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

Anne Lightbody

,

University of New Hampshire

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#### 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.

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## 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.

#### 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.

#### 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.

#### 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.

## 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.