Application of CFD Methods to the Upgrading of an Anaerobic Digester: the Case Study of the Ontinyent-agullent Wastewater Treatment Plant

Before realizing a description of fluid movement through mathematical tools in order to predict their evolution in time is necessary to assume a hypothesis of continuity.
We presume that the matter is a perfect continuous: under this assumption is possible to define the velocity field of any fluid particle which is function of the position x,y in a time t, the fluid density p(x,t) or the temperature T(x,y), as continuous and differentiable mathematical function.

Actually the fluctuations in molecular scales make wrong this hypothesis considering the smaller scales, but since we want a macroscopic description of the fluid, the observation of the fluid properties in each point are not affected by irregular variations present in the smaller scales. It means that if we focus on a fluid element, the physical properties, like the mass and momentum, are equally distributed throughout the volume.

We have two different approaches for the observation of a particle:

- Lagrangian point of view: with this approach each particle is individually followed: from a time t₀ the particle is identified with its position x₀(at the t₀ time). After a time t>0, it has a position x(x₀,t), velocity u = (∂x/∂t)x₀ and an acceleration a = (∂u/∂t)x₀. The sub index indicates that the derivative is referred only to this particle. This point of view usually is not used in fluid dynamics because it is impossible to follow entirely the particle along the movement.

- Eulerian point of view: with this approach it is considered the entire fluid, where are utilized global vectors and scalars in the whole domain of study: velocity vectors or temperature scalars are simultaneously solved over the whole domain, and no particle to particle as the Lagrangian point of view. This approach is the most used in this field of study.