Application of Frequency Response
Technique to the Analysis of Turbulent
Diffusion Phenomenon
JAMES R. HAYS, Graduate Student, Chemical Engineering
KARL B. SCHNELLE, Associate Professor of Chemical Engineering
PETER A. KRENKEL, Associate Professor of Sanitary and Water Resources Engineering
Vanderbilt University
Nashville, Tennessee
SUMMARY
This paper presents a review of the pulse testing method for dynamic analysis of systems. Experimental pulse data is reduced to frequency domain information by numerically computing the Fourier transform. A digital computer is
necessary for the data reduction. The procedure by which the frequency domain
information is found is presented in the paper, and a brief description of the
FORTRAN data reduction program for the IBM 7072 computer is given.
Simultaneous mass transport of dissolved or suspended material can be de-
described by the following equation:
dc
dt
=   D,
c)2C
Jc
dx
Two solutions of this equation are presented in this paper. One solution, due to
Levenspiel (1), is given for a mass of material rapidly injected at the origin of
the test system boundary. The second solution, by Clements (2), is applicable
when both the input and output concentrations are measured in the pipe. Under
such conditions the pulse testing method may be used for data reduction. Leven-
spiel's solution is in the time domain while Clements' of necessity presents a
frequency domain equation.
A mathematical modeling procedure is described which uses the integral of
the squared error as a measure of deviation. This parameter is minimized by a
nonlinear least squares procedure due to Marquardt (3). This technique simultaneously presents the statistical best values of the model parameters and the standard error of estimate,  a measure of how well the model describes the system.
Application of both the pulse testing techniques and the modeling procedure
was made to the problem of axial mixing in a 35-ft long, one-in. diameter,
Pyrex glass pipe. Sodium chloride pulses were injected and the input and output
recorded through electrical conductivity bridges. The results of the investigation are presented in this paper. The study showed that athigher Reynolds Number
the standard error of estimate is of the same order of magnitude as the experimental error. Thus, the model is a near perfect fit in the higher Reynolds Number
region^ It was found that over the range of Reynolds Number 3 x 103 through
3 x 105 the data could be correlated by the following expression:
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