1. HONR 297 Environmental Models Chapter 3: Air Quality Modeling 3.6: Two-Dimensional Diffusion

2. Two-Dimensional Diffusion Suppose we are in a situation where the concentration of diffusing material is a function of time t and position (x, y) in a two-dimensional coordinate system. As shown at right, a physical situation for which this may be an appropriate model is a point source in a room with a very low ceiling (or a source that is a column in the middle of a room – see Hadlock p. 65, Figure 3.4). 2 Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

3. Two-Dimensional Diffusion Just as with one-dimensional diffusion, we’d expect ◦ The material will diffuse outwards from the center. ◦ The center concentration will decrease. ◦ The concentration at points farther away from the center will gradually increase – it will be higher closer to the center, at least in the beginning. ◦ The concentration will eventually diffuse to zero. 3

4. Two-dimensional Diffusion Processes Recall that diffusion is caused by kinetic energy of individual molecules which causes them to move in random directions until they are more evenly distributed. This leads to lowering of a concentration in areas of higher concentration and raising of concentration in areas of lower concentration, at a rate proportional to the difference in concentration. 4

5. Two-dimensional Diffusion Processes There may also be other processes at work that cause diffusion – for example, the substrate may itself be agitated or mixed in some fashion (such as air currents or shaking of the substrate). For this reason, there may be a difference in the diffusion rate in different directions (i.e. the x- and y- direction). This can be taken into account by choosing different diffusion coefficients in a mathematical model! 5

6. Two-Dimensional Diffusion Equation To take into account diffusion in two dimensions, we modify the one-dimensional diffusion equation! Equation (2) is known as the two-dimensional diffusion equation! The main differences are that there are two diffusion coefficients and concentration units are [C] = mass/length^2 or weight/length^2. 6

7. Working with the Two-Dimensional Diffusion Equation Using Excel, try problems 1 – 3 on p. 85 of Hadlock. Using Mathematica, plot the concentration C(x, y, t) at the times t = 1, 2, …, 10 minutes for -3 < x < 3 and -3 < y < 3 (units are in meters for x and y). 7

8. Hadlock – p. 85 #1 x – direction is east/west y – direction is north/south D1 = 0.3 m^2/min D2 = 0.9 m^2/min M = 800 kg C(15, 20, 10) = 800/(4*pi*10*sqrt(0.3*0.9)) *exp(-(15^2)/(4*0.3*10)) *exp(-(20^2)/(4*0.9*10) )) = 1.32 x 10^(-12) kg/m^2 8

9. Hadlock – p. 85 # 2 9

10. Hadlock – p. 85 # 3 10

11. C(x, y, t) on [-3,3] x [-3,3] at fixed times! 11

12. Resources Charles Hadlock, Mathematical Modeling in the Environment – Chapter 3, Section 6 12