International Journal of Engineering Mathematics

Volume 2015, Article ID 278275, 9 pages

http://dx.doi.org/10.1155/2015/278275

## Effects of Temperature and Stirring on Mass Transfer to Maximize Biodiesel Production from *Jatropha curcas* Oil: A Mathematical Study

Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India

Received 8 May 2015; Revised 29 September 2015; Accepted 12 October 2015

Academic Editor: Viktor Popov

Copyright © 2015 Fahad Al Basir and Priti Kumar Roy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Biodiesel, the most promising renewable and alternative energy, is produced through transesterification of vegetable oils. One of the most cost effective sources of biodiesel is *Jatropha curcas* oil. Transesterification of Jatropha oil depends significantly on reaction parameters such as reaction time, temperature, molar ratio, catalyst amount, and stirrer speed. Among these parameters temperature and stirring have noteworthy effect on mass transfer. In this research article, we have shown the simultaneous effect of temperature and stirring on mass transfer by considering a mathematical model. The optimal profiles of temperature and stirring are determined as a combined parameter, for which maximum biodiesel can be obtained. Further, we have shown that this pair exists and is unique for the optimality of the system.

#### 1. Introduction

Esters of vegetable oils are known as biodiesel and used as an alternative fuel for diesel [1]. Biodiesel is a nonpolluting, locally available, accessible, sustainable, and reliable fuel obtained from renewable sources such as vegetable oils or animal fats by transesterification [2]. An alternative way of biodiesel synthesis is produced from vegetable oils.* Jatropha* oil is one of such nonedible oils, which has an estimated annual production potential of 200 thousand metric tons in India and it contains high amount of oil, that is, triglycerides, and the produced biodiesel has a similar property as that of diesel [3].

Transesterification or alcoholysis is adopted to convert* Jatropha curcas* oil, to biodiesel. Two types of transesterification have been developed and analysed for biodiesel production using either chemical catalyst or biological catalyst [4–7]. There are ample research articles where alkaline transesterification for biodiesel production is investigated [8–10]. Raw materials with a high water or free fatty acid content need pretreatment with an acidic catalyst in order to esterify FFA [11, 12]. It is evident that reaction parameters such as molar ratio between alcohol and oil, reaction time, catalyst concentration, and reaction temperature influenced the transesterification process [13–15].

Triglycerides are immiscible with alcohol due to polar and nonpolar nature of alcohol and oil, respectively. Thus, biodiesel production process suffers initial mass transfer limitations problem [16]. This problem can be avoided by applying stirring on the system. Roy et al. [4, 17] successfully observed the effect of stirring on transesterification by formulating a mathematical model. Noureddini and Zhu proposed an initial mass transfer controlled region followed by a kinetically controlled region for base catalytic transesterification of sunflower oil [18]. Hou et al. showed that the reaction is very slow initially due to mass transfer limitations between methanol and oil phase [19]. Peterson et al. [20] have studied the effect of stirrer speed on the transesterification of vegetable oil with alcohol. Rashid et al. also observed the stirring intensity of the transesterification of cotton seed oil using several catalyst under experimental setup and analytical chemistry of fuel properties [21]. Sharma et al. have observed that beyond an optimum stirrer speed production level decreases [22].

In a previous study, influence of temperature on mass transfer is investigated in case of biodiesel production from* Jatropha curcas* oil [23]. It is observed that, with an increase in reaction temperature, conversion of oil increased significantly; that is, mass transfer has been increased [5, 12, 24]. But after a certain level of temperature (50°C), biodiesel yield decreases [25, 26]. It was shown by many researchers that mass transfer is highly dependent on temperature. At fixed stirring, mass transfer rate is directly proportional to its temperature profile, that is, high mass transfer at high temperature [6, 27].

Hence, mass transfer limitation is one of the most deciding factors for optimum production of biodiesel. Optimization of mechanical agitation and temperature for the evaluation of the mass transfer resistance is essential in transesterification of* Jatropha* oil. In this research article, we have considered a mathematical model [17] for biodiesel production and unveil the combined effect of temperature and stirring on mass transfer in different phases of biodiesel production. Using mathematical control theory, optimal profiles for temperature and stirring are derived. Our aim is to minimize mass transfer limitation to get maximum biodiesel production and make the process cost effective. We have also shown the existence and uniqueness of control variables for the optimal system. Analytical results are shown numerically using Matlab.

#### 2. The Mathematical Model

In this research article, we have attenuated two key parameters of our model system, which has been studied in detail in our previous works [17, 23]. We have chosen temperature and stirring as a combined parameter and study the dynamical behaviour impacted on biodiesel production due to different stirring as well as fluctuations of reaction temperature. The base model is constructed from the previous works [17, 23] and here we assume the same reaction mechanism as described in previous work. For a detailed discussion of the model assumptions one may refer to our earlier works [17, 23].

Biodiesel (BD) can be produced by reacting* Jatropha* oil (i.e., triglycerides (TG)) with methanol (AL). The reactions happen in three consecutive reversible steps. During the course of reaction of triglycerides and methanol, some intermediates (diglyceride (DG) and monoglyceride (MG)) are formed. The schematic explanation of the reaction is given by We denote the concentration of biodiesel, triglycerides, diglycerides, monoglycerides, methanol (alcohol), and glycerol by , , , , , and , respectively. We have the following set of rate equations as described in [17, 23]:with initial conditions,Here, , , and are forward and , , and are backward reaction rates. The dependency of reaction rate constants on the temperature, to , is expressed by the Arrhenius equation [26]: is the reaction temperature, is the frequency factor, and in which is the activation energy for each component and is the universal gas constant. Using the values of and [26], we obtain the values of . is the reaction temperature, and are constants, and their values are given in Table 1. We use as the mass transfer rate constant due to stirring and it is defined as [17, 23] where is the speed of stirrer and , , and are constants. The term is used in our model by the expression . Here, represents maximum biodiesel production in an ideal situation, which is defined as a system having no mass transfer resistance.