Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
AN ANALYTICAL ESTIMATION OF THE CORIOLIS METER'S
CHARACTERISTICS BASED ON MODAL SUPERPOSITION
J. Kutin*, I. Bajsić
Laboratory of Measurements in Process Engineering, Faculty of Mechanical Engineering, University of
Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia
*Corresponding author: T: +386-1-4771-131, F: +386-1-4771-118, E: [email protected]
Abstract
The aim of this paper is to derive approximate, analytically expressed, theoretical
characteristics for a straight, slender-tube Coriolis meter, which can be applied to any of its
working modes. The mathematical model is based on the theories of the Euler beam and onedimensional fluid flow, and includes the effects of axial force, added masses, damping and
excitation. The analytical approximations are evaluated by applying a Taylor-series expansion
to the solutions of the Galerkin method, which are considered as a superposition of the Eulerbeam modal functions. On the basis of the obtained analytical expressions, the properties of
the meter's characteristics are discussed, with the emphasis being on particular nonidealities.
Keywords: Coriolis meter; Theoretical characteristics; Analytical approximations;
Nonidealities
1.
Introduction
The primary sensing element of the Coriolis meter, a device for measuring the mass
flowrate and the density of fluids, is a vibrating measuring tube that conveys the fluid to be
measured. The tube vibration is maintained at its natural frequency by using a motion sensor
and an exciter in a proper control loop. The mass flowrate is determined by measuring its
effect on the tube mode shape, mostly as the phase or time difference between the signals
from symmetrically located motion sensors. Furthermore, the fluid density is measured using
its influence on the tube's natural frequency.
The purpose of this paper's theoretical study is to present some basic characteristics of the
Coriolis meter in an analytical form, showing explicit relations to the measured quantities and
other effects. Such results (although possibly approximate) could be useful, not only for
giving a clearer understanding of the meter's behaviour, but also as a guideline for its design,
proposing the form of calibration functions, reducing nonidealities, etc. In any case, the
mathematical model has to be fairly simple in order that the analytical calculations are
reasonable.
The meter under discussion, with its straight and slender measuring tube, represents one
of the simplest (but not unusual) configurations of the Coriolis meter. It is assumed that it can
be adequately modelled by using the theory of the Euler beam and one-dimensional fluid
flow. Such a model has been already studied in several scientific works relating to Coriolis
meters. Some numerical results, including the effect of the added masses, were reported in
Refs. [1, 2] by following an exact solution procedure. Other studies have mostly been oriented
to analytical approximations. In Refs. [3-5], the theory of small perturbation was used to
analyze the meter's ideal characteristics, for which the measurements of mass flowrate and
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Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
fluid density are independent. This analysis was later extended to the added-masses effects of
the exciter [6] and the motion sensors [7].
The method of characterizing the Coriolis meter in this paper follows and builds upon the
authors' previous work (see, e.g. Refs. [8, 9]). The mathematical model is solved by utilizing
the Galerkin method, in which the general solution is treated as a superposition of the Eulerbeam modal functions (see Section 2). By applying a Taylor-series expansion to the results,
analytical approximations of the meter's characteristics are determined. They are presented in
the following sections: first, the ideal characteristics in Section 3; and then the nonideal
effects of large fluid velocity, axial force, added masses, and damping and excitation in
Sections 4-7, respectively.
We already discussed such nonidealities in Ref. [8], but without explicit analytical
derivations. The effect of a large fluid velocity, also termed the stability-boundary effect, has
been further (also analytically) studied in Ref. [9] – its analytical description is included in
this paper only for reasons of its integrity. However, the present analysis of the added-masses
effect, when compared with the previous accessible works, contributes with its relatively
simple analytical approximations, and the combined discussion of all three masses leads to
some interesting conclusions. Some new findings are also revealed by studying the extended
mathematical model with the effects of damping and excitation force.
All analytical expressions in this paper are given in the general form, valid for any of the
meter's working modes. However, the numerical results, which are mostly presented in the
diagrams, were calculated only for the first mode.
2.
Mathematical model
2.1 Basic assumptions
The model of the Coriolis meter under discussion is schematically presented in Fig. 1. It
consists of a straight, slender measuring tube of length L, mass per unit length Mt and flexural
rigidity EI, which is clamped at both ends and is modelled as Euler beam. The tube conveys
the incompressible, steady fluid flow of mass per unit length Mf and mean axial velocity V,
which is approximated by a one-dimensional, plug-flow model. The tube is generally loaded
with a compressive axial force P, which could be the consequence of tube prestress, internal
fluid pressurization or temperature tension. The effect of the exciter and two motion sensors,
which are attached to the tube, is introduced by three point masses mj at the locations xj,
where j = e, l and r, for the exciter, the left sensor and the right sensor, respectively. The
whole paper assumes that the sensors are located symmetrically with respect to the tube's
midpoint ( xl  L  xr ), and the exciter at the midpoint ( xe  L / 2 ). The mathematical model
also includes a linear viscous dissipation of vibrational energy with the equivalent damping
constant C, which is compensated by the external excitation force f(t) at x  xe . The assumed
damping model is mostly appropriate to represent the viscous-damping effect of the
surrounding fluid.
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Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
z
L
s
me
ml
P
V
w(x,t)
xl
mr
P
x
f( t)
xe
xr
Fig. 1.
Model of the straight, slender-tube Coriolis meter.
2.2 Equation of motion and boundary conditions
The derivation procedure of the mathematical model will not be presented in this paper.
(The reader is referred to the literature on fluid-structure interactions, e.g. [10], where the
models of such systems are also studied.) However, based on the model assumptions in
Section 2.1, the equation of motion and the boundary conditions, which describe the tube's
lateral deflections, may be written as:
  2
 4
 2
 2
 
2

















2
1

 2

j j  j   2       e  ,
 4

 

(1)

(2)

 0 at   0 ,   1 ,

where a dimensionless notation was applied by using the reduction on the tube's parameters L,
Mt and EI:

w
x
t
,  ,  2
L
L
L

CL2
fL2
, 
.
EI
M t EI
Mf
mj
Mt
EI
PL2
, 
,   VL
, j 
, 
,
Mt
EI
LM t
Mt
EI
(3)
Recall the meaning of the fluid's dimensionless parameters:  may be related to the fluid
density,  to the fluid velocity and  to the fluid mass flowrate.
2.3 Solution procedure
A harmonic excitation force () with frequency  leads to the harmonic steady-state
response of the tube (,) with the same frequency. Both functions may be written in
complex form as follows:
()   exp  i  , (, )   () exp  i  ,
(4)
where  and () are the dimensionless (generally complex) amplitudes. The dimensionless
frequency  is related to the dimensional circular one, , by:
  L2
Mt
.
EI
(5)
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Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
Utilizing the Galerkin method [10], () can be expressed as a linear superposition of a finite
number of the Euler-beam modal functions n(), which satisfy all the boundary conditions
(2):
 ()   An n () ,
(6)
n
where:
n ()  cosh  n   cos  n  
cosh  n  cos  n
 sinh  n   sin  n   ,
sinh  n  sin  n
(7)
and the n values result from the equation cos  cosh   1 .
Following the Galerkin method, Eqs. (4) and (6) are substituted into Eq. (1), which is
multiplied by m() and then integrated with respect to  from 0 to 1. The result is a
nonhomogenous system of linear equations:


YA  X , Ymn   m 4  i  1     2 cmn  2id mn   2    emn   2   j f mn ( j ) ,
j
X m    m ( e ) ,
(8)
where:
1
cmn   m n d , d mn
0
1
1
d n
d 2 n
d , emn    m
d ,
  m
d
d 2
0
0
f mn ( j )  m ( j ) n ( j ) .
(9)
(The reader is referred to Ref. [11], where the integrals in constants cmn, dmn and emn are also
analytically evaluated.) After solving Eq. (8) for Am, the actual tube deflections are
determined as the real part of the supposed complex solution (4):
(, )  () cos    ()  , ()  arg () .
(10)
The locations of symmetrically attached motion sensors can be given as l  1    / 2 and
r  1    / 2 , where  is their dimensionless separation. The phase and time difference, 
and , between the motion of these points are:
  ( r )  (l ) ,    /  .
(11)
Because the measuring tube of a regularly operating Coriolis meter vibrates at resonance, i.e.
with its natural frequency, we are mainly interested in a description of this state. The natural
frequencies k can be calculated as the real parts of the solutions of the equation det Y  0 . In
the following, all parameters evaluated at the particular natural frequency k will be denoted
by the corresponding index k.
If only the natural modes are of interest, the simplified, homogeneous mathematical
model, which neglects the effects of damping and excitation, may also be suitable. Such a
model will be assumed in the major part of this paper, except in Section 7. The solution
procedure is similar to that already described, only that the Galerkin method leads to a
homogeneous system of linear equations. For its nontrivial solutions, the condition det Y  0
has to be fulfilled, and so the natural frequencies k are given. The proportional constant of
an arbitrary value in the Am solutions can be determined by the mode normalization.
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Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
The solution method described in this section is approximate, with its results converging
to the exact values with an increasing number of terms in the Galerkin expansion. All
numerical calculations in this paper were realized with a sufficient number of terms, so that
the error of the Galerkin method can be ignored.
3.
Ideal characteristics
We begin with the solution of a simplified equation of motion (1), where the effects of
axial force, added masses, damping and excitation are eliminated (    j      0 ):
2
 4
 2
 2
2  







2
1
  2  0.
4
2


(12)
Fig. 2 and 3, respectively, show the variations of the first-mode natural frequency and the
time difference with the mass flowrate, calculated in the stable range (   cr ,1 ) for three
values of fluid density. The critical velocity cr ,k is associated with the onset of the static or
dynamic instabilities of the measuring tube, and can be estimated for a particular mode k as:
2

 k .
ekk 
cr ,k
(13)
For the simplified model under discussion, the ideal characteristics of the Coriolis meter can
be defined with the condition of an adequate distance from the stability boundary, i.e.
2
2  cr , k (see, e.g. Ref. [3, 4]). By considering this condition in the expansion of the
Galerkin-method solutions in the Taylor series with regard to the fluid velocity  at   0 , we
can estimate the natural frequency and time difference analytically:
2

( id )
k

 k ,
1 
(kid )   (kmid ) , (kmid )  4
m
(14)
d mk
 m ( l )
,
4
 m   k k (l )
4
(15)
where m represents a series of even integers, m = 2, 4, ...  (odd integers, m = 1, 3, ... ) for
an odd (even) mode k. As is evident from the above equations, the ideal characteristics enable
independent measurements of fluid density and mass flowrate, and the linear relation of the
time difference to the mass flowrate. The measuring effect of the fluid density is associated
with the translational inertial force   2  / 2 , and that of the mass flowrate with the Coriolis
inertial force 2  2  /  . In view of Eq. (15), the Coriolis-force effect can be understood
as a superposition of the anti-symmetric (symmetric) modes on the symmetric (antisymmetric) working mode.
4.
Stability-boundary effect
Variations in the ideal characteristics, which are the consequence of an inadequate
distance from the stability boundary, are termed the stability-boundary effect [9]. They can be
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Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
20
 = 0.5
=1
=2
1
15
10
5
0
Fig. 2.
0
1
2
3
4

5
6
7
8
9
Variation of the first-mode natural frequency with the mass flowrate in the stable range for three
values of fluid density ("exact" results).
0.12
 = 0.5
=1
1
0.09
=2
0.06
0.03
0.00
Fig. 3.
0
1
2
3
4

5
6
7
8
9
Variation of the first-mode time difference with the mass flowrate in the stable range for three values
of fluid density ("exact" results,  = 0.5).
analytically approximated by a further Taylor-series expansion in the fluid velocity at   0 ,
4
up to the condition 4  cr , k :

  2
 k   (kid ) 1   g cen , k  4 g cor , k
  ,
1  

(16)
 
  2
4
k   (kmid ) 1   hcen , km  4 k hcor ,km
   ,
1  
m
 

(17)
where:
2
g cen ,k
e
d
 kk4 , g cor , k   4 nk 4 ,
k
n n  k
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Flow Measurement and Instrumentation 12 (2002) 345–351
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hcen,km  
e
 ( ) 
d
emn
d nk
e
  4 pk 4  mp  p l   4 kk 4 ,
4
k (l )   m   k
 n   k d mk
p  p   k  d mk
hcor ,km  
d nk
4
n  k
n
n
4
4
1
 4
4
m  k

p
 d mp  p (l ) 



  k  d mk k (l ) 
d np
4
p
4
2
2
d nk
1
d
 ( ) 
n  4   4  3  n  4 nk 4 n (l )  .
k
l 
n
k
n
k

(18)
For the odd (even) mode k, the indices m and n represent a series of even integers, m = n = 2,
4, ...  (odd integers, m = n = 1, 3, ... ), and p a series of odd integers, p = 1, 3, ...  with p 
k (even integers, p = 2, 4, ...  with p  k). The meaning of the indices is kept the same in the
next sections. It is evident from Eqs. (16) and (17) (and from Figs. 2 and 3) that, for flowrates
that are too large, the natural frequency decreases, and the relation between the time
difference and the mass flowrate becomes nonlinear and simultaneously dependent on the
fluid density. These nonidealities can be interpreted by the effect of the centrifugal force
2  2  /  2 (terms with gcen,k and hcen,km) and the higher-order effect of the Coriolis force
42 2   2  /   (terms with gcor,k and hcor,km).
2
5.
Axial-force effect
The equation of motion (12) can be extended by taking into account the effect of the
compressive axial force that is acting on the measuring tube,   2  /  2 . The estimation of
this force's effect on the meter's characteristics can be analytically calculated using
linearization with respect to  at   0 (and also considering a negligible stability-boundary
2
effect, 2  cr , k ):
 k   (kid ) 1  g cen ,k  ,
(19)
k   (kmid ) 1  hcen ,km   ,
(20)
m
where the constants g cen , k and hcen , km are defined as in the previous section. Because the
  2  /  2 term is identical to the centrifugal force 2  2  /  2 , we would expect an
identical influence on the meter's characteristics, and consequently the identical condition to
2
be negligible,   cr ,k . In spite of the axial-force effect, introduced by Eqs. (19) and (20),
the mass flowrate and the density measurements are still independent, however, it can lead to
large changes in the calibration constants. Actually, the compression or tension in the axially
constrained tube, arising from temperature variations, is one of the demands for measuring
temperature in the Coriolis meter and correcting its characteristics.
6.
Added-masses effect
Furthermore, the equation of motion (12) can be extended by taking into account the
effect of the added masses that are attached to the measuring tube,  j  j      j   2  / 2 .
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Flow Measurement and Instrumentation 12 (2002) 345–351
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The influence of these masses on the natural frequency and the time difference will be
analytically approximated in the same manner as in the previous section, by using
2
linearization with respect to  j at  j  0 (and also considering 2  cr , k ). Following the
expected relation between the natural frequency and the mass of vibrating system, it is
physically correctly affecting the natural frequency by the added masses under the square root
of the denominator. Thus, we obtain:
1
k  
( id )
k


1
f kk ( j )  j  ,
1 

 1  j

k   
( id )
km
m
(21)
4


k
2hs ,km  s  he ,km  e   ,

1 
 1 

(22)
where  s   l   r is the sensors' mass,  e the exciter's mass, and the constants hs ,km and
he,km are given by:
hs ,km  
f (l )  d mp  p (l ) 
f mn (l ) d nk
f kk (l )
  pk

,
 4
4
4
4
4 
k (l )   m   k 4
 n   k d mk
p  p   k  d mk
he,km  
f pk (e )  d mp  p (l ) 
f kk (e )

.
 4
4
4 
4
 p   k  d mk k (l )   m   k
n
p
(23)
As expected, the added masses decrease the natural frequencies. Figs. 4 and 5 show the
comparision of the analytical approximations (Eqs. (21) and (22)) and the "exact" calculations
for the first natural frequency and the time difference, respectively, where the model with the
null exciter's mass has been studied in this case. Up to the observed values of the sensors'
masses, a very good analytical estimation is evident for the natural frequency, but also the
deviations of Eq. (22)'s results decrease rapidly with the smaller added masses.
1.00
0.97
1 / 1
(id)
l = r = 0.05
0.94
0.91
0.88
0.0
Fig. 4.
l = r = 0.10
0.2
"exact" results
Eq. (21)
0.4

0.6
0.8
1.0
Variation of the relative first-mode natural frequency with the sensors' distance for two values of the
sensors' masses and  e  0 , and for the "exact" and analytical calculations ( = 1,  = 0).
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Flow Measurement and Instrumentation 12 (2002) 345–351
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1.01
1.00
1 / 1
(id)
l = r = 0.05
0.99
0.98
l = r = 0.10
"exact" results
Eq. (22)
0.97
0.0
Fig. 5.
0.2
0.4

0.6
0.8
1.0
Variation of the relative first-mode time difference with the sensors' distance for two values of the
sensors' masses and  e  0 , and for the "exact" and analytical calculations ( = 1,  = 0.05).
A weak point of the added-masses effect on the time difference is that it becomes
generally dependent on the fluid density , with the major influence being for fluids with
small densities (see Eq. (22)). Fig. 6 shows the relative changes of the first-mode time
difference in terms of two different fluid densities,    1,1.5 / 1,0.5  1 (mass flowrate is
constant), for some combinations of the added masses. Although the masses' magnitudes are
not excessive (  j  0.05 means the added mass is 5 percent of the tube mass), looking
generally at the whole diagram, we find that the obtained variations of the time difference are
certainly not negligible for the new-generation Coriolis meters (with an accuracy better than 
0.1 % of the measured flowrate).
0.8
l = r = 0.05, e = 0
l = r = 0, e = 0.05
l = r = e = 0.05
 (%)
0.6
0.4
0.2
0.0
-0.2
0.0
Fig. 6.
densities,
0.2
0.4

0.6
0.8
1.0
Variation of the relative error of the first-mode time difference (in terms of two different fluid
   1,1.5 / 1,0.5  1 ) with the sensors' distance for some combinations of added masses
("exact" results,  = 0.05, e  0.5 ).
9
Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
However, the whole region of the sensors' distance in Fig. 6 is not expected to be used
with the Coriolis meters. In the first mode, for example, the time difference increases with ,
but the vibration amplitude decreases in the same direction. In one model, the optimum
distance was defined with the maximum of the product of time difference and amplitude [1],
which yields  = 0.43 for the first mode analysis. Furthermore, available information about
the sensors' distance in one of the commercial straight-tube Coriolis meters shows that  = 2/3
[12]. Based on these two  values, let us limit the expected distance region, for example, from
 = 0.4 to 0.7. Taking a close look at Fig. 6 in this region, we have an interesting finding: the
fluid-density effect is only relatively small if all three masses are (nearly) the same, but it is
not negligible for the null (relatively small) masses of the exciter or the sensors, respectively.
7.
Damping and excitation effect
Finally, the equation of motion (12) can be extended by taking into account the effects of
damping and excitation force,   /  and      e  . By solving the obtained linear
nonhomogenous model it is clear that the excitation amplitude  does not influence
parameters such as the natural frequency or the time difference. Nevertheless, this amplitude
defines the magnitude of the tube response, and has to be small enough to keep the system
nonlinearities negligible.
Following with a discussion of the viscous damping effects, it may be appropriate to
assume a modally damped system, where the damping is specified separately for a particular
mode k [13]. The damping constant  is replaced with k, and then expressed with the
damping ratio k:
  k  2 1    k  k .
(24)
It could be supposed that the metallic measuring tube conveying the incompressible fluid is
lightly damped, with k of the order of magnitude 10-4 up to 10-2, which is far from the critical
damping limit  k  1 . Analyzing such a damping influence on the meter's (static)
characteristics, we would find it relatively insignificant. It is clear that if the meter employs a
resonance control system based on the phase locking technique, which maintains the vibration
at the natural frequency of an undamped system k, the fluid density measurement is not
under the influence of damping. Even if the resonance point is, for example, determined by
2
 k 1  2 k , the influence of damping could be neglected
the maximum amplitude,  max
k
for the supposed values of k. Besides, the time difference is also independent of the damping
up to the linearization with regard to k at  k  0 .
Finally, the mathematical model discussed in this section can also be used for studying
the effect of excitation frequency or its deviation from the natural frequency. If the solution
for the time difference is successively linearized with respect to the damping ratio k at  k  0
2
and the excitation frequency  at    k (and also subjected to the condition 2  cr , k ),
it can be expressed in an analytical form as:

 

k   (kid ) 1  2h ,km  ( id )  1  ,
m
 k


(25)
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Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
where:
h,km 
4
 p (e )  k 4  d mp  p (l ) 
k

p  ( )  4   4  d   ( )  .
4
4
m  k
k
e
k
l 
p
k  mk
(26)
Fig. 7 represents the comparison between the Eq. (25)'s results and the "exact" calculations.
The diagram shows that a deviation from the natural frequency, e.g. by 1 %, alters the time
difference (the meter sensitivity to mass flowrate) by approximately 0.1 %. Considering
(kid )  (kid ) () in Eq. (25), it is evident that the mass-flowrate measurement generally
depends on the fluid density out of resonance.
1.010
"exact" results
Eq. (25)
 / 1
1.005
1.000
0.995
0.990
0.90
0.95
1.00
1.05
1.10
 / 1
Fig. 7.
Variation of the relative time difference with the relative excitation frequency in the vicinity of the
first natural frequency for the "exact" and analytical calculations ( = 1,  = 0.05,  = 0.5, e  0.5 ,
1  0.001 ).
8.
Conclusions
This paper presents the analytical approximations of some basic characteristics of the
Coriolis meter with a straight and slender measuring tube, in which the variations of the
natural frequency and the time difference are used for measuring the fluid density and the
mass flowrate, respectively. Besides the ideal characteristics, the nonideal effects of
inadequate stability-boundary distance, axial force, added masses, and damping and excitation
have been discussed. The analytical expressions are given in such a manner that we can
observe the influences of particular nonidealities on the meter's ideal operation. In general,
they could be used for studying any working mode of the tube, however, the numerical
calculations in the paper have been limited to the first mode. The constants in the analytical
expressions consist of the (superposed) parameters of the Euler-beam modal functions.
Although the numerical results have been calculated by using a large number of terms in the
modal summation, a good estimation can also be obtained in some cases, by considering only
the nearest modes with regard to the working mode.
We can summarize some of the main conclusions that result from such an analysis of the
Coriolis meter. The most significant property of the ideal characteristics is certainly the
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Flow Measurement and Instrumentation 12 (2002) 345–351
doi:10.1016/S0955-5986(02)00006-7 © 2002 Elsevier Science Ltd.
independence of the mass flowrate and the density measurements. However, it can change for
the worse if the fluid velocity is not small enough with respect to the critical velocity. A
negative effect can also be contributed by the added masses of the exciter and the motion
sensors, especially if they are not of nearly the same magnitude. Generally, a deviation of the
excitation frequency from the natural frequency can also lead to the fluid-density effect.
However, the axial force, up to the defined magnitude, only affects the meter's sensitivity, but
not the independence of both measurands.
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