Week 1:  Unit Conversion and Mathematical Functionality

Unit Conversion and Balancing Units:

You have probably learned in high school how to balance units in calculations and the use of unit balancing to inspect equations.  As engineers you are expected to be able to convert from CGS to SI to American Engineering Units.  We will work a number of problems dealing with unit conversion in this week's problem session.

Dimensional Homogeneity:

You will need to determine if the units in an equation are homogeneous and to calculate the dimensions for constants where dimensions are not given.

Unitless Numbers:

Engineers often use dimensionless numbers and dimensionless analysis to simplify equations and to bring out generalities in behavior.  The most popular example is use of the Reynold's Number, Re, to characterize the flow of fluids.  When Re is larger than 2000 the flow is turbulent and when Re is less than 2000 it is laminar.  Re is the product of the lateral size of the flow path (pipe diameter), the velocity of flow and the fluid density divided by the viscosity.  High viscosity liquids are more likely to display laminar flow, high velocity flows are likely to display turbulent behavior, small channel flows are more likely to display laminar flow.  Flow in microfluidic devices is almost always laminar and for this reason it is difficult to achieve mixing in microfluidic devices.  Many other dimensionless numbers have been defined in engineering to define simple rules that can be applied to a wide variety of systems.

Significant Figures:

Generally, with the use of calculators and computers it is tempting to carry far too many figures through your calculations.  This is a major error since it indicates the degree of certainty you have in a value.  If you report 6 digits of accuracy on a micrometer reading you will be indicating that you know the value to two orders smaller than the size of an atom.  This becomes a major problem in technical and scientific work.  We will run through exercises in carrying through the correct number of significant figures in the problem session this week.

Order of Magnitude Estimate:

Engineers are often required to quickly judge the reasonableness of a calculation based on an order of magnitude calculation.  This is important in cross checking results.

Statistics to Estimate Confidence in Values:

You will need to use statistical calculations to estimate your confidence in values obtained by measurements.  Generally this involves calculation of the standard deviation.  You need to also understand how you can use calculus to propagate errors through an equation.  The calculation and propagation of error is the means by which you determine the number of significant figures you should report in a value.  You also need to develop confidence in estimating a reasonable value for the error in a measured number.  This can usually be done by trying different error values and asking yourself if the actual value should be larger or smaller.  A reasonable error value can be propagated through equations to yield an error value for parameters that you can not guess the error for. 

Linear Interpolation and Linear Fits:

You have probably already learned how to interpolate a value for a series of x and y values.  You will need to write a program to perform a linear fit to a series of x, y data and how to include the error in the y values in such a fit.  Linear fit method of least squares (Perry)

Linearization of Plots for Arbitrary Functions:

We will work some problems looking at how to produce a linear plot from a complex function.  We will also work problems dealing with writing an equation for a log-lin, log-log and other types of linearized plots.  We will discuss how to find simple functional forms for measured values.  Example the Avrami Equation.

















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