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Scaling Analysis as a Pedagogical Tool in Teaching Transport and Reaction Processes Abstract

This paper describes the use of scaling analysis as a pedagogical tool in teaching courses in fluid dynamics, heat transfer, mass transfer, transport phenomena, mass transfer with chemical reaction, and process modeling. Scaling analysis is a systematic method for nondimensionalizing a system of describing equations for transport or reaction processes so that all the dimensionless variables as well as their derivatives have a magnitude no greater than approximately one. This permits assessing the importance of various terms on the basis of the values of the dimensionless groups that multiply them. As such, scaling analysis is an invaluable tool for educators since it provides a systematic way to arrive at model approximations. It thereby permits presenting disparate topics in transport and reaction processes in a unified and integrated manner. Highlights are given of how scaling analysis has been used in teaching graduate-level courses. An eight-step procedure is outlined for applying scaling analysis and then is applied to an illustrative problem involving interpreting data obtained from a permeation cell used to determine the properties of a membrane. This example illustrates how scaling analysis can be used to both interpret performance data as well as to extract useful characterization parameters from the data. As such, it underscores the advantages of scaling analysis as a learning tool to hone students’ intuition.

1. Introduction

Each successive generation of engineering students must learn the essence of what their predecessors mastered in addition to an ever-expanding body of new knowledge. This presents a pedagogical challenge to engineering educators who must develop effective means for teaching more material within essentially the same time frame. Obviously this requires developing learning tools that effectively integrate seemingly disparate concepts in our body of engineering knowledge. Scaling analysis provides a very effective pedagogical tool to address this challenge.

Scaling analysis is a systematic method for nondimensionalizing a system of describing equations for a physical process. The resulting dimensionless system of equations represents the minimum parametric representation of the process. By this we mean that the solution for any quantity that can be obtained from these equations will be at most a function of the dimensionless independent variables and the dimensionless groups generated by the scaling process. In conventional dimensional analysis there is no unique set of dimensionless dependent and independent variables and associated dimensionless groups; that is, for any system of describing equations, one set of dimensionless variables and dimensionless groups can always be obtained from any other set. In contrast, scaling analysis involves nondimensionalizing a system of describing equations in a unique way to ensure that the relevant dependent and independent variables and their derivatives are bounded of order one, denoted by ○(1). By this we mean that the magnitude of the particular dimensionless variable or its derivative is bounded between zero and approximately one. The utility of the systematic scaling procedure is that when all the relevant dependent and independent variables and their derivatives in the resulting dimensionless describing equations are bounded of ○(1), one can assess the importance of various terms on the

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Krantz, W. (2007, June), Scaling Analysis As A Pedagogical Tool In Teaching Transport And Reaction Processes Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--3061