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In several areas of engineering, it is possible to put real problems in mathematical functions; when we represent a problem with variables in the form of function, we were able to extract various information from it. This paper compared two different mathematical methods, being the finite difference method and the Fourth Order Range-Kutta method, to analyze the concentration and temperature of the water flow inside a tubular reactor. These results were compared with the analytical and experimental results of the problem, demonstrating that the Fourth Order Range-Kutta method was more advantageous than the finite difference method.

The Tubular Reactor describes chemical reactions in continuous flow systems, so that the main reactor variables, such as reactor dimensions, can be estimated [

A differential equation is called an equation in which the unknown is a function, and has a relation with the derivatives of this function [

In this context, this paper had the objective of comparing two mathematical methods and comparing them with the analytical and experimental solution.

For experimental analysis, a tubular glass reactor was used, with 10 thermocouples of type J to measure the temperature and 10 Capacitive density transmitters to measure the concentration. The reactor of approximately 100 mm internal diameter and 1000 mm in length can best be observed in

To do the numerical analysis of the experiment, we first have to consider the differential equation capable of confronting the resulting physical phenomena. A positioning scheme of the apparatus installed in the reactor can best be seen in

Consider a jacketed tubular reactor conducting a second order exothermic chemical reaction (2A → B). The diagram of this reactor can be better observed in the

The initial concentration of H_{2}O is 10 kmol/m^{3}. Assuming that the reactor

jacket temperature is constant, and that it does not lose heat to the environment, the mathematical model that describes the variation of reagent concentration A and the reactor internal temperature throughout the equipment is given by the following Equations ((1) and (2)) differentials successively:

d C A d x = − k 0 ⋅ e − E A R ⋅ T ⋅ C A 2 ⋅ A ⋅ ρ F ; C A 0 ( 0 ) = 10 kmol / m 3 (1)

d T d x = − U ⋅ 2 π ⋅ R R F ⋅ C p ⋅ ( T − T c ) + k 0 ⋅ e − E A R ⋅ T ⋅ C A 2 ⋅ A ⋅ ( − Δ H R ) F ⋅ C p (2)

As the fluid used in this study was water, recurrent water parameters were used at 295 K. These values can be better observed in

To solve numerically the system of differential equations 1 and 2, applying the Finite Differences Method and the Runge-Kutta Method, we have the following resolution equations:

d C A d x = − ( 50 ) ⋅ e − 30000 8.314 ⋅ T ⋅ C A 2 ⋅ π ⋅ ( 0.050 ) 2 ⋅ 1000 0.05 (3)

d T d x = − ( 100 ) ⋅ 2 π ⋅ ( 0.0127 ) 0.05 × 4187 ⋅ ( T − 315 ) + 50 ⋅ e − 30000 8.314 ⋅ T ⋅ C A 2 ⋅ π ⋅ ( 0.0127 ) 2 ⋅ [ − ( − 1 × 10 − 7 ) ] 0.05 × 4187 (4)

The method of finite differences is proposed by Equation (5):

f ( x 0 , u 0 ) = y 1 − y 0 h → y 1 = y 0 + h f ( x 0 , y 0 ) (5)

Variables | Values | Units |
---|---|---|

F | 0.05 | kg/s |

ρ | 1000 | kg/m^{3} |

R_{R} | 0.05 | m |

k_{0} | 50 | m^{3}/kmol∙s |

R | 8.314 | J/mol.K |

E_{A} | 30,000 | J/mol |

ΔH | −1 × 10^{−7} | J/mol |

U | 100 | J/m^{2}∙s∙K |

L | 1 | m |

Cp água | 4.187 | KJ/kg∙K |

T_{0} | 295 | K |

T_{c} | 315 | K |

Implementing Equations ((3) and (4)), which are the concentration and temperature equations, we obtain Equations ((6) and (7)) successively:

d C A d x = C A 1 − C A 0 h → C A 1 = C A 0 + h d C A d x (6)

d T d x = T 1 − T 0 h → T 1 = T 0 + h d T d x (7)

The fourth order Runge-Kuta method is defined by the following equations:

y 1 = y 0 + [ 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) ] h (8)

k 1 = f ( x 0 , y 0 ) (9)

k 2 = f ( x 0 + 1 2 h , y 0 + 1 2 h k 1 ) (10)

k 3 = f ( x 0 + 1 2 h , y 0 + 1 2 h k 2 ) (11)

k 3 = f ( x 0 + h , y 0 + h k 3 ) (12)

To perform these calculations, steps 10 cm, 5 cm and 1 cm were used to know which of these interactions converges faster to solve the system.

Applying the mathematical and experimental methods, a 10 cm step was initially used to perform the calculations. The results of concentration and temperature can be seen in

Analyzing

In the next analysis, the same parameters were compared with a step of 5 cm. These results can be seen in

Similar to

The concentration results of

approximate the experimental results. However, the Runge-kutta Method is now the one that gets values closer to reality, almost touching the same curve. In the results of temperature, the methods end up leaving the curve route at the end of the course of the reactor. And the curve that comes closest to reality is that of finite differences.

Analyzing the methods by looking at the graphs and calculations performed, we realize that the smaller the step used, the more precise the concentration and temperature values arrive, because with the big steps, the curves go out a little bit of reality and change the concentration drastically and increase the temperature beyond the final length.

The method that proved most efficient in wide steps was the finite difference method, and as the steps were narrowing the Runge-Kutta method began to show smoother and more accurate curves.

de Miranda, D.A., Cristofolini, R., Corazza, E.J., dos Santos, G.J. and do Amaral, C.E. (2018) Compa- rison of Mathematical Methods to Obtain Concentration and Temperature of New- tonian Fluids in Tubular Reactors. Open Access Library Journal, 5: e4329. https://doi.org/10.4236/oalib.1104329