Error Analysis of Flow Experiments
André Strupstad
Norwegian University of Science and Technology
Department of Petroleum Engineering and Applied Geophysics
Trondheim
May, 2009
ii
Contents
Contents
iii
List of Tables
v
1 Error Analysis of Flow Experiments
1.1 Measures of error . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Error estimate from measurements . . . . . . . . . .
1.1.2 Error estimate from measuring instruments . . . . .
1.2 Propagation of errors . . . . . . . . . . . . . . . . . . . . . .
1.3 Error in the di¤erence between two measurements . . . . .
1.4 Error in measured quantities . . . . . . . . . . . . . . . . .
1.4.1 Error in pipe diameters . . . . . . . . . . . . . . . .
1.4.2 Error in length of the di¤erential pressure section . .
1.4.3 Error in the gas mass ‡ow . . . . . . . . . . . . . . .
1.4.4 Error in pressure measurements . . . . . . . . . . . .
1.4.5 Error in absolute pressure . . . . . . . . . . . . . . .
1.4.6 Error in di¤erential pressure . . . . . . . . . . . . . .
1.4.7 Error in temperature . . . . . . . . . . . . . . . . . .
1.5 Error in calculated quantities . . . . . . . . . . . . . . . . .
1.5.1 Error in density . . . . . . . . . . . . . . . . . . . . .
1.5.2 Error in viscosity . . . . . . . . . . . . . . . . . . . .
1.5.3 Error in Reynolds number . . . . . . . . . . . . . . .
1.5.4 Error in friction factor and friction factor di¤erence
1.6 Data …tting . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Fitting experimental data . . . . . . . . . . . . . . .
1.6.2 Evaluating the goodness of …t . . . . . . . . . . . . .
1.6.3 Con…dence bounds . . . . . . . . . . . . . . . . . . .
References
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
2
2
2
3
3
3
3
4
5
5
5
6
6
6
6
7
8
9
10
11
11
13
15
iii
iv
CONTENTS
List of Tables
1.1
1.2
1.3
1.4
1.5
1.6
Error in the indirectly measured pipe diameters. . . . . . . . . . . . . . . . . . .
4
Relative error in the length of the two di¤erential pressure sections one each pipe.
4
Error in the gas Reynolds number for three typical gas mass ‡ows used in the ‡ow
experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Relative error, best estimate and absolute error in the friction factor for some
typically interdependent values of di¤erential pressure, mass ‡ow and temperature. 10
Di¤erent contributions to the relative error in the friction factor from the measured
and calculated quantities used to calculate the friction factor. . . . . . . . . . . . 10
Relative error, best estimate and absolute error in the friction factor di¤erence
for some typically interdependent values of di¤erential pressure, mass ‡ow and
temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
v
vi
LIST OF TABLES
1
Error Analysis of Flow
Experiments
This note concerning experimental uncertainty or experimental error is based on Chapter "Error
Analysis" in Strupstad (2009). It contains a short theoretical introduction to error analysis and
the use of this theory on the results from a pipe ‡ow experiment. For further information about
the experiment and theoretical background information consult Strupstad (2009).
When reporting experimental results it is important to include the experimental uncertainty,
also called the experimental error, of these results. An error in a scienti…c context is not a
blunder or mistake, but points to the uncertainty that a¤ects all measurement. Since errors are
inevitable, the best one can do is to ensure that the errors are as small as reasonably possible
and have estimates on how large they are [Taylor (1997)].
Experimental errors are divided into two types of errors; namely indeterminate (frequently
called random) and determinate (frequently called systematic) errors. Indeterminate errors are
present in all experimental measurements. There is no way to determine the size or sign of an
error in any individual measurement. The size of indeterminate errors can usually be reduced
somewhat by taking repeated measurements and then calculating their average values. This average value is generally considered to be a better representation of the true value. If measurements
have relatively small indeterminate errors they are said to have high precision.
Determinate errors have the same size and algebraic sign for every measurement and the size
and sign are determinable. A common case of determinate error is instrumental or procedural
bias, typical a miscalibrated scale or instrument. If measurements have small indeterminate
errors and small determinate errors they are said to have high accuracy. One should note that
precision does not necessarily imply accuracy; precise measurements may be inaccurate if they
have a determinate error.
While working on this chapter about error analysis the book by Taylor (1997) were repeatedly
consulted. For further insight the books by Rabinovich (2000) and International Organization
for Standardization (1995) were consulted. To get a wider pictures of the subject several internet
sources were also consulted, among them Simanek (2008). Experimental error and experimental
uncertainty represents the same concept and in this text we use the terms error and uncertainty
interchangeably.
1
2
1.1
1.1.1
1. ERROR ANALYSIS OF FLOW EXPERIMENTS
Measures of error
Error estimate from measurements
There are several di¤erent standard ways in which an experimental error ( x) in a distribution
of measurements can be expressed. Two frequently used methods are the absolute error method
and the statistical uncertainty concept.
The absolute error method use absolute limits as its measure of error; the measured quantity is
assumed to never fall outside these bounds. The random nature of errors in typical experiments,
results in a so-called normal, or bell-shaped distribution curves around the true values. Therefore,
it is not likely that the worst possible error will actually occur, making the absolute error method
too conservative.
In the statistical uncertainty concept we assume that the errors in the measured quantities
have normal distribution and are statistically independent. With this assumptions it is known
that the best estimate (x) for the true value (X) of a physical quantity based on n measurements
(x1 ; x2 ; :::; xn ) is the mean value of these measurements, given by Taylor (1997) as:
P
xi
x=
:
(1.1)
n
The standard deviation is an estimate of the average uncertainty of a measurement. The best
estimate for the standard deviation ( x ) of the above set of measurements is given by Taylor
(1997) as:
v
u
n
u 1 X
2
t
(1.2)
(xi x) :
x =
n 1 i=1
We use the factor n 1 instead of n, because this corrects for the error introduced by using our
best estimate (x), instead of the true value (X). With the assumption of normal distribution of
the measurements, approximately 68% of our measurements will fall within a distance of
x
from the best estimate x. The probability that a measurement will fall inside 2 x is 95%, and
within 3 x is 99:7%.
The fractional uncertainty of the uncertainty ( x ), or the fractional standard deviation of
the standard deviation is only dependent on the number of measurements n, and is given by
Taylor (1997) as:
1
:
(1.3)
x = p
2 (n 1)
This result indicates strongly the need for numerous measurements before the uncertainty can
be known reliably, and that the probabilities of …nding measurements within
; 2 and 3
quoted above can only be used as approximations in real experimental situations. One needs to
decide on the accepted percentage of the events that probably will falls inside the error estimate
( x), before one can use these values to calculate the error of a computed quantity.
1.1.2
Error estimate from measuring instruments
When using a measuring instrument to perform a measurement of a quantity, the experimental
error ( x) in the measurements is usually given in the instruction manual supplied by the manufacturer. In the manual it is usually distinguish between the accuracy and the precision, usually
called the repeatability, of the measurement. The repeatability (precision) has always a smaller
value than the accuracy.
1.2. PROPAGATION OF ERRORS
3
If the instrument is installed and used correctly, this error estimate is usually a bit conservative. Typically will all measurements fall well inside the error limits. In this thesis we have used
the information from the manufacturer of the instrument to estimate the error in the quantity
measured with a particular instrument.
1.2
Propagation of errors
A function q (x; :::; z) is computed from the measured quantities x; :::; z with uncertainties x; :::; z.
Assuming that the errors in the measured quantities are random and statistically independent,
the total uncertainty of a computed result ( q) is given by Doebelin (2004) as:
s
2
2
@q
@q
x + ::: +
z :
(1.4)
q=
@x
@z
If all the quantities (x; :::; z) enter into the function q (x; :::; z) raised to a power of orders nx ; :::; nz ,
i.e.
q (x; :::; z) / xnx ::: z nz ;
(1.5)
the relative uncertainty1 may be given as
s
q
x
=
nx
q
x
2
+ ::: + nz
z
z
2
;
(1.6)
where the x; :::; z is the best estimate, typically the mean value (x = x; :::; z = z), of the measured
quantities.
1.3
Error in the di¤erence between two measurements
If one is looking at the error in the di¤erence between two values calculated with the same
equations, based on the same quantities, which are measured with the same experimental setup
and procedures, one would expect the error in the di¤erence only to contain the indeterminate
errors and that the determinate errors will cancel out. It is only the precision and not the
accuracy of the measurements that are important in this case.
Since the main results from our measurements are the change in the friction factor from
measurements without and with particles, the determinate errors e¤ectively drop out of the
calculations, and we are left only with the indeterminate errors. Therefore, it is important to
identify the di¤erent error sources as either indeterminate or determinate, before calculating the
error in a di¤erence between two measurements.
1.4
1.4.1
Error in measured quantities
Error in pipe diameters
The mean diameters (Dpipe ) of the test pipes (pipe 5, 10, 11 and 14) used in the ‡ow experiments
in this thesis were calculated from measurements of the height (hwater ) of the water column given
1 The relative uncertainty (or error) is also called fractional uncertainty (or error). The uncertainty ( q) is
sometimes called the absolute uncertainty to avoid confusion with the fractional uncertainty.
4
1. ERROR ANALYSIS OF FLOW EXPERIMENTS
Table 1.1: Error in the indirectly measured pipe diameters.
hwater
Dpipe
Dpipe
Dpipe
hwater
Pipe
hwater
Dpipe
[mm]
[mm]
[mm]
5
3278
3 :051 10 4
24:144
3:67 10 4
0:009
4
10
3282
3:047 10
24:129
3:67 10 4
0:009
11
3275
3:053 10 4
24:155
3:67 10 4
0:009
14
3276
3:053 10 4
24:151
3:67 10 4
0:009
Table 1.2: Relative error in the length of the two di¤erential pressure sections one each pipe.
Ldp1
Ldp2
Ldp1
Ldp2
Pipe
Ldp1
Ldp2
[mm]
[mm]
5
1900
5:26 10 4
1900
5:26 10 4
4
10
1904
5:25 10
1904
5:25 10 4
4
11
1958
5:11 10
1851
5:40 10 4
4
14
1900
5:26 10
1900
5:26 10 4
by a speci…c volume of water (Vwater = 1500):
Dpipe =
r
4Vwater
:
hwater
(1.7)
The volume of the water was measured with a measuring glass with uncertainty of 0:75 ml.
Including also the fact that small amounts of water remained inside the measuring glass when
poured into the pipes, the uncertainty of the measured water volume was estimated to be
1
ml. This gave the relative uncertainty of the measured water volume:
1:0 ml
Vwater
=
Vwater
1500 ml
6:67
10
4
:
(1.8)
The height of the water columns were measured with a yardstick and the uncertainty of the
height measurement was estimated to be hwater = 1 mm.
Applying Equation 1.6 we found the following equation which can be used to calculated the
relative inaccuracy of the pipe diameter;
Dpipe
=
Dpipe
s
1 Vwater
2 Vwater
2
+
1 hwater
2 hwater
2
:
(1.9)
Table 1.1 shows the error in the indirectly measured pipe diameters.
1.4.2
Error in length of the di¤erential pressure section
The length between the pressure taps used for the di¤erential pressure measurements (Ldp )
were measured with a yardstick. On each pipe we had three pressure taps, which we used to
create two consecutive di¤erential pressure measurements sections. The uncertainty of the length
measurements were estimated to be less than Ldp = 1 mm. Table 1.2 shows the relative error
of the length of the two di¤erential pressure sections on each pipe.
1.4. ERROR IN MEASURED QUANTITIES
1.4.3
5
Error in the gas mass ‡ow
The gas mass ‡ow rate was measured with a Bronkhorst Hi-Tec ‡ow meter F-106B. The ‡ow me:
ter has according to the producer an accuracy of 0:5% of reading plus 0:1% of full scale mmax = 453:27 kg=h
:
:
:
:
m = 0:005m + 0:001mmax = 0:005m + 0:45327 kg=h:
(1.10)
The relative error in the gas mass ‡ow measurements were given by
:
:
m
mmax
:
: = 0:005 + 0:001
:
m
m
(1.11)
The relative error in the gas mass ‡ow instrument decreases continuously as the gas mass ‡ow
increases towards its maximum ‡ow:
:
m
:
m
=
0:005
min
453:27 + 0:45327
= 6:0
453:27
10
3
:
(1.12)
The repeatability of the Bronkhorst Hi-Tec ‡ow meter F-106B was according to the producer better than 0:2% of reading. This means that the precision and, consequently, also the
indeterminate error of the gas mass ‡ow measurements were
:
m
:
m
<2
10
3
;
(1.13)
indeterm inate
which will be important when we were calculating the error in the di¤erence between to friction
factors.
1.4.4
Error in pressure measurements
Wall tap error
The wall tap error was considered and the maximum error in the static pressure due to pressure
taps was found to be less than 0:041 %. It is assumed that the measured di¤erential pressure
should be independent of the geometry and the hole size as long as the wall taps were uniform,
because the same error then would occur at both wall taps. Burrs on the edge of the pressure
taps would however in‡uence both the absolute pressure and the pressure drop. Therefore, all
wall taps were inspected before use, and eventual burrs were removed prior to ‡ow experiments.
See Strupstad (2009) for a further discussion about wall tap error. Information about wall tap
error can be found in Shaw (1960), Benedict (1984) and McKeon and Smith (2002).
1.4.5
Error in absolute pressure
The accuracy in the absolute pressure ( p) measured with pressure sensor P15RVA2 from Hottinger Baldwin Messtechnic Gmbh were reported by the producer to be less than 1% of reading,
typical 0:5%. After performing a calibration of the pressure sensors, we choose to use the value
p
= 0:005 = 5
p
10
3
:
(1.14)
Even though the tap diameter sizes were slightly larger than the recommended size, the error
due to wall taps were insigni…cant compared with the sensor inaccuracy.
6
1.4.6
1. ERROR ANALYSIS OF FLOW EXPERIMENTS
Error in di¤erential pressure
The total operating performance for the di¤erential pressure transmitters (Rosemount 3051 CD)
were reported to be 0:15% of span. The span used was 62 mbar, resulting in an error in the
di¤erential pressure measurements ( ( p)) of
( p) = 0:0015
span = 0:0015
62 mbar = 0:093 mbar:
(1.15)
The relative error in the di¤erential pressure measurement depends on the value of the di¤erential
pressure
0:093 mbar
( p)
=
;
(1.16)
p
p
normally ranging from about 2 to 50 mbar, resulting in a relative error from
( p)
0:093
=
= 46:5
p
2
1.4.7
10
3
( p)
0:093
=
= 1:5
p
62
down to
10
3
:
(1.17)
Error in temperature
The temperatures were measured with PT100 elements, with an uncertainty in the temperature
measurements of T = 0:1 C. All the experiments were performed around a temperature of
about 20 C, resulting in a typical relative error in the temperature measurements of
T
0:1 C
=
=5
T
20 C
3
10
:
(1.18)
Using degrees Kelvin as the unit for temperature the relative error becomes
0:1 K
T
=
= 3:4
T
273:15K + 20K
1.5
1.5.1
10
4
:
(1.19)
Error in calculated quantities
Error in density
The density was calculated from the multiconstant equation of state for air given by Reynolds
(1979)
"
#
5
X
2
1=2
3 i
p =
Rdens T +
A1 T + A2 T
+
Ai T
i=3
+
3
+
+[
+
+
9
X
Ai T 7
i
4
Ai T 11
i
+
5
A13
i=6
i=10
6
(A14 =T + A15 =T 2 ) + 7 A16 =T + 8 (A17 =T + A18 =T 2 ) + 9 A19 =T 2
3
(A20 =T 2 + A21 =T 3 ) + 5 (A22 =T 2 + A23 =T 4 ) + 7 (A24 =T 2 + A25 =T 3 )
9
(A26 =T 2 + A27 =T 4 ) + 11 (A28 =T 2 + A29 =T 3 )
13
+
12
X
(A30 =T 2 + A31 =T 3 + A32 =T 4 )]e(
2
)
;
(1.20)
where p is the pressure [P a], is the density kg=m3 , T is the temperature [K] and Rdens =
287:0686 is a constant (not the universal gas constant). The coe¢ cients in Equation 1.20 are
given in Reynolds (1979) and can also be found in Strupstad (2009).
1.5. ERROR IN CALCULATED QUANTITIES
7
In Straus (2004) the Reynolds equation was compared with the experimental data in the
relevant temperature and pressure range. From this we found the relative error from the density
equation to be [Straus (2004)]
< 1:5
10
4
;
(1.21)
Equation
in the relevant temperature and pressure range. The dominant term in the density equation is
the …rst term on the right hand side in Equation 1.20 which is
=
p
Rdens T
;
(1.22)
where is the ‡uid density kg=m3 , p is the absolute pressure [P a], T is the temperature [K]
and Rdens is a constant. From this we found the relative error in the calculated density due to
the errors in the pressure and temperature measurements were approximately given by:
s
2
2
p
T
=
+
p
T
s
2
2
0:005p
0:1 K
+
=
p
T
s
2
0:1 K
2
(5 10 3 ) +
=
:
(1.23)
T
Inserting a typical temperature from the experiments (T = 20 C = 293:15 K) :
=
q
(5
5
10
10
3
3 )2
+ (3:4
10
4 )2
:
(1.24)
By comparing Equation 1.24 with Equation 1.21, it was evident that the relative error in the
calculated density was dominated by the error in the pressure measurements, while the error in
the temperature measurements and from the Reynolds density equation was negligible.
1.5.2
Error in viscosity
The viscosity was calculated from the Jones equation
=
1:70257 10 5 + 6:05434 10 8 T
+8:08321 10 10 p + 5:97259 10
1:33200
13 2
p ;
10
10
T2
(1.25)
where T is the temperature in [ C], p is the pressure in [bara], and is the viscosity in [P a s].
In Straus (2004) the Jones equation was compared with the experimental data in the relevant
temperature and pressure range. From this comparison we found that the relative error in the
Jones viscosity equation will be [Straus (2004)]
<5
Equation
10
3
;
(1.26)
8
1. ERROR ANALYSIS OF FLOW EXPERIMENTS
in the relevant temperature and pressure range. The error in calculated viscosity ( ) due to the
errors in the measured temperatures [ C] and pressures [bara] was given by Equation 1.4:
s
2
2
@
@
+
T
p
=
@T
@p
=
( 6:05434
+
8
10
8:08321
2:66400
10
10
+ 1:1945
10
10
10
12
T
T
2
2
p )1=2 :
p
Inserting typical values for temperature (T = 20 C), pressure (p = 1:5 bara) and viscosity
from our experiments:
=
=
5:5215
5:5215
9 2
10
10
9
+ 6:0758
10
(1.27)
= 1:818
1=2
12 2
P a s;
(1.28)
we found that the relative error in the calculated viscosity due to errors in temperature and
pressure measurements typically was
=
5:5215 10
1:818 10
9
5
= 3:0371
10
4
:
(1.29)
We observed that the error in viscosity is dominated by the error in the Jones viscosity equation
(see Equation 1.26), and that the error in the temperature measurements were almost insigni…cant
and that the error in the pressure measurements were totally insigni…cant (Equation 1.29). We
concluded that the relative error in the viscosity in our experiments was given by the error in
the Jones viscosity equation:
< 5:0
1.5.3
10
3
= 0:5%:
(1.30)
Error in Reynolds number
The Reynolds number can be given in the useful form
:
Re =
4 m
:
Dpipe
(1.31)
The relative error in the calculated gas Reynolds number was given by
s
:
2
2
2
Dp
Re
m
=
+
+
:
Re
D
m
:
=
[
0:005m + 0:4512 kg=h
:
m
+ 4:52
10
4 2 1=2
]
:
2
+ 5:0
10
3 2
(1.32)
Table 1.3 show the relative errors in, the best estimates of, and the absolute errors in the gas
Reynolds number for di¤erent gas mass ‡ows. From the numbers in the table it was evident that,
the main contribution to the error in the Reynolds number at low mass ‡ow was the uncertainty
in the gas mass ‡ow measurement. As the gas mass ‡ow increased, the relative error in the gas
mass ‡ow decreased towards its minimum value; 6:0 10 3 (see Equation 1.12), and the relative
error from the viscosity equation 5:0 10 3 became more signi…cant.
10
5
Pa s
1.5. ERROR IN CALCULATED QUANTITIES
9
Table 1.3: Error in the gas Reynolds number for three typical gas mass ‡ows used in the ‡ow
experiments.
:
:
Dp
Re
m
m
:
Re
D
m
Re
Re
3
3
3
[kg/h]
10
10 4
10
10
12:4
57110
709
70:82
11:4
5:0
4:52
10:1
97890
985
121:40
8:72
5:0
4:52
8:83
160250 1416
198:69
7:27
5:0
4:52
1.5.4
Error in friction factor and friction factor di¤erence
Assuming j pj
p one can use the Darcy-Weisbach friction factor (neglecting the acceleration
term (see Strupstad (2009) for further information)
2
f=
5
Dpipe
: (
8Lpipe m2
p) =
Dpipe
1
2
2
(
p) :
(1.33)
U Lpipe
Error in friction factor
The acceleration term in the momentum equation was small compared to the other term; therefore, it can be neglected in the error analysis for the friction factor. The main parameters in the
friction factor correlations is then given from Equation 1.33:
f/
p Dp5
:
m L
(1.34)
: 2
Applying Equation 1.6 on the friction factor gave the relative error in the calculated friction
factor:
s
:
2
2
2
2
2
f
( p)
Dp
L
m
=
+
+
+ 5
+
2 :
f
p
D
L
m
:
=
[ 2
0:005m + 0:4512 kg=h
:
m
+ 1:835
10
3 2
+ 5:40
2
10
+ 5
4 2 1=2
]
10
:
3 2
+
0:093 mbar
p
2
(1.35)
Table 1.4 show the relative errors in, the best estimates of, and the absolute errors in the
friction factor for some typically experiments with corresponding values of di¤erential pressure,
gas mass ‡ow and temperature. The relative errors in the pressures, which in‡uence the relative
density error, were constant. Table 1.5 shows the di¤erent contributions to the relative error in
the friction factor from the measured and calculated quantities used to calculate it. It was evident
that it was the error in the gas mass ‡ow measurement that gave the dominate contribution to the
error in the calculated friction factor. The error in the friction factor increased with decreasing
gas mass ‡ow.
Error in friction factor di¤erence
When calculating the error in the calculated friction factor di¤erence ( fdif f )one can remove
all determinate errors from the error estimate. Using the same pipe with and without particles,
the errors in the pipe diameter and lengths will, for the purpose of calculating the change in
10
1. ERROR ANALYSIS OF FLOW EXPERIMENTS
Table 1.4: Relative error, best estimate and absolute error in the friction factor for some typically
interdependent values of di¤erential pressure, mass ‡ow and temperature.
:
f =f
f
f
p
T
m
[mbar]
[ C]
[kg/h]
10 3
10 3
10 3
24:636
21:1
0:51982
70:82
11:9
19:61
18:552
18:7
0:34692
121:40
27:3
19:65
15:597
17:0
0:26515
198:69
52:5
19:45
Table 1.5: Di¤erent contributions to the relative error in the friction factor from the measured
and calculated quantities used to calculate the friction factor.
:
( )
( p)
D
L
(m)
f =f
5 Dp
2 m:
L
p
3
3
3
3
3
10
10 3
10
10
10
10
24:636
22:742
5
7:8151
1: 835
0:540
18:552
17:433
5
3:4066
1: 835
0:540
15:597
14:542
5
1:7714
1:835
0:540
friction factor, be considered as constants containing only determinate errors. Assuming that
the determinate errors were constant in all measurements, we were left with the indeterminate
:
errors in gas mass ‡ow m , di¤erential pressure ( p) and density ( (p; T )). According to
Equation 1.13, the indeterminate error in the gas mass ‡ow measurements were less than 0:2%.
It was more di¢ cult to distinguish between the determinate and indeterminate errors in the
di¤erential pressure and pressure measurements; therefore, we will use the full error values for
these parameters. Applying Equation 1.6 on the calculated friction factor di¤erence (fdif f ) we
get the relative error in the calculated friction factor di¤erence:
fdif f
fdif f
v
u
u
= t 2
=
[ 4
+
:
m
:
m
10
!2
3 2
( )
+
+ 5
0:093 mbar
p
10
2
+
( p)
p
2
3 2
2
1=2
]
:
(1.36)
Table 1.6 show the relative errors in, the best estimates of, and the absolute errors in the
friction factor di¤erence for some typically experiments with corresponding values of di¤erential
pressure, gas mass ‡ow and temperature. Comparing these values for the error in the friction
factor di¤erence with the values for the error in the friction factor from Table 1.4, the errors
were reduced with approximately 60% compared with the errors in the friction factors. The main
contributor to this reduction was the removing of the determinate errors in the gas mass ‡ow
measurements.
1.6
Data …tting
In this thesis the …tting of curves are preformed using either Microsoft Excel or Matlab. In
Matlab either a self-made script or the statistical GUI ’cftool’were used.
1.6. DATA FITTING
11
Table 1.6: Relative error, best estimate and absolute error in the friction factor di¤erence for
some typically interdependent values of di¤erential pressure, mass ‡ow and temperature.
:
f dif f =f
f dif f
f
p
T
m
[mbar]
[ C]
[kg/h]
10 3
10 3
10 3
10:103
21:1
0:23881
70:82
11:9
19:61
7:2529
18:7
0:16571
121:40
27:3
19:65
6:6436
17:0
0:14283
198:69
52:5
19:45
1.6.1
Fitting experimental data
Experimental data can be …tted using the least squares method. Having n experimental data
points (xi ; yi ) ; where i = 1; :::; n, that can be modeled by a function g (x; pm ) with m adjustable
coe¢ cients. We wish to …nd those coe¢ cients values for which the model "best" …ts the data. The
least squares method de…nes "best" as when the sum, Sres , of squared residuals is a minimum:
Sres =
n
X
i=1
ri2 =
n
X
(yi
2
g(xi ; pm )) :
(1.37)
i=1
A residual (ri ) is de…ned as the di¤erence between the values of the dependent variable (yi ) and
the predicted values from the estimated model (g(xi ; pm )). If n is greater than the number of
unknowns (m), then the system of equations is overdetermined.
Least squares problems fall into two categories, linear and non-linear. The linear least squares
problem has a closed form solution, but the non-linear problem does not and is usually solved
by iterative re…nement; at each iteration the system is approximated by a linear one, so the core
calculation is similar in both cases.
Because the least squares …tting process minimizes the summed square of the residuals, the
coe¢ cients are determined by di¤erentiating S with respect to each parameter, and setting the
result equal to zero:
@Sres
=
@pm
2
n
X
i=1
(yi
g(xi ; pm ))
@g(xi ; pm )
= 0;
@pm
f or all m:
(1.38)
Solving these equations simultaneously either directly or by regression gives the coe¢ cients which
give the best …t to the data set.
1.6.2
Evaluating the goodness of …t
In this thesis the goodness of the …t statistics were found using the Curve Fitting Toolbox ’cftool’
in Matlab. The goodness of the …t statistics is typically reported as R-square. Description of
R-square together with some other variables describing the goodness of …t statistics is given
below. The descriptions given below were mainly found in Matlab (2007).
SSE
SSE is the sum of squares due to error. This statistic measures the total deviation of the response
values (experimental data points) from the …t to the response values (points calculated from the
correlation). A value closer to 0 indicates a better …t. It is also called the summed square of
12
1. ERROR ANALYSIS OF FLOW EXPERIMENTS
residuals and is usually labeled as SSE. SSE is de…ned as
SSE =
n
X
2
ybi ) ;
wi (yi
i=1
(1.39)
where yi is the experimental data point and ybi (= g(xi ; pm )) is the point calculated from the
correlation. If the experimental data is not of equal quality, the …t might be unduly in‡uenced
by data of poor quality. In such data one can use weights (wi ) to improve the quality of the …t.
In this thesis we assume the quality of the data included in …ttings are of equal quality resulting
in wi = 1.
R-square
R-square R2 is the coe¢ cient of multiple determination. This statistic measures how successful
the …t is in explaining the variation of the data. A value closer to 1 indicates a better …t. Put
another way, R-square is the square of the correlation between the response values and the
predicted response values. It is also called the square of the multiple correlation coe¢ cients and
the coe¢ cient of multiple determination.
R-square is de…ned as the ratio of the sum of squares of the regression (SSR) and the
:
R2 =
SSE
;
SST
SSR
=1
SST
where SSR is de…ned as
SSR =
n
X
wi (b
yi
2
y) ;
(1.40)
i=1
and SST, also called the sum of squares about the mean, is de…ned as
SST
=
n
X
wi (yi
2
y) ;
(1.41)
i=1
SST
= SSR + SSE:
(1.42)
y is the overall mean de…ned as
y
n
P
i=1
n
yi
:
(1.43)
R-square can take on any value between 0 and 1, with a value closer to 1 indicating a better
…t. For example, a R-square value of 0.8234 means that the …t explains 82.34% of the total
variation in the data about the average.
If you increase the number of …tted coe¢ cients in your model, R-square might increase
although the …t may not improve. To avoid this situation, you should use the degrees of freedom
adjusted R-square statistic described below.
Note that it is possible to get a negative R-square for equations that do not contain a constant
term. If R-square is de…ned as the proportion of variance explained by the …t, and if the …t is
actually worse than just …tting a horizontal line, then R-square is negative. In this case, R-square
cannot be interpreted as the square of a correlation.
1.6. DATA FITTING
13
Adjusted R-square
2
Adjusted R-square Radj
is the degree of freedom adjusted R-square. The adjusted R-square
statistic can take on any value less than or equal to 1, with a value closer to 1 indicating a better
…t. It is generally the best indicator of the …t quality when you add additional coe¢ cients to
your model.
This statistic uses the R-square statistic de…ned above, and adjusts it based on the residual
degrees of freedom. The residual degrees of freedom (v) is de…ned as the number of response
values n (experimental points) minus the number of …tted coe¢ cients m (number of free coef…cients pm , that is, coe¢ cients which are not forced to one speci…c value) estimated from the
response values:
vr = n m:
(1.44)
vr indicates the number of independent pieces of information involving the n data points that is
required to calculate the sum of squares. Note that if parameters are bounded and one or more
of the estimates are at their bounds, then those estimates are regarded as …xed. The degree of
freedom is increased by the number of such parameters.
The adjusted R-square statistic is generally the best indicator of the …t quality when you add
additional coe¢ cients to your model. The adjusted R-square is de…ned as
2
Radj
=1
SSE (n
SST (vr
1)
:
1)
(1.45)
In this thesis we only reported the adjusted R-square when it has a value which was di¤erent
from the R-square value.
RMSE
RMSE is the root mean squared error. RMSE is de…ned as
p
RM SE = M SE;
(1.46)
where MSE is the mean square error or the residual mean square
M SE =
SSE
:
vr
(1.47)
A value closer to 0 indicates a better …t. This statistic is also known as the …t standard error
and the standard error of the regression. In this thesis we used RMSE instead of SSE when we
reported the goodness of the …t statistic.
1.6.3
Con…dence bounds
Con…dence (and prediction) bounds de…ne the lower and upper values of the associated interval,
and de…ne the width of the interval. The width of the interval indicates how uncertain you
are about the …tted coe¢ cients, the predicted observation, or the predicted …t. For example, a
very wide interval for the …tted coe¢ cients can indicate that you should use more data when
…tting before you can say anything very de…nite about the coe¢ cients. The level of certainty is
often 95%, but it can be any value such as 90%, 99%, 99.9%, and so on. The 95% con…dence
bound indicates that 95% of your observations are actually contained within the lower and upper
con…dence bounds.
14
1. ERROR ANALYSIS OF FLOW EXPERIMENTS
References
Benedict, P. (1984). Fundamentals of Temperature, Pressure, and Flow Measurements (3th ed.).
John Wiley and Sons Inc.
Doebelin, E. (2004). Measurement Systems - Application and Design (5. ed.). McGraw-Hill.
ISO (1995). Guide to the Expression of Uncertainty in Measurement. International Organization
for Standardization.
Matlab (2007). Matlab Help. The Mathworks Inc.
McKeon, B.McKeon, B.J. and Smith, A.J. (2002). Static pressure correction in high reynolds
number fully developed turbulent pipe ‡ow. Meas. Sci. Tech., 13, 1608–1614.
Rabinovich, S. (2000). Measurement Errors and Uncertainties - Theory and Practice (Second
ed.). Aip Press, Springer-Verlag New York.
Reynolds, W. (1979). Thermodynamic Properties in SI. Department of Mechanical Engineering,
Stanford University, Standford CA 94305.
Shaw, R. (1960). The in‡uence of hole dimensions on static pressure measurements. J. Fluid
Mech., 7, 550–564.
Simanek, D. (2008). Error analysis (non-calculus). http://www.lhup.edu/ dsimanek/errors.htm.
Straus, V. (2004). Air viscosity and air density. Technical report prepared at NTNU.
Strupstad, A. (2009). PRESSURE LOSS IN NATURAL GAS PIPELINES - Experimental Studies of Gas-Particle Flow, Wall Roughness and Drag Reduction. PhD thesis, Norwegian University of Science and Technology, Department of Petroleum Engineering and Applied Geophysics.
Taylor, J. (1997). Error Analysis (Second ed.). University Science Books.
15