INSTRUMENTATION AND
MEASUREMENTS
Hamid R. Rahai, Ph.D.
Professor
Mechanical and Aerospace Engineering Department
California State University, Long Beach
Spring 2013
1
ACKNOWLEDGEMENTS
The authors would like to acknowledge contributions from Dr Darrell W.
Guillaume of CSU Los Angeles and Dr. John LaRue of UC Irvine on
Measurement Uncertainty and Dr. Huy Hoang of CSU Long Beach in
reviewing the manuscript and providing additional examples for the chapter
on Probability and Statistics for Engineers.
2
1
Chapter 1
PROBABILITY AND STATISTICS FOR
ENGINEERS
3
1. PROBABILITY AND STATISTICS FOR ENGINEERS
Probability is a measure of occurrence. In repeated trials of an experiment, it represents
the probability of occurrence of a sample, a value or an event. The probability of
occurrence is represented by a Probability Density Function (PDF), p(x) .
Statistical theory enables us to estimate the value of p(x) .
A PDF p(x) of a sample x has the following properties:
p( x)  0

 p( x)dx  1

b
 p( x)dx  p(a  x  b)
a
Here p(a  x  b) represents the probability of occurrence of x between limits a and b .
For a sample A, the probability of occurrence is p(A) , where 0  p( A)  1 .
Within the experimental data, values can be exhaustive or exclusive. Exclusive events are
independent of each other.
p( E1  E2 )  p( E1 )  p( E2 )
When the events are not exclusive, then we have:
p( E1  E2 )  p( E1 )  p( E2 )  p( E1E2 )
Here p( E1E2 ) is the probability of joint event, both ( E1 and E2 ) occurring.
Example: If a coin is tossed, the probability of having a head or a tail is 50% or ½ .
If two coins are tossed, the probabilities are as follow:
head-head, head-tail, tail-head, and tail-tail.
Thus probability of each event is 25% or ¼ .
Example: What is the probability of rolling two dice to have a sum of 7 and the number
3?
Solution:
As shown in the following table, for rolling two dice, there are at least 36 events with 6
events having a sum of 7, 11 events with number 3 and 2 events having both number 3
and sum of 7.
4
11
21
31
41
51
61
71
p( E1 ) 
12
22
32
42
52
62
72
13
23
33
43
53
63
73
14
24
34
44
54
64
74
15
25
35
45
55
65
75
16
26
36
46
56
66
76
17
27
37
47
57
67
77
6
11
2
, p( E2 )  , and p( E1 E2 ) 
36
36
36
Then
p( E1  E2 ) 
6 11 2 15



36 36 36 36
Probability of an event E1 , given the event E 2 has occurred is:
p( E1 E2 )
p E1 E2 
p( E2 )
Thus the probability of the joint event can be calculated as:
p( E1E2 )  p( E2 ) p E1 E2
For independent events, p E1 E2  p( E1 ) and the above equation becomes:
p( E1E2 )  p( E2 ) p( E1 )
which is the product law for independent events.
Conditional probability is defined as the sum of the probabilities of the sample points in
the event:
p( E )   p(sample pointsin E )
Example:
If two dice is rolled, given that a dice shows number 3, what is the probability of having
the sum of the two numbers equal 7?
p E1 E2
2
2
 36 
11 11
36
Permutations deal with order of selecting samples within a population. Combinations
disregard this order. Permutations and combinations are part of the combinatorial theory.
5
Let‟s say we have three numbers, 1, 2, and 3. If we want to select two numbers at a time,
the number of permutations for these numbers are:
1 2, 1 3, 2 3, 2 1, 3 1, 3 2 (the number of permutation is 6)
In general the number of permutations can be calculated from the following equation:
n!
(n  r )!
For the above example, n is 3 and r is 2 which results in 6/1 = 6 which the results we
obtained before.
n
Pr 
The combinations for these numbers are:
1 2, 1 3, 2 3 (since we disregarded the order of selection, then 1 2 = 2 1, 1 3= 3 1, and 2 3
= 3 2 . The number of combinations is 3).
The following equation can be used to calculate the number of combinations:
n
Cr 
n
n!
P
 r
(n  r )! r! r!
Random variables are either discrete or continuous. Discrete random variables are finite
while continuous random variables include infinite (large) variables. Followings are some
statistical terms for random variables:
1. The population mean:
1
 xi
n
Here n and xi are sample size and individual sample respectively.

2. The population variance:
1
 ( xi   ) 2
n
Variance is a measure of dispersion. However, to have the same scale as the
population mean, we use the standard deviation which is the square root of the
 x2 
variance;  x   x2 .
3. The standard error of the mean:   
x
n
4. Variance of the sum and covariance:
6
.
Let‟s assume we have two sets of samples, xi and yi with n data points. The variance
of the sum of these samples is defines as:
1
 x2 y   ( xi   x )  ( y i   y ) 2
n
or




1
 ( xi   x ) 2  ( yi   y ) 2  2( xi   x )( yi   y )
n
The last term on the right hand side of the above equation, ( xi   x )( yi   y ) , is
 x2 y 
called covariance of x and y and is a measure of their interdependence, how they
move together. If the samples are independent, then their covariance is zero. The
above equation can also be written as:
 x2 y   x2   y2  2 covariance ( x, y)
In statistics, sampling techniques are used as a basis for inference about the entire
population. For samples of x i , the mean and variances are defined as:
1
 xi
n
1
s2 
( xi  x ) 2

n 1
x
The sample standard deviation is: s  s 2 .
Other terms and definitions are:
Range: xmax  xmin
X max  X min
2
Mode: The most frequent item in the population;
Median: The central term in the population when they are arranged in increasing and
decreasing order. For even number of samples, the average of the middle samples is
taken as the median.
The most probable error of a single measurement Es:
Midrange:
( xi  x ) 2
E s  0.6745 
i 1 ( n  1)
Average deviation from the mean:
1 n
 xi  x
n i 1
Root mean square (R.M.S.) Deviation from the mean:
n
1 n
( xi  x) 2

n i 1
7
Standard deviation from the mean:
Sx 
1 n
( xi  x ) 2

n  1 i 1
s
Standard error of the mean: s x 
n
Degrees of freedom: Degrees of freedom, df, or F of a sample is equal to the number
of observations n less the number of constraints k imposed on the data; F = n - k.
Sample and Population: Population is collection of all possible measurements that
relate to the same phenomenon. Sample is subset of a population, a group of
measurements drawn from a population. Probability of a sample point is the
proportion of occurrence of the sample point in a long series of experiments.
Coefficient of Variation: Coefficient of variation is a relative variation of the data
s
and is equal to the standard deviation divided by the mean of the data, . Coefficient
x
of variation is independent of the units in which the data is collected, as long as the
unit starts from zero. For example we can measure the weight of an object using
kilogram or pounds. Since both units start from zero, regardless of which system of
unit is used, the coefficient of variation would be the same.
CHEBYSHEV’S THEOREM
For a given number k>1 and a set of measurements that has a mean, x , and a standard
1
deviation , s, at least 1  2 of those measurements fall between  ks of the mean.
k
Example: How many samples of a population of 500 fall between  2 standard deviation
of the mean?
Solution:
1
1
1  2 = 1  2 =0.75,
k
2
0.75  500 = 375 samples fall within 2 standard deviation of the mean.
CORRELATION COEFFICIENT
A correlation coefficient is a measure of the strength of a linear relationship between two
variables. The correlation coefficient is defined as:
 x

n
r
i
 x yi  y
i 1
(n  1) s x s y
8

Using expressions for x , y , s x , and s y , the formula for r can be rewritten as:
r
n x
n  xi y i   xi  y i
2
i

  xi  n  yi2   yi 
2
2

The value of r range from -1 to 1 and at these limits the relationship is definitely linear,
having either a positive or negative slope. For r=0, no linear relationship exists.
2. THE METHOD OF LEAST SQUARES
If the test points are distributed normally and independently, the best line drawn through
them would be so drawn that the sum of the squares of the deviation from the line would
be a minimum. Assume the line is represented by:
y  a0  a1 x
and individual errors between the point and line can be calculated as:
 i  yi  (a0  a1 xi )
n
For the best line, then
  i2  minimum. For this to be true, we should have:
i 1
n
n
   i2
   i2
i 1
 0 , i 1
0
 a0
 a1
If we substitute for  i perform the differentiation, we get:
n
n
i 1
i 1
 yi   a1 x  na0  0
n
n
i 1
i 1
 xi yi  a1 
xi2
n
 a 0  xi  0
i 1
Solving for a0 and a1 results in the following equations:
xi  xi yi   yi  xi2

a0 
 xi 2  n xi2
 xi  y i  n  xi y i
a1 
 xi 2  n xi2
The standard error of the linear relationship which is the total error associated with fitting
a fist order polynomial to the experimental data can be obtained from the following
equation:
9
1
   yi  a1 xi  a0 2  2
Standard error = 

n2


The straight line relationships for various functions that are not linear can be obtained
with proper scales. Table 1 provides examples of values used to obtain the linear relations
for these non-linear functions.
________________________________________________________________________
Example: The following data fit an equation in the form of y  a0  a1  x . Find the
equation and its correlation coefficients.
X
1.0
1.6
3.4
4.0
5.2
Y
1.2
2.0
2.4
3.5
3.5
Solution: The coefficients of the equations are calculated from the following formulas:
xi  xi yi   yi  xi2

 x i  y i  n x i y i
a

a0 
and
1
 xi 2  n xi2
 xi 2  n xi2
summation, 
xi
yi
xiyi
x i²
yi²
1.0
1.2
1.20
1.00
1.44
1.6
2.0
3.20
2.56
4.00
3.4
2.4
8.16
11.56
5.76
4.0
3.5
14.00
16.00
12.25
5.2
3.5
18.20
27.04
12.25
15.2
12.6
44.76
58.16
35.70
With the substitution of the calculated data, the coefficients of the equations are
(15.2) * (44.76)  (12.6) * (58.16)
(15.2) * (12.6)  5 * (44.76)
 0.88 and a1 
 0.54
2
(15.2)  5 * 58.16
(15.2) 2  5 * 58.16
Thus, the equation of the line is y  0.88  0.54  x . The correlation coefficient between
the data set X and Y can be calculated as:
n xi y i   xi  y i
r
=0.94
2
(n xi2   xi  )(n yi2  ( yi ) 2
a0 
_______________________________________________________________________
10
Function
Abscissa
Ordinate
y  a0  a1 x
x
y
y  kxa
log x
log y
y  keax
x
log y
1
x
y
x
a  bx
x
x
y
y  a  bx  cx 2
x
y  y1
x  x1
x
a  bx  cx 2
x
x  x1
y  y1
x
log y  log y1
ya
y
y
b
x
2
y  kebx cx
Table 1.
________________________________________________________________________
Example:
The following data fits the equation of the form y  ae bx . Find a and b.
Solution:
Taking the natural logarithm for both sides of the equation yields:
ln y  ln(ae bx )  ln a  ln e bx  ln a  bx * ln e  ln a  bx
This equation is similar to the equation of the line y  a0  a1  x with y=lny, a0  ln a
and a1  b . Follow the same procedure as in the previous example, it gives
11
summation, 
a0
xi
yi
0.00
9.40
0.43
xi*lnyi
x i²
2.2407
0.0000
0.0000
7.10
1.9601
0.8428
0.1849
1.25
5.35
1.6771
2.0964
1.5625
1.40
4.20
1.4351
2.0091
1.9600
2.60
2.60
0.9555
2.4843
6.7600
2.90
1.95
0.6678
1.9367
8.4100
4.30
1.15
0.1398
0.6010
18.4900
9.0761
9.9703
37.3674
12.88
 x  x y   y  x =2.2

 x   n x
 x  y  n x y =-0.49

 x   n x
i
i
i
2
i
i
2
i
2
i
i
2
i
i
a1
lnyi
i
i
2
i
Since a0  ln a , so a=9.05
and a1  b =-0.49. Thus the equation of the curve is: y  9.05e 0.49x
3. PROBABILITY DENSITY FUNCTIONS
1. Binomial Distribution
Binomial distribution provides the probability of success of n events among N
independent populations. It is defined as:
p ( n) 
N!
p n (1  p) N  n
n! ( N  n)!
Here p is the probability of success and the quantity (1  p) is the probability of
failure. The limit of the Binomial distribution (when N  , p  0 ) is:
N  p = constant = a
which is called Poisson distribution and is given as:
p a ( n) 
a n e a
n!
The standard deviation of the Poisson distribution is a . Poisson distribution is used
when we are concerned with decay of substances and materials over a large period of
time.
12
2. Normal or Gaussian distribution
The mathematical representation of a normal or Gaussian distribution is:
p( x) 
1
x
 1

exp 
(x  )2 
2
2
 2 x

Most often, the “standard” normal distribution is used instead. To obtain the standard
normal distribution, let:
x
z
x
With this transformation, the equation for the standard normal distribution becomes:
1
z2
p( z ) 
exp(  )
2
2
For the standard normal distribution, the mean is zero, the standard deviation is one
and



p( z )dz  1.0
Table 2 provides the values of the standard normal probability distribution for
different z values. Please note that these values are for the half of the distribution.
Since the distribution is symmetric, for the total probability, the values from the table
should be multiplied by 2.
Example: What is the probability for z=0.21?
Solution: To find the probability for z=0.21 from Table 2, locate the corresponding
value for z=0.2 at the right of the first column and 0.01 at the top row. The
probability for z=0.21 found from Table 2 is 0.0832.
13
Table 2. Standard Normal Distribution
Example: What is the probability that an observation being more than two standard
deviations from the mean (p(x > µ + 2σ),p(x< µ-2σ))?
x
Solution: Substitute µ + 2σ and µ - 2σ for x in the equation z 
, then z

becomes >2 and <-2. From Table 2 for z=2, p(z) is 0.4772. Since the distribution is
symmetric, the probability of z> 2 is 0.5-0.4772=0.023 which is also equal to the
probability of z<-2. Thus, the total probability is 0.046 or approximately 0.05. 0.05
is equal to 5% or 1 in 20 which means in 20 observations; one can fall outside the two
standard deviation range.
14
Example: Calculate and plot the standard normal distribution for the following set of
data:
57.8, 24.8, 27.4, 36.5, 43.1, 44.0, 31.7, 40.1, 36.0, 47.2, 27.2
1 n
Solution: From the data, N=11, x   xi  37.8 and
n i 1
x  s 
1 n
( xi  x) 2 =9.96.

n  1 i 1
xx
Using the equations z 
and P( z ) 
s
data are shown in the table below:
1
e
2

z2
2
, the values of z and P(z) for each
x
z
P(z)
57.8
24.8
27.4
36.5
43.1
44.0
31.7
40.1
36.0
47.2
27.2
2.0080
-1.3052
-1.0442
-0.1305
0.5321
0.6225
-0.6124
0.2309
-0.1807
0.9438
-1.0643
0.0531
0.1703
0.2313
0.3957
0.3464
0.3288
0.3308
0.3885
0.3926
0.2556
0.2265
0.45
0.4
P(Z)
0.35
0.3
P(Z)
0.25
0.2
0.15
0.1
0.05
0
-1.5
-1
-0.5
0
0.5
Z
15
1
1.5
2
2.5
Confidence intervals are used to estimate the range of parameters of random variables
with a known degree of uncertainty. For a normally distributed random data, the range
can be calculated as:




(x  )
P  z  
 z   1  
x
 2
2 


N
Here α is the significant level and 1-α is the level of confidence. For 100% probability, α
is zero. α is the remaining area outside the probability.
_______________________________________________________________________
Example: Let α = 0.1. Find the range for the true mean, μ, and the corresponding
confidence level of the following set of data:
54.0, 19.5, 23.1, 33.0, 33.9, 39.0, 23.7, 35.2, 34.2, 42.5, 25.2.
Solution: From the data, N=11, x 
1 n
 xi  33.0 and  x  s 
n i 1
1 n
( xi  x ) 2

n  1 i 1
=10.0.
Level of confidence = 1- α = 0.9 or 90%.




(x  )
P  z 0.05 
 z 0.05   0.9
x




N
Since the distribution is symmetric about z=0, the probability of the range corresponds
to Z 0.05 =(1-α)/2=(1- 0.1)/2=0.45. From table 2, the corresponding Z value for P(z)=0.45
is 1.645. Substitute and rearrange, with 90% confidence level, the probability of the
range for the mean is:
(33.0   )
or
 1.645 
 1.645
28.04    37.96
10
11
16
3. Histogram:
Histogram provides the probability of events within each increment. Histogram can
be used to check if the data follows a standard distribution or not. The following steps
can be used to draw a histogram:
(a) Choose a number of class intervals (usually between 5 and 20) that covers
the data range. Select the class marks which are the mid-point of the class
intervals. If you arrange data in ascending order, the first data should fall
in the first class interval.
(b) For each class interval, determine the number of data that fall within that
interval. If a data falls exactly at the division point, then it is placed in the
lower interval.
(c) Construct rectangles with centers at the class marks and areas proportional
to class frequencies. If the widths of the rectangles are the same, then the
height of the rectangles represent the class frequencies.
_______________________________________________________________________
Example: Develop the histogram for the following 25 data:
3.0, 6.0, 7.5, 15.0, 12.0, 6.5, 8.0, 4.0, 5.5, 6.5, 5.5, 12.0
1.0, 3.5, 3.0, 7.5, 5.0, 10.0, 8.0, 3.5, 9.0, 2.0, 6.5, 1.0, 5.0
Solution:
The following guide is used to obtain the number of class intervals. It is recommended to
use more or fewer classes than the guide suggests if it makes the graph more descriptive.
Sample size (data)
Number of class intervals
10-20
5
20-50
6
50-100
7
100-200
8
200-400
9
400-700
10
In this example, the number of data is 25 so the number of classes is 6. The class width is
calculated as
𝑋
−𝑥
∗1.2
15.0−1.0 ∗1.2
∆𝑥 = 𝑚𝑎𝑥 𝑚𝑖𝑛
=
= 2.8. Here, the factor 1.2 is used to increase the
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠
6
coverage of data range. The first point of histogram is calculated as
x
2.8
x1  xmin 
 1.0 
 0.4
2
2
Next calculate the subsequent class interval points by adding ∆x.
x2  x1  x  0.4  2.8  2.4
x3  x2  x  2.4  2.8  5.2
x4  x3  x  5.2  2.8  8.0
x5  x4  x  8.0  2.8  10.8
x6  x5  x  10.8  2.8  13.6
x7  x6  x  13.6  2.8  16.4
The next step is to form the class subintervals. This is done using the method of left
incursion. For example, the first subinterval is from -0.4 to 2.4. The second subinterval is
from 2.4 up to 5.2. And so on. For each class subinterval, the class frequency is the
17
number of data (or tally) occurs in the subinterval and the class mark is a half of the class
interval.
The histogram is constructed to show the data distribution over the interval -0.4 to 16.4.
The vertical axis is the class frequency and the horizontal axis is the data.
Students may try different number of class sizes to see if the graph is more descriptive. In
some cases, the relative class frequencies are preferred over the class frequencies.
Relative class frequencies make it easier to understand the distribution of the data and to
compare different sets of data. Relative frequencies are found by dividing each class
frequency by the sum of the frequencies.
𝑐𝑙𝑎𝑠𝑠 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =
𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠
Example: Plot a histogram for the above example using MS Excel.
The steps to plot the histogram using the MS Excel are as follows:
1. Highlight the columns of class marks and class frequency to plot XY (scatter)
chart.
18
2. Click the X axis to change the scale of minimum, maximum, and major unit
(unchecked Auto scaling)
First point X1
Last point X7
Class width
19
3. Draw a rectangular box between the two endpoints of each class subinterval with the
height matching the dot of each class mark.
_____________________________________________________________________
4. Chouvenet’s Criterion.
Assuming the data follows a normal distribution, Chouvenet‟s Criterion is used to
eliminate bad data points. For any set of data, first the mean and standard deviation
of the entire population are calculated. Then the deviations of the individual data
points are compared with the standard deviation of the whole population according to
the following table, to accept or reject the data points. Here in the table, the ratio of
maximum allowable deviations to the standard deviation,
 max
 x are given
according to the number of data points, n. Once the bad data points are eliminated,
then mean and standard deviation of the population are recalculated.
The following procedure can be used to assess a data point:
(a) For a sample population, calculate x and  x .
(b) Using sample population n, find
 max
x .
(c) Knowing  x , find  max .
(d) Calculate x  x . Here x is the sample that you are assessing. If the difference is
larger than  max , the sample is discarded, otherwise it is retained.
20
 max
n
3
4
5
6
7
10
15
25
50
100
300
500
1000
x
1.38
1.54
1.65
1.73
1.8
1.96
2.13
2.33
2.57
2.81
3.14
3.29
3.48
Table 3. Chouvenet‟s values.
Example: Use the Chouvenet „s criterion to determine the bad data points from the
following set of data:
5.30, 5.73, 6.77,5.26, 4.33, 5.45, 6.09, 5.64, 5.81, 5.75.
Solution: From the data, we calculate
x
1 n
 xi  5.61 and  x  s 
n i 1
From Table 3, the value of
1 n
( xi  x) 2 =0.62.

n  1 i 1
 max
is 1.96 for n=10. Next, we find  max =1.96*  x .
x
 max =1.21. Next, calculate x  x from the data.
Data
1
2
3
4
5
6
7
8
9
10
xi
x-xi
5.3
5.73
6.77
5.26
4.33
5.45
6.09
5.64
5.81
5.75
0.31
-0.12
-1.16
0.35
1.28
0.16
-0.48
-0.03
-0.2
-0.14
21
Since the data number 5 has x  x =1.28, larger than the value of  max = 1.21,
Therefore, the data 4.33 is considered as bad data point.
5. Chi-square (  2 ) distribution.
It is used as a test of “goodness of fit”, to assess whether data matches a specific
distribution. It is defined as:
n
(observed - expected) i2
2  
(expected) i
i 1
Here, the expected values are from the specific distribution. The Chi-square statistics
is:
z2
2  
F
Here z is taken from standard normal distribution and F is degrees of freedom. The
Chi-square distribution is always positive. The smaller the  2 value the better the fit
between the experimental data and the expected distribution. If  2 is zero, then we
have a perfect match between our experimental data and the specific distribution. For
large number of samples, the Chi-square distribution assumes the bell shape of the
normal distribution. The  2 table (table 4) on the following page provides the
probability p that this value of  2 or higher values could occur. To use this table, first
the degrees of freedom and the  2 are calculated, then the probability p is obtained
from the table which is the probability of the goodness of fit.
22
DF 0.995
--1
2 0.010
3 0.072
4 0.207
5 0.412
6 0.676
7 0.989
8 1.344
9 1.735
10 2.156
11 2.603
12 3.074
13 3.565
14 4.075
15 4.601
16 5.142
17 5.697
18 6.265
19 6.844
20 7.434
21 8.034
22 8.643
23 9.260
24 9.886
25 10.520
26 11.160
27 11.808
28 12.461
29 13.121
30 13.787
40 20.707
50 27.991
60 35.534
70 43.275
80 51.172
90 59.196
100 67.328
0.99 0.975 0.95 0.90
--- 0.001 0.004 0.016
0.020 0.051 0.103 0.211
0.115 0.216 0.352 0.584
0.297 0.484 0.711 1.064
0.554 0.831 1.145 1.610
0.872 1.237 1.635 2.204
1.239 1.690 2.167 2.833
1.646 2.180 2.733 3.490
2.088 2.700 3.325 4.168
2.558 3.247 3.940 4.865
3.053 3.816 4.575 5.578
3.571 4.404 5.226 6.304
4.107 5.009 5.892 7.042
4.660 5.629 6.571 7.790
5.229 6.262 7.261 8.547
5.812 6.908 7.962 9.312
6.408 7.564 8.672 10.085
7.015 8.231 9.390 10.865
7.633 8.907 10.117 11.651
8.260 9.591 10.851 12.443
8.897 10.283 11.591 13.240
9.542 10.982 12.338 14.041
10.196 11.689 13.091 14.848
10.856 12.401 13.848 15.659
11.524 13.120 14.611 16.473
12.198 13.844 15.379 17.292
12.879 14.573 16.151 18.114
13.565 15.308 16.928 18.939
14.256 16.047 17.708 19.768
14.953 16.791 18.493 20.599
22.164 24.433 26.509 29.051
29.707 32.357 34.764 37.689
37.485 40.482 43.188 46.459
45.442 48.758 51.739 55.329
53.540 57.153 60.391 64.278
61.754 65.647 69.126 73.291
70.065 74.222 77.929 82.358
0.10
0.05 0.025
0.01 0.005
2.706 3.841 5.024 6.635 7.879
4.605 5.991 7.378 9.210 10.597
6.251 7.815 9.348 11.345 12.838
7.779 9.488 11.143 13.277 14.860
9.236 11.070 12.833 15.086 16.750
10.645 12.592 14.449 16.812 18.548
12.017 14.067 16.013 18.475 20.278
13.362 15.507 17.535 20.090 21.955
14.684 16.919 19.023 21.666 23.589
15.987 18.307 20.483 23.209 25.188
17.275 19.675 21.920 24.725 26.757
18.549 21.026 23.337 26.217 28.300
19.812 22.362 24.736 27.688 29.819
21.064 23.685 26.119 29.141 31.319
22.307 24.996 27.488 30.578 32.801
23.542 26.296 28.845 32.000 34.267
24.769 27.587 30.191 33.409 35.718
25.989 28.869 31.526 34.805 37.156
27.204 30.144 32.852 36.191 38.582
28.412 31.410 34.170 37.566 39.997
29.615 32.671 35.479 38.932 41.401
30.813 33.924 36.781 40.289 42.796
32.007 35.172 38.076 41.638 44.181
33.196 36.415 39.364 42.980 45.559
34.382 37.652 40.646 44.314 46.928
35.563 38.885 41.923 45.642 48.290
36.741 40.113 43.195 46.963 49.645
37.916 41.337 44.461 48.278 50.993
39.087 42.557 45.722 49.588 52.336
40.256 43.773 46.979 50.892 53.672
51.805 55.758 59.342 63.691 66.766
63.167 67.505 71.420 76.154 79.490
74.397 79.082 83.298 88.379 91.952
85.527 90.531 95.023 100.425 104.215
96.578 101.879 106.629 112.329 116.321
107.565 113.145 118.136 124.116 128.299
118.498 124.342 129.561 135.807 140.169
Table 4. Chi –square table.
23
Example: A random sample of furniture defects from a shipping company was recorded. The
observed and expected defects were classified into four types: A,B,C, and D.
Does the observed data differ significantly from the expected data with significant level α=0.05?
Solution:
In order to compare the goodness-of-fit test, the total counts of observed data must
agree with the total counts of expected data.
Then calculate the Chi-square:
(observed  exp ected ) 2 (89  82) 2 (18  20) 2 (12  8) 2 (81  90) 2
2  




exp ected
82
20
8
90
=3.698
The number of recording for each set of observed and expected data is 4. Thus, n=4.
Furthermore, we impose one restriction on the data: the number of recording is fixed. Thus k=1
and the degree of freedom is dF = n-k = 4-1=3. Assuming a significant level of 0.05, the critical
value for from Table 4 is 7.815. Since the calculated value 3.698 is less than the critical value
7.815. Therefore, the observed distribution is a good fit with the expected distribution.
6. Student t-distribution.
It is the ratio of the normal to chi-square distribution. It is used for small sample. It is flatter
than the normal distribution with more population towards the tails of the distribution. It is
written as:
x x
tF  i
s
As the sample size increases, the frequency around the mean increases, while the number of
data under its tails decreases, and it approaches the normal distribution curve. Table5
provides the values for the student t distribution.
24
Table 5. Student t distribution.
t-Test Comparison of Different samples
t-Test comparison is used to determine if significant difference exists between two samples.
The following steps describe the process for determining whether the two samples within a
confidence level are statistically the same or different:
1. Determine the number of sample, n, the mean x , and the standard deviation s for each
sample.
2. Calculate the t value from the following equation:
25
t
x1  x 2
 s12 s 22 
  
 n1 n2 
1
2
3. Determine the degrees of freedom, F, from the following equation:
F
 s12 s 22 
  
 n1 n2 
2
2
2
 s12 
 s 22 
 
 
 n1    n 2 
n1  1 n2  1
4. Select a confidence level
5. Use the F value and the proper confidence level corresponding to half of the significant
level selected to find a new t value from the student t table (table 5).
6. If the t value from step 2 is less than the t value from step 5, then the two samples are
statistically the same. Otherwise they are different.
Note: The significance level is the probability remaining under the tail of the distribution. For
90% confidence, the significance level is 10%.
__________________________________________________________________________
Example:
The following two sets of data provide the statistics of the production lines:
Set 1: 80, 86, 80, 86, 85, 78, 75, 91, 89, 81
Set 2: 74, 81, 73, 78, 79, 76, 78, 84, 80, 74
With 95% confidence level, decide (show the calculations) whether the two samples are the
same or different.
Solution:
For the set 1, n1=10, x 1 
1 n
 xi  83.1 and  x  s1 
n i 1
1 n
( xi  x) 2 =5.09.

n  1 i 1
1 n
1 n


s

( xi  x) 2 =3.50
and
x

77
.
7


x
2
i
n  1 i 1
n i 1
Calculate the t and F values from equation above:
t=2.77
F=15.95
The degree of freedom is rounded down to 15. With 95% confidence level, α=0.05 and
α/2=0.025, the t value from the student t table (table 5) is t=2.131.
Since the t value from the formula is higher than the t value from the table, the two sets are
statistically different.
For the set 2, n2=10, x 2 
______________________________________________________________
26
Exercises
1. For the following data, find the mean, median, variance and standard deviation.
21, 20, 8, 14, 6, 19, 24
2. Find mean, standard deviation, mode, median, and coefficient of variation of the following
data. Develop a histogram.
10.256, 10.855, 10.115, 9.995, 10.556, 10.188, 10.100, 10.656, 10.050, 9.995, 10.580,
10.655, 10.100, 10.650, 10.660, 10.755, 10.800, 10.456, 10.582, 10.338, 10.400, 10.455,
10.256, 10.588, 10.399
Note that the accuracy of the data is to 3rd decimal place and thus your answer should have a
maximum accuracy of the same significance.
3. For 800 respondents to an inquiry of their age, results show a mean of 42 years and a
standard deviation of 5 years. How many respondents fall between  2 standard deviation
from the mean?
4. Specific heat of any substance is defined as the change in heat per unit mass per units rise
in temperature. The following data represents the change in specific heat of a substance
with temperature:
T (C) = 40, 50, 60, 70, 80, 90, 100
Specific Heat = 1.58, 1.60, 1.63, 1.67, 1.7, 1.72, 1.78
Plot temperature vs. specific heat. Find a first order polynomial and the standard error for
the data. What is the correlation coefficient?
5. Given the following data, find the straight-line equation using the least square method.
Find the standard error.
X= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Y=2.5, 4.8, 6.5, 7.9, 9.7, 11.1, 12.7, 14.7, 16, 17.4.
6. The following data fit an equation in the form of y  e a bx . Find a and b.
X= 0, 0.5, 1.0, 1.5, 2.0
Y=1.3499, 1.7333, 2.2255, 2.8577, 3.6693
7. Calculate the correlation coefficient for the following data:
X=0, -1, -2, -3, -2, -2, 0, 1, 2, 3, 2, 1
Y=3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2
8. Calculate the correlation coefficient for the following data:
X=1, 5, 10, 15
Y=7, 19, 34, 49
Can the data be represented with a linear equation? If so, find the equation.
27
9. Given the following table for grades on a test, draw histogram and calculate mean and
standard deviation.
X
95-100
85-90
75-80
65-70
55-60
45-50
35-40
25-30
15-20
F
2
4
16
22
35
43
32
20
6
10. Two sets of samples for length are taken from a production line with the
following statistics
X1  3.632  1  0.06 cm n1  17
X 2  3.611  2  0.02 cm n 2  24
With 95% and 65% confidence levels, determine whether the two samples are
statistically the same or different.
11. Find the range for the population mean value with 95% and 65% confidence intervals for
each set of data.
X1  3.632  1  0.06 cm n1  17
X 2  3.611  2  0.02 cm n 2  24
12. A packaging company conducts a survey for the two new package designs for lunch boxes
in a supermarket. The results are as follows:
158.5 138.4 168.1 149.4 145.8 168.7 154.4 162.9
Design1
Design2 150.3 155.4 151.6 158.8 151.4 150.8 161.4 157.6 156.8 147.6
a. What is the range of the true (or population) mean of the data in each design for 80%
confidence level?
b. Calculate the standard normal distribution z and P(z) for each data set.
28
2
Chapter 2
MEASUREMENT UNCERTAINTY
29
2.1. MEASUREMENT UNCERTAINTY
The term uncertainty is used to describe variation in the measurement or control of any quantity
associated with an experiment. Thus, uncertainty is a measure of the accuracy of the
experimental data. Specifically, it is an estimate of the maximum error which might reasonably
be expected from a set of measurements. It is important to estimate the uncertainty associated
with a measurement and to determine ways to reduce it.
Similar to an experiment, numerical analysis of any problem also contains errors associated with
the assumption(s) made, the method of analysis, the type of grid and the number of iterations that
are used, etc. It is also very important that results from any numerical analysis also presented
with their uncertainty.
Measurement of any property contains errors. The errors are the differences between the
measured values and the true value. The true value is either given or estimated from previous
repeated experiments, or assumed to be the reasonable human error associated with the
limitations of the measuring instrument being used. For example, the uncertainty of an
instrument such as a digital pressure transducer will typically be given in the operating manual.
If it is not, then a set of experiments that compare the reading from the transducer to a known
input can be devised that allows estimation of the uncertainty of the transducer. In the case of a
simple measurement device, such as a ruler, the uncertainty can be estimated by determining
how closely an individual can reasonably read the graduations on the ruler.
Error in a measurement has two components: fixed or bias error and random or precision error
(repeatability). Bias error is the difference between the mean value, x , of the measurements and
the actual value of the quantity being measured. The mean value is estimated from the following
equation:
1 N
x   xi
N i 1
(2-1)
In this equation, N is the number of samples and xi is the value for each measurement. With
thirty or more samples, the mean is often used as an estimate of the true value of a measurement
Bias is the systematic error. In repeated measurements at the same conditions, bias error is the
same for all of them. The bias can not be determined unless we know the true value. Many
experimental apparatus can operate within a special range of experimental conditions. If the
environment changes, the accuracy of the equipment will affected, and thus the bias will change.
Bias errors can be eliminated by calibration if they are large and known. Small unknown biases
contribute to the bias limit.
Large unknown biases usually come from human errors in data processing, incorrect handling
and installation of instrumentation, and unexpected environmental disturbances such as changes
30
in humidity, temperature, or Barometric pressure. A well controlled environment should not have
large unknown biases.
The major sources of bias error are: 1) Nonlinearity, 2) poor calibration, 3) Hysteresis, and 4)
Drift.
The output of a device should be linearly proportional to its input. However, all devices deviate
from linearity. Nonlinearity can always be described as a combination of an offset from the ideal
linear response and a wobble about the ideal linear response. The departure from linearity may
be classified in one of two ways: Independent of reading and dependent on reading. When an
independent of reading classification is specified, it means that the maximum deviation from
linearity is less than a certain percentage of full scale. The a dependant on reading classification
is specified, it means that the maximum deviation from the measurement reading is less than a
certain percentage of that particular reading.
Calibration is used to determine the response of a device to a known input. Once this response is
known, the device can be used to determine the value of an unknown input. The accuracy of the
relationship of the response determined during calibration directly affects the accuracy of the
measurement.
The response of a measurement device has hysteresis when the output of the device is dependent
on the direction from which the corresponding input is approached. Specifically, a reading near
the midrange of an instrument will have two different values depending on whether the reading
is achieved from descending from full scale or achieved from ascending from zero. The usual
cause of hysteresis is friction or inertia.
Drift is used to describe an error that manifests itself as a change in output for a fixed input over
time. Drift can be observed by maintaining a steady input while monitoring the output.
Generally, drift is due to the effect of temperature change during the warm up of the electronic
components in a device.
Precision error is the variation between repeated measurements. The estimated standard
deviation from the mean, S, is used as a measure of the precision error. A large standard
deviation means a large scatter in the measurements. The standard deviation is defined as:
1
 N  x  x 2  2
(2-2)
S   i

 i 1 N  1 
Here, xi , x and N are respectively the individual, the estimated average, and the number of data
points. Figure 2-1 shows precision and bias errors of a set of measurements.
If the number of data points is large and they follow a normal distribution curve, and if there is
no bias, the mean is the estimated average of the data points and 2S is the uncertainty interval
with 95% confidence level or 20:1 odd. The 20:1 odd means that if the experiment is repeated 20
times, there is a possibility that 19 times the data falls within the range x  2S and there is only
once that it will be out of this range.
31
Experiments are either single sample or multiple samples. In single sample experiments, test
point is run once. In multiple sample experiments, each test point is run more than once.
Research in fluid mechanics and heat transfer are usually single sample experiments.
In single sample experiments, the precision error is dominant. In contrast, in multiple samples
experiments, the fixed error is dominant since in multiple measurements of a point, the random
errors are averaged out.
Figure 2-1
2.2. UNCERTAINTY IN EXPERIMENTAL RESULTS
In general, the results of experiments are calculated from a set of measurements:
R  R( x1 , x2 ,.., xN )
(2-3)
The uncertainty in experimental results should have the same odd or confidence level as the
measurements. The uncertainty in the result is expressed as:


 N  R
xi 
x


 i 1  i
R   
2
1
2




(2-4)
This method is called Root Sum Square (RSS). The partial derivative of the result, R, with
R
respect to the measurement xi ,
, is called the sensitivity coefficient. xi is the uncertainty in
x i
the variable xi . The sensitivity coefficient weights the value of the uncertainty of the variable
corresponding to the dominance of the variable in the fundamental equation.
The RSS method can be used to estimate uncertainty in experimental results if: (a) measurements
are independent and repeated measurements of each quantity follow a normal distribution and (b)
the same odds are used for estimating the uncertainty of each measurement. The RSS equation is
generally used in the normalized form as:
32
u
R

2

 x


R
  1
u 
 R x 1 

1 

x

 2
 R

2




2
 
R u   ..   x N R u 
N
x 2 
 R x
2
N


 

1
2
(2-5)
Here, uR and ui are, respectively, the uncertainties in the result and in the measurement of xi .
Example: The following equation is used in an orifice plate to calculate the volume flow rate of
the fluid:
Q  CA
P

Here Δ𝑃 is pressure differential, ρ is density and Q is the volume flow rate. If ΔP = 250±5 Pa
and ρ=1.195± 0.01 kg/m³, find the volume flow rate and its variations. C and A are constants at
0.7 and 0.01m² respectively.
Solution:
The volume flow rate is calculated using the average or nominal values of the given data.
P
250 N m 2
Q  CA
 (0.7) * (0.01m ) *
 0.101 m 3 s
3

1.195 kg m
In this problem, the uncertainty for volume flow rate is a function of the pressure and density
where these two variables have its measurement uncertainty. The uncertainty equation for Q is:
2
1
2
2
 P Q
   Q
 2
U Q   
U P   
U   
 Q P
  Q 
 
Taking the partial derivative and then substitution yields
Q CA  P 



P
2   

Q  CA  P 




2   
1
2
1
→


1
2
P

2
→
P Q

Q P
 Q

Q 
P

CA

CA 1

P  2  2

 
2


CA

CAP
1

2
2
P  2 

 
2
5
(2%)
 0.02
250
0.01
U 
 0.0084 (0.84%)
1.195
Thus, the uncertainty equation for Q becomes:
U P 
33
1
2
2
 1
  1
 2
U Q    U P     U     0.01084 or 1.08%
  2
 
 2
If both bias and precision errors are considered, first the root sum square of the biases and the
precision are calculated separately as:
1
2
 N  x R
 2
i
S R   
 xi  
 i 1  R xi
 

 x i R
 
 xi  
 i 1  R xi
 
N
 R   
2
1
2
(2-6)
Then, depending on the confidence level, one of the following equations is used to estimate the
total uncertainty, uR .
uR  (  R  tS R ) for 95% confidence level
(2-7)
Here, S R and  R are respectively the precision and bias errors in the final result, and t is from
the student t distribution. The student t distribution is used because the distribution of the
precision error does not have a normal distribution. It has a distribution that is dependent on the
number of samples and, therefore, the degrees of freedom. Thus, t in Equation 2-7 is dependent
on the number of samples collected. The shape of the student t distribution is very similar to the
normal distribution, but has a greater width to its bell shape.
For samples higher than 30, t is assumed to be 2. For samples less than 30, t is obtained from the
student t distribution table, using the required confidence level and the number of degrees of
freedom.,  . For single sample experiments, the sample number is the number of degrees of
freedom. Table 2-1 presents the values for t in terms of the degrees of freedom for the 95%
confidence level.
34
Figure for Table 2-1
Table 2-1
Equation 2-7 can be used to estimate the overall uncertainty, provided that the bias limit is
symmetrical about the measurement. If the bias limit is nonsymmetrical, then the upper and
lower limits of the uncertainty interval should be specified separately. Figure 2-2 shows the
measurement uncertainty for symmetrical and nonsymmetrical biases.
35
Figure 2-2
If the number of independent sets of measurements for calculating a quantity is more than one,
then first the cumulative degrees of freedom for the measurements is calculated from the WelchSatterthwaite equation as:
36
( S i2 ) 2
vR  N 4
Si

i 1 vi
(2-8)
Then, from the student t distribution the t values are found for the 95% confidence level. The
overall bias and precision errors are found from the following equations:
S

 Si
N
(2-9)
  i2
Here, S i ,  i , and N are respectively the overall precision and bias errors and the number of
independent sets of measurements. Assuming symmetrical bias, the uncertainty interval is:
u  (   tS ) (2-10)
37
REFERENCES
1.
Abernethy, et al, 1973, “Handbook, Uncertainty in Gas Turbine Measurements,” Arnold
Engineering Development Center Report No. AEDCTR-73-5.
2.
Abernethy, R.B., Benedict, R.P., and Dowdell, R.B., 1985, “ASME Measurement
Uncertainty,” ASME Journal of Fluids Engineering, Vol. 107,
pp. 161-164.
3.
Kline, S.J., 1985, “The Purpose of Uncertainty Analysis,” ASME Journal of
Fluids Engineering, Vol. 107, pp. 153160.
4.
Kline, S.J., and McClintock, F.A., 1953,
“Describing Uncertainties in Single
Sample Experiments,” Mechanical Engineering, pp. 3-8.
5.
Moffat, R.J., 1988, “Describing the Uncertainties in Experimental Results,”
Experimental Thermal and Fluids Science, Vol. 1, pp. 3-17.
38
EXAMPLES
1.
The Bernoulli‟s equation given below has been used extensively to obtain the mean
velocity from dynamic pressure measurement. What is the uncertainty in the U, given that the
uncertainties in U, P , and  are respectively u , p and  .
1
 2P  2

U  
  
Solution:
2
2
 P U
   U  
uU   
p   
  
U  P   U 
 

1
2
But,
1
1
P U P 1  2  2



U  P U 2  P 
 2P  2


1   
1


2 U
2
1
1
 U  1  2P  2



U  U 2   
 2P  2


1
1   
1



2 U
2
Then,
 1
  1 
uU    P     
  2

 2
2
2.
2



1
2
What is the uncertainty in the Reynolds number given as:
Re D 
UD

The uncertainties in  ,U , D, and  are respectively  , U , D, and  .
39
Solution:
u ReD
2
2
2
2
   Re
  U  Re D
  D  Re D     Re D  
D
  
   
U   
D   
  
 Re D 
  Re D U
  Re D D
  Re D 
 
1
2
But,

Re D
 Re D
UD 1

1

 Re D
U  Re D
UD 1

1
Re D U
 Re D
D  Re D
UD 1

1
Re D D
 Re D
  Re D UD  1

 1
Re D 
 Re D
then,

u ReD      U   D     
2
2
2
2

1
2
4. What is the uncertainty of the increase in length L, with a wire with diameter D, and a
load of F?
Uncertainty in diameter is d, the length is L, and the uncertainty in F is F. Assume
that the E modulus of elasticity has no uncertainty (E=0).
L
 e  L  L  e
L
40
2
2
 L
  L
 
L  
 L   
 e  
  e
 
 L
e
1/ 2

E
2
 e

 e

e  
    
 E  

 E


 


1
2

F
A
2
2
 
  
 
  
 F   
 A  
  A
 
 F
1/ 2
Substitute partial derivatives and rewrite,
2
2
 1
 F
 
    F    2  A  
  A
 
 A
1/ 2
2
2
1  1
 F
 
e     F    2  A  
E  A
  A
 
1/ 2
Hence,

L2
L  e  L 2  2
E

2
 F
 F
 
 
 F    2  A  
  A
 
 A
41
1/ 2
e
 

4.
A hook carries a load of F pounds. What is the uncertainty of the calculated maximum
stress in the vertical portion of the hook? The hook has an a x a square cross-section and the
radius in the bend of the hook is R. The uncertainty in length measurement is a, in radius is R
and the uncertainty in force measurement is F.
Substituting into the uncertainty equation, :

F My

A
I
2
2
2
2
2
 
  
  
  
  
 
  
 A   
 F   
 y   
 M   
 I  
  F
  y
  I
 
 A
  M
A = a2
 A  2 
A    
 a  
y
1/ 2
 2  a  a
a
2
2
 y
 
y    a  
 
 a
1/ 2

1
 a
2
M  FR
2
2
 M
  M
 
M  
 F   
 R  
  R
 
 F
I
1/ 2

 R  F   ( F  R) 2
2
b  h3 a 4

12
12
42

1/ 2
1/ 2
2
 I
 
I    I  
 
 a
1/ 2

4  a3
 a
12
2
2
2
 2  F  a
 1
 M 1
  a
2
2
  
 a     F      a   
 R  F   F  R 
2
 A
  I 2
  2 I
 A


1/ 2
3

  M a 4a

 a 
  
2
12
  2 I

2
2 1/ 2



Rearrange,
2
2
2
2
2
 2  F  a
 1 M  a4
 
 1
 M 1
  a 
2
2
  
 a     F   
  a   
 a  
  R  F   F  R    
2
I2
 A
  I 2
  2 I 
 A
6
 

43

1
2
3
Chapter 3
ELECTRICAL TRANSDUCERS
44
3.0. ELECTRICAL TRANSDUCERS
There are two general types of electrical transducers; active and passive transducers. Active
transducers generate an electromotive force (emf) when the physical property of interest is being
sensed. The sensitivity of the active transducers is expressed as the change in electrical output to
the change in physical input. Examples of the active transducers are thermocouples and
piezoelectric transducers.
Passive transducers produce a change in resistance, inductance, or capacitance by the physical
effect of the property being measured. The change is then being sensed by a voltage device. The
relative sensitivity of the passive transducers is expressed as
Z
Z
to the change in physical input.
Here Z is the internal impedance of the system. Examples of passive transducers are resistance,
inductive and capacitive transducers.
3.1. Active Transducers
3.1.1. Thermocouple
When two dissimilar metals are in contact with each other, they form a thermocouple. When
thermocouple is exposed to a temperature variation, an emf is generated. The net voltage
generated is:
Enet 
TL
 (e1  e2 )dT
T0
Here e1 , e2 , T0 and TL are voltages generated along each wire and the reference temperatures.
The above equation can be used if only two wires are used and the materials are
homogeneous and voltage output is not a function of position along the wire. Each wire
begins at temperature T0 and ends at temperature TL .
For construction “positive” and a “negative” metal should be used. A “positive” metal is the
one that the emf increases with temperature along its length. On the other hand a “negative”
metal is the one that the emf decreases with the temperature along its length. Figure 1 shows
characteristics of alloys most commonly used for thermocouple constructions.
Figure 3.1. Characteristics of metal alloys with temperature.
45
Figure 2 shows a thermocouple that measures Thot relative to ambient temperature Tamb as
the reference temperature.
Figure 3.2.
However, if other reference temperatures (i.e. ice bath) are required, then the configurations
shown in Figure 3 can be used. Here the copper extension wires do not affect the emf output.
Most electronic devices that are used for temperature measurements in conjunction with
thermocouples have already been calibrated to display temperature with reference to the icebath condition.
Figure 3.3.
Depending on the temperature range and sensitivity different materials are used for the wires.
Table 1 shows different thermocouples with their corresponding range and sensitivity.
Factors that can introduce errors in temperature measurements are:
(a) Wire inhomogeneties from defect, plastic deformation, and a change in chemical
composition;
(b) Straining of the wires due to variations in cold working of the wires or from vibration;
(c) The exposure of the thermocouple to electrolytes, which results in significant addition in
the emf output.
46
Type
Materials
Range
Sensitivity
C
C
Nickel-10%Chromium(+)
E
Vs
-100 to 1,000
0.5
0 to 760
0.1
0 to 1,370
0.7
0 to 1,000
0.5
0 to 1,750
1.0
-160 to 400
0.5
Constantan(-)
Iron(+)
J
Vs
Constantan(-)
Nickel-10%Chromium(+)
K
Vs
Nickel-5% (-)
Platinum-13% Rhodium (+)
R
Vs
Constantan(-)
Platinum-10% Rhodium (+)
S
Vs
Constantan(-)
Copper(+)
T
Vs
Constantan(-)
Table 3.1. Different thermocouples.
47
3.1.2. Law of Thermoelectricity
1. The thermal emf of a thermocouple with junctions at T1, 4 and T3 is not affected by
temperature elsewhere in the circuit (i.e. T2 ), provided that the two metals used are
homogeneous.
Figure 3.4.
2. If a third homogeneous wire C is inserted into either metal A or B, as long as the two new
junctions (2) and (4) are at the same temperatures, the net emf of the circuit is unchanged,
even if C is at different temperature than the measuring and reference temperatures.
Figure 3.5.
3. If wire C is inserted between A and B at one of the junctions, as long as the new junctions
(2) and (4) are at the same temperature, the net emf of the circuit will be the same as
before.
Figure 3.6.
3.1.3. Piezoelectric Transducers
The piezoelectric effect is the ability of a material to generate an electric potential when
subjected to a mechanical straining or to change dimension when subject to a voltage
potential. The sensing elements have high sensitivity and resonant frequency and respond
both to static and dynamic loadings. Examples of piezoelectric materials are quartz, ceramic,
48
sugar, and ammonium dihydrogen phosphate. Figure 7 shows various deformations of a
piezoelectric plate. The material that is widely used is quartz due to its stability. However,
quartz generates a very low voltage output. Generally a miniature voltage amplifier is packed
with the device to increase its sensitivity. Quartz is usually shaped into a silvered face thin
disk with electrodes attached to each face. The disk thickness depends on the required
frequency response since different thick nesses resonate at different frequencies.
Piezoelectric devices are used for force, pressure, stress, acceleration, and sound and noise
measurements, among others. Piezoelectric pressure transducers with quartz crystal can
measure pressure up to 100,000 psi while the ones with the ceramic can measure up to a
maximum pressure of 5,000 psi.
The piezoelectric effect is not present, until the element is subjected to a polarizing treatment.
The usual procedure is to heat the element to a temperature above 120 C and then apply a
high voltage (in the order of 10,000 Volts/cm thickness) to the element and then cool it down
to room temperature.
Figure 3.7. Configurations used for different measurements.
All piezocrystals have a Curie-point temperature, above which the structure of the crystal
changes and the piezoelectric effect is lost. Thus, care should be made not to expose the
crystal to a temperature above the Curie-point temperature.
The performance of these transducers can be improved by stacking multi-crystals together
under the same load.
3.2.0 Passive Transducers
3.2.1. Variable Resistance Transducers
The equation governing the operation of variable resistance transducers is:
49
R L
A
Here R,  , A are resistance, thermal resistivity of resistance material, and cross-sectional
area of conductor respectively, and L is the length of the conductor. The transducers can
sense any changes in the quantities given by the above equation. A common transducer is a
strain gage pressure transducer. It incorporates four active strain gage elements. When
pressure is applied, resistance of two of the elements decreases from tension and the other
two increase from compression. The difference creates an electric potential that can be
related to the measured pressure. Other types of strain-gage pressure transducers are:
1. Gage diaphragm transducers. Good for wind tunnel testing of different models since the
diaphragm with strain gages could be attached directly to the surface for local pressure
measurement. It has a fast time response.
Figure 3.8. A gage diaphragm transducer.
2. Cantilever type transducers are used for low pressure measurement. The cantilever
material should have higher stiffness than the applied load.
Figure 3.9. A bounded strain-gage cantilever transducer.
50
3. Pressure vessel transducers. Good for high pressure measurements. The pressure range is
1,000 to 100,000 psi. It consists of a cylindrical element with one end closed and the
pressure is applied to the open end. It has two active strain gages and two dummy ones
(to complete the bridge), attached to the outside surface.
Figure 3.10. A pressure vessel transducer.
4. Embedded strain-gage transducers. It consists of a shell with epoxy resin as embedding
material. The strain gages are placed inside the embedding material. When pressure is
applied, the strain gage resistances change which can be calibrated to measure applied
pressure. It has a fast response and has a maximum pressure not exceeding 750 psi.
Figure 3.11. An embedded pressure transducer.
5. Unbounded strain-gage transducers. It operates on the same principle as bounded
transducers (ex: gage diaphragm transducers). When pressure is applied, it causes
changes in strain, which results in changes in resistance of the wire or filament. It has a
slow response and is good for static pressure measurements.
51
Figure 3.12. An unbounded pressure transducer.
3.2.2. Inductive Transducers
Inductance is the property of a device that reacts against a change in the current that flows
through the device. The reaction would be a drop in voltage which can be calibrated to
represent measured value. The equation governing the operation of the inductive transducers
is:
L E
dI
Dt
Here E and I are the applied voltage and the current and L is the inductance. Inductance
changes with the change in the magnetic field. As the following figures show, a change in the
gap or position of the magnetic core results in a change in the magnetic field which changes
the inductance and subsequently the output. Inductive reactance, X L , is a measure of the
inductive effect and can be calculated from the following equation:
X L  2fL
Here f is the frequency of the applied voltage.
52
Figure 3.13. An inductive pressure transducer
3.2.3. Capacitive Transducers
The basic principle of operation of these transducers is governed by the following equation:
C  KA( N  1)
d
Here C is the capacitance, K is the dielectric constant, N is the number of plates, d is spacing
between plates, and A is the area of one side of a plate. When the physical property being
sensed changes one of the parameters on the right hand side of the above equation, the
capacitance changes which can be calibrated to represent the measured value. The following
figures show two typical methods for designing capacitive transducers. For method (a)
electrical capacitance exists between the diaphragm and the metal plates on each side. When
pressure is applied, the diaphragm is deflected which results in a change in corresponding
capacitance. Care should be made to keep dielectric clean.
In method (b), when the pressure is applied, a traveling wave through the dielectric causes a
change in the capacitance. The device should be used for measuring pressure changes that are
faster that the time required for the reflected wave in the bar to reach the electrodes location.
53
(a)
(b)
Figure 3.14. Examples of capacitive transducers.
54
4
Chapter 4
PRESSURE PROBES FOR FLUID
VELOCITY AND VOLUME
MEASUREMENTS
55
4.0. PRESSURE PROBES FOR FLUID VELOCITY MEASUREMENTS
4.1. Pitot Tube
In 1732, Henri Pitot offered the first description of a tube that was used to measure mean
velocity in a river and for this reason the pitot tube is named after him. Generally pitot tubes are
classified according to the quantities that they measure. The word pitot tube refers to a
cylindrical tube with its open end pointed upstream opposite to the flow direction which
measures impact or total pressure. Another type is pitot static tube which is two concentric tubes
that measure both static and total pressures. The difference between the two pressures are used to
obtain the mean velocity.
When total pressure alone is measured, local wall static pressure from static pressure hole is used
to obtain the pressure differential for calculation of the mean velocity.
Figure 4.1 shows example of a pitot static tube along with the corresponding variation of the
pressure coefficient as a function of the distance from the probe tip. The tube can be in cantilever
or straight forms and its tip can be hemispherical, square or elliptical shape. For pitot static
tubes, the static pressure holes are located at some distances downstream of the probe tip. Very
small error in the location of the static pressure holes results in considerable errors in measured
pressure.
The location of the static pressure holes is where the vacuum pressure is asymptotically
approaching zero. For cantilevered pitot static tube, the stem causes increase in pressure
upstream and it is comparatively easy to find a location where the vacuum due to the nose of the
cylinder is equal to the increase in pressure due to the stem. The static pressure holes are placed
at this location.
Figure 4.1.
56
If D is the diameter of the cylinder, d1 and d2 are respectively the diameters of total and static
pressure holes, and 1 and 2 are respectively the distances between the nose and static pressure
holes and the static pressure holes and the stem, Prandtl offers the following criteria for
designing a pitot static tubes:
1  3D
2  8  10 D
d1  0.3D
d2  01
.D
(8 holes equally spaced)
Some other designers have used 8D and 16D for respectively 1 and 2 .
Assuming there is no significant variation in the mean velocity across the nose of the tube, the
Bernoulli's equation for one dimensional flow is used to obtain the mean velocity from pressure
differential. For steady flow, along a streamline, the Euler's equation can be written as:
dp
1
 gdZ  dV 2  0

2
Here dp is the pressure difference,  is the fluid density, g is the acceleration of gravity, dZ
is elevation difference and V is fluid velocity along the streamline. Integration of above equation
leads to:
p
1
 gZ  V 2  Constant

2
If flow is incompressible (  is constant) and variation in potential energy is insignificant, and
knowing that velocity is zero at the surface, then we have:
pt  ps 
1
V 2
2
Here pt and ps are respectively the total and static pressures.
Pitot tube nose geometry should be such that it does not deflect the streamlines and causes flow
separations. The size of the tube introduces errors in the measurements. When the local pressure
in the vicinity of a body is reduced significantly, cavitation occurs. For a pitot tube, the
occurrence of cavitation beyond the tip does not change total pressure reading. However, for
pitot static tube, the presence of cavitation results in appreciable change in static pressure
measurement.
57
An error of 2% in measured pressure is introduced if the Reynolds number based on tube
diameter is higher than 300.
Static pressure holes are used for surface pressure measurements. From various experiments,
accurate static pressure measurements are obtained if the hole is perpendicular to the surface and
if its diameter is around 0.25 mm. If the hole diameter is 1mm or if it is off by as much as 45
degrees from the perpendicular line, an error equal to 1% of the dynamic pressure is introduce in
the measurements.
In a two dimensional flow field, a two dimensional probe measures average magnitude and
direction of the velocity in a small area rather than at a point. The effective center, yc of total
pressure of a square ended circular impact tube is given by Young and Mass (1936) as:
yc
d
 0131
.  0.82 1
D
D
Results of Hemke (1926) shows that if the total pressure hole is less than or about 0.15 mm,
accurate pressure measurements are obtained.
For tubes used in a rake, results of Krause (1951) show that correct static pressure measurements
can be obtained if the center distance between heads is greater than 6 head diameter. However, if
the Mach number is greater than 0.6, greater distance is required.
4.1.1. Yaw Angle Sensitivity and Tube Inlet Geometry
The impact pressure is very sensitive to the yaw angle. In order to have a probe which is
insensitive to the angularity, a shielded total pressure probe known as a Kiel probe is used.
Figure 4.2 shows a Kiel probe. Results of Gracey et al (1951) show that the total pressure from
kiel probe is insensitive to the angularity up to Mach number 0.3 and then decreases with
increasing Mach number. Those probes with curved entry have less sensitivity to the angularity
than those who have straight conical entry. Figure 4.2 also shows a minimum type probe
proposed by markowski and Moffat (1948) which is used to obtain total pressure in
turbomachinary.
58
Figure 4.2
Dudziniski and Krause (1971) studied the effects of inlet tube geometry of miniature total
pressure tubes on pressure measurements. Their studies are limited to five inlet geometry of
circular, flattened oval and internal bevel at 15, 30, and 45 degrees. Tables 4.1 and 4.2 show the
tube tested and their dimensions. Testing criteria is that up to what flow angle there is only 1%
error in the total pressure as compared to the impact pressure. The tubes are tested in air in the
Mach number range of 0.3 to 0.9 and Reynolds numbers of 500 to 80,000. The flow angle is
changed between -45 degrees and 45 degrees. Their results show that the internally beveled tube
with 15 degrees bevel angle has the highest flow angle ( 27 ) for producing accurate total
pressure. The flow angle decreases with increasing the bevel angle. Minimum flow angle of 12
degrees is obtained for the flattened oval geometry.
59
Tables 4.1 and 4.2.
60
4.1.2. Turbulence Effects
Generally turbulence in a fluid stream causes a pressure reading that is higher than the true
pressure. Goldstein (1936) offers the following expressions for true total ( Pt ) and static ( Ps )
pressures in isotropic turbulent flow:
Pt  Ptmeasured
q2
q 2
 
2
2
Ps  Psmeasured
q 2

6
Here, q and q are respectively mean and fluctuating components of measured velocity.
In a pipe flow, based on experimental results of Townsend (1932), Fage (1936) offer the
following expressions for obtaining true static pressure:
Ps  Psmeasured
q 2

4
Christiansen and Bradshaw (1981) studied the effects of free stream turbulence intensity of 24
percent, on the mean pressure coefficients of several static pressure probe and yaw meter, for a
range of yaw angles and ratios of tube size to typical eddy size. The probes tested are shown in
Figure 4.3 are a standard static pressure probe, a disc static pressure probe, a Conrad three-hole
yaw meter, a Gupta three hole yaw meter, and a single hole yaw meter.
61
Figure 4.3
Their results show that the multi-hole pressure probe yaw meters are not much affected by the
ratio of the eddy size to the probe size. However, the effects of free stream turbulence can
become large if the instantaneous yaw angle is larger than 10 degrees. The yaw angle calibration
in free stream turbulence shows significant difference from the corresponding results for the
laminar free stream beyond this value.
The turbulence effect on readings of Gupta probe was much severe than others. Their overall
results show that the yaw meter probes are excellent device for obtaining the mean pressure
coefficient, especially if they are used as a null reading devices.
The disc static pressure probe was sensitive to the flow angles and the free stream turbulence.
Small flow angles and free stream turbulence caused significant change in its calibration and it
was recommended not to use this probe in turbulent flow.
For standard static pressure probe, results show that the variation of the static pressure
coefficient with eddy size was significant. However, the error caused due to the presence of free
stream turbulence for the range of ratios of tube size to eddy size investigated, was about  2
percent of the dynamic pressure.
4.1.3 Measurement at Low Reynolds Number
The accuracy of the total pressure tube is not only depends on the geometry of the tube and the
impact opening but also depends on Reynolds number based on tube external head diameter. For
Reynolds number less than 500, the magnitude of the pressure coefficient define as:
62
Ci 
Pi  P
1
U 2
2
becomes increasingly higher than unity with decreasing Reynolds number. Here Pi and P are
respectively impact and free stream static pressures and U  is the free stream mean velocity.
Table 4.3 (Folsom (1956)) gives theoretical pressure coefficient for impact tubes with different
probe head shapes at low Reynolds numbers.
Table 4.3
4.1.4 Measurements in Supersonic Flows
In supersonic flow the pitot static tube is usually a cone-cylinder type with 15 degrees total
included angle with static holes are located about 13 tube outside diameter downstream of the
cone base. Results of the experiments show that for Reynolds numbers based on tube outside
diameter of higher than 3500, accurate static and total pressure measurements are obtained.
Some results show that the accuracy decreases with decreasing Reynolds number.
63
4.2 Five Holes Probe
Five hole probes are mostly used to obtain the components of the mean velocity vector in a three
dimensional flow field. As pointed out by Treaster and Yocum (1979), the first application of the
five hole probe dates back to 1915 by Taylor and followed by Pien (1958) who found that a set
of calibration charts are necessary before the probe can be used in any experiments. Figure 4
shows a five hole probe along with the corresponding equations for obtaining mean velocity
components. Depending on the mode of calibration, the equations for obtaining the velocity
vectors are different. The main idea is to use the right equations for obtaining the mean velocity
components.
Figure 4.4.
Based on the geometry shown, for obtaining the calibration charts, the pressure coefficients and
average pressure are defined as:
P2  P3
P1  P
Cp yaw 
Cp pitch 
Cptotal 
64
P4  P5
P1  P
P1  Ptotal
P1  P
Cp yaw 
P
P  Pstatic
P1  P
P2  P3  P4  P5
4
Figure 4.5 shows examples of the calibration charts for different pitch and yaw angles.
Figure 4.5.
When measurements are performed at a point in a flow field, first the Cp yaw and Cp pitch are
calculated from above equations. Then from the calibration charts the pitch and yaw angles are
found and then Cptotal and Cpstatic are obtained. The magnitude of the velocity vector is then
obtained from the following equation:
V

P  P 1  C

2
1
pstatic
 C ptotal

The mean velocity components are obtained from the corresponding equations developed from
the flow configuration and method of calibration. Here,
65
P1  P1  Pref
and
P 
P2  P3  P4  P5
4
For obtaining the three components of the mean velocity in low speed flows, Ligrani et al (1989)
presents description along with calibration charts for a miniature five hole pressure probe. Their
results compared well with the corresponding results from a Kiel probe in a curved channel.
Measurements of Treaster and Yocum (1979) show week dependency of
Reynolds number and dependency of
Cpstatic
on flow
Cp yaw and Cpstatic on the distance between the probe and
the wall. The Reynolds number dependency can be avoided if the calibration charts are obtained
at or in the range of Reynolds numbers that the experiments are performed.
When the center hole of the probe is within two probe diameters from the wall and is outside the
boundary layer, the measured pressure coefficients are inaccurate. However, when the probe is
used for measurements in boundary layers, the main errors are due to the spacing between the
probe off center pressure taps and the presence of the velocity gradient upstream of the probe.
All measured pressures from off center pressure taps and resulting velocity components should
be corrected for the corresponding pressure or velocity gradient in spanwise and vertical
directions (i.e. Sitaram et al (1981), Eibeck and Eaton (1985), and Westphal et al (1987)). The
procedure may be to subtract addition of the mean velocity components due to the velocity
gradient upstream of the probe from the corresponding measured values.
Another approach for correcting for spatial resolution and downwash velocity for multiple-hole
probes is given by Ligrani et al (1989) where for pressure corrections, cubic spline curves are fit
to the measured pressures at different spanwise but constant vertical locations and then the
values of off center pressure taps are corrected by interpolating them to the center pressure hole
location. For correcting velocity components in spanwise and vertical directions, addition of
velocity gradient in corresponding direction multiplied by a constant is proposed. In their case
which was measurements in a curved channel flow, the constant was 20% of the probe diameter.
Westphal et al (1987) from calibration results of Treaster and Yocum (1979) show that for
variation in pitch and yaw angles of  20 degrees,
Cp yaw and Cp pitch are nearly independent of
pitch and yaw angles respectively and it is possible to express variation of these coefficients with
respectively yaw and pitch angles in terms of polynomials. This approach is preferred in flow
configurations where the flow angle does not go beyond  20 degrees.
66
Huffman et al (1980) used slender body theory to obtain theoretical relationships for variation in
average, pitch, yaw, static and total pressure coefficients for a conical probe. Figure 4.6 shows
their probe configuration and dimensions. Their theoretical results for average pressure
coefficient compares well with the corresponding experimental results up to Mach number 0.9.
The theoretical calibration charts for the variation in Cp yaw and Cp pitch with pitch and yaw angles
also have similar characteristics as the corresponding experimental results. However, the theory
becomes incapable of predicting correct calibration charts for pitch and yaw angles beyond  20
degrees.
Figure 4.6.
In order to increase the range of flow angles for the five hole probes, Ostowari and Wentz
(1983) present new calibration approach by replacing the center pressure tap with the upwind
port pressure as the reference total pressure and the stalled downwind pressure tap with the
center pressure tap. Figure 4.7 shows their five tube probe. This approach extends the calibration
range for the pitch angles while the yaw angle remains in null position. Their results show good
calibration curves up to the flow angles of  85 degrees.
4.3. Preston Tube
Prston (1954) based on the law of the wall proposed a method for determining the wall
shear stress and subsequently the skin friction coefficient in boundary layers. His method was
based on measuring the total pressure by placing a pitot tube on the surface and the static
pressure which is measured on the surface at the same location. In non-dimensional form the
pressure difference is related to the friction velocity (U ) and subsequently the wall shear stress
as:
67
P
 U d 
 f

2
U 
  
Figure 4.7.
Near the surface, the boundary layer thickness depends on the wall shear stress,   , density of
the fluid,  , kinematics viscosity,  , and a length scale. For a pipe flow Preston proposed the
following relation between the wall shear stress and the measured dynamic pressure:
 P0  Pw 2 
 d 2
 f 
d 
2
2
4 
 4 

Here, P0 and Pw are respectively total and wall static pressures and d is the diameter of the pitot
tube. From measurements of flow through pipes, he found the following relationship between the
measured pressure and the wall shear stress:
 d 2
log 10 
2
4



 P  Pw d 
7
  2.628  log 10  0

2

8
4




68
Patel(1964) performed calibration of the preston tube in three different pipes using fourteen
different pitot tubes for three cases of zero, adverse and favorable pressure gradients. His results
show that the calibration results of Preston are not very accurate. Using McMillan relation for
1
effective center of the pitot tube defined as y  Kd where in the sublayer region K is defined
2
as 0.3, he offered the following equations for the case of Zero pressure gradient:
in linear sublayer:
y* 
1 * 1
1

x  log 10  K 2  for y *  1.4
2
2
2

in transition region:
1 *
 A
y

 B 
x  y  2 log 10 
log 10  K  10 2  C  

2
 2



for 1.4  y * < 3.56
*
*
and finally in the fully developed turbulent region
B 
 A
x *  y *  2 log 10 
log 10 y *  2 log 10 K 

2
2 2
for 3.56  y *


The constants A and B are determined to be 5.5 and 5.45 respectively. The expressions for
x*and y* are:
y  log 10
*
P0  Pw d 2
4  2
 d 2
x  log 10
4  2
*
In transition region the constant C is difficult to determine and instead the following cubic
polynomial used:
69
y  0.8287  01381
.
x  01437
.
x  0.006 x
*
*2
*
*3
15
.  y*  35
. .
Preston also offered the same equation for the range
Patel also showed that in the case of adverse pressure gradient, large diameter preston tube
register larger errors and in the case of favorable pressure gradient the preston tube register
nearly correct readings. Since for large pressure gradient, departure from the law of the wall is
seen, he offered the following restrictions in obtaining the wall shear stress using the Preston
tube:
i) Adverse Pressure Gradient
maximum error 3%
0    0.01,
U d

 200
maximum error 6%
U d
0    0.015,   250

ii) Favorable pressure gradient
maximum error 3%
0    0.005,
U d

 200
maximum error 6%
U d
0    0.007,   250

Here  which is obtained from dimensional analysis defined as
70
 dP
U3 dx .
4.3. Volume Flow Measurements
4.3.1. The Venturi Meter
The venturi meter uses reduction in area and the resulting changes in pressure to measure the
volume or mass flow passing through the tube. Figure 4.8 shows two design consideration, one
for inline continuous measurements and the other using a bypass valve for periodic monitoring of
the flow volume. The pressure taps placed upstream and downstream of the venturi are used for
pressure measurements.
2 gc
Qa  C * M * A2

( P1  P2 )
Qa  Actual Volumetric Flow
C  Discharge Coefficien t 
Q a Actual Volumetric Flow

QI
Ideal Volumetric Flow
 A 
M  Velocity of Approach factor  1 -  2 
  A1 
2




1
2
Figure 4.8. The venturi meter and its set-up design.
71
4.3.2 The Orifice Plate
The orifice plate is used for differential pressure measurements. It is used for both liquid and gas
applications. The differential pressure between upstream and downstream sides of the orifice
plate is measured using pressure taps located on the corresponding sides. The pressure taps
should be located far enough from the orifice plate where there is no flow recirculation.
2gc
Qa  KA2

( P1  P2 )
Qa  Actual Volumetric Flow
K  Flow Coefficien t
Figure 4.9. The orifice plate and its flow configuration.
4.3.3. Rotometers (variable Area Meter)
The position of the float or Bob is as a result of a balance between the drag force on Bob and
Bob‟s weight and the buoyancy force. The position can be calibrated to show mean velocity, U m ,
volume flow rate, Q , or mass flow rate, m .
Taking D= diameter of the tube at inlet, d=diameter of Bob, C= meter‟s constant (found through
calibration),  B ,  f , A B, and VB are respectively the density of the Bob, fluid density, projected
area and the volume of the Bob, The equations for U m , Q , and m are:
72
 2 gV   B   f
B 
Um  

 CD AB   f
Q  U m A, where A 

m  CZ (  B   f )  f





1/ 2

( D  Z )
4

2
d2

1/ 2
Here C D and  are respectively the drag coefficient of Bob and a correction parameter related
to the tube taper.
Figure 4.9. The Rotameter (the Variable Area Meter)
73
4.3.4. Doppler Flowmeter (Meters Liquids in Pipe Flow)
It is used for measuring mass flow rate, volume flow rate and/or mean flow velocity.
An ultrasonic beam from the piezoelectric crystal is transmitted through the pipe wall into the
fluid at an angle to the flow stream. Reflected signals of the flow disturbances are detected by a
second piezoelectric crystal. The two signals are compared and their differences are calibrated to
indicate the required measurement.
Limitations:




Pipe wall thickness should be less than 1.92 cm.
Liquid mean velocity should be higher than 0.15 m/sec.
The metered liquid should have at least 2% suspended solids by volume.
Pipe material should be homogeneous composition (For example it can not be used for
concrete or composite pipes).
Figure 4.10. The Doppler Flowmeter
74
References
Cheremisinoff, N.P., Applied Fluid Flow Measurement, fundamentals and technology, 1979,
Marcel Dekker, INC, New York.
Christiansen, T. and Bradshaw, P., 1981, ``Effect of Turbulence on Pressure probe,” J. Phys. E.:
Sci. Instrum., Vol. 14, pp.992-997.
Eibeck, P.A., and Eaton, J.K., 1985, ``An Experimental Investigation of the Heat Transfer
Effects of a Longitudinal Vortex Embedded in a Turbulent Boundary Layer,” Report MD-48,
Mechanical Engineering Department, Stanford University, Stanford, California.
Fage, A., 1936, ``On the Static Pressure in Fully-Developed Turbulent Flow,” Proceedings of the
Royal Society of London, Series A., Vol. 155, pp. 576-596.
Folsom, R.G., 1956, ``Review of the Pitot Tube,” Transaction of the ASME, pp. 1447-1460.
Fox, R.W., and McDonald, A.T., Introduction to Fluid Mechanics, Fourth Edition, Wiley, 1992.
Goldstein, S., 1936, `` A Note on Measurement of Total Head and Static Pressure in a Turbulent
Stream,” Proceedings of the Royal Society of London, Series A., Vol. 155, pp. 570-575.
Gracey, W., Letko, W., and Russel, W.R., 1951, ``Wind Tunnel Investigation of a Number of
Total Pressure Tubes at High Angles of Attack-Subsonic Speeds,” NACA TN 2331.
Hemke, P.E., 1926, ``Influence of the Orifice on Measured Pressures,” NACA TN 250.
Huffman, G.D., Rabe, D.C., and Poti, N.D., 1980, ``Flow Direction Probes From a Theoretical
Point of View,” J. Phys. E.: Sci. Instrum. Vol. 13, pp. 751-760.
Krause, L.N., 1951, ``Effects of Pressure-Rake Design Parameters on Static Pressure
Measurement for Rakes Used in Subsonic Free Jet,” NACA TN 2520.
Ligrani, P.M., Singer, B.A., and Baun, L.R., 1989, ``Spatial Resolution and Downwash Velocity
Corrections for Multiple-Hole Pressure Probes in Complex Flows,” Experiments in Fluids, Vol.
7, pp. 424-426.
Markowski, S.J., and Moffatt, F.M, 1948, ``Instrumentation for Development of Aircraft Power
Plant Components Involving Fluid Flow,” Trans. SAE, Vol. 2, pp. 104-116.
Patel, V.C., 1965, ``Calibration of the Preston Tube and Limitations on Its Use in Pressure
Gradients,” J. Fluid Mech. Vol. 23, Part 1, pp. 185-208.
Pien, P.C., 1958, ``Five-Hole Spherical Pitot Tube,” David Taylor Model Basin Report 1229.
75
Preston, J.H., 1954, ``The Determiation of Turbulent Skin Friction by Means of Pitot Tubes,” J.
Roy. Aero. Soc. 58,109.
Sitaram, N., Lakshminarayana, B., and Ravindranath, A., 1981, ``Conventional Probes for the
Relative Flow Measurement in a Turbomachinary Rotor Blade Passage,” Transactions of the
ASME, Vol. 103, pp. 406-414.
Treaster, A.L., and Yocum, A.M., 1979, ``The Calibration and Application of Five-Hole Probes,
” ISA Transactions, Vol. 18, No.3, pp. 23-34.
Townsend, H., 1934, ``Statistical Measurements of Turbulence in the Flow of Air Through a
Pipe,” Proceedings of the Royal Society of London, Series A., Vol. 145, pp. 180-211.
Westphal, R., Pauley, W.R., and Eaton, J.K.,1987, ``Interaction Between a Vortex and a
Turbulent Boundary Layer, Part I: Mean Flow Evolution and Turbulence Properties,” NASA TM
88361.
Young, A.D., Mass, J.N., 1936, ``The Behaviour of a Pitot Tube in a Transverse Total-Pressure
Gradient,” British ARC, R & M 1770.
76
5
Chapter 5
TEMPERATURE MEASUREMENTS_____
Compiled and written by: Huy Hoang, Ph.D.
77
Temperature measurements have countless applications in the world today.
Typical examples of application ranges from the thermostat controlled temperature for
the air conditioning inside the house during the summer season to the heat rejection of
a spacecraft in space.
Temperature of an object is defined as a measure of thermal potential of that
object. It cannot be measured by using basic standard methods for direct comparison.
It can only be determined through the some standardized form of calibrated devices.
Various instruments have been developed for different applications of temperature
measurement. The basic operations of these instruments are designed to accompany
its primary effects of temperature change. The effects include the changes in physical
state, in chemical state, electrical properties, mechanical properties, and optical
properties of thermometers.
I.
Liquid-in-Glass Thermometers
Mercury-in-glass thermometer is an example of liquid filled thermometer. The
thermometer consists of a relative large bulb at the lower end, a capillary tube with
scale, and mercury filling both the bulb and a portion of the capillary as shown in Figure
5.1. The volume enclosed in the capillary above the liquid may contain a vacuum or be
filled with air or inert gas. As the temperature is raised, the expansion of the mercury
causes it to rise in the capillary of the thermometer. The height of the rise is used as a
measure of the temperature. By calibrating the expansion of the mercury at different
temperature, the thermometer can provide accurate temperature reading.
Figure 5.1. Mercury-in-glass thermometer
There are several desirable properties for liquid-in-glass thermometer. These
properties include:
1. The relationship between temperature and dimensional change should be
linear.
78
2. The liquid should have a coefficient of expansion as large as possible. Large
coefficient of expansion makes possible larger capillary bores and easier
reading.
The coefficient of expansion for alcohol is larger than mercury.
3. The liquid should operate in a reasonable temperature range without change
of state. Mercury is limited at the low-temperature end by its freezing point,
-39ºC, and at the high-temperature end at its boiling point, 357ºC. Alcohol is
limited at the low-temperature end by its freezing point, -75ºC, and at the
high-temperature end at its boiling point, 127ºC.
4. The liquid should be safe to human health and environments. Mercury has
very high toxicity and chemical hazardous. For this reason, alcohol filled
thermometer is the preferable device.
II.
Thermocouples
A thermocouple consists of two wires of dissimilar metals coupled at the probe tip
(sensing/or measuring junction) and extended to the reference junction of known
temperature. When the probe tip is heated, the temperature difference between the
sensing junction and the reference junction is measured the change in voltage at the
reference junction. The change in voltage is caused by the Seebeck effect named after
Thomas Seebeck discovered this phenomenon in1821. The temperature gradients
along the conductors between the two junctions generate a small voltage which is
known as the Peltier electromotive force (e.m.f). The voltage is usually measured in
milivolts (1/1000th of Volt). The thermocouple wires are chosen according to the
maximum temperature measurement and linearity of its output (e.m.f) versus
temperature.
Figure 5.2. Thermocouple circuit and probe
Thermocouple Types
There are several thermocouple types available for a wide range of temperature
from -250ºC to 3000ºC. Each type has a useful temperature range and sensitivity for
different applications. American Society for Testing and Materials (ASTM), which is
79
recognized in the United States as the authority for temperature measurements, has
established the guidelines for different thermocouple types. The ASTM E-230
guidelines cover the wire composition, color codes, and manufacturing specifications.
Commercial thermometers are specified by Instrument Society of America (ISA)
types. The ISA types are recognized by the letter designations. Thermocouple with
letter types E, J, K, R, and T are common used. Type E, J, K, N and T are base metal
thermocouples and can be used up to 1000C. Base metal thermocouples compose of
common and inexpensive metal wires such as nickel, iron, and copper. Type S, R, and
B are noble-metal thermometer and can be used up to about 2000C. Noble metal
thermocouples are manufactured with precious metal wires of Platinum and Rhodium.
Type C is refractory metal thermocouple and can be used up to 2320C. Refractory
metal thermocouples are very expensive, difficult to manufacture, and wires made from
the exotic and brittle metals like tungsten and Rhenium.
The following table summarizes the basic properties of different thermocouple
types. Each manufacturer has its own thermocouple’s specifications.
Table 5.1. Different thermocouple types
80
Conversion of Voltage to Temperature
The measured voltage that a thermocouple produces can be converted to
temperature. The voltage and temperature are not quite linear regression of the firstorder. A polynomial equation that best describes the relationship between the voltage
(V) versus temperature (T) for a wide range of temperature is:
T
n N
a V
n 0
n
n
 a0  a1V  a2V 2  ..  anV n
A typical 𝑎𝑛 coefficient for thermocouple types E, J, K, R, S and T is listed the
Table 5.2.
Table 5.2. Coefficients of polynomial equation for voltage to temperature conversion.
Measuring Junction Types
Figure 3 shows the layout of measuring junction types. The choice of measuring
junction types depends on the application. Generally, the grounded junction offers the
best compromise between performance and reliability.
Bare wire (exposed) junction provides the fastest response time but leaves the
thermocouple wires unprotected against corrosion or mechanical damage.
Insulated (ungrounded) junction provides the protection for thermocouple junction
and avoids the ground loops between instruments, power supplies, and sensors. The
81
lead wire is fully insulated from the shield. The insulated junction is excellent for
applications where stray electromotive forces would affect the reading and for rapid
temperature cycling. Response time is slow due to insulation.
Grounded junction provides very fast response time approach of the bare wire
junction. The shield and the lead wires are welded together, forming a complete shield
and integrated junction. The grounded junction is recommended for applications with
the presence of liquids, moisture, gas, or high pressure.
Figure 5.3. Layout of measuring junction types
III.
Thermistors
A thermistor is a thermally sensitive variable resistor which is made from semiconducting material. It has a very fast non-linear response to the temperature change
and the majority of the thermistors has negative response coefficients due to their
resistances decrease with increasing temperature. It is used widely in the laser diode
and detector cooling applications because of their high sensitivity, small size, fast
response times, and low cost. A thermistor is usually restricted to a temperature range
of -100°C to 250° C. Figure 5.4 shows typical thermistor sensors and probes.
82
Figure 5.4. Typical thermistor sensors and probes.
The resistance-temperature characteristics for most thermistors are described by
the Steinhart-Hart equation:
1
 A  B * (ln R)  C * (ln R)3
T
where T is the absolute temperature (in Kelvin), R is the resistance in Ohm, and
A, B, and C are constants which can be determined from measured values of resistance
and temperature. Typical resistance-temperature response curves are shown in Figure
5. For good calibration data, the Steinhart-Hart equation introduces errors of less than
0.1°C over a temperature range of -30°C to 150°C.
.
83
Figure 5.5. Resistance-Temperature response curves
IV.
Resistance Thermometers
Resistance temperature detector (RTD) contains a resistor that changes the
resistance value as its temperature change. Unlike the thermistor uses ceramic
semiconducting materials which response inversely with temperature, the RTD probe
has the resistance increase with temperature. RTD is available in a tube and wire and
mineral construction. The platinum resistance thermometer is a common device of the
RTD. It consists of a length of platinum which has been trimmed in length to give an
accurate resistance of 100 at 0 C. The wire is wounded and protected with insulation. It
is then further protected with a metallic shell. The resistance changes with a change in
temperature. It provides a nearly linear temperature resistance relationship which is
stable over a long time period. Its thermal response suffers due to degree of
mechanical protection. Platinum resistance thermometer is typically used for
temperature sensitivity and laboratory applications.
84
Figure 5.6. Platinum resistance thermometer
V.
Bi-metal Thermometer
The bi-metal thermometer consists of two thin metal strips with different
coefficient of linear expansions that are fastened together. The result is a strip that
bends significantly when heated. One end of the strip is fixed and the other end is
connected to a pointer on a calibrated dial that deflects with changes in temperature.
When the strip is formed into a coil, its sensitivity is increased.
Figure 5.7. Bi-metal Thermometer
Bi-metal thermometers are used in industrial process, food processing and waste
water. A typical bi-metal thermometer has maximum pressure of 125 psi (861 KPa) and
the ambient operating conditions from -50°C to 120°C. The selection of bi-metal
thermometer is based on the media and the ambient operation conditions. The bimetal
should not expose continuously to process temperature over 425°C.
85
VI.
Pyrometers
Pyrometer is a non-contacting thermometer device to measure temperature in
the form of thermal radiation. Figure 5.8 shows a typical pyrometer instrument. There
are three different instruments referred to as pyrometers: total radiation, the optical, and
the infrared pyrometers.
Figure 5.8. A typical pyrometer instrument
The total radiation pyrometer has a thermal sensor that infers the temperature of
an object by detecting its naturally emitted thermal radiation. The optical pyrometer
collects the visible and infrared energy from an object and focuses through lenses on a
detector. The infrared pyrometer is similar to the total radiation except that the
measurements are restricted to the infrared spectrum from 0.7 to 1000m wavelengths.
The detectors of pyrometers convert the collected radiation energy into an electrical
signal to drive a temperature display as shown in Figure 5.9.
Figure 5.9. Infrared measuring system pyrometer.
86
The pyrometers offer the following advantages:
1. It is very fast and time savings.
2. Measurements can be taken for very high temperature or high voltage objects
without in contact with them. Typical applications are the temperature
measurements for the molten glass or molten metal during the smelting and
forming operations.
3. Measurements can be taken for moving target or object.
4. There is no energy lost or interference between the instrument and the object.
5. There is no hazardous risk or contamination on the surface of the object.
THERMAL RESPONSE
The most important characteristic for selecting a thermometer for a particular application
is its thermal response. The thermal response is defined by the time it takes to change
63.2% of the step change. This is defined as the “time constant” and is irrespective of
the step change in temperature. Figure 5.10 shows a temperature-time plot for a
thermometer with two different step changes. The time constant remains the same,
irrespective of the difference in temperature step changes.
Figure 5.10. Time constant
87
References
1. http://dtec.net.au/Thermocouple%20Probes,%20EGT%20Exhaust%20Gas%20Tempera
ture%20%E2%80%98pyro%E2%80%99,%20K%20type.htm
2. http://instrumentationandcontrollers.blogspot.com/2010/06/thermocouple-withcircuit.html
3. Thermocouple_Theory
http://www.sensortecinc.com/docs/technical_resources/Thermocouple_Theory.pdf
4. Measuring Temperature with Thermocouple
http://www.noise.physx.u-szeged.hu/digitalmeasurements/sensors/thermocouples.pdf
5. http://www.efunda.com/designstandards/sensors/thermocouples/thmcple_intro.cfm
6. http://www.accessscience.com/overflow.aspx?SearchInputText=Thermometer&ContentT
ypeSelect=10&term=Thermometer&rootID=797745
7. http://www.npl.co.uk/educate-explore/factsheets/temperature
8. http://www.winters.com/pdfs/Cut%20Sheets/Thermometer/BimetalThermometer.pdf
9. http://www.circletrack.com/chassistech/ctrp_0801_wheel_tire_products/photo_06.html
10. Principles of Non-contact Temperature Measurement
http://support.fluke.com/rayteksales/Download/Asset/IR_THEORY_55514_ENG_REVB_
LR.PDF
11. http://www.globalspec.com/reference/10956/179909/chapter-7-temperaturemeasurement-radiation-pyrometers
12. http://www.efunda.com/designstandards/sensors/thermistors/thermistors_intro.cfm
13. http://assets.newport.com/webDocuments-EN/images/AN02_Select_Thermistor_IX.pdf
14. Beckwith, Thomas G. and Magrangoni, Roy D., Mechanical Measurements, 4th Edition,
Addison-Wesley Publishing,1990.
15. Rahai, Hamid R., MAE 300 Experiments, CSU Long Beach, CA. Spring 2011.
88
6
Chapter 6
DATA ACQUISITION
Compiled and written by: Huy Hoang, Ph.D.
89
Data collection by hand for a single or multiple sample experiments is a tedious
task and time consuming. In addition to the experiment uncertainties, human errors are
difficult to avoid due to the lack of patience for doing the repetitive tasks. Also, some
complex measurement tasks could not easily be done by hand. With the advance of
modern technology, data can be easily recorded using automate systems with the use
of personal computers (PC). Processed data can be display on the screen of
computers instantaneously for most real time applications. The method of acquiring
and recording data using computer controlled is called data acquisition.
Data acquisition is a process of acquiring signals from a sensor, digitizing the
signals with a data acquisition device, and application software. Figure 6.1 shows the
schematic of data acquisition process.
Sensors
Data acquisition card
Computer with application software
Hardware drivers
Data acquisition storage
Application software program
Figure 6.1 Schematic of data acquisition
1. Sensors
A typical sensor used for acquiring signals from real world phenomena is a
transducer. The transducer is a device that converts any measurement
parameters from mechanical or electrical energy into electrical signals. For
temperature measurements, a thermocouple produces a change in voltage emf
between the sensing junction and the reference junction when it is heated up.
The voltage is measured in milivolt (mV). For pressure and displacement
measurement, a strain gauge is an example of resistance transducers that
senses the strain produced by the forces applying on the wires of the gauge.
The resistance is proportional to the change in the length of the wires and is
measured in ohms.
90
Tension causes resistance increase
Resistance measured
between these points
Gauge is insensitive to
lateral forces
Compression causes resistance decrease
Figure 6.2. Strain gauge measurement
2. Data acquisition(DAQ) Card
A device that reads the analog signal and converts it into digital signal is known
as data acquisition card. Figure 6.3 shows a typical multi-function DAQ card for
desktop PC. Figure 6.4 shows an external USB type DAQ device for laptop and
desktop PC with the DAQ card inside. DAQ card acts as an interface between
the sensor and the computer. It has the following features:
- Analog Input
- Analog Output
- Digital Input/ Output(I/O)
- Counter/Timers
91
Figure 6.3. Multifunction data acquisition card ADLINK ACL-8112
Figure 6. 4 NI USB-6008 DAQ device for laptop and desktop computer
a. Analog Input
An analog input is an input port to the DAQ card specifically for continuous,
analog signals. Typical features of analog inputs are the number of channels,
sampling rate, multiplexing, signal filtering, resolution, and input range.
 Number of channels-This parameter specifies the number of channel
inputs on boards of DAQ card. There are two types of channel inputs:
single-end and differential. Single-ended inputs are used when the
input signals are greater than 1V, the leads from the signal source to
the analog input hardware are less than 15 ft, and all input signals
share a common ground point. For differential inputs, each input has
its own ground reference.
92





Sampling rate-The effective sampling rate is inversely proportional to
the number of channels. A fast sampling rate acquires more input
data points for a given time and can form a better representation of
original signals.
Multiplexing-This is a common process when the DAQ card has to
sample signals from many channels instead of one. The DAQ
samples one channel at a time before switching to the next channel.
Signal filtering-The use of filtering is to remove the noise and
unwanted signals.
Resolution-The number of bits to represent the analog signal is called
a resolution. The higher the number of divisions of the input gives the
higher the resolution of the signals
Input range- The electrical signals from the sensors or transducers
must be optimized for the input ranges of the DAQ card. The low-level
signals can be amplified to increase the resolution and to reduce
noise.
b. Analog Output
The measured electrical signal is often analog and must be converted into a
digital signal so that it can be processed by a computer. The digital
representation of an analog signal offers the following advantages:
1) digital signals can be stored and accessed in the computer RAM and hard
drive,
2) digital signals can be reproduced without errors.
3) digital signals can be imported or transferred between computers for
processing and computational analysis.
In general, the digital representation is not quite a good representation of the
original analog data because the datum point between each digital sample is
lost in the analog-to-digital conversion.
The process of analog-to-digital converter (ADC) is shown in Figure 6.5.
Figure 6.5. Analog-to-digital converter
93
Figure 6.6 shows a sine wave and its corresponding 3-bits digital image.
A 3-bit ADC divides the analog range into 8 divisions. Each division is
represented by a binary code between 000 and 111.
Figure 6.6 Digitized sine wave with a resolution of 3 bits
c. Digital I/O
The use of digital I/O is to transfer the data between a computer and
equipment such as data processors. The important parameters of digital I/O
are the number of digital lines available, the rate at which the equipment can
accepts, and the source and the drive capacity of digital lines.
d. Counter/timers
The counter/timers circuitry is used for counting the occurrence of a digital
event and generating the wave forms. The significant operations of
counter/time are the resolution and clock frequency. The resolution is
determined by the number of bits the counter uses. The clock frequency
determines how fast the digital source input can be toggled.
3. Computer with application software
Application software transforms the computer and data acquisition hardware into
a complete data acquisition system for analysis and display. The driver software
controls the functions of DAQ hardware such analog I/O, digital I/O, and
counter/timers and manages the integration between the hardware and the
computer.
Labview and Matlab are the most common data acquisition software use worldwide by researchers and laboratories. The manufacturers of software provide
DAQ drivers to integrate seamlessly with DAQ hardware, signal conditioning
equipment, and the capability of computer operating system (OS).
A display of data acquisition system with Labview is shown in Figure 6.7.
94
Figure 6.7. Temperature measurement using Lab View
Figure 6.8 shows setting up parameters for DAQ configuration. It requires the
input of parameters such as the number of channels, the number of samples per
channel, sampling mode, and rates.
Figure 8. Configuration parameters for Labview DAQ.
The tutorials for Labview Data Acquisition can be downloaded from the website:
http://www.ni.com/gettingstarted/labviewbasics/. For MATLAB, the tutorials can
be found at http://www.mathworks.com/help/pdf_doc/daq/adaptorkit.pdf.
Referefences:
1. http://www.neutronusa.com/prod.cfm/1016190/glb?gclid=CNKlkP6KsbQCFSF
yQgod-2oATA
2. http://www.eng.uwaterloo.ca/~tnaqvi/downloads/DOC/sd292/introduction_DA
Q_LV.pdf
3. http://physweb.bgu.ac.il/COURSES/SignalNoise/data_aquisition_fundamental
.pdf
4. http://sine.ni.com/nips/cds/print/p/lang/en/nid/1460
95
7
Chapter 7
FLOW VISUALIZATION METHODS
Compiled and written by Huy Hoang, Ph.D.
All photographs in this chapter are used for illustration purposes only. Copyrights are
reserved by various listing sources.
96
Flow visualization is the art of making the fluid patterns visible. Most fluid
such as liquid and gases are invisible. If the flow patterns are visible, researchers
and engineers can gain the insight knowledge of fluid characteristics and flow
phenomenon. There are two main reasons for the needs of flow visualization.
First, experimental measurements for fluid motion in complex geometry or high
speed flows are not always easy and accurate. An introduction of a device likes a
pitot static tube or a five-hole probe near the surface of an object may cause
some interference or disturbance with the free stream velocity. In addition,
measurements of velocity profiles at around the object are very tedious jobs with
the cost of time duration, expensive equipment, and data processing. On the
other hand, the flow visualization provides the visibility of the whole flow field
with low cost and short amount of time. Second, the use of flow visualization
along with experimental measurements can provide the accurate turbulent
models for computational fluid dynamics (CFD). CFD simulation can help
engineers to design a model and to predict more realistic results in time saving
manner.
Flow visualization methods can be divided into three main groups: surface flow
visualization, particle tracer, and optical methods.
1. Surface flow visualization method.
In this method the flow pattern makers or chemical substrate such as paint
dots, pressure sensitive paints, ink dots, mineral oil, turfs, or shear sensitive liquid
crystals are applied on the exterior surface of a test model. The test model is then
placed inside the wind tunnel. The fluid flow patterns will be formed due to the
response of the substrate substance to the air free stream pressure applying on
the test model.
Figure 1 shows a typical open-circuit wind tunnel. The wind tunnel consists of
a motor, diverging section, honeycomb filter, converging section and a test
section. When the wind tunnel is turned on, the motor fan draws the air from
outside into the chamber. The air is going through the diverging section to reduce
its speed. Then it goes through a set of honeycomb filters to make the flow more
uniform, breaking up any turbulence scales. After that, it goes through the
converging section to increase the speed of the flow before entering to the test
section.
97
Figure 1. Open circuit Wind Tunnel
Figure 2 shows the use of pressure sensitive paints to study the boundary layer
flow. The paints were dotted on the test model before placing in the wind tunnel.
Figure 2. Paint dots on the test model
(Source: httt://flowmetrics.com/services/prosphere/paintdot.htm)
A visual observation on the test surface provides valuable information about
the state of boundary layer flow such as laminar or turbulent, transition, and
regions of separation flows.
98
An interesting technique with the use of Ink dots on photo paper for flow
visualization was presented by William Jana in 2012 ASEE Southeastern Section
Conference paper. In this method the ink dot matrices are applied on the photo
paper with the use of a stencil. The photo paper is then placed underneath of a
test model as shown in Figure 3. Both the photo paper and model are then put in
the test section of the wind tunnel running at low speed.
Figure 3. Flow Visualization Using Ink Dot Matrix on Photo Paper
(Source: Image from 2012 ASEE Southeastern Section Conference paper “Ink Dot
Technique of Flow Visualization for the Undergraduate Fluid Mechanics
Laboratory” by William Janna.)
99
A similar technique is the application of mineral oil on the surface normal to a
test model. Figure 4 shows the flow visualization of the flow past a cylindrical test
model.
Figure 4. Side view of wind tunnel test at Re = 78,500 using mineral oil.
(Source: http://www2.le.ac.uk/departments/engineering/people/academicstaff/
aldorona/research/applications/subsonicaerodynamics/oilflowvisualizationofbluff
body)
Photochemical technique is a capable method of quantitatively visualizing the
concentration and distribution of chemical species under the fluid shear stress.
Shear sensitive liquid crystals are the materials that reveal the changes of color
when they are sheared. Typical liquid crystals are cholesteric crystals that
naturally occurring sterols or chiral-nematic crystals derived from non-sterols
synthetic cholesterics mixtures. The liquid crystals are normally coated on the
mate black surface of test model to minimize the reflection of light. The test
model is placed in the wind tunnel. The equipment required for flow visualization
is an illumination source and a photo detector. When the illumination source is
incidence on the test model, the surrounding oxygen molecules are permeated
into the polymer binder of liquid crystals. The oxygen molecules become
quenching and the crystals will illuminate. The change of color will be recorded
using the photo detector.
100
Illumination source
photo detector
Figure 5. Shear sensitive liquid crystals.
Figure 6 shows a result of visualization of laminar-turbulent transition on a wing
by the use of shear stress sensitive liquid crystals at angles of attack 0 degree.
Figure 6. Visualization of laminarturbulent transition on a wing
by shear stress sensitive liquid
crystals at angles of attack 0°
and flow velocity V=60m/s.
(Source: http://www.itmo.by/pdf/isfv/ISFV15-049.pdf)
101
2. Particle tracer methods
Particles, such as a dye or smoke, can be added to the flow to trace the fluid
motion. It is important that the particles have the same specific density
as working fluid. There are five main methods: dye visualization, hydrogen
bubble, smoke flow visualization, planar laser induced fluorescence, and
particle image velocimetry. Dye visualization and hydrogen bubble are used
for water tunnel test only.
A. Dye visualization method
Dye indicators is the most common method used to visualize the fluid in the
water tunnel test. Most common dye indicators for flow visualization are
food dye, condensed milk, and florescent dye. Two typical methods of
introducing dye into the flow are followed.
i.
ii.
Dye is releasing through a probe of stainless steel tubing or a needle of 1.52.0 mm diameter by gravity or pressurized reservoir.
Dye is releasing through dye ports which are usually fabricated as part of the
test model
Figure 7. Dye indicators are releasing through needles in front of Reversed
Delta wing.
(Source: http://fluidsengineering.asmedigitalcollection.asme.org/
article.aspx?articleid=2478867)
102
Figure 8. Colored dye emitted from tiny ports on this 1/48 F-18 model
displays the vortex flow field during tests in the Dryden Flow Visualization
Facility. (NASA Photo ECN 33298-36)
(Source: https://www.nasa.gov/centers/dryden/about/Organizations/
Technology/Facts/TF-2004-05-DFRC.html)
B. Hydrogen bubble method
Another method to visualize the visualize the fluid motion in the water
tunnel test is the Hydrogen bubble method. It consists of using a fine wire,
placed in water, as one end of a DC circuit to electrolyze the water. Electricity is
pulsed in the wire which cause the water to disassociate and generate hydrogen
bubble. The bubble motion can be photographed with a camera. Time of bubble
observation in the flow is limited by the dissolution of bubble in the fluid.
The basic requirements for this method is a variable voltage DC supply (50-70V)
with current capacity approximately 1amp and a single platinum wire with
diameter about 25-50 μm.
103
Figure 9. Visualization of flow between two parallel plates Re=150.
C. Smoke visualization method.
Smoke flow visualization method is the most common method used in the
wind tunnel testing for laboratories and industrial applications. In automobile
industries, smoke flow visualization provides information about the aerodynamics
and ground effects of a new car model to help engineers improve their design for
fuel efficiencies and styling. Smoke for flow visualization is generated by
burning wood, straw or vaporizing hydrocarbon oils. It is necessary that the
individual smoke particles be of small mass so they are carried freely at the flow
velocity. Commercial smoke generators produce high quantity of smoke with
particle size of 0.5 μm. Figure 10 shows a commercial vaporized kerosene mist
generator. The machine has a heating element and oil reservoir. Smoke is
generated by vaporizing the hydrocarbon oils inside the reservoir. Figure 11
shows a photograph of flow visualization of a car.
104
Figure 10. Preston-Sweeting Mist Generator
Figure 11. Smoke flow visualization.
(Source: http://300sl.org/2012/06/20/dreihunderter-im-windkanal-300s-inthe-wind-tunnel/)
105
A simple smoke wire technique produces smoke filaments by vaporizing oils
from a fine wire heated by an electric current. This technique requires to brush
oils constantly on the wire.
D. Planar Laser Induced Fluorescence method.
In this method, a laser sheet is passed through some flow field that generates
excited atoms/molecules. The excited atoms or molecules are spontaneously
emitting lights and the fluorescence is captured on a digital camera. This method
has the following steps:
i. The flow field is seeded with molecular species such as NO, O2, or acetone
which act as fluorescer to produce visible light.
ii. The laser beam is expanded into a thin sheet, which is passed through the
flow field. The laser beam is either directly scattered from molecules or is
absorbed by molecules species.
iii. A camera is placed to see a cross sectional image of the flow field and the
intensity of the light pattern.
Figure 12. Setup of Planar Laser Induced Fluorescence
106
Figure 13. Study of vortex core using planar laser induced fluorescence.
(Source https://www.grc.nasa.gov/www/k-12/airplane/tunvlaser.html)
E. Particle Image Velocimetry method.
Particle image velocimetry (PIV) is a non intrusive measurement technique
used to obtain instantaneous velocity measurements of particles in some type of
flow (gas, viscous fluid).
PIV method has the following steps:
i. The medium is seeded with tracer particles such as polystyrene or oils with
mean diameter 0.5 m.
ii. The medium is illuminated periodically by some high power light source,
which is often a laser.
iii. Successive digital images are obtained from charged coupled device (CCD)
cameras. These images can be analyzed by a computer to determine the
velocity of the tracer particles.
107
Figure 14. Schematic setup of PIV method
Figure 15 shows an image of Particle Image Velocimetry of the wake flow of a
cylinder at Re=250
Figure 15. Particle Image Velocimetry of the wake flow of a cylinder.
(Source http://fluidsengineering.asmedigitalcollection.asme.org/
article.aspx?articleid=1429418)
108
3. Optical methods
In some flows, the flow patterns are revealed by ways of changing their
optical index of refraction. The optical index of refraction of the gas is a
function of the gas density. In the case of compressible flow, the gas density
is varied. The flow will produce an optical disturbance to light rays passing
through the flow field.
Figure 16
There are three basic optical techniques as follow:
 Shadowgraph
 Schlieren
 Interferometry
The shadowgraph technique
The shadowgraph is created by passing a parallel light beam through a
moving fluid.
1. The density variations will cause some of the light rays to be
refracted.
2. If the light beam is focused on the photographic plate, the refraction
of some of the light beams results in dark and light areas.
109
Figure 17. The variation of flow field density will affect the index of
refraction.
Figure 18. Schematic setup of shadowgraph system
110
Figure 19 shows an image of a shadowgraph of a bullet in supersonic flow.
Figure 19. Shadowgraph of a bullet in supersonic flow.
The Schlieren technique
The Schlieren technique resembles the shadow graph technique. It also uses
a parallel light beam passing through the flow region.
 The light beam is then focused onto the edge of a razor blade which is
mounted in front of the photographic plate.
 Light refracted and intercepted by the blade gives the illusion of a shadow
and visualization of density gradients.
Figure 20 Schematic of Schlieren system
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Figure 21 shows an image of Schlieren photography of supersonic shock
waves of an aircraft.
Figure 21. Schlieren photography of supersonic shock waves of an aircraft.
The interferometer technique
The interferometer makes use of two beams, a reference beam passing
around the flow and an object beam passing through the flow field.
1. The wave front of the object beam are deformed due to the phase
shift caused by the variations of the density and the changes of local
index of refraction.
2. By adjusting the interferometer, the interference fringes will form to
map out the density variations of the flow field.
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Figure 22. Schematic of interferometer system
Figure 23 shows an image of interferometer photograph of a small bullet at
Mach 3.
Figure 23. Interferometer photograph of a small bullet at Mach 3
Source: http://www.onera.fr/en/daap/interferometry
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