Modelo Del Termino De Reacción De La Ec. Advección-Difusión-Reacción

MODELLING OF REACTION TERM OF THE ADVECTION-DIFFUSION-REACTION EQUATION FOR ENVIRONMENTAL PROBLEMS.
Introduction
The Advection Diffusion Reaction Equation (ADRE), ∂φ ∂φ ∂ ⎛ ∂φ ⎞ ⎜ ⎟ + Γφ , +Ui = KT (1) ∂t ∂xi ∂xi ⎜ ∂xi ⎟ ⎝ ⎠ where KT is the transference coefficient and Гφ is the reaction term, is not only used to evaluate the hydrodynamics (velocity vectors) of a system, it’s also used tocalculate scalars, such as Temperature, Salinity, Concentration of a Pollutant, Population Dynamics, Equilibrium Variables of the system (BOD, OD), etc. The main difference in the treatment of the reaction term in the ADRE is that it’s represented by an Ordinary Differential Equation (ODE). Since the methods to solve the ODE’s are different from those to solve the former parts of the ADRE, a new chooseof method should be made.

Problems of interest
1. Population Dynamics.
The choose model for this dynamics is the Lotka-Volterra model, which consists in differential equations that are used to model predator-prey interactions. The system considers of two entities. The Lotka-Volterra equations for the two dimensional system with exponential growth is defined by the following differentialequations: ⎧ x = Ax − Bxy = x( A − By ) F ( x(t ), y (t )) = ⎨ (2) ⎩ y = −Cy + Dxy = y (−C + Dx) Where x is the prey population, y is the predator population, A, B, C, and D are positive constants representing the growth and decay rate, and the interspecies interactions given in day-1. In order to extend the two entities system above into a three entities system, a third specie is added to the system.In this case the third specie feeds exclusively on the first predator population. The three species Lotka-Volterra system is defined by:

⎧ x = Ax − Bxy = x( A − By ) ⎪ F (x(t ), y (t )) = ⎨ y = −Cy + Dxy − Eyz = y (−C + Dx − Ez ) (3) ⎪ z = − Fz + Gzy = z (− F + Gy ) ⎩ Where z is the new predator population, E, F and G are positive constants representing decay rate and the new interspeciesinteractions given in day-1.

2. Model for BOD – DO.
For water quality modeling these Biological Oxygen Demand (BOD) and Dissolved Oxygen (DO) are variables of mean importance. Nearly all water quality models characterize BOD decay with first order kinetics represented by:
d [BOD ] = − k [BOD ] dt Where k is the first order decay rate (day-1). Streeter and Phelps developed the kinetics of DO as:A) Deoxygenation: d [DO ] = − k1 [DBO ] dt

(8)

(9)

B) Reareation: d [DO ] = k 2 [C s − C ] dt Where Cs is the concentration of saturation of DO and C is the actual DO.

(10)

From equations 9 and 10 in terms of oxygen deficit (D) the resultant equation is: dD (11) = k1 [DBO ] − k 2 [D ] dt The analytical solutions for these equations are:

[DBO] = [DBO]0 e − kt k [DBO ]0 − k t (e −e −k t )+ [D ] e −k t D= 1
i

(12)

k 2 − k1

1

2

0

2

3. Consecutive Reactions.
In many cases the pollutants in ecosystems are reactive compounds that with themselves, or with other pollutants, or with the system compounds. In order to make a closure to this problem, a system of ODE’s is developed for a pollutant that reacts twice. The system can be set as: (4) A ⎯k1 B ⎯k 2 C⎯→ ⎯→ Where A, B, and C are the concentration of the compound, k1 and k2 are the constant kinetics coefficient given in hr-1. An example of this kind of reactions nitrification:
− NH 3 − N ⎯OXD → NO 2 − N ⎯OXD → NO3− − N ⎯ ⎯ ⎯ ⎯

(5)

For this kind of reactions the equation system can be set as:

d [A] = − k1 [A] dt d [B ] = k1 [A] − k 2 [B ] dt d [C ] = k 2 [C ] dt

(6)

Which analyticalsolutions are:

[A] = [A]0 e − k t [B ] = [B ]0 e −k t + [A]0 k1 (e −k t − e −k t )
1
2

⎛ ⎞ [C ] = [A]0 + [B]0 + [C ]0 − [B]0 e −k t + ⎜ [A]0 ⎟(k1e −k t − k 2 e −k t ) ⎜ ⎟
2 2 1

k 2 − k1

1

2

(7)

⎝ k 2 − k1 ⎠

Where [A]0, [B]0, and [C]0 are the initial concentrations. In order to demonstrate the preservation of mass, the total mass of the system is calculated in all...