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Measurement 44 (2011) 1117–1127
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Measurement jou r n a l h o m e pa ge: w ww . el s evi er . c o m/ lo c at e /m e asu r em en t
Analytical calculation of sensitivity for Coriolis mass flowmeter L.J. Wang a, L. Hu a,⇑, Z.C. Zhu a, P. Ye b, X. Fu a a b
The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, People’s Republic of China Tancy Instrument Corporation, Cangnan, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 10 March 2010 Received in revised form 2 March 2011 Accepted 8 March 2011 Available online 12 March 2011 Keywords: Coriolis mass flowmeter Sensitivity Analytical calculation U-shaped Narrowed-U-shaped
a b s t r a c t A simple analytical method is introduced to calculate the sensitivity of Coriolis mass flowmeter (CMF). The definition of the sensitivity is further developed based on the reciprocity principle, through employing two approaches: selecting the moment when the displacement of drive point is zero as the calculation moment; reducing the degree of high-order static indeterminacy by taking a half of the structure due to symmetry in structure and anti-symmetry in load. With these approaches, the method can be used to calculate sensitivities for the flowmeters with any shaped tubes and anywhere detected positions; thus, provides theory basis for tube shape design and detected positions determination. Detail analytical calculations for typical Straight-Circle-joint-shaped CMFs are illustrated. The method is validated on a published U-CMF and then, is further illustrated and experimentally validated through predicting the most sensitive detected point for a narrowed-U-CMF. 2011 Elsevier Ltd. All rights reserved.
1. Introduction Mass flow rate measurement is very important in many occasions, such as measurements of salinity in determination of the dielectric of the water [1], ingredients in chemical reaction [2], and compressed nature gas in energy trading [3], etc. Being an instrument to directly measure the true mass flow rate, Coriolis mass flowmeter (CMF) has attracted more and more interests in the past few decades, for its high accuracy, high rangeability and high repeatability [4]. Typical CMF is composed of electromagnetic driver (EMD), vibrating tube (VT), electromagnetic velocity sensor (EMVS two EMVSs are usually installed symmetrically at upstream and downstream with the EMD respectively) and signal transmitter (ST). At zero flow, the whole tube vibrates synchronously (in phase) as alternating current passing through the EMD. While there is flow passing through the tube, Coriolis force will be acted on the tube wall by the flow. The force has opposite directions on the upstream and downstream tube walls with the EMD, ⇑
Corresponding author. Tel./fax: +86 057187953395. E-mail address:
[email protected] (L. Hu).
0263-2241/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2011.03.011
which distorts the tube and causes phase difference between the output signals of the two EMVSs. Thus, the mass flow Qm can be determined by ST through measuring the time difference DT between the EMVSs’ movements (proportional to the phased difference). The ratio DT/Qm is commonly defined as the sensitivity S of CMF [5,6], which is a very important performance parameter for a CMF, especially for the CMFs used to measure small mass flow rate (such as used in perfume industry [2]) and flows of small density fluids (such as used in compressed nature gas trading [3]). By now, various studies have been developed to improve CMF performance in this regard, which can be formulated to two approaches: – Designing more effective VT shapes to magnify the effect of the Coriolis force: U-shaped [7], X-shaped [4], D-shaped [8], narrowed-U-shaped [9], and straight-shaped [10] have been developed. – Finding more suitable (larger DT) EMVS installation positions (detected points) on the VT [11]. At same time, sensitivity calculation methods (including analytical [7,12] and numerical [13–15] ones) are also studied to exploit the operation of the CMF and also
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Nomenclature a mean radius of tube, a = (r + r0 )/2 dDP displacement of drive point dMP displacement of measured (detected) point l distance of detected point from end point of VT lx position of detected point on VT lST length of the straight segment of tube r inner radius of tube r0 outer radius of tube rCT curve radius of the curved segment of tube s position parameter along the VT v velocity of fluid flow vMP velocity of measured (detected) point x angle velocity of tube x position parameter of VT straight section x0 position of detected point on VT straight section E modulus of elasticity EI flexural rigidity (E I) F additional force added on drive point FC coriolis force F(s) load force on VT including f and fc
provide theory base for above sensitivity improvement works on VT shapes and detected positions. By now, analytical models for U-CMF [7] and straight-CMF [12] have been built based on the theory of vibrating beams in interaction with the 1D fluid flow. However, in order to exploit the entire vibrating procedure of the whole VTs, the calculations are very complex eigenvalues problems; thus, the analytical models prove to be efficient only for these special shaped VTs. Recent advance in mathematical modeling and computational algorithms have provided a rationale basis for numerical approach to solve the problem. In [14], the uncertainties associated with finite element modeling of CMF are discussed in 1D or 2D. A 3D fluid–structure coupled numerical model is built in [15] to simulate the operation of CMF. A common disadvantage of the numerical approaches is different models should be built for different size parameters (even for same VT shape). To optimize the performance of CMF (for example improve its sensitivity), researchers usually have to built large amount of models to find best size parameters and detected points. So, it is still very attractive if we can supply a simple analytical method to solve the problem. As mentioned, providing simple analytical method to exploit the entire vibrating procedure of the whole VTs is still very difficult. What studied in this paper is an analytical method to calculate the CMF sensitivity simply. The method is based on a definition of the time difference (the most important parameter to calculate the CMF sensitivity) given by [16], which relates the time difference to the relative displacement of two detecting points. According to [16], the time difference can be calculated through dividing the magnitude of relative displacement DdMP by the magnitude of two EMVSs velocity vMP. So the sensitivity S = Dt/Qm can be deduced to following equation:
S¼
DdMP
v MP
1 Qm
ð1Þ
G GJ I J Mf M(s) MðsÞ Qm T T(s) T ðsÞ S DdMP Dt
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shear modulus torsional rigidity (G J) second moment polar moment J = 2pa3t fluid mass per unit length moment on VT produced by load force moment on VT produced by unit force mass flow rate additional torque added on drive point torque on VT produced by load force torque on VT produced by unit force sensitivity of CMF, S = DT/Qm relative displacement between detected points time difference between detected points narrowed degree of narrowed-U-CMF angle between tangent of VT and direction of moment angle parameter of VT curve section angle position of detected point on VT curve section
u u(s) h h0
Eq. (1) will be still applicable if the sensitivity is calculated at the moment when the displacement of drive point dDP = 0, for relative displacement and velocity of two EMVSs reach their maximum (equal the magnitudes of these two parameters) at this moment. The parameters DdMP and vMP at this moment can further be calculated as following equations, according to the basic formulations in typical mechanics of materials
mk DdMP ¼
b
ks
v MP ¼ x
l
ð2Þ ð3Þ
where mk is torque produced by Coriolis force and drive force, ks is torsional rigidity of the whole tube structure, mk/ks is the torsional angle according to a basic hooke’s law of torsion in typical mechanics of materials, b is the distance between two EMVSs, x is angle velocity of VT and l is the distance of detected point from end point of VT. However, it is still very difficult to solve the DdMP determination problem when dealing with the complex geometry in practical sensitivity calculations, for it is not very easy to find the inexplicit value of the ks, which even changes as different temperatures and pressures in VT [17,18]. The definition in [16] has a very clear physical meaning and thus, provides a basis for further sensitivity calculations, which also include the method in this paper. The method in this paper further develops the analytical sensitivity calculation method by solving the DdMP determination problem based on the reciprocity principle proposed in paper [19]. This reciprocity principle, which states that the alteration of the VT at the detecting points can be calculated as the integral of the Coriolis force field multiplied by the mode shape driven by unit forces acting at the sensing points, is particularly useful in analytical modeling and analyzing of the CMFs (such as evaluating the velocity profile effects in straight tube CMF [20,21]). By further employing two approaches introduced below,
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the method in this paper solves the DdMP determination problem, making the sensitivity definition and the reciprocity principle applicable in actual calculation; then, provides a basis for both designing more effective vibrating tube shapes to magnify the effect of the Coriolis force and finding more sensitive detected positions on the tube. The remainder of this page offers the followings: 1. In the presentation of the generally principle, a solvable calculation model is built based on two approaches employed to solve the DdMP determination problem, which include selecting the moment dDP = 0 as the calculation moment and reducing the degree of high-order static indeterminacy by taking a half of the structure to a statically determinate structure. Then, general expressions of moment and torque in the model for any shape VT and any detected positions are introduced in implement considerations, to explain how the method calculate the relative displacement at any point on any shaped tube in theory at same time. Detail derivation process for exact analytical expressions of StraightCircle-joint-shaped type CMF sensitivities are given as an illustration of the model calculation. The error sources are also discussed at the last of this section. 2. Due to its typical in structure and available (published) experiment data, a U-CMF introduced in [7] is selected as a special verification example to illustrate and validate the method and its analytical calculation expressions. As will be shown, the comparison between the calculated sensitivity and published experimental data show excellent agreement. 3. The method is further illustrated and experimentally validated by predicting the most sensitive detected point for a narrowed-U-CMF. This also gives a direct introduction of the exact industrial application of our method. 2. General principle 2.1. Calculation model The following discussion explains the generally principle of our method, with the following assumptions: (1) The VT vibrates in the fundamental mode with small
amplitude, which is the general situation in the commercially available CMFs; (2) The VT consists of rigid sections, which assumes that the whole VT vibrates with the same angular velocity; (3) It assumes an incompressible fluid and a uniform (flat profile) flow within the VT ignoring the effect of the distribution of velocity. A straight VT is chosen here as an example to illustrate the general principle of our method. As shown in Fig. 1, the VT is clamped at two end points. When fluid flowing through the VT, its shape will be distorted from straight due to the Coriolis force. As mentioned above, to calculate the sensitivity, the relative displacement DdMP between the two EMVSs should be obtained at first. This is exactly the insurmountable problem in [16]. In our method, two approaches are employed to solve the problem and then, the displacement of any point on the whole VT can be calculated. 2.1.1. Approach I: as shown in Fig. 1a, the moment when the displacement of drive point dDP = 0 is selected as the calculation moment Generally speaking, there are two kinds of forces being acted on the VT during the vibrating procedure, which include the drive force from EMD and Coriolis forces from the measured flow. In most CMFs, the drive force is changing at real-time according to the flow conditions [22], so it is very difficult to obtain the actual value of this force whether through measurement or calculation, and the VT displacement caused by this force is also difficult to be calculated. In our study, the output force performances of EMD for different dDP are analyzed and utilized to select the suitable calculation moment. The selection of the moment when dDP = 0 is exactly based on the analyzed feature that the displacement of the whole VT caused by the drive force from EMD is equal to zero at this moment. Further more, values of the relative displacement and velocity of the detecting points reach maximums (equal the magnitudes of these two parameters) at this selected moment, which assures that the definition in [16] and the Eq. (1) are still applicable in our method. This approach greatly easies the DdMP calculation not only through excluding the influence caused by the drive force, but also simplifying the Coriolis force calculation to be a simple expression.
Coriolis forces Fc1
End point
VT length
VT at its neutral position (without the flow effect) Drive point
Coriolis forces F c2
(a) Illustration of the distribution of Coriolis forces (ignoring the small VT distortion caused by the Coriolis forces) Coriolis forces Fc1 F VT length End point
VT at its neutral position (without the flow effect)
T Drive point
(b) Calculated half of the VT Fig. 1. Calculation model of VT displacement.
End point
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~ ~ Fc ¼ 2Mf x
~ v
including Coriolis force and additional force; the moment Mðs; lxÞ and torque T ðs; lxÞ of every point on integration path are caused by unit force acted on the detected point. Utilizing the symmetry of the structure, the relative displacement of the two EMVSs can be calculated by DdMP = 2dMP and then, substituting it into (1), a general calculation equation about the sensitivity can be given as follow.
ð4Þ
where Mf is the fluid mass per unit length and ~ v is the velocity of measured fluid flow. According to this equation, we can find that in the selected moment, for straight VT, the Coriolis forces acted on it not only have opposite directions on the upstream and downstream halves divided with the drive point (where EMD is installed), but also have sinusoidal distributed magnitudes (as shown in Fig. 1a) after ignoring the small displacement due to the
R
2
MðsÞMðs;lxÞ
ds þ
R
EI
T ðsÞT ðs;lxÞ
ds
GJ
distortion caused by the Coriolis forces (generally, the ratio of the displacement to the VT length is as small as 10 3).
SðlxÞ ¼
2.1.2. Approach II: as show in Fig. 1b, the degree of static indeterminacy is reduced by taking a half of the structure to a statically determinate structure based on the symmetrical feature in structure and anti-symmetrical feature in load, to make the model solvable Although selecting dDP = 0 as the calculation moment, the displacement of VT still can not be calculated directly. As shown in Fig. 1a, the whole VT model is a high-order statically indeterminate system, which is over-constrained and unsolvable. Here, we reduce the degree of static indeterminacy by taking a half of the structure as a statically determinate structure and adding an additional force F and torque T at the dividing point (drive point). This is based on the symmetrical feature in structure and antisymmetrical feature in load. Utilizing the precondition that the twisted angle and bended displacement of the drive point is zero at the selected calculation moment, the additional force F and torque T can be calculated through solving the following canonical equation:
2.2. Implementation considerations
d11
d12
T
d21
d22
F
D1F þ
D2F
¼0
Fig. 2 is used here to introduce the implementation considerations to obtain exact formulations of M(s), T(s), Mðs; lxÞ and T ðs; lxÞ and then, illustrate how the model can be used to design more effective VT shapes and finding more suitable EMVS installation positions (detected points) to improve the CMF performance in sensitivity. The calculated VT in Fig. 2 has a random shape. A curve coordinate s is defined along the VT. Take the point s = s0 to be an illustration, the loaded force on it is F(s0 ); the chord length from it to the moment and torque calculated point is given by a function d(s, s0 ); the angle between the chord and the normal direction at the calculated point is u(s, s0 ); thus, the moment and torque at the calculated point caused by the loaded force at the point s = s’ can be calculated as
DMðs; s0 Þ ¼ Fðs0 Þdðs; s0 Þ sin½uðs; s0 Þ DTð s; s0Þ ¼ Fð s0Þ dð s; s0Þ cos ½ uð s; s0Þ
respectively
(
¼
T
6Þ D1F d22 D2F d12
d11 d22 d21 d12
A simplified calculation of the additional force F and torque T is also provided in Appendix. Through above two approaches, the displacement of any point on the whole VT can be calculated according to the reciprocity principle proposed in paper [19], employing the typical unit-loaded method in mechanics [23].
dMPðlxÞ ¼
Rs 0
Rs 0
Fðs0 Þdðs; s0 Þ sin½uðs; s0 Þ ds
Fðs0 Þdðs; s0 Þ cos½uðs; s0 Þ ds
0
0
ð10Þ
At the same calculated point, the moment and torque caused by the unit force at the detected point, can be calculated according to (9), by setting the parameter s0 = lx
D2F þd21 T d22
MðsÞ ¼ T ðsÞ ¼
and bended displacement caused by unit force, D1F and D2F are twisted angle and bended displacement caused by load force. We have
F¼
ð9Þ
The total moment M(s) and toque T(s) can be calculated by integrations of DM(s, s0 ) and DT(s, s0 ) at a range 0 6 s0 6 s
ð5Þ
where d11 and d21 are twisted angle and bended displacement caused by unit torque, d12 and d22 are twisted angle
(
ð8Þ
Q m xl
ð
and the loaded force F(s0 )=1 N
(
Mðs; lxÞ ¼ dðs; lxÞ sin½uðs; lxÞ T ðs; lxÞ ¼ dðs; lxÞ cos½uðs; lxÞ
ð11Þ
Substituting Eqs. (10) and (11) into Eq. (8), we can obtain entire formulations to calculate the CMF sensitivity. The
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MðsÞMðs;
lxÞ
ds
EI þ
Z
7
T ðsÞT ðs; lxÞ
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ds
GJ ð Þ
As will be given below, s means the position parameter of the curve coordinate along the VT, which is related to the shape of VT; lx means the detected point position; the integration path is from the detected point to the end point along the VT; the moment M(s) and torque T(s) of every point on integration path are caused by a total load force
Fig. 2. Illustration of implementation considerations for random shaped VT.
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ar r' Cross-section of VT x0 z
x
0
x
End point y
v
EMD with vibration angle velocity w
rCT lST
Fig. 3. Parameters of narrowed-U-CMF.
remained parameters are determined by the shape of the VT and the position of detected point respectively or together. In other words, the formulations can be used to calculate sensitivity for any shaped CMFs through giving the basic shape function of its VT, and to finding more suitable detected points through giving different values of lx.
along the circle section. The angle turning from the detected point (as locates on the circle section) to the joint point is h0. Thus, parameters mappings can be given in Table 1, which will be utilized in following calculation:
2.3. Illustration of the calculation with Straight-Circle-jointshaped CMF
2.3.2. When detected point locates on the straight section Utilizing the mapping relationship in Table 1, the moment M(s) and torque T(s) at any point on the straight section can be deviated from Eq. (10) to
Many typical VTs, such as U-shaped, narrowedU-shaped, X-shaped, and D-shaped, can be considered exactly or approximately to be composed of several straight
8 2 > MðxÞ ¼ Fðx þ x0 þ r CT cos /Þ þ 21 F Cðx þ x 0Þ > R p2 þ/ > > þ 0 F Cr CT sinða /Þ½ðx þ x 0 Þ þ r CT cosða > > ¼ Fðx þ x0 þ rCT cos /Þ þ 12 F C ðx þ x0 Þ 2 > <
/Þ da
(
and circle sections. This kind of CMFs are defined as >
þ
) F C r CT ðx þ x0 Þ cos / þ 1
2
4 1
F C rCT cosð2/Þ 1
cos 2 / þ 2 p
2
p
Straight-Circle-joint-shaped CMFs here. To illustrate the
calculation process, a narrowed-U-shaped VT proposed by [9] as shown in Fig. 3 is chosen to be introduced in detail. The narrowed-U-VT was proposed in [9] as its combination of the advantages of U-shaped and D-shaped structures. Its torsional rigidity of the whole structure is smaller than the U-shaped structure, and integration path is longer than the U-shaped structure, but the vibrating frequency of this structure is smaller than the D-shaped structure [9]. The reason we select this VT as example here is its generality in shape function, which can be deformed to a series of VT shapes through changing several parameters. As shown in Fig. 3, the VT is composed of two straight sections with length lST and one circle section with turning angle p + 2u and radius rCT. It can be verified that the VT can be transformed to U-shaped, X-shaped and straightshaped VTs respectively as u = 0, lST = 0 and rCT = 0, and remains narrowed-U shape in other time. 2.3.1. Parameters mapping
> F r cos / sin / sin 2 / þ 2 > 2 C CT > R p2 þ / > 2 > T ðxÞ ¼ Fr CT ð1 þ sin /Þ þ 0 F C rCT sinða /Þ½1 p ¼ Fr CT ð1 þ sin /Þ þ F C r2 CT 1 4
sinða
/Þ da
ð12Þ
As shown in Fig. 3, a coordinate system (with variable x) whose origin locates at the detected point is established along the straight part of VT. The distance from the detected point (as locates on the straight section) to the joint point of straight and circle sections is assumed as x0. A sim-
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In the equation of M(x), the 1st item means the moment caused by the additional force F (its formula and detail derivation is presented in Appendix), the 2nd and 3rd item mean the moment caused by the Coriolis force FC acted on the straight and circle sections respectively. In the equation of T(x), the 1st item means the torque caused by the additional force F, the 2nd item mean the torque caused by the Coriolis force FC acted on the circle section, and the torque caused by the Coriolis force FC acted on the straight section is zero. ilar coordinate system with variable h is also established
The moment Mðs; lxÞ and torque T ðs; lxÞ caused by the unit force can be calculated according to
Table 1 Parameters mapping. Parameters
On straight section
On curve section
s lx
x + x0 x0
rCT (h + h0) r CT h0
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( MðxÞ ¼ x T ðxÞ ¼ 0
The moment Mðs; lxÞ and torque T ðs; lxÞ caused by the unit force can be calculated according to
ð13Þ
( MðhÞ ¼ r CT sin h
Then, the second item in (8) can be reduced and the sensitivity formula becomes to
1 1 Sðx0 Þ ¼
( 2F
l 8 ST
EI
Q x
5
C
2
1
þ 24 x0 þ 3 r CT cos / ðlST
FC
2
þ 4EI ð1 þ cos 2/ÞrCT ðlST
m
x0 Þ þ
FC 1
x0 Þ þ EI 2F EI
1
2
h2
)
2
x0 þ r CT cos /x0 ðlST
1
x0 Þ
ð14Þ
1
l þ 6 x0 þ 2 r CT cos / ðlST 3 ST
The parameter x can be reduced after substituting the equations of FC (Eq. (4)) and F (Eq. (30) in Appendix) into (14). The distance parameter l of detected point from end point of VT can be calculated as l = lST x0. Other parameters are defined in the nomenclature table in Appendix.
8 R pþ/ > MðxÞ ¼ Fðx þ r CT cos /Þ þ 21 F C x 2 þ 2 F C r CT sinða
ð16Þ
2
T ðhÞ ¼ 2r 2CT sin
x0 Þ
According to the Eq. (8), to calculate the displacement of the detected point which is located on the circle section, it is still required to know the moment M(s) and torque T(s) of any point on straight section. The value of the moment M(s) and torque T(s) can be calculated simply according to Eq. (12), making the x0 in Eq. (12) be zero:
/Þ½x þ r CT cosða
/Þ da
0
> ¼ Fðx þ r CT cos /Þ þ 12 F C x2 þ F C r CT x cos / þ 14 F C r 2CTð1 þ cos 2/Þ
ð17Þ
p /
R 2þ
> T ðxÞ ¼ FrCT ð1 þ sin /Þ þ F C r 2 sinða 0 CT > ¼ FrCT ð1 þ sin /Þ þ F C r 2CT 1 p4
/Þ½1
sinða
/Þ da
where the physical meanings of the items in the equation of M(x) and equation of T(x) are of the same with Eq. (12). The moment Mðs; lxÞ and torque T ðs; lxÞ of any point on the straight section caused by the unit force can be calculated according to
2.3.3. When detected point locates on the circle section Also utilizing the mapping relation in Table 1, the moment M(s) and torque T(s) at any point on the circle section can be deviated from Eq. (10) to
8 R hþh 2 sinða /Þ sinðh þ h 0 aÞda > MðhÞ ¼ FrCT sinðh þ h 0 Þ þ 0 0 F Cr CT > ( ) 3 > cos / sin ðh þ h Þ cos / cosðh þ h Þ½h þ sin 2ðh þ h Þ h > 0 0 0 0 > ¼ Fr sinðh þ h Þ þ 1 F r 2 CT 0 C > CT 2 < sin / sinðh þ h0 Þ½h þ h0 þ 1 sin 2ðh þ h Þ þ sin / cosðh þ h Þ sin ðh þ h Þ 2
> T ðhÞ ¼ 2Fr CT sin > > > >
2
¼ 2Fr CT sin
hþh0
hþh0
2
hþh0 2
þ
R 0
2
(
2
2
2F C r CT sinða
/Þ sin
0
2 hþh0 a
2
0
cos /½1
cosðh þ h0 Þ
þ cosðh þ h0 Þ sin /
h h þ 0 2
ð15Þ
2
da 1
þ F C r CT
0
sin / sinðh þ h0 Þ þ 1 sin 2ðh þ h0 Þ 4
<
2
)
2
sin ðh þ h0 Þ cosðh þ h0 þ /Þ
sinðh þ h0 Þ cos /
h h þ 0
1
2
4
sin 2ðh þ h0 Þ
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Same with (12), the 1st item in the equation of M(x) means the moment caused by the additional force F, the 2nd item mean the moment caused by the Coriolis force FC acted on the circle section. The 1st item in the equation of T(x) means the torque caused by the additional force F, the 2nd item mean the torque caused by the Coriolis force FC acted on the circle section.
8 :
MðxÞ ¼ x ð18Þ T ðxÞ ¼ r CT ½1
sinðh0
/Þ
Then, the sensitivity formula of any detected point h0 on the circle section becomes to (19)
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2
hR
p h 0 MðhÞ MðhÞ
lST MðxÞ MðxÞ 0
dx þ
EI
R /þ 2 0
rCT dh þ
EI
Sðh0 Þ ¼ 8
R lST
T ðxÞ T ðxÞ
0
GJ
dx þ
R /þ 2 0
i
h0 T ðhÞ T ðhÞ
r CT dh
GJ
p
Q m xl FC
9
Fr2CT
3
r CT
1
1
l þ 3 r CT cos / þ 6 cosðh0 /Þ lST þ GJ ðlST þ r CT cos /Þð1 þ sin /Þ ½1 sinðh0 /Þ > > EI 8 ST > > > h i > F C r 3 lST FlST 1 2 1 1 2 > > CT þ l þ r cosðh /Þl þ r l cos / þ r cos / cosðh /Þ þ > > cosðh /Þð1 þ cos 2/Þ CT 0 ST CT ST 0 0 4EI CT > EI 3 ST 2 2 > > >
2 ¼
> Fr 3 h > þ CT / þ p 2EI 2 > <
p 1 2
h0 cos h0
FC r 4
2
1
cos h0 sin 2ð/ þ 2
i h0 Þ þ
F C r 2 l3 CT ST 8EI
FC r4
>
3p
CT
>
þ
GJ
l 4 ST
6
2
2
þ sin 2/
2
r
2
ST
sin /
2
where the parameter l in Eq. (19) means the distance of detected point from end point of VT. When the detected point located on the circle section, the parameter l can be calculated as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
½lST þ r CT cosðh0
sin 2/ þ 2 Þ
/Þ þ ½rCT
> ½r CT
r CT sinðh0
/Þ
3
r 1
GJ
CT
þ 4 lST sin 2/
3
> > > > > >
ST
p cos / þ
/l
CT
þ lST cos /
2
2/ cos /
3h0 þ 3/
ð19Þ
p
1 2
Tl
> þ 8EI cos / cos h0 > > FC r2 r p : CT CT
> > > =
ð1 þ cos 2/Þ
FC r 4
2
2
p
h0 Þ þ sin h0 sin ð/ þ 2
CT CT Q m xl > 8EI sin / cos h0 ð/ þ 4 p þ /p h0 þ / sin 2/ þ 2 sin 2/Þ 8EI sin / sin h0 ð/ þ h0 þ > F r4 F C r 2 l2 2 > þ C CT cos / sin h ð/ 2 þ 1 p 2 CT ST h0 þ 3 þ /p 3 cos2 /Þ þ 2EI cosðh0 /Þ cos / 0 > 4 8EI >
>
l¼
2
p
> > > ; >
sin /
(b) As mentioned above, our method assumes a uniform (flat profile) flow within the VT and ignores the effect of the distribution of velocity, which causes the calculated results differ from the experimental results, as discussed in paper [19].
2
r CT sinðh0
/Þ
ð20Þ
2.4. Discussion about error sources The major error sources of above calculation include: (a) In above study, the moment when dDP = 0 is selected as the calculation moment. In this calculation, the magnitudes of Coriolis forces acting on VT is assumed to be sinusoidal distributed and unchanging. However, in actual VT vibrating procedure, the Coriolis forces lead VT to distort, and the distortion of VT lead the Coriolis force to change in turn due to the changed fluid flow. In other words, the fluid–structure interaction arises at all times. The neglect of this interaction in above sensitivity calculation method brings some errors.
(c) When calculating the second moment I according to the equation I = pr3t, the braces on VT and the added mass of the EMVSs and EMD are ignored in this equation. So, as will be shown in following calculation examples, the calculated value of I will be smaller than the true value, which causes the calculated S larger than the true value. (d) Errors are induced due to imprecise determination of physical parameters, e.g. the value of E is assumed to be constant, but actually it changes with different temperatures and pressures.
3. Validation on a U-CMF CMF with U-shaped VT is one of the most widely used in industry; furthermore, experimental data and analytical solution for sensitivity of an actual U-CMF has been given
EMVS
EMD with vibration angle velocity w
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
z
lST rCT
x y
End point
EMVS v
Fig. 4. Parameters of a U-CMF.
1128
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
Table 2 Parameters of the validated U-CMF. Parameters
rCT (m)
lST (m)
r0 (mm)
r (mm)
a (m)
E (GPa)
G (GPa)
EI (N/m)
GJ (N/m)
Mf (kg/m)
Values
0.15
0.64
0.0254
0.0236
0.0245
208
80
17,288
13,299
1.7488
h
F C ðlST þr CT Þ ðlST þr CT Þ3
in [7]; thus, we select this example to be a good illustration
3
and validation of our calculation. The VT structure of the
T¼
3EI
þ
2
r CT
h r CT
2EI
ðlST þr CT Þ
2 ð lST þr CT Þ
GJ
EI
3EI
Þ4
F ðl þr C ST
CT
8EI r2 l
3
lST þr CT
þ 2EI
i
3 þ 3EI
GJ
2
ðlST þr CT Þ
CMF is illustrated in Fig. 4, where the EMD is installed in
r CT ðlST þr CT Þ
þ
CT
r
ð ST þ
CT
r3 Þ
þ
GJ
the middle of the curve section, and two EMVSs are installed on the two joint points of the straight and the curve
1
The experiment data of time difference verses different velocity of the measured fluid (water) have been given in [7]. We plot them again in Fig. 5. In the experiment system of [7], other velocity flowmeter was adopted to measure the velocity of fluid in VT, and the time difference was obtained by counting the pulses in ST. The experiment curve is approximately linear, according to which the experiment sensitivity can be calculated. As
2F C 1
Q x
1 FC 2 2 r lST lST þ r CT lST þ 2EI CT 8 3
EI
3EI
ð23Þ
sections respectively. The space and physical parameters are listed in Table 2. As stated in section 2, the narrowed-U-CMF becomes UCMF when u = 0, so following sensitivity expression can be deduced from Eq. (14) by setting the parameter u = 0 and x0 = 0
S¼
i
CT
shown in Fig. 5, the time difference reaches 0.72e 5 s
m
when the velocity of the measured fluid equals 2 m/s,
2F 1 þ
1 lST þ r CT lST EI 3 2
ð21Þ
then the sensitivity of this defined U-CMF can be calculated according to its definition. Assuming the experiment sensitivity as S0u , we have
where FC is defined in Eq. (4). F can be calculated as following equation by setting u = 0 in Eq. (30) in Appendix
CT
p
1
1 3
2
3 2
r CT 4 p
3
r4
ST
F¼
pr
þ
3
r
2
CT
4EI
3
1
þ EI
1
l 3 ST
2
þ r CT lST þ r CT lST þ
r CT
lST þ
GJ
ð22Þ
CT
GJ
ðlST þ r CT Þ
Dt
S0 U
according to
x 10
þ
2
Also setting u = 0 in (32) of Appendix, T can be calculated
1.5
6
TlST r CT
¼
Qm
Dt ¼
v Mf
0:20585
10
5
s2 =kg
¼
-5
12
Experiment data (adapted from [7])
Most sensitive detected position
x 10
10 8
Δt (s)
2
sitivity S (s /kg)
1
0.75
Þ
Using Eqs. (21)–(23), the time difference at the same velocity points are calculated. The calculated curve is plotted in
-5
Calculated data 1.25
24 ð
6 0.5
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
4 0.25 2 0
Circle section
Straight section
0 0
0.5
1
1.5
2
2.5
3
3.5
v (m/s)
Fig. 5. Time difference versus velocity of measured fluid.
4
0
0.025 0.05 0.075 0.1 0.125 0.15 0.175
0.2
0.225 0.28
s (m)
Fig. 6. Sensitivity along the half-tube of narrowed-U-CMF.
1130
Fig. 5. According to our calculation, the time difference changes linearly with fluid velocity, and the linear calculated curve is very close to the experiment curve as shown in Fig. 5. Then the calculation of the sensitivity can be carried out according to the calculated curve or according to Eqs. (21)–(23) with the parameters in Table 2. Assuming the calculated sensitivity as Su, we have
SU ¼
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
Dt Dt ¼ ¼ 0:20802 Qm v Mf
10
5
s2 =kg
4.1. Predicting the most sensitive detected positions The parameters of the narrowed-U-shaped VT are listed in Table 3. Using the parameter mapping in Table 1 and Eqs. (14) and (19), the change of sensitivity S versus different detected position can be calculated and shown in Fig. 6. We can see that the most sensitive detected position locates on the circle section as s = 0.12246m (h0 = p/2). The sensitivity S = 1.1237e 4 s2/kg.
ð25Þ
4.2. Sensitivity performance experiment of narrowed-U-CMF product and comparison
The relative error between the calculated and experiment sensitivities is
error ¼
According to the optimization result, we fix the EMVSs
0 US
SU
¼ 1:06%
S0U
on the optimized locations. An experiment system is further built to test the sensitivity performance of the narrowed-U-CMF:
ð26Þ
4. Application of the method in predicting the most sensitive detected positions for narrowed-U-CMF
1. The tested narrowed-U-CMF is shown in Fig. 7a, which includes a transducer (with inner structure shown in the left half of Fig. 7a) and a transmitter (with signal processing method shown in the right half of Fig. 7a and whole Fig. 7b). This narrowed-U-CMF is already calibrated and the accuracy is estimated to be within ±0.2%. 2. As shown in left of Fig. 7a, the transducer is composed of two parallel narrowed-U-shaped VTs (with same parameter listed in Table 3), a EMD located at the middle point of the VT, a temperature sensor located at the end of VT and two EMVSs located at the most sensitive detected position (h0 = p/2). 3. As shown in Fig. 7a and b, the transmitter based on DSP is used to drive the VT to vibrate at its fundamental
As mentioned above, the narrowed-U-VT was proposed in [9] as its combination of the advantages of U-shaped and D-shaped structures. Its torsional rigidity of the whole structure is smaller than the U-shaped structure, and integration path is longer than the U-shaped structure, but the vibrating frequency of this structure is smaller than the D-shaped structure [9]. For U-EMF, the most sensitive detected position is the joint point of the straight and the circle sections. However, we found this rule is not correct as used in narrowed-U-CMFs. Thus, we use the proposed analytical calculation method to find the most sensitive detected positions for the narrowed-U-CMF.
Table 3 Parameters of narrowed-U-CMF. rCT (mm)
lST (mm)
r (mm)
r0 (mm)
a (mm)
Mf (kg/m)
u
Values
78
81.98
2.2
3
2.6
1.5197e 2
0.09956p
Drive Upstream EMVS EMD
Frequency
Density
VT
Temp Sensor
Phase
Rate
Temperature
Transducer
Transmitter
(a) Inner structure and principle of Narrowed-U-CMF
Phase (dgree)
Downstream EMVS
Amplitude (mv) EMVS signal (v)
Parameters
( )
E (GPa)
G (GPa)
EI (N/m)
GJ (N/m)
208G
80G
9.1834
7.064
Signal form upstream EMVS Signal form downstream EMVS 0.2 0.1 0 -0.1 0
10
20
30
40
50
Time (ms) 20 frequency
10
0
80
82
84
86
Frequency (HZ)
88
90 phase 1
2 0 -2
phase 2
-4 80
82
84
86
Frequency (HZ)
88
90
(b) Digital signal processing of EMVS signals based on FFT
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
Fig. 7. Illustration of the experiment setup of narrowed-U-CMF.
1131
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
reducing the degree of high-order static indeterminacy by taking a half of the structure; thus, makes the relative displacement DdMP and sensitivity S solvable by unit-load method. The method can calculate the relative displacement at any point on any shaped tube in theory and then, provides a basis for both designing more effective vibrating tube shapes to magnify the effect of the Coriolis force and finding more sensitive detected positions on the tube. Detail calculation process and exact analytical expressions
lST End point rCT Narrowed-U-VT Simplified VT
φ Drive point
Fig. 8. Simplification of the VT structure to calculate the T easily.
frequency (feed-back control technology is employed to realize precise and dynamical tracking of the frequency even the velocity of measured fluid is changing rapidly) and realize the mass flow calculation using signal processing method based on FFT method. 4. A data acquisition card (not shown in Fig. 7) is used to sample the two signals produced by EMVSs respectively. With the FFT method, the vibrating frequency f of these two signals can be obtained (the middle sub-figure) and then, the phase difference Df at this frequency is obtained to calculate the mass flow (the bottom sub-figure). Then, the time difference can be calculated as Dt = Df/(360 f). 5. A water circulation system (not shown in Fig. 7) is used to supply and recovery water. An electromagnetic flowmeter is selected as a reference meter in the water circulation system, to monitor the flow rate from the water circulation system online. The accuracy of the reference meter is estimated to be within ±0.2%. Experiment has been conducted to test the sensitivity performance of this narrowed-U-CMF, with water as the measured fluid passing through the VT, and keeping the temperature constant in the whole experiment process. When the mass flow rate reach 100 kg/h in each VT, the frequency of the VT with the fluid passing equals 91 Hz, and the phase difference from two signals produced by the electromagnetic EMVSs equals 0.1 , then the time difference can be calculated following Eq. (27).
Dt ¼ ð27Þ
0:1 360
1 ¼ 3:0525e 91 Hz
1132
are illustrated with Straight-Circle-joint-shaped type CMFs. The major error sources of the method are as follows: I. ignoring the fluid–structure interaction during the calculation; II. assuming zero VT displacement at the calculated moment dDP = 0, i.e. ignoring the time lag between the displacement and output force of the EMD; III. simplifications of the tube structure as calculating the additional torque T; IV. imprecise determination of physical parameters. The calculation method is validated by calculating sensitivity of a U-CMF introduced in [7] and comparing the calculated result with the published experimental data. The relative error between calculated and experimental data is only 1.06%, which means excellent agreement. The method is further illustrated and experimentally validated by predicting the most sensitive detected point for a narrowed-U-CMF. Different with conventional U-CMF, whose most sensitive detected position is the joint point of the straight and the curve sections, through calculation we find that the most sensitive detected position of the narrowed-U-CMF locates on the circle section (h0 = p/2 for the studied narrowed-U-CMF). An experiment system is built to test the sensitivity performance of the narrowed-U-CMF, and the actual sensitivity of the prototype agrees well with the prior calculated data (relative error is 2.47%). As mentioned above, the proposed method can calculate the relative displacement at any point on any shaped tube in theory. Thus, further work is still needed to calculate the sensitivity of various shaped VT which can not be deformed from the narrowed-U-VT, such as S-shaped VT, L-shaped VT and B-shaped VT, etc. The most sensitive detected positions are also needed to be researched on these VTs.
6s Acknowledgements
So, the result for product testing of sensitivity performance can be calculated as follows 4
Dt Dt 3:0525e ¼ Q m v f M f ¼ 6s 100 kg=h
The authors are grateful to the financial support of the
ð28Þ
National Basic Research Program (973) of China (No. 2006CB705400), the International S&T Cooperation Pro-
According to the experiment data given by Eq. (28) and the calculated data in Fig. 6, the relative error can be calculated
gram of China (No. 2008DFR70410) and Zhejiang Provincial Natural Science Foundation of China (No. R105008).
S0 ¼
10
S0
S error ¼
¼ 1:0968
S
0
¼ 2:47%
ð29Þ
Appendix A.1. Derivation of additional forces F and T
5. Conclusion
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
An analytical sensitivity calculation method is provided in this paper, which adopts two key approaches: I. selecting the moment dDP = 0 as the calculation moment; II.
1133
According to Eq. (6), the additional force F can be calculated by the fact that the displacement of the drive point equals zero, so the F on the Straight-Circle-joint-shaped VT can be calculated easily from Eq. (19) as follows
1134
8 F
1
C > > EI > >
l þ 3 r CT cos 8 ST
/þ
cos / lST þ
6
9
F C r3CT lST
3
r CT
1
F C r CT lST
cos /ð1 þ cos 2/Þ þ
4EI
CT
8EI
2
FC r2
p
2
CT
p
cos /
cos /2 3p þ sin 2/ þ 3/
r2
2
CT ST
2EI
2
1
2/ cos2 /
TlST GJ ðr CT
> > >
> þ r CT sin /Þ > =
3
ð30Þ
CT
þ 4 lST sin 2/
2
3
cos / þ r
ST
sin /
2
F C r 2 l2
/l
CT
4 lST þ lST
6
CT
2
r
r
CT
>þ > GJ > > > F r4 : C þ 8EI
1
sin /ð/ þ 4 p þ /p þ / sin 2/ þ 2 sin 2/Þ þ
ð1 þ cos 2/Þ
8EI 2
FC r 4
> <
F¼
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
3
> > > > >
sin /
p cos2 / r
l
ðl þ r CT cos /Þð1 þ sin /Þ þ EI GJ ST
l þ r CT lST cos / þ r CT cos / þ ð2/ þ p þ sin 2/Þ 3 ST 4EI
CT
1
ST
2
2
;
3
CT
According to Eq. (6), the additional torque T can be calculated based on the precondition that the twisted angle and bended displacement of the drive point is zero at the selected calculation moment. However, a simplified method can be adopted to calculate the T by simplifying the VT structure as follows, which avoiding the complicated angle calculation of the Straight-Circle-joint-shaped VT (see Fig. 8): In the simplified VT, the parameters d11, d21, d12, d22, D1F and D2F in Eq. (5) can be calculated easily as follows
[3] Furio Cascetta, Giuseppe Rotondo, Marilena Musto, Measuring of compressed natural gas in automotive application: a comparative analysis of mass versus volumetric metering methods, Flow Meas. Instrum. 19 (2008) 339–341. [4] Martin Anklin, Wolfgang Drahm, Alfred Rieder, Coriolis mass flowmeters overview of the current state of the art and latest research, Flow Meas. Instrum. 17 (2006) 317–323. [5] J. Hemp, Calculation of the sensitivity of a straight tube Coriolis mass flowmeter with free ends, Flow Meas. Instrum. 12 (2002) 411–420. [6] G. Bobovnik, J. Kutin, I. Bajsic, The effect of flow conditions on the sensitivity of the Coriolis flowmeter, Flow Meas. Instrum. 15 (2004)
8 2 r þr CT sin / cos /Þ þ lST þr CTEI cos / d12 ¼ ðlST þr CT d11 ¼ CT GJ 2EI > <
69–76. [7] G. Sultan, J. Hemp, Modelling of the Coriolis mass flowmeter, J. Sound Vib. 132 (3) (1989) 473–489.
lST r CT cos /
>
d21 ¼ ð
Þ2
ðl ST þr CT cos /Þ
þ
d22 ¼
2EI 3
ðr þr sin /Þ /ÞCT CT
þ
3EI
F C ðl ST þr CT cos /Þ
F C ðl ST þr CT cos /Þ
D1F ¼
3
D2F ¼
2
ðl ST þr CT cos
ðr CT þr CT sin /Þ
þ
GJ
3
3EI
[8] Robert Cheesewright, Ali Belhadj, Colin Clark, Effect of mechanical
4
ð31Þ
vibrations on Coriolis mass flow meters, J. Dyn. Syst. Meas. Contr. 125/103 (2003). [9] Chen Kaiyun, Qiu Liang, Chen Aimin, Xiong Chuxiong, A Coriolis mass
Then the additional torque T can be calculated easily according to Eq. (6) as follows
flowmeter with narrowed-U-shape vibrating tube, Chinese Patent, CN200420081684.4, 2004. [10] M. Anklin, A. Wenger. A new slightly bent single tube CMF for
6EI
8EI
8 9 < F C ðlST þrCT cos /Þ4 þ F C ðrCT þrCT sin /Þ2 ðlST þrCT cos /Þ2 = 9EI
: /Þ
3
F C ðr CT þr CT sin /Þ ðl ST þr CT cos
þ
T¼
corrosive fluids, in: Proceedings FLOMEKO 2003, 2003. [11] Pradeep Gupta, K. Srinivasan, S.V. Prabhu, Tests on various
3GJ
F C ðl ST þr CT cos /Þ
9EI
4
; configurations of
8EI
8 r CT þr CT sin / lST þr CT cos / ðl ST þr CT cos /Þ2 < ð þ Þ 2EI GJ EI
9
2ðlST þr CT cos /Þ
(2006) 296–307.
=
3
3
[12] G. Sulatan, Single straight-tube coriolis mass flowmeter, Flow Meas.
2
:
r CT þr CT sin /
2EI
þ ðl
ST þr CT
2
cos /Þ
ð
GJ
þ
lST þr CT cos / cos /Þ
ðr CT þr CT sin /Þ
Þ½
EI
GJ
ðl ST þr CT
;
ðr CT þr CT sin /Þ
þ
Instrum. 3 (4) (1992) 241–246.
3EI
[13] T. Wang, R.C. Baker, Manufacturing variation of the measuring tube in a Coriolis flowmeter, Sci. Meas. Technol. 151 (3) (2004). [14] R. Cheesewright, Simon Shaw, Uncertainties associated with finite element modelling of Coriolis mass flow meters, Flow Meas. Instrum. 17 (2006) 335–347. [15] N. Mole, G. Bobovnik, J. Kutin, B. Stok, I. Bajsic, An improved threedimensional coupled fluid–structure model for Coriolis flowmeters, J. Fluids Struct. 24 (2008) 559–575. [16] R.C. Baker, Flow Measurement Handbook, Cambridge University Press, 2000. [17] F. Cascetta, S. Della Valle, A.R. Guido, P. Vigo, A Coriolis mass
ð32Þ Obviously, calculation of the additional torque T in this simplified structure is not so precise, although very convenient and simple. This will make the calculation of the additional force F imprecise, because F is related to T according to Eq. (30). This error is evaluated in the validation of the sensitivity calculation method in Section 3 F0 error ¼
F
flowmeter based on a new type of elastic suspension, Measurement 7 (4) (1989) 182–191.
F l ST r CT
GJ
¼ r4
Coriolis mass flowmeters, Measurement 39
T
1%
[18] F. Cascetta, S. Della Valle, A.R. Guido, S. Pagano, P. Vigo, A new r r3
Tl
r
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L.J. L.J.Wang Wangetetal.al./ Measurement / Measurement4444(2011) (2011)1117–1127 1117–1127
ð33Þ CT
p
16EI
1
þ 2EI
1 3
2
3 2
3
l þ r CT lST þ 2 r CT lST þ r CT 4 ST
CT
lST þ GJ
4
4
p
lST þ
CT
6
1135
ST CT
þ
GJ
Substituting the parameters listed in Table 2 into the Eq. (3), we will find that 1% uncertainty of T only induces 0.23% uncertainty of F, so the simplification of the structure is assumed reasonable and the error induced by this simplification is assumed acceptable.
[19] [20]
[21]
References [22] [1] Jannicke Hilland, Simple sensor system for measuring the dielectric properties of saline solutions, Meas. Sci. Technol. 8 (1997) 901–910. [2] Richard W. Miller, Flow Measurement Engineering Handbook, third ed., McGraw-Hill International Edition, 1996.
[23]
straight pipe Coriolis mass flowmeter: the mathematical model, Measurement 9 (3) (1991) 115–123. John Hemp, The weight vector theory of Coriolis mass flowmeters, Flow Meas. Flow Meas. Instrum. 5 (1994) 247–253. J. Kutin, J. Hemp, G. Bobovnik, I. Bajsic, Weight vector study of velocity profile effects in straight-tube Coriolis flowmeters employing different circumferential modes, Flow Meas. Instrum. 16 (2005) 375–385. J. Kutin, G. Bobovnik, J. Hemp, I. Bajsic, Velocity profile effects in Coriolis mass flowmeters: recent findings and open questions, Flow Meas. Instrum. 17 (2006) 349–358. M.P. Henry, D.W. Clarke, N. Archer, J. Bowles, M.J. Leahy, R.P. Liu, J. Vignos, F.B. Zhou, A self-validating digital Coriolis mass-flow meter: an overview, Control Eng. Practice 8 (2000) 487–506. S. Timoshenko, J. Gere, Mechanics of Materials, Van Nostrand Reinhold Co., New York, 1972.