# Using Excel to plot numerical and analytical forms of the diffusion equation

Anne Lightbody

, University of New Hampshire

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This page first made public: Jun 29, 2010

#### Summary

This computer-based assignment forces students to compare and contrast integral and differential forms of the conservation of mass equation, as well as analytical and numerical approaches to solution. Students are given a text description of a simple environmental problem (a conservative tracer diffusing in a one-dimensional system with no-flux boundaries) and are then required to first write equations that describe the system and then implement these equations in an Excel spreadsheet or Matlab m-file. Students then use their spreadsheets/m-files to compare different solution methods and must communicate these results in short text answers.

## Context

#### Audience

Undergraduate hydrology class introducing students to basic physical transport processes (advection, diffusion, dispersion) and chemical reactions (first-order reactions, boundary sources and sinks) in surface water, ground water, and atmospheric systems. The class has 3 hours of lecture and one hour of recitation per week; there is no associated laboratory.

#### Skills and concepts that students must have mastered

Before beginning this exercise, students must be able to:

Translate word problems into equations.

Recognize integral and differential forms of the conservation of mass equation.

Define and use timescales to describe diffusive mass transport

Write and understand Fick's Law for diffusive transport.

Use simple computer programs (Excel & Matlab) to construct spreadsheet models, including the use of $ notation in Excel.

Translate word problems into equations.

Recognize integral and differential forms of the conservation of mass equation.

Define and use timescales to describe diffusive mass transport

Write and understand Fick's Law for diffusive transport.

Use simple computer programs (Excel & Matlab) to construct spreadsheet models, including the use of $ notation in Excel.

#### How the activity is situated in the course

One out of eight homework assignments, occurring somewhere near the middle of the class. Previous homeworks included developing a two-box numerical model of a lake and plotting Gaussian curves. One variation is to assign the problem over two weeks, allowing students to receive feedback on their proposed approach before using those equations to develop a computer model.

## Goals

#### Content/concepts goals for this activity

Conceptualize mass transport via diffusion.

Evaluate applicability and use of timescales for diffusive transport.

Account for boundaries in systems with diffusion.

Evaluate applicability and use of timescales for diffusive transport.

Account for boundaries in systems with diffusion.

#### Higher order thinking skills goals for this activity

Compare and contrast integral and differential forms of the conservation of mass equation.

Improve understanding and methodology of numerical integration.

Compare and contrast numerical integration and analytical solutions.

Describe equations and numerical results in prose.

Evaluate appropriateness of simplifying assumptions.

Improve understanding and methodology of numerical integration.

Compare and contrast numerical integration and analytical solutions.

Describe equations and numerical results in prose.

Evaluate appropriateness of simplifying assumptions.

#### Other skills goals for this activity

Interpret text descriptions of environmental systems and use quantitative tools to understand these systems.

Use differential equations to describe environmental systems.

Use simple computer programs (Excel or Matlab) to model environmental systems.

Troubleshoot spreadsheets/m-files in these programs.

Ask classmates and the instructor for assistance but not the answer.

Use differential equations to describe environmental systems.

Use simple computer programs (Excel or Matlab) to model environmental systems.

Troubleshoot spreadsheets/m-files in these programs.

Ask classmates and the instructor for assistance but not the answer.

## Description of the activity/assignment

This computer-based assignment forces students to compare and contrast integral and differential forms of the conservation of mass equation, as well as analytical and numerical approaches to solution. Students are given a text description of a simple environmental problem (a conservative tracer diffusing in a one-dimensional system with no-flux boundaries) and are then required to first write equations that describe the system and then implement these equations in an Excel spreadsheet or Matlab m-file. Students then use their spreadsheets/m-files to compare different solution methods and must communicate these results in short text answers.

## Determining whether students have met the goals

Examine student written answers and computer spreadsheet/m-file to determine whether they (a) wrote the correct set-up equations (Parts A-C), (b) completed the assigned computer tasks correctly (Parts D & E), and (c) provided reasonable answers to follow-up questions asking for further reflection.

More information about assessment tools and techniques.## Download teaching materials and tips

**Instructors Notes:**

## Other Materials

- Lightbody sample spreadsheet (Excel 1.4MB May25 10)