Diffusion coefficients of methane in liquid hydrocarbons

ngineering 53 (2006) 47–60www.elsevier.com/locate/petrol

Journal of Petroleum Science and E

Diffusion coefficients of methane in liquid hydrocarbons at highpressure and temperature

M. Jamialahmadi a, M. Emadi b, H. Müller-Steinhagen c,d,⁎

a University of Petroleum Industry, Ahwaz, Iranb Department of Chemical and Process Engineering, University of Surrey, UK

c Institute for Technical Thermodynamics, German Aerospace Centre (DLR), Germanyd Institute for Thermodynamics and Thermal Engineering, University of Stuttgart, Germany

Received 27 January 2005; received in revised form 30 January 2006; accepted 31 January 2006

Abstract

The accurate prediction of diffusion coefficients of methane in liquid hydrocarbons is one of the key parameters for improvingthe prediction of compositional oil reservoir simulators, for designing surface facilities, and for high pressure gas/liquid masstransfer operations. In this investigation, a precision high pressure and temperature diffusion cell apparatus has been used tomeasure diffusion coefficients of methane in dodecane and in a typical Iranian crude oil at high pressures and temperatures. Thediffusion cell is modelled using a finite-domain moving boundary method. Extensive experimental results are presented at highpressures and temperatures, especially for regions where published data were insufficient. The mechanism of mass transfer duringthe incubation period is analysed by solving the equation of continuity for the diffusion cell numerically and considering that thediffusion coefficient is changing with time and concentration. The effects of pressure, liquid phase viscosity and operatingtemperature on the diffusion coefficient of the solute gas in the solution are discussed. The results indicate that uncertainties whichgenerally exist in the physical and thermodynamic properties of the crude oils may have a significant effect on the diffusioncoefficient of a solute gas in a complex mixture like crude oil.© 2006 Elsevier B.V. All rights reserved.

Keywords: Diffusion coefficients; High pressure; High temperature; Secondary oil recovery

1. Introduction

When a gas and a liquid phase which are notthermodynamically in equilibrium are brought intoclose contact, transfer of one or more components mayoccur from the gas phase to the liquid or, vice versa, bythe mechanism of molecular diffusion. Mass transfer by

⁎ Corresponding author. Institute for Technical Thermodynamics,German Aerospace Centre (DLR), Germany.

E-mail addresses: [email protected] (M. Jamialahmadi),[email protected] (H. Müller-Steinhagen).

0920-4105/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2006.01.011

molecular diffusion is the basic physical mechanismunderlying many important areas of soil science,petroleum engineering, chemical engineering, biotech-nology and nuclear engineering. In recent years, theproblem of mass transfer by molecular diffusion in oilreservoirs, when a non-equilibrium gas is injected intothe reservoir, has become increasingly important andreceived significant attention (i.e. Burger and Mohanty,1995; LeGallo et al., 1997). Nguyen and Faroug Ali(1998) measured the solubility and diffusivity of carbondioxide in Aberfeldy heavy oil using impure carbon dio-xide gas containing nitrogen as the main contaminant. It

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http://dx.doi.org/10.1016/j.petrol.2006.01.011



48 M. Jamialahmadi et al. / Journal of Petroleum Science and Engineering 53 (2006) 47–60

was noted that increasing the concentration of nitrogen inthe carbon dioxide stream decreased the solubility andthe diffusivity of carbon dioxide in the oil. Riazi (1996)developed a method for determining diffusion coeffi-cients of gases in liquids at constant volume andtemperature using a PVT cell. No compositionalmeasurements are necessary for this method of measur-ing diffusion coefficients. Results obtained by thismethod for binary systems yielded diffusion coefficientswithin ±5% of literature values. Zhang et al. (2000)discussed the importance of molecular diffusion for thestudy of petroleum recovery techniques. The mainobjective of their work was to develop an experimentaltechnique for measuring gas diffusion coefficients inheavy oils. They used Riazi's (1996) method andmeasured the diffusion coefficients of methane andcarbon dioxide in heavy oil at 21°C and pressures up to3500kPa. Comparing their results with published valuesof the diffusion coefficients of methane and carbondioxide, they found that deviations were much larger formethane, probably because of the different oils and testconditions. Upreti and Mehrotra (2002) measured thediffusivity of carbon dioxide, methane, ethane andnitrogen in Athabasca bitumen in the range 25–90°Cat 4 and 8MPa. The diffusivity of these gases has beendetermined as a function of gas concentration in thebitumen using a non-intrusive experimental method.The diffusivity of gases was found to increase withtemperature and pressure.

In oil recovery projects, gas is injected into oilreservoirs for different reasons such as maintenance ofreservoir pressure and enhanced recovery of oil. In thefirst case, some of the gas, which is injected into thedome above the oil zone, dissolves in the oil, thuslowering its viscosity, surface tension and density. Thesepossibly favorable changes resulting from re-pressuringof the oil reservoir create interest in the rate at which thegas may be expected to dissolve under differentprevailing conditions. Previous investigators (i.e. Hilland Lacy, 1934; Bertram and Lacy, 1935; Reamer et al.,1956) have shown that the rate of dissolution of a gassuch as methane in a quiescent body of hydrocarbon oilis controlled primarily by the rate of diffusion of thedissolved gas from the gas–oil interface into the body ofthe liquid phase.

In the second case, when gas is injected for thepurpose of enhanced oil recovery, understanding andmodelling of diffusion processes may be of greatimportance for the planning and evaluation of gasinjection projects in oil reservoirs. The most importantproperty required for determining the rate of masstransfer between the gas and oil phase is the molecular

diffusion coefficient at reservoir conditions. Kinetictheory has been shown to be adequate for estimationof molecular diffusion in low pressures gases throughthe Chapman–Enskog theory (1939). For liquidsystems, most correlations are based on the Stokes–Einstein equation, which relates inversely the diffusioncoefficient to liquid viscosity. Descriptions of thesecorrelations and conditions for which their applicationhas been recommended are summarized by Taylor andKrishna (1993) and by Poling et al. (2001).

For high-pressure systems, most theory-based corre-lations fail to perform properly, and for this reasonempirical correlations have been developed. Slatteryand Bird (1961) developed a chart for calculation ofself-diffusion coefficients of dense gases. This chartextends to a reduced pressure of four and is based onfew experimental data. Subsequently these charts wererecommended in the API Technical Data Book(American Petroleum Institute, 1997) for estimation ofbinary diffusion coefficients of dense hydrocarbonsystems. Mathur and Thodos (1965) developed ageneralized correlation for the self-diffusivity of sub-stances in the gaseous and liquid states in terms ofreduced density and reduced temperature. Sigmund(1976) reported some experimental data and a correla-tion for the prediction of binary diffusion coefficients ofdense hydrocarbons for both liquids and gaseoussystems. Even though most data used for developingthis correlation for liquid systems were at atmosphericpressure, it has found practical application in masstransfer calculations in the oil reservoirs at highpressure. Riazi and Whitson (1993) showed that theSigmund correlation predicts diffusivities for gasesdissolved in liquid hydrocarbons at high pressures,which are 80–100% higher than reported experimentalvalues. Denoyelle and Bardon (1983) reported diffusioncoefficients for carbon dioxide in oil at reservoirconditions, which are 5 to 10 times higher than thosemeasured at atmospheric conditions. They concludedthat measured diffusion coefficients at atmosphericconditions cannot be used at reservoir conditions.Renner (1986) measured diffusion coefficients ofhydrocarbon gases in consolidated porous media atpressures up to 6MPa (900psi). He also developed acorrelation as a function of liquid viscosity andthermophysical properties of the diffusing gases. Riaziand Whitson (1993) proposed a generalized correlationin terms of viscosity and molar density of solutions forboth gases and liquids. Although the proposed correla-tion is based on experimental data of hydrocarbonsystems, they claimed that it may also be used for non-hydrocarbon systems.

49M. Jamialahmadi et al. / Journal of Petroleum Science and Engineering 53 (2006) 47–60

Unfortunately, none of the available correlations issuitable for the prediction of diffusion coefficients ofhydrocarbon gases in hydrocarbon liquids at the highpressures and temperatures which are commonly foundin oil reservoirs. The reason for this discrepancy is thelack of reliable high-pressure and temperature moleculardiffusion experimental data. The data at high pressureare scarce and fragmentary, at least in part, for the reasonthat performing these measurements in the laboratory isdifficult, expensive and time consuming. The aim of thepresent investigation is to study systematically themechanisms of molecular diffusion of gases in liquidsby measuring diffusion coefficients of methane indodecane, which has physical properties similar tocrude oil and in a typical Iranian crude oil, up to apressure of 40MPa at different temperatures. In thispaper, experimental results are presented over a widerange of pressure and temperature, especially for regionswhere previously published data were insufficient. Alltests are performed in an accurate high-pressurediffusion cell with “finite-domain” moving boundarybehaviour. The effect of solute concentration andviscosity on the diffusion coefficient is demonstratedexperimentally. At present, the data acquired are used toassess the predictions of various available correlations

Fig. 1. Schematic drawing of th

for diffusion coefficients. A Radial Basis Function(RBF) neural network is trained for the prediction ofdiffusion coefficients of methane in hydrocarbonliquids. Ultimately, the trained network will be used todevelop a simple approximate parametric correlation forthe prediction of diffusion coefficients of a solute gas inliquid hydrocarbons.

2. Experimental equipment and procedure

2.1. Test apparatus

The complete diffusion apparatus is shown in Fig. 1.The diffusion cell is surrounded by a temperature-controlled air bath. To increase the temperature of thediffusion cell from ambient to the desired values up to180°C, several thermostatically controlled electricalheaters have been installed in various parts of the airbath. The mean body of the bath consists of a heavyangle iron frame, covered with aluminum alloy sheetand insulated with a 5-cm thickness of compressedglass wool. The interior is lined with insulation board.The bath is partitioned such that a duct is separatedfrom the area provided for the cell. The duct housesthe heaters, the thermostat and the blower, which

e experimental apparatus.


provides forced air circulation through the bath.Louvers installed at the duct outlet permit adjustingthe air stream. Furthermore, the bath is also equippedwith a high-pressure stainless steel reservoir cylinderwith a volume of 500cm3. This cylinder, which isused to store methane gas at the desired temperatureand pressure, is connected to the top of the diffusioncell. The diffusion cell is a cylinder with a volume of400cm3. Three glass windows are arranged in the faceof the cell so that any liquid level may be determinedvisually. The readings are obtained with a precisecathetometer, which is equipped with a sliding scopeand illumination system, allowing the operator thesafety of observing volume changes through a mirrorimage rather than facing the pressurized windowedcell directly. The cross-sectional area of the cell borehas been designed so that a 1-mm fluid level changecorresponds to a volume change of 1cm3. Thecathetometer permits readings to 0.01cm3. The cellis supported in the air bath by a journal, about whichit can be revolved from a vertical position with oneend of the cell up through the horizontal to a verticalposition with the other end of the cell up. Two high-pressure stainless steel tubing spirals connect the inletand outlet of the cell to the inlet valves of the bathassembly. Three shallow holes are drilled into the cellwall for inserting thermocouples to determine whentemperature equilibrium has been reached. Thediffusion cell permits examination of gas and liquidsamples for pressures up to 70MPa (10000psi) andtemperatures up to 180°C. The injection of gas andliquid phase at the desired pressure was performedusing fully automatic twin mercury pumps. The flowrate in each pump is variable from 1cm3/h to about1500cm3/h. The operating pressure of each pump isup to 70MPa (10000psi) and a calibrated ruler isinstalled between the two pumps, to determine theoutlet volume with an accuracy of 0.01cm3.

2.2. Experimental procedure

In preparation for a typical diffusion test, the cell wascleaned, dried and the system connected to a vacuum toremove air or any other gases present in the system.Once the pressure of the system has reached approxi-mately 0.025kPa (0.2mm Hg), the liquid is introducedby injection from the bottom upward and filled up toabout 35% of the cell volume. Pure methane is alsointroduced into the reservoir cylinder of the air bath atthe desired pressure. Then, the main power of the airbath heaters is switched-on and the temperature of thesystem allowed to rise very gently up to the desired

temperature. Various combinations of main heaters mayhave to be used in order to reach this temperature. Fivethermocouples, three on the diffusion cell and two in themain body of the bath, record the temperatures invarious parts of the diffusion cell and the air bath. Thesystem was degassed and then left at the desiredpressure and temperature for about 10h to ensure ahomogeneous condition throughout the system. Now,the top three-way valve of the cell was connected to thehigh-pressure gas reservoir cylinder. When the pressurein the cell reached the desired value, the top valve wasclosed and the liquid level recorded. As time passes,methane diffuses into the liquid phase, therefore, theliquid level rises and its volume increases. The pressurein the diffusion cell was kept constant by continuousinjection of mercury to the bottom of the cell. From themovement of the interface and the position of the pistonrod of the mercury pump, the volume of dissolvedmethane in the liquid phase may be readily determinedas a function of time. Acquisition of a complete set ofvolume–time data usually requires 2–6days. Thediffusion coefficient may then be determined from thevolume–time data. At the end of each experimentsamples of the liquid and gas phases were withdrawnfrom the bottom and top parts of the cell, respectively.The liquid sample was collected in a cylinder; thevolume of gas separated from the liquid was measuredin a gasometer. The composition of the gas phase wasanalyzed by a gas chromatograph to check thepossibility of evaporation of the liquid phase, whichproved to be negligible. All experimental runs wereperformed in random fashion and some runs wererepeated later to check the reproducibility of theexperiments, which proved to be good.

2.3. Test fluids

Diffusion experiments were performed using puremethane as gas phase, pure normal dodecane and atypical Iranian crude oil as liquid phase. Accurateknowledge of physical properties and phase behavior ofmethane-dodecane and methane-crude oil under variousoperating temperatures and pressures is an essentialprerequisite for determination of diffusion coefficientsfrom volume–time data. Limited gas solubility mea-surements were performed using a standard PVT cellapparatus. The system is based on the principle that gasdissolved in the liquid phase evolves when the pressureis released and the temperature of the system isincreased. The solubility of methane in the liquidphase was measured at several pressures and tempera-tures. In each run, the equilibrium was detected by

Fig. 3. Diffusion coefficient of CO2 in decane as function of pressure.

Fig. 2. (a) Solubility of the methane in dodecane and molar volume ofsolution as a function of pressure and temperature. (b) Solubility ofmethane-crude oil and molar volume of solution as a function ofpressure at different temperatures.


noting a constant value on the pressure gauge of thePVT cell over a period of 1week. Then, a sample ofliquid was withdrawn from the bottom of the cell.Knowing the volume of gas liberated from the sampleand the weight of the liquid phase, the solubility of thegas phase was calculated. These data were used to tunethe Peng–Robinson equation of state in the range ofoperating pressures and temperatures of this investiga-tion. This equation was then used to determine liquiddensity, viscosity, molar volume and saturation concen-tration of methane in the solution. Fig. 2a and b presentthe results of solubility of methane and molar volume ofthe solutions versus pressure for methane-dodecane andmethane-crude oil at different temperatures, respective-ly. The solubility of methane in the liquid phaseincreases as pressure is increased or temperature of thesystem is decreased. Contrariwise, the molar volume ofthe solution decreases with pressure and increases withincreasing temperature.

3. Experimental results and discussions

To check the calibration of the test rig, some controlruns were performed using the CO2/decane system. Inrecent years, several investigators (i.e. Renner, 1986;Grogan et al., 1988) have used this system to studydiffusion of gases in liquids up to pressures of 10MPa(1500psi). Fig. 3 shows a comparison between reporteddiffusion coefficients and those obtained in thisinvestigation, as a function of pressure. For the wholerange of pressure, the agreement between the presentresults and the values reported by other investigators isvery good, confirming the reliability of the equipmentused in this investigation.

Fig. 4a and b show the typical variation of the heightof the liquid phase as a function of time for pressuresranging from 3.5 to 35MPa (500 to 5000psi) formethane-dodecane and methane-crude oil, respectively.Similar results have also been obtained for othertemperatures. The rate of swelling of the liquid phaseis increased significantly with increasing operatingpressure. The general shapes of these curves arecharacterized by an initially sharp increase in heightfollowed by gradual increase towards the saturationconcentration of methane in the liquid phase, at thespecified conditions. The extent of the variation of theliquid height is strongly affected by the adjusted pressureand temperature. Liquid height in the diffusion cell athigher pressures increases faster and to a larger extentthan at lower pressures. The variation of the liquid heightwith time, which results from the diffusion of methaneacross the interface, is equal to the rate of variation of theliquid phase volume:dVV

¼ d υACP

Að Þ ð1Þ

Fig. 4. (a) Height of the liquid phase as a function of time and pressurefor the methane-dodecane system. (b) Height of the liquid phase as afunction of time and pressure for the methane-dodecane system.

Fig. 5. (a) Concentration of methane in dodecane as a function of timeand pressure for T=65°C. (b) Concentration of methane in crude oil asa function of time and pressure for T=25°C.


where C̄A is the mean concentration of methane in theliquid phase defined by

CP

A ¼

Z V

V0

CAdV

V−V0ð2Þ

Integration of Eq. (1) between the limits of V0 to Vt

gives:

CP

A ¼ 1υA

lnVt

V0¼ 1

υAln

ZtZ0

ð3Þ

When the dimensionless product υACA≪1, Eq. (3)may simplify to:

CP

A ¼ Zt−Z0ZtυA

ð4Þ

Eq. (3) is used to calculate the average methaneconcentration in the liquid phase as a function of timeat various operating pressures and temperatures.

Typical results are plotted in Fig. 5a and b forpressures ranging from 3.5 to 35MPa, for methane-dodecane and methane-crude oil, respectively. Therate of dissolution of methane in the liquid phase isincreased significantly with increasing pressure, anddecreasing temperature. The trend of these curves isconcordant with the variation of the liquid heightwith time (Fig. 4a and b). The concentration ofmethane in the liquid at high pressures increasesfaster and to a larger extent than at low pressures.The dissolution of the methane results in loweringviscosity, surface tension and density of the solution.Fig. 6a and b display the relationship betweenviscosity and density of the solution as a functionof pressure for methane-dodecane and methane-crudeoil systems at different temperatures, respectively. Thedissolution of methane in the liquid phase caused asubstantial decrease in the viscosity and density ofthe solution. Maximum reduction in viscosity and

Fig. 6. (a) Variation of viscosity and density of the solution withpressure for the methane-dodecane system. (b) Variation of viscosityand density of the solution with pressure for the methane-crude oilsystem.

Fig. 7. One-dimensional diffusion process in a test diffusion cell.


density of the solution is observed when all methanegas had dissolved in the liquid phase. Furtherincrease in the pressure increases the density andviscosity of the solution gradually. The early stages ofdissolution are often affected by convective mixingarising from initially high mass transfer rates andsurface tension driven instabilities. This initial periodor “incubation period” (Chukwuma, 1993; Renner,1986) increases as operating pressure increases. Sincethe effects of the incubation period decay withincreasing contacting time, middle and late timedata are more reliable for estimation of diffusioncoefficients from the experimental data.

3.1. Determination of diffusion coefficient fromexperimental data

A number of mathematical models have beenproposed to determine the diffusion coefficients fromexperimental volume–time profiles (Civan and Ras-

mussen, 2002). All these models are developed fromthe equation of continuity for the solute component:

ACA

AtþjdCAu

V ¼ −jdJVA þ rA ð5Þ

Referring to Fig. 7, for a one-dimensionaldiffusion cell in the absence of chemical reaction,and including the movement of the interface in theboundary conditions of the system, Eq. (5) takesthe form:

ACA

At¼ D

A2CA

Ax2ð6Þ

The use of Fick's second law to describe thediffusion at high pressure is justified if it isconsidered that the diffusion coefficient can beeasily modified by a thermodynamic factor for non-ideal mixtures (Riazi, 1996):

D ¼ Da 1þ Aln/Alnx

� �ð7Þ

Initial calculations showed that ignoring this correc-tion for the present systems leads to an error of less than2.5%.

Before the start of the diffusion process, theconcentration of solute in the liquid phase is neg-ligible. Thus, the initial condition of the diffusion cellis:

CA ¼ 0 for t ¼ 0 and 0VxVZt ð8ÞAccording to the film theory of Whitman (1923), the

gas and liquid phases at the interface (i.e. x=Zt) arethermodynamically in equilibrium, i.e. the interfacial


concentration of solute, CAi, remains unchanged as longas temperature and pressure of the system are keptconstant. Hence, the first boundary condition of Eq. (6)is:

CA ¼ CAi for t > 0 and x ¼ Zt ð9ÞAfter methane and liquid phase dodecane are brought

into close contact mass transfer commences, andmethane diffuses across the interface into the liquidphase. As a result, the liquid phase swells and theinterface moves to a new position Zt. Depending oncontacting time, height of liquid, and rate of masstransfer, the system may be considered as “semi-infinite” or “finite-domain” moving boundary system.

The semi-infinite system assumption is valid as longas the solute does not penetrate to the bottom of thediffusion cell during the course of the experiment. Thiscondition prevails when the diffusion cell is long or thetime of contact between gas and liquid phase is short.Hence, the second boundary condition of Eq. (6) can bedefined as:

CA ¼ 0 for x ¼ 0 and tz0 ð10ÞThe solution of Eq. (6) subject to the initial and

boundary conditions defined by Eqs. (8)–(10) is givenby Crank (1975):

CA x; tð Þ ¼ CAi 1−erfx

2ffiffiffiffiffiDt

p� ��

¼ CAierfcx

2ffiffiffiffiffiDt

p� �

ð11Þ

The total mass of methane transferred into the liquidphase after time t can be calculated from the integrationof Eq. (11) over the volume of the diffusion cell:

mA ¼Z V

−lCA x; tð ÞdV ¼ S

Z Z

−lCA x; tð Þdx

¼ 2SdCAi

ffiffiffiffiffiDt

p

r¼ Kd

ffiffit

p ð12Þ

where

K ¼ 2SdCAi

ffiffiffiffiDp

rð13Þ

Eq. (12) shows that a plot of mA versus the squareroot of time should provide a straight line with a slope ofconstant K. Hence, the diffusivity of the solute in theliquid phase can be obtained from Eq. (13) bycalculating the value of K from the experimental data.Most investigators (i.e. Renner, 1986; Grogan et al.,1988) used this method for determination of diffusion

coefficients. The mass average concentration of solutecan be obtained from the combination of Eq. (12) withEq. (1):

C̄A ¼ mA

V¼ mA

SdZt¼ 2CAi

Z0expðυAC̄AÞ

ffiffiffiffiffiDtp

rð14Þ

The application of the semi-infinite solution limitsthe experimental data which can be used to determinediffusion coefficients to those at early times of theexperiments. This limitation is a serious drawback ofthis method as early time experimental data areaffected by convective mixing arising from initiallyhigh mass transfer rates, especially at high pressurewhere the solute has a high solubility in the liquidphase. This initial mixing effect decreases withincreasing contacting time, hence middle and latetime data are less affected and provide better estimatesof diffusion coefficients.

3.2. Finite-domain moving boundary model

From the experimental point of view, finite systemsprovide conditions for more precise estimates of diffusioncoefficients. These systems allow design and constructionof small and compact apparatus, which is a great advantagefor working at high pressure and temperature. When liquidin the diffusion cell comes into close contact with the gasphase, diffusion commences. However, since the cell issealed and is of finite length, there can be no mass transferacross the ends of the cell. Therefore, the second boundarycondition of Eq. (6) for finite systems is:

ACA

Ax¼ 0 for x ¼ 0 ð15Þ

The solution of Eq. (6) in conjunction with its finiteboundary conditions may be performed assuming thatthe diffusion coefficient itself is either independent ordependent on solute concentration in the liquid phase.Therefore, the diffusivity may remain constant over theentire course of the diffusion process, or alternatively, itmay change as diffusion time increases and the solutionbecomes more concentrated with the solute.

3.2.1. Concentration-independent diffusion coefficientIntegration of Eq. (6) with the assumption of constant

diffusivity between x=0 and x=Zt yields:

Z Z

0

ACA

Atdx ¼ D

Z Z

0

A2CA

Ax2dx ¼ D

ACA

Ax

��Z

ð16Þ


Applying Leibnitz's integration rule (Stroud, 1996),Eq. (16) simplifies to:

ddt

Z Z

0CA x; tð Þdx ¼ D

ACA

Ax

��Zt

þ CAiAZtAt

ð17Þ

Differentiating Eq. (1) with respect to time leads to:

dC̄A

dt¼ 1

Ztd1dt

Z Z

0CA x; tð Þdx− C̄A

ZtddZtdt

ð18Þ

which can be combined with Eq. (17) to give:

ZtdC̄A

dt¼ CAi−C̄A

� dZtdt

þ DACA

Ax

��Zt

ð19Þ

Substituting for Zt and its derivative from Eqs. (2)and (3) yields:

dC̄A

dt¼ CAi−C̄A

� υA þ D

Z0exp υAC̄A

� d ACA

Ax

��Zt

ð20Þ

Eq. (20) relates the rate of diffusion of the soluteinto the liquid phase to the concentration profile at theinterface. The main problem in the solution of thisequation is that the interfacial concentration profile∂CA /∂x|Zt

is unknown. Do and Pinczewski (1991)have shown that for moving boundary systems it issatisfactory to assume a variable power concentrationprofile for the solute throughout the diffusion process:

CA ¼ b0 þ b1xn ð21Þ

where the power n is a function of time and given by:

n ¼ ax where x ¼ CAi

C̄Að22Þ

The parameter a is positive and independent of timeand should be determined from experimental data. Theinterfacial and mean solute concentrations at any timecan be calculated from Eq. (21) and its combination withEq. (2), respectively:

CAi ¼ b0 þ b1Znt ð23Þ

and

C̄A ¼ b0 þ b1nþ 1

Znt ð24Þ

Substituting for Zt and n from Eqs. (4) and (22) intoEq. (24) yields:

C̄A ¼ b0 þ b1axþ 1

Z01−υC̄A

� �ax

ð25Þ

The parameters b0 and b1 can be obtained in terms ofCAi and CA from the simultaneous solution of Eqs. (23)and (25):

b0 ¼ C̄A þ C̄A

a1−xx

� �ð26Þ

and

b1 ¼ CAi−C̄A

� axþ 1ax

� �1−υC̄A

Z0

� �ax

ð27Þ

The interfacial concentration gradient can also becalculated by differentiating Eq. (21) with respect to x:

ACA

Ax

��Zt

¼ b1dndZn−1t

¼ ðCAi−C̄AÞðaCAi−C̄AÞð1−υAC̄AÞZ0C̄A

ð28Þ

Substituting Eq. (28) into Eq. (20) gives:

dC̄A

dt¼ Dð1−υAC̄AÞ3ðCAi−C̄AÞðaCAi−C̄AÞ

Z20 C̄Að1−υCAiÞ

ð29Þ

Integrating Eq. (29) using the separation of variablesmethod yields:

1a7

lnðaCAi þ C̄AÞa1

ðCAi þ C̄AÞa2ð1−aC̄AÞa3=aþ a42aa7ð1−aC̄AÞ2

þ a5aa7ð1−aC̄AÞ

þ a6a7

¼ Ddt ð30Þ

The parameters α1 to α7 are constant and given inTable 1.

As already stated by Do and Pinczewski (1991), theconcentration profiles defined by Eq. (14) are not yetaffected by the non-flux boundary condition during theearly stages of the diffusion process. Hence, at thebeginning of the diffusion process, the prediction of Eq.(30) should coincide with the semi-infinite solution ofEq. (14). The validity of the variable profile concentra-tion of the solute is demonstrated in Fig. 8, where themass of carbon dioxide transferred into the liquid phasepredicted from Eq. (30) is compared with valuesobtained from Eq. (14), for several values of theadjustable parameter a. Excellent agreement betweenEqs. (14) and (30) is obtained with a value of a=0.56 atshort contacting times, where the finite solution of Eq.(30) is identical to the semi-infinite solution. Thus, a plotof the left-hand side of Eq. (30) versus time should give astraight line and the slope of such a plot is the diffusivity.

Table 1Coefficients in Eq. (30)

Parameters Equation

Parameter α1a1 ¼ −aCAi

CAið1þ aCAiÞð1þ aÞ

Parameter α2a2 ¼ 1

ð1þ aÞð1þ υACAiÞ3

Parameter α3 a3 ¼ aða1−a2Þ

Parameter α4 a4 ¼ υA

ðυACAi−1ÞðaυACAi þ 1Þ

Parameter α5a5 ¼ −

a1aCAi

þ a2CAi

þ a3 þ a4

� �

Parameter α6 a6 ¼ −a1ln aCAið Þ þ a2ln CAið Þ− a42a

−a5a

Parameter α7a7 ¼ 1

Z20 ð1−υACAiÞ

Fig. 9. (a) Left-hand side of Eq. (29) versus time for methane-dodecanesystem at different pressure. (b) Left-hand side of Eq. (29) versus timefor methane-crude oil system at different pressure.


Fig. 9a and b show typical plots for several test runs ofmethane-dodecane and methane-crude oil systems,respectively. The experimental data conform well tothe straight line predicted by Eq. (30) after the incubationperiod. The diffusion coefficients are determined fromthe slopes of these straight lines by a linear least squarestechnique. The results of these calculations are given inFig. 10a and b as a function of pressure for methane-

Fig. 8. Predictions of CO2 concentration from Eq. (29) for variousvalues of parameter a.

dodecane andmethane-crude oil at various temperatures,respectively. It can be seen that the diffusion coefficientof methane in the liquid phase increases steadily withincreasing operating pressure. The maximum value ofthe diffusion coefficient appears at a pressure of about31MPa. This pressure corresponds to the point at whichthe gas phase has disappeared and all methane gas isliquefied. As the pressure is increased further, theviscosity and the density of the solution are alsoincreased and, as a result, the diffusion coefficientdecreases steadily.

3.2.2. Concentration-dependent diffusion coefficientIn the literature, few attempts are made to describe

the variation of the diffusion coefficient with time as thesolution becomes concentrated with solute gas duringthe incubation period. A number of methods areavailable to solve the diffusion cell model numericallywhen the diffusion coefficient varies with time and

Fig. 10. (a) Diffusion coefficient of methane in dodecane as a functionof pressure for different temperatures. (b) Diffusion coefficient ofmethane in crude oil as a function of pressure for differenttemperatures.


solute concentration. The diffusion cell model may berearranged as:

ACA

At¼ d

AxDACA

Ax

� �ð31Þ

An implicit finite-difference method is used to solveEq. (31) subject to the following finite domain movingboundary conditions:

at x ¼ zt; CA ¼ CAi and

at x ¼ 0;ACA

Ax¼ 0

ð32Þ

The finite difference approximations are:

ACA

At

��CJi

¼ Cjþ1i −Cj

i

Dtand

A2CA

Ax2

��Cji

¼ Cjþ1iþ1−2C

jþ1i þ Cjþ1

i−1

Dx2ð33Þ

Replacing the derivatives by the mean of their finite-differences, Eq. (31) can be approximated by:

kCjþ1i−1 −ð1þ 2kÞCjþ1

i þ kCjþ1iþ1 ¼ −Cj

i ð34Þwhere the variable λ is defined as:

k ¼ Ddt

ðDxÞ2 ð35Þ

Solving Eq. (34) for Cij+1 result in:

Cjþ1i ¼ 1

1þ 2kCji þ k Cjþ1

i−1 þ Cjþ1iþ1

�h ið36Þ

Since all the values of Ci0 are known, all the values

of Ci1 can be calculated and in this manner the

remaining concentrations can be computed. However,there are two unknowns in Eq. (36), namely theconcentration profile at time t+Δt and the diffusioncoefficients at these concentrations, D(Ci

j+1). Toovercome this problem, the successive substitutionmethod is used. In this technique, the diffusioncoefficients at the new concentrations, as a firstapproximation, are set equal to the previous value,i.e. D(Ci

j+1)=D(Cij). With this assumption the new

concentration profile can then be calculated. Once thenew profile is computed, the volume average soluteconcentration can also be obtained from Eq. (3). Next,the values of D(Ci

j+1) can be computed and the sametime step must be repeated to calculate the value of Ci

j+1

more accurately. The computations have been per-formed in the MatLab environment and the iterativesubstitution continued until the resulting volume-average concentrations coincided with the valuesobtained from the experiment. In the programming,non-equal time steps have been used. At the beginning,where the concentration profiles are steep, small timesteps are used; as the diffusion process proceeded, thetime steps have been increased. This method requiresthe volume-average concentration versus time, theboundary conditions and the first estimate for thediffusion coefficient.

Typical results of the numerical calculations areillustrated in Fig. 11a and b, where the computeddiffusion coefficients are plotted as a function of timepressures of 13.8MPa and 28.9MPa, respectively, at65°C. The computed volume-average concentrationprofile calculated from the predicted diffusion coeffi-cients is also depicted in these figures. As expected, theexperimental data and the predicted values are perfectlycoinciding. The general shape of the diffusion coefficientversus time curves is characterized by an initial sharpdecrease in diffusion coefficient (Region 1), followed by

Fig. 12. (a) Diffusion coefficient of methane in dodecane as a functionof liquid phase viscosity and temperature. (b) Diffusion coefficient ofmethane in crude oil as a function of liquid phase viscosity andtemperature.

Fig. 11. (a) Variation of diffusion coefficient and methane concentra-tion with time for p=13.8MPa. (b) Variation of diffusion coefficientand methane concentration with time for p=28.9MPa.


a gradual decrease towards a constant value (Region 2).Region 1 corresponds to the incubation period wheremass transfer occurs by both molecular and convectivediffusion mechanisms. Convective diffusion decaysquickly with increasing contact time, even though thisis strongly affected by the adjusted pressure. Withincreasing pressure, the extent of Region 1 and theoperating time within this region increases significantly.The convective diffusion at high pressures decays slowerand to a smaller extent than at low pressures. In Region 2,mass transfer from the gas phase into the liquid phase iscontrolled by molecular diffusion. In this regime, thedependency of diffusion coefficient on contacting timeand solute concentration is weak, which is in agreementwith the advanced kinetic theory of Hirschfelder et al.(1954), which predicts only a small effect of compositionon diffusion in binary mixtures. Near the end of thediffusion operation, when the entire gas phase hasdissolved in the liquid phase, the rate of mass transferdecreases towards zero and, consequently, the soluteconcentration in the liquid phase approaches a constant

value. The diffusion coefficient predicted from Eq. (30)is also displayed in Fig. 11a and b. Excellent agreementexists between the diffusion coefficients obtained fromEq. (30) and the mean molecular diffusion coefficients inRegion 2 where mass transfer is controlled by themechanism of molecular diffusion.

3.3. Effect of liquid phase viscosity and temperature

Almost all correlations available for the prediction ofdiffusion coefficients of gases in liquids show that at agiven temperature the diffusion coefficient dependsprimarily on the liquid phase viscosity. For this reasonseveral investigators (Riazi and Whitson, 1993; Hayduket al., 1973; Swapan and Buttler, 1996; McManameyand Woollen, 1973; Grogan et al., 1986) have tried tocorrelate their experimental data with respect to theliquid phase viscosity. The variation of the diffusion

Table 2Coefficients of Eq. (36)

Coefficients Methane-dodecane system Methane-crude oil system

T=45°C T=65°C T=81°C T=25°C T=50°C

a0 0.24719×10−8 0.26309×10−8 0.27494×10−8 0.2304×10−8 0.25028×10−8

a1 −1.0006 −1.00012 −1.00024 −1.009 −1.00717


coefficient with liquid phase viscosity is shown in Fig.12a and b for methane-dodecane and methane-crude oilsystems at various temperatures, respectively. At eachtemperature, the diffusion coefficient decreases steadilyas the liquid phase viscosity increases. Liquid phaseviscosity also decreases with increasing liquid phasetemperature. Therefore, the diffusion coefficient alsoincreases as liquid phase temperature increases. Thedependency of diffusion coefficient on liquid phaseviscosity may be correlated as:

DðT ;PÞ ¼ a0la1 ð37Þ

where a0 and a1 are coefficients which can be obtainedfrom the experimental data. The reported values for thecoefficient a0, varies from 0.5×10−10 to 0.4×10−11. Onthe other hand, almost all investigators reported that thecoefficient a1 is negative and varies from −0.45 to −1.2.This indicates that the diffusion coefficient has aninverse relationship with liquid phase viscosity. Theexperimental data for methane-dodecane fit nicely toEq. (37) at each temperature, as shown in Fig. 12a. Thecoefficients of the correlation are summarized in Table 2for the methane-dodecane and methane crude oilsystems.

Although Eq. (37) fits the diffusion coefficient ofmethane in dodecane very well, its predictions forthe crude oil are subject to considerable error. Thereason for this inconsistency arises from the presenceof various components in the crude oil and the tracesof heavy metals which may have a significant effecton the diffusion coefficient of methane in the crudeoil. The uncertainties which exist in the physical andthermodynamic properties of this solution may alsohave a significant effect on the determination ofdiffusion coefficients of methane in crude oil.

4. Conclusions

The diffusion coefficients of methane and carbondioxide in liquid hydrocarbons have been measuredover a wide range of pressures and temperatures, using aprecision high pressure diffusion apparatus. The diffu-sivities are determined using the continuity equationwith finite-domain boundary conditions, with the

assumption that the diffusion coefficient itself is eitherindependent or dependent on the solute concentration inthe liquid phase. The results indicate that during theincubation period mass transfer occurs by both convec-tive and molecular diffusion mechanisms and that theextent of this period increases significantly as operatingpressure increases. After the incubation period, masstransfer is mostly controlled by the mechanism ofmolecular diffusion, and the diffusion coefficient of thesolute gas in the liquid hydrocarbons increases asoperating pressure and temperature increase. Further-more, in this period, the diffusion coefficient is almostindependent of solute concentration. The maximumvalue of diffusion coefficient appears at a pressurecorresponding to the point at which the gas phase hasdisappeared and all gas has dissolved in the liquid phase.As the pressure is increased further, viscosity anddensity of the solution increase and, as a result, thediffusion coefficient decreases.

Nomenclaturea0, a1 Coefficients of Eq. (37)b0 and b1 ConstantsCA Mass concentration of solute (kg/m3)CA(x,t) Solute concentration at position x at time t (kg/

m3)D Diffusion coefficient (m2/s)Da Activity-corrected diffusion coefficient (m2/s)K Defined by Eq. (12)jA Mass transfer by the mechanism of molecular

diffusion (kg/m2s)mA Mass of solute (kg)MW Molecular weight (kg/kmol)n Variable powerp Pressure (Pa)rA Rate of reaction (kg/m3s)S Cross-section area of diffusion cell (m2)t Time (s)T Temperature (K)u Velocity (m/s)V Volume (m3)Z Position of interface (m)w Synaptic weightsx Coordinate direction (m)


Greek symbolsα1–α7 Constantsλ Defined by Eq. (34)ϕ Fugacity coefficientμ Viscosity of solution (kg/m s)ν Molar volume (m3)ρ Density (kg/m3)ω Ratio of interface and average solution

concentration

Subscript and superscriptA SoluteG Gas phaseL Liquid phasex Coordinatei Initialt At time, t0 Standard condition

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Diffusion coefficients of methane in liquid hydrocarbons