1. Advection – Diffusion Equation
Lecture 15
Advection – Diffusion Equation

2. Objectivos
This chapter aims to present resolution methods for the Advection- Diffusion Equation in a 1D system and its application to predict the evolution of properties in a River involving heat and/or mass transfer.
The Chapter gives continuity to the problem of diffusion solved in Fluid Mechanics using knowledge about heat transfer acquired already in TEM.
It uses a VBA code and is an introduction to the work to be done in the Environmental Modelling course.

3. Program of this Chapter
Reassessment of the finite-volume approach to quantify the conservation principle “The rate of accumulation inside a control volume is equal to the entrance minus the leaving rates, plus production minus consumption”.
Assessment of the numerical difficulties of advection. Upwind, central differences and “Quick” methods for advection. Numerical diffusion.
Time discretization and stability: Explicit, implicit and semi-implicit methods (Crank-Nicholson). Courant and Diffusion numbers.
Deduction of the algebraic equations from differential equations, using Taylor series. Accuracy and truncation error.

4. Fluxes
The Flux of a Property is the amount of property that crosses a Area per unit of time.
The Property flows by Advection when the average velocity of the molecules along that Area is not null, i.e. when there is net velocity
The property flows by diffusion if there is interaction between fluid parcels (molecules or eddies) at a scale not described by the velocity.

5. Rate of Accumulation
When the time interval tends to zero, one gets:

6. Constant volume In case of Heat:
The rate of mass accumulation per unit of volume:
In case of Heat:
The rate of accumulation per unit of volume:

7. Fluxes Advective: How much volume of fluid is crossing the area?
Why the internal product?
Why the integral?
Is this what is flowing in minus what is flowing out?

8. Fluxes Diffusive: How much volume of fluid is crossing the area?
Why the internal product?
Why the integral?
Is this what is flowing in minus what is flowing out?

9. The Conservation Equation
If the volume is small and does not change in time:

10. Advection
Dividing by the volume:

11. Differential Equation