PROCESS DATA REPRESENTATION AND

ANALYSISLOGARITHMIC COORDINATES

Log-log and semi-log plots useful for when experimental data span many orders of

magnitude determining functional relationships of data determining constants for fitting experimental

data too exponential function y=aebx

o power law function y=a xb

Large Range of Experimental Data

Experimentally determined relationship between friction factor and Reynolds number for sphere moving through a fluid given below. Analyse the data by plotting

Reynolds Number 0.9 0.33 0.12 0.074 0.023 0.0091 0.0065 0.0027 0.0011Friction Factor 27 7 3 200 3 2 4 3 0 0 2 635 3 700 8 888 21 900

Plot on normal graph paper not very useful

Data crowded near origin and lie very near to the axes.

Graph not very useful for interpolating and checking suspicious data.

Plot the logarithm of the data

Plot shows a linear correlation and reveals that point measured at Re=0.023 is suspect

Relationship can be expressed as f=aℜb (a power law). Constants a and b can be determined from graph

Instead of computing the base-10 logs plot the data on a log-log graph paper

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

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Gradations of both abscissa and ordinates increase logarithmically.

Gridlines not evenly spaced - distance bet 0.1 and 0.2 greater than bet 0.9 and 1.01 000 is halfway bet 100 and 10 000 bec. log103 is halfway bet log102 and log104

Determining Functional Relationship

Consider the three functionsy = 5x y = x3 and y = ex

plot of these functions on normal graph paper

5x is linear, difficult to distinguish x3 from ex

plot on log-log paper

both 5x and x3 are linear

any power law function plotted on log-log paper is linear

plot on semi-log paper

ex is linear

any exponential function plotted on semi-log paper is linear

Determining the Constants in the Equations

Consider the Reynolds number vs friction factor plot - ignoring erroneous data point

0.01 0.1 1

100

1000

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Reynolds number

Friction factor

straight line on log-log plot means the functional relationship is a power law f=aℜb or log10 f=b log10 ℜ+ log10a

b=slope= riserun

=log1010000−log10100

log10 0.0024−log10 0.23=−1.009

to find a use any set of given dataa= f

ℜ−1.009= 3700

0.0065−1.009=22.98

final relationshipf=22.98 ℜ−1.009

The absorption of radiation by material can be modeled by

R ( x )=Ro eβx

R(x) is the count rate (Geiger counter clicks in one minute)Ro is count rate with no shieldingx is thickness of shielding materialβ is a constant

The table shows measurements of the rate at which radiation particles emitted by 55Fe are detected when a Geiger counter is shielded by aluminium sheets of various thicknesses.

Al Thickness (cm) 0.00162 0.00324 0.00486 0.00648 0.00810 0.00972

Count rate (counts/min) 1 850 1 250 800 450 310 165

Determine the constants Ro and β for this case.