Archives of Mining Sciences 50, Issue 3 (2005) 327–341

Archives of Mining Sciences 50, Issue 3 (2005) 327–341

Archives of Mining Sciences 50, Issue 3 (2005) 327–341
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PAWEŁ LIGĘZA*, KATARZYNA SOCHA*
MODIFICATION OF AN ALGORITHM FOR DETERMINING THE GAS FLOW VELOCITY
COMPONENTS IN HOT-WIRE MEASUREMENTS
MODYFIKACJA ALGORYTMU ZNACZENIA SKŁADOWYCH WEKTORA PRĘDKOŚCI PRZEPŁYWU
GAZU W POMIARACH ANEMOMETRYCZNYCH
Measurements of flow velocity are of major importance in the studies of mine ventilation systems. One
of applied methods of measuring velocity is hot wire anemometry which is based on heat transfer between
the medium and a hot element. It is a indirect measurement method, utilising thermal phenomena.
Presented in the paper is the experimental setup for measuring the velocity vectors. The main
component is a three-wire sensor with orthogonally placed wires. The sensor is supplied from
a digitally–controlled four-channel CTA system. This configuration enables the simultaneous measurement
of three signals.
The paper outlines the conventional method of determining the velocity vector. The calculation
procedure involves two stages: determination of the effective velocity, and obtaining the velocity vector
components. Of particular interest is the modified method whereby the velocity vector components are
obtained directly from the measurement signals. Due to a large number of parameters in this method,
they were determined using genetic algorithms. They are better in such applications then traditional
optimisation methods.
An experiment was conducted to compare the accuracy of the methods. In order that the results be
easier to interpret, the parameters in each case were obtained for the traditional sensor configuration during
the calibration. The evaluation of results is supported by the qualitative and quantitative analysis.
It is readily apparent that the proposed modified method of determining the velocity vector components
yields similar or even better results that the conventional two-step method. Careful selection of parameters
in the genetic algorithms should further improve the accuracy of the procedure.
Keywords: hot-wire anemometry, three-wire sensor, velocity vector components, genetic algorithms
Pomiar prędkości przepływu ma duże znaczenie przy badaniu procesu przewietrzania wyrobisk
kopalnianych. Jedną ze stosowanych metod pomiaru prędkości jest termoanemometria. Polega ona na
określaniu intensywności wymiany ciepła między medium, a umieszczonym w nim grzanym elementem.
Jest to metoda pomiaru pośredniego oparta o wykorzystanie zjawisk cieplnych.
*
INSTYTUT MECHANIKI GÓROTWORU PAN, UL. REYMONTA 27, 30-059 KRAKÓW, POLAND
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W artykule omówiono stanowisko pomiarowe umożliwiające wykonanie pomiaru wektora prędkości.
Głównym jego elementem jest trójwłóknowy czujnik rezystancyjny, z ortogonalnie rozmieszczonymi
włóknami. Sonda pomiarowa zasilana jest sterowanym cyfrowo czterokanałowym układem temperaturowym. Taki system umożliwia jednoczesny pomiar trzech sygnałów.
W pracy przedstawiono klasyczną metodę wyznaczania wektora prędkości. Obliczenia w tej metodzie
można podzielić na dwa etapy: wyznaczenie prędkości efektywnych, a na ich podstawie składowych
wektora prędkości. Następnie zaprezentowano metodę zmodyfikowaną. Polega ona na bezpośrednim
wyznaczeniu z sygnałów pomiarowych składowych wektora prędkości. Ze względu na dużą liczbę parametrów w proponowanej metodzie, do ich wyznaczenia zastosowano algorytmy genetyczne. Lepiej radzą
sobie one z tego typu zadaniami niż klasyczne metody optymalizacji.
Przedstawiono również eksperyment wykonany w celu porównania obu metod. Dla łatwiejszej
analizy otrzymanych wyników parametry, dla każdej z metod wyznaczono przy klasycznym ustawieniu
sondy w trakcie wzorcowania. Do oceny otrzymanych wyników przeprowadzono analizę jakościową i
ilościową.
Na podstawie dokonanej analizy można stwierdzić, że zaproponowana modyfikacja metody wyznaczania wektora prędkości doprowadziła do otrzymania podobnych, a nawet lepszych wyników w porównaniu
z metoda dwukrokową. Odpowiednie dobranie parametrów algorytmów genetycznych powinno zwiększyć
dokładność wyznaczenia składowych wektora prędkości.
Słowa kluczowe: termoanemometria, czujnik trójwłóknowy, składowe wektora prędkości, algorytmy
genetyczne
1. Introduction
The research work at the Laboratory of Flow Metrology of the Strata Mechanics
Research Institute of The Polish Academy of Sciences focuses on gas flow velocity
measurements. Of particular interest are those methods that might be applied in the
mining sector. Flow velocity measurements are the key element in the investigations of
the mine ventilation conditions and processes. In order that the underground mines be
aired effectively, it is required that the information about the relevant parameters in the
selected points of the mine be regularly acquired and updated.
One of the available methods of measuring airflow velocity is hot-wire anemometry,
particularly the thermal anemometry whereby a hot element (a sensor) is placed inside the
investigated medium. The desired quantity is then obtained from the heat balance. The
most important advantages of this method are: point velocity measurements (the sensor
is small in size), there are no mobile elements which might lead to the field distortions,
a wide frequency transmission band and electric output signals. The main drawback,
however, is that the measurement results depend not only on velocity, but also on temperature and chemical composition of the investigated medium (Poleszczyk, 2002).
This study outlines the conventional method of determining the spatial components
of the velocity vector and its suggested modifications.
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2. Hot-wire measurements of the velocity vector
An experimental setup was specially designed in the Laboratory of Flow Metrology,
for the purpose of hot–wire measurements, particularly the measurements of gas flow
velocity vector’s components (Gawor et al., 1994; Ciombor, 2004). It comprises a multiwire sensor, a supply system and two measurement cards.
2.1. Experimental setup
The key element in presented system is a three-wire sensor with sensitive elements
(wires) normal to one another. The wires are fitted on supports such that they form the
cube edges whilst the sensor axis coincides with the cube diagonal and is inclined at
54.7ş with respect to the wires. The wires are 2 mm long, made from tungsten wire 5
m in diameter. Fig. 1 shows the photo of a sensor utilised to determine the absolute
value of velocity vector components.
Fig. 1. Three-wire sensor
Rys. 1. Trójwłóknowy czujnik
The sensor is supplied from a four-channel digitally-controlled system CCC2002
enabling the constant-temperature and constant-current mode of operation (Ligęza,
2003).
Data acquisition from the digital system CCC2002 utilises two two-channel A/D
cards, which enable simultaneous measurement of three signals from the sensor as well
as on-line processing of measured voltages (Gawor, 1999).
The system comprises also a wind tunnel, where the flow velocities 0-35 m/s can be
achieved. There is also a coordinate positioning device enabling the sensor movements
in the horizontal plane and a rotor allowing the sensor rotation at any angle with respect
to the flow direction.
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The control function is realised via a PC with a specialist software developed in the
Laboratory of Flow Metrology (Poleszczyk, 2002; Gawor 2003; Ciombor 2004).
The experimental setup is shown schematically in Fig. 2.
Fig. 2. Conceptual design of the measuring circuit
Rys. 2. Schemat ideowy systemu pomiarowego
2.2. Spatial orientation of a three-wire sensor
Knowing the geometric configuration of the sensor wires, we find the theoretical
decomposition of the velocity vector, depending on the preset velocity and the angles
associated with the sensor’s position with respect to the wind tunnel.
An analysis of spatial interrelations between the preset velocity vector and the sensor’s rotating motion leads us to the conclusion that the mathematical description and
interpretation became more understandable once the coordinate systems is transferred
from the immobile tunnel to the sensor. The sensor will be thus immobile in space whilst
the preset velocity vector should move along the generator of a cone whose apex becomes the origin of the coordinate system associated with the sensor. The coordinate axes
will be contained in the generator of the cone (Fig. 3). The sensor’s rotating motion is
therefore changed into the rotating motion of a velocity vector over the cone surface.
This vector can be decomposed as follows:
v' x = v | cos g (cos q sin b cos a + sin q cos b ) - sin g sin b sin a |
v' y = v | sin g (cosq sin b cos a + sin q cos b ) + cos g sin b sin a |
v' z = v | (sin q cos b - cos q sin b cos a ) |
(1)
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where:
β = 54.74°
θ = 180° – β
γ = –45°
α
v
—
—
—
—
—
angle between the wires and the sensor’s axis,
angle of rotation around the OX axis,
angle of rotation around the OZ axis,
angle of sensor’s rotation with respect to its axis,
preset velocity of flow.
The absolute value of the velocity components, introduced in the model, is associated
with the sensor type. It may be utilised only to determine the values of these components
whist sense of the vector still remains unknown.
Fig. 3. Conceptual design of a sensor (indicated is the area in which the velocity vector can move)
Rys. 3. Schemat ideowy czujnika z zaznaczonym obszarem, po którym porusza się wektor prędkości
Fig. 4 shows a case where a velocity vector is decomposed while the sensor is rotated
around its own axis (angle α) for the flow velocity 1 m/s. Three characteristic points
are readily apparent. For the angle α = 0°, 120°, 240° two components are 0 and one
assumes the value of the preset velocity. For such angles α two wires will be always
normal to the flow and one will be parallel to it. This spatial configuration is utilised in
sensor calibration.
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Fig. 4. Angular characteristics of a theoretical three-wire sensor at the preset velocity 1 m/s
Rys. 4. Charakterystyka kątowa teoretycznej sondy trójwłóknowej przy zadanej prędkości 1 m/s
3. Methods of determining the velocity vector
There are several methods of finding the flow velocity vector components. This study
outlines the conventional two-step method and the look-up table approach.
3.1. Conventional two-step method
Conventional method of determining the velocity vector involves two stages. In the
first stage the effective velocity responsible for wire cooling effect is obtained on the
basis of voltage readouts. Thus obtained effective velocities yield the velocity vector
components in the second stage.
3.1.1. Mathematical model
For a single wire the spatial velocity vector has three components, as shown in Fig. 5,
governed by the Jorgensen formula (Jorgensen, 1971; Poleszczyk, 2002; Rachalski,
2003):
(2)
where:
vef — effective velocity responsible for wire cooling effect,
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vN — vector component normal to the plane determined by a wire and its supports,
vT — vector components parallel to the wire,
vBN — vector component normal to the wire, situated in the plane determined by
the wire and its supports,
k — factor expressing the impacts of the vector component parallel to the wire
axis,
l — actor expressing the effects of a binormal component, its value approaching
unity.
Fig. 5. Decomposition of a velocity vector for a single wire
Rys. 5. Rozkład wektora prędkości na trzy składowe dla pojedynczego włókna
In the present configuration the sensor wires are orthogonal, so they can be related
to the Cartesian coordinate system: wire I – axis x, II – axis y, III – axis z (see Fig. 3).
Accordingly, we get a system of equations:
(3)
The effective velocity for individual wires is related to the measured signals, in accordance with the King’s law given by the formula:
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(4)
where:
U — voltage being the measured signal from the sensor,
A, B, n — factors obtained from calibration.
The equation (4) directly yields the effective velocity:
(5)
Parameters of the sensor being determined, equations (3) and (5) yield a simple
formula:
(6)
where:
It is readily apparent that velocity components depend on the matrix K containing
the factors k, associated with individual wires, and the effective velocity.
3.1.2. Calibration procedure
In order that velocity vector components are derived on the basis of measurement
signals, it is required that individual values of parameters of A, B, n, k be first determined
for each wire. The sensor is configured such that each wire in turn should be parallel to
the flow whilst the others are normal to it.
The calibration procedure utilises the two-step method. Firstly, the relationship (5) is
fitted basing on the characteristics U = f(v) of each wire, obtained for the flow direction
normal to the wire. Accordingly, we get the desired parameters A, B, and n.
In the second step the factor k was obtained for the flow direction parallel to the wire.
Accordingly, the Jorgensen formula (2) can be simplified, yielding:
(7)
The effective velocity and the preset velocity in a wind tunnel being known, the above
equation yields the set of k values for each wire, for various velocities. The relationship
k(ν) is then fitted with a straight line, to find the desired parameter ki in the matrix K.
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3.2. Look-up table method
Another method widely applied to find the velocity vector components is referred
to as the look-up table technique (Elsner & Drobniak, 1995) where calibration data are
tabulated and stored in computer memory. Velocity vector components are table elements,
the voltages acting as indices.
Major advantages of the look-up table method are: minimisation of the measurement
errors and reducing the number of operations required to derive the effective velocity
from (5) and the velocity vector from (6) so the calculation procedure gets much shorter.
On the other hand, the table to be stored in the computer memory might be large in size
and the sensor calibration becomes a most complex procedure.
4. Modified method of deriving velocity vector components
The modified method involves the direct derivation of velocity vector components
from the measured signals, without any intermediate procedures.
4.1. Mathematical model
The simplified two-step method utilises transformed equations (5) and (6), such that
one step should directly yield velocity vector components. The matrix equation is solved
using the method of determinants:
(8)
where: the parameter ki2 is substituted by hi, and effective velocity vef – by the transformed King’s law (5) for each wire.
Accordingly, velocity vector components are derived from the formulas:
(9)
assuming the following substitutes:
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b1x =
b1 y =
b1z =
a1x =
a1 y =
h2 h3 - 1
,
B12 n1W
1 - h3
,
B12 n1W
1- h2
,
B12 n1W
A1 ,
A1 ,
a1z = A1 ,
1 - h3
,
B22 n2W
h h -1
,
b2 y = 1 2 3n
B2 2W
1- h
b2 z = 2 n 1
B2 2W
a2x = A2 ,
a2 y = A2 ,
b2 x =
a2z = A2 ,
b3 x =
b3 y =
b3 z =
a3x =
a3 y =
1 - h2
,
B32 n3W
1 - h1
,
B32 n3W
h1h2 - 1
,
B32 n3W
A3 ,
A3 ,
(10)
a3z = A3 ,
W = h1h2 h3 - h1 - h2 - h3 + 2
Hence we get a mathematical model relating the velocity vector components to the
voltages from the three-wire sensor. This model has 27 parameters describing the sensor
behaviour in the investigated flow. Parameter values are obtained from calibration.
4.2. Calibration – measurement data processing algorithm
The modified algorithm for obtaining velocity components on the basis of measured
signals has 27 parameters. Determining such a number of parameters on the basis of calibration data is far from being a trivial task that is why genetic algorithms are employed.
They are mostly used to solve multi-dimensional optimisation problems which are hard
to solve by conventional methods as they calculation procedure might prove difficult and
time-consuming or the relationships describing the problem are not available.
Genetic algorithms are procedures that employ the mechanisms of natural selection
and natural genetics. In order to solve an optimisation problem, it is required that an
individual in a population and the fitness function be determined first. The individual
contains full information about the investigated problem, retains the values of all variables in the form of binary string of adequate length. For each variable the following
parameters are defined: variability range, number of bites and the coding mode. An
individual acts as a single solution to an optimisation problem. Individuals make up
a population, being the set of solutions.
The fitness function is the key element in optimisation. It is indicative of the individual’s quality. The value of the fitness is a comparative index expressing how close
optimisation criteria are fulfilled. A major advantage of genetic algorithms is that there
are no specific requirements with respect to the fitness function, such as continuity or
differentiability.
A genetic algorithm applies the calculation procedure to the whole population of
individuals. In subsequent generations the population tends to maximise the fitness
function through the selection of individuals, based on the principles of natural selection.
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Improvement of the population is achieved through the processes of selection (of the best
fitted individuals from the populations), crossovers (exchange of information between
individuals) and mutation (adding new information to some individuals).
Genetic algorithms employ the function of random selection to create the initial population, to select the individuals for the crossovers or mutation, etc. If the parameters are
appropriate, genetic algorithms prove to be insensitive to local extremes. For sufficiently
large populations genetic algorithms usually find nearly optimal solutions.
The operating principle of genetic algorithms is shown schematically in Fig. 6.
Fig. 6. Schematic diagram of a genetic algorithm
Rys. 6. Schemat działania algorytmu genetycznego
In our problem the normalised fitness function is the inverse of the sum of squared
differences between the preset velocity and that derived from the model (9). The purpose
of the optimisation procedure is to achieve the best possible fit of the preset characteristic.
Because of the nature of genetic algorithms, the task of minimisation had to be changed
to that of maximisation, by applying the inverse of the objective function.
5. Experimental
The adequacy of the traditional and modified method was compared and results
were verified in the course of an experiment involving the series of measurements taken
with a three-wire sensor. Velocity vector components were then established by the two
methods.
5.1. Experimental setup and data acquisition
The sensor was placed inside a wind tunnel. The probe was placed inside the rotor seat
and fixed at the angle 54,7° with respect to the flow path. During the whole experiment
measurements were taken for the full revolution of the probe, every 10°, the velocity being
varied from 1 to 30 m/s every 1 m/s. The results were written in the form of files.
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Voltages obtained for the angle α = 0°, 120°, 240° were considered during the calibration as only in these configurations the one wire would be parallel and two should
be normal to the flow. The remaining data were utilised to compare the conventional
method with the modified one.
5.2. Results
The comparison between the two methods is supported by a qualitative and quantitative analysis.
5.2.1. Qualitative evaluation
Fig. 7 shows the decomposition of a velocity vector, obtained by the two methods for
the flow velocity 14 m/s. Theoretically predicted patterns (1) are indicated on graphs for
comparison. For this value of preset velocity there is no major difference between the two
determine characteristics and the discrepancy between the experimental and theoretical
Fig. 7. Comparison of the conventional and modified method against the theoretically predicted
decomposition of a velocity vector
Rys. 7. Porównanie metody klasycznej i zmodyfikowanej z teoretycznym rozkładem
wektora prędkości na składowe
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data is readily apparent. A velocity increase whilst the sensor rotates between the two configurations normal to the flow (vx – 180°, vy – 300°, vz – 60°) is associated with the fact that
the wire is obscured by supports. The “insensitivity zone’ observed in the perpendicular
configuration might be attributable to inaccurate sensor calibration for small velocities.
The relative error in the form of surface areas for each vector component and the
absolute value is shown in Fig. 8. This error is determined with respect to values derived
from the theoretical model (1).
Fig. 8. Relative error in the conventional and modified method (vt – velocity in tunnel).
Rys. 8. Błąd względny dla metody klasycznej i metody zmodyfikowanej
(vt – prędkość zadawana w tunelu).
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It appears that the modified method gives better results in the case of this particular
sensor, particularly for the vector components vy and vz. Improved results can be seen
in plots of absolute value. The graph obtained for the modified method is flatter. The
optimisation did not involve the absolute value of the velocity vector but its components
only, therefore it is reasonable to suppose that the modified method should give better
results – the modulus of velocity is approaching the preset value.
5.2.2. Quantitative evaluation
The comparison between the two methods utilises the quality index:
where:
v’ — predicted velocity derived from the velocity vector decomposition
model (1),
v — velocity calculated from the measurement data,
Nx = Ny = Nz = N — the number of measurements.
Accordingly, we get:
• for the two-step method: Q = 1.83%,
• for the modified method: Q = 1.66%.
It is readily apparent that the variations of the quality index in the two methods are
minor, though in the modified methods its value is smaller. In relation to the conventional
method the quality index improves by about 10%. It appears that further optimisation
of the data processing algorithm might yield still more favourable values of the quality
index.
6. Conclusions
The paper outlines the conventional and modified method of obtaining the velocity
vector. An experiment was conducted to reliably compare the accuracy of the two methods. The experimental data were then utilised in the sensor calibration to obtain the
relevant parameters such that velocity vector components should be determined. The
comparison of experimental data and the reference (theoretical) model reveals that the
modified method gives slightly better results that the conventional two-step method.
A major shortcoming of the conventional method is that sensor position with respect to
the flow has to be most carefully controlled during calibration. In the modified approach
parameters are not determined for each wire separately, but for the sensor as a whole,
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which is considered to be a major advantage. Any configuration of sensors wires during
the calibration is permissible as long as the mathematical model of this configuration is
known. It vastly simplifies the calibration procedure and helps to more precisely determine the velocity vector components.
The proposed modified method utilises 27 parameters describing three sensor wires.
The parameters are determined using the genetic algorithms. It appears that the optimisation results can be further improved provided the parameters in the genetic algorithms
are carefully chosen.
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REVIEW BY: PROF. DR HAB. JAN KIEŁBASA, KRAKÓW
Received: 18 July 2005