1.061 | Fall 2008 | Undergraduate

# Transport Processes in the Environment

## Lecture Notes

TOPICS ANIMATIONS EXAMPLE PROBLEMS
1. Conceptual model of diffusion (PDF) Random walk animation

Problems (PDF)

Solutions (PDF)

2. Conservation of mass (PDF) Total derivative animations

Problems (PDF)

Solutions (PDF)

3. Diffusion of an instantaneous point release (PDF) Animation of anisotropic diffusion of a point release

Problems (PDF)

Solutions (PDF)

4. Boundary conditions (PDF) Animation comparing no-flux and absorbing boundaries

Problems (PDF)

Solutions (PDF)

5. Advection and diffusion of an instantaneous release (PDF) Animation showing simultaneous advection and diffusion

Problems (PDF)

Solutions (PDF)

6. Continuous point source (PDF) Animation of a continuous source discharging into a uniform flow

Problems (PDF)

Solutions (PDF)

7. Velocity profiles and turbulence (PDF)

Problems (PDF)

Solutions (PDF)

8. Dispersion processes (PDF)

Problems (PDF)

Solutions (PDF)

9. Reaction and interfacial exchange (PDF)

Problems (PDF)

Solutions (PDF)

10. Transport of particles (PDF)

Problems (PDF)

Solutions (PDF)

The animation is generated with MATLAB software.

Directions: click on the image to begin the animation.

### Random Walk Animation

Diffusion animation.

This animation shows the motion of 500 particles in a one-dimensional random walk with step size X = 1 in time t = 1. This random walk animation mimics the effect of Fickian Diffusion. For some background and thought problems pertaining to this animation, review the theory (page 4) behind this conceptual model of diffusion.

* Note to Unix users: the animation opens up behind active windows.

Please note that animations A and B use an older Indeo video compression not supported by most multimedia players. If you have problems playing these files, we recommend using the VLC multimedia player.

The animations were generated with MATLAB software.

Directions: Click on the images to begin the animation.

### A) Illustration of Total Derivative: Steady Temperature Field

Illustration of total derivative: steady temperature field.

This animation shows a one-dimensional system with a spatial gradient of temperature, T(x). A temperature probe (white dot) moves with the flow, making a Lagrangian observation. The probe records the material (total) derivative. Go to the theory section (page 6) for some background material on the concept of the total derivative.

* Note to Unix users: the animation opens up behind active windows

### B) Illustration of Total Derivative: Unsteady Temperature Field

Illustration of total derivative: unsteady temperature field.

This animation shows a one-dimensional system with an unsteady temperature field, T(x,t). A temperature probe is moving through the flow at velocity u, and it records the material derivative. Three additional probes are located at the fixed positions A, B, and C. Go to the theory section (page 6) for some background material on the concept of the total derivative.

* Note to Unix users: the animation opens up behind active windows

### C) Steady, Spatially Accelerating Flow in a Pipe

Steady, spatially accelerating flow in a pipe.

In this animation, flow through a pipe accelerates downstream as the pipe cross-section decreases. The velocity and displacement of a probe moving with the flow are shown versus time. For some background and thought problems pertaining to this animation, review the theory (page 6) behind spatially accelerating flows.

* Note to Unix users: the animation opens up behind active windows.

The animations were generated with MATLAB software.

Directions: Click on the images to begin the animation.

Anisotropic diffusion in two dimensions output of MATLAB software.

This animation depicts the diffusion of a discrete mass released at (x = 0, y = 0, t = 0). The diffusion is anisotropic, Dx = 4 Dy. The length scales grow in proportion to the square root of the diffusion, such that the dimensions of the cloud are anisotropic, with Lx = 2 Ly. Note that the profiles of concentration along the x- and y-axes are Gaussian in shape. For some background and thought problems pertaining to this animation, review the theory (page 6) behind the diffusion of instantaneous, point releases.

* Note to Unix users: the animation opens up behind active windows.

The animations were generated with MATLAB software.

Directions: Click on the images to begin the animation.

Boundary conditions animation output of MATLAB software.

The following animation examines the evolution of concentration after a slug mass is released mid-way between solid, parallel boundaries. Two scenarios are considered, perfectly absorbing and no-flux boundaries. For each system the concentration field is displayed in the plane z = 0. In addition, the concentration profile C(x = 0, y, z = 0) for each system is plotted for comparison on a single graph. For some background and thought problems pertaining to this animation, review the theory (page 8) behind modeling the effect of no-flux and perfectly absorbing boundaries.

* Note to Unix users: the animation opens up behind active windows.

The animations were generated with MATLAB software.

Directions: Click on the images to begin the animation.

### Temporal Records of Concentration as a Diffusing Cloud Passes By

Advection and Diffusion of an Instantaneous Point Source

The following animation displays temporal records of concentration at three points after a mass is instantaneously released into a uniform flow. The concentration is Gaussian about the center of mass, which travels at a velocity U. The Peclet number of all measurement locations is high, so the maximum concentration is observed at roughly the advection time-scale.

* Note to Unix users: the animation opens up behind active windows.

The animations were generated with MATLAB software.

Directions: Click on the images to begin the animation.

### Continuous Release into a Uniform Flow

This animation depicts the evolution of the concentration field downstream of a continuous point source in a channel with steady flow (U = 1 cm/s). The concentration is measured at three points. At each point the center of the front, defined by C = 0.5*Cfinal, arrives at the advection time scale, x/U. The duration of the front, which is the time required for the concentration to rise from C=0 to Cfinal, is 4 σ/U, where σ is the length-scale of the front at t = x/U, i.e. σ= sqrt(2Dx/U).

* Note to Unix users: the animation opens up behind active windows.

## Course Info

Fall 2008
##### Learning Resource Types
Course Introduction
Simulation Videos
Problem Sets
Lecture Notes