establishing the maximum carbon number for reliable quantitative gas chromatographic analysis of...

Establishing the Maximum Carbon Number for Reliable QuantitativeGas Chromatographic Analysis of Heavy Ends Hydrocarbons. Part 2.Migration and Separation Gas Chromatography ModelingDiana M. Hernandez-Baez,† Alastair Reid,† Antonin Chapoy,†,* Bahman Tohidi,† and Roda Bounaceur‡

†Hydrates, Flow Assurance & Phase Equilibria Group, Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH144AS, Scotland, United Kingdom‡Laboratoire Reactions et Genie des Procedes (LRGP), UPR 3349 CNRS, Nancy Universite, ENSIC, 1, rue Grandville, BP 20451,54001 NANCY Cedex, France

ABSTRACT: Reservoir fluid characterization by high-temperature gas chromatography (HTGC) extends the range of singlecarbon number (SCN) groups in oil analyses by temperature programming up to 450 °C. However, the reliability of HTGCanalyses is questionable for two main reasons: first, possible pyrolysis of the injected oil inside the GC column which couldinduce overestimation of light and intermediate fractions; and second, possible incomplete elution of heavy fractions, which inturn would induce under-estimation. The former has been treated in the first paper of this series,1 which focused on predictingthe pyrolysis temperature of n-alkanes (nC14H30−nC80H162) at GC conditions. The latter is the focus of this second paper whichintroduces a gas chromatography migration and separation model for the n-alkane range nC12H26−nC62H126 in an HT5 column,using as main input the in-house distribution factors derived from isothermal GC retention time measurements. On the basis ofthe developed model, the concentration and velocity of the above n-alkanes were determined at every point and time throughoutthe GC column, for typical temperature-programmed analyses. Retention times were then predicted, and validated againstexperimental values, with an overall relative error within 2%. This study gives an insight into the components’ behaviorthroughout the GC column, allowing preliminary assessment of elution, by proposing a new approach for determining the non/incomplete elution of every component by introducing: the degree of elution, defined as the amount of component which has beeneluted in relation to the amount injected. Thus, the degree of elution of each of the heavy n-alkanes studied in this work:(nC12H26−nC62H126) has been calculated for a typical temperature program. This new approach can be applied, in order todetermine the analytical conditions required for ensuring maximum elution of a given component, with the possibility ofimproving the practice of HTGC by optimizing the separation process.

1. INTRODUCTION

Gas chromatography is a separation technique for compounds,which also provides information regarding their concentrationsin a mixture. The components are required to be sufficientlyvolatile and thermally stable in order to perform a reliable gaschromatography analysis.The thermal stability of heavy hydrocarbons during High

Temperature Gas Chromatography (HTGC) has been a concernfor some authors2 based on the results of thermal gravimetricanalysis (TGA) published by Schwartz et al.3 who reportedthermal decomposition of the sample starting at ca. 370 °C.It is therefore very important to be able to model the thermal

cracking of heavy n-alkanes at HTGC condition in order to verifythese findings requiring as input data: carrier gas pressure,temperature, and concentration of every component through theGC column. This aspect has been reported in the first paper ofthis series,1 analyzing a range of concentrations rather than anspecific concentration.Therefore, computer simulation of gas chromatography

becomes necessary to complete the previous study of HTGClimits by predicting the precise concentration of everycomponent inside the GC column which can then serve asinput to the Pyrolysis model.Gas chromatography modeling also provides an insight into

the migration/separation of the sample at each point of the

column, for both isothermal and temperature programmed GCanalysis, and thus potentially to optimize the partitioningprocess.Twomodels have been developed in the present study: one for

solving the diffusion-convection equation4 using finite elementssolved by COMSOL,5 which enables the concentration profile tobe obtained; and another, solving a simplified iterativeconvection equation6 using MATLAB, which allows theretention times to be obtained more quickly. The retentiontimes obtained with the two models have been compared withthe experimental results, and due to the superior results, whichhighlighted the superior performance of the convection model,and led to its selection for optimizing the calculation time of theconvection-diffusion model.The main input used in the GC modeling is the database of

distribution factors derived from isothermal GC analyses of then-alkanes (nC12H26−nC62H126)

7 on an HT5 capillary column.On the basis of the developed gas chromatography model, a

new approach for determining the non/incomplete elution ofevery component has been proposed in this study by introducinga new approach: the degree of elution, defined as the amount of

Received: December 7, 2012Revised: March 5, 2013Published: March 9, 2013

Article

pubs.acs.org/EF

© 2013 American Chemical Society 2336 dx.doi.org/10.1021/ef302009n | Energy Fuels 2013, 27, 2336−2350

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component which has been eluted in relation to the amountinjected. Thus, the degree of elution of each of the heavy n-alkanes studied has been calculated for a typical temperatureprogram.

2. BASIC APPROACHS AND TERMS IN GASCHROMATOGRAPHY

Capillary gas chromatography is a well-established technique forseparating constituent compounds in a mixture between twophases: a gaseous “mobile phase” assumed to behave ideally inmost GC applications;8 and a “stationary phase” consisting of aliquid bonded to, and distributed on the interior surface of theopen tubular column. The mobile phase transports the mixturedownstream within the column, while each component re-equilibrates between the two phases, after every displacement at agiven temperature and pressure. The differences in thecomponents’ partitioning ratios thus permit their separation.When the separate analytes elute from the column in

combination with the mobile phase, the mixture passes througha detector [generally a flame-ionization detector (FID)]generating a response which indicates the presence of the solute.The FID response to each solute should be ideally proportionalto the solute amount or concentration, which is normally the casefor hydrocarbons.Ideally, the chromatograms (plot of detector signal) should

represent each solute as a vertical line, but as it migrates along thecolumn it instead occupies a zone (or band) whose widthgradually increases with time due to the dispersion of thecomponent in the mobile and stationary phases.Blumberg9 has well explained two important approaches that

will be used in the next section of this document: the solute zone,which corresponds to the space occupied by a solute migrating ina column; and the solute peak, which corresponds to the timethat the solute zone will take for eluting from the column.Ideally, using the probability theory, a solute injected very

sharply (as a delta function), under the action of moleculardiffusion, migrates in accordance with the random walk model,which states that at every time-step, each particle will travel thesame space-step, either forward or backward with equalprobability. Then, at the limit of many steps, using the CentralLimit Theorem the probable location of each particle approachesa normal distribution. Thus, the distribution of molecules alongthe column may be represented by a Gaussian zone (particles/unit length) which elutes from the column as a nearly Gaussianpeak (particles/unit time). Therefore, the width of the solutezone and solute peak may be described by its standard deviationsmeasured in units of length and time, respectively.The specific mass profile (particles/unit length)4,9 for every

analyte can be obtained from the Gaussian distribution of theanalytes through the column10,11 (probability density function[particles/unit length]) and the transverse area of the GCcolumn, yielding at (t = 0) the eq 1:

σ π σ=

·−

−⎡⎣⎢

⎤⎦⎥m x t

x x( , )

12

exp( )

200

2

2(1)

Here, σ corresponds to the standard deviation (in space units)of the amount of component throughout the GC column, x0corresponds to the centroid of its Gaussian distribution and xcorresponds to the position of the component’s dispersal aroundthe centroid x0.Since the analytes initially present in the mixture injected into

the GC column will not only diffuse but travel at the flow velocity

of the carrier gas by advection throughout the column, theconcentration profile of the analytes will vary with time and spaceaccording to the convection/diffusion conservation of massequation, explained in the following section.

3. MASS BALANCE (DIFFUSION-CONVECTION)EQUATION IN GAS CHROMATOGRAPHY

Zone broadening under time-variant and nonuniformconditions(coordinate dependent: such as the density gradientof the carrier gas caused by the pressure drop), which changefrom the inlet to the outlet of the column can be described by aone-dimensional convective-diffusion mass-balance equation,after the Taylor14 reduction of the cylindrical coordinate, mass-conserving equation for solute migrating in a capillary tube.11 Assuch, the resulting equation is applicable to either isothermal ortemperature programmed gas chromatography.This approach was developed by Golay,4 taking into account

the presence of a retentive layer, and became the most widelyused equation in the theoretical analysis of chromatography in anonuniform time-invariant linear medium. The nonuniformity ina chromatographic medium was considered a few years later byGiddings12−15 by dividing a column into small, equal segments,and assuming that the local conditions within each approachuniformity. They are then represented with any requiredprecision when the number of segments becomes sufficientlylarge.13 The mass balance of the solute8,9 in an infinitely thin sliceof column is described by eq 2.

υ∂∂

= − ∂∂

·

+ ∂∂

· ∂∂

⎡⎣⎢

⎤⎦⎥

m x tt x

m x t x t

xD x t

m x tx

( , )[ ( , ) ( , )]

( , )( , )

eff

eff(2)

Here, m(x,t) is the mass per unit length; veff and Deff representthe effective cross-sectional average velocity and apparentdiffusion obtained after multiplying the original values of vMand D by the frontal ratio “R = 1/(1+k)” or fraction of moleculesin the mobile phase to those in the stationary phase. D is theapparent diffusion coefficient (assumed to be of physical interestonly in the x-direction4) which represents all factors causingdispersion in a zone;16 and vM is the velocity of migration of thecarrier gas. The retention factor k is the ratio of moles of solute inthe stationary phase to moles in the mobile phase.The separation is assumed to be linear, i.e., the diffusivity and

velocity of the solute are independent of concentration.17

Another consequence of the linearity assumption is thepossibility of treating individually, each component of a complexmixture, enabling its migration to be studied separately.17

Although both the velocity of the analyte and its dispersion ateach specific location are functions of the coordinate of thelocation, the distance traveled, x is insufficient for predictionpurposes as the mass balance will not be conservative. Therefore,a general theory of chromatography in a nonuniform, time-variant medium has been introduced, based on a more generalequation of convective diffusion in a one-dimensionalmedium.10,17

The relationship of band broadening to the kinetics of masstransfer in gas chromatography, has been described and validatedin open tubular columns by Golay,4 who expressed the columnplate height (H(x,t)) as a spatial rate of dispersion of a zone (eq3), and the apparent diffusivity D, as a representation of thezones’ temporal dispersion rate (eq 4).

Energy & Fuels Article

dx.doi.org/10.1021/ef302009n | Energy Fuels 2013, 27, 2336−23502337

σ =x

H x tdd

( , )2

(3)

σ υ= · = ·t

H x t x t D x tdd

( , ) ( , ) 2 ( , )2

M (4)

Thus, Golay4 derived an exact equation to relate the bandbroadening and the kinetics of mass transfer in gas chromatog-raphy for open tubular columns with a smooth retentive coating,in very good agreement with experimental results.

υυ= · +

+ · + ·· +

·

+ ·· +

·

⎪

⎪

⎪

⎪

⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥⎫⎬⎭

H x tD x t

x tx t

k T t k T tk T t

rD x t

k T tk T t

wD x t

( , ) 2( , )( , )

( , )

1 6 ( ( )) 11 ( ( ))24 [1 ( ( ))] ( , )

2 ( ( ))3 [1 ( ( ))] ( , )

M

MM

2

202

M

2

2

S (5)

Golay,4 compared his chromatographic expression for thecolumn plate height of circular cross-section, coated tubularcolumns (eq 5), with the van Deemter equation for the HETP(Height Equivalent to a Theoretical Plate) of packed columns(eq 6). The Eddy-diffusion term, A which represents thediffusion caused by the multiple paths taken by the carrier gasflowing through a packed column is eliminated, there being but asingle flow-path option in a coated tubular column.

υυ= + + ·H x t A

Bx t

C x t( , )( , )

( , )M

M(6)

The first term corresponds to the B term in the van Deemterequation (eq 6), which represents the static longitudinaldiffusion; the second term related to DM is absent in the vanDeemter equation (eq 6), and represents the dynamic diffusionof the sample; and the last term, related toDS represents the masstransfer, and corresponds to the C term.Golay4 called this term the “hysteresis diffusion” of the sample,

representing the diffusion of the sample between the gas−liquidinterface and within the liquid phase.4

Therefore, by virtue of (eqs 4 and 5), it is possible to derive theequation representing the local dispersion term which can beexpressed by eq 7, and which depends on the static longitudinaldiffusion, the dynamic diffusion and the diffusion by forwardmass transfer in the stationary phase:

υ= +

+ · + ·· +

·

+ ·· +

·

⎪

⎪

⎪

⎪

⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥⎫⎬⎭

D x t D x tx t

k T t k T tk T t

rD x t

k T tk T t

wD x t

( , ) ( , )( , )

2

1 6 ( ( )) 11 ( ( ))24 [1 ( ( ))] ( , )

2 ( ( ))3 [1 ( ( ))] ( , )

MM

2

2

202

M

2

2

S (7)

In summary, the gas chromatographic migration andseparation of a sharply injected sample, can be described bythe diffusion-convection mass balance equation (eq 2) by meansof effective diffusion, and velocity of the sample as a function ofretention factor; diffusivity in both the stationary and mobilephases; and column specification as internal ratio, and filmthickness.

4. ITERATIVE RETENTION TIME PREDICTION BYCONVECTIVE MIGRATION ONLY

The use of discretization methods for calculating the retentiontimes has been introduced by Snijders.6 In his approach thediffusion effects are considered to be negligible in determiningthe peak position, enabling it to be described only byconvection.18

The convection can be expressed by the effective velocity ofthe analyte in the carrier gas (eq 19), which leads to thefollowing:

υ υυ

= = ·+

=+

β

x txt

x tN

N Nx t

( , )dd

( , )( , )

1i

i

i iK T teff, M

,M

,M ,S

M( ( ))i

(8)

Then, discretization of the velocity into finite time-steps leadsto (eq 9), which can be used to track the average position of theanalyte at every time step, and hence prediction of the retentiontime of the analyte, when it reaches the column outlet.6

υ= +

+·Δ

β

+x xx t

t( , )

1i i

i iK T t1

M( ( ))i i

(9)

5. TIME AND COORDINATE-DEPENDENTPARAMETERS IN GC CALCULATIONS

The application and solution of the transient diffusion-convection mass balance (eq 2) for temperature-programmedgas chromatography require that all the parameters involvedpreviously should be expressed as a function of time andcoordinate.5 The calculation of these parameters is treated in thefollowing section.In all of the simulations carried out, each of the parameters has

been related to the two main dependent variables, time and x-coordinate.

5.1. Coordinate-Dependent Pressure. By virtue of Boyle’sLaw, the average carrier gas flow velocity under steady state(constant mass flow of carrier gas through any cross-section ofthe column at any given time interval), can be expressed asfollows:

υ υ υ= · = = · = ·P x x t P t P x x t( 0) ( 0, ) (0) (0, ) ( ) ( , )M M M(10)

The steady-state motion of the carrier gas in capillary gaschromatography is described by the differential form of theHagen−Poiseuille equation (left-hand part of (eq 11)).19,20 It isobtained by relating the carrier gas velocity at any position in thecolumn, to the pressure gradient at that point21 by a proportionalconstant q. Substituting the expression of velocity (vM) from (eq10) into the left-hand part of (eq 11), we obtain the right-handpart of (eq 11) which relates the local pressure drop at position x,with the initial value of velocity and pressure:

υυ

= − · = − ··P x

xq x t q

P tP x

d ( )d

( , )(0) (0, )

( )MM

(11)

Thus, the Hagen−Poiseuille equation can be applied to adifferential element in gas chromatography by the assumption ofincompressibility of the gas in such an element at position x, dueto its extremely low pressure drop.20

By integrating (eq 11), in the inlet and outlet position, with Pinand Pout, respectively, and rearranging we obtain the expression



(eq 12), which permits calculation of pressure at any position inthe column:

= − − ·P x P P PxL

( ) ( )in2

in2

out2

2(12)

For the purpose of this study, an SGE HT5 column of 12 mlength has been used (Table 1). The profile of pressure drop with

x-coordinate has been calculated assuming the outlet pressure tobe atmospheric and the inlet pressure has been set to be 119.6 ±0.5 kPa, in accordance with gas hold-up time measurements, t0for methane.5.2. Time-Dependent Temperature. For a temperature

programmed analysis, eq 13 describes the temperature rampfollowed by the GC oven.

= + ·ΔT t T T t( ) ramp0 (13)

This has been used for the purpose of the simulationspresented in this article, e.g., the basic temperature programshown in Table 2.

5.3. Viscosity of the Carrier Gas (ηm). The carrier gasviscosity can be assumed to be dependent only on temperatureand therefore independent of pressure as long as density changescaused by the pressure drop are negligible.

The expression used in the case where the carrier gas is helium,has been introduced by Kestin22 and simplified by Hawkes,23

giving the viscosity in μPa·s.

η

α

αα

α

= · ·Ω

= + · ·

= − *

* =

= −= −=

= +·Ω

− ·Ω + · ·

− ·*

− ·*

− ·*

Ω = · · +*

+*

+*

−

⎪

⎪

⎪

⎪

⎧⎨⎩

⎡⎣⎢

⎤⎦⎥⎫⎬⎭

⎡⎣⎢

⎤⎦⎥

T t T tf

f

T

T T t

abc

E

aT

bT

cT

aT

bT

cT

( ( )) 0.7840374 ( )

13

196(8 10 )

13.65299 ln( )

( ( ))/10.4

0.1265161.230553

2.171442

11

42

0.00635209

2ln( )

3ln( )

4ln( )

0.00635209 1.04ln( ) ln( )

ln( )

1/2

7 2

2

3 4 5

22 3

4(14)

This algorithm can be applied for temperatures above 104 K(−169 °C), where viscosity predictions for the HTGCtemperature range show a maximum deviation of about 0.5%.However, a correction can be made over the range of 300−700 Kor 25−425 °C.23 The derived values from eq 14 may beoptimized by multiplying a correction factor,{0.995 + (T-300)·2.5 × 10−5}, to match experimental data within 0.1%.23

The carrier gas viscosity is a function of temperature, which inturn is a function of time when temperature programming isinvolved (Table 2). Figure 1 illustrates the increase in viscosity ofHelium with temperature and therefore with time, using atemperature ramp of 15 °C/min. Its viscosity increases from 19.4× 10−6 Pa·s at 10 °C (at time 0) reaching a maximum of 36.3 ×10−6 Pa·s at the upper temperature limit of 425 °C (27.67 min).

Table 1. Column Dimensions of in-House HTGC

SGE HTS GC column

length 12 mdiameter 0.53 mmfilm thicknes 0.15 μm

Table 2. Temperature Programming

n-alkanes

T0 (°C) 10hold at T0 (min) 0ramp of T (°C/min) 15Tmax(°C) 425hold at Tmax (min) 12

Figure 1. Viscosity of Helium as a function of time. [Temperature programming (Table 2): (From 10 to 425 °C, ramp of temperature:15 °C/min).]



5.4. Velocity of the Mobile Phase (VM). The proportion-ality constant q of (eq 11) for circular cross-section columns4 is:

η= ·q

T tr

8 ( ( ))

02

(15)

Then by integrating eq 11 from the inlet to the outlet and usingBoyle’s Law (eq 10), we obtain an expression describing thevelocity profile of the mobile phase as a function of temperature,

and therefore of time; and also as a function of pressure, andtherefore of the x-coordinate position.5,8,19−21

υη

=· −

· · ·x t

r P PT t L P x

( , )( )

16 ( ( )) ( )M02

in2

out2

(16)

On the basis of the data in Table 1, the pressure has beencalculated as a function of x-coordinate for the 12 m, HT5column; and similarly the viscosity has been expressed as a

Figure 2. Velocity of Mobile phase as a function of time and x-coordinate. [Temperature programming (Table 2): (From 10 to 425 °C, ramp oftemperature: 15 °C/min)].

Figure 3.Diffusivity of n-alkanes (SCN: 20−60) in Helium. [Temperature programming (Table 2): (From 10 to 425 °C, ramp of temperature: 15 °C/min).]



function of temperature and hence time, using temperatureprogramming (Table 2).In relation to temperature, the maximum velocity of the

mobile phase (helium) is found at the lowest values, and hence atthe earliest times in the temperature program since the velocity isinversely proportional to its viscosity, which in turn increaseswith temperature (eq 14).In relation to the x-coordinate (i.e., distance traveled in the

column), maximum velocity of the mobile phase is found at thehighest value, at the column outlet, i.e., at x = L. This is becausevelocity is also inversely proportional to the pressure (eq 16), andthe pressure drop is at maximum at the outlet of the column, atatmospheric pressure.

Thus, the velocity may vary from 0.35 m/s at the highesttemperature (latest time) and the highest pressure (at the GCcolumn inlet, and therefore the highest pressures) to 0.76 m/s atthe lowest temperatures and lowest pressures, approaching thecolumn outlet, as shown in Figure 2.Therefore, carrier gas velocity at the column inlet decreases as

temperature increases with analysis time, a consequence of whichis that the rate of desorption of the heavier components retainedat the column inlet reduces as the analysis proceeds.

5.5. Diffusion Constant, Mobile Phase (DM). Thediffusion constant in the mobile phase may be calculated fromthe empirical method of Fuller, Schettler, and Giddings.24

Figure 4. Effective Velocity of nC20H42 and nC60H122 in (He) in a 12 m × 0.53 mm × 0.15 μm HT5 column. [Temperature programming (Table 2):(From 10 °C to 425 °C, ramp of temperature:15C/min).]



= =· · +

· ∑ + ∑D x t D x t

T t

P x v v( , ) ( , )

0.00100 ( )

( ) [( ) ( )]

MW MWM AB

1.75 1 1

A

1/3

B

1/32

A B2

(17)

Here, MWA and MWB are the molecular weight of thecomponent in the sample, and of the carrier gas, respectively, andυA and υB are the special atomic diffusion volumes calculated as asum of all of the atomic diffusion volume increments (reportedby Fuller et al.24) of the atoms involved in the molecule ofinterest. Thus, the greater the number of carbon atoms, thegreater is the value of atomic diffusion volume.

The variation of the diffusivity in helium of n-alkanes; nC20H42and nC60H122, has been analyzed under the temperature programdescribed in Table 2.According to eq 17, the greater the temperature, the greater is

the diffusivity of n-alkanes in helium; and conversely, the greaterthe pressure, and the heavier the n-alkane, the lower is thediffusivity in helium.Therefore, the highest values of diffusivities apply to the

lightest n-alkanes, at the highest temperatures (latest elutiontimes) and the lowest pressures (approaching the GC columnoutlet), as shown in Figure 3.Thus, nC20H42 at 425 °C (time >27.67 min) and atmospheric

pressure (when approaching the GC column outlet) has the

Figure 5. Effective Diffusivity of nC20H42 and nC60H122 in (He) in a 12m × 0.53 mm × 0.15 μmHT5 column. [Temperature programming (Table 2):(From 10 to 425 °C, ramp of temperature: 15 °C/min).]



highest diffusivity of 6.06 × 10−5 m2/s; whereas nC60H122 has thelowest diffusivity value of 7.85 × 10−6 m2/s at the lowesttemperature of 10 °C (initial time) and greatest pressure, at theGC column inlet.Nevertheless, temperature is the most influential factor in the

diffusivity of n-alkanes in He, as can be seen in Figure 3, where atthe lowest temperature of 10 °C there is little evidence ofvariation of diffusivity with pressure and SCN(single carbonnumber) groups, compared with higher temperatures.5.6. Diffusion Constant, Stationary Phase (DS). The

diffusion constant in the stationary phase is a very importantparameter in gas chromatography, even although there is nomodel, to date, providing good accuracy for all systems whichinclude a liquid solvent.5 For that reason an approximate value iscalculated from its relationship with the diffusion constant in themobile phase6 by the following expression:

=·

D x tD x t

( , )( , )

5 10SM

4 (18)

Accordingly, the diffusivity of n-alkanes in the stationary phaseis directly proportional to their diffusivity in helium, andtherefore the same correlations apply with temperature, the x-coordinate, and SCN.The highest value of diffusivity in the stationary phase is 1.21×

10−9 m2/s and corresponds to nC20H42 at 425 °C (time >27.67min), and atmospheric pressure (approaching the GC columnoutlet).Thus, the lowest value of diffusivity in the stationary phase is

1.57 × 10−10 m2/s, corresponding to nC60H122 at the lowesttemperature (at initial time), and greatest pressure (approachingthe GC column inlet).5.7. Effective Velocity (veff). The effective velocity is an

average of the fraction of sample which flows in the mobile phase,equal to 1/(1 + k), andmoving at the velocity of themobile phasevM, and the fraction of sample which has been retained by thestationary phase with zero velocity, equal to k /(1 + k).

υυ

=+

x tx t

k T t( , )

( , )1 ( ( ))i

ieff,

M

(19)

The effective velocities of nC20H42 and nC60H122, have beenanalyzed under the temperature program shown in Table 2.Figure 4, clearly shows that the effective velocity has a low

dependency on pressure and therefore on the x-coordinate. Thiscontrasts with its high dependency on temperature, and thereforetime, due to the retention factor being a function only oftemperature (time). Consequently, its values are high comparedwith the values of velocity of mobile phase, which are a functionof both pressure and temperature. Thus, temperature is thepredominant influential variable on a component’s effectivevelocity in the mobile phase because of its powerful effect onretention factor.The retention factor of a component determines its effective

velocity, since the fraction of the component moving at thevelocity of the mobile phase is given by the former. Therefore, ata given temperature, the greater the retention factor of acompound, the more strongly it will be retained in the stationaryphase, and therefore the lower will be its fractions in mobilephase, making the effective velocity lower.This explains why nC20H42 achieves a higher effective velocity

at a lower temperature more quickly than nC60H122, as shown inFigure 4.

5.8. Effective Diffusivity (Deff). In order to obtain theeffective diffusivity (eq 20), an analogous averaging method isused as with Veff. The effective diffusivity correspond to thefraction of sample which is found in the mobile phase, equal to 1/(1 + k), with a local dispersion D. The local dispersion takes intoaccount its static longitudinal diffusion, the dynamic diffusionand the diffusion by mass transfer forward the stationary phasesaccording to (eq 7):

=+

D x tD x t

k T t( , )

( , )1 ( ( ))i

ieff,

(20)

The effective diffusivities of two n-alkanes: nC20H42 andnC60H122 have been analyzed under the temperature programdescribed in Table 2.As in the case of the effective velocity, Figure 5, shows that the

effective diffusivity exhibits a low dependency on pressure, andtherefore on the x-coordinate, compared with its high depend-ency on temperature and therefore time.Nevertheless, the local dispersion D (eq 7), is not the same for

every component, as is the case of the effective velocity, wherethe velocity of the mobile phase is independent of the proportionof the components flowing throughout the column. Rather, localdispersion takes into account the diffusivity of every componentin both the mobile and stationary phases, resulting in largevariations between different components, depending on temper-ature, and hence retention factor.Similarly to the effective velocity, the effective dispersion

depends on the fraction of component which dwells in themobile phase “1/(1 + k)”. Since this fraction is greater for thelightest components at a given temperature, then the effectivedispersion will also be greater.Thus, the heavier the component, the lower is its effective

dispersion at a given temperature, which explains why nC60H122

takes longer to elute than nC20H42, since a higher temperature isrequired to release it from the stationary phase.The large difference in dispersion between that observed for

nC20H42 compared with nC60H122, simply reflects the fact thatnC60H122 is retained longer on the stationary phase withsignificant vaporization not occurring until temperature ismuch nearer the isothermal maximum temperature hold uptime (Table 2). As a result, smoother changes in diffusivity areevident. Conversely, large changes occur in greater measure inthe case of nC20H42, where the temperature at which thestationary phase starts to release the component is achievedduring the temperature ramping period.

5.9. Retention and Distribution Factor. Knowledge ofhow the distribution factor varies with temperature is an essentialrequirement in gas chromatography when temperature-pro-gramming is the most common practice in order to accelerateanalysis of solutes with a wide range of boiling points.Application of a time-dependent function of distribution factor

enables calculation of retention factors, and hence prediction ofretention times7 (eq 9). It also permits simulation of theconcentration profile inside the column by solving (eq 2), andtherefore optimization of the separation of complex mixtures.

5.10. Thermodynamic Equilibrium of the Solvation inGC. The solvation of a solute in the bulk (three-dimensional)14

solvent can be expressed at thermodynamic equilibrium by thelogarithm of the solute molecule’s numerical density ratio in bothphases:25,26



ρρ

= = − Δ·

⎡⎣⎢⎢

⎤⎦⎥⎥ K

G TR T

ln ln( )L

AG

A (21)

The distribution coefficientK involves the ideal behavior of thegas phase and infinite dilution, with assumptions of negligibleinteraction between solute−solute and solute-carrier gas, withthe main interaction occurring between the solute and stationaryphase. In addition, interfacial and extra-column effects on themass transfer, which lead to nonequilibrium conditions, areexpected to be negligible.27

Under the above conditions, the isothermal retention timescan be expressed by eq 22, where the distribution factor has beenreplaced by the first two terms of the Taylor series expansionwhich has been treated in terms of thermodynamic properties byCastells et al.28 yielding a semiempirical model.19,29

Here, ΔH and ΔS, represent the changes in enthalpy, andentropy associated with the transfer of solute from the stationaryphase to the mobile phase at a given temperature T.

β β= · + = + · + ·

= Δ = − Δ

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

t tK T

tt

a aT t

aS TR

aH T

R

1( ) 1

( )

( );

( )

r ii

, M MM

0 1

0 1

(22)

Aldaeus18 has proposed two retention mechanisms accordingto the nature of the separation hold between the analyte and thestationary phase, based on the semiempirical values of thethermodynamic properties of eq 22.The entropy-driven mechanism (e.g., size exclusion chroma-

tography), is dominated by the loss of the molecules’translational, rotational, and vibrational degrees of freedom,being retained in the absence of proper interaction by thestationary phase. However, the enthalpy-driven mechanism (e.g.,partition chromatography) is dominated by the differencebetween the dissolution energies of the analyte in the mobilephase and stationary phase.This study has used the values reported by Hernandez-Baez et

al.,7 (Figure 6) of the thermodynamic properties for the n-

alkanes in the range of nC12H26−nC64H130 based on the linearfitting of numerous isothermal measurements carried out attemperatures up to 430 °C in an HT5 GC column,corresponding to eq 22.

6. VALIDATIONOF THE PREDICTED RETENTION TIMESA model in MATLAB R2010bSP1 has been developed forpredicting retention times, (eq 9) which contains the distributioncoefficients of every compound7 and the correspondingequations for the calculations of viscosity, pressure, and velocitythrough the GC column, as explained in the previous sections.It is important to note that all GC analyses have been carried

out using constant flow mode for the column carrier gas supply,and therefore the algorithm used calculated the variation of theinlet pressure required for maintaining the flow constant atreference conditions, while the temperature increased, andcarrier-gas viscosity did likewise.Validation of this model has been carried out using both

literature solvation thermodynamic properties of a series of n-alkanes from C12−C40, and PAHs from C10−C22 in a DB-1 and aDB-5 column,19 and the thermodynamic properties obtainedwith the in-house experimental data (Figure 6), for an HT-5capillary column7 for the n-alkanes from C12H26 throughnC62H126.For the DB-1 column, relative to the published19 measured

retention times, average deviations of 1.9% for n-alkanes and2.0% for PAHs were obtained with the in-house model; and forthe published19 predicted retention times, the correspondingaverage deviations were 1.2% for n-alkanes and 0.7% for PAHsfor the in-house predictions. (Figure 7).

In the case of the DB-5 column, relative to measured retentiontimes predicted19 average deviations for the in-house modelpredictions were 2.2% for n-alkanes and 2.6% for PAH’s; and forthe predicted retention times published,19 the correspondingerrors were 0.8% for n-alkanes and 0.3% for PAH’s (Figure 8)Figure 9 shows a comparison of the retention times predicted

by the in-house model and the in-house experimental valuesobtained with an HT5 column, based on the temperature

Figure 6. Distribution factors in a HT5 capillary column, based on theretentions times of every compound and hold-up time for every constanttemperature in the range of temperature 80−430 °C, using ln((tr/tm −1)·β) = ln (K(T)).

Figure 7. Validation of the retention times predicted with the in-housemodel developed, compared with literature data (Aldaeus19) in a DB-1column for PAH’s (from C10-C22) and n-alkanes (from C12−C40), usingtheir retention factors.



program shown in (Table 2), but also applying ramp rates of 10°C/min and 20 °C/min.The average deviations, relative to in-house measured values,

for the retention times predicted with the in-house model, were1.3%, 1.1%, and 2.2% for a temperature ramp of 10 °C/min,15°C/min and 20 °C/min, respectively.As regards the observed deviations between measured and

predicted retentions times in Figure 9, two features are notable:first, for the data relating to the lowest ramp rate(10 °C/min) thefive highest retention times are overpredicted; and second, forthe highest ramp rate (20 °C/min), virtually all measuredretention times are greater than predicted. Two distinct causesare suggested for these observations. In the case of theoverpredicted retention times at the lowest ramp rate, themeasured hold-up times in the region of the upper temperaturelimit are subject to higher deviations as temperatures increased,affecting both the back calculated inlet pressure and thecalculated distributions factors for the associated alkanes. Andin the case of the under-predicted retention times for the highestramp rate of 20 °C/min, it is certain that the true column

temperature is lagging the apparent ramp set-point value. (Assuch, the effect could be confirmed by applying a higher ramprate of say 25 °C/min, in which case even larger deviations wouldbe evident).However, means exist for correcting for such temperature

differentials, and can be applied retrospectively and for futurework.Finally, it is notable that the accurate retention time

predictions have been obtained for the three temperatureramps, which initiated from 10 °C up to 430 °C, even when thetemperature range for which the distribution factors have beenderived, related to isothermal measurements in the range 80 to430 °C.

7. MEASUREMENT OF N-ALKANE ISOTHERMALRETENTION TIMES

In isothermal gas chromatography, components of a homologousseries exhibit a rapid increase in retention time and peak widthwith increasing boiling point, in a generally linear plot of log(RT)vs Carbon Number. As a consequence, only a limited number ofalkanes’ isothermal RTs can be reliably measured from a singleinjection at a given temperature before increasing peakbroadening necessitates their rejection.Another constraint is that single alkanes above nC40 are not

readily commercially available with adequate purity with theexception of nC44, nC50, and nC60 and Polywaxes are generallyutilized for retention time measurements to generate boiling-point/RT calibration plots for HTGC analyses. However, thelatter are mixtures comprising polyethylene oligimers of evencarbon number intervals, and are qualitative mixtures only.Hence the weight fraction of each oligimer in a particularPolywax distribution is not readily known, although accurateestimation is possible if the complete distribution can bechromatographed and total elution can be demonstrated, e.g., byspiking.While qualitative alkane or Polywax mixtures or a combination

of the two are suitably adequate for both isothermal andtemperature programmed retention time measurements, gravi-metric dilutions in CS2 of the ASTMD5442 Linearity Standard30

were also used in this study, covering the alkanes nC12−nC14−nC16−nC18−nC20−nC22−nC24−nC26−nC28−nC30−nC32−nC36−nC40−nC44−nC50−nC60.In such cases, fairly accurate calculations are possible of the

molar quantities of each alkane injected in a given volume.However, this is not the case where a gravimetric blend of thisstandard is made with a Polywax solution in CS2 except for thosealkanes which elute before the lightest oligomers present in thePolywax range.The injection technique use was FVI (Flash Vaporization

Injection), in order to have the same conditions in all isothermalsat the GC column inlet.

8. DEGREE OF ELUTION

In order to determine the degree of elution of every component,their retention factors (ratio of moles of “i” in the stationaryphase to the moles of “i” in the mobile phase) have been analyzedduring a GC analysis using the temperature-program of Table 2.(Tmax, 425 °C).The period of movement has been defined as the time elapsing

from the moment at which a component starts to travel throughthe GC columnincreasing the amount of moles available in thegas phase by re-establishing its equilibrium with increasing

Figure 8. Validation of the retention times predicted with the in-housemodel developed, compared with literature data (Aldaeus19) in a DB-5column for PAH’s(from C10−C22) and n-alkanes (from C12−C40),),using their retention factors.

Figure 9.Validation of the model developed with in-house experimentaldata for Alkanes in aHT5 column, using three ramps of temperatures 10,15, and 20 °C/min in the range of 10−430 °C.



temperature (lower retentions factors) and changing its effectivevelocity as pressure decreasesuntil the time when thecomponent is completely eluted as it reaches the column outlet,and achieves 100% elution (eq 24).The minimum effective velocity at which a component starts

to move has been set here at 0.25 mm/sec, corresponding to 1mm/°C for the 15 °C/min ramp for the oven temperature-program of Table 2.The interval of elution of a component is defined as the time

elapsing from the first moment when some of its molecules passthe GC outlet, until the component has fully eluted.During the period of movement depicted in Figures 10 and 11,

each component “i” travels through the GC column, increasing

the number of moles in the gas phase by re-establishing theirequilibrium with the increasing temperature (lower retentionsfactors) and changing their effective velocity with the decreasingpressure, until all the components are eluted, reaching the GCcolumn outlet.Second, the interval of elution, is defined as the period from

the first moment where some molecules of a given component

pass the GC outlet, to the time when all the molecules of thegiven component has entirely eluted.Figure 10 depicts the period of movement of n-alkanes in the

range C12 through C62 as each travels through the GC column,with the rose colored band depicting their retention factorsduring that interval. The band shows that the round average,minimum retention factor to initiate movement for the n-alkanesstudied is 2000 [moles “i” in the stationary phase per moles “i” inthe gas phase].The round average elution retention factor is 2 over the nC12−

nC62 range, being lower for the heaviest component since theyelute at higher temperatures, and therefore their solubility in thestationary phase is lower at elution.Nevertheless, it is important to note that the elution

temperature of these components is lower than the maximumtemperature (Tmax, 425 °C) reached in the GC column. Thismeans that components heavier than nC62 which elute at Tmaxwill re-establish equilibrium until total elution occurs during thefinal isothermal period and therefore with a constant retentionfactor. Conversely, during the temperature ramping period thegreater the temperature the lower is the retention factor.Making use of retention factor during the moving period of a

component inside the GC column until elution, themole fractionremaining in the gas phase relative to the total amount of molesinjected, can be determined by eq 23.

=+

= +

⎡⎣⎢

⎤⎦⎥X

k T ti

i( )

11 ( ( ))

(moles“ ”)(moles“ ”)i

i

M

S M(23)

Figure 11 depicts (in green) the fraction of each component inthe gas phase during its moving period until elution; and (inblack), the fraction of component in the gas phase during itselution period.The average initial elution fraction of the n-alkanes studied,

relative to the corresponding amount injected is 0.3 mols in thegas phase at elution per total moles injected. The lowest valuesoccur with the most volatile components since they elute at lowertemperature, where solubility in the stationary phase is stillconsiderable, in relation to those eluting at higher temperatures,but below the maximum for the analysis.Therefore, from Figure 11, only a fraction of respectively 0.24

at 110 °C and 0.35 at 420 °C of injected moles of nC12 and nC62are in the gas phase available to elute initially, and only thepercentage which passes the GC outlet, will elute at thattemperature. Thus, the number of moles remaining inside thecolumn can be recalculated, being the difference between molesinjected and moles eluted at the given temperature. (However, itshould be recalled that estimated concentrations have beenapplied here for the alkanes which are not present in the ASTMD5442 linearity standard, as these components derive from aqualitative Polywax standard, or a blended mixture of one withthe ASTM standard.)At the next time step, (1 °C higher from the initial elution

temperature), the retention factor of each component decreases(lower retention in the stationary phase), and the fraction in thegas phase increases in relation to the amount of moles remainingin the column.Again, only a percentage of the moles available will pass the

GC outlet, and the total amount of component remaining canagain be recalculated, as before.Thus, the equilibrium is re-established at every time step (i.e.,

per °C from initial elution) and the amount of moles inside thecolumn is recalculated, until total elution for each component.

Figure 10. Retention Factor vs Temperature (blue), Interval ofRetention Factors, which allows movement to every analyte until itselution, reaching the GC outlet (* rose) for the n-alkanes from nC12 tonC64 in a HT5 capillary GC column, under a temperature-programmed15 °C/min in the range of 10−425 °C.

Figure 11. Fraction of moles of “i” in mobile phase to the total moles of“i” vs temperature (blue) covers all the range of temperature, (green)covers the temperature up to elution temperature of every component“i”. The component “i”, corresponding the n-alkanes with Single CarbonNumber (SCN) from nC12H26 to nC62H126.



The degree of elution can then be calculated at every time stepusing eq 24.

=∑

=+

· ·

⎡⎣⎢⎢

⎤⎦⎥⎥

i T ti

i T t

k T ti T t

degree of elution(moles“ ”) ( ( ))(moles“ ”)

(moles“ ”) ( ( ))1

1 ( ( ))(moles“ ”) %moles elute( ( ))

i

elute

injected

elute

inside

(24)

Figure 12 depicts the degree of elution of every componentstudied, using the temperature program of Table 2, as a function

of time, showing that all components from nC12 to nC62completely elute from the GC column, reaching 100% of degreeof elution before the end of the GC analysis time.It is important to note, that the degree of elution increases

sharply once the elution starts, producing sharp peaks during thetemperature programmed used.Knowing the retention factors of components heavier than

nC62, whose elution is very difficult to identify in a chromato-gram, will allow their degree of elution to be determined, as wellas the extent of nonelution of the components which are unableto elute completely. This subject will be treated in a furtherpublication, covering an analysis extended to much heavier n-alkanes.

9. CONCENTRATION AND TEMPERATURE PROFILESBy solving eqs 1 and 2, it is possible to determine the distribution(assumed to be normal Gaussian), of moles of each componentduring the GC analysis, taking account of the increasingtemperature, decreasing pressure, and movement of thecomponent through the column. In this way, the dispersionand movement of the components at every moment can bedescribed by their standard deviations and centroid, respectively.Figure 13, shows the position of the centroid of every

component with the variation of temperature, using thetemperature-program of Table 2. It is noticeable here thatevery component remains at the column inlet until it reaches aminimum temperature at which the stationary phase starts torelease it.

The minimum temperatures can be seen clearly in Figure 13,as in the case of nC12 where movement starts from the beginningof the analysis at 10 °C, and for nC64 for which the temperature isabout 330 °C. It is evident as expected that the heavier thecomponent, the higher its elution temperature, and the higherthe minimum temperature to initiate the movement inside theGC column.In order to calculate the total moles of gas phase(carrier gas +

component “i”) which occupies a volume covering 95% of thecomponent in the gas, ideal gas behavior is assumed and thepercentile equivalence of normal Gaussian distributions, whichstates that 95% of a distribution occupies (4·σ(T(t)). Thus, thevolume where 95% of molecules of component “i” are located,mixed with carried gas, can be calculated, multiplying the crosssectional area of the tube (ignoring the retentive layer) by fourtimes the standard deviation at the given temperature.The gas molar fraction of “i” and the distribution of moles of

component “i” inside the GC column are respectively depicted inFigures 14 and 15 during the period of time before elution.Therefore, the molar fraction of component “i” available in the

gas phase relative to the total number of moles of gasphase(carrier gas + component “i”) depicted in Figure 14, hasbeen determined, based on the total amount of moles of ideal gasin the corresponding volume, equal to 4·σ(T(t)·Free Transverse

Figure 12.Degree of Elution vs time of each component “i”: n-alkanes inthe range of C12H26 to nC62H126. Degree of elution = moles of “i” insidethe GC column at time (t)/moles injected of “i”.

Figure 13.Centroid Position of every component “i” vs Temperature upto elution from the GC column, using the temperature programmed ofTable 2.

Figure 14.Molar fraction of component “i” in the gas phase vs centroidposition of the moving component “i”, using the temperatureprogrammed of Table 2. The period depicted correspond at the timebefore elution.



Area, and calculating the amount of moles of component “i” inthe gas phase at the given temperature as: moles injected·(1/(1 +k(T))).The higher molar fraction is found before the beginning of the

interval of elution of every component, as the molecules of thecomponent have not started to be released from the columnoutlet. Hence, the higher the temperature, the greater thefraction of component available in the gas phase relative to themoles injected, as explained previously by eq 23 and Figure 11.Thus, nC20 shows the higher value of ca. 1.8 × 10−5 moles in

the gas phase per total moles of gas, and a lower correspondingvalue is seen for nC64 of ca. 0.18× 10−5 moles. These proportionscorrespond to those of their injected values according to Table 3.This confirms that their elution fractions retain the sameproportion as when injected.

Finally, Figure 15, depicts the distribution of moles [mol/m],with time and position throughout the GC column, showing thatnC12 starts to move from the beginning of the analysis, and elutesat about 6 min, when nC20 has barely started to move for elutingat about 12 min; while in turn, nC30 has just started to move, foreluting at about 17 min. For, nC44 start of movement occursaround 16 min, for elution at about 23 min; and nC50 starts tomove at about 19 min, when nC44 was located at about 1 m fromthe GC inlet for elution at 24 min. Lastly, nC64 is located at 1 mfrom the GC inlet, eluting at about 27 min.It is important to note that every component travels singly, and

there is no mixing of components through the GC column, sincethey travel the same distance, and pass through the samepositions but at different times, and hence do not meet eachother during their journey. In this way, good separation of thecomponents occurs during the analysis.It may be seen that the amount of moles per unit length of

column increases in the case of nC12, nC20, and nC30, butdecreases in the case of nC44, nC50 and nC64, which correspondsto the expected behavior, retaining the same proportion as thecorresponding amount injected, as described in Table 3.As expected, the standard deviations of every component, and

therefore their dispersion, increase with time, and thereforetemperature through the column, from the inlet to the outlet.The sharper the eluting peak, the lower the dispersion.

10. CONCLUSIONSThis study provides further insight into the limits of hightemperature gas chromatography (HTGC), proposing a newapproach for determining the non/incomplete elution of everycomponent by introducing: the degree of elution, defined as theamount of component which has been eluted in relation to theamount injected. The degree of elution of the n-alkanehydrocarbons in the range, nC12H28 to nC62H126 has beencalculated based on the continuous equilibrium re-establishedduring the interval of elution for every component, using theircorresponding retention factors, and assuming no cracking insidethe GC column.This new approach will be applied in a future work for n-

alkanes heavier than nC62H126 in order to determine theanalytical conditions required for ensuring maximum elution ofa given component, allowing the possibility of improving thepractice of HTGC by optimizing the separation process.

Figure 15. Distribution of moles of component “i” per unit of lengthregarding the position inside the GC column, and the analysis time untilelution, using the temperature programmed of Table 2.

Table 3. Composition of Injected n-Alkanes (Mixture ofASTM 54179 and Polywax, Assumed Values for CalculationPurposes Only)

moles injected in 0.3 μL

C12H26 3.65 × 10−11 C40H82 2.75 × 10−11

C14H30 4.69 × 10−11 C42H86 2.75 × 10−11

C16H34 5.47 × 10−11 C44H90 2.00 × 10−11

C18H38 6.09 × 10−11 C46H94 2.00 × 10−11

C20H42 6.58 × 10−11 C48H98 2.00 × 10−11

C22H46 7.98 × 10−11 C50H102 1.32 × 10−11

C24H5D 9.15 × 10−11 C52H106 1.32 × 10−11

C26H54 1.69 × 10−10 C54H110 1.32 × 10−11

C28H58 1.61 × 10−10 C56H114 1.32 × 10−11

C30H62 7.34 × 10−11 C58H118 1.32 × 10−11

C32H66 5.50 × 10−11 C60H122 7.36 × 10−12

C34H70 5.50 × 10−11 C62H126 7.36 × 10−12

C36H74 3.67 × 10−11 C64H130 7.36 × 10−12

C38H78 3.67 × 10−11



The study introduces a preliminary method of calculating, ateach moment during a temperature-programmed analysis, themolar fraction of the components in the gas phase, in accordancewith the standard deviations of their Gaussian distribution at thepoint where 95% of the molecules are traveling through thecolumn.The Pyrolysis model developed in the previous paper1 and the

Gas Chromatography Migration and Separation Modeldeveloped in this paper, will be combined to form the basis ofa further study, in order to complete the analysis of the crackingrisk of heavy n-alkanes.This study also provides a deeper understanding of the

separation of components in a gas chromatographic column, andprovides a basis for further analysis of nonelution of componentsheavier than nC62H126, which will be treated in a later publication.

■ AUTHOR INFORMATIONCorresponding Author*Fax: +44 (0)131 451 3127; e-mail: [email protected] .NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors wish to thank the members of our joint industryproject Marathon Oil Corporation, Schlumberger and TOTALfor their technical and financial support for this project. DianaM.Hernandez-Baez wishes to thank the Institute of PetroleumEngineering for financial support, and her family for theircontinual encouragement.

■ LIST OF SYMBOLSDeff = effective average Diffusivity (unit length2/ unit time)D = apparent diffusion coefficient, represents all factorscausing dispersion (unit length2/ unit time)DM = diffusion constant, mobile phase (unit length2/ unittime)DS = diffusion constant, stationary phase (unit length2/ unittime)H(x,t)) = column plate height, spatial rate of dispersion of azone (unit length)K = distribution factor of a compound (mol/volume) instationary phase/ (mol/volume) in gas phase)k = retention factor of a compound.(moles in stationaryphase/mol in gas phase)L = length of the GC column (unit length)m(x,to) =mass profile for every analyte (particles/unit length)N i,M = moles of component “i” in the mobile phaseN i,S = moles of component “i” in the stationary phaseP(x) = pressure at position x (Pa)Pin = pressure at the GC colum inletPout = pressure at the GC column outlet (Pa)r0 = internal radius of GC column. (unit length)rampT = ramp of temperature of the temperatureprogrammedT(t) = temperature at the time tT0 = initial temperature of the temperature programmedt = time (unit time)veff = effective cross-sectional average velocity (unit length/unit time)vM = velocity of migration of the carrier gas (unit length/unittime)

w = film thicknes (unit length)Xi = fraction of component i in the gas phase relative to themoles in both stationary and gas phasex0 = centroid of Gaussian distribution of distribution ofcomponent inside the GC column (unit length)x = position of the component’s dispersal around the centroidx0 (unit length)

Greek Lettersσ = standard deviation of the distribution of component insidethe GC column (unit length)β = phase ratio (volume of mobile phase in the column to thevolume of stationary phase)ηm = viscosity of the carrier gas (μPa·s)Δt = time step (unit time)

■ REFERENCES(1) Hernandez-Baez, D. M.; Tohidi, B.; Chapoy, A.; Reid, A.;Bounaceur, R. Energy Fuels 2012, 26, 2600−2610.(2) Dulaurent, A.; Dahan, L.; Thiebaut, D.; Bertoncini, F.; Espinat, D.Oil Gas Sci. Technol.- Rev. IFP. 2007, 62 (1), 33−42.(3) Schwartz, H. E.; Brownlee, R. G.; Boduszynski, M. M.; Su, F. Anal.Chem. 1987, 59 (10), 1393−1401.(4) Golay, M. J. E. Gas Chromatogr., Proc. Symp. 1958, 36−53.(5) Aldaeus, F.; Thewalim, Y.; Colmsjo, A. J. Chromatogr., A 2009,1216 (1), 134−139.(6) Snijders, H.; Janssen, H. G.; Cramers, C. J. Chromatogr., A 1995,718 (2), 339−355.(7) Hernandez-Baez, D. M.; Reid, A.; Chapoy, A.; Tohidi, B.Establishing the Maximum Carbon Number for Reliable QuantitativeGas Chromatographic Analysis of Heavy Ends Hydrocarbons. Part 3.(Determination of Distribution Factors (nC12-nC98) in an HT5 GCcolumn). In unpublished manuscript.(8) Guiochon, G.; Guillemin, C. L., Book Title: Quantitative GasChromatography for Laboratory Analysis and On-lin Control; Elsevier:Amsterdam: 1988.(9) Blumberg, L. M. Temperature-Programmed Gas Chromatography;Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, 2010.(10) Blumberg, L. M. J. Chromatogr. 1993, 637 (2), 119−128.(11) Thewalim, Y.; Aldaeus, F.; Colmsjo, A. Anal. Bioanal. Chem. 2009,393 (1), 327−334.(12) Giddings, J. C. Anal. Chem. 1963, 35 (3), 353−356.(13) Giddings, J. C. J. Gas Chromatogr. 1964, 2, 167−169.(14) Giddings, J. C. Par I, Principles and theory. In Dynamics ofChromatography; Edward Arnold/Marcel Dekker, Inc.: London/NewYork, 1965.(15) Giddings, J. C.; Seager, S. L.; Stucki, L. R.; Stewart, G. H. Anal.Chem. 1960, 32 (8), 867−870.(16) Taylor, G. Proc. R. Soc. London. Ser. A. Math. Phys. Sci. 1953, 219(1137), 186−203.(17) Blumberg, L. M.; Berger, T. A. J. Chromatogr. 1992, 596 (1), 1−13.(18) Aldaeus, F. Doctoral thesis in analytical chemistry, StockholmUniversity: Stockholm,2007.(19) Aldaeus, F.; Thewalim, Y.; Colmsjo, A. Anal. Bioanal. Chem. 2007,389 (3), 941−950.(20) Gonzalez, F. R.; Nardillo, A. M. J. Chromatogr. A 1999, 842 (1−2),29−49.(21) Davankov, V. A. Chromatographia 1997, 44 (5−6), 279−282.(22) Kestin, J.; Knierim, K.; Mason, E. A.; Najafi, S. T.; Ro, S. T.;Waldman, M. J. Phys. Chem. Ref. Data 1984, 13 (1), 229.(23) Hawkes, S. J. Chromatographia 1993, 37 (7−8), 399−401.(24) Fuller, E. N.; Schettler, P. D.; Giddings, J. C. Ind. Eng. Chem. 1966,58 (5), 19−27.(25) Gonzalez, F. R.; Alessandrini, J. L.; Nardillo, A. M. J. Chromatogr.,A 1999, 852 (2), 583−588.(26) Ben Naim, A. Solvation Thermodynamics; Plenum Press: NewYork, 1987.



mailto:[email protected]

mailto:[email protected]

(27) Gonzalez, F. R. J. Chromatogr., A 2000, 873 (2), 209−219.(28) Castells, R. C.; Arancibia, E. L.; Nardillo, A. M. J. Chromatogr.1990, 504 (1), 45−53.(29) Gonzalez, F. R. J. Chromatogr., A 2002, 942 (1−2), 211−221.(30) ASTM® 5442 (C12-C60) Quantitative Linearity Standard 0.01%(w/w) each component in cyclohexane, analytical standard. In ASTMAnalytical standard (http://www.sigmaaldrich.com/catalog/product/supelco/502235?lang=en&region=GB).



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