Biodiesel Production in Stirred Tank Chemical Reactors: A Numerical Simulation

Abstract

The biodiesel production was performed in stirred tank chemical reactor by numerical simulation. The main results are that the percentage of conversion from triglyceride to biodiesel is approximately of 82 % when the molar flow ratio between triglyceride/alcohol is 1:5. This system displays only one equilibrium point. Since there are imaginary eigenvalues in the Jacobian matrix analysis, the equilibrium point is unstable. The biodiesel production in stirred tank chemical reactor is good because the settling time is short, and has higher conversion.

Appendix

The particular model for the case study is governed by the next set of ordinary differential equations.

$$ \begin{array}{l}\frac{d{x}_1}{dt}={\gamma}_1-\theta {x}_1-{\beta}_1{x}_1{x}_2+{\beta}_2{x}_3{x}_4\\ {}\frac{d{x}_2}{dt}={\gamma}_2-\theta {x}_2-{\beta}_1{x}_1{x}_2+{\beta}_2{x}_3{x}_4\\ {}\kern2.4em -{\beta}_3{x}_3{x}_2+{\beta}_4{x}_5{x}_6-{\beta}_5{x}_5{x}_2+{\beta}_6{x}_7{x}_8\\ {}\frac{d{x}_3}{dt}={\gamma}_3-\theta {x}_3+{\beta}_1{x}_1{x}_2-{\beta}_2{x}_3{x}_4\\ {}\kern2.4em +{\beta}_3{x}_3{x}_2-{\beta}_4{x}_5{x}_6\kern1.56em \\ {}\frac{d{x}_4}{dt}={\gamma}_4-\theta {x}_5+{\beta}_1{x}_1{x}_2-{\beta}_2{x}_3{x}_4\\ {}\frac{d{x}_5}{dt}={\gamma}_5-\theta {x}_5+{\beta}_3{x}_3{x}_2-{\beta}_4{x}_5{x}_6\\ {}\kern2.4em +{\beta}_5{x}_5{x}_2-{\beta}_6{x}_7{x}_8\\ {}\frac{d{x}_6}{dt}={\gamma}_6-\theta {x}_6+{\beta}_3{x}_3{x}_2-{\beta}_4{x}_5{x}_6\\ {}\frac{d{x}_7}{dt}={\gamma}_7-\theta {x}_7+{\beta}_5{x}_5{x}_2-{\beta}_6{x}_7{x}_8\\ {}\frac{d{x}_8}{dt}={\gamma}_8-\theta {x}_8+{\beta}_5{x}_5{x}_2-{\beta}_6{x}_7{x}_8\\ {}\frac{d{x}_9}{dt}={\gamma}_9-\theta {x}_9+\varDelta {H}_{rxn}\Big[{\beta}_1{x}_1{x}_2-{\beta}_2{x}_3{x}_4\;\\ {}\kern2.4em +{\beta}_3{x}_3{x}_2-{\beta}_4{x}_5{x}_6+{\beta}_5{x}_5{x}_2-{\beta}_6{x}_7{x}_8\Big]+u\left({T}_c-{x}_9\right)\end{array} $$

(6)

Where: β _i  = k ^o _i  exp(−E _A, i /Rx ₉) and γ _i  = θx _i,0

Using the expansion Taylor series the dynamic system governing by (6) can be represented by (7).

$$ \begin{array}{l}\frac{d\underset{\sim }{x}}{dt}=\underset{\sim }{f}\\ {}\underset{\sim }{f}=\underset{\sim }{f}{\underset{\sim }{x}}_e+{\left.A\right|}_{{\underset{\sim }{x}}_e}\left[\underset{\sim }{x}-{\underset{\sim }{x}}_e\right]\end{array} $$

Reordering the before equation

$$ \frac{d\underset{\sim }{x}}{dt}=\underset{\underset{\sim }{A}{\underset{\sim }{x}}_e}{\underbrace{\left.\left[\begin{array}{ccc}\hfill \frac{\partial {f}_1}{\partial {x}_1}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial {f}_1}{\partial {x}_9}\hfill \\ {}\vdots & \ddots & \vdots \\ {}\hfill \frac{\partial {f}_9}{\partial {x}_1}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial {f}_9}{\partial {x}_9}\hfill \end{array}\right]\right|}}\underset{\underset{\sim }{x}}{\underbrace{\left[\begin{array}{c}\hfill {x}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {x}_9\hfill \end{array}\right]}}+\underset{B}{\underbrace{{\underset{\sim }{f}}_{{\underset{\sim }{x}}_e}+{\left.\underset{\sim }{A}\right|}_{{\underset{\sim }{x}}_e}\underset{{\underset{\sim }{x}}_e}{\underbrace{\left[\begin{array}{c}\hfill {x}_{1,e}\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {x}_{9,e}\hfill \end{array}\right]}}}} $$

(7)

In deviation variables $\underset{\sim }{\xi }=\underset{\sim }{x}-{\underset{\sim }{x}}_e$ (7) have the next form:

$$ \frac{d\underset{\sim }{\xi }}{dt}={\left.\underset{\sim }{A}\right|}_{{\underset{\sim }{x}}_e}\underset{\sim }{\xi }+{\underset{\sim }{f}}_{{\underset{\sim }{x}}_e} $$

(8)

For the case study the function Jacobian matrix values are:

$$ \underset{\sim }{f}{\underset{\sim }{x}}_e=\left[\begin{array}{c}\hfill -19.43\hfill \\ {}\hfill -166.33\hfill \\ {}\hfill -27.48\hfill \\ {}\hfill 19.44\hfill \\ {}\hfill -68.06\hfill \\ {}\hfill 46.93\hfill \\ {}\hfill 99.97\hfill \\ {}\hfill 99.97\hfill \\ {}\hfill -194461.33\hfill \end{array}\right]\times {10}^{-15}\cong \underset{\sim }{0} $$

(9)

$$ \begin{array}{c}{\left.\underset{\sim }{A}\right|}_{{\underset{\sim }{x}}_e}=\left[\begin{array}{ccccccccc}\hfill -872.98\times {10}^{-3}\hfill & \hfill -45.97\times {10}^{-3}\hfill & \hfill -17.29\times {10}^{-3}\hfill & \hfill -927.18\times {10}^{-6}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\times {10}^{-6}\hfill \\ {}\hfill -692.98\times {10}^{-3}\hfill & \hfill -06.97\times {10}^{-3}\hfill & \hfill -3.15\hfill & \hfill 927.18\times {10}^{-6}\hfill & \hfill -1.97\hfill & \hfill 26.09\times {10}^{-3}\hfill & \hfill 61.28\times {10}^{-3}\hfill & \hfill 61.28\times {10}^{-3}\hfill & \hfill -1.23\times {10}^{-6}\hfill \\ {}\hfill 692.98\times {10}^{-3}\hfill & \hfill 97.16\times {10}^{-3}\hfill & \hfill 3.12\hfill & \hfill -927.18\times {10}^{-6}\hfill & \hfill -283.93\times {10}^{-3}\hfill & \hfill -26.09\times {10}^{-3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1.95\times {10}^{-9}\hfill \\ {}\hfill 692.98\times {10}^{-3}\hfill & \hfill 3.77\times {10}^{-3}\hfill & \hfill -17.29\times {10}^{-3}\hfill & \hfill -927.18\times {10}^{-6}\hfill & \hfill -150\times {10}^{-3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\times {10}^{-6}\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -3.14\hfill & \hfill 0\hfill & \hfill 1.82\hfill & \hfill -26.09\times {10}^{-3}\hfill & \hfill -61.28\times {10}^{-3}\hfill & \hfill -61.28\times {10}^{-3}\hfill & \hfill 198.43\times {10}^{-9}\hfill \\ {}\hfill 0\hfill & \hfill 51.19\times {10}^{-3}\hfill & \hfill 3.14\hfill & \hfill 0\hfill & \hfill -283.93\times {10}^{-3}\hfill & \hfill -26.09\times {10}^{-3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 957.55\times {10}^{-3}\hfill \\ {}\hfill 0\hfill & \hfill 59.82\times {10}^{-3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 59.82\times {10}^{-3}\hfill & \hfill 0\hfill & \hfill -61.28\times {10}^{-3}\hfill & \hfill -61.28\times {10}^{-3}\hfill & \hfill 960.03\times {10}^{-3}\hfill \\ {}\hfill 0\hfill & \hfill 59.82\times {10}^{-3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2.26\hfill & \hfill 0\hfill & \hfill -61.28\times {10}^{-3}\hfill & \hfill -61.28\times {10}^{-3}\hfill & \hfill 960.03\times {10}^{-3}\hfill \\ {}\hfill -839.72\hfill & \hfill -190.22\hfill & \hfill -3.78\times {10}^3\hfill & \hfill 1.12\hfill & \hfill -2.39\times {10}^{-3}\hfill & \hfill 31\hfill & \hfill 74.26\hfill & \hfill 74.26\hfill & \hfill. \mathrm{5.15}\hfill \end{array}\right]\\ {}\\ {}\end{array} $$

(10)

The eigenvalues of Jacobian matrix can be find by (11).

$$ \det \left({\left.\underset{\sim }{A}\right|}_{{\underset{\sim }{x}}_e}-\lambda \underset{\sim }{I}\right)=0 $$

(11)

The eigenvalues are displays in (12).

$$ \underset{\sim }{\lambda }=\left[\begin{array}{c}\hfill -5.15\hfill \\ {}\hfill 3.57\hfill \\ {}\hfill 481.58\times {10}^{-3} + 2.44\mathrm{i}\hfill \\ {}\hfill 481.58\times {10}^{-3} - 2.44\mathrm{i}\hfill \\ {}\hfill -834.84\times {10}^{-3}\hfill \\ {}\hfill -59.79\times {10}^{-3}\hfill \\ {}\hfill -1.082\times {10}^{-3}\ \hfill \\ {}\hfill -516.2\times {10}^{-6}\hfill \\ {}\hfill 99.54\times {10}^{-18}\ \hfill \end{array}\right] $$

(12)