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HONR 297 Environmental Models

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

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HONR 297 Environmental Models

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  1. HONR 297Environmental 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). 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.

  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.

  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!

  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.

  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).

  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 • C = 800 kg • C(15, 20, 10) = 800/(4*pi*10*sqrt(0.3*0.9)) *exp(-(15^2)/(4*0.3*10^2)) *exp(-(20^2)/(4*0.9*10^2) )) = 1.32 x 10^(-12) kg/m^2

  9. Hadlock – p. 85 # 2

  10. Hadlock – p. 85 # 3

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

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

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