HONR 297 Environmental Models Chapter 3: Air Quality Modeling 3.5: One-Dimensional Diffusion

Diffusion in a Long Tube Suppose we inject a total mass M of dispersible material into an infinitely long tube at the point x = 0, at time t = 0. 2 Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

Diffusion in a Long Tube Note that for this situation, we are assuming the tube is infinitely long to avoid the issue of modeling a tube with ends that are blocked off. This works fine if the tube is very long relative to the distance scale used. It also works well for outdoor air pollution problems! 3

Diffusion in a Long Tube Question: What will be the concentration of material at any point at any time after the moment of initial injection? 4

Expected Behavior … Intuitively, what should we expect? ◦ The material will diffuse to the left and right. ◦ 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. 5

One-Dimensional Diffusion Equation To quantify the above qualitative description of what will happen, we can use the following model equation. Equation (1) is known as the one- dimensional diffusion equation! 6

One-Dimensional Diffusion Equation In this equation, ◦ C is the concentration of contaminant, with units [C] = mass/length or weight/length (for example gm/cm or lb/ft), since we are looking at one- dimensional diffusion! ◦ M is the amount of material injected instantly into the center of the tube, with units [M] = mass or weight (for example gm or lb). ◦ x is the position along the tube, with units [x] = length (for example meters or feet). ◦ t is the time after the contaminant is initially released (release time is t = 0), with units [t] = time (for example seconds or hours). 7

One-Dimensional Diffusion Equation Finally, ◦ D is the diffusion coefficient, with units [D] = (length^2)/time (check that this makes sense in equation (1) dimensionally). ◦ Example units might be (m^2)/sec or (ft^2)/hour. ◦ D gives an indicator of how fast (or slow) the diffusing material moves through the substrate (underlying medium) in the tube. 8

One-Dimensional Diffusion Equation Values for D depend both on the diffusing material and the substrate. For example, think of perfume in air vs. sugar in water vs. ink in gel – which will have a higher diffusion coefficient? In general, we will need to know D for a given situation to compute C! 9

Computing Concentration Question: If we know diffusion coefficient D, how can we evaluate the concentration C at any position x, at any time t > 0? Answer: Use equation (1) with a choice of x and t. For example, if D = 0.05 (cm^2)/sec, M = 10 gm, what is C at x = 2 cm, when t = 5 sec? 10

Computing Concentration Another Question: Suppose we want to know the concentration at one hundred different points, for every minute, for one hour after initial release. Would this be “fun” to do by hand? Answer: NO! In order to compute many values of C, it is better to use a computer program, such as Excel or Mathematica. 11

Concentration via Excel Using Excel, for the example above, find concentration levels C at positions x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 cm and times t = 1, 10, 20, 30, 40, 50, 75, 100 sec. Then plot the concentration at fixed time t = 10 sec and fixed position x = 2 cm. How could we plot concentration as a function of both position and time? One way is via Mathematica! 12

Excel Table for Concentration 13

Concentration at t = 10 sec 14

Concentration at x = 2 cm 15

Concentration C(x,t) 16

Resources Charles Hadlock, Mathematical Modeling in the Environment – Chapter 3, Section 5 17