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Alternative energy sources are supposed to be the most challenging job of today’s world. Among the alternative renewable energy sources, biodiesel attracts considerable attention as an environmental friendly and green energy sources. Biodiesel can be produced from different sources but most suitable and cost effective approach goes to nonedible vegetable oil of fruits of Jatropha Curcas plant through transesterification reaction with methanol. The productivity of biodiesel preparation through transesterification reaction is mainly affected by molar ratios, reaction temperature, catalyst concentration and stirrer speed or mixing effect. Among these parameters, mixing effect through mass transfer limitations has a profound influence for optimal completion of reaction. In the present research investigation, attempts have been made to develop a mathematical model to discuss about the effect of mass transfer in different phases using variation of mixing intensity in transesterification reaction along with temperature. A control theoretic approach is applied to administer the said dynamics for the maximum production of biodiesel
Effect of mass transfer kinetics for maximum production of biodiesel from Jatropha Curcas oil: A mathematical approach q Priti Kumar Roy a, ⇑ , Siddhartha Datta b , Sumit Nandi c , Fahad Al Basir a a Department of Mathematics, Jadavpur University, Kolkata 700 032, India b Department of Chemical Engineering, Jadavpur University, Kolkata 700 032, India c Department of Chemistry, Narula Institute of Technology, Kolkata 700 109, India highlights Biodiesel can be produced through transesterification of Jatropha oil with methanol. Mass transfer limitations are emphasized mathematically through stirring dynamics. Impeller speed and temperature influence the mass transfer resistance rate. Maximum biodiesel can be obtained when system has no mass transfer resistance. Control approach on the stirrer speed has an impact for economic production of BD. article info Article history: Received 13 February 2014 Received in revised form 2 May 2014 Accepted 8 May 2014 Available online 2 June 2014 Keywords: Biodiesel Jatropha Curcas oil Transesterification Stirring effect Optimal control abstract Alternative energy sources are supposed to be the most challenging job of today’s world. Among the alternative renewable energy sources, biodiesel attracts considerable attention as an environmental friendly and green energy sources. Biodiesel can be produced from different sources but most suitable and cost effective approach goes to non-edible vegetable oil of fruits of Jatropha Curcas plant through transesterification reaction with methanol. The productivity of biodiesel preparation through transeste- rification reaction is mainly affected by molar ratios, reaction temperat ure, catalyst concentration and stirrer speed or mixing effect. Among these parameters, mixing effect through mass transfer limitations has a profound influence for optimal completion of reaction. In the present research investigation, attempts have been made to develop a mathematical model to discuss about the effect of mass transfer in different phases using variation of mixing intensity in transesterification reaction along with temper- ature. A control theoretic approach is applied to administer the said dynamics for the maximum produc- tion of biodiesel. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction Scarcity of non-renewable energy sources encourages research- ers and academicians to think about alternative renewable green energy sources for the future world. In this aspect, biodiesel creates a lot of attention as it is environmental friendly, biodegradable, nontoxic and renewable green energy fuel [1,2]. It also reduces car- bon dioxide emission by 78% [3,4]. Chemically, biodiesel is defined as fatty acid methyl esters produced from transesterification reaction between vegetable oil and alcohol in the presence of cat- alyst [5,6]. Different oils are used for this purpose such as Jatropha Curcas, Rapeseed, Palm, Sunflower oil. But the most promising oil is from Jatropha Curcas as it is non-edible and thus does not com- promise the edible oils and moreover, this plant can be cultivated in adverse environment also. Due to these reasons, there is a growing interest in using Jatropha Curcas L. oil as the feedstock for biodiesel production [7–9]. Furthermore, Jatropha Curcas seed has a high content of oil and the biodiesel produced has similar properties to that of petroleum-based diesel. Thus, Jatropha Curcas oil has been highlighted as a potential biodiesel feedstock among the non-edible oils [10,11]. There are many research articles based on modeling strategies for biodiesel production. The development of kinetic model for transesterification reactions were established by Noureddini and http://dx.doi.org/10.1016/j.fuel.2014.05.021 0016-2361/Ó 2014 Elsevier Ltd. All rights reserved. q The research work is supported by UGC Major Research Project, F. No. 41-768/ 2012 (SR), dated: 18 July, 2012. ⇑ Corresponding author. Tel.: +91 9432095603 (M), +91 3324572736 (O); fax: +91 3324146584. E-mail address: pritiju@gmail.com (P.K. Roy). Fuel 134 (2014) 39–44 Contents lists available at ScienceDirect Fuel journal homepage: www.elsevier.com/locate/fuel Zhu [12], Bambase et al. [13], Berchmans et al. [14], Stamenkovic et al. [15], Vicente et al. [16], Paola et al. [17] and Diwekar and Benavides [18] for biodiesel production and they showed that suc- cessful production of biodiesel depends on different reaction parameters like temperature, molar ratio of oil and alcohol, mixing effect and catalyst concentration. Among the reaction parameters, mixing effect or mass transfer limitations is one of the most deciding factors for optimum produc- tion of biodiesel as in transesterification reaction, oils or fats are immiscible with alcohol due to polar and non-polar nature of alco- hol and oil respectively. Biodiesel production in a batch reactor may be considered as a pseudo-homogeneous one with no mass transfer limitations [19]. Nonetheless, an initial mass transfer controlled region followed by a kinetically controlled region is generally pro- posed for the mechanism of biodiesel production [12]. The conven- tional base catalysed transesterification reaction is characterized by slow reaction rates at both initial and final stages limited by mass transfer between polar methanol/glycerol phase and non-polar oil phase [20]. It was also shown by Hou et al. [21] that initially the reaction is very slow due to mass transfer limitations between methanol and oil phase. Peterson et al. [22] has studied the effect of stirrer speed on the transesterification of vegetable oil with alco- hol. Ataya et al. [23] has observed the absence of the catalyzed reac- tion between canola oil and methanol by applying stirring. Ma et al. [24] has demonstrated the importance of mixing in the base cata- lyzed methanolysis of beef tallow. The effect of the stirring intensity of the transesterification of cotton seed oil using several catalysts has been analyzed by Rashid et al. [25] and shown that beyond an optimum stirrer speed production level decreases. Sharma et al. [26] has shown that biodiesel production increases with the increase of stirrer speed up to a maximum level at which the pro- duction stabilizes. Recently, in an experimental work, Kafuku et al. [27], observed that an optimum stirring is needed for maxi- mum biodiesel production. It was also experimentally detected by Zhang et al. [20], Berrios et al. [28] and Paola et al. [17] that depen- dency of the transesterification reaction on the stirrer speed for a fixed time of reaction plays a vital role for optimal completion of reaction. Thus, the transesterification reaction for biodiesel produc- tion is initially mass transfer limited because the two reactants are immiscible with each other and later because the glycerol phase separates together with most of the catalyst [29]. In our research article, a mathematical model has been devel- oped to describe the effect of mixing intensity through stirring dynamics in transesterification kinetics of Jatropha Curcas oil and methanol in the presence of catalyst with the help of some basic assumptions. Here emphasis was given on the analysis of mass transfer limitations between polar/non-polar phases. A mathemat- ical control approach on stirring has also been studied to obtain maximum biodiesel production at an optimum stirring speed. Here, the model equations are analyzed in two different ways, analytical and numerical. For the determination of optimal control approach, Pontryagin Minimum Principle has been adopted and it has been solved using Hamiltonian. Numerical analysis has been done to find out the system parameters, for which the product can be optimized. Numerical findings are in agreement with the experimental data. 2. Basic assumptions and formulation of the mathematical model To describe a simple mathematical model for the transesterifi- cation reaction of Jatropha Curcas oil, the following assumptions have been made: (A1) Transesterification of vegetable oil with methanol consists of three stepwise and reversible reactions. (A2) Since there is only little water (0.2%, w/w) in the reaction mixtures and negligible free fatty acid is detected in the sys- tem, the hydrolysis reaction could be negligible [30]. (A3) The catalyst used in this study has no positional specificity, so it acts concurrently on any acyl-group. (A4) As mixing intensity in the reaction system directs the mass transfer limitations between phases, so mechanical stirring is one of the most effective factors in transesterification reaction. Here, we use k S as the mass transfer rate constant and its unit is min À1 and the term has been defined as below [31]: k S ¼ a ð1 þ expðÀbðN À cÞÞ ; ð1Þ where N is the speed of stirrer and its unit is rpm. a; b and c are dif- ferent parameters of this expression. It is given in our model in logistic fashion that is using the term k S x B 1 À x B B max in developing the model. This term represents the influence of different mixing conditions on the overall reaction rate because the experimental evidence clearly indicates the dependence of transesterification reaction velocity on stirrer speed. Here x B denotes the concentration of biodiesel and B max represents maximum biodiesel production in an ideal situation which is defined as system having no mass trans- fer resistance. The unit of x B ; B max is represented by moles/L. The term has been used logistically because with the increase of stirrer speed, the mass transfer resistance decreases and beyond a certain stirrer speed the mass transfer resistance is negligible. This is also evident from experimental observations of other workers [32]. (A5) Each reaction step is assumed to be of first order with respect to each reacting component. Here we denote triglycerides, diglycerides, monoglycerides, bio- diesel (methyl ester), methanol and glycerol by TG, DG, MG, BD, AL and GL respectively. BD can be produced by transesterification of triglycerides and methanol in the presence of alkaline catalyst. TG reacts with methanol to produce DG, which further reacts with AL to produce MG. Finally, MG reacts with AL to give GL as byprod- uct. At each reaction step, one molecule of methyl ester (BD) is pro- duced for each molecule of methanol consumed. The intermediate steps are shown by the following schematic diagram TG þ AL k 1 k 2 DG þ BD; DG þ AL k 3 k 4 MG þ BD; MG þ AL k 5 k 6 GL þ BD: ð2Þ Overall reaction: TG þ 3AL 3BD þ GL. Here k 1 ; k 3 and k 5 are forward reaction rates and k 2 ; k 4 and k 6 are backward reaction rates. The dependency of reaction rate con- stants on the temperature k i , where i ¼ 1—6, is expressed by the Arrhenius equation [18]: k i ¼ a i e Àb i T : ð3Þ T is the reaction temperature, a i is the frequency factor, and b i ¼ E a i R : in which Ea i is the activation energy for each component and R is the universal gas constant. Using the values of a i and b i [18],we obtain the values of k 1 to k 6 shown in Table 1. We denote the con- centration of TG, DG, MG, BD, AL and GL by x T ; x D ; x M ; x B ; x A and x G respectively. Based on the above assumptions, the differential equa- tions characterizing the stepwise reactions are as follows: 40 P.K. Roy et al. /Fuel 134 (2014) 39–44 dx B dt ¼ k 1 x T x A À k 2 x D x B þ k 3 x D x A À k 4 x M x B þ k 5 x M x A À k 6 x G x B þ k S x B 1 À x B B max ; dx T dt ¼Àk 1 x T x A þ k 2 x D x B ; dx D dt ¼ k 1 x T x A À k 2 x D x B À k 3 x D x A þ k 4 x M x B ; dx M dt ¼ k 3 x D x A þ k 4 x M x B À k 5 x M x A þ k 6 x G x B ; dx A dt ¼Àk 1 x T x A þ k 2 x D x B À k 3 x D x A þ k 4 x M x B À k 5 x M x A þ k 6 x G x B ; dx G dt ¼ k 5 x M x A þ k 6 x G x B : ð4Þ The last term, in expression dx B dt , represents only the mass transfer rate or indirectly the contribution of the mass transfer resistance to the overall resistance. 3. The optimal control problem Our main objective is to select the stirrer speed so that maxi- mum production of biodiesel can be obtained. Also it is our object to keep the cost function as measured in terms of time as low as possible. We use a control variable u ðtÞ, which represents the stir- ring activator input at a time t satisfying 0 6 uðtÞ 6 1. Here uðtÞ represents control input with values normalized to be between 0 and 1 [33]. Also uðtÞ¼1 represents the maximal use of stirrer and uðtÞ%0, which signifies no stirrer. Based on the above assumptions, our optimal control problem corresponding to Eq. (2) would be: dx B dt ¼ k 1 x T x A À k 2 x D x B þ k 3 x D x A À k 4 x M x B þ k 5 x M x A À k 6 x G x B þ k S x B 1 À x B B max uðtÞ; dx T dt ¼Àk 1 x T x A þ k 2 x D x B ; dx D dt ¼ k 1 x T x A À k 2 x D x B À k 3 x D x A þ k 4 x M x B ; dx M dt ¼ k 3 x D x A þ k 4 x M x B À k 5 x M x A þ k 6 x G x B ; dx A dt ¼Àk 1 x T x A þ k 2 x D x B À k 3 x D x A þ k 4 x M x B À k 5 x M x A þ k 6 x G x B ; dx G dt ¼ k 5 x M x A þ k 6 x G x B ; ð5Þ with initial conditions: x B ð0Þ¼x B 0 ; x T ð0Þ¼x T 0 ; x D ð0Þ¼x D 0 ; x M ð0Þ¼x M 0 ; x A ð0Þ¼x A 0 ; x G ð0Þ¼x G 0 . Here B max is the maximum BD production and k S is the mass transfer rate constant caused by stirrer rotation. The cost function is thus formulated as J½uðtÞ ¼ Z t f t 0 Pu 2 ðtÞÀQx 2 B ðtÞ hi dt: ð6Þ In words, maximizing BD production, and minimizing the cost of production are the main objectives of the present study. The param- eter Pð> 0Þ is the weight constant on the benefit of the cost produc- tion and Q is the penalty multiplier. Now, attempts have been made to find the optimal control u à such that Jðu à ޼minfJðuÞ : u 2 Ug, subject to the system (5). Where U is the admissible control set defined by U ¼fuðtÞ : uðtÞis measurable; 0 6 uðtÞ 6 1; t 2½t i ; t f g: Here Pontryagin Minimum Principle [34,35] has been used to find u à ðtÞ. The Hamiltonian is given by H ¼ Pu 2 ðtÞÀQx 2 B ðtÞ þ n 1 & k 1 x T x A À k 2 x D x B þ k 3 x D x A À k 4 x M x B þ k 5 x M x A À k 6 x G x B þk S x B 1 À x B B max uðtÞ ' þ n 2 fÀk 1 x T x A þ k 2 x D x B g þ n 3 fk 1 x T x A À k 2 x D x B À k 3 x D x A þ k 4 x M x B gþn 4 fk 3 x D x A þ k 4 x M x B À k 5 x M x A þ k 6 x G x B gþn 5 fÀðk 1 x T x A À k 2 x D x B þ k 3 x D x A À k 4 x M x B þ k 5 x M A À k 6 x G x B Þg þ n 6 fk 5 x M x A þ k 6 x G x B g; ð7Þ where n 1 ; n 2 ; n 3 ; n 4 ; n 5 and n 6 are adjoint variables. Theorem. If the given optimal control u à ðtÞ and the solution x à B ; x à T ; x à D ; x à M ; x à A ; x à G ÀÁ of the corresponding system (5) minimize JðuÞ over U, then there exists the adjoint variables n 1 ; n 2 ; n 3 ; n 4 ; n 5 and n 6 which satisfying the following equations dn 1 dt ¼À ÀQx B þ n 1 ðÀk 2 x D À k 6 x G À k 4 x M þ k S 1 À 2x B B max þ n 2 k 2 x D þn 3 ðÀk 2 x D þ k 4 x M Þn 4 ðÀk 4 x M þ k 6 x G Þþn 5 ðk 2 x D þ k 6 x G Þ þn 6 ðÀk 6 x G ÞÞ ! ; dn 2 dt ¼À½n 1 k 1 x A þ n 3 k 1 x A À n 5 k 1 x A ð8Þ dn 3 dt ¼ÀÀn 1 k 4 x B þ n 1 k 3 x A þ n 2 k 2 x B À n 3 k 2 x B À n 3 k 3 x A þ n 4 k 3 x A ½ þn 5 ðk 2 x B À k 3 x A Þ; dn 4 dt ¼ÀÀn 1 k 4 x B þ n 3 k 4 x B þ n 4 ðÀk 5 x A À k 4 x B Þþn 5 ðÀk 5 x A þ k 4 x B Þ;½ þn 6 k 5 x A ; dn 5 dt ¼Àn 1 ðk 1 x T þ k 3 x D þ k 5 x M ÞÀn 2 k 1 x T À n 3 k 3 x A þ n 4 k 3 x D À k 5 x M ðÞ½ þn 5 ðÀk 1 x T À k 3 x A À k 5 x M Þþn 6 k 5 x M ; dn 6 dt ¼ À½Àn 1 k 6 x B þ n 4 k 6 x B þ n 5 k 6 x B À n 6 k 6 x B ; along with the transversality condition n i ðt f Þ¼0 for i ¼ 1; 2; 3; 4; 5; 6. Further, u à ðtÞ is represented by u à ðtÞ¼max 0; min 1; k S x B 1 À x B B max ðÀn 1 Þ 2P 0 @ 1 A 0 @ 1 A : Proof. The Hamiltonian (7) can be written as: H ¼ Pu 2 ðtÞþn 1 k S x B 1 À x B B max uðtÞþterms without uðtÞ: ð9Þ According to the Pontryagin Minimum Principle, the unconstrained optimal control variables u à ðtÞ satisfies Table 1 Values of parameters used for models dynamics calculations [12,18] at 50 °C. Parameters Definition Value (mole À1 L min À1 ) k 1 Rate constant for forward reaction 0.0500 k 2 Rate constant for backward reaction 0.1099 k 3 Rate constant for forward reaction 0.1220 k 4 Rate constant for backward reaction 0.2147 k 5 Rate constant for forward reaction 0.2420 k 6 Rate constant for backward reaction 0.0070 P.K. Roy et al. /Fuel 134 (2014) 39–44 41 @H @u à ¼ 0: ð10Þ Thus from (9) and (10), we have @H @u à ¼ 2Pu à þ k S x B 1 À x B B max ðn 1 Þ¼0: Solving we get, u à ðtÞ¼ k S x B 1 À x B B max ðÀn 1 Þ 2P : ð11Þ Due to the boundedness of the standard control, u à ðtÞ¼ 0; k S x B 1À x B B max ðÞ ðÀn 1 Þ 2P 6 0; k S x B 1À x B B max ðÞ ðÀn 1 Þ 2P ; 0 < k S x B 1À x B B max ðÞ ðÀn 1 Þ 2P < 1; 1; k S x B 1À x B B max ðÞ ðÀn 1 Þ 2P P 1: 8 > > > > < > > > > : ð12Þ Hence the compact form of u à ðtÞ is u à ðtÞ¼max 0; min 1; k S x B 1 À x B B max ðÀn 1 Þ 2P 0 @ 1 A 0 @ 1 A : ð13Þ According to Pontryagin Minimum Principle [33], it can be written as follows: dn i dt ¼À @H @x i ; i ¼ 1; 2; 3; 4; 5; 6; ð14Þ where x i ðx B ; x T ; x D ; x M ; x A ; x G Þ and the necessary condition satisfy- ing the optimal control u à ðtÞ are Hðx i ðtÞ; u à ðtÞ; n i ðtÞ; tÞ¼min u2U ðHðx i ðtÞ; uðtÞ; n i ðtÞ; tÞÞ; i ¼ 1; 2; 3; 4; 5; 6: ð15Þ So, the adjoint equation corresponding to the system (6) and (7) are dn 1 dt ¼À @H @x B ; dn 2 dt ¼À @H @x T ; dn 3 dt ¼À @H @x D ; dn 4 dt ¼À @H @x M ; dn 5 dt ¼À @H @x A ; dn 6 dt ¼À @H @x G : Therefore, dn 1 dt ¼À ÀQx B þ n 1 ðÀk 2 x D À k 6 x G À k 4 x M þ k S 1 À 2x B B max þ n 2 k 2 x D þn 3 ðÀk 2 x D þ k 4 x M Þn 4 ðÀk 4 x M þ k 6 x G Þþn 5 ðk 2 x D þ k 6 x G Þ þn 6 ðÀk 6 x G ÞÞ ! ; dn 2 dt ¼À½n 1 k 1 x A þ n 3 k 1 x A À n 5 k 1 x A ð16Þ dn 3 dt ¼ÀÀn 1 k 4 x B þ n 1 k 3 x A þ n 2 k 2 x B À n 3 k 2 x B À n 3 k 3 x A þ n 4 k 3 x A ½ þn 5 ðk 2 x B À k 3 x A Þ; dn 4 dt ¼ÀÀn 1 k 4 x B þ n 3 k 4 x B þ n 4 ðÀk 5 x A À k 4 x B Þþn 5 ðÀk 5 x A þ k 4 x B Þ;½ þn 6 k 5 x A ; dn 5 dt ¼Àn 1 ðk 1 x T þ k 3 x D þ k 5 x M ÞÀn 2 k 1 x T À n 3 k 3 x A þ n 4 k 3 x D À k 5 x M ðÞ½ þn 5 ðÀk 1 x T À k 3 x A À k 5 x M Þþn 6 k 5 x M ; dn 6 dt ¼ À½Àn 1 k 6 x B þ n 4 k 6 x B þ n 5 k 6 x B À n 6 k 6 x B ; where n i ðt f Þ¼0; ði ¼ 1; 2; 3; 4; 5; 6Þ, are transversality conditions and x B ð0Þ¼x B 0 ; x T ð0Þ¼x T 0 ; x D ð0Þ¼x D 0 ; x M ð0Þ¼x M 0 ; x A ð0Þ¼x A 0 ; x G ð0Þ¼x G 0 are initial conditions. 4. Numerical simulation Numerically the model equations have been solved for better understanding of the dynamical behaviour of the transesterifica- tion reaction in the perspective of mass transfer kinetics. Pontrya- gin Minimum Principle is used for observing the effect of mixing intensity in the system dynamics with the help of Hamiltonian. The kinetics of the transesterification reaction has been analyzed using numerical methods in presence and absence of the control input. Here, mass transfer restriction is employed in order to have a better understanding of the effect of stirrer speed on biodiesel production. For better realization of its effect on the reaction kinet- ics, the variability of stirring is applied during a time period of first 60 min and two sets of concentration profiles are compared. At the start of the reaction, alcohol and Jatropha oil are in oppo- site phases due to their polar and non-polar nature respectively. This immiscibility of two reactants leads to a mass transfer resis- tance in the transesterification kinetics. It is evident from Fig. 1 that during the very early stage of the reaction, conversion rate for BD is slow at low mixing intensity. There is a delay or time lag for BD production due to immiscibility of opposite phases. Fig. 1 also reveals that delay time is decreasing as the process moves from lower to higher stirrer speed. This is due to the fact that increasing mixing intensity leads to higher to lower mass transfer limitations which ultimately decreases time lag and it has been observed that at least a stirrer speed of 600 rpm is required for minimizing delay for BD production at temperature of 10 °C. Subsequently, when time lag comes to an end, a constant production is obtained without mass transfer resistance. So production of BD through transesterification reaction depends importantly with mixing intensity. Apart from that, reac- tion kinetics also depends on other vital reaction parameters like temperature, molar ratio of reactants and catalyst concentration. Temperature is one of the most important deciding factors in this regard. In fact both reaction rate and mass transfer rate will vary with the change in temperature. But the change of mass transfer rate (diffusion coefficient of liquid) with the change of temperature will be negligible compared to the effect of temperature change on reaction rate. Hence Fig. 2 represents only the effect of tempera- ture on reaction rate for the conversion of biodiesel production at 600 rpm. In our analysis, the temperature dependency of BD for- mation is also studied and presented in Fig. 2. We see from Fig. 2 that increasing temperature for a fixed stirred speed increases the reaction rate and ultimately gives a constant production. This is due to the fact that increasing temperature at a definite mixing 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 Time (min) biodiesel (mol/L) 100 rpm 200 rpm 300 rpm 400 rpm 500 rpm 600 rpm 700 rpm Fig. 1. Effect of stirrer speed on conversion of biodiesel production at 10°C. 42 P.K. Roy et al. / Fuel 134 (2014) 39–44 intensity (600 rpm) increases the energy levels of molecules which results in more fruitful diffusion into continuous phase along with enhancing rate constant of the reaction. The improved solubility of Jatropha oil in alcohol at elevated temperature is also partially responsible for this behaviour. It reveals from Fig. 2 that at 600 rpm, increasing temperature to 50 °C rapidly increases the reaction rate. So this temperature is the acceptable temperature for optimum production of BD. Here, it may be mentioned that at 50 °C, the loss of alcohol due to vaporization would be negligible but at the same time it should be kept in mind that higher temper- ature should be avoided because of higher volatility of alcohol. Now, a comparative study has been made with regard to mixing intensity by employing three stirring speeds e. g. 300, 600 and 900 rpm for the reaction system keeping the reaction temperature at 50 °C and molar ratio of AL to TG 6:1 with an objective to iden- tify a suitable stirring speed for the transesterification reaction under these reaction conditions. It has been observed from Fig. 3 that at 300 rpm, the concentration of BD production is not satisfac- tory within stipulated time due to mass transfer limitations. Enhancement of stirrer speed up to 900 rpm increases the BD pro- duction initially but after 20.56 min of reaction, the production of BD is declining and concentration of BD falls undesirably with time. This is due to the fact that increasing stirrer speed enhances the volatility of alcohol which acts as a reactant as well as medium for the transesterification reaction. Vaporization of alcohol from the reaction system not only decreases the amount of one of the reactants but also hampers the proper ratio of alcohol to TG in the reaction mixture. This opposing effect prevents the formation of BD which is evident from the magnified version of little part of Fig. 3. So it can be predicted from this phenomenon that too low (300 rpm) or too high (900 rpm) mixing intensity is not desir- able for this type of reaction. It has also been observed from the fig- ure that at the stirrer speed of 600 rpm, the yield of BD is maximum at about 24.73 min and after that yield is decreasing. So a fixed application of mixing intensity (600 rpm) during a defi- nite time period for transesterification reaction requires control approach for optimal completion of the reaction. So, from this study, it can be said that though mixing intensity has a major influence on the BD production along with other kinetic parameters but controlling stirring speed during the reac- tion has a significant impact for optimizing desired product. From Fig. 4, it is clear that initially higher control ðu à ðtÞÞ on mechanical stirring is needed in transesterification reaction for minimizing mass transfer resistance as initially there is a heterogeneous mix- ture of two immiscible phases e. g. Jatropha oil and methanol. So these two reactants cannot be properly mixed or collide with each other until a higher stirrer condition is applied for dynamic diffu- sion. So increasing rate of mixing intensity from the start of the reaction is essential and after 2.653 min of reaction (evidenced from Fig. 4), it is observed that lesser control on stirring is required after a definite time of reaction. This can be explained based on the fact that the BD which is produced after 2.653 min can act as a medium for mixing of the two reactants. This BD actually works as a continuous phase for reactants and minimizes mass transfer resistance. This phenomenon helps to progress for further reaction with decreasing stirring speed. Finally, reaction system requires very low mixing intensity at nearly 60 min as during this time, reaction reaches at completion. After this stage, continued stirring should enhance the backward reaction between BD and GL which ultimately oppose optimality of the system kinetics. Fig. 5 pro- claimed parameter dependent reaction kinetics in a nutshell. Fig. 6 describes the comparative study for the concentration of BD with and without control approach on stirring dynamics. It is observed from Fig. 6 that formation of BD is improved by about 13.66% under this control operation. Initial higher mixing intensity is obtained by the higher stirring rate but applying control on stir- rer speed only produces BD where time and energy saving are pos- sible. So, it can be predicted that initial higher control density is 0 10 20 30 40 50 60 0 0.5 1 1.5 2 2.5 3 Time (min) Biodiesel (mol/L) 300 rpm 600 rpm 900 rpm 40 50 60 2.8 2.9 3 Time (min) Biodiesel (mol/L) Fig. 3. Prediction of the effect of the stirrer speed on the evaluation of BD production. 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 Time (min) Biodiesel (mol/L) 20 o C 30 o C 40 o C 50 o C 60 o C Fig. 2. Effect of temperature on conversion of biodiesel production at 600 rpm. 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 Reaction Time (min) Control Profile Control u*(t) Fig. 4. Controlling the effect of stirring is plotted as a function of time. 0 10 20 30 40 50 60 0 500 1000 1.5 2 2.5 3 3.5 Temperature ( o C) Stirrer speed (rpm) Biodiesel (mol/L) Fig. 5. Concentration of biodiesel for different temperatures and mixing intensities. P.K. Roy et al. / Fuel 134 (2014) 39–44 43 vital for a definite time period until the production of continuous phase and lesser control approach is needed afterwards for the proper characterization of BD production through transesterifica- tion reaction. 5. Discussion and conclusion In our current research article, a mathematical model of transe- sterification reaction for BD production has been developed with the help of some assumptions and exhibit that the progress of the reaction system depends significantly on mixing intensity and temperature. The study reveals that optimization of mechani- cal agitation and evaluation of mass transfer resistance is essential in the transesterification reaction for BD production. Hence the effects of impeller speed along with temperature are predominant in influencing the mass transfer determined rate of BD production process in batch reactor. By implying control approach on the stir- rer speed, it is seen that administering stirrer action on the system kinetics has a noteworthy impact for economic production of BD. In conclusion, it is suggested that the proposed model of BD production and its effect of control on stirring can be successfully applied to experimental cases also. Along with this, the numerical analysis offers a better understanding of optimal control theory on mass transfer limitations for the optimization of BD production. Thus the proposed model of transesterification is more functional and provides an idea of describing the effect of mass transfer resis- tance for faster rate of BD production. In this way, accurate prior prediction of system dynamics by analytical analysis and numeri- cal simulation would be realistic to experimental researchers. 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Roy et al. / Fuel 134 (2014) 39–44 . 0.1099 k 3 Rate constant for forward reaction 0.1220 k 4 Rate constant for backward reaction 0.2147 k 5 Rate constant for forward reaction 0.2420 k 6 Rate constant for backward reaction 0.0070 P .K. . n 4 ð k 5 x A À k 4 x B Þþn 5 ð k 5 x A þ k 4 x B Þ;½ þn 6 k 5 x A ; dn 5 dt ¼Àn 1 k 1 x T þ k 3 x D þ k 5 x M ÞÀn 2 k 1 x T À n 3 k 3 x A þ n 4 k 3 x D À k 5 x M ðÞ½ þn 5 ð k 1 x T À k 3 x A À k 5 x M Þþn 6 k 5 x M ; dn 6 dt ¼. n 3 k 4 x B þ n 4 ð k 5 x A À k 4 x B Þþn 5 ð k 5 x A þ k 4 x B Þ;½ þn 6 k 5 x A ; dn 5 dt ¼Àn 1 k 1 x T þ k 3 x D þ k 5 x M ÞÀn 2 k 1 x T À n 3 k 3 x A þ n 4 k 3 x D À k 5 x M ðÞ½ þn 5 ð k 1 x T À