Research Article Application of Mathematica Software to...

Hindawi Publishing CorporationISRN Chemical EngineeringVolume 2013 Article ID 765896 6 pageshttpdxdoiorg1011552013765896

Research ArticleApplication of Mathematica Software toSolve Pulp Washing Model

Jitender Kumar1 Ishfaq A Ganaie2 and Vijay K Kukreja2

1 Department of Applied Sciences BGIET Sangrur 148001 Punjab India2Department of Mathematics SLIET Longowal 148106 Punjab India

Correspondence should be addressed to Vijay K Kukreja vkkukrejagmailcom

Received 29 August 2013 Accepted 9 October 2013

Academic Editors J J Rodriguez and A M Seayad

Copyright copy 2013 Jitender Kumar et alThis 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

The removal of the bulk liquor surrounding the pulp fibers using less concentrated liquor is known as pulp washing In the presentstudy a pulpwashingmodel involving diffusion-dispersion through packed beds of finite length is presented Separation of variablesis applied to solve system of governing partial differential equations and the resulting equations are solved using MathematicaResults from the present case are compared with those of previous investigators The present case is giving better results than theprevious investigators

1 Introduction

The objective of pulp and paper industry to produce itstarget productionwith high efficiency and less environmentalload can only be met by initiating a meticulously plannedresearch on mathematical methods Pulp washing plays animportant role in reduction of black liquor solids in the pulpbeing carried forward for further processingThe efficiency ofwashing depends on the degree of mixing rate of desorptiondiffusion-dispersion of dissolved solids and chemicals fromthe fibrous matrix Modeling of pulp washing is done mainlyusing three approaches namely (a) process modeling (b)physical modeling and (c) statistical modeling

A complete review of the various process models usedto describe pulp washing has been presented by [1] Initiallyresearchers like in [2 3] proposed the models based on axialdispersion Pellett [4] introduced amathematicalmodel com-bining the effects of particle diffusion and axial dispersion Adetailedmodel related tomass transfer in fibrous particle wasgiven by [5] it was also restricted for axial dispersion onlyComprehensive models involving physical features of thefibers such as fiber porosity and fiber radius were presentedby [6 7]

Extensive study of axial dispersion model has beencarried out by [8ndash26] The model has been solved usinganalytic and numerical techniques like Laplace transform

technique [2ndash4 10 15 23 26] finite difference technique[25] orthogonal collocation method [5 7 12] orthogonalcollocation on finite elements [6 20 21] GalerkinPetrovGalerkin method [8 19] Hermite collocation method by[11 17 24] and Spline collocation method [13]

The accuracy of the analytic solution undoubtedlyexceeds the limit of applicability of the theory to realsituations Moreover it is highly desirable to have a simpleand consistent model of the transport phenomenon based onessential features of real situation Keeping this modest goalin mind axial dispersion model is solved along with linearadsorption isothermThemethod of separation of variables isfirst applied on partial differential equation and then Laplacetransform is taken of the equations Finally the mathematicalexpressions are solved using Mathematica software to obtainsolute concentration at any location and time

2 Model Based on Axial Dispersion

The displacement washing model based on the axial disper-sion and particle diffusion describing the washing zone isgiven by

120597119888

120597119905+ 119906

120597119888

120597119911+ 120583

120597119899

120597119905= 119863119871

1205972119888

1205971199112(1)

2 ISRN Chemical Engineering

with adsorption isotherm

119899 = 119896119888 (2)

This equation represents the basis for the mathematicalmodels of displacement washing where 119905 is the time from thecommencement of the displacement 119911 is the distance fromthe point of introduction of the displacing fluid 119888 = 119888(119911 119905)

is the solute concentration 119863119871is longitudinal dispersion

coefficient 119906 is the average interstitial velocity of the fluidand 119871 is thickness of the packed bed

On account of unusual nature of displacement pro-cess appropriate boundary conditions have been extensivelydiscussed in the literature [2 6 27 28] Accordingly theboundary conditions at the inlet and outlet of the bed are

119888 = 119888119904

at 119911 = 0

120597119888

120597119911= 0 at 119911 = 119871

(3)

and initial condition is given by

119888 (119911 0) = 119899 (119911 0) = 119888119894

for 0 lt 119905 lt119871

119906 (4)

Conversion of Model into Dimensionless Form Equations (1)to (4) can be put in dimensionless form using dimensionlessvariables

119862 =119888 minus 119888119904

119888119894minus 119888119904

119873 =119899 minus 119888119904

119888119894minus 119888119904

119885 =119911

119871 119879 =

119906119905

(1 + 120583119896) 119871

(5)

The dimensionless time 119879 corresponds physically to thenumber of pore displacements introduced into the mediumsince the start of the experiment

By these means (1) reduces to

120597119862

120597119879+120597119862

120597119885=

1

Pe1205972119862

1205971198852 (6)

where Pe = 119906119871119863119871is the Peclet number The boundary

conditions are now of the form

119862 (0 119879) = 0 at 119885 = 0 for 119879 gt 0 (7)

120597119862 (1 119879)

120597119885= 0 at 119885 = 1 for 119879 gt 0 (8)

while the initial condition is

119862 = 1 at 119879 = 0 for 0 lt 119885 le 1 (9)

Now our main aim is to estimate 119862 = 119862(119885 119879) satisfying (7)ndash(9) which will eventually lead to exit solute concentration119862119890= 119862119890(119879) = 119862(1 119879)

3 Solution of Model

Method of separation of variables is applied to solve (6) Thismethod transforms the PDE into a system of ODEs each ofwhich depends only on one of the functions and the solutionis given as product of the functions

Equation (6) can be separated in terms of variables 119885and 119879 by assuming that 119862(119885 119879) = 119883(119885)119884(119879) and thensubstituting 120597119862120597119885 = 119883

1015840119884 and 120597119862120597119879 = 119883119884

1015840 in it as follows

11988310158401015840minus Pe1198831015840

119883=Pe1198841015840

119884= minus1199012(constant) (10)

Individual solutions of expression (10) are given by

119883(119885) = exp (Pe1198852

) [1198881cos (120572119885) + 1198882 sin (120572119885)] (11)

119884 (119879) = 1198883exp(minus

1199012119879

Pe) (12)

Application of boundary condition 119883 = 0 at 119885 = 0 in (11)gives 119888

1= 0 and the boundary condition 120597119883120597119885 = 0 at 119885 = 1

gives Pe tan120572 + 2120572 = 0Therefore the solution 119862(119885 119879) = 119883(119885)119884(119879) is given by

119862 (119885 119879) = 119860 exp (Pe1198852

) sin (120572119885) exp(minus1199012119879

Pe) (13)

Equation Petan120572 + 2120572 = 0 is a transcendental equationit will have infinite many root therefore the solution (13) willdepend on 119899 that is

119862 (119885 119879) sim 119862119899 (119885 119879) = 119883

119899 (119885) 119884119899 (119879)

= 119860119899exp (Pe119885

2) sin (120572

119899119885)

times exp(minus1199012

119899119879

Pe)

(14)

Using the principle of superposition we get

119862 (119885 119879) =

infin

sum

119899=1

119862119899 (119885 119879)

=

infin

sum

119899=1

119860119899exp(Pe119885

2) sin (120572

119899119885) exp(minus

119901119899

2119879

Pe)

(15)

Applying the initial condition 119862(119885 0) = 1 we find thatinfin

sum

119899=1

119860119899sin (120572

119899119885) = exp (minusPe119885

2) (16)

therefore 119860119899represents the Strum-Liouville problem for

exp(minusPe1198852) and is given by

119860119899=int1

0119890minusPe1198852 sin (120572

119899119885) 119889119885

int1

0sin2 (120572

119899119885) 119889119885

(17)

ISRN Chemical Engineering 3

Finally the solute concentration at any location and time inthe bed can be written as

119862 (119885 119879) =

infin

sum

119899=1

int1

0119890minusPe1198852 sin (120572

119899119885) 119889119885

int1

0sin2 (120572

119899119885) 119889119885

times exp(Pe1198852

minus1199012

119899119879

Pe) sin (120572

119899119885)

(18)

where 120572119899(119899 = 1 2 3 ) are the positive roots taken in order

of increasing magnitude of the transcendental equationPe tan120572

119899+ 2120572119899= 0 and 119901

119899= radic(4120572

1198992 + Pe2)4

It is important to mention that (6) is also solved analyti-cally by Grahs [5] and Kukreja [14] Their solutions are givenbelow

Applying separation of variables Grahs [5] found theconcentration of solute as

119862 (119885 119879) = exp Pe2(119885 minus

119879

2)

times [

infin

sum

119899=1

41205722

119899sin (120572

119899119885) exp (minus1205722

119899119879Pe)

2120572119899minus sin (2120572

119899) ((Pe4) + 1205722

119899)]

(19)

where 120572119899are the positive roots of Pe tan120572

119899+ 4120572119899= 0

Using Laplace transform Kukreja [14] found the concen-tration of solute as

119888 minus 119888119904

119888119894minus 119888119904

=

infin

sum

119899=1

41205722

119899exp(119901

119899119879 +

Pe1198852

)

times

infin

sum

119899=1

[Pe sin (119885 minus 1) 120572119899 minus 2120572119899 cos (119885 minus 1) 120572119899]

Pe119901119899(Pe2 + 41205722

119899+ 2Pe) sin (120572

119899)

(20)

where Pe tan120572119899+ 2120572119899= 0 and 119901

119899= minus(4120572

2

119899+ Pe2)4Pe

The solutions given by (18)ndash(20) are complex and timeconsuming These equations are evaluated using Mathemat-ica software This technique is novel and elegant and can beconveniently handled by any nonmathematician also

4 Results and Discussion

Solution of (18)ndash(20) are obtained using Mathematica bytaking 22mesh points within the dimensionless time and alsoin dimensionless distance119909 For Pe = 1 the surface plots of thedimensionless concentration with respect to dimensionlessdistance (0 le 119909 le 1) and dimensionless time (0 le 119879 le 12)are shown in Figures 1 2 and 3 respectively for the presentcase (18) Grahs (19) and Kukreja (20) It can be observedfrom Figure 1 that the surface plot of (18) is very smooth at119879 = 0 whereas for (19) and (20) fluctuation is observed

In Figures 4 to 6 exit solute concentration obtained from(18)ndash(20) is plotted respectively In Figure 5 the solutionprofile of (19) is starting from 13 approximately which ishighly impossible in pulp washing system as the dimension-less concentration can never exceed 1 Smoothest profile canbe seen in case of Figure 4

10

10

Dim

ensio

nles

s

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

conc

entra

tion

Figure 1 Surface plot for the present case represented by (18)

1010

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

Dim

ensio

nles

s co

ncen

tratio

n

1 0 Dimen

sionl

ess d

Dimensionless time

05

05

Figure 2 Surface plot for Grahs [5] represented by (19)

1010

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

Dim

ensio

nles

s co

ncen

tratio

n

1 0 Dimen

sionl

ess d

i

Dimensionless ti

05

0 05

Figure 3 Surface plot for Kukreja [15] represented by (20)

4 ISRN Chemical EngineeringD

imen

sionl

ess c

once

ntra

tion

10

08

06

04

02

Dimensionless time02 04 06 08 10 12

Figure 4 Exit solute concentration profile for present case


02

04

06

08

10

12

Dim

ensio

nles

s con

cent

ratio

n

Figure 5 Exit solute concentration profile by Grahs [5]

In Figures 7 and 8 the absolute error obtained from (19)and (20) from (18) is plotted The magnitude of error incase of (19) is 4 to 5 times higher the (20) This indicatesmore deviation between the present case and Grahs [5] ascompared to Kukreja [14]

The results from the three investigators are summarizedin Table 1 At T = 0 the error with Kukreja [14] is 292whereas with Grahs [5] it is 2913 ideally these should havebeen 00 Similarly at T = 01 the error is found to be 625while comparing the results with Kukreja [14] and for Grahs[5] it is 1223 which is again very high It can further beseen in Table 1 that at T = 05 onwards the error is 00 withKukreja [14] whereas with Grahs [5] it is reducing but persistscontinuously Hence this comparison shows that results ofKukreja [14] are matching with present case but those ofGrahs [5] are showing significant error

5 Conclusion

The investigation based on diffusion model of longitudinalmixing in beds of finite length is applicable to displacementwashing with axial dispersion and particle diffusion Thepresent solution of themodel as well as the solution proposedby Grahs [5] and Kukreja [14] involves complicated expres-sions The application of Mathematica makes it convenient


02

04

06

08

Dim

ensio

nles

s con

cent

ratio

n

Figure 6 Exit solute concentration profile by Kukreja [14]


Dim

ensio

nles

s con

cent

ratio

n 025

020

015

010

005

Figure 7 Absolute error between present case and Grahs [5]

Table 1 Comparison of solution of Grahs [5] and Kukreja [14] withthe present case

Time Presentcase

Grahs[5]

Kukreja[14]

errorwith [5]

errorwith [14]

000 10000 12913 09708 2913 292010 09864 01087 09239 1223 625020 06866 08477 06805 1611 061030 04779 06203 04773 1424 006040 03326 04512 03326 1186 000050 02315 03279 02315 0964 000060 01611 02382 01611 0771 000070 01122 01731 01122 0609 000080 00781 01258 00781 0477 000090 00543 00914 00543 0371 000100 00378 00664 00378 0286 000110 00263 00482 00263 0219 000120 00183 00350 00183 0167 000

to evaluate these expressions The present results are moreaccurate than the output provided by Grahs [5] and Kukreja[14]The algorithms based onMathematica are novel and easyto set up Therefore the present technique provides a good



Dim

ensio

nles

s con

cent

ratio

n 007

006

005

004

003

002

001

Figure 8 Absolute error between the present case and Kukreja [14]

alternate to the available techniques for dealing with suchtype of problems

Nomenclature

119888 Concentration of the solute in the liquor kgm3119888119894 Initial solute concentration kgm3119888119904 Solute concentration of incoming fluid kgm3

119888119890 Exit solute concentration from the bed kgm3

119863119871 Longitudinal dispersion coefficient m2s

119871 Cake thickness m119899 Concentration of the solute in the fiber kgm3119905 Time s119906 Liquor speed in cake pores ms119911 Distance from point of introduction of solvent m120576 Porosity of packed bed120583 Ratio of porosity (1 minus 120576)120576

Acknowledgments

This work is supported by NBHM Mumbai India in theform of research projects 248(14)2009RampD-II2806 TheJRF provided to Mr Ishfaq A Ganaie by NBHM Mumbai isthankfully acknowledged

References

[1] M Pekkanen and H V Norden ldquoReview of pulp washingmodelsrdquo Paperi Ja Puu vol 67 pp 689ndash696 1985

[2] H Brenner ldquoThe diffusion model of longitudinal mixing inbeds of finite length Numerical valuesrdquo Chemical EngineeringScience vol 17 no 4 pp 229ndash243 1962

[3] W R Sherman ldquoThemovement of a solublematerial during thewashing of a bed of packed solidsrdquo AIChE Journal vol 10 pp855ndash860 1964

[4] G L Pellett ldquoLongitudinal dispersion intra particle diffusionand liquid-phase mass transfer during flow through multiparticle systemsrdquo TAPPI Journal vol 49 pp 75ndash82 1966

[5] L E Grahs Washing of cellulose fibres analysis of displacementwashing operation [PhD dissertation] Department ofChemicalEngineering Chalmers University of Technology GothenburgSweden 1974

[6] S Arora S S Dhaliwal and V K Kukreja ldquoSimulation of wash-ing of packed bed of porous particles by orthogonal collocationon finite elementsrdquo Computers amp Chemical Engineering vol 30no 6-7 pp 1054ndash1060 2006

[7] N S Raghavan and D M Ruthven ldquoNumerical simulation ofa fixed-bed adsorption column by the method of orthogonalcollocationrdquo AIChE Journal vol 29 no 6 pp 922ndash925 1983

[8] M Al-Jabari A R P van Heiningen and T G M van de VenldquoModeling the flow and the deposition of fillers in packed bedsof pulp fibresrdquo Journal of Pulp and Paper Science vol 20 no 9pp J249ndashJ253 1994

[9] S Arora and F Potucek ldquoModelling of displacement washing ofpacked bed of fibersrdquo Brazilian Journal of Chemical Engineeringvol 26 no 2 pp 385ndash393 2009

[10] B V Babu and A S Chaurasia ldquoPyrolysis of biomass Improvedmodels for simultaneous kinetics and transport of heat massand momentumrdquo Energy Conversion and Management vol 45no 9-10 pp 1297ndash1327 2004

[11] K D Edoh R D Russell and W Sun ldquoComputation ofinvariant tori by orthogonal collocationrdquo Applied NumericalMathematics vol 32 no 3 pp 273ndash289 2000

[12] L T Fan G K C Chen and L E Erickson ldquoEfficiencyand utility of collocation methods in solving the performanceequations of flow chemical reactors with axial dispersionrdquoChemical Engineering Science vol 26 no 3 pp 378ndash387 1971

[13] B Gupta and V K Kukreja ldquoModelling amp simulation ofpacked bed of porous particles by orthogonal spline collocationmethodrdquo Applied Mathematics amp Computation vol 219 pp2087ndash2099 2012

[14] V K Kukreja Modeling of washing of brown stock on rotaryvacuum washer [PhD dissertation] Department of Pulp andPaper Technology University of Roorkee Roorkee India 1996

[15] V K Kukreja andA K Ray ldquoMathematicalmodeling of a rotaryvacuumwasher used for pulpwashing a case study of a lab scalewasherrdquo Cellulose Chemistry amp Technology vol 43 no 1ndash3 pp25ndash36 2009

[16] A Kumar D K Jaiswal and N Kumar ldquoAnalytical solutionsof one-dimensional advection-diffusion equation with variablecoefficients in a finite domainrdquo Journal of Earth System Sciencevol 118 no 5 pp 539ndash549 2009

[17] C P Leao and A E Rodrigues ldquoTransient and steady-statemodels for simulated moving bed processes numerical solu-tionsrdquo Computers amp Chemical Engineering vol 28 no 9 pp1725ndash1741 2004

[18] H-T Liao and C-Y Shiau ldquoAnalytical solution to an axialdispersion model for the fixed-bed adsorberrdquo AIChE Journalvol 46 no 6 pp 1168ndash1176 2000

[19] F Liu and S K Bhatia ldquoApplication fo Petrov-Galerkinmethodsto transient boundary value problems in chemical engineeringadsorption with steep gradients in bidisperse solidsrdquo ChemicalEngineering Science vol 56 no 12 pp 3727ndash3735 2001

[20] Y Liu and E W Jacobsen ldquoOn the use of reduced order modelsin bifurcation analysis of distributed parameter systemsrdquo Com-putersampChemical Engineering vol 28 no 1-2 pp 161ndash169 2004

[21] Z Ma and G Guiochon ldquoApplication of orthogonal collocationon finite elements in the simulation of non-linear chromatog-raphyrdquo Computers amp Chemical Engineering vol 15 no 6 pp415ndash426 1991

[22] F Potucek ldquoWashing of pulp fibre bedrdquo Collection of Czechoslo-vak Chemical Communications vol 62 no 4 pp 626ndash644 1997


[23] A Rasmuson and I Neretnieks ldquoExact solution of a modelfor diffusion in particles and longitudinal dispersion in packedbedsrdquo AIChE Journal vol 26 pp 686ndash690 1980

[24] A La Rocca and H Power ldquoA double boundary collocationHermitian approach for the solution of steady state convection-diffusion problemsrdquo Computers and Mathematics with Applica-tions vol 55 no 9 pp 1950ndash1960 2008

[25] L M Sun and F Meunier ldquoAn improved finite differencemethod for fixed-bed multicomponent sorptionrdquo AIChE Jour-nal vol 37 no 2 pp 244ndash254 1991

[26] Y Zheng and T Gu ldquoAnalytical solution to a model for thestartup period of fixed-bed reactorsrdquo Chemical EngineeringScience vol 51 no 15 pp 3773ndash3779 1996

[27] F Shiraishi ldquoHighly accurate solution of the axial dispersionmodel expressed in S-system canonical form by Taylor seriesmethodrdquo Chemical Engineering Journal vol 83 no 3 pp 175ndash183 2001

[28] M K Szukiewicz ldquoNew approximate model for diffusion andreaction in a porous catalystrdquo AIChE Journal vol 46 no 3 pp661ndash665 2000

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with adsorption isotherm

119899 = 119896119888 (2)

This equation represents the basis for the mathematicalmodels of displacement washing where 119905 is the time from thecommencement of the displacement 119911 is the distance fromthe point of introduction of the displacing fluid 119888 = 119888(119911 119905)

is the solute concentration 119863119871is longitudinal dispersion

coefficient 119906 is the average interstitial velocity of the fluidand 119871 is thickness of the packed bed

On account of unusual nature of displacement pro-cess appropriate boundary conditions have been extensivelydiscussed in the literature [2 6 27 28] Accordingly theboundary conditions at the inlet and outlet of the bed are

119888 = 119888119904

at 119911 = 0

120597119888

120597119911= 0 at 119911 = 119871

(3)

and initial condition is given by

119888 (119911 0) = 119899 (119911 0) = 119888119894

for 0 lt 119905 lt119871

119906 (4)

Conversion of Model into Dimensionless Form Equations (1)to (4) can be put in dimensionless form using dimensionlessvariables

119862 =119888 minus 119888119904

119888119894minus 119888119904

119873 =119899 minus 119888119904

119888119894minus 119888119904

119885 =119911

119871 119879 =

119906119905

(1 + 120583119896) 119871

(5)

The dimensionless time 119879 corresponds physically to thenumber of pore displacements introduced into the mediumsince the start of the experiment

By these means (1) reduces to

120597119862

120597119879+120597119862

120597119885=

1

Pe1205972119862

1205971198852 (6)

where Pe = 119906119871119863119871is the Peclet number The boundary

conditions are now of the form

119862 (0 119879) = 0 at 119885 = 0 for 119879 gt 0 (7)

120597119862 (1 119879)

120597119885= 0 at 119885 = 1 for 119879 gt 0 (8)

while the initial condition is

119862 = 1 at 119879 = 0 for 0 lt 119885 le 1 (9)

Now our main aim is to estimate 119862 = 119862(119885 119879) satisfying (7)ndash(9) which will eventually lead to exit solute concentration119862119890= 119862119890(119879) = 119862(1 119879)

3 Solution of Model

Method of separation of variables is applied to solve (6) Thismethod transforms the PDE into a system of ODEs each ofwhich depends only on one of the functions and the solutionis given as product of the functions

Equation (6) can be separated in terms of variables 119885and 119879 by assuming that 119862(119885 119879) = 119883(119885)119884(119879) and thensubstituting 120597119862120597119885 = 119883

1015840119884 and 120597119862120597119879 = 119883119884

1015840 in it as follows

11988310158401015840minus Pe1198831015840

119883=Pe1198841015840

119884= minus1199012(constant) (10)

Individual solutions of expression (10) are given by

119883(119885) = exp (Pe1198852

) [1198881cos (120572119885) + 1198882 sin (120572119885)] (11)

119884 (119879) = 1198883exp(minus

1199012119879

Pe) (12)

Application of boundary condition 119883 = 0 at 119885 = 0 in (11)gives 119888

1= 0 and the boundary condition 120597119883120597119885 = 0 at 119885 = 1

gives Pe tan120572 + 2120572 = 0Therefore the solution 119862(119885 119879) = 119883(119885)119884(119879) is given by

119862 (119885 119879) = 119860 exp (Pe1198852

) sin (120572119885) exp(minus1199012119879

Pe) (13)

Equation Petan120572 + 2120572 = 0 is a transcendental equationit will have infinite many root therefore the solution (13) willdepend on 119899 that is

119862 (119885 119879) sim 119862119899 (119885 119879) = 119883

119899 (119885) 119884119899 (119879)

= 119860119899exp (Pe119885

2) sin (120572

119899119885)

times exp(minus1199012

119899119879

Pe)

(14)

Using the principle of superposition we get

119862 (119885 119879) =

infin

sum

119899=1

119862119899 (119885 119879)

=

infin

sum

119899=1

119860119899exp(Pe119885

2) sin (120572

119899119885) exp(minus

119901119899

2119879

Pe)

(15)

Applying the initial condition 119862(119885 0) = 1 we find thatinfin

sum

119899=1

119860119899sin (120572

119899119885) = exp (minusPe119885

2) (16)

therefore 119860119899represents the Strum-Liouville problem for

exp(minusPe1198852) and is given by

119860119899=int1

0119890minusPe1198852 sin (120572

119899119885) 119889119885

int1

0sin2 (120572

119899119885) 119889119885

(17)



119862 (119885 119879) =

infin

sum

119899=1

int1

0119890minusPe1198852 sin (120572

119899119885) 119889119885

int1

0sin2 (120572

119899119885) 119889119885

times exp(Pe1198852

minus1199012

119899119879

Pe) sin (120572

119899119885)

(18)



119899+ 2120572119899= 0 and 119901

119899= radic(4120572

1198992 + Pe2)4



119862 (119885 119879) = exp Pe2(119885 minus

119879

2)

times [

infin

sum

119899=1

41205722

119899sin (120572

119899119885) exp (minus1205722

119899119879Pe)

2120572119899minus sin (2120572

119899) ((Pe4) + 1205722

119899)]

(19)


119899+ 4120572119899= 0


119888 minus 119888119904

119888119894minus 119888119904

=

infin

sum

119899=1

41205722

119899exp(119901

119899119879 +

Pe1198852

)

times

infin

sum

119899=1


Pe119901119899(Pe2 + 41205722

119899+ 2Pe) sin (120572

119899)

(20)

where Pe tan120572119899+ 2120572119899= 0 and 119901

119899= minus(4120572

2

119899+ Pe2)4Pe





10

10

Dim

ensio

nles

s

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

conc

entra

tion


1010

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

Dim

ensio

nles

s co

ncen

tratio

n

1 0 Dimen

sionl

ess d

Dimensionless time

05

05


1010

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

Dim

ensio

nles

s co

ncen

tratio

n

1 0 Dimen

sionl

ess d

i

Dimensionless ti

05

0 05



imen

sionl

ess c

once

ntra

tion

10

08

06

04

02




02

04

06

08

10

12

Dim

ensio

nles

s con

cent

ratio

n




5 Conclusion



02

04

06

08

Dim

ensio

nles

s con

cent

ratio

n



Dim

ensio

nles

s con

cent

ratio

n 025

020

015

010

005



Time Presentcase

Grahs[5]

Kukreja[14]

errorwith [5]

errorwith [14]

000 10000 12913 09708 2913 292010 09864 01087 09239 1223 625020 06866 08477 06805 1611 061030 04779 06203 04773 1424 006040 03326 04512 03326 1186 000050 02315 03279 02315 0964 000060 01611 02382 01611 0771 000070 01122 01731 01122 0609 000080 00781 01258 00781 0477 000090 00543 00914 00543 0371 000100 00378 00664 00378 0286 000110 00263 00482 00263 0219 000120 00183 00350 00183 0167 000




Dim

ensio

nles

s con

cent

ratio

n 007

006

005

004

003

002

001



Nomenclature





Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119862 (119885 119879) =

infin

sum

119899=1

int1

0119890minusPe1198852 sin (120572

119899119885) 119889119885

int1

0sin2 (120572

119899119885) 119889119885

times exp(Pe1198852

minus1199012

119899119879

Pe) sin (120572

119899119885)

(18)



119899+ 2120572119899= 0 and 119901

119899= radic(4120572

1198992 + Pe2)4



119862 (119885 119879) = exp Pe2(119885 minus

119879

2)

times [

infin

sum

119899=1

41205722

119899sin (120572

119899119885) exp (minus1205722

119899119879Pe)

2120572119899minus sin (2120572

119899) ((Pe4) + 1205722

119899)]

(19)


119899+ 4120572119899= 0


119888 minus 119888119904

119888119894minus 119888119904

=

infin

sum

119899=1

41205722

119899exp(119901

119899119879 +

Pe1198852

)

times

infin

sum

119899=1


Pe119901119899(Pe2 + 41205722

119899+ 2Pe) sin (120572

119899)

(20)

where Pe tan120572119899+ 2120572119899= 0 and 119901

119899= minus(4120572

2

119899+ Pe2)4Pe





10

10

Dim

ensio

nles

s

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

conc

entra

tion


1010

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

Dim

ensio

nles

s co

ncen

tratio

n

1 0 Dimen

sionl

ess d

Dimensionless time

05

05


1010

10

05

00

Dimen

sionl

ess d

istan

ce

Dimensionless time

05

00 05

00

Dim

ensio

nles

s co

ncen

tratio

n

1 0 Dimen

sionl

ess d

i

Dimensionless ti

05

0 05



imen

sionl

ess c

once

ntra

tion

10

08

06

04

02




02

04

06

08

10

12

Dim

ensio

nles

s con

cent

ratio

n




5 Conclusion



02

04

06

08

Dim

ensio

nles

s con

cent

ratio

n



Dim

ensio

nles

s con

cent

ratio

n 025

020

015

010

005



Time Presentcase

Grahs[5]

Kukreja[14]

errorwith [5]

errorwith [14]

000 10000 12913 09708 2913 292010 09864 01087 09239 1223 625020 06866 08477 06805 1611 061030 04779 06203 04773 1424 006040 03326 04512 03326 1186 000050 02315 03279 02315 0964 000060 01611 02382 01611 0771 000070 01122 01731 01122 0609 000080 00781 01258 00781 0477 000090 00543 00914 00543 0371 000100 00378 00664 00378 0286 000110 00263 00482 00263 0219 000120 00183 00350 00183 0167 000




Dim

ensio

nles

s con

cent

ratio

n 007

006

005

004

003

002

001



Nomenclature





Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










imen

sionl

ess c

once

ntra

tion

10

08

06

04

02




02

04

06

08

10

12

Dim

ensio

nles

s con

cent

ratio

n




5 Conclusion



02

04

06

08

Dim

ensio

nles

s con

cent

ratio

n



Dim

ensio

nles

s con

cent

ratio

n 025

020

015

010

005



Time Presentcase

Grahs[5]

Kukreja[14]

errorwith [5]

errorwith [14]

000 10000 12913 09708 2913 292010 09864 01087 09239 1223 625020 06866 08477 06805 1611 061030 04779 06203 04773 1424 006040 03326 04512 03326 1186 000050 02315 03279 02315 0964 000060 01611 02382 01611 0771 000070 01122 01731 01122 0609 000080 00781 01258 00781 0477 000090 00543 00914 00543 0371 000100 00378 00664 00378 0286 000110 00263 00482 00263 0219 000120 00183 00350 00183 0167 000




Dim

ensio

nles

s con

cent

ratio

n 007

006

005

004

003

002

001



Nomenclature





Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Dim

ensio

nles

s con

cent

ratio

n 007

006

005

004

003

002

001



Nomenclature





Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Application of Mathematica Software to...

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