An Analytical Model for Real Gas Flow in Shale Nanopores
with Non-Circular Cross-Section
Wenxi Ren,Gensheng Li,Shouceng Tian,Mao Sheng,and Xin Fan
State Key Laboratory of Petroleum Resources and Prospecting,China University of Petroleum,
Beijing 102249,China
DOI 10.1002/aic.15254
Published online April 6,2016in Wiley Online Library ()
An analytical model for gas transport in shale media is propod on the basis of the linear superposition of convective flow and Knudn diffusion,which is free of tangential momentum accommodation coefficient.The prent model takes into the effect of pore shape and real gas,and is successfully validated against experimental data and Lattice–Boltz-mann simulation results.Gas flow in
noncircular nanopores can be accounted by a dimensionless geometry correction factor.In continuum-flow regime,pore shape has a relatively minor impact on gas transport capacity;the effect of pore shape on gas transport capacity enhances significantly with increasing rarefaction.Additionally,gas transport capacity is strongly dependent of average pore size and streamline tortuosity.We also show that the prent model without using weighted factor can describe the variable contribution of convective flow and Knudn diffusion to the total flow.As pressure and pore radius decrea,the number of molecule-wall collisions gradually predominates over the number of intermolecule collisions,and thus Knudn diffusion contributes more to the total flow.The parameters in the prent model can be determined from independent laboratory experiments.We have the confidence that the prent model can
provide some theoretical support in numerical simulation of shale gas production.V C 2016American Institute of Chemi-cal Engineers AIChE J ,62:2893–2901,2016
Keywords:shale nanopores,pore shape,real gas,Knudn diffusion
Introduction
Shale gas,due to its clean-burning and efficient nature,is becoming an increasingly promising alterna
tive energy resource.1,2Understanding gas transport in shale media pos a long-standing challenge to scientists and engineers.Recently,the unlocking of shale gas through advanced hori-zontal drilling and hydraulic fracturing technologies rais the necessity for a deeper understanding of gas transport in shale media.
The challenges that are faced in the study of transport prop-erties of shale media are twofold.The first part involves nano-scale phenomena,and the cond part involves complex microstructure of shale media.Shale is referred to as extraor-dinarily fine-grained diments that mainly consist of nano-pores.3,4In the ca of gas flow through nanopores,gas molecules not merely have intermolecular collisions,but also collide with the wall.The state of the gas is referred to as rare-fied,5which caus deviation from continuum flow.Further this deviation ris with increasing rarefaction.The Knudn number,defined as the ratio of the molecular mean free path to a reprentative physical length scale,which quantifies the rarefaction of fluid.In terms of Knudn number,the flow regimes can be classified as 6:continuum regime (Kn <1023),slip flow regime (1023<Kn <1021),transition flow regime
(1021<Kn <10),and free molecular regime (Kn >10).Due to the coexistence of various flow regimes,a variety of studies have been performed in an effort to cover all situations.The rearches can be b
roadly divided into two categories:micro-scopic methods bad on dilute gas kinetic theory and modi-fied continuum approaches.The former includes molecular dynamics (MD),direct simulation Monte Carlo,and Lattice–Boltzmann method (LBM).For the microscopic method,it is not currently feasible to simulate gas flow in field-scale porous media due to computational time and memory constraints.7The modified continuum approaches mainly include slip mod-els and volume diffusion hydrodynamic models.For slip mod-els,the main inconvenience is the estimation of slip coefficient,which is a function of the tangential momentum accommodation coefficient (TMAC).There is no universal methodology which can estimate or predict the TMAC.8In engineering application,TMAC is usually t as an empirical constant or ud as an adaption parameter for fitting experi-mental data.For example,underlying assumption that TAMC equals unity,the general slip boundary condition was propod by Karniadakis and Beskok.9Following this assumption,a ries of transport models bad on Karniadakis and Beskok general slip boundary condition 9have been developed,as summarized in Table 1.The volume diffusion hydrodynamic model is on the basis of the superposition of convection and diffusion.10To avoid the uncertainty of TMAC,the convective flow can be described by a solution of the Navier–Stokes equa-tions with no-slip boundary condition,which is justified in the
Correspondence concerning this article should be addresd to G.Li at ligs@
V
C 2016American Institute of Chemical Engineers AIChE Journal 2893
August 2016Vol.62,No.
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slip regime and the beginning of the transition regime.11The two regimes are dominant in shale gas production.12,13In this ca,slipflow can be considered as a part of Knudn diffu-sion.14In this work,we also adopt the volume diffusion hydro-dynamic approach and describe gas transport in shale media as a linear superposition of convectiveflow and Knudn diffusion. The pores orfluid channels in shale media are usually tortu-ous and noncircular.15However,the majority of existing trans-port models for shale media,are derived bad on the assumption of nanopores with circular cross-ction.This assumption may not be appropriate for accurately describing gas transport in shale media,becau the effect of cross-ction shape on gasflow in nanochannels is not negligi-ble.16,17To circumvent this problem,Singh and Javadpour18 developed an apparent permeability model by assuming that the permeability follows a log-normal distribution,which is the statistical sum of the individual permeability from cylin-drical tubes and rectangular channels.To determine the per-c
entages of cylindrical tubes and rectangular channels in rervoir rock,numerous image analys(Scanning Electron Microscopy(SEM)and Atomic Force Microscopy(AFM) images)are needed,which is tedious and time consuming. There are only a few studies that have investigated the effect of both pore shape and rarefaction on gas transport in shale media.Therefore,a robust and simple model capturing effect of pore shape and rarefaction is urgently needed.
The prent work is dedicated to dealing with the two chal-lenges mentioned above.To achieve this goal,a gas transport model is developed which requires no empirical parameters(such as TAMC)and considers the possibility of noncircular nanopores. The remainder of this article is organized as follows.Wefirst put forward the formulation of the prent model.Next,we validate the model by comparing its predictions with experimental data and LBM simulation results.Then we u the prent model to study gas transport dynamics in shale media.We also study the combined effects of pore shape and streamline tortuosity on gas transport capacity.The work ends with concluding remarks. Model
We modified the Veltzke and Th€o ming formulation10for rarefied gasflow in microducts to account for shale nanopores by incorporating tortuosity,porosity,and geometry correction factor of shale media.
The influence of streamline convolutedness onflow can be characterized by tortuosity25
s5
L a
L0
(1)
Where s is tortuosity,dimensionless;L a is the actual length of capillaries,m;L0is the length of capillaries,m.For shale res-ervoirs,the tortuosity value ranges from2.3to11.9,as stated by Katsube et al.26
For considering real gas effect,the ideal gas law is corrected by the compressibility factor
pV5ZnRT(2)
Where Z is the gas compressibility factor,dimensionless;R is universal gas constant,PaÁm3/(molÁK);V is volume,m3;n is the number of moles of gas,mol;T is temperature,K;p is pressure,Pa.
The compressibility factor can be estimated as a function of pudo-reduced pressure and pudo-reduced temperature27
Z50:702e22:5T r p2r25:524e22:5T r p r10:044T2r20:164T r11:15
ÀÁ
(3)
p r5
p
p c
(4)
T r5
T
T c
(5)
Where p r is pudo-reduced pressure,dimensionless;T r is pudo-reduced temperature,dimensionless;p c is the critical pressure of gas,Pa;T c is the critical temperature of gas,K. Gas viscosity can be estimated using the following correlation28
l5a0e b0q1000
ðÞc01027(6) a05
9:37910:01607M
ðÞ1:8T
ðÞ1:5
(7)
b053:4481
986:4
1:8T
10:01M(8)
c052:44720:2224b0(9) Where q is the gas density,kg/m3;M is gas molar weight,kg/ mol.
Table1.The Main Features of the Existing Gas Transport Models
Model Description Cons
Javadpour19(2009)Model developed bad on Maxwellfirst-order slip
boundary condition;Linear superposition of slip
flow and Knudn diffusion Only for straight capillary tube;Needs TMAC values; ideal gas
Civan20(2010)Model developed bad on Beskok and Karniadakis
general slip boundary condition Only for circular capillary;TMAC is assumed to be1; ideal gas
Darabi et al.21Bad on the Javadpour model;The effect of surface
roughness on Knudn diffusion is considered Only for circular capillary;Needs TMAC values;ideal gas
Rahmanian et al.22Model developed bad on Beskok and Karniadakis
general slip boundary condition;Weighted superpo-
sition of slipflow and Knudn diffusion Complex numerical model;TMAC is assumed to be1; Several empirical parameters;ideal gas
Singh and Javadpour18Model developed without the assumption of wall slip;
Linear superposition of convectiveflow and Knud-
n diffusion The model is a probabilistic combination bad on gas transport model for circular and rectangular capillaries
Wu et al.23Model developed bad on Beskok and Karniadakis
general slip boundary condition;Weighted superpo-
sition of slipflow and Knudn diffusion The model is a probabilistic combination bad on gas transport model for circular and rectangular capilla-ries;TMAC is assumed to be1
Yuan et al.24Model developed bad on Beskok and Karniadakis
general slip boundary condition;Considering the
fractal characters of nanopores in shale media Only for circular capillary;Needs the maximum and the minimum pore sizes;TMAC is assumed to be1; ideal gas
2894DOI10.1002/aic Published on behalf of the AIChE August2016Vol.62,No.8AIChE Journal
In gas kinetic theory,the mean free path can be defined as11
k5
1
ffiffiffi
2
p
pr2m n v
(10)
Where r m is the molecular diameter,m;n v is the number of molecules per unit volume,m23.
The number of molecules per unit volume can be determined from Avogadro’s number and the real gas law,leading to:
n v5nN A
V
5
nN A p
ZnRT
5
N A p
ZRT
(11)
Where N A is the Avogadro constant,mol21.Substituting Eq. 11into Eq.10,the mean free path for real gas can be expresd as:
k5
ZRT
ffiffiffi
2
p
pr2m N A p
5
Zk B T
ffiffiffi
2
p
pr2m p
(12)
The mean Knudn number for real gas can be written as:
Kn m5k
s
5
Zk B T
s
ffiffiffi
2
p
pr2
m
p m
(13)
p m5p in1p out
(14)
Where k is the mean free path for real gas,m;Kn m is the mean Knudn number for real gas,dimens
ionless;k B is the Boltz-mann constant,J/K;s is the reprentative physical length scale, m;p m is the average pressure,Pa;p in is the inlet pressure,Pa; p out is the outlet pressure,Pa.When Z51,Eq.13reduces to the expression of mean Knudn number for ideal gas.11
In this work,we adopt the volume diffusion hydrodynamic approach and describe gas transport in shale media as linear superposition of convectiveflow and Knudn diffusion.18For convectiveflow term,the pressure-driven,steady-stateflow through microchannels and nanochannels of arbitrary cross-ctional shape X with boundary@X can be described by the Poisson Equation.29
@2u @x21
@2u
@y2
5
1
l
d p
d z
(15)
Thefirst-order Maxwell boundary condition for slip velocity is
u5r22
r
k
@u
@n
at@X(16)
Where u is bulk velocity,m/s;k is the gas mean free path,m; r is the TMAC,dimensionless;l is gas visc
osity,PaÁs. Bahrami et al.30converted the nonhomogeneous boundary condition(Eq.16)to homogeneous boundary condition by intro-ducing a relative velocity,and got the average velocity for ellip-tical microchannel.Bad on the analytical solution of elliptical duct,they successfully predicted the Poiuille number of slip flow in microchannels of arbitrary“regular”cross-ction.The average axial velocity for elliptical microchannels is
u m5
c2e
411e2
ðÞ
D p
l L a
1
r22
r
k
p c2
e
D p
C el L a
(17) C54b e E
ffiffiffiffiffiffiffiffiffiffiffi
12e2
p
(18)
E5
ðp2
ffiffiffiffiffiffiffiffiffiffiffiffiffi
12e x2
p
d x(19)
Where u m is the average axial velocity,m/s;b e and c e are the major and minor mi-axes of the cross-ction,m;e is the ratio of the channel major and minor mi-axes,dimension-less;C is the perimeter of elliptical cross-ction,m;E(e)is the complete elliptical integral of the cond kind, dimensionless.
Theflow rate in elliptical duct can be expresd as
q es5u m A5
p c e4
4e11e2
ðÞ
11
r22
r
k
4p11e2
ðÞ
C e
D p
l L a
(20)
Where q es is theflow rate considering slipflow in elliptical duct,m3/s;A is cross-ctional area,m2.Bad onfirst-order and cond-order slip boundary condition and Eq.15,Duan and Muzychka29,31formulated an analytical solution for slip flow through rectangular channel.The complete solution is complex and space consuming,and not be repeated here. Using the solution,they successfully predicted the pressure drop of slip slow in microchannels and nanochannels of arbi-trary“regular”cross-ction.The solutions of Baharmi and Duan all include TMAC,which is a controversial issue.10To avoid the uncertainty of TMAC,we consider slipflow as a part of Knudn diffusion in nanopores.14Thus,without dou-ble considering slip term,the convectiveflow in nanochannel can be described by Eq.15with no-slip boundary condition. Accordingly,Eq.20becomes
q e5
p c e4
4e11e2
ðÞ
D p
l L a
(21)
Where q e is theflow rate without considering slipflow in ellip-tical duct,m3/s.
Similarly,the convectiveflow rate in circular channel can be expresd as
q cir5
p d4D p
128l L a
(22)
Where q cir is the convectiveflow rate in circular channel,m3/ s;d is channel radius,m.Equation22is the classical Hagen–Poiuille Equation.
Forfluidflow in natural porous media,Pickard32propod a modified Hagen–Poiuille Equation for noncircular capillaries.
q cirm5G
p D h4D p
128l L a
(23)
Where q cirm is the convectiveflow rate in noncircular capilla-ries,m3/s;D h is characteristic length scale or hydraulic diame-ter,m.The characteristic length scale is estimated for ellip
(51.4b e),for rectangle(52b r),and for circular(5d).Where
b r is the short axis of the rectangle,m;G is the shape factor dependent on the shape of channel,dimensionless,G51for circular.For detailed discussion of the values,interested readers can consult the available literature.33,34
The morphology of nanopores in shale media is diver and irregular.Becau of the diversity of nanopores in shale media, trying to capture the exact shape of each pore might be imprac-tical and daunting.23Without unduly complicating the analysis, we absorb G into a more general parameter a that quantifies the effect of pore shape onflow properties(as was done by Cai et al.35for natural porous media).Rearrange Eq.23to get
q cirm5
p ar e
ðÞ4
8l
D p
L a
(24)
Where a is the geometry correction factor,dimensionless, with a51for a circular;r e is equivalent pore radius,m.
AIChE Journal August2016Vol.62,No.8Published on behalf of the AIChE DOI10.1002/aic2895
Considering the effect of pore shape,the intrinsic perme-ability can be written as
k d 5
u a 4r e 28s
(25)
Where u is the porosity,dimensionless;s is the tortuosity,
dimensionless.Note that we u the term “intrinsic perme-ability”to refer to gas transport capacity computed with a con-tinuum model.In the following ction,we u the term “apparent permeability”to refer to gas transport capacity when rarefaction effect is considered.
Bad on Eq.24,the convective mass flow rate in shale media compod of tortuous noncircular nanopores can be written as
J C 5q
n t A A
q c 5q n t p ar e ðÞ48l D p s L 05q k d l D p
L 0(26)
Where J C is the convective mass flow rate,kg/(m 2Ás);n t is the total number of capillaries or pores per unit area,m 22.
Here our model is bad on the linear superposition of con-vective flow and Knudn diffusion.The Knudn diffusion term is capable of capturing the slip and Knudn flow,which supports the applicability of Eq.26to describe convective flow through noncircular nanopores in our study.
There are five well-known types of diffusion in porous media,including gaous (or molecular)diffusion,Knudn diffusion,surface diffusion,liquid diffusion,and configuration diffusion.36Configuration diffusion occurs when the molecu-lar diameter is comparable to the channel diameter.For shale media,the pores which are believed to transport and store gas are greater than the diameter of methane molecules.12Thus,configuration diffusion in shale nanopores i
s negligible.21Liq-uid diffusion is out of the scope of this article,becau we are primarily interested in dry gas flow.In addition,we rerve the investigation of surface diffusion for future work.
In shale nanopores,the diffusive transport is governed by Knudn diffusion.37According to Fick’s first law of diffu-sion,the diffusive flow through a single nanochannel can be expresd as
J 5D D C L a
(27)
Where J is the diffusive flow rate,mol/(m 2Ás);D is diffusion coefficient,m 2/s;C is concentration of gas,mol/m 3.
For Knudn flow in noncircular nanochannels,simply u of hydraulic diameter to modify Knudn diffusion coefficient for considering the effect of pore shape may lead to error of diffu-sive flux prediction.38,39A more accurate approach to calculate Knudn diffusion coefficient is to u the correlations estab-lished between shape factor and hydraulic diameter.16To handle diver and irregular nanopores in shale media,we u the geometry correction factor to quantify the deviation from circu-larity of the cross-ction of the pore that controls the flow prop-erties and express Knudn diffusion coefficient for ideal gas as:
D ki 523ar e ðÞu mi 523ar e ðÞ
ffiffiffiffiffiffiffiffiffi
8RT
p M r (28)Where D ki is the Knudn diffusion coefficient for ideal gas,
m 2/s;u mi is the molecular mean speed for ideal gas,m/s.For real gas,the molecular mean speed can be written as 40
u mr 5
ffiffiffiffiffiffiffiffiffiffiffi8ZRT
p M r (29)Where u mr is the molecular mean speed for real gas,m/s.
The combination of Eqs.28and 29gives Knudn diffusion coefficient for real gas
D kr 523ar e ðÞ
ffiffiffiffiffiffiffiffiffiffiffi8ZRT
p M r (30)The concentration C in Eq.27can be expresd by the amount
of substance n per unit volume V .Roy et al.6showed that Knud-n diffusion in nanopores can be written in the form of pressure gradient.The diffusive mass flow rate for shale media compod of tortuous noncircular nanoporesis described as
J D 5
n t A p r e 2A
MJ 5n t p r e 2MD kr D C s L 05u s MD kr 1L 0D p ZRT 5
u s q p D kr D p
L 0(31)
Where J D is the diffusive mass flow rate,kg/(m 2Ás).
Equation 31agrees with the model derived by Lira and Paterson 41for Knudn diffusion through nanoporous anodic alumina membranes when Z 51and s 51.Furthermore,for the special ca that a
51,the Eq.31reduces to the Knudn diffusion model for real gas flow through the reconstructed shale propod by Chen et al.42On the basis of linear superpo-sition of convective flow and Knudn diffusion,the total mass flow rate can be written as:
J T 5J C 1J D 5k d 11
u s D kr l pk d
q l D p
L 0(32)Where J T is the total mass flow rate,kg/(m 2Ás).
Bad on Eq.32,the apparent permeability can be written as:
k a 5k d 11
u s D kr l
pk d
(33)Where k a is the apparent permeability,m 2.
The correction factor of intrinsic permeability can be expresd as:
f c 511u s D kr l pk d 511163l a 3r e p ffiffiffiffiffiffiffiffiffiffiffi
8ZRT p M r (34)
Where f c is the correction factor of intrinsic permeability,
dimensionless.
Model Validation
We first compare the model results with experimental data of gaous flow in microtubes.The experiments referred to were performed in isothermal conditions around room temper-ature (Table 2)with nitrogen as working gas (Table 3).For convenience of comparison,a reduced mass flow rate is intro-duced,which is defined as the ratio of the total mass flow rate Eq.32to the convective mass flow rate Eq.26:
S 5J T J C 511
163l a 3r e p m ffiffiffiffiffiffiffiffiffiffiffi
8ZRT
p M r (35)Where S is the reduced mass flow rate,dimensionless.
Figure 1plots the comparison between the experimental data and the predictions by the prent model.The predictions agree satisfactorily with the experimental data.With increas-ing pressure,the dimensionless mass flow rate gradually con-verges to 1.This is becau that convective flow dominates gas transport at high pressure.
In addition,experimental data was collected for nitrogen,helium,carbon monoxide,and carbon dioxide flow through
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DOI 10.1002/aic
Published on behalf of the AIChE
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AIChE Journal
the nanoporous anodic aluminum oxide membranes,46which consist of straight pores normal to the planar surface of the membranes.The materials are uful for mass transport stud-ies becau their relatively uniform pore geometry or surface chemistry.47Table4lists the values of the membrane parame-ters.Due to the uncertainty of pore size and shape,41,48we treated the pore radius and the geometry correction factor as fitting parameters tofit the measured data.Figure2a depicts the comparison between the experimental data and the predic-tions by our model.The610%error lines are also shown in the plot.Values a51.07and r e54.65nm,were ud tofit
the experimental data successfully.Also note that most of the data fall within the610%error lines.Furthermore,we com-pared the model predictions assuming a circular cross-ction (a51and r e54.65nm)with the experimental data,as shown in Figure2b.As can been e in Figure2,the predictions assuming a circular cross-ction show more scatter than that of our model.The comparisons show that the good agree-ment between our model and experimental data can be ascribed to the geometry correction factor ud.
For further verification of the prent model,we compare the model predictions with Lattice–Boltzmann simulation results from Chen et al.42In their work,porous structure of shale was reconstructed using Markov Chain Monte Carlo method on SEM images.Then they ud LBM to sim
ulate real gasflow in reconstructed samples.Pore size distribution of the reconstructed samples is shown in Figure3.For testing the prent model,the equivalent pore radius and the geometry correction factor must be determined.We ud the pore size at 20%cumulative pore volume to reprent the porous media.49,50Bad on the Eq.34,the geometry correction fac-tor was regarded as an adjustable parameter,and determined byfitting the predictions to the simulations.Fairly good matches are obtained between the simulation results and the predictions,as shown in Figure4.The comparisons indicate that the prent model,which allows forflexible pore shape, can describe real gas transport in shale nanopores.
As shown in Figure4,with decreasing pressure,the correc-tion factor increas slowly,and then increas sharply to over 100at a pressure of0.068MPa.It indicates that as rervoir pressure depletes,the apparent permeability of shale matrix increas.In shale gas production,the loss of production due to depleted rervoir pressure is compensated by the incread apparent permeability,which is one of reasons that the shale matrix provides stable long-term production.51,52Results and Discussion
In the previous ction,a gas transport model was derived by considering the effect of pore shape and streamline tortuos-ity in shale nanopores.The model us the concept of a linear superposition o
f convectiveflow and Knudn diffusion.To understand the prent model more thoroughly,we compare the results from the prent model with four previous models in the literature,as shown in Figure5.For the sake of compari-son,we take gas compressibility factor as1.For all models, with increasing pressure,apparent permeability approaches intrinsic permeability.The model propod by Civan is bad on Beskok and Karniadakis general slip boundary condition,it only considers slip effect.Thus,compared with other models, the Civan model usually underestimates the apparent perme-ability,especially for low pressure(high Knudn number regime).The model propod by Javadpour takes into account the slip effect by bringing in correction factor F,which is a function of TMAC.The value of TMAC has a significant effect on the predicted apparent permeability.When TMAC increas,the apparent permeability decreas,as shown in Figure5.Thus the limitation in the Javadpour model is that it requires knowledge of TMAC.The measurement of TMAC for shale may be hampered by complex pore network topolo-gies,tortuous transport paths,and poorly characterized pore wall structures.The model propod by Wu et al.is on the basis of weighted superposition of slipflow and Knudn dif-fusion,where the weighted factor is a function of pressure.In the high pressure regime(2MPa<p<30MPa),the Wu et al. prediction is slightly lower than the Civan prediction.This is becau that,in Wu del,the slipflow rate is propor-tional to pressure,and its weighted factor is also proportional to pressure,which further reduces the contribution of slipflow to the totalflow when pressure decreas.In the low pressure
Table2.Values of Parameters in Gas Flow Experiments
References Geometry
Size
Gas
T
(K) Diameter
(m)
Length
(m)
Ewart
et al.43
Microtube25.2731026 5.331022N2298 Yamaguchi
et al.44
Microtube32031026 5.92531022N2293
Table3.Physical Properties of Working Gas Ud in the
Cited Literature45
Gas M(kg/mol)r(nm)
l(1025PaÁs)
290K293K298K
N228.01310230.3798 1.732 1.746
1.769
Figure1.Comparisons of the prent model predictions
and experimental data in microtubes.
The geometry correction factor and gas compressibility
factor ud in Eq.35are all equal to1.
Table4.Dimensions of the Membranes46
Anodic aluminum
oxide membranes Nominal radius(nm)$5
N*(m22) 4.631014
Length(m)6531026 *The total number of capillaries per unit area.
AIChE Journal August2016Vol.62,No.8Published on behalf of the AIChE DOI10.1002/aic2897
