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Research ArticlePressure Transient Analysis and FluxDistribution for Multistage Fractured Horizontal Wells inTriple-Porosity Reservoir Media with Consideration ofStress-Sensitivity Effect

Jingjing Guo1 Haitao Wang1 Liehui Zhang1 and Chengyong Li2

1State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum UniversityChengdu Sichuan 610500 China2College of Energy Resources Chengdu University of Technology Chengdu Sichuan 610059 China

Correspondence should be addressed to Haitao Wang 907088352qqcom

Received 17 October 2014 Accepted 18 March 2015

Academic Editor Cecile Autret-Lambert

Copyright copy 2015 Jingjing Guo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Triple-porosity model is usually adopted to describe reservoirs with multiscaled pore spaces including matrix pores naturalfractures and vugs Multiple fractures created by hydraulic fracturing can effectively improve the connectivity between existingnatural fractures and thus increase well deliverability However little work has been done on pressure transient behavior ofmultistage fractured horizontal wells in triple-porosity reservoirs Based on sourcesink function method this paper presents atriple-porosity model to investigate the transient pressure dynamics and flux distribution for multistage fractured horizontal wellsin fractured-vuggy reservoirs with consideration of stress-dependent natural fracture permeability The model is semianalyticallysolved by discretizing hydraulic fractures and Pedrosarsquos transformation perturbation theory and integration transformationmethod Type curves of transient pressure dynamics are generated and flux distribution among hydraulic fractures for a fracturedhorizontal well with constant production rate is also discussed Parametric study shows that major influential parameters ontransient pressure responses are parameters pertinent to reservoir properties interporosity mass transfer and hydraulic fracturesAnalysis of flux distribution indicates that flux density gradually increases from the horizontal wellbore to fracture tips and the fluxcontribution of outermost fractures is higher than that of inner fractures The model can also be extended to optimize hydraulicfracture parameters

1 Introduction

The concept of dual-porosity media was first proposed byBarenblatt et al [1] to describe naturally fractured mediaWarren and Root [2] then extended this model to analyzepressure transient dynamics for vertical wells in naturallyfractured oil reservoirs Recently Cai et al [3 4] analyzedthe imbibition mechanism in fractured-porous media basedon fractal geometry Although the dual-porosity model candescribe most naturally fractured reservoirs there are stillsome drawbacks when applying the dual-porosity modelto describe reservoirs with multiple pore media such as

reservoirs with natural fractures and vugs (different frommatrix pores)

In order to better describe reservoir heterogeneity in thistype of naturally fractured-vuggy reservoirs the concept oftriple-porosity model was proposed The first triple-porositymodel in the literature was proposed by Abadassah andErshaghi [5] in which the reservoir was represented by acombination of two matrix systems with different propertiesand uniformly distributed natural fractures The model wasthen further extended by Al-Ghamdi and Ershaghi [6] torepresent reservoirs with two fracture systems (microfrac-tures and macrofractures) and one matrix system After that

Hindawi Publishing CorporationJournal of ChemistryVolume 2015 Article ID 212901 16 pageshttpdxdoiorg1011552015212901

2 Journal of Chemistry

Hydraulic fractures

Horizontal well

Triple-porosity reservoir

Matrix

Vug

Natural fracture

Figure 1 Schematic of a multistage fractured horizontal well in atriple-porosity reservoir

over the past several decades numerous efforts have beenmade to investigate transient pressure behavior in so-calledtriple-porosity reservoirs However almost all the researchfocuses on vertical wells [7ndash19] or horizontal wells [20ndash22]in triple-porosity reservoirs and few studies have been doneon complex well-reservoir configuration such as multistagefractured horizontal well in triple-porosity reservoirs

Field practices have proven that horizontal wellborecombining multistage hydraulic fracturing can effectivelyenhance the permeability in the vicinity of horizontal well-bore Almost all the triple-porosity models [23 24] forfractured horizontal wells assume linear flow which cannotcompletely characterize the pressure responses during allflowing periods (such as pseudoradial flow period) On theother hand in most existing models for fractured horizontalwells in triple-porosity reservoirs three porous media denotehydraulic fractures natural fractures and matrix poreswhich means the reservoir itself is actually considered as adual-porosity medium

This study develops a semianalytical model based onsourcesink function theory for analyzing transient pressureresponses and flux distribution of naturally fractured-vuggyreservoirs in which the reservoir itself is conceptualized astriple-porosity media and the interaction between hydraulicfractures and reservoir is considered The model presentedhere can completely reflect transient pressure characteristicsduring all the possible flowing period as well as flux dis-tribution among multiple fractures In addition this modelalso takes into account the stress-sensitivity of permeabilitycaused by closure of natural fractures during production

2 Physical Model of a MultistageFractured Horizontal Well ina Triple-Porosity Reservoir

As shown in Figure 1 a horizontal well with multiple hydra-ulic fractures producing at a constant rate (119902) in a reservoirwith multiscale storage spaces is considered here Both thecap rock and underlying rock formation of the reservoir areusually shales with extremely low permeability which can beconceived as no-flow boundariesThe reservoir thickness andoil viscosity are assumed to be ℎ and 120583 separately Other basicassumptions involved in the derivation of the triple-porositymodel are as follows

(1) The reservoir is assumed to be composed of three con-tiguous porous media which are matrix natural

Natural fractures

Vugs Matrix

Hyd

raul

ic fr

actu

re

Figure 2 Illustration of fluid flowing paths in a triple-porosity res-ervoir

fractures and vugs (representing larger voids in thereservoir which are not part of natural fractures)

(2) Natural fractures are assumed to be directly con-nected to hydraulic fractures or the horizontal well-bore while the fluid stored in matrix pores or vugsdoes not have direct access to hydraulic fractures orthe horizontal wellbore Figure 2 presents a simpleillustration of the fluid flowing path in triple-porosityreservoirs

(3) The permeability of natural fractures is consideredstress-sensitive to incorporate the effect of partial orcomplete closure of natural fractures during produc-tion

(4) Both oil and rock are considered slightly compress-ible and the reservoir has uniform pressure distribu-tion at time zero

(5) Consider single-phase isothermal flow and negligiblegravity and capillary effects

To obtain the pressure responses caused by the well-reservoir configuration shown in Figure 1 a mathematicalmodel together with corresponding boundary and initialconditions is required In the problem addressed here themultiple fractures inner boundary condition is so complexthat it is rather difficult to directly describe it with a math-ematical equation In order to solve this problem sourcefunction theory a powerful tool to solve complex transientflow problems in reservoirs is adopted in this work Wefirst studied the transient pressure responses caused by acontinuous line-sink in triple-porosity reservoirs and thenadopted the line-sink solutionwith superposition principle toobtain the pressure transient dynamics and flux distributionof multistage fractured horizontal wells in triple-porosityreservoirs

3 Line-Sink Model in Stress-SensitiveTriple-Porosity Reservoirs

31 Physical Model for a Line-Sink in Triple-Porosity Reser-voirs As shown in Figure 3 a continuous line-sink in atriple-porosity reservoir with stress-dependent fracture per-meability is considered in this section The strength of theline-sink is represented by 119902(119905)

Journal of Chemistry 3

Top boundary

Triple-porosity reservoir Line-sink

Bottom boundaryZw

h

re

Figure 3 A line-sink in a stress-sensitive triple-porosity reservoir

32 Mathematical Model for a Line-Sink inTriple-Porosity Reservoirs

(1) Governing Equations Imposing mass-balance equationover a control volume in the triple-porosity reservoir andsubstituting equation of state and equation of motion we canget the governing equations for matrix natural fracture andvug systems

For natural fracture systemwith stress-dependent perme-ability the governing equation is as follows (ie (A26b) inAppendix A)

1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2= eminus120574(119901fminus119901i) [

120601f0120583119862f t119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(1)

For matrix system the governing equation is as follows(ie (A30) in Appendix A)

120601m0119862mt120597119901m120597119905

+1205721119896m120583

(119901m minus119901f) = 0 (2)

For vug system the governing equation is as follows (ie(A34) in Appendix A)

120601v0119862vt120597119901v120597119905

+1205722119896v120583

(119901v minus119901f) = 0 (3)

(2) Boundary Conditions According to the derivation inAppendix B the inner boundary condition of the line-sinkmodel is (ie (B5) in Appendix B)

lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (4)

Assuming infinitely large reservoirs the outer boundarycondition can be expressed as

119901f (119903 119905)1003816100381610038161003816119903rarrinfin = 119901i (5)

(3) Initial Condition The initial pressure distribution in thereservoir is assumed to be uniform which is

119901f (119903 119905)1003816100381610038161003816119905=0 = 119901m (119903 119905)

1003816100381610038161003816119905=0 = 119901v (119903 119905)1003816100381610038161003816119905=0 = 119901i (6)

33 Dimensionless Form of the Line-Sink Model For thepurpose of simplifying derivation and comparing betweendifferent reservoirs the mathematical model is derived andsolved in dimensionless form in this workThepressure drops

involved in themathematical model areΔ119901f = 119901iminus119901f Δ119901m =

119901i minus 119901m and Δ119901v = 119901i minus 119901v and relevant dimensionless vari-ables used in the mathematical model are listed in Table 1

Obviously 120596f 120596m and 120596v in Table 1 satisfy the followingequation

120596f +120596m +120596v = 1 (7)

With the definitions in Table 1 the dimensionless formsof (1)sim(6) can be obtained as follows

1205972119901fD

120597119903D2 +

1119903D

120597119901fD120597119903D

minus 120574D (120597119901fD120597119903D

)

2

= e120574D119901fD [120596f120597119901fD120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

]

(8)

120596m120597119901mD120597119905D

+120582mf (119901mD minus119901fD) = 0 (9)

(1minus120596f minus120596m)120597119901vD120597119905D

+120582vf (119901vD minus119901fD) = 0 (10)

lim120576Drarr 0

eminus119901fD120574D 119903D120597119901fD120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (11)

119901fD1003816100381610038161003816119903Drarrinfin

= 0 (12)

119901fD1003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0 (13)

34 Solution of the Dimensionless Line-Sink Model

341 Pedrosarsquos Linearization The stress-sensitivity of natu-ral fracture permeability makes the above seepage modelstrongly nonlinear which cannot be solved analytically Herefollowing Pedrosa Jr [25] we introduced the followingequation to alleviate the nonlinearity of (8) and (11)

119901fD (119903D 119905D) = minus1120574D

ln [1minus 120574D119880D (119903D 119905D)] (14)

where 119880D is an intermediate dimensionless parameter whichis also a function of radial distance and time

With (14) (8)sim(13) become

1205972119880D

1205971199032D+

1119903D

120597119880D120597119903D

=1

1 minus 120574D119880D120596f

120597119880D120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

(15)

120596m120597119901mD120597119905D

+120582mf [119901mD +1120574D

ln (1minus 120574D119880D)] = 0 (16)


+120582vf [119901vD +1120574D

ln (1minus 120574D119880D)]

= 0(17)


Table 1 Definitions of dimensionless variables

Dimensionless pressure 119901119891119863

=2120587119896119891119894

ℎ

119902119861120583Δ119901119891

119901119898119863

=2120587119896119891119894

ℎ

119902119861120583Δ119901119898

119901V119863 =2120587119896119891119894

ℎ

119902119861120583Δ119901V

Dimensionless radial distance 119903119863

=119903

119871 120576119863

=120576

119871

Dimensionless time 119905119863

=119896119891119894

119905

120583 (1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905) 1198712

Dimensionless storativity ratio of natural fracture system 120596119891

=1206011198910

119862119891119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)

Dimensionless storativity ratio of matrix system 120596119898

=1206011198980

119862119898119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)

Dimensionless storativity ratio of vug system 120596V =120601V0119862V119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)

Dimensionless interporosity flow coefficient between matrix andnatural fractures

120582119898119891

=120572119898119891

119896119898

1198712

119896119891119894

Dimensionless interporosity flow coefficient between vugs andnatural fractures

120582V119891 =120572V119891119896V119871

2

119896119891119894

Dimensionless permeability modulus 120574119863

=119902119861120583

2120587119896119891119894

ℎ120574

Dimensionless production rate of the line-sink 119902119863

= 119902119863

(119905119863

) =119902 (119905)

119902

Note 119871 is an arbitrary reference length which can be 119903w 119871h and so forth

lim120576Drarr 0

119903D120597119880D120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (18)

119880D1003816100381610038161003816119903Drarrinfin

= 0 (19)

119880D1003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0 (20)

342 Perturbation Technique According to the perturbationtheory the terms119880D minus(1120574D)ln[1minus120574D119880D(119903D 119905D)] and 1(1minus120574D119880D(119903D 119905D)) in (15) through (20) can be expanded as powerseries in dimensionless permeabilitymodulus andwe can getthe following equations

119880D = 119880D0 + 120574D119880D1 + 1205742D119880D2 + sdot sdot sdot

minus1120574D

ln [1minus 120574D119880D (119903D 119905D)]

= 119880D (119903D 119905D) +12120574D119880D

2(119903D 119905D) + sdot sdot sdot

11 minus 120574D119880D (119903D 119905D)

= 1+ 120574D119880D (119903D 119905D) + 1205742D119880D (119903D 119905D) + sdot sdot sdot

(21)

Given that the dimensionless permeability modulus 120574D isusually small (much smaller than 1) the zero-order approx-imate solution can satisfy the requirements of engineeringprecision so (15)sim(20) become

1205972119880D01205971199032D

+1119903D

120597119880D0120597119903D

= 120596f120597119880D0120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

120596m120597119901mD120597119905D

+120582mf (119901mD minus119880D0) = 0


+120582vf (119901vD minus119880D0) = 0

lim120576Drarr 0

119903D120597119880D0120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D

119880D01003816100381610038161003816119903Drarrinfin

= 0

119880D01003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0

(22)

Equation (22) is the linearized line-sink model in triple-porosity reservoirs

343 Laplace Transformation Take the following Laplacetransformation

119880D0 = int

+infin

0119880D0eminus119906119905Dd119905D (23)

where 119906 is the Laplace transformation variable


Then (22) becomes

d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 120596f119906119880D0 +120596m119906119901mD +120596v119906119901vD (24)

120596m119906119901mD +120582mf (119901mD minus119880D0) = 0 (25)

120596v119906119901vD +120582vf (119901vD minus119880D0) = 0 (26)

lim120576Drarr 0

119903Dd119880D0d119903D

100381610038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (27)

119880D010038161003816100381610038161003816119903Drarrinfin

= 0 (28)

344 Line-Sink Solution Equations (25) and (26) can berewritten as

119901mD =120582mf

120596m119906 + 120582mf119880D0

119901vD =120582vf

120596v119906 + 120582vf119880D0

(29)

Substituting (29) into (24) yields

d2119880D0d1199032D

+1119903D

d119880D0d119903D

= (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906119880D0

(30)

If we define

119892 (119906) = (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906 (31)

then (30) can be expressed as follows

d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 119892 (119906)119880D0 (32)

The general solution of (32) can be easily obtained asfollows

119880D0 = 11988811198700 (radic119892 (119906)119903D)+ 11988821198680 (radic119892 (119906)119903D) (33)

where 1198700

and 1198680

are Bessel functions and 1198881

and 1198882

areunknown coefficients which can be determined by corre-sponding boundary conditions

If the line-sink is located at the origin of coordinates (cen-ter of the triple-porosity reservoir) then the dimensionlessradial distance 119903D in (31) can be calculated by the followingequation

119903D = radic119909D2 + 119910D

2 (34a)

If the line-sink is located at (119909w 119910w) instead then 119903D iscalculated by the following equation

119903D = radic(119909D minus 119909wD)2+ (119910D minus 119910wD)

2 (34b)

where

119909D =119909

119871 (35)

119910D =119910

119871 (36)

119909wD =119909w119871

(37)

119910wD =119910w119871

(38)

Substitution of (33) into the inner boundary condition(27) yields

lim120576Drarr 0

radic119892 (119906)119903D

sdot [minus11988811198701 (radic119892 (119906)119903D)+ 11988821198681 (radic119892 (119906)119903D)]1003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D

(39)

According to the properties of Besselrsquos function which arelim119909rarr 01199091198701(119909) rarr 1 and lim

119909rarr 01199091198681(119909) rarr 0 (39) becomes

1198881 = 119902D (40)

Similarly with the outer boundary condition given in (28)and the properties of Besselrsquos function lim

119909rarrinfin

1198700(119909) rarr 0and lim

119909rarrinfin

1198680(119909) rarr infin the coefficient in (33) 1198882 shouldsatisfy the following equation

1198882 = 0 (41)

With (40) and (41) we can get the final form of (33) asfollows

119880D0 = 119902D1198700 (radic119892 (119906)119903D) (42)

Equation (42) is the basic zero-order perturbation solu-tion for a line-sink in a stress-sensitive triple-porosity reser-voir Asmentioned above zero-order perturbation solution isenough to approximate the exact solution of (15) to (20) thatis119880D asymp 119880D0 With the definition of 119902D in Table 1 (42) can beexpressed as

119880D =119902

1199021198700 (radic119892 (119906)119903D) (43)

Equation (43) can be rewritten as

119880D =119902

119902Ω (119909D 119910D 119909wD 119910wD) (44)

where

Ω(119909D 119910D 119909wD 119910wD) = 1198700 (radic119892 (119906)119903D) (45)

Equation (44) is the basic line-sink solution for stress-sensitive triple-porosity reservoirs With the basic line-sinksolution and the superposition principle we can obtain thepressure responses caused by multistage fractured horizontalwells in stress-sensitive triple-porosity reservoirs


x

y

1 i M

O

(0 0) (0 y1)(XiN YiN)

(Xi1 Yi1)

(xiN yiN)

(xi1 yi1)(xi2 yi2)

(xiN+2 yiN+2)(XiN+1 YiN+1)

(xiN+1 yiN+1)

(Xi2N Yi2N) (xi2N yi2N)

(0 yM)(0 yi)

(xi2N+1 yi2N+1)

Figure 4 Schematic of discrete fracture segments

4 Pressure Responses for MultistageFractured Horizontal Wells inStress-Sensitive Triple-Porosity Reservoirs

Figure 4 shows the schematic of a horizontal well withmultiple (a total of119872) hydraulic fractures in a stress-sensitivetriple-porosity reservoir The 119910-axis is set to be aligned withthe horizontal wellbore The intersection of the 119894th (119894 =

1 2 119872) hydraulic fracture and the 119910-axis is representedby (0 119910

119894

) and the distance between every two adjacentfractures is Δ119910

119894

To successfully obtain the pressure responses caused by

the multiple fractures with the line-sink solution derived inthe above section it is necessary to discretize the multiplehydraulic fractures As shown in Figure 4 each hydraulicfracture is divided into 2119873 segments along its length Thecoordinate of the midpoint (ie discrete node) of the 119895thsegment on the 119894th fracture is denoted by (119883

119894119895

119884119894119895

) and thecoordinates of the two corresponding endpoints are denotedby (119909119894119895

119910119894119895

) and (119909119894119895+1 119910119894119895+1)

With the discretization scheme shown in Figure 4 thecoordinates of endpoints of discrete segments can be calcu-lated by the following equations

119909119894119895

= minus119873 minus 119895 + 1

119873119883fL119894

119910119894119895

= 119910119894

1 le 119895 le 119873

119909119894119895

=119895 minus 119873 minus 1

119873119883fR119894

119910119894119895

= 119910119894

119873 + 1 le 119895 le 2119873 + 1

(46)

The coordinates of discrete nodes (midpoint of eachsegment) can be calculated by the following equations

119883119894119895

= minus2119873 minus 2119895 + 1

2119873119883fL119894

119884119894119895

= 119910119894

1 le 119895 le 119873

119883119894119895

=2 (119895 minus 119873) minus 1

2119873119883fR119894

119884119894119895

= 119910119894

119873 + 1 le 119895 le 2119873

(47)

where119883fL119894 is the length of the leftwing of 119894th fracturem119883fR119894is the length of the right wing of 119894th fracture m In the modelpresented in this paper119883fL119894 and119883fR119894 can be different and canvary from fracture to fracture

The distance between 119910119894

and 119910119894minus1

is

Δ119910119894

= 119910119894

minus119910119894minus1 (48)

It is obvious that the flux strength is different alongthe fracture length However in the same discrete segment(assuming the discrete segment is small enough) the fluxstrength can be considered as constant If we use 119902

119894119895

torepresent the flux density per unit length then with thesuperposition principle the pressure response at (119909D 119910D) inthe reservoir caused by the discrete segment (119894 119895) can beobtained by integrating the basic line-sink along the discretesegment which is

119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906)

119902int

119909119894119895+1

119909119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909w(49)

With the definitions of 119909D and 119909wD that is (35) and (37)(49) can be changed into

119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906) 119871

119902int

119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(50)

If we define dimensionless flux density per unit length as

119902D119894119895 (119905D) =119902119894119895

(119905D) 119871

119902(51)

and take Laplace transformation of (51)

119902D119894119895 (119906) =

119902119894119895

(119906) 119871

119902 (52)


119880D119894119895 (119909D 119910D)

= 119902D119894119895 int119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(53)

The pressure response at (119909D 119910D) caused by 2119873 times 119872

segments can be obtained by applying the principle ofsuperposition over all discrete fracture segments

119880D (119909D 119910D) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D 119910D) (54)


Thus the pressure response at the discrete segment(120594 120581) (1 le 120594 le 119872 1 le 120581 le 2119873) can be obtained as follows

119880D (119909D120594120581 119910D120594120581) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D120594120581 119910D120594120581) (55)

Since the permeability of hydraulic fractures is alwaysmuch higher than the original reservoir permeability thepressure drop in hydraulic fractures is much smaller than thepressure drop caused by oil flow in the reservoir Thus thehydraulic fractures can be considered as infinitely conductivemeaning the pressure in hydraulic fractures is equal to thebottom-hole pressure

119880D (119883D120594120581 119884D120594120581) = 119880wD (56)

Combining (55) and (56) we can get

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119883D120594120581 119884D120594120581) = 119880wD (57)

The subscripts 120594 and 120581 in (57) denote that this equationis written for discrete segment (120594 120581) By writing (57) for alldiscrete segments that is letting (120594 = 1 2 119872 120581 =

1 2 2119873) we can get 2119873times119872 equations It should be notedthat there are 2119873times119872+1 unknowns in these 2119873times119872 equationswhich are 119880wD and 119902D119894119895 (119894 = 1 2 119872 119895 = 1 2 2119873)To compose a closed system of equations the assumption ofconstant total flow rate can give the following equation

119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] = 119902 (58)

whereΔ119883119894119895

is the length of the discrete segment (119894 119895) and canbe calculated by

Δ119883119894119895

= 119883119894119895+1 minus119883

119894119895

(59)

Taking Laplace transformation of (58) one can get

119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] =119902

119906 (60)

The dimensionless form of (60) is

119872

sum

119894=1

2119873sum

119895=1[119902D119894119895Δ119883D119894119895] =

1119906 (61)

where

Δ119883D119894119895 =Δ119883119894119895

119871 (62)

Equations (57) and (61) represent a system of (2119873times119872+1)linear algebraic equations relating (2119873 times 119872 + 1) unknownswhich can be solved by direct methods such as Gaussianelimination method Gauss-Jordan reduction method

119880wD calculated by (57) and (61) does not take into accountthe effects of wellbore storage and skin According to vanEverdingen [26] and Kucuk and Ayestaran [27] the effectsof wellbore storage and skin can be incorporated in thecalculated bottom-hole pressure response by

119880wDS =119906119880wD + 119878

119906 + 119862D1199062 (119906119880wD + 119878)

(63)

where 119862D is the dimensionless wellbore storage coefficientand 119878 is the skin factor defined by the following respectively

119862D =119862

2120587 (120601f119862f t + 120601m119862mt + 120601v119862vt) ℎ1198712 (64)

119878 =2120587119896fiℎ119902sf120583

Δ119901119904

(65)

where 119862 is the wellbore storage coefficient m3Pa 119902sf is thesandface flow rate of the multistage fractured horizontal wellin the triple-porosity reservoir m3s Δ119901

119904

is an extra pressuredrop near the wellbore Pa

119880wDS in (63) is the bottom-hole pressure responseobtained in the Laplace domain and by Stehfest [28]numerical inversion algorithm we can calculate the pressureresponses 119880wDS(119905D) in real time domain Then with (66) wecan obtain the bottom-hole pressure response for multistagefractured horizontal wells in triple-porosity reservoirs incor-porating the stress-sensitivity of natural fracture system

119901wD (119905D) = minus1120574D

ln [1minus 120574D119880wDS (119905D)] (66)

5 Results and Discussion

In this section a horizontal well with three hydraulic frac-tures (ie119872 = 3) in a triple-porosity reservoir is consideredand the model introduced above is applied to investigatethe dimensionless pressure responses of this well-reservoirconfiguration

51 Flow Periods and Effect of Stress-Sensitivity Figure 5presents the dimensionless pressure responses and corre-sponding pressure derivatives of a multistage fractured hori-zontal well in an infinitely large triple-porosity reservoir withconsideration of stress-sensitivity of natural fractures It canbe observed from the figure that as many as ten flow periodscan be identified at different time scales Different flowperiods and their corresponding characteristics are explainedas follows

Period 1 It is the early-time wellbore storage period Inthis period both pressure curves and their derivative curvesexhibit unit slope on the log-log plots

Period 2 It is the transition flow period after wellbore storageperiod The pressure derivative curves exhibit a ldquohumprdquorepresenting the effect of skin

Period 3 It is the first linear flow period This periodcorresponds to the linear flow in the direction perpendicular


1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05

Slope = 05 Dip 2Dip 1

120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D

Figure 5 Log-log plots of dimensionless pressures and pressurederivatives of a multistage fractured horizontal well in an infinitelylarge triple-porosity reservoir

to hydraulic fracture length as illustrated in Figure 6(a) Thepressure derivative curves exhibit a straight line with slopeequal to ldquo12rdquo

Period 4 It is the first pseudoradial flow period This periodcan occur when the distances between adjacent fractures arerelatively large Oil flows in the reservoir in a way as shownin Figure 6(b) and pseudoradial flow occurs around eachfracture This period corresponds to a nearly horizontal linewith a value of ldquo1(2119872)rdquo in the derivative curve where 119872 isthe number of hydraulic fractures

Period 5 It is the second linear flow period This periodcorresponds to linear flow perpendicular to the horizontalwellbore as shown in Figure 6(c) Another straight line witha slope of about ldquo12rdquo can be observed in the derivative curveWhen the permeability of natural fracture system remainsconstant during the whole production corresponding to theno stress-sensitivity case the slope of the derivative curvesduring this period is ldquo12rdquo When taking into account thestress-sensitivity of natural fracture system the correspond-ing pressure derivative curve slightly deviates from the one-half-slope line and exhibits a slight upward tendency

Period 6 It is the second pseudoradial flow period corre-sponding to radial flow in natural fracture system Duringthis period multiple hydraulic fractures and the horizontalwellbore behave like an enlarged wellbore The derivativecurves exhibit a horizontal line with an upward tendencydepending on the value of dimensionless permeability mod-ulus In the case that 120574D = 0 the derivative curves are hori-zontal with a value of 05 on the 119910-axis With the increase ofthe value of 120574D the derivative curves turn upward gradually

Period 7 It is the first interporosity flow period This periodreflects mass transfer between vugs and natural fracturesystem Due to the oil flowing from vugs to natural fracturesthe bottom-hole pressure drops slower and an obvious ldquodiprdquocan be observed in the derivative curves

Period 8 It is the third pseudoradial flow period correspond-ing to compound radial flow in natural fracture system and

vugs Due to the existence of stress-sensitivity of naturalfracture system the pressure derivative curves are no longerhorizontal The position of the derivative curves becomeshigher with larger value of 120574D

Period 9 It is the second interporosity flow period Thisperiod reflects oil flowing from matrix to natural fracturesystem Another characteristic ldquodiprdquo can be observed in thederivative curves

Period 10 It is the late-time compound pseudoradial flowperiod This period reflects compound radial flow in naturalfracture system matrix and vugs as shown in Figure 6(d)Similarly the derivative curves exhibit upward tendencies dueto the existence of stress-sensitivity of natural fracture system

52 Effect of Skin Factor Figure 7 shows the effect of skinfactor on dimensionless pressure curves and correspondingderivative curves As stated above the skin factor mainlyaffects the shape of type curves during the transition period(Period 2) after wellbore storage period With the increaseof the value of skin factor the position of the ldquohumprdquo inthe derivative curves becomes higher representing largerpressure drop in the formation

53 Effect of Storativity Ratio of Natural Fracture SystemFigure 8 shows the effect of storativity ratio of natural fracturesystem on dimensionless pressure and pressure derivativecurves It is obvious in Figure 8 that when the storativity ratioof matrix (ie 120596m) and other parameters are kept constantthe storativity ratio of natural fracture system 120596f mainlyaffects the transient pressure dynamics during the first linearflow period the first pseudoradial flow period the secondlinear flow period second pseudoradial flow period and thefirst interporosity flow period On the one hand the positionof the dimensionless pressure curves becomes higher duringthe abovementioned flowing periods with the decrease of 120596f On the other hand with the decrease of120596f the position of thepressure derivative curves during the first linear flow periodthe first pseudoradial flow and the second linear flow periodbecomes higher and the first ldquodiprdquo in pressure derivativecurves becomes wider and deeper

54 Effect of Storativity Ratio of Matrix Figure 9 shows theeffect of storativity ratio of matrix on dimensionless pressureand derivative curves It can be observed that with fixed120596f the storativity ratio of matrix 120596m has primary effect onpressure dynamics during two interporosity flowing periodsWith the decrease of the value of 120596m the fist ldquodiprdquo in thepressure derivative curves becomes wider and deeper whilethe second ldquodiprdquo in the pressure derivative curves exhibits theopposite trend

55 Effect of Interporosity Flow Coefficient between Vugs andNatural Fractures Figure 10 shows the effect of interporosityflow coefficient between vugs and natural fractures (120582vf ) ondimensionless pressure and pressure derivative curves Asexpected 120582vf mainly affects the occurrence time of the first


(a) (b)

(c) (d)

Figure 6 (a) First linear flow period (b) First pseudoradial flow period (c) Second linear flow period (d) Late-time compound pseudoradialflow period

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095

120582mf = 5E minus 11 120582 f = 1E minus 9 120574D = 008

120596f = 0015

pw

Dp

wD998400middott

DC

D v

v

Figure 7 Type curves with different skin factors

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v

Figure 8 Type curves with different storativity ratios of naturalfracture system

interporosity flow period that is the appearance time of thefirst ldquodiprdquo in pressure derivative curves The smaller the valueof120582vf is the later the first ldquodiprdquo can be observed in the pressurederivative curves When 120582vf decreases to a certain extent thethird pseudoradial flow period (Period 8 in Figure 5) maynot be observed in the type curves In addition when 120582vfincreases to a certain extent the second pseudoradial flow

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v

Figure 9 Type curves with different storativity ratios of matrix

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008

120582 f = 3E minus 8 4E minus 9 1E minus 9

pw

Dp

wD998400middott

DC

D

Figure 10 Type curves with different 120582vf

period (Period 6 in Figure 5) and even the second linear flowperiod (Period 5 in Figure 5) may bemasked by the followingfirst interporosity flow period

56 Effect of Interporosity FlowCoefficient betweenMatrix andNatural Fractures Figure 11 shows the effect of interporosityflow coefficient between matrix and natural fractures (120582mf )on dimensionless pressure and pressure derivative curves Itcan be seen that 120582vf mainly affects the occurrence time of


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008

120582mf = 5E minus 105E minus 118E minus 12

pw

Dp

wD998400middott

DC

D

Figure 11 Type curves with different 120582mf

the second interporosity flow period that is the appearancetime of the second ldquodiprdquo in pressure derivative curves Thesmaller the value of 120582mf is the later the second ldquodiprdquo can beobserved in the pressure derivative curves With the increaseof 120582mf the reflection of the third pseudoradial flow periodon the derivative curves (Period 8 in Figure 5) may notbe observed When 120582mf increases to a certain degree thesecond ldquodiprdquo may merge with the first ldquodiprdquo meaning thatsimultaneous interporosity mass transfers happen betweennatural fractures and matrix as well as natural fractures andvugs

57 Effect of Length of Hydraulic Fractures Figure 12 presentsthe effect of half-length of hydraulic fractures on type curvesFor the convenience of discussion 119883fL119894 = 119883fR119894 = 119883f isassumed here It can be observed that the length of hydraulicfracture has primary effect on the first linear flow periodand the subsequent first pseudoradial flow period With theincrease of fracture length the first linear flow period lastslonger with lower position of pressure derivative curvesand the first pseudoradial flow period occurs later withshorter durationWhen the half-length of hydraulic fracturescontinues to increase the horizontal line in the pressurederivative curves reflecting the first pseudoradial flow periodaround each fracture will gradually disappear

58 Effect of Hydraulic Fracture Spacing Figure 13 shows theeffect of the distance between adjacent hydraulic fracturesThe case discussed in Figure 13 assumes equal fracture spac-ing for the convenience of discussion It can be observedthat the distance between adjacent hydraulic fractures Δ119910

119894

mainly affects the first pseudoradial flow period and thesubsequent second linear flow periodThe larger the distancebetween adjacent fractures the longer the duration of the firstpseudoradial flow period and the later the occurrence of thesecond linear flowperiod Another point should be addressedhere is that whenΔ119910

119894

increases to a certain degree the secondpseudoradial flow period may not be observed in the typecurves that is the first interporosity flow period appears afterthe second linear flow period

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D

Figure 12 Type curves with different 119883f

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9120582mf = 5E minus 11

120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D

Figure 13 Type curves with different Δ119910119894

59 Flux Distribution Figure 14 presents the dimensionlessflux distribution along hydraulic fractures at a given timebased on the calculation results obtained with the modelproposed in this paper In the case presented in Figure 14three hydraulic fractures with equal half-length and fracturespacing are considered and each hydraulic fracture is dis-cretized into ten segments It can be observed that for thesame hydraulic fracture the flux distribution is symmetricalwith respect to the horizontal wellbore and the flux density atfracture tips is larger than that in themiddle of the fracture Inaddition for different hydraulic fractures it can be found thatthe flux contribution from hydraulic fracture in the middle(Fracture 3 in Figure 14) is always smaller than that fromouter fractures (Fractures 1 and 2 in Figure 14)

6 Conclusion

Based on the mathematical model and discussion in thispaper the following conclusions can be warranted

(1) A mathematical model is presented for investigatingtransient pressure dynamics as well as flux distri-bution of multistage fractured horizontal wells instress-sensitive triple-porosity reservoirs The modelcan better describe fluid flow in naturally fracturedreservoirs with multiple types of storage spaces andcomplex well type


0 1 2 3 4 5 6 7 8 9 10 11

Fractures 1 and 3Fracture 2

j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075

Figure 14 Dimensionless flux distribution along hydraulic frac-tures

(2) Semianalytical solution is developed in the Laplacedomain by the integral transformation method anddiscretization of hydraulic fractures With the semi-analytical solution and the numerical inversion algo-rithm proposed by Stehfest dimensionless pressureresponses and corresponding pressure derivatives aswell as flux distribution among all the fractures canbe obtained in real time domain Computation resultsprove that the model presented in this paper is stableand accurate

(3) Analysis of transient pressure and derivativeresponses indicates that as many as ten flow periodscan be identified for a multistage fractured horizontalwell in triple-porosity reservoirs Major factorsaffecting the transient pressure responses includeskin factor storativity ratios of different porousmediatype interporosity flow coefficients and parametersrelated to the geometry and placement of hydraulicfractures With different combinations of theseparameters the reflection of some flow periods maybe masked in the pressure derivative curves

(4) Two characteristic ldquodipsrdquo can be observed in the pres-sure derivative curves for triple-porosity reservoirsreflecting the mass transfer between natural fracture-vugs and natural fracture-matrix separately

(5) The stress-sensitivity of natural fracture system resultsin larger pressure drop during intermediate andlate flowing periods which is reflected by upwardtendencies in both dimensionless pressure curves andcorresponding derivative curves

(6) Analysis of flux distribution among multiple hydrau-lic fractures indicates that the flux contribution fromouter fractures is higher and for a given fracture theflux contribution from fracture tips is higher

(7) The presentedmodel can be used to interpret pressuresignal for fractured horizontal wells in triple-porosityreservoirs and provide some important dynamicparameters for reservoir development

Appendices

A Derivation of GoverningEquations for Natural Fracture SystemMatrix System and Vug System inTriple-Porosity Reservoirs

Governing equations for natural fracture system matrix sys-tem and vug system in triple-porosity reservoirs can beobtained by combining equation of motion equation of stateand mass conservation equation

(1) Equation of MotionAssuming radial flow in triple-poros-ity reservoirs the oil flow velocity in natural fracture systemis given by

Vf =119896f120583

120597119901f120597119903

(A1)

where Vf is the radial oil velocity in natural fracture systemms 119896f is the permeability of natural fracture system underpressure 119901f m

2 120583 is the oil viscosity Pasdots 119901f is the pressureof natural fracture system Pa 119903 is the radial distance m

During the production process the reservoir pressuregradually decreases and the effective stress increases Conse-quently the aperture of natural fractures gradually decreasesresulting in the reduction in the permeability of naturalfracture system which is so-called stress-sensitivity of per-meability of natural fracture system To account for thisa stress-dependent permeability is adopted in the naturalfracture flow model

Following Kikani and Pedrosa [29] the permeabilitymodulus 120574 which is used to describe the stress-sensitivity ofnatural fracture system is defined by

120574 =1119896f

d119896fd119901f

(A2)

where 120574 is the permeability modulus Paminus1Equation (A2) can be further written as

int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)

where 119896fi is the permeability of natural fracture system underinitial condition m2 119901i is the reservoir pressure under initialcondition Pa

Solving (A3) one can get

119896f = 119896fieminus120574(119901iminus119901f ) (A4)

Equation (A4) is one of the most common relationshipswhich are used to describe stress-dependent permeabilityThis exponential relationship corresponds to Type I rocksdiscussed by Raghavan and Chin [30]

Substituting (A4) into (A1) yields the equation ofmotionin natural fracture system with consideration of stress-sensitivity effect as follows

Vf =119896fieminus(119901iminus119901f )120574

120583

120597119901f120597119903

(A5)


(2) Equation of State

(a) Equation of State for OilOil in natural fracture system canbe described by the following equation of state

120588f = 1205880e119862o(119901fminus1199010) (A6a)

where 120588f is the oil density in natural fracture system kgm31205880 is the oil density at the reference pressure 1199010 kgm

3 119862o isthe oil compressibility Paminus1

The term e119862o(119901fminus1199010) in (A6a) can be expanded intoMaclaurin series Considering that 119862o ≪ 1 the second andhigher orders of the expansion can be neglected thus we canget

120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)

Similarly oil in matrix system can be described by thefollowing equation of state

120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)

where 119901m is the pressure of matrix system PaOil in vug system can be described by the following

equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)

where 119901v is the pressure of vug system Pa

(b) Equation of State for Porous Media The equation of statefor natural fracture system can be written as

120601f = 120601f0e119862f (119901fminus1199010) (A9a)

where 120601f is the porosity of natural fracture system fraction120601f0 is the porosity of natural fracture system at the referencepressure 1199010 fraction 119862f is the compressibility of naturalfracture system Paminus1

Maclaurin series can also be adopted to approximate(A9a) Considering that119862f ≪ 1 the second and higher orderterms can be neglected and one can get

120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)

By following the sameprocedure one can get the equationof state for matrix system as follows

120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)

where 120601m0 is the porosity of matrix system at the referencepressure 1199010 fraction 119862m is the compressibility of matrixsystem Paminus1

Similarly the equation of state for vug system can bedescribed as follows

120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)

where 120601v0 is the porosity of vug system at the referencepressure 1199010 fraction 119862v is the compressibility of matrixsystem Paminus1

(3) Mass Conservation Equation By applying the mass con-servation lawon a representative elemental volume in a triple-porosity reservoir we can get the following equation fornatural fracture system

1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)

Equation (A12a) can also be written in the followingequivalent form

120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)

where 119902c1 is the mass flow rate between matrix system andfracture system per unit-volume reservoir kg(m3sdots) and itcan be expressed by (A13) 119902c2 is the mass flow rate betweenvug system and fracture system per unit-volume reservoirkg(m3sdots) and it can be expressed by (A14)

119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)

where 120572mf is the shape factor of matrix blocks mminus2 120572vf isthe shape factor of vug system mminus2 119896m is the permeabilityof matrix system m2 119896v is the permeability of vug systemm2

By using the same method we can get the following massconservation equations for matrix system and vug system

1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)

Here following the assumption adopted in the dual-porosity model proposed by Warren and Root [2] naturalfractures are assumed to be oil flowing channels and matrixand vugsmainly serve as storage space thus the flowing termsin (A15) and (A16) can be reduced to

1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)


With (A17) (A15) and (A16) can be rewritten as

120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)

Equations (A12a) (A12b) (A18) and (A19) are massconservation equations for natural fracture system matrixsystem and vug system in triple-porosity reservoirs sepa-rately

(4) Governing Equation for Natural Fracture System Substitu-tion of (A15) into the mass conservation equation for naturalfracture system that is (A12b) yields

120597 (120588f (119896fieminus(119901iminus119901f )120574120583) (120597119901f120597119903))

120597119903

+1119903120588f

119896fieminus(119901iminus119901f )120574

120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)

With (A6a) (A6b) and the transformation formulae119862o(119901fminus1199010)(120597119901f120597119903) = (120597120597119903)[e119862o(119901fminus1199010)119862o] the first term in theleft-hand side of (A20) can be written as


120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)

Following the same procedure the second term in theleft-hand side of (A20) changes into

1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)

With (A6b) and (A9b) the product term within thebracket in the right-hand side of (A20) can be expanded asfollows

120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)

Given that both 119862f and 119862o are very small terms theirproduct will yield a much smaller term which is negligible(119862f119862o asymp 0) and (A23a) can be simplified as

120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)

Thus the first term in the right-hand side of (A20) can bewritten as

120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)

where 119862f t is the total compressibility of natural fracturesystem including the compressibility of oil and naturalfractures and

119862f t = 119862f +119862o (A25)

Substituting (A13) (A14) (A21) (A22) and (A24) into(A20) one can get

1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2

= eminus120574(119901fminus119901i) [120601f0120583119862f t

119896fi

120597119901f120597119905

minus120572mf119896m

119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)

After deriving governing equations formatrix system andvug system (A26a) can be further written as

1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)

Equation (A26b) is the final form of governing equa-tion for natural fracture system incorporating the pressure-dependent permeability that is (1) in the main body of thispaper

(5) Governing Equation for Matrix System With (A7b) and(A10b) the product of oil density and porosity of matrixsystem can be written as

120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)

Analogously because both 119862m and 119862o are very smallterms their product 119862m119862o can be neglected and (A27a) canbe simplified to obtain

120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)

where 119862mt is the total compressibility of matrix systemincluding the compressibility of oil and matrix and

119862mt = 119862m +119862o (A29)

Substitution of (A13) and (A28) into the mass conserva-tion equation for matrix system that is (A18) yields

120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)

Equation (A30) is the final form of governing equationfor matrix system in triple-porosity reservoirs that is (2) inthe main body of this paper

(6) Governing Equation for Vug System The derivation ofgoverning equation for vug system is similar to that applied


to matrix system With (A8b) and (A11b) the product of oildensity and porosity of vug system can be written as

120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)

Given that both 119862v and 119862o are much smaller than 1 theirproduct 119862v119862o is negligible and (A31a) reduces to

120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)

where119862vt is the total compressibility of vug system includingthe compressibility of oil and vug and

119862vt = 119862v +119862o (A33)

Substituting (A14) and (A33) into mass conservationequation for vug system that is (A19) one can get

120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)

Equation (A34) is the final form of governing equationfor vug system in triple-porosity reservoirs that is (3) in themain body of this paper

B Derivation of Inner BoundaryCondition for the Line-Sink

A vertical line-sink can be treated as the limiting case of avertical cylinder-sink with radius 120576 rarr 0The flow rate acrossthe cylinder wall with radius 120576 (120576 rarr 0) can be expressed as

lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)

where119860 is the lateral area of the cylinder-sink m2 which canbe calculated by (B2) Vf is oil velocity at the cylinder wallms which can be calculated by (B3) 119902(119905) is flow rate of thecontinuous line-sink under standard condition m3s 119861 is theoil formation volume factor dimensionless

119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)

Substitution of (A4) into (B3) yields

Vf1003816100381610038161003816119903=120576 =

119896fieminus120574(119901iminus119901f )

120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)

Substituting (B2) and (B4) into (B1) we can get

lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)

Equation (B5) is the inner boundary condition for a line-sink in triple-porosity reservoirs that is (4) in themain bodyof this paper

Nomenclature

119861 Oil formation volume factor m3sm3119862 Wellbore storage coefficient m3Pa119862D Dimensionless wellbore storage

coefficient dimensionless119862o Oil compressibility Paminus1119862f Compressibility of natural fracture system

Paminus1119862m Compressibility of matrix system Paminus1119862v Compressibility of vug system Paminus1119862f t Total compressibility of natural fracture

system Paminus1119862mt Total compressibility of matrix system

Paminus1119862vt Total compressibility of vug system Paminus1ℎ Reservoir thickness m119896f Permeability of natural fracture system

m2119896fi Permeability of natural fracture system

under initial condition m2119896m Permeability of matrix system m2119896v Permeability of vug system m21198700(119909) Modified Bessel function of the second

kind zero order1198701(119909) Modified Bessel function of the second

kind first order1198680(119909) Modified Bessel function of the first kind

zero order119871 Reference length m119871h Length of horizontal wellbore m119872 Total number of hydraulic fractures119873 Number of discretized segments for each

fracture119901f Pressure of natural fracture system Pa119901m Pressure of matrix system Pa119901v Pressure of vug system Pa119901fD Dimensionless pressure of natural fracture

system dimensionless119901mD Dimensionless pressure of matrix system

dimensionless119901vD Dimensionless pressure of vug system

dimensionless119901i Initial reservoir pressure Pa1199010 Reference pressure PaΔ119901s Extra pressure drop caused by skin effect

near the wellbore Pa119902(119905) Production rate of the line-sink under

standard condition m3s119902D Dimensionless production rate of the

line-sink under standard conditiondimensionless

119902 Constant surface production rate of themultistage fractured horizontal well m3s

119902119894119895

Flux density of the 119895th segment in the 119894thhydraulic fracture m3(ssdotm)

119902119894119895

(119906) Laplace transformation of 119902119894119895


119902D119894119895 Dimensionless flux density of the 119895thsegment in the 119894th fracture dimensionless

119902D119894119895(119906) Laplace transformation of 119902D119894119895119902sf The sandface flow rate of the multistage

fractured horizontal well m3s119902c1 Mass flow rate between matrix and natural

fractures per unit-volume reservoirkg(m3sdots)

119902c2 Mass flow rate between vug system andfracture system per unit-volume reservoirkg(m3sdots)

119903w Radius of horizontal wellbore m119903 Radial distance 119903 = radic1199092 + 1199102 m119903D Dimensionless radial distance

dimensionless119878 Skin factor dimensionless119905 Time s119905D Dimensionless time dimensionless119906 Variable of Laplace transformation

dimensionlessVf Radial velocity for oil flow in natural

fracture system ms119909 119910 119909- and 119910-coordinates m119909D 119910D Dimensionless 119909- and 119910-coordinates

dimensionless119909w 119910w 119909- and 119910-coordinates of the line-sink m119909wD 119910wD Dimensionless 119909- and 119910-coordinates of

the line-sink dimensionless119883119894119895

119884119894119895

119909- and 119910-coordinates of the 119895th discretesegment in the 119894th fracture m

119883fL119894 Length of the left wing of the 119894th fracturem

119883fR119894 Length of the right wing of the 119894thfracture m

Δ119883119894119895

Length of the discrete segment (119894 119895) mΔ119883D119894119895 Dimensionless length of the discrete

segment (119894 119895) dimensionless119910119894

119910-coordinate of the intersection betweenthe 119894th fracture and 119910-axis m

Δ119910119894

Distance between 119910119894

and 119910119894minus1

Δ119910119894

= 119910119894

minus 119910119894minus1

1205721 Shape factor of matrix blocks mminus2

1205722 Shape factor of vug system mminus2120588f Oil density in natural fracture system

kgm31205880 Oil density at the reference pressure 1199010

kgm3120601f Porosity of natural fracture system

fraction120601m Porosity of matrix system fraction120601v Porosity of vug system fraction120601f0 Porosity of natural fracture system at the

reference pressure 1199010 fraction120601m0 Porosity of matrix system at the reference

pressure 1199010 fraction120601v0 Porosity of vug system at the reference

pressure 1199010 fraction

120596f Storativity ratio of natural fracture systemdimensionless

120596m Storativity ratio of matrix systemdimensionless

120596v Storativity ratio of vug systemdimensionless

120582mf Interporosity flow coefficient betweenmatrix and natural fracturesdimensionless

120582vf Interporosity flow coefficient betweenvugs and natural fractures dimensionless

120574 Permeability modulus Paminus1120574D Dimensionless permeability modulus

dimensionless120583 Oil viscosity Pasdots120587 Circumference ratio 31415926

dimensionless

Subscript

D Dimensionless

Superscript

minus Laplace transformation

Conflict of Interests

Theauthors declare that there is no conflict of interests relatedto this paper

Acknowledgments

Support for this work is provided by the National NaturalScience Foundation of China (Grants nos 51404206 and51304165) the National Science Fund for DistinguishedYoung Scholars of China (Grant no 51125019) the Project ofNational First-Level Discipline in Oil and Gas Engineeringfrom Chinarsquos Central Government for Development of LocalColleges and Universities and SWPU Science amp TechnologyFund (Project no 2013XJZ005)The authors alsowould like tothank the reviewers and editors for their helpful comments

References

[1] G E Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocksrdquo Journal of Applied Mathematics and Mechanics vol 24no 5 pp 1286ndash1303 1960

[2] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo Society of Petroleum Engineers Journal vol 3 no 3pp 245ndash255 2013

[3] J-C Cai S-L Guo L-J You and X-Y Hu ldquoFractal analysis ofspontaneous imbibition mechanism in fractured-porous dualmedia reservoirrdquo Acta Physica Sinica vol 62 no 1 Article ID014701 2013


[4] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal of Modern Physics C vol 24 no 8 ArticleID 1350056 2013

[5] D Abadassah and I Ershaghi ldquoTriple-porosity systems forrepresenting naturally fractured reservoirsrdquo SPE FormationEvaluation vol 1 no 2 pp 113ndash127 1986

[6] A Al-Ghamdi and I Ershaghi ldquoPressure transient analysis ofdually fractured reservoirsrdquo SPE Journal vol 1 no 1 pp 93ndash100 1996

[7] S Q Cheng and X F Qu ldquoWell testing analysis method for thetriple porosity reservoirrdquoWell Testing vol 6 no 1 pp 5ndash11 1997

[8] J Yao W-H Dai and Z-S Wang ldquoWell test interpretationmethod for triple medium reservoir with variable wellborestoragerdquo Journal of the University of Petroleum China vol 28no 1 pp 46ndash51 2004

[9] F H Escobar N F Saavedra G D Escorcia and J H PolanialdquoPressure and pressure derivative analysis without type-curvematching for triple porosity reservoirsrdquo in Proceedings of theSPE Asia Pacific Oil and Gas Conference and Exhibition PaperSPE 88556 Perth Australia October 2004

[10] J Yang J Yao and Z S Wang ldquoStudy of pressure-transientcharacteristic for triple-medium composite reservoirsrdquo Journalof Hydrodynamics Series A vol 20 no 4 pp 418ndash425 2005

[11] B Mirshekari H Modarress Z Hamzehnataj and E Momen-nasab ldquoApplication of pressure derivative function for well testanalysis of triple porosity naturally fractured reservoirsrdquo in Pro-ceedings of the SPE Saudi Arabia Section Technical SymposiumPaper SPE 110943 Dhahran Saudi Arabia May 2007

[12] Y-S Wu C A Ehlig-Economides G Qin et al ldquoA triple-continuum pressure-transient model for a naturally fracturedvuggy reservoirrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition (ATCE rsquo07) Paper SPE 110044 pp1811ndash1820 Anaheim Calif USA November 2007

[13] L Zhang D-K Tong and X-D Ma ldquoPressure dynamicanalysis of triple permeabilitymodel in deformed triple porosityreservoirsrdquo Engineering Mechanics vol 25 no 10 pp 103ndash1092008

[14] H T Wang and L H Zhang ldquoSolution to the unsteady flowproblem in anisotropic triple-porosity reservoir with imper-meable top and bottom boundaries and a partially penetratingvertical well based on the source function approachrdquo Journal ofDaqing Petroleum Institute vol 32 no 3 pp 29ndash33 2008

[15] R S Nie Y L Jia J Yu B Liu and Y L Liu ldquoThe transientwell test analysis of fractured-vuggy triple-porosity reservoirwith the quadratic pressure gradient termrdquo in Proceedings ofthe SPE Latin American and Caribbean Petroleum EngineeringConference Paper SPE 120927 Cartagena Colombia May-June2009

[16] C L Xing and J Yan ldquoTriple medium reservoir model and theLaplace space solutionrdquoMathematical Theory and Applicationsvol 29 no 2 pp 14ndash18 2009

[17] Y-L Zhao L-H Zhang and S-L Qing ldquoWell test model andtypical curve analysis of non-Newtonian fluidrsquos transient flowin triple media reservoirsrdquo Chinese Journal of HydrodynamicsA vol 25 no 2 pp 254ndash261 2010

[18] D-L Zhang J-L LiW-J Du andY-SWu ldquoTriple-continuumnumerical well test interpretation method of oil-water flowfor a naturally fractured vuggy reservoirrdquo Chinese Journal ofHydrodynamics A vol 25 no 4 pp 429ndash437 2010

[19] Y-S Wu Y Di Z Kang and P Fakcharoenphol ldquoA multiple-continuum model for simulating single-phase and multi-phase flow in naturally fractured vuggy reservoirsrdquo Journal ofPetroleum Science and Engineering vol 78 no 1 pp 13ndash22 2011

[20] C-Y Li Q-G Liu R Zhang K Yang and L-X Sun ldquoTheresearch of well test of horizontal well in triple mediumreservoirrdquo Journal of Southwestern Petroleum Institute vol 28no 4 pp 32ndash35 2006

[21] S-Q Cheng L-J Zhang and X-F Li ldquoWell-test analysis ofmulti-branched horizontal wells in a triple medium reservoirrdquoChinese Journal of Hydrodynamics Series A vol 24 no 2 pp127ndash132 2009

[22] H Wang L Zhang and J Guo ldquoA new rod source model forpressure transient analysis of horizontal wells with positivenegative skin in triple-porosity reservoirsrdquo Journal of PetroleumScience and Engineering vol 108 pp 52ndash63 2013

[23] H A Al-AhamadiA triple-porositymodel for fractured horizon-tal wells [MS thesis] Texas AampM University College StationTex USA 2010

[24] H A Al-Ahamadi S Aramco and R AWattenbarger ldquoTriple-porosity models one further step toward capturing frac-tured reservoirs heterogeneityrdquo in Proceedings of the SPEDGSSaudi Arabia Section Technical Symposium and Exhibition SPE149054 pp 15ndash18 Al-Khobar Saudi Arabia May 2011

[25] O A Pedrosa Jr ldquoPressure transient response in stress-sensitiveformationsrdquo in Proceedings of the SPE 56th Annual CaliforniaRegional Meeting Paper SPE 15115 pp 203ndash214 Oakland CalifUSA April 1986

[26] A van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 06 pp 171ndash176 2013

[27] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well test-ing (includes associated papers 13937 and 14693)rdquo Journal ofPetroleum Technology vol 37 no 2 pp 323ndash334 1985

[28] H Stehfest ldquoNumerical inversion of Laplace transformrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[29] J Kikani andOA Pedrosa ldquoPerturbation analysis of stress-sen-sitive reservoirs (includes associated papers 25281 and 25292)rdquoSPE Formation Evaluation vol 6 no 3 pp 379ndash386 1991

[30] R Raghavan and L Y Chin ldquoProductivity changes in reservoirswith stress-dependent permeabilityrdquo SPE Reservoir Evaluationand Engineering vol 7 no 4 pp 308ndash315 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


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Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


Hydraulic fractures

Horizontal well

Triple-porosity reservoir

Matrix

Vug

Natural fracture

Figure 1 Schematic of a multistage fractured horizontal well in atriple-porosity reservoir

over the past several decades numerous efforts have beenmade to investigate transient pressure behavior in so-calledtriple-porosity reservoirs However almost all the researchfocuses on vertical wells [7ndash19] or horizontal wells [20ndash22]in triple-porosity reservoirs and few studies have been doneon complex well-reservoir configuration such as multistagefractured horizontal well in triple-porosity reservoirs

Field practices have proven that horizontal wellborecombining multistage hydraulic fracturing can effectivelyenhance the permeability in the vicinity of horizontal well-bore Almost all the triple-porosity models [23 24] forfractured horizontal wells assume linear flow which cannotcompletely characterize the pressure responses during allflowing periods (such as pseudoradial flow period) On theother hand in most existing models for fractured horizontalwells in triple-porosity reservoirs three porous media denotehydraulic fractures natural fractures and matrix poreswhich means the reservoir itself is actually considered as adual-porosity medium

This study develops a semianalytical model based onsourcesink function theory for analyzing transient pressureresponses and flux distribution of naturally fractured-vuggyreservoirs in which the reservoir itself is conceptualized astriple-porosity media and the interaction between hydraulicfractures and reservoir is considered The model presentedhere can completely reflect transient pressure characteristicsduring all the possible flowing period as well as flux dis-tribution among multiple fractures In addition this modelalso takes into account the stress-sensitivity of permeabilitycaused by closure of natural fractures during production

2 Physical Model of a MultistageFractured Horizontal Well ina Triple-Porosity Reservoir

As shown in Figure 1 a horizontal well with multiple hydra-ulic fractures producing at a constant rate (119902) in a reservoirwith multiscale storage spaces is considered here Both thecap rock and underlying rock formation of the reservoir areusually shales with extremely low permeability which can beconceived as no-flow boundariesThe reservoir thickness andoil viscosity are assumed to be ℎ and 120583 separately Other basicassumptions involved in the derivation of the triple-porositymodel are as follows

(1) The reservoir is assumed to be composed of three con-tiguous porous media which are matrix natural

Natural fractures

Vugs Matrix

Hyd

raul

ic fr

actu

re

Figure 2 Illustration of fluid flowing paths in a triple-porosity res-ervoir

fractures and vugs (representing larger voids in thereservoir which are not part of natural fractures)

(2) Natural fractures are assumed to be directly con-nected to hydraulic fractures or the horizontal well-bore while the fluid stored in matrix pores or vugsdoes not have direct access to hydraulic fractures orthe horizontal wellbore Figure 2 presents a simpleillustration of the fluid flowing path in triple-porosityreservoirs

(3) The permeability of natural fractures is consideredstress-sensitive to incorporate the effect of partial orcomplete closure of natural fractures during produc-tion

(4) Both oil and rock are considered slightly compress-ible and the reservoir has uniform pressure distribu-tion at time zero

(5) Consider single-phase isothermal flow and negligiblegravity and capillary effects

To obtain the pressure responses caused by the well-reservoir configuration shown in Figure 1 a mathematicalmodel together with corresponding boundary and initialconditions is required In the problem addressed here themultiple fractures inner boundary condition is so complexthat it is rather difficult to directly describe it with a math-ematical equation In order to solve this problem sourcefunction theory a powerful tool to solve complex transientflow problems in reservoirs is adopted in this work Wefirst studied the transient pressure responses caused by acontinuous line-sink in triple-porosity reservoirs and thenadopted the line-sink solutionwith superposition principle toobtain the pressure transient dynamics and flux distributionof multistage fractured horizontal wells in triple-porosityreservoirs

3 Line-Sink Model in Stress-SensitiveTriple-Porosity Reservoirs

31 Physical Model for a Line-Sink in Triple-Porosity Reser-voirs As shown in Figure 3 a continuous line-sink in atriple-porosity reservoir with stress-dependent fracture per-meability is considered in this section The strength of theline-sink is represented by 119902(119905)


Top boundary


Bottom boundaryZw

h

re





1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2= eminus120574(119901fminus119901i) [

120601f0120583119862f t119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(1)


120601m0119862mt120597119901m120597119905

+1205721119896m120583

(119901m minus119901f) = 0 (2)


120601v0119862vt120597119901v120597119905

+1205722119896v120583

(119901v minus119901f) = 0 (3)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (4)


119901f (119903 119905)1003816100381610038161003816119903rarrinfin = 119901i (5)


119901f (119903 119905)1003816100381610038161003816119905=0 = 119901m (119903 119905)

1003816100381610038161003816119905=0 = 119901v (119903 119905)1003816100381610038161003816119905=0 = 119901i (6)





120596f +120596m +120596v = 1 (7)


1205972119901fD

120597119903D2 +

1119903D

120597119901fD120597119903D

minus 120574D (120597119901fD120597119903D

)

2

= e120574D119901fD [120596f120597119901fD120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

]

(8)

120596m120597119901mD120597119905D

+120582mf (119901mD minus119901fD) = 0 (9)


+120582vf (119901vD minus119901fD) = 0 (10)

lim120576Drarr 0

eminus119901fD120574D 119903D120597119901fD120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (11)

119901fD1003816100381610038161003816119903Drarrinfin

= 0 (12)

119901fD1003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0 (13)



119901fD (119903D 119905D) = minus1120574D

ln [1minus 120574D119880D (119903D 119905D)] (14)



1205972119880D

1205971199032D+

1119903D

120597119880D120597119903D

=1

1 minus 120574D119880D120596f

120597119880D120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

(15)

120596m120597119901mD120597119905D

+120582mf [119901mD +1120574D

ln (1minus 120574D119880D)] = 0 (16)


+120582vf [119901vD +1120574D

ln (1minus 120574D119880D)]

= 0(17)




=2120587119896119891119894

ℎ

119902119861120583Δ119901119891

119901119898119863

=2120587119896119891119894

ℎ

119902119861120583Δ119901119898

119901V119863 =2120587119896119891119894

ℎ

119902119861120583Δ119901V


=119903

119871 120576119863

=120576

119871


=119896119891119894

119905

120583 (1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905) 1198712


=1206011198910

119862119891119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


=1206011198980

119862119898119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


120582119898119891

=120572119898119891

119896119898

1198712

119896119891119894


120582V119891 =120572V119891119896V119871

2

119896119891119894


=119902119861120583

2120587119896119891119894

ℎ120574


= 119902119863

(119905119863

) =119902 (119905)

119902


lim120576Drarr 0

119903D120597119880D120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (18)

119880D1003816100381610038161003816119903Drarrinfin

= 0 (19)

119880D1003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0 (20)



minus1120574D

ln [1minus 120574D119880D (119903D 119905D)]

= 119880D (119903D 119905D) +12120574D119880D


11 minus 120574D119880D (119903D 119905D)


(21)


1205972119880D01205971199032D

+1119903D

120597119880D0120597119903D

= 120596f120597119880D0120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

120596m120597119901mD120597119905D

+120582mf (119901mD minus119880D0) = 0


+120582vf (119901vD minus119880D0) = 0

lim120576Drarr 0

119903D120597119880D0120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D

119880D01003816100381610038161003816119903Drarrinfin

= 0

119880D01003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0

(22)



119880D0 = int

+infin

0119880D0eminus119906119905Dd119905D (23)



Then (22) becomes

d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 120596f119906119880D0 +120596m119906119901mD +120596v119906119901vD (24)

120596m119906119901mD +120582mf (119901mD minus119880D0) = 0 (25)

120596v119906119901vD +120582vf (119901vD minus119880D0) = 0 (26)

lim120576Drarr 0

119903Dd119880D0d119903D

100381610038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (27)

119880D010038161003816100381610038161003816119903Drarrinfin

= 0 (28)


119901mD =120582mf

120596m119906 + 120582mf119880D0

119901vD =120582vf

120596v119906 + 120582vf119880D0

(29)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906119880D0

(30)

If we define

119892 (119906) = (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906 (31)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 119892 (119906)119880D0 (32)


119880D0 = 11988811198700 (radic119892 (119906)119903D)+ 11988821198680 (radic119892 (119906)119903D) (33)

where 1198700

and 1198680


and 1198882



119903D = radic119909D2 + 119910D

2 (34a)



2 (34b)

where

119909D =119909

119871 (35)

119910D =119910

119871 (36)

119909wD =119909w119871

(37)

119910wD =119910w119871

(38)


lim120576Drarr 0

radic119892 (119906)119903D


= minus 119902D

(39)



1198881 = 119902D (40)


119909rarrinfin

1198700(119909) rarr 0and lim

119909rarrinfin


1198882 = 0 (41)


119880D0 = 119902D1198700 (radic119892 (119906)119903D) (42)


119880D =119902

1199021198700 (radic119892 (119906)119903D) (43)


119880D =119902

119902Ω (119909D 119910D 119909wD 119910wD) (44)

where

Ω(119909D 119910D 119909wD 119910wD) = 1198700 (radic119892 (119906)119903D) (45)



x

y

1 i M

O

(0 0) (0 y1)(XiN YiN)

(Xi1 Yi1)

(xiN yiN)

(xi1 yi1)(xi2 yi2)


(xiN+1 yiN+1)


(0 yM)(0 yi)

(xi2N+1 yi2N+1)





119894


119894



119894119895

119884119894119895


119910119894119895

) and (119909119894119895+1 119910119894119895+1)


119909119894119895

= minus119873 minus 119895 + 1

119873119883fL119894

119910119894119895

= 119910119894

1 le 119895 le 119873

119909119894119895

=119895 minus 119873 minus 1

119873119883fR119894

119910119894119895

= 119910119894

119873 + 1 le 119895 le 2119873 + 1

(46)


119883119894119895

= minus2119873 minus 2119895 + 1

2119873119883fL119894

119884119894119895

= 119910119894

1 le 119895 le 119873

119883119894119895

=2 (119895 minus 119873) minus 1

2119873119883fR119894

119884119894119895

= 119910119894

119873 + 1 le 119895 le 2119873

(47)



and 119910119894minus1

is

Δ119910119894

= 119910119894

minus119910119894minus1 (48)


119894119895


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906)

119902int

119909119894119895+1

119909119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909w(49)


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906) 119871

119902int

119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(50)


119902D119894119895 (119905D) =119902119894119895

(119905D) 119871

119902(51)


119902D119894119895 (119906) =

119902119894119895

(119906) 119871

119902 (52)


119880D119894119895 (119909D 119910D)

= 119902D119894119895 int119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(53)



119880D (119909D 119910D) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D 119910D) (54)



119880D (119909D120594120581 119910D120594120581) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D120594120581 119910D120594120581) (55)


119880D (119883D120594120581 119884D120594120581) = 119880wD (56)


119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119883D120594120581 119884D120594120581) = 119880wD (57)



119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] = 119902 (58)

whereΔ119883119894119895


Δ119883119894119895

= 119883119894119895+1 minus119883

119894119895

(59)


119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] =119902

119906 (60)


119872

sum

119894=1

2119873sum

119895=1[119902D119894119895Δ119883D119894119895] =

1119906 (61)

where

Δ119883D119894119895 =Δ119883119894119895

119871 (62)



119880wDS =119906119880wD + 119878

119906 + 119862D1199062 (119906119880wD + 119878)

(63)


119862D =119862

2120587 (120601f119862f t + 120601m119862mt + 120601v119862vt) ℎ1198712 (64)

119878 =2120587119896fiℎ119902sf120583

Δ119901119904

(65)


119904



119901wD (119905D) = minus1120574D

ln [1minus 120574D119880wDS (119905D)] (66)








1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05


120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D
















(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Top boundary


Bottom boundaryZw

h

re





1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2= eminus120574(119901fminus119901i) [

120601f0120583119862f t119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(1)


120601m0119862mt120597119901m120597119905

+1205721119896m120583

(119901m minus119901f) = 0 (2)


120601v0119862vt120597119901v120597119905

+1205722119896v120583

(119901v minus119901f) = 0 (3)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (4)


119901f (119903 119905)1003816100381610038161003816119903rarrinfin = 119901i (5)


119901f (119903 119905)1003816100381610038161003816119905=0 = 119901m (119903 119905)

1003816100381610038161003816119905=0 = 119901v (119903 119905)1003816100381610038161003816119905=0 = 119901i (6)





120596f +120596m +120596v = 1 (7)


1205972119901fD

120597119903D2 +

1119903D

120597119901fD120597119903D

minus 120574D (120597119901fD120597119903D

)

2

= e120574D119901fD [120596f120597119901fD120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

]

(8)

120596m120597119901mD120597119905D

+120582mf (119901mD minus119901fD) = 0 (9)


+120582vf (119901vD minus119901fD) = 0 (10)

lim120576Drarr 0

eminus119901fD120574D 119903D120597119901fD120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (11)

119901fD1003816100381610038161003816119903Drarrinfin

= 0 (12)

119901fD1003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0 (13)



119901fD (119903D 119905D) = minus1120574D

ln [1minus 120574D119880D (119903D 119905D)] (14)



1205972119880D

1205971199032D+

1119903D

120597119880D120597119903D

=1

1 minus 120574D119880D120596f

120597119880D120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

(15)

120596m120597119901mD120597119905D

+120582mf [119901mD +1120574D

ln (1minus 120574D119880D)] = 0 (16)


+120582vf [119901vD +1120574D

ln (1minus 120574D119880D)]

= 0(17)




=2120587119896119891119894

ℎ

119902119861120583Δ119901119891

119901119898119863

=2120587119896119891119894

ℎ

119902119861120583Δ119901119898

119901V119863 =2120587119896119891119894

ℎ

119902119861120583Δ119901V


=119903

119871 120576119863

=120576

119871


=119896119891119894

119905

120583 (1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905) 1198712


=1206011198910

119862119891119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


=1206011198980

119862119898119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


120582119898119891

=120572119898119891

119896119898

1198712

119896119891119894


120582V119891 =120572V119891119896V119871

2

119896119891119894


=119902119861120583

2120587119896119891119894

ℎ120574


= 119902119863

(119905119863

) =119902 (119905)

119902


lim120576Drarr 0

119903D120597119880D120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (18)

119880D1003816100381610038161003816119903Drarrinfin

= 0 (19)

119880D1003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0 (20)



minus1120574D

ln [1minus 120574D119880D (119903D 119905D)]

= 119880D (119903D 119905D) +12120574D119880D


11 minus 120574D119880D (119903D 119905D)


(21)


1205972119880D01205971199032D

+1119903D

120597119880D0120597119903D

= 120596f120597119880D0120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

120596m120597119901mD120597119905D

+120582mf (119901mD minus119880D0) = 0


+120582vf (119901vD minus119880D0) = 0

lim120576Drarr 0

119903D120597119880D0120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D

119880D01003816100381610038161003816119903Drarrinfin

= 0

119880D01003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0

(22)



119880D0 = int

+infin

0119880D0eminus119906119905Dd119905D (23)



Then (22) becomes

d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 120596f119906119880D0 +120596m119906119901mD +120596v119906119901vD (24)

120596m119906119901mD +120582mf (119901mD minus119880D0) = 0 (25)

120596v119906119901vD +120582vf (119901vD minus119880D0) = 0 (26)

lim120576Drarr 0

119903Dd119880D0d119903D

100381610038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (27)

119880D010038161003816100381610038161003816119903Drarrinfin

= 0 (28)


119901mD =120582mf

120596m119906 + 120582mf119880D0

119901vD =120582vf

120596v119906 + 120582vf119880D0

(29)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906119880D0

(30)

If we define

119892 (119906) = (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906 (31)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 119892 (119906)119880D0 (32)


119880D0 = 11988811198700 (radic119892 (119906)119903D)+ 11988821198680 (radic119892 (119906)119903D) (33)

where 1198700

and 1198680


and 1198882



119903D = radic119909D2 + 119910D

2 (34a)



2 (34b)

where

119909D =119909

119871 (35)

119910D =119910

119871 (36)

119909wD =119909w119871

(37)

119910wD =119910w119871

(38)


lim120576Drarr 0

radic119892 (119906)119903D


= minus 119902D

(39)



1198881 = 119902D (40)


119909rarrinfin

1198700(119909) rarr 0and lim

119909rarrinfin


1198882 = 0 (41)


119880D0 = 119902D1198700 (radic119892 (119906)119903D) (42)


119880D =119902

1199021198700 (radic119892 (119906)119903D) (43)


119880D =119902

119902Ω (119909D 119910D 119909wD 119910wD) (44)

where

Ω(119909D 119910D 119909wD 119910wD) = 1198700 (radic119892 (119906)119903D) (45)



x

y

1 i M

O

(0 0) (0 y1)(XiN YiN)

(Xi1 Yi1)

(xiN yiN)

(xi1 yi1)(xi2 yi2)


(xiN+1 yiN+1)


(0 yM)(0 yi)

(xi2N+1 yi2N+1)





119894


119894



119894119895

119884119894119895


119910119894119895

) and (119909119894119895+1 119910119894119895+1)


119909119894119895

= minus119873 minus 119895 + 1

119873119883fL119894

119910119894119895

= 119910119894

1 le 119895 le 119873

119909119894119895

=119895 minus 119873 minus 1

119873119883fR119894

119910119894119895

= 119910119894

119873 + 1 le 119895 le 2119873 + 1

(46)


119883119894119895

= minus2119873 minus 2119895 + 1

2119873119883fL119894

119884119894119895

= 119910119894

1 le 119895 le 119873

119883119894119895

=2 (119895 minus 119873) minus 1

2119873119883fR119894

119884119894119895

= 119910119894

119873 + 1 le 119895 le 2119873

(47)



and 119910119894minus1

is

Δ119910119894

= 119910119894

minus119910119894minus1 (48)


119894119895


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906)

119902int

119909119894119895+1

119909119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909w(49)


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906) 119871

119902int

119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(50)


119902D119894119895 (119905D) =119902119894119895

(119905D) 119871

119902(51)


119902D119894119895 (119906) =

119902119894119895

(119906) 119871

119902 (52)


119880D119894119895 (119909D 119910D)

= 119902D119894119895 int119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(53)



119880D (119909D 119910D) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D 119910D) (54)



119880D (119909D120594120581 119910D120594120581) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D120594120581 119910D120594120581) (55)


119880D (119883D120594120581 119884D120594120581) = 119880wD (56)


119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119883D120594120581 119884D120594120581) = 119880wD (57)



119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] = 119902 (58)

whereΔ119883119894119895


Δ119883119894119895

= 119883119894119895+1 minus119883

119894119895

(59)


119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] =119902

119906 (60)


119872

sum

119894=1

2119873sum

119895=1[119902D119894119895Δ119883D119894119895] =

1119906 (61)

where

Δ119883D119894119895 =Δ119883119894119895

119871 (62)



119880wDS =119906119880wD + 119878

119906 + 119862D1199062 (119906119880wD + 119878)

(63)


119862D =119862

2120587 (120601f119862f t + 120601m119862mt + 120601v119862vt) ℎ1198712 (64)

119878 =2120587119896fiℎ119902sf120583

Δ119901119904

(65)


119904



119901wD (119905D) = minus1120574D

ln [1minus 120574D119880wDS (119905D)] (66)








1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05


120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D
















(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




=2120587119896119891119894

ℎ

119902119861120583Δ119901119891

119901119898119863

=2120587119896119891119894

ℎ

119902119861120583Δ119901119898

119901V119863 =2120587119896119891119894

ℎ

119902119861120583Δ119901V


=119903

119871 120576119863

=120576

119871


=119896119891119894

119905

120583 (1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905) 1198712


=1206011198910

119862119891119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


=1206011198980

119862119898119905

(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


(1206011198910

119862119891119905

+ 1206011198980

119862119898119905

+ 120601V0119862V119905)


120582119898119891

=120572119898119891

119896119898

1198712

119896119891119894


120582V119891 =120572V119891119896V119871

2

119896119891119894


=119902119861120583

2120587119896119891119894

ℎ120574


= 119902119863

(119905119863

) =119902 (119905)

119902


lim120576Drarr 0

119903D120597119880D120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (18)

119880D1003816100381610038161003816119903Drarrinfin

= 0 (19)

119880D1003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0 (20)



minus1120574D

ln [1minus 120574D119880D (119903D 119905D)]

= 119880D (119903D 119905D) +12120574D119880D


11 minus 120574D119880D (119903D 119905D)


(21)


1205972119880D01205971199032D

+1119903D

120597119880D0120597119903D

= 120596f120597119880D0120597119905D

+120596m120597119901mD120597119905D

+120596v120597119901vD120597119905D

120596m120597119901mD120597119905D

+120582mf (119901mD minus119880D0) = 0


+120582vf (119901vD minus119880D0) = 0

lim120576Drarr 0

119903D120597119880D0120597119903D

10038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D

119880D01003816100381610038161003816119903Drarrinfin

= 0

119880D01003816100381610038161003816119905D=0

= 119901mD1003816100381610038161003816119905D=0

= 119901vD1003816100381610038161003816119905D=0

= 0

(22)



119880D0 = int

+infin

0119880D0eminus119906119905Dd119905D (23)



Then (22) becomes

d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 120596f119906119880D0 +120596m119906119901mD +120596v119906119901vD (24)

120596m119906119901mD +120582mf (119901mD minus119880D0) = 0 (25)

120596v119906119901vD +120582vf (119901vD minus119880D0) = 0 (26)

lim120576Drarr 0

119903Dd119880D0d119903D

100381610038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (27)

119880D010038161003816100381610038161003816119903Drarrinfin

= 0 (28)


119901mD =120582mf

120596m119906 + 120582mf119880D0

119901vD =120582vf

120596v119906 + 120582vf119880D0

(29)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906119880D0

(30)

If we define

119892 (119906) = (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906 (31)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 119892 (119906)119880D0 (32)


119880D0 = 11988811198700 (radic119892 (119906)119903D)+ 11988821198680 (radic119892 (119906)119903D) (33)

where 1198700

and 1198680


and 1198882



119903D = radic119909D2 + 119910D

2 (34a)



2 (34b)

where

119909D =119909

119871 (35)

119910D =119910

119871 (36)

119909wD =119909w119871

(37)

119910wD =119910w119871

(38)


lim120576Drarr 0

radic119892 (119906)119903D


= minus 119902D

(39)



1198881 = 119902D (40)


119909rarrinfin

1198700(119909) rarr 0and lim

119909rarrinfin


1198882 = 0 (41)


119880D0 = 119902D1198700 (radic119892 (119906)119903D) (42)


119880D =119902

1199021198700 (radic119892 (119906)119903D) (43)


119880D =119902

119902Ω (119909D 119910D 119909wD 119910wD) (44)

where

Ω(119909D 119910D 119909wD 119910wD) = 1198700 (radic119892 (119906)119903D) (45)



x

y

1 i M

O

(0 0) (0 y1)(XiN YiN)

(Xi1 Yi1)

(xiN yiN)

(xi1 yi1)(xi2 yi2)


(xiN+1 yiN+1)


(0 yM)(0 yi)

(xi2N+1 yi2N+1)





119894


119894



119894119895

119884119894119895


119910119894119895

) and (119909119894119895+1 119910119894119895+1)


119909119894119895

= minus119873 minus 119895 + 1

119873119883fL119894

119910119894119895

= 119910119894

1 le 119895 le 119873

119909119894119895

=119895 minus 119873 minus 1

119873119883fR119894

119910119894119895

= 119910119894

119873 + 1 le 119895 le 2119873 + 1

(46)


119883119894119895

= minus2119873 minus 2119895 + 1

2119873119883fL119894

119884119894119895

= 119910119894

1 le 119895 le 119873

119883119894119895

=2 (119895 minus 119873) minus 1

2119873119883fR119894

119884119894119895

= 119910119894

119873 + 1 le 119895 le 2119873

(47)



and 119910119894minus1

is

Δ119910119894

= 119910119894

minus119910119894minus1 (48)


119894119895


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906)

119902int

119909119894119895+1

119909119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909w(49)


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906) 119871

119902int

119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(50)


119902D119894119895 (119905D) =119902119894119895

(119905D) 119871

119902(51)


119902D119894119895 (119906) =

119902119894119895

(119906) 119871

119902 (52)


119880D119894119895 (119909D 119910D)

= 119902D119894119895 int119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(53)



119880D (119909D 119910D) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D 119910D) (54)



119880D (119909D120594120581 119910D120594120581) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D120594120581 119910D120594120581) (55)


119880D (119883D120594120581 119884D120594120581) = 119880wD (56)


119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119883D120594120581 119884D120594120581) = 119880wD (57)



119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] = 119902 (58)

whereΔ119883119894119895


Δ119883119894119895

= 119883119894119895+1 minus119883

119894119895

(59)


119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] =119902

119906 (60)


119872

sum

119894=1

2119873sum

119895=1[119902D119894119895Δ119883D119894119895] =

1119906 (61)

where

Δ119883D119894119895 =Δ119883119894119895

119871 (62)



119880wDS =119906119880wD + 119878

119906 + 119862D1199062 (119906119880wD + 119878)

(63)


119862D =119862

2120587 (120601f119862f t + 120601m119862mt + 120601v119862vt) ℎ1198712 (64)

119878 =2120587119896fiℎ119902sf120583

Δ119901119904

(65)


119904



119901wD (119905D) = minus1120574D

ln [1minus 120574D119880wDS (119905D)] (66)








1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05


120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D
















(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Then (22) becomes

d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 120596f119906119880D0 +120596m119906119901mD +120596v119906119901vD (24)

120596m119906119901mD +120582mf (119901mD minus119880D0) = 0 (25)

120596v119906119901vD +120582vf (119901vD minus119880D0) = 0 (26)

lim120576Drarr 0

119903Dd119880D0d119903D

100381610038161003816100381610038161003816100381610038161003816119903D=120576D

= minus 119902D (27)

119880D010038161003816100381610038161003816119903Drarrinfin

= 0 (28)


119901mD =120582mf

120596m119906 + 120582mf119880D0

119901vD =120582vf

120596v119906 + 120582vf119880D0

(29)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906119880D0

(30)

If we define

119892 (119906) = (120596f +120582mf

120596m119906 + 120582mf120596m +

120582vf120596v119906 + 120582vf

120596v)119906 (31)


d2119880D0d1199032D

+1119903D

d119880D0d119903D

= 119892 (119906)119880D0 (32)


119880D0 = 11988811198700 (radic119892 (119906)119903D)+ 11988821198680 (radic119892 (119906)119903D) (33)

where 1198700

and 1198680


and 1198882



119903D = radic119909D2 + 119910D

2 (34a)



2 (34b)

where

119909D =119909

119871 (35)

119910D =119910

119871 (36)

119909wD =119909w119871

(37)

119910wD =119910w119871

(38)


lim120576Drarr 0

radic119892 (119906)119903D


= minus 119902D

(39)



1198881 = 119902D (40)


119909rarrinfin

1198700(119909) rarr 0and lim

119909rarrinfin


1198882 = 0 (41)


119880D0 = 119902D1198700 (radic119892 (119906)119903D) (42)


119880D =119902

1199021198700 (radic119892 (119906)119903D) (43)


119880D =119902

119902Ω (119909D 119910D 119909wD 119910wD) (44)

where

Ω(119909D 119910D 119909wD 119910wD) = 1198700 (radic119892 (119906)119903D) (45)



x

y

1 i M

O

(0 0) (0 y1)(XiN YiN)

(Xi1 Yi1)

(xiN yiN)

(xi1 yi1)(xi2 yi2)


(xiN+1 yiN+1)


(0 yM)(0 yi)

(xi2N+1 yi2N+1)





119894


119894



119894119895

119884119894119895


119910119894119895

) and (119909119894119895+1 119910119894119895+1)


119909119894119895

= minus119873 minus 119895 + 1

119873119883fL119894

119910119894119895

= 119910119894

1 le 119895 le 119873

119909119894119895

=119895 minus 119873 minus 1

119873119883fR119894

119910119894119895

= 119910119894

119873 + 1 le 119895 le 2119873 + 1

(46)


119883119894119895

= minus2119873 minus 2119895 + 1

2119873119883fL119894

119884119894119895

= 119910119894

1 le 119895 le 119873

119883119894119895

=2 (119895 minus 119873) minus 1

2119873119883fR119894

119884119894119895

= 119910119894

119873 + 1 le 119895 le 2119873

(47)



and 119910119894minus1

is

Δ119910119894

= 119910119894

minus119910119894minus1 (48)


119894119895


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906)

119902int

119909119894119895+1

119909119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909w(49)


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906) 119871

119902int

119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(50)


119902D119894119895 (119905D) =119902119894119895

(119905D) 119871

119902(51)


119902D119894119895 (119906) =

119902119894119895

(119906) 119871

119902 (52)


119880D119894119895 (119909D 119910D)

= 119902D119894119895 int119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(53)



119880D (119909D 119910D) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D 119910D) (54)



119880D (119909D120594120581 119910D120594120581) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D120594120581 119910D120594120581) (55)


119880D (119883D120594120581 119884D120594120581) = 119880wD (56)


119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119883D120594120581 119884D120594120581) = 119880wD (57)



119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] = 119902 (58)

whereΔ119883119894119895


Δ119883119894119895

= 119883119894119895+1 minus119883

119894119895

(59)


119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] =119902

119906 (60)


119872

sum

119894=1

2119873sum

119895=1[119902D119894119895Δ119883D119894119895] =

1119906 (61)

where

Δ119883D119894119895 =Δ119883119894119895

119871 (62)



119880wDS =119906119880wD + 119878

119906 + 119862D1199062 (119906119880wD + 119878)

(63)


119862D =119862

2120587 (120601f119862f t + 120601m119862mt + 120601v119862vt) ℎ1198712 (64)

119878 =2120587119896fiℎ119902sf120583

Δ119901119904

(65)


119904



119901wD (119905D) = minus1120574D

ln [1minus 120574D119880wDS (119905D)] (66)








1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05


120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D
















(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


x

y

1 i M

O

(0 0) (0 y1)(XiN YiN)

(Xi1 Yi1)

(xiN yiN)

(xi1 yi1)(xi2 yi2)


(xiN+1 yiN+1)


(0 yM)(0 yi)

(xi2N+1 yi2N+1)





119894


119894



119894119895

119884119894119895


119910119894119895

) and (119909119894119895+1 119910119894119895+1)


119909119894119895

= minus119873 minus 119895 + 1

119873119883fL119894

119910119894119895

= 119910119894

1 le 119895 le 119873

119909119894119895

=119895 minus 119873 minus 1

119873119883fR119894

119910119894119895

= 119910119894

119873 + 1 le 119895 le 2119873 + 1

(46)


119883119894119895

= minus2119873 minus 2119895 + 1

2119873119883fL119894

119884119894119895

= 119910119894

1 le 119895 le 119873

119883119894119895

=2 (119895 minus 119873) minus 1

2119873119883fR119894

119884119894119895

= 119910119894

119873 + 1 le 119895 le 2119873

(47)



and 119910119894minus1

is

Δ119910119894

= 119910119894

minus119910119894minus1 (48)


119894119895


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906)

119902int

119909119894119895+1

119909119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909w(49)


119880D119894119895 (119909D 119910D)

=

119902119894119895

(119906) 119871

119902int

119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(50)


119902D119894119895 (119905D) =119902119894119895

(119905D) 119871

119902(51)


119902D119894119895 (119906) =

119902119894119895

(119906) 119871

119902 (52)


119880D119894119895 (119909D 119910D)

= 119902D119894119895 int119909D119894119895+1

119909D 119894119895

Ω(119909D 119910D 119909wD 119910wD) d119909wD(53)



119880D (119909D 119910D) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D 119910D) (54)



119880D (119909D120594120581 119910D120594120581) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D120594120581 119910D120594120581) (55)


119880D (119883D120594120581 119884D120594120581) = 119880wD (56)


119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119883D120594120581 119884D120594120581) = 119880wD (57)



119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] = 119902 (58)

whereΔ119883119894119895


Δ119883119894119895

= 119883119894119895+1 minus119883

119894119895

(59)


119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] =119902

119906 (60)


119872

sum

119894=1

2119873sum

119895=1[119902D119894119895Δ119883D119894119895] =

1119906 (61)

where

Δ119883D119894119895 =Δ119883119894119895

119871 (62)



119880wDS =119906119880wD + 119878

119906 + 119862D1199062 (119906119880wD + 119878)

(63)


119862D =119862

2120587 (120601f119862f t + 120601m119862mt + 120601v119862vt) ℎ1198712 (64)

119878 =2120587119896fiℎ119902sf120583

Δ119901119904

(65)


119904



119901wD (119905D) = minus1120574D

ln [1minus 120574D119880wDS (119905D)] (66)








1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05


120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D
















(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



119880D (119909D120594120581 119910D120594120581) =

119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119909D120594120581 119910D120594120581) (55)


119880D (119883D120594120581 119884D120594120581) = 119880wD (56)


119872

sum

119894=1

2119873sum

119895=1119880D119894119895 (119883D120594120581 119884D120594120581) = 119880wD (57)



119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] = 119902 (58)

whereΔ119883119894119895


Δ119883119894119895

= 119883119894119895+1 minus119883

119894119895

(59)


119872

sum

119894=1

2119873sum

119895=1[119902119894119895

Δ119883119894119895

] =119902

119906 (60)


119872

sum

119894=1

2119873sum

119895=1[119902D119894119895Δ119883D119894119895] =

1119906 (61)

where

Δ119883D119894119895 =Δ119883119894119895

119871 (62)



119880wDS =119906119880wD + 119878

119906 + 119862D1199062 (119906119880wD + 119878)

(63)


119862D =119862

2120587 (120601f119862f t + 120601m119862mt + 120601v119862vt) ℎ1198712 (64)

119878 =2120587119896fiℎ119902sf120583

Δ119901119904

(65)


119904



119901wD (119905D) = minus1120574D

ln [1minus 120574D119880wDS (119905D)] (66)








1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05


120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D
















(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


1 23 4

5 6 7 8 91(2M)

10

1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

Slope = 05


120574D = 011

120574D = 006120574D = 0

CD = 10 S = 02 M = 3N = 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015

120596m = 089 120596 = 0095 120582mf = 5E minus 11

120582 f = 1E minus 9

Horizontal

vv

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

pw

Dp

wD998400middott

DC

D
















(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


(a) (b)

(c) (d)


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

S = 06S = 04S = 02

CD = 10 M = 3 N = 3 Δyi = 250m

X f L i = Xf R i = 20m L = 01m120596m = 089 120596 = 0095


120596f = 0015

pw

Dp

wD998400middott

DC

D v

v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596f = 0035

120596f = 0012

120596f = 0006

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596m = 089


pw

Dp

wD998400middott

DC

D

v



1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

120596m = 065

120596m = 080

120596m = 089

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m 120596f = 0015


pw

Dp

wD998400middott

DC

D v


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

XfLi = Xf Ri = 20m L = 01m120596m = 089 120596v

v

= 0095

120596f = 0015

120582mf = 5E minus 11 120574D = 008


pw

Dp

wD998400middott

DC

D





1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 4E minus 9 120574D = 008


pw

Dp

wD998400middott

DC

D





119894


119894


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 2

1Eminus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3 Δyi = 250m

L = 01m 120596v

v

= 0095120596f = 0015 120596m = 089


Xf = 15mXf = 25mXf = 50m

pw

Dp

wD998400middott

DC

D


1E + 2

1E + 1

1E + 0

1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

1E+8

1E+9

1E+10

1E+11

1E minus 21E

minus2

1E minus 1

1Eminus1

tDCD

CD = 10 S = 02 N = 3M= 3

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015


120574D = 008

Δyi = 150m 250m 450m

pw

Dp

wD998400middott

DC

D



6 Conclusion




0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 1 2 3 4 5 6 7 8 9 10 11


j

250E minus 03

200E minus 03

150E minus 03

100E minus 03

500E minus 04

000E + 00

qD

CD = 10 S = 02 N = 5M= 3 Δyi = 250m

Xf Li = Xf Ri = 20m L = 01m120596m = 089 120596v = 0095

120596f = 0015

120582vf = 1E minus 9

120582mf = 5E minus 11

120574D = 008 tD = 9075








Appendices




Vf =119896f120583

120597119901f120597119903

(A1)





120574 =1119896f

d119896fd119901f

(A2)


int

119896fi

119896f

1119896fd119896f = int

119901i

119901f

120574 d119901f (A3)







120583

120597119901f120597119903

(A5)




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




120588f = 1205880e119862o(119901fminus1199010) (A6a)




120588f = 1205880 [1+119862o (119901f minus1199010)] (A6b)


120588m = 1205880e119862o(119901mminus1199010) (A7a)

or

120588m = 1205880 [1+119862o (119901m minus1199010)] (A7b)


equation of state

120588v = 1205880e119862o(119901vminus1199010) (A8a)

or

120588v = 1205880 [1+119862o (119901v minus1199010)] (A8b)



120601f = 120601f0e119862f (119901fminus1199010) (A9a)



120601f = 120601f0 [1+119862f (119901f minus1199010)] (A9b)


120601m = 120601m0e119862m(119901mminus1199010) (A10a)

or

120601m = 120601m0 [1+119862m (119901m minus1199010)] (A10b)



120601v = 120601v0e119862v(119901vminus1199010) (A11a)

or120601v = 120601v0 [1+119862v (119901v minus1199010)] (A11b)



1119903

120597 (119903120588fVf)120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12a)


120597 (120588fVf)120597119903

+1119903120588fVf =

120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2 (A12b)


119902c1 =120572mf119896m1205880

120583(119901m minus119901f) (A13)

119902c2 =120572vf119896v1205880

120583(119901v minus119901f) (A14)



1119903

120597 (119903120588mVm)

120597119903=

120597 (120588m120601m)

120597119905+ 119902c1 (A15)

1119903

120597 (119903120588vVv)120597119903

=120597 (120588v120601v)

120597119905+ 119902c2 (A16)


1119903

120597 (119903120588mVm)

120597119903asymp 0

1119903

120597 (119903120588vVv)120597119903

asymp 0

(A17)



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



120597 (120588m120601m)

120597119905+ 119902c1 = 0 (A18)

120597 (120588v120601v)

120597119905+ 119902c2 = 0 (A19)




120597119903

+1119903120588f


120583

120597119901f120597119903

=120597 (120588f120601f)

120597119905minus 119902c1 minus 119902c2

(A20)



120597119903

=1205880119896fi120583

[120574e120574(119901fminus119901i) (120597119901f120597119903

)

2+ e120574(119901fminus119901i)

1205972119901f

1205971199032]

(A21)


1119903120588f


120583

120597119901f120597119903

=1119903

1205880119896fi120583

e120574(119901fminus119901i)120597119901f120597119903

(A22)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010)

+ 120601f01205880119862f119862o (119901f minus1199010)2

(A23a)


120588f120601f = 120601f01205880 +120601f01205880 (119862f +119862o) (119901f minus1199010) (A23b)


120597 (120588f120601f)

120597119905= 120601f01205880119862f t

120597119901

120597119905 (A24)


119862f t = 119862f +119862o (A25)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905


119896fi(119901m minus119901f)

minus120572vf119896v119896fi

(119901v minus119901f)]

(A26a)


1205972119901f

1205971199032+1119903

120597119901f120597119903

+ 120574(120597119901f120597119903

)

2


119896fi

120597119901f120597119905

+120601m0120583119862mt

119896fi

120597119901m120597119905

+120601v0120583119862vt

119896fi

120597119901v120597119905

]

(A26b)



120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010)

+ 120601m01205880119862m119862o (119901m minus1199010)2

(A27a)


120588m120601m = 120601m01205880 +120601m01205880 (119862m +119862o) (119901m minus1199010) (A27b)

Therefore

120597 (120588m120601m)

120597119905= 120601m01205880119862mt

120597119901m120597119905

(A28)


119862mt = 119862m +119862o (A29)


120601m0119862mt120597119901m120597119905

+120572mf119896m

120583(119901m minus119901f) = 0 (A30)





120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010)

+ 120601v01205880119862v119862o (119901v minus1199010)2

(A31a)


120588v120601v = 120601v01205880 +120601v01205880 (119862v +119862o) (119901v minus1199010) (A31b)

Therefore120597 (120588v120601v)

120597119905= 120601v01205880119862vt

120597119901v120597119905

(A32)


119862vt = 119862v +119862o (A33)


120601v0119862vt120597119901v120597119905

+120572vf119896v120583

(119901v minus119901f) = 0 (A34)




lim120576rarr 0

119860Vf1003816100381610038161003816100381610038161003816119903=120585

= 119902 (119905) 119861 (B1)


119860|119903=120576

= 2120587119903ℎ|119903=120576

(B2)

Vf1003816100381610038161003816119903=120576 =

119896f120583

120597119901f120597119903

10038161003816100381610038161003816100381610038161003816119903=120576

(B3)


Vf1003816100381610038161003816119903=120576 =


120583

120597119901f120597119903

100381610038161003816100381610038161003816100381610038161003816119903=120576

(B4)


lim120576rarr 0

eminus120574(119901iminus119901f ) (119903120597119901f120597119903

)

119903=120576

=119902 (119905) 120583119861

2120587119896fiℎ (B5)


Nomenclature



















119902119894119895


119902119894119895














119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of













119884119894119895




Δ119883119894119895




Δ119910119894


and 119910119894minus1

Δ119910119894

= 119910119894

















dimensionless

Subscript

D Dimensionless

Superscript




Acknowledgments


References









































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of






































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

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