Environmental Data Analysis with MatLab by William Menke & Joshua Menke

Author:William Menke & Joshua Menke
Language: eng
Format: epub
ISBN: 9780123918871
Publisher: Elsevier
Published: 2012-06-16T16:00:00+00:00

Note that each formula involves only current and past values of the heat. This is an expression of causality—the future cannot affect the present. The relationship embodied in Equation (7.1) is called a convolution, and is denoted by the asterisk, * (which does not mean multiplication when used in this context).

The gs in Equation (7.1) are coefficients that express the linear proportionality between heat and temperature. Note that exactly the same coefficients, g1, g2, g3, g4 … are used in all the formulas in Equation (7.1). This pattern implements the time-shift invariance that we discussed earlier—only the relative time between the application of the heat and the measurement of the temperature matters. Of course, this is only an idealization. If, for instance, the plate were oxidizing during the experiment, then its thermal properties would change with time and the set of coefficients that linked θ(t) to h(t) at one time would be different from that at another.

For Equation (7.1) to be of any practical use, we need to assume that only the recent past affects the present, that is, the coefficients, g1, g2, g3, g4, …, eventually diminish in magnitude, sufficiently for the sequence to be approximated as having a finite length. This notion implies that the underlying physics has a characteristic time scale over which equilibration occurs. Heat that was supplied prior to this characteristic response time has negligible effect on the current temperature. In the case of the flame in the laboratory, we would not expect that yesterday's experiment would affect today's experiment. The temperature of the plate has adequate time to equilibrate overnight.

As Equation (7.1) is linear, it can be arranged into a matrix equation of the form d = Gm:

(7.2)

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.