway ‡SINTEF Energy Research, NO-7465 Trondheim, Norway E-mail: [email protected]

Abstract

tended corresponding state EoS has a large potential for improvement. The molecular-based

Equations of state (EoS) are essential in the

SAFT family of EoS are preferred when predic-

modeling of a wide range of industrial and nat-

tive ability is important, e.g. for systems with

ural processes. Desired qualities of EoS are ac-

strongly associating uids or polymers where

curacy, consistency, computational speed, ro-

few experimental data are available. We discuss

bustness and predictive ability outside of the

some of their benets and present challenges.

domain where they have been tted.

In this

A discussion is presented on why predictive

work, we review present challenges associated

thermodynamic models for reactive mixtures

with established models, and give suggestions

such as CO2 -NH3 and CO2 -H2 O-H2 S must be

on how to overcome them in the future.

developed in close combination with phase- and

The most accurate EoS available, multipa-

reaction equilibrium theory, regardless of the

rameter EoS, have a second articial Maxwell

choice of EoS. After overcoming present chal-

loop in the two-phase region that gives prob-

lenges, a next-generation thermodynamic mod-

lems in phase-equilibrium calculations and ex-

eling framework holds the potential to improve

clude them from important applications such as

the accuracy and predictive ability in a wide

treatment of interfacial phenomena with mass

range of applications such as process optimiza-

based density functional theory.

tion, computational uid dynamics, treatment

Suggestions

are provided on how this can be improved.

of interfacial phenomena and processes with re-

Cubic EoS are among the most computation-

active mixtures.

ally ecient EoS, but they often lack sucient accuracy. We show that extended correspond-

Introduction

ing state EoS are capable of providing significantly more accurate single-phase predictions than cubic EoS with only a doubling of the

An accurate and robust thermodynamic frame-

computational time.

In comparison, the com-

work is at the base of most higher-level model-

putational time of multiparameter EoS can be

ing and simulation tools. Such a framework is

orders of magnitude larger. For mixtures in the

central for handling problems of practical rele-

two-phase region, however, the accuracy of ex-

vance such as two-phase ow of CO 2 -mixtures

ACS Paragon Plus Environment 1

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and design of heat exchangers

24

as well as

Page 2 of 25

will be capable of describing all components or

9

problems of a more academic character, like re-

uids, or be suitable for all applications.

search on the thermal state of the earth's core

process simulations and time-consuming opti-

5

In

or discussions about water cavitation at highly

mization studies, it is advantageous to use com-

negative pressures.

putationally ecient cubic EoS such as Peng

6

The typical framework has

10

11

a collection of several equations of state (EoS),

Robinson

suitable for describing the properties of uids

higher accuracy is needed, corresponding state

or materials of interest, and algorithms to pro-

methods like LeeKesler, extended correspond-

cess the information in the EoS to provide the

ing state methods like SPUNG

appropriate properties for a particular appli-

rameter EoS like GERG-2008 are taken ad-

cation.

vantage of.

For instance, modeling of multiphase

or SoaveRedlichKwong.

1320

12

When

or multipa-

Models with enhanced predic-

heat exchangers or distillation columns requires

tive ability such as Statistical Associating Fluid

phase equilibrium calculations to be performed

Theory (SAFT) are preferred for certain appli-

to obtain the relevant properties.

cations, e.g. for systems with strongly associ-

The accuracy of the thermodynamic model

ating uids or polymers where few experimen-

2123

employed can have a large impact on higher-

tal data are available.

level modeling, and even on the nal conclu-

work is needed to develop EoS for new uids,

sions made in a study.

especially polar and associating components

For instance, the ice-

24

It is clear that more

like structures called hydrates can form and ob-

such as electrolytes.

struct natural gas pipelines in the presence of

we shall discuss some of the fundamental chal-

even trace amounts of water. It is crucial for the

lenges associated with the EoS and the basic

petroleum industry to know the maximum al-

routines found in current state-of-the-art ther-

lowable water content before hydrates can form

modynamic modeling frameworks.

to specify the appropriate dehydration require-

we shall discuss how to overcome these chal-

ments. However, the predicted maximum water

lenges. By overcoming them, a next-generation

content depends strongly on the choice of EoS.

thermodynamic framework holds the potential

7

In this work however,

Moreover,

Hydrate formation is a good example of the

to improve the accuracy and predictive abil-

type of challenges a modern state-of-the-art

ity in a wide range of applications such as

thermodynamic framework should be able to

process optimization, computational uid dy-

handle.

The basis of hydrate equilibrium cal-

namics, treatment of interfacial phenomena and

culations with the van der WaalsPlatteeuw

processes with reactive mixtures such as in dis-

model is an EoS capable of describing non-

tillation columns.

polar, polar and associating components.

8

The

Add-on/utility

EoS should be easy to extend and improve when

Model-indep. routines

new experimental data are available to accommodate for new components and conditions.

Generic EoS interface

Moreover, the EoS should be thermodynam-

Model-specific routines

ically consistent, incorporate known physical constraints, and be able to predict accurately

Temporary storage

outside of the domain where experimental data are available.

Next, phase equilibrium calcu-

Database

lations must be performed. Phase equilibrium calculations can be challenging and often time-

Figure 1:

consuming. In hydrate equilibria, it is possible

structure of a typical thermodynamic frame-

to have simultaneous coexistence of vapor, solid

work

An illustration of the layer-based

and one or more liquid phases. It then becomes challenging to identify the thermodynamically most stable conguration. It is generally accepted that no single EoS


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Structure of the thermody-

pressions, but it may also include modeloptimized iterative routines such as den-

namic framework

sity solvers.

Generic EoS interface

From a programming point of view, a modern thermodynamic framework should be modular and easy to maintain and update.

It is

tines.

important to test the accuracy and robustness

then added in.

EoS-

This layer enables the

writing of generic algorithms that can be

Moreover, it should be straight-

ignorant about the underlying model.

forward to add parameters for new EoS or new components.

They simply re-direct a prop-

independent ideal gas contributions are

gramming errors and development of inconsis-

25

wrapper

from the currently chosen model.

Automatic routines

for consistency checks reduce the risk of protent EoS.

are

erty request to the appropriate routine

of EoS to ensure a thermodynamically consistent implementation.

These

routines around the model-specic rou-

Model-independent routines

Post-processing routines to

tines

data are natural parts of a modern thermody-

to do calculations based on principles

namic framework. A natural structure of such a

which are common to all consistent EoS.

framework is illustrated in Fig. 1, which repre-

Examples include iteratively solving for

sents a layer-type structure. Starting from the

bubble/dew points or multi-phase equi-

inner layer, the dierent layers can be described

librium states, or checking for phase sta-

as follows:

bility.

EoS

interface

This layer in-

cludes, but is not limited to:

all available models and components in the framework are stored. This includes

•

model constants, component-specic pa-

Routines for result collection and plotting.

rameters for EoS models and ideal gas

•

correlations, and interaction parameters

Add-on models such as checking for possible

for pairs of components for each model. Since

generic

Add-on and utility routines

This is where the parameters for

Temporary parameter storage

the

rou-

plot phase envelopes and other thermodynamic

Database

utilize

These

formation

of

hydrates

or

other solids.

•

the

database layer can grow large, it is more

Testing routines such as consistency checks.

practical to work with a more compact

•

structure for the components and the

Framework for tting model parameters to experimental data.

models currently selected, since only a small subset of the database is likely

A thermodynamic framework can be used as

needed for a given application. This layer

part of a process simulation or as part of a

is structured to enable convenient writing

code to perform computational uid dynam-

of high performance property calculation

ics (CFD). When accurate and computationally

routines.

expensive thermodynamic models are needed,

Model-specic routines

e.g. in CFD dealing with compressible ows, These routines im-

thermodynamic calculations can become the

plement the specic thermodynamic mod-

bottleneck of the overall computation.

els.

therefore important that a modern thermody-

Based on the temporary param-

eter storage,

they calculate thermody-

namic framework lends itself easily to paral-

namic properties such as pressure, en-

lelization. Herein, one must consider both

tropy, Gibbs energy, and their derivatives, using a specic model.

It is

1. Enabling simultaneous calls to serial rou-

These routines

tines in the library without causing mem-

are mostly based on explicit analytical ex-

ory collisions between separate instances.



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2. Enabling routines in the library to utilize

highlight

parallelization internally to solve its prob-

Page 4 of 25

some

important

last

multiparameter EoS must overcome.

lem.

Computational speed/robustness

For the rst point,

hurdles

At present,

a modular structure is

cubic EoS are preferred if high computa-

needed with careful control of any shared mem-

tional eciency is needed. However, they

ory.

This can usually be achieved in a non-

can be inaccurate for predicting certain

intrusive manner, with few implications to the

properties such as the density and the

actual core routines. For the second point, par-

speed of sound in the liquid phase.

allelizing routines requires choosing algorithms

explore the possibility of enhancing the

that scale well in a parallel environment while

accuracy by exploiting the corresponding

minimizing overhead when run on a single pro-

state concept, and also consider how this

cessor. Here, shared memory approaches such

impacts computation time.

as OpenMP oers exibility, low code intrusive-

tions such as steady-state heat exchanger

ness and ease of implementation.

modeling, where the mixture composition is

Models

in

the

constant,

another

We

For applica-

feasible

approach

is to tabulate the values from a high-

thermody-

accuracy EoS in set of grid points (e.g. in

namic framework

the

T P -plane)

and then interpolate.

Predictive capability

Desired qualities of thermodynamic models are

29

For associating uids

accuracy, thermodynamic consistency, compu-

and polymers with few available experi-

tational speed, robustness and predictive abil-

mental data, it becomes necessary to em-

ity outside the region with the data that has

ploy EoS with enhanced predictive abili-

been used for tting the model.

ties such as SAFT and group contribution

The relative

importance of these qualities depend on the

approaches.

application.

dictive thermodynamic models for mix-

Fiscal metering of CO 2 ow has

Moreover, to arrive at pre-

extremely high accuracy demands, while com-

tures such as CO 2 -NH3 and CO2 -H2 O-

putational speed and predictive ability are ir-

H2 S, EoS must be developed in close

relevant. For integrated process optimizations

combination

where the solvent used is also optimized, predic-

equilibrium theory.

tiveness is key since data are scarce for most sol-

must explicitly incorporate the underly-

vents.

ing physical mechanisms such as chem-

26

For large-scale CFD calculations where

with

phase-

and

reaction

A predictive model

the uncertainties in the ow model dominate

ical reactions and association.

those in the thermodynamic model, and where

tively, ignoring association interactions in

ash calculations are called millions of times,

associating mixtures is similar to ignor-

computational speed and robustness take pri-

ing chemical reactions in reacting mix-

ority.

tures.

Since no single thermodynamic model

Qualita-

Although one is often able to

possesses all of these qualities, a exible ther-

t a simple cubic EoS to an associating

modynamic framework caters to these diering

(or reacting) binary system, models ex-

needs by implementing a range of thermody-

plicitly accounting for association (reac-

namic models.

The state-of-the-art modeling

tions) are needed to correctly predict the

approaches within each of the areas are as fol-

behavior of ternary or higher-order mix-

lows:

tures, due to cross eects such as several

Accuracy/consistency thermodynamic

molecules associating (reacting) with the The

models

most for

accurate

same molecule.

predicting

for some

properties of gases and liquids are multiparameter EoS. sistency

27,28

criteria

fullled,

we

We shall see that even mixtures, such as NH 3 -

CO2 , the performance of present EoS is

Although most con-

are

binary

very poor if reactions are neglected.

shall


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Industrial & Engineering Chemistry Research Cubic EoS Equation (SRK, PR) + α correlations (Classic, Twu, Mathias-Copeman,…) + Mixing rule (Quadratic, Huron-Vidal, Wong-Sandler,…) + Volume shift

In the following, we discuss present research challenges associated with some of the main components framework:

of

a

modern

thermodynamic

thermodynamic models and rou-

tines for calculating phase and reaction equilibria.

Multiparameter EoS (EoS-CG, BWR,…)

In this discussion, we shall deal with

challenges associated with each of the desired

Association contribution

qualities elaborated above.

Corresponding states EoS

An overview of some common thermodynamic models and how they are connected is illus-

PC-SAFT

CPA

trated in Fig. 2. The gure shows that there are

Phase and chemical reaction equilibria

many choices associated with cubic EoS, such as the type of alpha correlations, mixing rules, and incorporation of so-called volume shifts.

Thermodynamic properties

30

Further, the dashed lines elucidate that cubic

Figure 2: Overview of models. Dashed arrows

EoS are used as input in many thermodynamic

represent submodels, and names in parentheses

models such as corresponding state (CSP) mod-

represent choices.

els and the cubic plus association (CPA) EoS. Multiparameter EoS and Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT),

curacy. Multiparameter EoS have been devised

however, do not rely on cubic EoS. Incorporat-

for single-component uids

ing the association contribution in PC-SAFT

The real uid properties are in multiparameter

and CPA involves choosing the number of asso-

EoS dened in terms of the following reduced

ciation sites, as the gure shows.

Helmholtz energy function:

Finally we mention that an alternative to computing

thermodynamic

EoS, is to use the so-called

properties

γ−φ

αr (δ, τ ) =

from

approach:

Nk δ dk τ tk +

k

X

Here, the vapor phase is modeled with a conventional EoS (e.g.

X

SRK), while an activity

1318

and mixtures.

20

27

X

Nk δ dk τ tk exp −δ lk +

k

Nk δ τ exp −ηk (δ − k )2 − βk (τ − γk )2 dk tk

k (1)

coecient model is used for the liquid phase. These models will not be treated here due to

dk , tk , lk , Nk , k , γk , βk , ηk δ and τ are the

their severe thermodynamic inconsistency, e.g.

where

their inability to predict the existence of a

rameters, and

vapor-liquid critical point.

sity and temperature.

are tted pareduced den-

In addition, Gaussian

terms with prefactors that diverge at the critical point, called non-analytical terms, must

Present

challenges

with

es-

be introduced to reproduce experimental data very close to the critical point.

tablished methods

gions where thermodynamic data are available; however, they have challenges that restrict their

rameter EoS

popularity.

For instance, they give substan-

tially longer computational time than simpler

The desire to represent the available experi-

EoS,

mental data in a compact and precise manner

31

in particular for phase equilibrium cal-

culations, where it is challenging to achieve ro-

has motivated the development of state-of-the-

27

Multiparam-

eter EoS have unparalleled accuracy in the re-

Multiple Maxwell loops in multipa-

art multiparameter EoS.

27

bust and time-ecient calculations.

These equations are

32

The main reason for most of their current

founded on a comprehensive analysis of exper-

drawbacks is the second articial Maxwell loop

imental data and a diligent optimization pro-

in the two-phase region.

cedure, with functional forms optimized for ac-


27

Figure 3 demon-


20 10

;geq ;gs

P [MPa]

;leq

;ls

stable homogeneous phase within the spinodals, ρgs and ρ`s , where subscript s refers to the spinodals and superscripts g and l refer to gas

A Metastable regions ! 1

A

0 -10

@P @;

2

Peq

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