Chapter 7 Reaction Advection Dispersion First Order Reactions

Chapter 7 Reaction – Advection – Dispersion First Order Reactions

First Order Degradation Reactions �Advection-Dispersion Equation with degradation �Or in 1 d �Guess what – use Fourier transforms to study infinite domain solution…

�In Fourier space Same as conservative �Therefore, �Greens in real space Function? ? Random Walks?

Random Walks � Update position in space and time as before � But now, during a time step a particle can die with Probability Is this consistent – say if g=0 � How to implement numerically?

Steady State Plumes � One of the interesting features of degrading systems is that they allow for steady state plumes (that is the system can reach a point where you are putting in the same as is dying at a given time) Steady State We have a continuous source of concentration C 0 being released from x=0 that advects, disperses and degrades as it moves downstream (sound familiar to anyone – Arial? ) � How to solve (can’t formally use Fourier on semi-infinite domain) �

Let’s start easy – set D=0 How to solve?

Let’s start easy – set D=0 Integrate and impose BC at x=0 What does this mean and does it make sense to you?

Ok – now other extreme – v=0 How to solve now? Can’t just integrate as we did before, but look at the equation carefully. What function when you differentiate it twice comes back with the same structure? Also 2 nd order ODE – which means two solutions

Ok – now other extreme – v=0 Find A 1 and A 2 from boundary conditions – use your physical reasoning also

Now keep v and D Now what? Use what you learned from both cases above to look for an appropriate solution here i. e. use the solutions to guess an appropriate solution structure

Now keep v and D Assume and substitute into ODE What is a? Solve the quadratic

Now keep v and D Assume and substitute into ODE Again, use BCs to fix A 1 and A 2

By the way �How might you know when you can neglect diffusion and when you can neglect advection?