research article biodiesel production by reactive...

Research ArticleBiodiesel Production by Reactive Flash A Numerical Simulation

Alejandro Regalado-Meacutendez123 Sigurd Skogestad2 Reyna Natividad1 and Rubiacute Romero1

1Centro Conjunto de Investigacion en Quımica Sustentable UAEMex-UNAM Universidad Autonoma del Estado de MexicoCarretera Toluca-Atlacomulco Km 145 Unidad San Cayetano 50200 Toluca MEX Mexico2Department of Chemical Engineering Norwegian University of Science and Technology (NTNU) 7034 Trondheim Norway3Universidad del Mar Ciudad Universitaria SN Puerto Angel 70902 San Pedro Pochutla OAX Mexico

Correspondence should be addressed to Alejandro Regalado-Mendez alejandroregalado33gmailcom

Received 26 December 2015 Revised 11 March 2016 Accepted 4 April 2016

Academic Editor Deepak Kunzru

Copyright copy 2016 Alejandro Regalado-Mendez et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Reactive flash (RF) in biodiesel production has been studied in order to investigate steady-state multiplicities singularities andeffect of biodiesel quality when the RF system approaches to bubble pointThe RF was modeled by an index-2 system of differentialalgebraic equations the vapor split (120601) was computed by modified Rachford-Rice equation and modified Raoultrsquos law computedbubble point and the continuation analysis was tracked on MATCONT Results of this study show the existence of turning pointsleading to a unique bubble point manifold (119909Biodiesel 119879) = (046 47841K) which is a globally stable flashing operation Also theresults of the simulation in MATLAB of the dynamic behavior of the RF show that conversion of triglycerides reaches 97 for aresidence time of 58 minutes and a methanol to triglyceride molar flow ratio of 5 1

1 Introduction

In the last decade biofuels production has been worldwidemotivated by the need of reducing greenhouse gases inorder to slow down climate change In addition biofuelsproduction has also been encouraged by the unstable oilprices the reduction of petroleum reserves and environmen-tal penalties in stationary and mobile sources of pollution Inthis context biodiesel emerges as a viable renewable energyalternative to petroleum diesel This clean renewable fuel issuperior to diesel oil in terms of sulfur and aromatic contentAlso it is environmentally safe nontoxic and biodegradableIn general biodiesel is derived from a transesterificationreaction of triglycerides in vegetable oils or animal fats withalcohol (ie methanol and ethanol) under the presence ofcatalyst [1] Biodiesel manufacturing at large scale demandsthe development of innovative and efficient processes In thissense reactive distillation (RD) seems to offer interesting anddesirable features RD is a hybrid process where a chemicaltransformation and a separation take place in the samevessel The product is removed at the same time that it is

formed This characteristic makes it possible to overcomethe thermodynamic equilibrium of the reaction as well asthe vapor-liquid equilibrium (VLE) [2] RD have manyadvantages which include reduced capital cost completeconversion in equilibrium-limited reactions heat integrationand reduced waste generation [3 4] However this makes ofRD a rather complex hybrid process because of the interac-tions between reaction and phase equilibrium This processis highly nonlinear and at time postures challenges in termsof operation and control Multiple steady states exist inconventional chemical reactors operated in continuousmode(ie continuous stirred tank reactors) and in nonreactive andreactive distillation columns Investigations of the physicalphenomena that causemultiplicities of steady states andHopfbifurcations in RD have been reported by many researchersaround the word [5ndash17] These studies show that there arevarious reasons for the occurrence of output and inputmultiplicity which include multiple steady states caused byheat effects or kinetic instabilities [6 18 19] or presence ofsingularities due to nonlinear transformation and decouplingof energy balance [20] Existence of azeotropes can also lead

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2016 Article ID 7843081 8 pageshttpdxdoiorg10115520167843081

2 International Journal of Chemical Engineering

to multiplicity in both nonreactive and reactive distillationcolumns [21] as well as the presence of reaction hysteresis dueto the interaction between nonreactive and reactive sections[12] Moreover according to Waschler et al [22] existenceof multiple steady states has been identified for light-boilingreactant sufficiently large difference in boiling points andreaction orders lower than the physical parameter 120572 (the120572 parameter is a measure of the phase-equilibrium-drivenself-inhibition of the reaction mechanism) and Purohit etal [17] propose a new relatively simple method to identifymultiplicity in reactive distillation due to the interactionof reaction and distillation Their method compared tothe bifurcation technique makes the analysis simpler andprovides more insight into the influences of different factorson multiplicity behavior Despite the many investigations ofthese important and interesting phenomena in RD only afew efforts to understand the basic causes of steady statesmultiplicity have been reported in the literature Few inves-tigators studied the multiplicity of steady states behavior ofthe reactive flash in order to understand these phenomenain RD systems Rodrıguez et al [23] demonstrated that inputand output multiplicity lead to an isobaric adiabatic reactiveflash for a binary mixture due to the presence of vapor-liquidequilibrium Lakerveld et al [24] studied the steady-statebehavior of a reactive flash by singularity theoryThe reactivesystem was characterized by an exothermic isomerizationreaction with first-order kinetics and a light-boiling reactantBy using the Damkohler number as continuation parametertwenty-five bifurcation diagramswere found exhibiting threesteady states and five feasibility boundaries when the heatof reaction the activation energy or the relative volatilityis increased Rodrıguez et al [7] investigated an isobaricreactive flash with controlled kinetics and a constant splitfraction They found that in this situation the coupling ofthe energy balance does not drive multiplicities with thematerial balances and phase equilibrium equations Theirsolutions revealed multiplicity when the heat of reaction wassmall compared to the heat of vaporization This providedsome insight into causes of multiplicity in RD columnswhen the constant molal overflow approximation works wellFurthermore Ruiz et al [25] and Ruiz et al [8] demonstratethe existence of Hopf bifurcation limit points and isolas withintersecting branches for an equilibrium and nonequilibriumreactive separation process when this is modeled by a systemof ordinary differential equations and a set of differentialalgebraic equations

Jaime-Leal et al [9] introduced a new approach and con-ditions to identify input multiplicity in reactive flash basedon the application of reaction-invariant composition vari-ables Finally Harney et al [26] demonstrate that dynamicbehavior of reactive flash and reactive distillation representedby index-2 system of differential algebraic equations (DAEs)can be reduced to an ordinary differential equations sys-tem (ODE) by single differentiation the resulting Jacobianmatrix will have a null eigenvalue at every steady statewith multiplicity of at least the dimension y These nulleigenvalues must be accounted for when determining thestability of a steady state Moreover Alvarez-Ramirez [27]found singular dynamics in a simple reactive flashmodel and

described different steady-state operating scenarios with atleast one of them belonging to the one-phase operation ofthe system including a globally stable flashing to unfeasibleoperation leading to emptiness of liquid phase As notedearlier steady-states multiplicity can occur in RD columnsdue to high nonlinearity [4 22] These multiplicities haveimpact on desired high conversion at steady state quality ofproducts dynamic behavior and performance and controlsystem design for RD systems Recent studies on control inRD systems have referred to the need to avoid controllingoutputs exhibiting multiplicity [28 29] A short review ofstudies of steady-state multiplicity focused on reactive flashand distillation shows that few investigators found steady-state multiplicity when the temperature of separation processor temperature of reactive system leads to boiling point andthe split fraction is constant In the present work authorssearch for dynamic singularity by dimensionless heat ofreaction rate when the system temperature achieves thebubble point temperature This issue is accomplished bystudying an isobaric reactive flash without a constant splitfraction and a biodiesel production from triglycerides withalkali catalyst Specifically reactive flash model is developedto analyze the operating conditions and is numerically solvedto illustrate possible steady-state multiplicity The dynamicbehavior is represented by an index-2 system of differentialalgebraic equations

2 Simulation Methodology

The dynamic simulations were numerically carried out inMATLAB by solving the model represented by (1)ndash(4) usingPetzoldrsquos method [30] where the consistent initial conditions[31] are given for 119909CIC isin [0 1] and bubble point temperatureIn the steady state the system of DAEs forms a set ofnonlinear algebraic equations To solve them in MATLABthe Newton-Raphson method was applied Besides theidentification of multiple steady states was tracked usingMATCONT from MATLAB toolbox [32] Also the vaporsplit fraction (120601) and the bubble point temperature werecomputed using the modified Rachford-Rice equation [25]and modified Raoultrsquos law by Newton-Raphson methodrespectively For simulation purposes the fluid package wasset as Wilson equation to compute the activity coefficientsas recommended by Carlson [33] and Suthar and Joshipura[34] Also extended Antoine equation to compute the vaporpressure was used Finally the Antoine constants and binarycoefficients for Wilson equation were obtained from ASPENProperties PLUS software An algorithm flowchart to solvethe system of DAEs is shown in Figure 1

21 Model The reason for modeling the reactive flash asa DAE system is that the equations for the vapor-liquidequilibriumcalculations are implicitHence the reactive flashmodel is obtained from mass and heat balances in dynamicconditions Figure 2 illustrates the reactive flash process Theassumptions for the employed model are as follows (i) thechemical reaction is accomplished in the homogeneous liquidphase (ii) vapor and liquid phases are well mixed and (iii)

International Journal of Chemical Engineering 3

Compute bubble point

Compute split fraction 120601

continuation parameter

Solve DAE system

dynamically

Steady-statesolution for DAE system

MATCONT

Continuation analysis

Initial guess

120573rxn

Tbb

xCIC

Read parametersF T0 P zi a Cp qext ki

k0i ΔHvap E i R ΔHRA

Figure 1 Algorithm flowchart reactive flash solutions

the vapor holdup is negligible compared to the liquid holdup(119867) The reactive flash model is represented by the set ofdifferential algebraic equations given in (1)ndash(4)

The dynamic representation model of a 119875-119876 reactiveflash with 119877 reactions 119899 components nonconstant molarholdup119867 liquidmolar feed rate119865 and nonideal vapor-liquidequilibrium behavior can be written as follows

As the liquid mixture achieves its bubble point (119879bbp)evaporation rises spontaneously The total mass balance isgiven by

119889120591

119889119905= 1 minus 120579

119871minus 120601 (1)

The component mass balance is described as follows

119889 (120591119909119894)

119889119905= 119911119894minus 120579119871119909119894minus 120601119910119894+ 120591119877119894

119894 = 1 119899 minus 1 (2)

By considering heating properties the energy balance is givenby

119889 (120591119879)

119889119905= 119879in minus (120579

119871+ 120601) 119879 + 120573rxn120591119877 + 120601120573vap + 119876 (3)

The restrictions of the system are represented by the algebraicequations (4) which are the thermodynamic considerations(ie equilibrium equation and the119870-value prediction)

0 =119899

sum119894=1

119875sat119894120574119871

119894119909119894minus 119875

0 =120574119871119894119875sat119894

119875minus 119870119894

0 = 119870119894119909119894minus 119910119894

(4)

The saturation pressure (119875sat119894) is given by the extended

Antoinersquos equation and activity coefficients (120574119871119894) are computed

by Wilson model as recommended by Carlson [33] andSuthar and Joshipura [34] The vapor split fraction (120601) wascomputed by using the modified Rachford-Rice equation[25]

119888

sum119894=1

(119870119894minus 1)

119911119894+ 119877119894119865

[(1 minus Θ minus 120601) + 120601119870119894]= 0 (5)

22 Kinetic Model The particular case considered here is anadaptation of the transesterification of triglycerides (oilsfats)by reaction with alcohol in the presence of NaOH as catalystto produce fatty acid alkyl esters and glycerol The reactionproceeds in three steps as shown in the following reactions[36]

Triglyceride + ROH1198961

9994459994681198962

Diglyceride + RCOOR1

Diglyceride + ROH1198963

9994459994681198964

Monoglyceride + RCOOR2

Monoglyceride + ROH1198965

9994459994681198966

Glycerol + RCOOR3

(6)

The reaction rates (1199031 1199032 and 119903

3) of each reaction are given as

follows

1199031= 1198961119862TG119862ROH minus 119896

2119862DG119862AE

1

1199032= 1198963119862DG119862ROH minus 119896

4119862MG119862AE

2

1199033= 1198965119862MG119862ROH minus 119896

6119862G119862AE

3

(7)

where 119862TG 119862DG 119862MG 119862G 119862AE and 119862ROH represent theconcentration of triglyceride diglyceride monoglycerideglycerol alkyl esters (biodiesel) and alcohol respectivelyAlso 119896

1 1198962 and 119896

6are forward and backward rate


F

P

T0z1 = a

Q T

Hz2 = 1 minus a

FC

LC

PC

Set

Set

Set

L x

V y

A + B 997888rarr C + D

Figure 2 Diagram of reactive flash adapted from [23 24]

Table 1 Kinetic parameters [35]

Component 1198960119894(Lmolmin) 119864

119860(Jmol)

Triglyceride 1469 times 108 587826720Alcohol 105100 449620452Diglyceride 119 times 1010 671939532Alkyl ester 1725 times 108 582258276Monoglyceride 24940 300319164Glycerol 1469 times 108 460422396

constants The temperature dependency of the rate constant(119896119894) is expressed by Arrheniusrsquo law [37]

119896119894= 1198960

119894exp(minus

119864119860119894

119877119879) (8)

The kinetic parameters are shown in Table 1 The heat ofthe reaction is Δ119867

119877= minus507 times 103 Jmol for each reaction

calculated from the formation heat data [36]

23 Global Regularity Condition onManifold Point Consideran index-2 system of differential algebraic equations given by

= 119891 (119909 119910)

0 = 119892 (119909 119910) (9)

where 119909 isin 119877119899 (119899 ge 2) 119910 isin 119877119898 and 119891 119906 rarr 119877119899 and 119892

119906 rarr 119877119898 are both 1198622 in an open neighborhood 119906 of (119909lowast 119910lowast)in 119877119899+119898

Definition 1 Let 119909119894isin [0 1] and 119875 be the liquid mol fraction

of component 119894th and the systempressure respectively Givena thermodynamic model based on activity coefficients the

bubble point temperature 119879bbp is given in general as implicitequation of the form

119872(119879bbp) = (119909 119879) isin 119877119899+119898

| 119892 (119909 119879) minus 119879bbp119897 119879bbp

= constant (10)

119872 is a regular manifold point of dimension 119899 if

rank [120597119892

120597119909

120597119892

120597119879] = 119898 or 119872 (11)

The structure of119872 depends of course on the parameter119879bbpEven for a simple system of DAEs it may not be satisfied forsome values of 119879bbp

The manifold point 119872 is the state space for the dynamicsystem of DAEs defined by (9) which induces a vector fieldon119872

The vector field may not be well defined at all points of119872 At any point (119909 119879) isin 119872 we have = (119891(119909 119879) minus 119879bbp119892)

and = (119892(119909 119879) minus 119879bbp119897) is not singular the is uniquelydefined by

= minus [120597119892

120597119879]minus1

[120597119892

120597119909] [119891 (119909 119879) minus 119879bbp119897] (12)

If = (119892(119909 119879) minus 119879bbp119897) is singular at point (119909 119879) isin 119872 thenthe vector field is notwell defined at that point Typically suchsingular point lies in codimension 1 submanifolds of119872 [24]The next theorem is written in adapted form from previousworks such as Rheinboldt [38] Sudarsan and Keerthi [39]and Okay et al [40]

Theorem 2 Suppose 119872 is regular manifolds point for all 120583near 120583lowast and that det[(119892(119909 119879) minus 119879

119887119887119901119897)] = 0 at point 120583 = 120583lowast

(119909 119879) = (119909lowast 119879lowast) isin 119872 Then (119909lowast 119879lowast 120583lowast) is said to be causal


Lemma 3 A singular point of the systems of the DAEs insteady state (119909ss 119879ss) isin 119878(120583ss) at given parameter 120583 = 120583lowast isalso an equilibrium point

Proof Suppose (119909ss 119879ss) is a singular point of the decoupledDAEs at the parameter value 120583 = [120583119879

119892120583119879119897]119879 such that

= 119891 (119909ss 119879ss) minus 120583ss119892 = 0

0 = 119892 (119909ss 119879ss) minus 120583ss119897

det [ (119892 (119909 119879)) minus 120583ss119897] = 0

(13)

Therefore a singular point (119909ss 119879ss) isin 119878(120583ss) at the parameter120583 = [120583119879

119892120583119879119897]119879 is an equilibrium point

At 119879 = 119879bbp the evaporation at the bubble pointspontaneously occurs when the excessive heat in the liquidphase is captured by transferring it into vapor phase Itis worth pointing out that (14) is rather important in themodeling of reactive flash since the reactive separationprocess can operate in one or two phases In consequencethe point (119909 119879) will be called bubble point manifold

119872 = (119909 119879) (14)

3 Results

The set of DAEs system is conformed by 8 differential equa-tions and 12 algebraic constraints which come frommass andenergy balances and thermodynamic considerations (equi-librium equation and 119870-value prediction) respectively Themain variables are residence time (120591) liquid and vapor molefractions and temperature Constants are activity coefficientsthat come from Wilson model equation Also vapor splitfraction can be computed from modified Rachford-Riceequation Finally 120573rxn is selected as bifurcation parametersince it can induce large steady-state changes in compositionand temperature as shown in Figure 4

The index-2 system of DAEs was solved by using ode15iMATLAB toolbox when 119876 = minus100Kmolminus1 A perfectholdup is assumed (119867 is constant) Hence 119889120591119889119905 = 0implies that 120579

119871= 1 minus 120601 Also the molar flow ratio 1 5

of triglyceridealcohol was chosen to perform the dynamicbehavior and the phase maps because this relationship leadsto the maximum of biodiesel molar fraction [36] Figure 3depicts the dynamic behavior of the molar fraction for themost important components Also the settling time 120591

119860 of the

reactive flash and the steady-statemolar fraction for biodieselwere determined to be 58 minutes and 046 respectivelyIn addition the molar fraction of triglyceride at 120591 = 58minutes is around 0004 This implies that a 97 conversionwas reached

The two-phase operating mode does not exhibit steady-state multiplicity The overall steady-state multiplicity isintroduced by the multiplicity of the one-phase operatingmode Figure 4 shows the continuation path for the biodieselmolar fraction and steady-state stability properties evaluatedat large values of the dimensionless reaction enthalpy 120573rxn

Time (minutes)

BiodieselMethanol

Steady state

Triglyceride

1086420

Mol

ar fr

actio

nx

06

08

04

02

00

xBiodiesel = 046

120591A asymp 58min

Figure 3 Dynamic behavior of the reactive flash process

where a region of steady-state multiplicity exists The dashedline indicates the transition from unstable to stable steadystates which is delineated by turning points respectivelyIn addition note that the two-phase branch emerges in anondifferential way from the one-phase branch In otherwords the bifurcation diagram is not differentiable at 120601 = 0This fact is a consequence of the discontinuous nature of thereactive flash process because the system operation goes fromone- to two-phasemodeAlso the blue line leads to lowmolarfraction biodiesel meanwhile the red dashed line leads tohigh molar fraction biodiesel corresponding to liquid phaseoperation and black line reaches a unique equilibrium pointcorresponding to a globally stable flashing operation

The singular trajectories of the system of DAEs arereflected in approaches to the bubble pointmanifold Figure 5illustrates the phenomena by showing the behavior of thesystem trajectories (119909(119905) 119879(119905)) as the bubble point manifoldis approached It can be observed that the dynamics tra-jectories converge to a stable equilibrium (046 47841 K)The dynamics trajectories display a sliding behavior as theyachieve the bubble point manifold to converge to a stableequilibrium The singular nature of these dynamics as thebubble point manifold is attained by interaction betweenliquid-vapor separation and chemical transformation

The reactive flash drawing has an asymptotic dynamicbehavior such as that exhibited by continuous stirred tankchemical reactors

The topological form of the bifurcation shapes of thiswork is of type ldquo119878rdquo such as typically displayed in continuousstirred tank chemical reactor and reactive flash This issupported by other studies (Rodrıguez et al [7] Ruiz et al[8] Alvarez-Ramirez [27] Jaime-Leal et al [9] and Harneyet al [26])

The shapes of the trajectories of the bifurcation mapdisplayed in Figure 5 are similar to spiral points (stable node)


Tem

pera

ture

T(K

)

minus400 minus200 0 800600400200

120573rxn (K)minus400 minus200 0 800600400200

120573rxn (K)

Liquid unstable phase Liquid unstable phaseVapor-liquid stable phase Turning point

Liquid stable phase Liquid unstable phaseVapor-liquid stable phase Turning point

300

330

360

390

420

450

480

00

01

02

03

04

05M

olar

frac

tion

xBi

odie

sel

Figure 4 Bifurcation diagram for 119876 = minus100Kmolminus1 as a function of the parameter 120573rxn

500

480

460

440

420

400

Tem

pera

ture

T(K

)

Molar fraction xBiodiesel

08 1006040200

Manifold point

(x T) = (046 47841)

Figure 5 Behavior of the system trajectories as the manifold isattained

[41 42] as is sketched in autonomous differential equationsystem for example in stirred tank chemical reactors

Finally modeling and analyzing a reactive flash provideimportant insights for understanding the design operationand control of higher order process For example in thiscase study the feasible flashing region is not too sensitiveto dimensionless reaction enthalpy 120573rxn at large valuesbecause it implies relatively small changes of composition andtemperature Therefore implementing composition controland temperature control in reactive stages should be avoidedon reactive distillation

4 Conclusions

The singularities in reactive flash when the system tem-perature approaches to the bubble point were investigated

It is shown that the differential algebraic equations displaysteady-state multiplicities revealing the existence of turningpoints and leading to the bubble point manifold this uniqueequilibrium point corresponding to a globally stable flashingoperation Results indicate that reactive distillation stages candisplay discontinuous dynamics by the large changes in theheat of reaction rate

Nomenclature

119865 Liquid feed rate (molminminus1)119867 Liquid molar holdup (mol)119871 Liquid exit flow (molminminus1)120579119871 Specific liquid molar exit rate (=119871119865)

119881 Vapor exit flow (molminminus1)119879 Temperature (K)119909119894 Liquid mole fraction of component 119894

119911119894 Feed mole fraction of component 119894

119910119894 Vapor mole fraction of component 119894

119875 Total pressure119875sat119894 Vapor pressure of component 119894 (Pa)

119870119894 119870-value prediction

119877119894 Rate of reaction for component 119894 in

reaction 119895 (molminminus1)119877 Total rate of reaction 119877 = sum

119862

119894=1119867]119894119895120576119877119894

119862119901 Heat capacity (J Kminus1)

119902ext External heat input to flash 119902ext divided byfeed rate 119865 (Jmolminus1)

119876 Total heat input to flash (=119902ext119862119901)(Kmolminus1)

Greek

120591 Residence time (=119867119865)120601 Vapor split fraction (=119881119865)]119894119895 Stoichiometric coefficient of component 119894 in reaction 119895


120576 Reaction volumeΔ119867rxn Enthalpy of reaction (J)Δ119867vap Enthalpy of vaporization (J)120573rxn Modified enthalpy of reaction (=Δ119867rxn119862119901) (K)120573vap Modified enthalpy of vaporization (=Δ119867vap119862119901)

(K)

Competing Interests

The authors declare that they have no competing interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to PRODEP (before PROMEP) forfinancial support through PROMEP1035105552

References

[1] A C Dimian C S Bildea F Omota andA A Kiss ldquoInnovativeprocess for fatty acid esters by dual reactive distillationrdquoComputers amp Chemical Engineering vol 33 no 3 pp 743ndash7502009

[2] R A Pai M F Doherty and M F Malone ldquoDesign of reactiveextraction systems for bioproduct recoveryrdquoAIChE Journal vol48 no 3 pp 514ndash526 2002

[3] KAlejski and FDuprat ldquoDynamic simulation of themulticom-ponent reactive distillationrdquo Chemical Engineering Science vol51 no 18 pp 4237ndash4252 1996

[4] R Taylor and R Krishna ldquoModelling reactive distillationrdquoChemical Engineering Science vol 55 no 22 pp 5183ndash52292000

[5] R Baur R Taylor and R Krishna ldquoBifurcation analysis forTAME synthesis in a reactive distillation column comparisonof pseudo-homogeneous and heterogeneous reaction kineticsmodelsrdquo Chemical Engineering and Processing Process Intensifi-cation vol 42 no 3 pp 211ndash221 2003

[6] F Chen R S Huss M F Doherty and M F Malone ldquoMultiplesteady states in reactive distillation kinetic effectsrdquo Computersamp Chemical Engineering vol 26 no 1 pp 81ndash93 2002

[7] I E Rodrıguez A Zheng and M F Malone ldquoParametricdependence of solution multiplicity in reactive flashesrdquo Chemi-cal Engineering Science vol 59 no 7 pp 1589ndash1600 2004

[8] G Ruiz M Diaz and L N Sridhar ldquoSingularities in reac-tive separation processesrdquo Industrial amp Engineering ChemistryResearch vol 47 no 8 pp 2808ndash2816 2008

[9] J E Jaime-Leal A Bonilla-Petriciolet J G Segovia-HernandezS Hernandez and H Hernandez-Escoto ldquoAnalysis and predic-tion of input multiplicity for the reactive flash separation usingreaction-invariant composition variablesrdquo Chemical Engineer-ing Research and Design vol 90 no 11 pp 1856ndash1870 2012

[10] R Monroy-Loperena and J Alvarez-Ramirez ldquoOn the steady-state multiplicities for an ethylene glycol reactive distillationcolumnrdquo Industrial amp Engineering Chemistry Research vol 38no 2 pp 451ndash455 1999

[11] A Kumar and P Daoutidis ldquoModeling analysis and control ofethylene glycol reactive distillation columnrdquoAIChE Journal vol45 no 1 pp 51ndash68 1999

[12] M G Sneesby M O Tade and T N Smith ldquoReaction hystere-sis a new cause of output multiplicity in reactive distillationrdquo

Developments in Chemical Engineering and Mineral Processingvol 7 no 1-2 pp 41ndash56 1999

[13] M A Al-Arfaj and W L Luyben ldquoComparative control studyof ideal and methyl acetate reactive distillationrdquo ChemicalEngineering Science vol 57 no 24 pp 5039ndash5050 2002

[14] A P Higler R Taylor and R Krishna ldquoNonequilibrium mod-elling of reactive distillation multiple steady states in MTBEsynthesisrdquo Chemical Engineering Science vol 54 no 10 pp1389ndash1395 1999

[15] T E Guttinger and M Morari ldquoPredicting multiple steadystates in equilibrium reactive distillation 2 Analysis of hybridsystemsrdquo Industrial amp Engineering Chemistry Research vol 38no 4 pp 1649ndash1665 1999

[16] T E Guttinger and M Morari ldquoPredicting multiple steadystates in equilibrium reactive distillation 1 Analysis of nonhy-brid systemsrdquo Industrial amp Engineering Chemistry Research vol38 no 4 pp 1633ndash1648 1999

[17] J L Purohit S M Mahajani and S C Patwardhan ldquoAnalysisof steady-state multiplicity in reactive distillation columnsrdquoIndustrial amp Engineering Chemistry Research vol 52 no 14 pp5191ndash5206 2013

[18] A UppalWH Ray andA B Poore ldquoOn the dynamic behaviorof continuous stirred tank reactorsrdquo Chemical EngineeringScience vol 29 no 4 pp 967ndash985 1974

[19] L F Razon and R A Schmitz ldquoMultiplicities and instabilities inchemically reacting systemsmdasha reviewrdquo Chemical EngineeringScience vol 42 no 5 pp 1005ndash1047 1987

[20] E W Jacobsen and S Skogestad ldquoMultiple steady states in idealtwo-product distillationrdquoAIChE Journal vol 37 no 4 pp 499ndash511 1991

[21] N Bekiaris G A Meski C M Radu andM Morari ldquoMultiplesteady states in homogeneous azeotropic distillationrdquo Industrialand Engineering Chemistry Research vol 32 no 9 pp 2023ndash2038 1993

[22] R Waschler S Pushpavanam and A Kienle ldquoMultiple steadystates in two-phase reactors under boiling conditionsrdquoChemicalEngineering Science vol 58 no 11 pp 2203ndash2214 2003

[23] I E Rodrıguez A Zheng and M F Malone ldquoThe stability of areactive flashrdquo Chemical Engineering Science vol 56 no 16 pp4737ndash4745 2001

[24] R Lakerveld C S Bildea and C P Almeida-Rivera ldquoExother-mic isomerization reaction in a reactive flash steady-statebehaviorrdquo Industrial and Engineering Chemistry Research vol44 no 10 pp 3815ndash3822 2005

[25] G Ruiz L N Sridhar and R Rengaswamy ldquoIsothermal isobaricreactive flash problemrdquo Industrial amp Engineering ChemistryResearch vol 45 no 19 pp 6548ndash6554 2006

[26] DAHarney T KMills andN L Book ldquoNumerical evaluationof the stability of stationary points of index-2 differential-algebraic equations applications to reactive flash and reactivedistillation systemsrdquo Computers and Chemical Engineering vol49 pp 61ndash69 2013

[27] J Alvarez-Ramirez ldquoSingular reactive flash dynamicsrdquo Chem-ical Engineering and Processing Process Intensification vol 69pp 119ndash125 2013

[28] S-B Hung M-J Lee Y-T Tang et al ldquoControl of differentreactive distillation configurationsrdquo AIChE Journal vol 52 no4 pp 1423ndash1440 2006

[29] M V Pavan Kumar and N Kaistha ldquoRole of multiplicity inreactive distillation control system designrdquo Journal of ProcessControl vol 18 no 7-8 pp 692ndash706 2008


[30] K E Brenan and L R Petzold ldquoThe numerical solutionof higher index differentialalgebraic equations by implicitmethodsrdquo SIAM Journal on Numerical Analysis vol 26 no 4pp 976ndash996 1989

[31] S L Campbell ldquoConsistent initial conditions for singularnonlinear systemsrdquo Circuits Systems and Signal Processing vol2 no 1 pp 45ndash55 1983

[32] A Dhooge W Govaerts Y A Kuznetsov H G Meijerand B Sautois ldquoNew features of the software MatCont forbifurcation analysis of dynamical systemsrdquo Mathematical andComputer Modelling of Dynamical Systems Methods Tools andApplications in Engineering and Related Sciences vol 14 no 2pp 147ndash175 2008

[33] E C Carlson ldquoDonrsquot gamble with physical properties forsimulationsrdquo Chemical Engineering Progress vol 92 no 10 pp35ndash46 1996

[34] K Suthar and M Joshipura ldquoA comparative study on pre-dictions of vapor liquid equilibrium of biodiesel systemsrdquo inProceedings of the 2nd International Conference on CurrentTrends in Technology (NUiCONE rsquo11) pp 1ndash6 Institute ofTechnology Nirma University Ahmedabad India December2011

[35] H Noureddini and D Zhu ldquoKinetics of transesterification ofsoybean oilrdquo Journal of the American Oil Chemistsrsquo Society vol74 no 11 pp 1457ndash1463 1997

[36] M Agarwal K Singh and S P Chaurasia ldquoSimulation and sen-sitivity analysis for biodiesel production in a reactive distillationcolumnrdquoPolish Journal of Chemical Technology vol 14 no 3 pp59ndash65 2012

[37] N de Lima da Silva E Ccopa Rivera C B Batistella D Ribeirode Lima R Maciel Filhob and M R Wolf Maciel ldquoBiodieselproduction from vegetable oils operational strategies for largescale systemsrdquo in 18th European Symposium on ComputerAided Process EngineeringmdashESCAPE 18 B Braunschweig andX Joulia Eds pp 1001ndash1007 Elsevier Lyon France 2008

[38] W C Rheinboldt ldquoSolving algebraically explicit DAEs with theMANPAK-manifold-algorithmsrdquo Computers and Mathematicswith Applications vol 33 no 3 pp 31ndash43 1997

[39] R Sudarsan and S S Keerthi ldquoNumerical approaches forsolution of differential equations on manifoldsrdquo Applied Math-ematics and Computation vol 92 no 2-3 pp 153ndash193 1998

[40] I Okay S L Campbell and P Kunkel ldquoCompletions ofimplicitly defined linear time varying vector fieldsrdquo LinearAlgebra and its Applications vol 431 no 9 pp 1422ndash1438 2009

[41] S E LeBlanc and D R Coughanowr Process Systems Analysisand Control McGraw-Hill New York NY USA 3rd edition2009

[42] Z Bilicki C Dafermos J Kestin G Majda and D L ZengldquoTrajectories and singular points in steady-state models of two-phase flowsrdquo International Journal of Multiphase Flow vol 13no 4 pp 511ndash533 1987

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

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VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Advances inOptoElectronics


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



to multiplicity in both nonreactive and reactive distillationcolumns [21] as well as the presence of reaction hysteresis dueto the interaction between nonreactive and reactive sections[12] Moreover according to Waschler et al [22] existenceof multiple steady states has been identified for light-boilingreactant sufficiently large difference in boiling points andreaction orders lower than the physical parameter 120572 (the120572 parameter is a measure of the phase-equilibrium-drivenself-inhibition of the reaction mechanism) and Purohit etal [17] propose a new relatively simple method to identifymultiplicity in reactive distillation due to the interactionof reaction and distillation Their method compared tothe bifurcation technique makes the analysis simpler andprovides more insight into the influences of different factorson multiplicity behavior Despite the many investigations ofthese important and interesting phenomena in RD only afew efforts to understand the basic causes of steady statesmultiplicity have been reported in the literature Few inves-tigators studied the multiplicity of steady states behavior ofthe reactive flash in order to understand these phenomenain RD systems Rodrıguez et al [23] demonstrated that inputand output multiplicity lead to an isobaric adiabatic reactiveflash for a binary mixture due to the presence of vapor-liquidequilibrium Lakerveld et al [24] studied the steady-statebehavior of a reactive flash by singularity theoryThe reactivesystem was characterized by an exothermic isomerizationreaction with first-order kinetics and a light-boiling reactantBy using the Damkohler number as continuation parametertwenty-five bifurcation diagramswere found exhibiting threesteady states and five feasibility boundaries when the heatof reaction the activation energy or the relative volatilityis increased Rodrıguez et al [7] investigated an isobaricreactive flash with controlled kinetics and a constant splitfraction They found that in this situation the coupling ofthe energy balance does not drive multiplicities with thematerial balances and phase equilibrium equations Theirsolutions revealed multiplicity when the heat of reaction wassmall compared to the heat of vaporization This providedsome insight into causes of multiplicity in RD columnswhen the constant molal overflow approximation works wellFurthermore Ruiz et al [25] and Ruiz et al [8] demonstratethe existence of Hopf bifurcation limit points and isolas withintersecting branches for an equilibrium and nonequilibriumreactive separation process when this is modeled by a systemof ordinary differential equations and a set of differentialalgebraic equations

Jaime-Leal et al [9] introduced a new approach and con-ditions to identify input multiplicity in reactive flash basedon the application of reaction-invariant composition vari-ables Finally Harney et al [26] demonstrate that dynamicbehavior of reactive flash and reactive distillation representedby index-2 system of differential algebraic equations (DAEs)can be reduced to an ordinary differential equations sys-tem (ODE) by single differentiation the resulting Jacobianmatrix will have a null eigenvalue at every steady statewith multiplicity of at least the dimension y These nulleigenvalues must be accounted for when determining thestability of a steady state Moreover Alvarez-Ramirez [27]found singular dynamics in a simple reactive flashmodel and

described different steady-state operating scenarios with atleast one of them belonging to the one-phase operation ofthe system including a globally stable flashing to unfeasibleoperation leading to emptiness of liquid phase As notedearlier steady-states multiplicity can occur in RD columnsdue to high nonlinearity [4 22] These multiplicities haveimpact on desired high conversion at steady state quality ofproducts dynamic behavior and performance and controlsystem design for RD systems Recent studies on control inRD systems have referred to the need to avoid controllingoutputs exhibiting multiplicity [28 29] A short review ofstudies of steady-state multiplicity focused on reactive flashand distillation shows that few investigators found steady-state multiplicity when the temperature of separation processor temperature of reactive system leads to boiling point andthe split fraction is constant In the present work authorssearch for dynamic singularity by dimensionless heat ofreaction rate when the system temperature achieves thebubble point temperature This issue is accomplished bystudying an isobaric reactive flash without a constant splitfraction and a biodiesel production from triglycerides withalkali catalyst Specifically reactive flash model is developedto analyze the operating conditions and is numerically solvedto illustrate possible steady-state multiplicity The dynamicbehavior is represented by an index-2 system of differentialalgebraic equations

2 Simulation Methodology

The dynamic simulations were numerically carried out inMATLAB by solving the model represented by (1)ndash(4) usingPetzoldrsquos method [30] where the consistent initial conditions[31] are given for 119909CIC isin [0 1] and bubble point temperatureIn the steady state the system of DAEs forms a set ofnonlinear algebraic equations To solve them in MATLABthe Newton-Raphson method was applied Besides theidentification of multiple steady states was tracked usingMATCONT from MATLAB toolbox [32] Also the vaporsplit fraction (120601) and the bubble point temperature werecomputed using the modified Rachford-Rice equation [25]and modified Raoultrsquos law by Newton-Raphson methodrespectively For simulation purposes the fluid package wasset as Wilson equation to compute the activity coefficientsas recommended by Carlson [33] and Suthar and Joshipura[34] Also extended Antoine equation to compute the vaporpressure was used Finally the Antoine constants and binarycoefficients for Wilson equation were obtained from ASPENProperties PLUS software An algorithm flowchart to solvethe system of DAEs is shown in Figure 1

21 Model The reason for modeling the reactive flash asa DAE system is that the equations for the vapor-liquidequilibriumcalculations are implicitHence the reactive flashmodel is obtained from mass and heat balances in dynamicconditions Figure 2 illustrates the reactive flash process Theassumptions for the employed model are as follows (i) thechemical reaction is accomplished in the homogeneous liquidphase (ii) vapor and liquid phases are well mixed and (iii)





Solve DAE system

dynamically


MATCONT


Initial guess

120573rxn

Tbb

xCIC







119889120591

119889119905= 1 minus 120579

119871minus 120601 (1)


119889 (120591119909119894)

119889119905= 119911119894minus 120579119871119909119894minus 120601119910119894+ 120591119877119894

119894 = 1 119899 minus 1 (2)


119889 (120591119879)

119889119905= 119879in minus (120579

119871+ 120601) 119879 + 120573rxn120591119877 + 120601120573vap + 119876 (3)


0 =119899

sum119894=1

119875sat119894120574119871

119894119909119894minus 119875

0 =120574119871119894119875sat119894

119875minus 119870119894

0 = 119870119894119909119894minus 119910119894

(4)




119888

sum119894=1

(119870119894minus 1)

119911119894+ 119877119894119865

[(1 minus Θ minus 120601) + 120601119870119894]= 0 (5)



9994459994681198962



9994459994681198964



9994459994681198966

Glycerol + RCOOR3

(6)



follows

1199031= 1198961119862TG119862ROH minus 119896

2119862DG119862AE

1

1199032= 1198963119862DG119862ROH minus 119896

4119862MG119862AE

2

1199033= 1198965119862MG119862ROH minus 119896

6119862G119862AE

3

(7)


1 1198962 and 119896



F

P

T0z1 = a

Q T

Hz2 = 1 minus a

FC

LC

PC

Set

Set

Set

L x

V y

A + B 997888rarr C + D




119860(Jmol)



119896119894= 1198960

119894exp(minus

119864119860119894

119877119879) (8)





= 119891 (119909 119910)

0 = 119892 (119909 119910) (9)






119872(119879bbp) = (119909 119879) isin 119877119899+119898

| 119892 (119909 119879) minus 119879bbp119897 119879bbp

= constant (10)


rank [120597119892

120597119909

120597119892

120597119879] = 119898 or 119872 (11)





= minus [120597119892

120597119879]minus1

[120597119892

120597119909] [119891 (119909 119879) minus 119879bbp119897] (12)



119887119887119901119897)] = 0 at point 120583 = 120583lowast





119892120583119879119897]119879 such that

= 119891 (119909ss 119879ss) minus 120583ss119892 = 0

0 = 119892 (119909ss 119879ss) minus 120583ss119897

det [ (119892 (119909 119879)) minus 120583ss119897] = 0

(13)




119872 = (119909 119879) (14)

3 Results





119860 of the



Time (minutes)

BiodieselMethanol

Steady state

Triglyceride

1086420

Mol

ar fr

actio

nx

06

08

04

02

00

xBiodiesel = 046

120591A asymp 58min








Tem

pera

ture

T(K

)

minus400 minus200 0 800600400200

120573rxn (K)minus400 minus200 0 800600400200

120573rxn (K)



300

330

360

390

420

450

480

00

01

02

03

04

05M

olar

frac

tion

xBi

odie

sel


500

480

460

440

420

400

Tem

pera

ture

T(K

)


08 1006040200

Manifold point

(x T) = (046 47841)




4 Conclusions



Nomenclature









119862

119894=1119867]119894119895120576119877119894




Greek




(K)

Competing Interests


Acknowledgments


References















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Solve DAE system

dynamically


MATCONT


Initial guess

120573rxn

Tbb

xCIC







119889120591

119889119905= 1 minus 120579

119871minus 120601 (1)


119889 (120591119909119894)

119889119905= 119911119894minus 120579119871119909119894minus 120601119910119894+ 120591119877119894

119894 = 1 119899 minus 1 (2)


119889 (120591119879)

119889119905= 119879in minus (120579

119871+ 120601) 119879 + 120573rxn120591119877 + 120601120573vap + 119876 (3)


0 =119899

sum119894=1

119875sat119894120574119871

119894119909119894minus 119875

0 =120574119871119894119875sat119894

119875minus 119870119894

0 = 119870119894119909119894minus 119910119894

(4)




119888

sum119894=1

(119870119894minus 1)

119911119894+ 119877119894119865

[(1 minus Θ minus 120601) + 120601119870119894]= 0 (5)



9994459994681198962



9994459994681198964



9994459994681198966

Glycerol + RCOOR3

(6)



follows

1199031= 1198961119862TG119862ROH minus 119896

2119862DG119862AE

1

1199032= 1198963119862DG119862ROH minus 119896

4119862MG119862AE

2

1199033= 1198965119862MG119862ROH minus 119896

6119862G119862AE

3

(7)


1 1198962 and 119896



F

P

T0z1 = a

Q T

Hz2 = 1 minus a

FC

LC

PC

Set

Set

Set

L x

V y

A + B 997888rarr C + D




119860(Jmol)



119896119894= 1198960

119894exp(minus

119864119860119894

119877119879) (8)





= 119891 (119909 119910)

0 = 119892 (119909 119910) (9)






119872(119879bbp) = (119909 119879) isin 119877119899+119898

| 119892 (119909 119879) minus 119879bbp119897 119879bbp

= constant (10)


rank [120597119892

120597119909

120597119892

120597119879] = 119898 or 119872 (11)





= minus [120597119892

120597119879]minus1

[120597119892

120597119909] [119891 (119909 119879) minus 119879bbp119897] (12)



119887119887119901119897)] = 0 at point 120583 = 120583lowast





119892120583119879119897]119879 such that

= 119891 (119909ss 119879ss) minus 120583ss119892 = 0

0 = 119892 (119909ss 119879ss) minus 120583ss119897

det [ (119892 (119909 119879)) minus 120583ss119897] = 0

(13)




119872 = (119909 119879) (14)

3 Results





119860 of the



Time (minutes)

BiodieselMethanol

Steady state

Triglyceride

1086420

Mol

ar fr

actio

nx

06

08

04

02

00

xBiodiesel = 046

120591A asymp 58min








Tem

pera

ture

T(K

)

minus400 minus200 0 800600400200

120573rxn (K)minus400 minus200 0 800600400200

120573rxn (K)



300

330

360

390

420

450

480

00

01

02

03

04

05M

olar

frac

tion

xBi

odie

sel


500

480

460

440

420

400

Tem

pera

ture

T(K

)


08 1006040200

Manifold point

(x T) = (046 47841)




4 Conclusions



Nomenclature









119862

119894=1119867]119894119895120576119877119894




Greek




(K)

Competing Interests


Acknowledgments


References















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










F

P

T0z1 = a

Q T

Hz2 = 1 minus a

FC

LC

PC

Set

Set

Set

L x

V y

A + B 997888rarr C + D




119860(Jmol)



119896119894= 1198960

119894exp(minus

119864119860119894

119877119879) (8)





= 119891 (119909 119910)

0 = 119892 (119909 119910) (9)






119872(119879bbp) = (119909 119879) isin 119877119899+119898

| 119892 (119909 119879) minus 119879bbp119897 119879bbp

= constant (10)


rank [120597119892

120597119909

120597119892

120597119879] = 119898 or 119872 (11)





= minus [120597119892

120597119879]minus1

[120597119892

120597119909] [119891 (119909 119879) minus 119879bbp119897] (12)



119887119887119901119897)] = 0 at point 120583 = 120583lowast





119892120583119879119897]119879 such that

= 119891 (119909ss 119879ss) minus 120583ss119892 = 0

0 = 119892 (119909ss 119879ss) minus 120583ss119897

det [ (119892 (119909 119879)) minus 120583ss119897] = 0

(13)




119872 = (119909 119879) (14)

3 Results





119860 of the



Time (minutes)

BiodieselMethanol

Steady state

Triglyceride

1086420

Mol

ar fr

actio

nx

06

08

04

02

00

xBiodiesel = 046

120591A asymp 58min








Tem

pera

ture

T(K

)

minus400 minus200 0 800600400200

120573rxn (K)minus400 minus200 0 800600400200

120573rxn (K)



300

330

360

390

420

450

480

00

01

02

03

04

05M

olar

frac

tion

xBi

odie

sel


500

480

460

440

420

400

Tem

pera

ture

T(K

)


08 1006040200

Manifold point

(x T) = (046 47841)




4 Conclusions



Nomenclature









119862

119894=1119867]119894119895120576119877119894




Greek




(K)

Competing Interests


Acknowledgments


References















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119892120583119879119897]119879 such that

= 119891 (119909ss 119879ss) minus 120583ss119892 = 0

0 = 119892 (119909ss 119879ss) minus 120583ss119897

det [ (119892 (119909 119879)) minus 120583ss119897] = 0

(13)




119872 = (119909 119879) (14)

3 Results





119860 of the



Time (minutes)

BiodieselMethanol

Steady state

Triglyceride

1086420

Mol

ar fr

actio

nx

06

08

04

02

00

xBiodiesel = 046

120591A asymp 58min








Tem

pera

ture

T(K

)

minus400 minus200 0 800600400200

120573rxn (K)minus400 minus200 0 800600400200

120573rxn (K)



300

330

360

390

420

450

480

00

01

02

03

04

05M

olar

frac

tion

xBi

odie

sel


500

480

460

440

420

400

Tem

pera

ture

T(K

)


08 1006040200

Manifold point

(x T) = (046 47841)




4 Conclusions



Nomenclature









119862

119894=1119867]119894119895120576119877119894




Greek




(K)

Competing Interests


Acknowledgments


References















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Tem

pera

ture

T(K

)

minus400 minus200 0 800600400200

120573rxn (K)minus400 minus200 0 800600400200

120573rxn (K)



300

330

360

390

420

450

480

00

01

02

03

04

05M

olar

frac

tion

xBi

odie

sel


500

480

460

440

420

400

Tem

pera

ture

T(K

)


08 1006040200

Manifold point

(x T) = (046 47841)




4 Conclusions



Nomenclature









119862

119894=1119867]119894119895120576119877119894




Greek




(K)

Competing Interests


Acknowledgments


References















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(K)

Competing Interests


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 biodiesel production by reactive...

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