Performance optimization of aluminium (U) type vibration based electromechanical Coriolis mass flow sensor using response surface methodology/Aliumininio tipo vamzdzio svyravimams optimizuoti naudojamo elektromechaninio koriolio srauto mases jutiklio charakteristiku tyrimas taikant pavirsiaus reakcijos metodologija.
Patil, Pravin P. ; Sharma, Satish C. ; Mishra, Roshan 等
1. Introduction
The electromechanical Coriolis mass flow sensor has a tube through
which a fluid to be measured flows is supported at one end or both ends
thereof , and vibration is applied to a portion of the tube around the
supporting point in a direction vertical to the flowing direction of the
tube. The Coriolis mass flow sensor utilizes the fact that the Coriolis
forces applied is referred to the flow tube when vibration is thus
applied thereto are proportional to a mass flow rate [1]. Successful
operating performance of mass flow sensor depends on the selection of
suitable design variables and conditions [2]. Therefore it is important
to determine the operating design parameters at which the response
reaches its optimum. The optimum could be either a maximum or a minimum
of a function of the design parameters. One of the methodologies for
obtaining the optimum results is response surface methodology (RSM).
Performance optimization requires many tests. However, the total number
of experiments required can be reduced depending on the experimental
design technique. It is essential that an experimental design
methodology is very economical for extracting the maximum amount of
complex information while saving significant experimental time, material
used for analyses and personnel costs [3].
This methodology is actually a combination of statistical and
mathematical techniques and it was primarily proposed by Box and Wilson
[4] to optimize operating conditions in the chemical industry. RSM has
been further developed and improved during the past decades with
applications in many scientific realms. Myers et al [5, 6] present
reviews of RSM in its basic development period and a comparison of
different RS metamodels with different applications is given by
Rutherford et al [7]. A comprehensive description of RSM theory can be
found in [3]. Apart from chemistry and other realms of industry, RSM has
also been introduced into the reliability analysis and model validation
of mechanical and civil structures [8, 9]. This methodology has been
widely employed in many applications such as design optimization,
response prediction and model validation. But so far the literature
related to its application in Coriolis mass flow sensing is scarce.
Thus, the primary objectives of this study was therefore to use RSM
in conjunction with central composite design, which requires fewer tests
than a full factorial design to establish the functional relationships
between three operating variables namely sensor location, drive
frequency and mass flow rate, and phase shift for optimum performance of
Coriolis mass flow sensor. These relationships can then be used to
determine the optimal operating parameters. In the following sections,
the application of RSM and CCD to modeling and optimization of the
influence of three operating design variables on the performance of
Coriolis mass flow sensor is discussed.
2. Response surface methodology (RSM)
RSM is a collection of statistical and mathematical methods that
are useful for modeling and analyzing engineering problems. In this
technique, the main objective is to optimize the response surface that
is influenced by various process parameters. RSM also quantifies the
relationship between the controllable input parameters and the obtained
response surfaces [3].
The design procedure for RSM is as follows: 1. Performing a series
of experiments for adequate and reliable measurement of the response of
interest. 2. Developing a mathematical model of the second-order
response surface with the best fit. 3. Determining the optimal set of
experimental parameters that produce a maximum or minimum value of
response. 4. Representing the direct and interactive effects of process
parameters through two and three-dimensional (3D) plots.
If all variables are assumed to be measurable, the response surface
can be expressed as follows
y = f ([x.sub.1], [x.sub.2], ..., [x.sub.i]) (1)
where y is the answer of the system, and [x.sub.i] the variables of
action called factors.
The goal is to optimize the response variable y. An important
assumption is that the independent variables are continuous and
controllable by experiments with negligible errors. The task then is to
find a suitable approximation for the true functional relationship
between independent variables and the response surface [3].
3. Central composite design (CCD)
As mentioned above, the first requirement for RSM involves the
design of experiments to achieve adequate and reliable measurement of
the response of interest. To meet this requirement, an appropriate
experimental design technique has to be employed. The experimental
design techniques commonly used for process analysis and modeling are
the full factorial, partial factorial and central composite designs. A
full factorial design requires at least three levels per variable to
estimate the coefficients of the quadratic terms in the response model
[4]. A partial factorial design requires fewer experiments than the full
factorial design. However, the former is particularly useful if certain
variables are already known to show no interaction [10]. An effective
alternative to factorial design is central composite design (CCD),
originally developed by Box and Wilson [4] and improved upon by Box and
Hunter [11]. CCD gives almost as much information as a three-level
factorial, requires many fewer tests than the full factorial design and
has been shown to be sufficient to describe the majority of steady-state
process responses. Hence in this study, it was decided to use CCD to
design the experiments. The number of tests required for CCD includes
the standard 2k factorial with its origin at the center, 2k points fixed
axially at a distance, say [beta], from the center to generate the
quadratic terms, and replicate tests at the center; where k is the
number of variables. The axial points are chosen such that they allow
rotatability [11], which ensures that the variance of the model
prediction is constant at all points equidistant from the design center.
Replicates of the test at the center are very important as they provide
an independent estimate of the experimental error. For three variables,
the recommended number of tests at the center is six [11]. Hence the
total number of tests required for the three independent variables is
[2.sup.3] + (2 x 3) + 6 = 20. Once the desired ranges of values of the
variables are defined, they are coded to lie at 1 for the factorial
points, 0 for the center points and [+ or -] [beta] for the axial points
[11]. The codes are calculated as functions of the range of interest of
each factor as shown in Table 1. When the response data are obtained
from the test work, a regression analysis is carried out to determine
the coefficients of the response model ([b.sub.1], [b.sub.2], ...,
[b.sub.n]), their standard errors and significance. In addition to the
constant ([b.sub.0]) and error ([epsilon]) terms, the response model
incorporates [10]:
* linear terms in each of the variables ([x.sub.1], [x.sub.2], ...,
[x.sub.n]);
* squared terms in each of the variables ([x.sub.1.sup.2],
[x.sub.2.sup.2], ..., [x.sub.n.sup.2]; x1 , x2 , ..., xn ;
* first order interaction terms for each paired combination
([x.sub.1], [x.sub.2], [x.sub.1], [x.sub.3], ..., [x.sub.n-i]
[x.sub.n]).
Thus for the three variables under consideration, the response
model is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
The b coefficients are obtained by the least squares method.
In general Eq. (2) can be written in matrix form
Y = bX + [epsilon] (3)
where Y is defined to be a matrix of measured values and X to be a
matrix of independent variables. The matrices b and [epsilon] consist of
coefficients and errors, respectively.
The solution of Eq. (3) can be obtained by the matrix approach [3]
b = [(X' x X ).sup.-1] X' x Y (4)
where X' is the transpose of the matrix X and [(X' x
X).sup.-1] is the inverse of the matrix X' x X .
The coefficients, i.e. the main effect ([b.sub.i]) and two-factor
interactions ([b.sub.ij]) can be estimated from the experimental results
by computer simulation programming applying the method of least squares
using the mathematical software package design expert [12].
4. Experimental design
CCD was used to design the experiments to the reason mentioned
above. In order to obtain the required data, the range of values of each
of the three variables was defined as follows: sensor location of 60 -
120 mm, drive frequency of 62 - 64 Hz, and mass flow rate of 0.1 - 0.3
kg/s. Applying the relationships in Table 1, the values of the codes
were calculated as shown in Table 2. These were then used to determine
the actual levels of the variables for each of the 20 experiments (Table
3).
5. Experimental setup and procedure
Trials were conducted in an indigenously developed setup based on
the Coriolis technology for vibration based aluminium U tube CMFS. A
brief description of the set-up and the Coriolis action is presented as
follows. The Experimental set up used in the present study has been
designed on Pro Engineer Wildfire modelling software and later
manufactured at the Instrumentation laboratory of Mechanical and
Industrial Engineering Department, IIT, Roorkee.
The actual photograph of the experimental setup has been shown in
Fig. 1, which consists of the several functional elements such as:
Hydraulic bench for providing regulated water supply to the flowmeter.
Test bench for supporting the tubes of the Coriolis mass flow sensor.
Excitation system for providing mechanical excitation to the Coriolis
mass flow sensor, consists of an electrodynamics shaker, control unit,
accelerometer and vibration sensor. Virtual instrumentation comprising
of noncontact optical sensors, and a signal conditioning unit as shown
in Fig. 1. A Coriolis mass flow sensor measures mass flow directly,
which is based on the conservation of angular momentum, as it applies to
the Coriolis acceleration of a given fluid.
[FIGURE 1 OMITTED]
In principle, as shown in Fig. 2, a Coriolis mass flow sensor
consists of a tube with a fixed inlet and outlet, which is vibrated
about the axis, formed by the inlet and outlet ends. The tube used in
this study is Aluminium U shaped vibrating tube, and is made to vibrate
using an electrodynamics vibration shaker attached at point as shown in
figure. Optical analog displacement sensors are mounted as indicated in
figure and is labeled as SL on two limbs of tube to measure displacement
signals from the vibrating tube. This means that liquid flow is measured
by transferring vibrational energy from the meter tubing to the flowing
liquid and back to the meter. To appreciate this princepple, imagine a
vibrating tube shown in Fig. 2. If no liquid is flowing, the excitation
in the middle of the tube will cause both arms to vibrate in phase. Mass
flowing into the tube starts to receive vibrational energy from the tube
walls as it enters the first bend. In this process, the tube loses the
same amount of energy. The result is that the phase of the vibrational
cycle lags at sensor location of one limbs, the reverse will happen at
the location of another limbs. The liquid is vibrating as it enters the
bend, but transfers this energy to the pipe. The result is that the mass
flow advances the vibrational phase at the sensor location of another
limb. When combined, these two changes in vibrational phase produce a
twisting of the flow tube. The amplitude of this twist is directly
proportional to the mass flow rate and is nearly independent of the
temperature, density, or viscosity of the liquid involved.
[FIGURE 2 OMITTED]
The details of the experimental procedure used to conduct the
present study have been described in Fig. 3 as shown below. The
hydraulic unit for providing regulated water supply to the mass flow
sensor. The hydraulic unit derives its power from the constant voltage
transformer (CVT) to maintain constant flow rate. The U-tube is made to
vibrate using an electronic shaker. An accelerometer is attached to the
shaker which measures the velocity, amplitude and acceleration of the
vibration induced by the electronic shaker. The accelerometer gives the
feedback to the vibration meter which is observed for maintaining the
constant amplitude. A pair of optical displacement sensors has been
placed on the mechanical positioning attachment facing the two limbs of
the U-tube. The output terminals of the sensors have been connected to
the input of the NIDAQ through a signal conditioner. The processing of
the signals is processed in Labview to extract phase shift from the two
acquired signals using FFT. Accuracy and repeatability for each
experiment was achieved with the same input conditions until stabilized
output was achieved.
[FIGURE 3 OMITTED]
6. Model development and results
Results from the experiments are summarized in Table 3. Considering
the effects of main factors and the interactions between two factors,
Eq. (2) takes the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
The coefficients, i.e. the main effect ([[beta].sub.i]) and
two-factor interactions ([[beta].sub.ij]) were estimated from the
experimental results using a computer simulation applying the method of
least squares in design expert simulation software. From the
experimental results in Table 3 and Eq. (4), the second-order response
functions representing phase shift can be expressed as a function of the
three operating parameters of CMFS, namely the sensor location, drive
frequency, mass flow rate.
The relationship between response (phase shift) and operating
parameters were obtained for coded unit as follows:
In coded variables
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
The response factors at any regime in the interval of our
experimental design can be calculated from Eq. (6). The predicted values
for phase shift with observed values are given in Table 4. The observed
values and predicted values of phase shift obtained using model Eq. (6)
is presented in Fig. 4 as can be seen, there is a good agreement between
predicted values and the observed data points ([R.sup.2] value of 0.97
for phase shift).
[FIGURE 4 OMITTED]
6.1. Effect of variables on phase shift
In order to gain a better understanding of the results, the
predicted models are presented in Fig. 5 through 7 as the 3D response
surface plots. Fig. 5 shows the effect of the sensor location and the
Drive frequency at the high level of mass flow rate. As can be seen, a
higher phase shift can be achieved maintaining a optimum level of drive
frequency and sensor location. Fig. 6 shows the effect of the mass flow
rate and sensor location at the center level of Drive frequency. The
general form of three-dimensional relationship is similar to the
previous figure, i.e. a higher phase shift is obtained with optimum
level of sensor location but maximum level of mass flow rate.
Fig. 7 shows the effect of the mass flow rate and the drive
frequency at the center level of sensor location. It can be seen that a
higher phase shift can be obtained with minimum level of mass flow rate
but center level of drive frequency.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
It is clear from the 3D response surface plots that drive
frequency; sensor location and mass flow rate have a significant effect
on phase shift. A centre level of sensor location is determined as
optimum to achieve maximum phase shift, whereas a maximum level of mass
flow rate is determined to achieve maximum phase shift.
7. Conclusions
The application of response surface methodology (RSM) in
conjunction with central composite design (CCD) to modeling and
optimizing the performance of a Coriolis mass flow sensor was discussed.
CCD was used to design an experimental program for modeling the effects
of sensor location, drive frequency and mass flow rate on the
performance of CMFS. The range of variables of CMFS used in the design
were SL 60-120 mm, DF of 62-64 Hz and mass flow rate of 0.1-0.3 kg/s. A
total of 20 tests including center points were conducted. A mathematical
model equation was derived for phase shift by using the experimental
data and the mathematical software package design expert. A predicted
value from the model equations was found to be in good agreement with
observed values ([R.sup.2] value of 0.97 for phase shift). In order to
gain a better understanding of the three variables for optimal CMFS
performance, the model was presented as 3D response surface graphs. The
model allow confident performance prediction by interpolation over the
range of data in the database, it was used to construct response surface
graphs (Figs. 5-7) to describe the effect of the variables on the
performance of a CMFS.
The results show that the all the three variables have a
significant effect on phase shift. This study demonstrates that RSM and
CCD can be successfully applied to modeling and optimizing CMFS and that
it is the economical way of obtaining the maximum amount of information
in a short period of time and with the least number of experiments.
Acknowledgement
The authors would like to thank the Department of Science and
Technology (DST) Government of India for providing the necessary funding
to carry out this research work.
Received December 21, 2010
Accepted June 07, 2011
References
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1165-1172.
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Table 1
Relationship between coded and actual
values of a variable [3]
Code Actual value of variable
-[beta] [x.sub.min]
-1 [([x.sub.max] + [x.sub.min])/2] -
[([x.sub.max] - [x.sub.min])/2[alpha]]
0 ([x.sub.max] + [x.sub.min])/2
+1 [([x.sub.max] + [x.sub.min])/2] +
[([x.sub.max] - [x.sub.min])/2[alpha]]
+[beta] [x.sub.min]
Table 2
Independent variables and their levels for CCD
Design parameter Symbol unit -1 0 +1
Low center high
Mass flow rate [X.sub.1] Kg/s 0.1 0.2 0.3
Sensor location (SL) [X.sub.2] mm 60 90 120
Drive frequency (DF) [X.sub.3] Hz 62 63 64
Table 3
Central composite design consisting of experiments for
the study of three experimental factors in coded and
actual levels with experimental results
Coded level of
variables
Test
run [X.sub.1] [X.sub.2] [X.sub.3]
1 1 -1 -1
2 -1 1 -1
3 0 0 0
4 0 1 0
5 1 1 1
6 1 -1 1
7 0 0 0
8 -1 0 0
9 -1 -1 1
10 0 0 0
11 0 0 0
12 0 -1 0
13 1 0 0
14 0 0 1
15 0 0 0
16 -1 -1 -1
17 1 1 -1
18 0 0 0
19 0 0 -1
20 -1 1 1
Actual level of Observed
variables phase
shift
Test Sensor Drive Mass
run location frequency flow
rate
mm Hz kg/s degrees
1 120 62 0.1 1.542886
2 60 64 0.1 1.238847
3 90 63 0.2 4.01569
4 90 64 0.2 2.420494
5 120 64 0.3 2.920386
6 120 62 0.3 2.449727
7 90 63 0.2 4.618508
8 60 63 0.2 2.85711
9 60 62 0.3 1.848543
10 90 63 0.2 4.42002
11 90 63 0.2 4.16356
12 90 62 0.2 1.449934
13 120 63 0.2 4.34051
14 90 63 0.3 4.911829
15 90 63 0.2 3.90024
16 60 62 0.1 0.835637
17 120 64 0.1 1.609594
18 90 63 0.2 4.152364
19 90 63 0.1 3.893454
20 60 64 0.3 3.068856
Table 4
Experimental and predicted values of phase shift
Actual level of variables Observed
value of Predicted
Test Sensor Drive Mass phase value of
run location frequency flow rate shift phase shift
mm Hz kg/s degrees
1 120 62 0.1 1.542886 1.509698
2 60 64 0.1 1.238847 1.291726
3 90 63 0.2 4.01569 4.192984
4 90 64 0.2 2.420494 2.514256
5 120 64 0.3 2.920386 2.923913
6 120 62 0.3 2.449727 2.263899
7 90 63 0.2 4.618508 4.192984
8 60 63 0.2 2.85711 3.123008
9 60 62 0.3 1.848543 1.766002
10 90 63 0.2 4.42002 4.192984
11 90 63 0.2 4.16356 4.192984
12 90 62 0.2 1.449934 1.887966
13 120 63 0.2 4.34051 4.606408
14 90 63 0.3 4.911829 4.800876
15 90 63 0.2 3.90024 4.192984
16 60 62 0.1 0.835637 0.699161
17 120 64 0.1 1.609594 1.559185
18 90 63 0.2 4.152364 4.192984
19 90 63 0.1 3.893454 3.585092
20 60 64 0.3 3.068856 2.969095
Pravin P. Patil, Department of Mechanical Engineering, Graphic Era
University, Dehradun, Uttarakhand, India, E-mail:
pravinppatil2004@gmail.com
Satish C. Sharma, Department of Mechanical and Industrial
Engineering, Indian Institute of Technology Roorkee, Roorkee, Dist.
Haridwar, Uttarakhand-247667, India, E-mail: sshmedme@iitr.ernet.in
Roshan Mishra, BHEL, Hydrabad, India, E-mail:
roshangreat1@gmail.com
