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.

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