kazahaya 2011

8/10/2019 Kazahaya 2011

1/12

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 4, APRIL 2011 1163

A Mathematical Model and Error Analysis ofCoriolis Mass Flowmeters

Masahiro Kazahaya,Life Senior Member, IEEE

AbstractManufacturers empirically made Coriolis mass flow-meters (CMFs) for more than 25 years, prior to formulation of acomprehensive, fundamental theory. In this paper, we analyticallydevelop a general model of CMFs that supports the designs ofthe flowmeters and are referenced for their improvement, as wellas their production method. We first make a model that fullyexplains CMFs with twin U-shaped flow tubes. Then, we discussits general application to CMFs that use different arrangements offlow tubes, e.g., a single tube or straight tubes. We further explain,by using the developed model, errors in the measurement causedby external vibration, errors by temperature and process pressure,and their error correction.

Index TermsAcceleration, Coriolis mass flowmeters (CMFs),error analysis, mathematical model, meter factor, straight,U-shape.

I. INTRODUCTION

INDUSTRIAL processes widely use Coriolis mass flowme-

ters (CMFs), because their measurement is in the unit of

mass flow rate, which is accurate and independent of the piping

in the upstream and downstream. CMFs measure mass flow

rate, whereas most of flowmeters do volumetric flow rate.

CMFs can calculate and display fluid density as a by-product

of flow measurement, without any additional hardware that is

special for density measurement.The ABB Group, Endress+Hauser Corporation, Foxboro of

Invensys plc, Krohne Inc., Micro Motion of Emerson Elec-

tric Company, Yokogawa Corporation, and other organizations

make and market CMFs with product differentiation. They

make CMFs in various configurations of flow tubes, which are

of twin U-shape, single U-shape, twin straight, single straight,

complex bending, or their variations.

Manufacturers can produce CMFs, without a theoretical

model, on the empirical premise that flow in a vibrating tube

causes Coriolis acceleration, because they individually calibrate

each flowmeter on a flow test stand.

The measuring principles of volumetric flowmeters are clearwith mathematical models. For example, the measuring prin-

ciple of a flowmeter with an orifice plate/differential pressure

transmitter is Bernoullis law, and the measuring principle

of electromagnetic flowmeters (magmeters) is Faradays law.

Oddly, the general theory of a CMF is not explained in the

Manuscript received March 15, 2010; revised September 9, 2010; acceptedSeptember 11, 2010. Date of publication November 9, 2010; date of currentversion March 8, 2011. The Associate Editor coordinating the review processfor this paper was Dr. Jerome Blair.

The author is with MKK International, Inc., Southampton, PA 18966 USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIM.2010.2086691

technical literature by manufacturers and in textbooks. Some

manufacturers partially do by the animation of the flow tubes

through which the flow goes, and others simply say that flow

in a vibrating tube causes the Coriolis force proportional to

the mass flow rate, without showing the relation between the

flow rate and the Coriolis acceleration with exact formulas.

In addition, patentees of the variations of CMFs described

their knowledge of the meters only enough to obtain their

patents.

The oddness may have occurred, because the Coriolis accel-

eration is not widely known. People know the term Corioliswith the wind direction of hurricanes and typhoons, as well

as ballistics on the earth, and wrongly think that the Coriolis

phenomenon is earthbound. It is difficult for them to associate

the Coriolis phenomenon with a stationary flowmeter. Because

the Coriolis phenomenon is not a core subject in science and

engineering, as mechanics, sounds and optics, electricity, and

magnetism are, textbooks of basic science do not necessarily

contain it, except a few that qualitatively explain it [1]. In

advanced books, e.g., [2] and [3], the general discussion of

Coriolis acceleration appears in the chapters of Rate of change

of a vector with respect to a rotating frame.

We use the general discussion to develop a mathematical

model of CMFs in this paper.Academically, researchers look at CMFs as if they try to

bring up their studies to the level of the production and ap-

plication of CMFs by manufacturers and users. Researchers

have made effort to developed CMF models in specific con-

figurations of twin U-shape [4], of straight tubes [5], or of

other shapes and to analyze specific parts of CMFs. A study

[6] assumed the flow tube as a beam and the flow as a moving

string to develop a model, instead of naturally handling them as

tube and fluid, respectively. Other researchers evaluated the ac-

curacies of CMFs in different conditions [7][9]. Furthermore,

readers who are interested in the current state of CMFs may

read through the article Coriolis mass flowmeters: Overview ofthe current state of the art and latest research[10]. No general

model that covers CMFs in all tube configurations has been

developed, and no error analysis of CMFs by using such a

model has yet been made.

The purpose of this paper is to explain CMFs with a mathe-

matical model. We first discuss the model of CMFs with twin

U-shaped flow tubes, which have been manufactured for more

than 25 years. The flow tube lengths are from a few millime-

ters of micromechanical flowmeters to about 1.5 m long of a

300-mm size. The discussion covers vibrations and rotations

(in partial arcs) of the flow tubes, the generation of the Coriolis

acceleration, the force exerted to the flow tube by fluid mass and

0018-9456/$26.00 2010 IEEE

8/10/2019 Kazahaya 2011

2/12

1164 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 4, APRIL 2011

Fig. 1. Block diagram of a Coriolis mass flowmeter.

the acceleration, the electric signals proportional to the flow,

and the electronics that calculates the mass flow rate from the

signals.

Then, we discuss its application to other CMFs that use

different tube forms, e.g., a single tube or straight tubes.

We further analyze the effects of external vibration to the

measurement, errors caused by temperature and internal pres-

sure of the flow tubes, and their error correction.

The model should support the designs of the flowmeters

and be referenced for the improvement of their design, their

production methods, and their applications.

Needless to say, electronic technology around a microproces-

sor enabled making a CMF a highly accurate flowmeter among

all types of industrial instruments, although this paper mostly

discusses the behavior of fluid flow and flow tubes.

II. STRUCTURE OFC MF S

Fig. 1 is a block diagram of CMFs and shows the generalstructure of CMFs. A CMF consists of the transducer, shown in

the upper half of the figure, and the electronics is shown in the

lower half. Manufacturers call the electronics transmitters.

The structure is also vertically divided into the oscillator and

the measuring sections.

The main structure of the transducer section is the flow tube

(or flow tubes) through which the fluid flows at the mass flow

rate M, in the unit of kilograms per second (kg/s). Manufac-turers use different configurations of tubes, straight, U-shape,

spiral, and other sophisticated bending; the number of the tubes

is one or two. Among these configurations, we first take on the

common configuration with twin U-shaped tubes in depth.

The driver continuously and sinusoidally vibrates the tips of

the flow tubes in a pushpull manner. The driver frequency may

be any frequency f, although most CMFs are operated at theresonance frequency of the electromechanical structure of the

transducer to measure the fluid density. When the fluid goes

through the vibrating tubes, it produces the Coriolis accelera-

tion as shown later, and the acceleration and fluid mass together

generate the force that affects the tube motions. The effect is

proportional to the mass flow rate. To determine the relation

among them is the core of the model development.

Two identical sensors, S1 and S2, are fabricated onto the

flow tubes, and they detect the movement of the flow tubes.

They are motion-to-electric signal converters, in generic terms.

S1 and S2 generate the sinusoidal electric signals, v1 and v2,respectively, whose phase difference is denoted as 2 in thispaper. The v1 and v2 voltage signals are the output of thetransducer section, which are sent to the electronics.

As will be calculated later, the mass flow rate is given by

the mathematical model, M=K(1/f)tan , whereK is theconstant, called the meter factor of the transducer. Because

tan , when is small, M is proportional to the phasedifference.

The meter factor of the individual transducer is measured in

wet calibration before completion in manufacturers. The term

wet calibration is common among all flowmeters. It calibrates

a flowmeter to determine its meter factor by placing it onto a

flow test stand. Even if air flow is used for calibration, it is

still customarily called wet calibration. Note that the meter

factors of some flowmeters are determined by the dimensions

of the transducer, such as radii and thickness, without using

a flow test stand. One example is the manufacturing of ori-

fice plates. This approach is called the dry calibration of a

flowmeter.The mechanical dimensions and characteristics of the flow

tubes change by fluid temperature (=tube temperature), as willbe shown later. Therefore, CMFs need and have temperature

compensation.

The electronics works as follows. The zero-crossing detec-

tors detect the timings of the zero crossing of v1 and v2.The pulse counters and the clock pulse are used to measure

the phase difference 2 between v1 andv2, as well as f. Themicroprocessor (P) or digital signal processor (DSP) intowhich the model is programmed calculatesMfrom tanor.The output circuit gives the mass flow signal in the standard

format of 420 mA [11], HART [12], or Fieldbus [13]. The

zero-crossing detection, pulse counters, and frequency mea-surement have been well used in electronics engineering, and

8/10/2019 Kazahaya 2011

3/12

KAZAHAYA: MATHEMATICAL MODEL AND ERROR ANALYSIS OF CORIOLIS MASS FLOWMETERS 1165

Fig. 2. Conceptual view of a transducer with U-shaped flow tubes.

the formats of the flow signal are common among industrial

instruments.

In the electromechanical oscillator section, the adder circuit

sums the v1 and v2 signals to eliminate the effect of theirphase shifts in its output. Then, the output is sent to the

driver after amplification in the polarity of positive feedback.In this arrangement, the driver vibrates at the electromechanical

resonance frequency.

Most CMFs provide the density of the flow fluid, because

manufacturers can provide density information without adding

hardware to the instrument. We record the algorithm that may

be used to calculate fluid density in Appendix B. If density

measurement is not required, a pulse generator or 60-Hz (or

50-Hz) power supply may be used to actuate the driver at the

selected frequency in place of the electromechanical oscillator

mechanism.

III. STRUCTURE OF AT RANSDUCERW IT HTWI NU -S HAPEDF LOWT UBES

We illustrate a conceptual view of the structure of transducers

with twin U-shaped flow tubes in Fig. 2. The OPJUS tube is

flow tube 1, and RQKVW is flow tube 2, both of which are in

an identical U-shape in one transducer with the same lengths

and radii and of the same material, e.g., Stainless Steel 316L.

The radii and the lengths are different by the meter size; the

larger the radii, the larger the meter size. They are vertically

mounted and secured on the rigid nest, of the same material, at

O and S, and R and W, respectively. The legs of the U-shaped

tubes, OP, SU, RQ, and WV, are straight.

The driver is made of a magnet and a voice coil, similar toa driver of a dynamic speaker. It is connected to the tubes at J

and K. When we apply ac current to the voice coil, the driver

oscillates and drives the tops of the tubes in a pushpull manner.

In use, the tips of the U-shaped tubes, J and K, always move in

the opposite directions.

In the figure, we draw each section PJU and QKV in a

half torus (arc): They are straight in some CMFs. They simply

transmit the vibration at J and K to points P and U and pointsQ and V, respectively. The sections should be long enough,

allowing the four points to deflect. As we will later show,

their dimensions do not come into the mathematical model,

however.

Although the driver oscillates point P, point O is fixed at the

nest. Consequently, the straight leg OP deflects like a cantilever,

as well as the motions of the legs SU, RQ, and WV. We will

later apply the formula of cantilever in the calculation of the

displacement of P, as well as U, Q, and V.

We connect one motion sensor S1 between the tips of the

straight legs, P and Q, and the other identical sensor S2 between

U and V to obtain the signals, v1 and v2, respectively. Themounting of the driver and the sensors should allow the tubes

to flexibly vibrate. The sensors may be capacitive sensors,

Hall-effect sensors, magnet-pickup coil sensors, or other types.

In modeling, we use a capacitive type whose mass is low.

Its electric output is proportional to and in phase with its

mechanical input. We explain its structure and characteristics in

Appendix A. The sensors, S1 and S2, measure the relative mo-

tions of the tips of the legs and generate v1and v2, respectively.Fig. 2 may give an impression that the space between the

two tubes is large, which is only for convenience of illustration.

In actual products, the tubes are closely placed, because the

amplitudes of the tube vibrations are small, and the driver and

the sensors are made thin.Now, observe the flow of the fluid. The fluid, whose mass

flow rate is measured, comes into the CMF from the inlet

(inflow) and then equally splits to tubes 1 and 2 by the tube

configuration. The mass flow rate in each tube is m (kg/s),

wherem = (1/2)M. The flows from the two tubes merge andgo out to the outlet (outflow). Because we measure the mass

that moves in the tubes, the flow patterns in themlaminar,

turbulent, or obscureddo not affect the measurement. Conse-

quently, the piping in the upstream and the downstream of the

CMF and the flow splitter in the transducer do not affect the

measurement.

Manufacturers use enclosures with their transducers. On theone hand, an enclosure protects the mechanical structure of

the transducer from the outside. On the other hand, it prevents

the leak of the process fluid to the environment when a flow

tube ruptures. Some designers use the enclosure for reinforcing

the nest.

IV. MODELING

Fig. 3 is a side view of the flow tubes, showing the half

sections, OPJ and RQK. We assign the fixed frame OXYZ on

the nest with theX-,Y-, andZ-axes, whose origin is at O inFigs. 2 and 3; The X-axis agrees with the line OR, theY-axis

coincides with the center line of the OP leg, and the Z-axiscoincides with the line OS. For vector analysis, we assign

8/10/2019 Kazahaya 2011

4/12


Fig. 3. Modeling of Coriolis mass flowmeters.

the unit vectors, i, j, and k to the respective axes in Fig. 3.

Flow measurement is studied in the fixed frame, because the

observers who measure the flow are also in the same fixed

frame, i.e., the ground. When the driver oscillates the tubes,

the OP and the RQ legs oscillate in theOX Yplane and the SUand WV legs in the plane that is parallel to theOX Y plane. Inother words, they vibrate only in the x-direction.

The length of OP, as well as RQ, is l, and the fluid flows

in the tubes at the speed u, upward in both OP and RQ.One input of the sensor S1 is connected to the top point P

of the straight leg OP, and the other input is to Q. Then, the

output signalv1from S1 is proportional to the displacement ofP and Q.

Observe the leg OP in Fig. 3. It is a cantilever. When the

driver drives the flow tubes, the leg rotates around O, although

the rotation is not in full circle but in an arc, i.e., a very small

arc. The movements of the driver and the tube are so small thatnaked eyes barely see them moving.

When flow moves in the leg, the movement and the rotation

together generate accelerations to the fluid, one of which is the

Coriolis acceleration, as explained in the following discussion.

The acceleration and the fluid mass generate force which, in

turn, works on the flow tube.

When the tubes oscillate at the angular velocity , thedisplacement of P, not J, in the figure, is given by D11 and is

expressed as follows:

D11= i D0sin t, and = 2f (1)

where i expresses the displacement is in the direction of thex-axis.D0is the amplitude of the displacement.

Denoting as the deflection angle of the leg, we write

D11= l , and = D11/l. (2)

Then, the angular velocity11of the leg around point O is,

based on (1) and (2) and with vector considerations, given as

follows:

11= d

dt =k

D0l

cos t. (3)

Next, we take on the motion of a particle in the leg OP.Its position vector is r, which rotates in theOX Yplane. For ageneral approach to such systems, vector mechanics teaches us

about plane motion of a particle to a rotating frame to consider

two frames of reference, both centered at O and both in the

plane of the figure (=the page surface), a fixed frame OXY ,and a rotating frame [14]. We draw the rotating frame Oxy in

Fig. 3; the plane Oxyis in the plane OX Y. The rotating leg OP

agrees with the Oyaxis, and Oxy rotates at the angular velocity11. In other words, we consider the rotating coordinate Oxyin the fixed coordinateOX Y.

In such frames, the absolute acceleration of,a in the fixedframeOX Yis given by [15]

a =a + a/Oxy+ ac (4)

where

a=acceleration of an assumed point

of moving frame Oxy coinciding with

a/Oxy=acceleration ofrelative to moving frame Oxy and

ac=complementary or Coriolis acceleration

=211xv/Oxy (5)

wherev/Oxyis the velocity ofrelative to theOxy frame.According to (4), we should note that the Coriolis acceler-

ation is not only the acceleration but two other accelerations

occur in the fluid. The effect of these accelerations to the

measurement should be studied, although technical discussions

by manufacturers and their citations have not discussed these

accelerations in a CMF.

Because point

is in a circular motion around O, a

has the two components: one component is tangential to the

motion and the other component is toward the center O. The

acceleration of the same magnitude in the opposite direction

of the first component also occurs in the leg RQ, and both

slightly and equally squeeze the sensor S1 without affecting

the measurement. The second component is in the direction

of the leg (toward the nest) and does not affect the deflection.

We should note that the U-shaped tubes in the plane cancel the

effect of accelerationa .

The second acceleration ofrelative to moving frame Oxy ,a/Oxy, means the acceleration of fluid in the tube OP, which

is fixed to the frame Oxy. On the other hand, the same mag-

nitude of the acceleration occurs in the tube SU. They canceleach other within the tube. Further investigation, however, is

8/10/2019 Kazahaya 2011

5/12


individually needed with tubes in different configurations other

than the U-shape.

Next, we calculate the last Coriolis acceleration ac11 of the

leg OP. Because the particle moves in the y-direction at thespeedu, we have

v/Oxy= j u. (6)

After the substitution of (3) and (6) into (5), we write

ac11= 2

k

D0l

cos t

[j u] = 2 i

D0l

u cos t.

(7)

Thus, the direction ofac11is i, perpendicular to the plane of

the U-shaped tube, but it changes by cos t and off-phase by/2(90) from the driver that moves in sin t.

This Coriolis acceleration and the fluid mass cause a force

onto the tube. The massof the unit length of fluid is

= S (8)

where is the fluid density, and S is the internal area of theflow tube.

Then, the force fc11 thatandac11 exert on the unit lengthof the leg OP is (we denotedffor the frequency andffor theforce)

fc11= ac11. (9)

The forcefc11per unit length uniformly works on the leg OPover its total length. Consequently, the deflection of the point

P may be obtained by a formula in the strength of materialthat gives the deflection of the tip of a cantilever by a uniform

force over its length. That is, the displacement of the tip of acantilever caused by uniform force w in a static mode is given

by =wl4/8EI, where l is the length, Eis Youngs modulus,andIis the moment of inertia of the tube OP [16].

Although the leg OP is in a dynamic mode by vibration, the

equation of may adequately be precise for the calculation ofthe deflection of the point P for the purpose of this paper. The

reasons are the amplitude of the deflection is small in an order

of a few 1/1000th mm, and the flow tube, including the leg OP,

and the driver are steadily an electrically resistive load to the

oscillator amplifier at the resonance frequency.

Then, the displacementdc11of point P by the Coriolis accel-eration is: dc11= fc11 l4/8EI. By substitution with (7)(9),we write

dc11= l4

8EI ac11= i

l3

4EI D0 Su cos t

= i l3

4EI D0m cos t (10)

becauseSu is the mass flow rate min the flow tube.The displacementd11of point P is then

d11= D11+ dc11= i D0

sin t +

l3

4EIm cos t

.(11)

By repeating the same calculation, we obtain the displace-

mentd12of point Q,d21of point U, andd22of point V, i.e.,

d12= i D0

sin t +

l3

4EIm cos t

(12)

d21= i D0

sin t

l3

4EIm cos t

(13)

d22= i D0

sin t

l3

4EIm cos t

. (14)

Equations (11) and (12), which have the opposite signs,

indicate that the forces by the Coriolis accelerations slightly

twist the top torus section PJU of the U-shaped flow tube 1.

In addition, (13) and (14) indicate that the acceleration does the

torus section QKV of tube 2, similarly by the same amount.

Manufacturers show the twists in the animation of the move-

ment of the tope section of the flow tubes in their sales and

technical literature that explains CMFs.

Sensor S1 receives the displacements d11 andd12 and gen-erates the electric voltage signal v1. Their relation is given by(39) of the capacitive sensor, which is explained in Appendix A:

v1= (2U3/g)(d11+ d12), whereU3 is the power supply volt-age to the sensor.

By the substitution of (11) and (12) into (39), we obtain

v1=4U3

g D0

sin t +

l3

4EIm cos t

(15)

=4U3

g D0

sin t +

l3

8EIM cos t

. (16)

By trigonometry, we have

v1=4U3

g D0 Hsin(t + ) (17)

where tan = l3

8EIM (18)

H2 = 1 +

l3

8EIM

2. (19)

Equation (17) means that the phase difference between the

driver wave form(sin t)andv1is .In the same way, we calculate the S2 outputv2to obtain

v2=4U3

g D0

sin t

l3

8EIM cos t

=4U3

g D0 Hsin(t ). (20)

In Section II, we stated that the adder circuit sums the v1andv2signals to eliminate the effect of their phase shifts in itsoutput. By adding (16) ofv1and (20) ofv2, the second terms ofboth equations cancel each other. It is clear thatv1+ v2are thefunctions of sint, without any phase change. The clean sine-wave signal actuates the driver. When only v1 orv2 is used to

actuate the driver, the phase of the driver signal may be slightlychanged by flow rate change.

8/10/2019 Kazahaya 2011

6/12


A. Mathematical Model by Phase Difference

Next, we formulate the mathematical model of CMF by using

(18). In comparison between (17) and (20), we know that the

phase difference between the sinusoidal signals, v1 and v2,is2. The phase angle is accurately measurable with knownelectronic circuits by detecting the zero-crossing timings ofv1and v2 and measuring the time difference between them. Wealso measure the one-cycle period offwith the same circuit tocalculatein radians.

Here, we should credit the measurement of mass flow rate

by phase difference to J. E. Smith, who discussed the method

in his patent disclosure [17] more than 25 years ago, using

timing of the midplane instead of timing of zero crossing. He

electromechanically detected the midplane. In the modeling in

this paper, we choose to electronically detect the midplane for

higher resolution and stability.

Based on (18), we write

M=8EI

l31

tan = 4

EI

l31

f tan . (21)

Because is small, tan .This expression is the mathematical model of CMF with

twin U-shaped flow tubes. Because the microprocessor in the

electronics can easily calculate tan , we may use tan withoutreplacing it with.

Of the term(4/)(EI/l3), Youngs modulus Eis constant ata given temperature, and the moment of inertia Iis determinedby the inner and outer radii of the tube. Therefore, this value is

constant for each CMF transducer. We write

4

EI

l3 K. (22)

By substituting (22) into (21), we obtain

M =K1

f tan or M=K

1

f, when is small (23)

where K is the meter factor of each transducer in the phase-

difference model.

Manufacturers calibrate CMFs by wet calibration with a flow

test stand and determine the meter factor for each transducer.

The temperature and the pressure of the test fluid may be

designed to be in the reference condition, i.e., 20 C and less

than 500 kPa. We denote theKvalue obtained in the referencecondition K0. K0is loaded into the electronics that is used withthe specific transducer in the application.

Up to (22), we used the moment of the inertia of the straight

legs of the U-shaped tubes for the variable I. The sensors S1and S2 are, however, attached to the legs and may change I[18]. Their effect may not be small in the transducers with small

tube sizes (1/10 and 1/8 in or 2 and 4 mm in size), but it is

negligible with medium and large tube sizes. The aggregate

moment of inertia of the tube legs and the sensors does not

change after each meter is assembled. Because the meter factor

is determined by wet calibration for each CMF that operates

with the aggregate moment of inertia while being calibrated,

the measured meter factor includes the effect of the sensors, ifthere is any.

Next, we study the characteristics of the phase-difference

model that make the design of CMFs easy and their operation

stable. Note that based on (22), the meter factor is determined

by the characteristics and dimension of the U-shaped flow tube.

The driving amplitude D0, the supply voltage U3, and thesensors linearity and sensitivity do not change the meter factor.

These factors only should be large enough to generate adequateamplitudes of signals,v1 andv2, to operate the electronics. Inaddition, we need only the timing signals of zero crossing to

measure for the phase-difference model. Consequently, wedo not need to make the sensor linear in its entire dynamic

range but make it precise around the center of the range. This

step enables the electronics (transmitters) to interchangeably

match with different transducers when the user changes the

meter factor in the memory of the electronics to the specific

value of the transducer to be matched. This approach is useful

for both manufacturers and users.

Micro Motion makes CMFs with flow tubes bent in a pen-

tagon, instead of in a rectangle. The developed mathematical

model should be applicable to the flow tubes; theoretically, the

shape allows making the equivalent leg length longer in a given

form factor, increasing the sensitivity of the transducer.

The CMFs with pentagon-shaped flow tubes has marketing

significance; it is a clever marketing maneuvering of the man-

ufacturer. In the design, the bottoms of the flow tubes O, S, R,

and W in Fig. 2 are brought closer each other. This condition

makes the face-to-face distance of the flowmeters equal to or

shorter than those of other flowmeters, such as magmeters,

vortex flowmeters, and turbine flowmeters. The face-to-face

distance is the distance between the surfaces of the inlet and

outlet flanges of a flowmeter. A user who uses a volumetric

flowmeter, e.g., a magmeter, may remove it from the processpipe and place a CMF with pentagon-shaped flow tubes for

mass flow measurement.

Although the temperature and pressure of fluid do not af-

fect the mass flow rate, its measured value changes, because

Youngs modulus E, the moment of inertia I, and the length lof the flow tubes change by temperature and slightly by pres-

sure in (22). We will later discuss the changes in the error

analysis.

B. Mathematical Model by Amplitude

Equations (17), (19), and (20) indicate that the amplitudeH of v1 and v2 contain the variable of mass flow rate M.Therefore, it is simple to develop a mathematical model of

CMFs by using (19). We call it the amplitude model of CMFs.

Manufacturers had started using the amplitude model mostly

to go around the Smith patents of the phase-difference model.

(The patents expired in the mid-1990s in most countries.)

To use the amplitude model, we need to measure the voltage

ofHwith a high-speed analog-to-digital converter and also thesupply voltage of the sensorU3. For the model, the sensors andthe electronics become demanding for their accuracy, linearity,

and stability to make a CMF with the accuracy of 0.05%or 0.1% of the reading. Not only all of the requirements

increase material and labor cost of the meter but also electronicsmay need consideration for interchangeability. Therefore, we

8/10/2019 Kazahaya 2011

7/12


avoid recording the not-much-beneficial amplitude model in

this paper.

Hereafter, we continue this paper with the phase-difference

model.

V. ERROR A NALYSIS

A. Uneven Flow Rates in Two Flow Tubes

One natural question is whether the CMF measurement is

correct, even if the total flow unevenly splits between twin

tubes. Next, we study for this question.

Assume that the total flow Mis unevenly split to two flowtubes, i.e.,m1 to tube 1 and m2to tube 2, by some reason; forexample, the flow pattern in the tube and construction of the

splitter,M=m1+ m2, and m1=m2. Then, based on (11)and (12), we write

d11= i D0

sin t +

l3

4EIm1 cos t

d12= i D0

sin t +

l3

4EIm2 cos t

.

By substitutingd11and d12into (39), we obtain

v1=4U3

g D0

2sin t +

l3

4EI(m1+ m2) cos t

=8U3

g D0

sin t +

l3

8EIM cos t

. (24)

Then, similar to the aforementioned calculation of (17) and

(18), we write

tan = l3

8EI M. (25)

Because (25) is the same as (18), we may state that CMF

correctly measures the mass flow rateM, even when the flowunevenly splits.

The statement may not be universally true if an uneven flow

split occurs by clogging or scaling in a flow tube. A scale or

clog may change the effective inner radius and the aggregate

Youngs modulus E, changing the meter factor. Old CMFs maycause some error by clogging, scaling, or both in long use in

installations.

B. Vibration Effects

A CMF vibrates its flow tubes to measure the mass flow.

Naturally, it is inherently prone to vibration effects from its

upstream and the downstream process pipes, in comparison

to magmeters, differential pressure transmitters, and other

flowmeters. We study, in this section, vibration effects to find

that CMFs with twin U-shaped flow tubes have a mechanism

for reducing the effects of external vibration.

First, we should realize that the frequencies of the vibration

from the external pipes are generally lower than the operational

frequency, 801000 Hz, of the transducers. Because of the

difference, the transducer does not absorb much energy fromthe external vibration.

Second, vector mechanics teaches us that the rate of change

of a vector is the same with respect to a fixed frame and with

respect to a frame in translation [19]. Therefore, vibrations

in translation in the plane OXY, OZY, or OXZ, whichare parallel vibrations, do not change the displacements, d11,

d12, d21, and d22 and, consequently, do not cause an error in

measurement.Third, rotational motions need further study. The center of

gravity of a CMF is off from the center line of the upstream

and the downstream pipes. Then, the external vibration tends to

generate rotational movement of the transducer.

When the transducer rotates around the X-axis, the vectordirections of the angular acceleration of both legs, OP and SU,

are i. The accelerations cause the Coriolis accelerations on both

OP and SU, in addition to the Coriolis accelerations that we

studied. They have, however, the same magnitudes but in the

same directions. Thus, such a rotation does not affect the output

of sensor S1. The same is true with S2.

When the transducer rotates around the Y-axis, the vectordirection of the angular accelerations of both legs, OP and SU,

arej. Because the direction of the flow is also j, no net force is

generated on the fluid. When the transducer rotates around the

Z-axis, the vector directions of the angular acceleration of fourlegs are k. Then, both U-shaped flow tubes sway in the same

magnitude in the same direction. The movements do not affect

d11,d12,d21, andd22and, thus,v1and v2.As aforementioned, the advantage of the twin U-shaped flow

tubes is because the tops of the tubes are not constrained and are

free to move. This condition enables the transducer to reduce

the external effect.

Because the model is developed assuming that the nest is

fixed, the model itself does not have power to analyze the effectof the deformation of the nest by the external vibration. Such

deformation should be prevented by its mechanical design, e.g.,

a thick nest.

C. Error Equations

We develop an error equation of CMF by taking the natural

logarithm of (23) and differentiating it as follows:dMM =

dKK +

dff +

d (26)

where |dM/M| is the error (accuracy) of a CMF, which is typi-cally specified to be 0.05% (=1/2000) of the measurementor some CMFs are 0.1%. Then, the total of the right-hand sideof (26) must be less than 0.05%.

Frequencyfand phase angle are measured with countersand clock pulse in the electronics. When we use 16-b counters,

they are measured with an accuracy of better than 0.002% (=1/216 0.05%). Therefore, we may ignore |df/f| + |d/|in (26). Still, we need to keep |dK/K|better than 0.05% toachieve the accuracy specification. On the other hand, the accu-

racy of the flow test stand, certified by the National Institute

of Standards and Technologies (NIST) in the U.S., is about

0.05%. This condition leaves no margin of error to CMF.

Consequently, we need to eliminate the effects of temperatureand pressure to the meter factor by compensation.

8/10/2019 Kazahaya 2011

8/12


D. Effect of Temperature

Next, we study the relation betweenKand the temperature.The moment of inertia I of a tube, whose inner and outer

radii areri andro, is known in vector mechanics and strengthof materials as

I=

4

r4o r

4i

. (27)

Based on (22) and (27), we rewrite Kas

K= 4

E

l3

4

r4o r

4i

=

E

l3

r4o r4i

. (28)

Then

K0= E0

l30

r4o0 r

4i0

(29)

whereK0,E0,l0,ro0, andri0are K,E,l,ro, andri at 20 C,

respectively.

When we writeE= E0 {1 + (T 20)}

ro = ro0 {1 + (T 20)}

ri = ri0 {1 + (T 20)} , and

l= l0 {1 + (T 20)}

where

T fluid temperature (=tube temperature) in C; thermal coefficient of Youngs modulus; thermal coefficient of expansion.

Kcan be expressed as

K=E0l30

r4o0 r4i0

{1 + ( + )(T 20)}

= K0 {1 + ( + )(T 20)} . (30)

We find that the temperature changes the meter factor bycoefficient( + )and not simply by.

We have obtainedE= 206.0{1 0.000389 (T 20)}bythe regression analysis of Youngs modulus of stainless steelat different temperatures, published by metal manufacturers.Then, = 3.89 104. (The exact value is slightly different,depending on stainless steel, e.g., 316, 316 L, or other alloysused for the flow tubes.) The of stainless steel= 16.4 106. At the application limit of CMF, 350 C,( + ) (T

20) = 0.11.Thus, the measured value of the mass flow rate changes

by 11% when the tube temperature increases from 20 C to350 C: We conclude that CMFs need temperature compensa-tion on the meter factor. Manufacturers mount resistance tem-

perature detector (RTD) on the flow tube or on the nest, closeto the tube to measure the temperature. The microprocessorcalculates the compensation with (30).

Note that the thermal coefficient of Youngs modulus is

about 20 times larger than the thermal coefficient of expansion.Consequently, the error analysis reveals that a change in Eby temperature affects the matter factor far more than thermalexpansion does. This condition suggests that stringent quality

management of metal composition and heat treatment of flowtubes is needed.

E. Effect of Inner Pressure

The theory of thin-walled pressure vessel may be used for

studying the effect of inner pressure to the flow tubes. Theprocess pressure of fluid in a tube causes stress in the tube

in the longitudinal (along the tube length) and the tangential

directions. Tangential stressstis given byst = (r/t)P, where

ris the radius of the tube, and t is the wall thickness of the tube.Pis the internal pressure [20], and we use P = 10 MPa, whichis the maximum pressure rating of most of the commercial

CMFs. Because the tangential stress is two times larger than

the longitudinal stress, we use values of the tangential stress forthis paper.

By Hooks law, the elongation L of metal length Lo isgiven by L= (st/E)L0. For typical cases, we assume thatthe material of the flow tubes is stainless steel 316L, whose

Youngs modulus at 20 C is 195 GPa.

Then, the relation of K and P is obtained from (22), (27), andst = (r/t)P. We have

K=K0(1 + P) where = r

tE (31)

whereK0is the meter factor whenP = 0kPa or P is low, andthe temperature is 20 C.

We calculate in Table I the percent changes of K by the

pressure P (= Pin %) for different nominal sizes of stainlesssteel tubes that are commercially available. (Note that the

nominal sizes of tubes are different from the meter size of

CMFs, because a CMF with twin U-shaped flow tubes uses twotubes in it.)

The rightmost column of Table I indicates that errors ofCMFs with tubes that are 1/4, 1/2, and 3/4 in size are lessthan the instrument accuracy (0.05%). In addition, mostflowmeters are used at less than the maximum pressure rating.Consequently, the CMFs of these tube sizes do not require

pressure compensation.

The errors of CMFs with tubes of 1 and 2 in size are aboutthe accuracy specification of the flowmeters. The errors of

CMFs with tubes that are equal to or larger than 3 in size

slightly exceed the specification. If application conditions areproperly considered, such CMFs may not need continuous

pressure compensation that uses a pressure gauge or a pressure

sensor. If the process pressure is far less than the pressure rating(

8/10/2019 Kazahaya 2011

9/12


TABLE IDEFORMATION OFT UBES OF STAINLESS STEEL316L I N DIFFERENTS IZES

Although we have not explicitly discussed the requirement

to U-shaped flow tubes, the following observations are clear

through the model development. Each U-shaped tube should be

in a flat plane. A pair of the flow tubes in one transducer should

be identical in shape, dimension, and characteristics. Because

the composition and heat treatment of tubes affect Youngs

modulus, a pair of the tubes should be cut from one material

pipe and should be heat treated in the same batch, at the location

next each other in the same oven, to achieve the same E value.

Such manufactured parts should be kept in the pair in inventory

until assembling.

Micro Motion has been making, selling, and offering ser-

vices for many models of CMFs with U-shaped flow tubes for

more than 25 years. Yokogawa Corporation and a few othersalso do.

VI. OTHERC ONFIGURATIONS OFF LOWT UBES

A. Single U-Shaped Flow Tube

A few manufacturers make CMFs with a single U-shaped

flow tube without a flow splitter. Users who measure the flow

of fluid that is slurry or contains foreign particles may avoid a

flowmeter with a splitter, because it is likely clogged.

Theoretically, we may consider that flow tube 2 is replaced

with a solid plate or block in Fig. 2 to make a transducer with a

single tube. The flow goes into tube 1 and out from the outlet.The solid plate is connected to the original nest in Fig. 2 to form

the new reference frame. We call it an extended nest. Then,

points Q and V become fixed points, and d12= d22= 0.We obtain the mathematical model of CMFs with a single

U-shaped flow tube by settingd12= d22= 0in the aforemen-tioned model development. Then, the model of single U-shaped

tube is

M= 2

EI

l31

f tan . (32)

By comparing (32) to (21), we know that the meter factor of

CMFs with a single U-shaped flow tube is 1/2 of that of theflowmeter with twin flow tubes of the same tube size. Before

the mathematical proof, you may have correctly guessed that

the meter factor of a CMF with one tube should be half of that

with two tubes.

We have explained with (4) that there are two other accel-

erations aside from the Coriolis acceleration in the rotating

legs of the flow tubes. They do not affect the sensor outputs

from twin U-shaped tubes, because they counterreact each

other. In a transducer with a single tube, the counteracting

accelerations from tube 2, which the extended nest replaces,

do not exist. Then, a may cause a constant zero shift in themeasurement. The zero shift needs to manually be nullified by

a zero adjustment in the electronics

A CMF with a single U-shaped flow tube is more prone to

vibration effects than a CMF with twin U-shaped tubes. Anextended nest itself tends to bend or vibrate more than the short

nest. The vibration of the driver may reach both sensors through

the extended nest, directly disturbing the sensor outputs.

The aforementioned analysis of the effects of external vi-

bration on CMFs with twin tubes is applicable to those with

a single tube, except for the rotation around the Z-axis. Withthe former condition, we studied that the resulting motions of

two tubes by the rotation around the Z-axis does not affectsensor S1. With the latter condition, the canceling motion from

tube 2 does not exist. Therefore, CMFs with a single tube may

be more sensitive to external vibration.

To internally reduce disturbance by vibration, designers ofsuch CMF may need to beef up the extended nest, the housing,

or both to stabilize the reference frame. Users should more

rigorously reduce external vibration that reaches the flowmeter

than CMFs with twin flow tubes.

B. Straight-Flow Tubes and Variations

Manufacturers make and sell CMFs with straight-flow tubes.

Some CMFs have twin tubes, and other CMFs use a single

tube. The appearances of CMFs with straight-flow tubes in long

styles are largely different from CMFs with U-shaped tubes

in boxy style, as if both are different flowmeters. The strong

feature over CMFs with U-shaped tubes is that straight tubesself-drain process fluid.

8/10/2019 Kazahaya 2011

10/12


Fig. 4. Conversion from the U-shape tubes to the straight tubes.

Hereafter, we investigate CMFs with straight tubes by using

the study result of CMFs with U-shaped tubes.

Fig. 4 illustrates a side view of a transducer with twin

straight-flow tubes, in solid lines. In the figure, tube 1 hides

tube 2; the flow tube RQVW is behind OPUS. Theoretically,

it is the conversion from a transducer with twin U-shaped tubes.

Therefore, we superimpose the configuration of the latter in

Fig. 2 onto Fig. 4, in chained lines. The markings of the key

points, O, P, Q, R, S, U, V, and W and the driving points J

and K correspond to those in Fig. 2. The sensors are connected

between P and Q, which is behind P in the figure and between

U and V. The reference points, O, R, S, and W, are anchored to

the nest.

Assume in Fig. 4 that we split the original nest and turn the

legs OP, RQ, SU, and WV by 90 to make the tubes straight,

moving the split nests with the tubes, as shown with the arrows.

The half tori PJQ and QKV are also straightened or flattened.

Then, we obtain the configuration of the transducer with twin

straight tubes OPUS and RQVW. The split nests are con-

nected with a bracket of chained lines; we call the connected-

split nests an elongated nest. In Fig. 4, the flow tubes verticallyvibrate to the figure (the page surface). When vibrated, the

straight-flow tube OPUS bends in arc. Because the amplitude

of the vibration is very small, we approximate the tube bents in

a trapezoid, keeping the legs OP and SU straight.

With the aforementioned mechanical changes, the fixed

frame OXYZ and the rotating frame Oxyz also change from

those used for the study of the transducers with twin U-shaped

tubes. By turning the legs by 90, theY- and theZ-axes coin-cide with each other. TheX-axis is split with the split nests. Invector analysis, however, we may consider both as the X-axis.

In the previous analysis, we considered the fixed reference

frame OXY and a rotating frame Oxy that coincides with theleg. When the extended nest is stable to be the reference frame,

we may apply the aforementioned analysis on the split nests

and find that the mathematical model (21) and the meter factor

(22) of twin U-shaped tubes are applicable to CMFs with twin

straight tubes.

Therefore, we may state that the mathematical model that

we first developed for CMFs with twin U-shaped tubes is a

general model of CMFs. The aforementioned error analyses by

the model are also applicable.

It is easy to assume that, in theory, the extended nest is

stable. It is, however, a demanding task to design and construct

the extended nest stable and free from vibration. If not stable,

the reference frame changes, and the angular accelerations change, causing error in the Coriolis accelerations. Designers

of such CMFs add stiff brackets to the extended nest between

the split nests, anchor them to the housing of the flowmeter,

or do both approaches to improve mechanical stability. Walls

of the housing are usually thick and heavy for the stability of

measurement.

The thermal expansion of the tubes and the reinforced

extended nest should be the same. Otherwise, the extendednest compresses the flow tubes (buckling) or elongates them,

causing a measuring error. Designers may choose the same

materials for both conditions.

Some manufacturers use tubes that are slightly bowed or

kinked at the center sections of the length, PJU and QKV, to

avoid or reduce buckling, keeping still straight tubes in PO,

US, QR, and VW. The direction of deformation becomes

predictable if the tube is bowed or kinked by design; then,

deformation may not affect the accuracy with such flowmeters.

A CMF with a single straight tube may be considered a

conversion from a CMF with a single U-shaped tube, similar

to the illustration in Fig. 4. The mathematical model is also

applicable.

CMFs with single or twin straight-flow tubes commonly have

the inherent drawback in the market place, i.e., long face-to-

face distances. They are long because of the long flow tubes and

are mostly longer than other types of flowmeters. To install such

a flow meter, the user has to cut out a long portion of the process

pipe. No CMF with straight-flow tubes is readily replaceable to

volumetric flowmeters of the similar pipe size.

Endress+Hauser, Krohne, and Micro Motion make CMFswith a single and twin straight tubes. ABB makes similar

CMFs, whose flow tubes are in elongated S-shape or kinked,

to which the mathematical model is also applicable. Other

manufacturers made CMFs with spiral flow tubes [21] or withtubes in more sophisticated bending. The mathematical model

is also applicable to these CMFs in approximation.

VII. CONCLUSION

In this paper, we have arrived at the following ten

observations.

1) A general mathematical model of CMFs has been devel-

oped. There are two types of models: a phase-difference

model and an amplitude model. In this paper, we record

and use the only the phase-difference model. The model

is applicable to CMFs with U-shaped flow tubes, straightones, or slightly curved (bowed) ones and to CMFs with

twin flow tubes or a single flow tube.

The model is applicable to from micromechanical

CMFs of a millimeter flow tube size and to large ones

with a 1-meter-long tube.

2) The general model is applicable to both CMFs with

U-shaped tubes and those with straight tubes, although

both appearances are largely different, as if they are

different flowmeters.

3) A CMF does not need the measurement of the density of

the fluid to get the mass flow rate. This condition suggests

that the flow tube may be driven at a predetermined

frequency or power line frequency if density data are notneeded. On the one hand, micromechanical CMFs may

8/10/2019 Kazahaya 2011

11/12


be driven at a high frequency (in kilohertz ranges). On

the other hand, drives with the line power may enable

engineers to make CMFs much larger in tube size than

the current tubes.

4) In the phase-difference model, the meter factor is deter-

mined only by the characteristic and dimension of the

flow tube. Therefore, the type and characteristics of thedriver and the type and sensitivity of the sensor, do not

change the meter factor.

5) Because of the aforementioned characteristic, the elec-

tronics (a transducer) of the phase-difference model may

interchangeably be used with any transducers.

6) Error analysis points out that a CMF with twin flow

tubes correctly measures the mass flow rate, even when

the inflow unevenly splits to two tubes. When uneven

splitting occurs because of scaling or clogging within a

flow tube, its effective Youngs modulus, radii, or both

may change, causing an error, however.

7) Error analysis reveals that all CMFs need temperature

compensation against changes in flow tube characteristics

by temperature, i.e., the change in Youngs modulus and

the thermal expansion of radii and length. A temperature

compensation equation (30) was developed.

8) The thermal coefficient of Youngs modulus of stainless

steel tubes is about 20 times larger than the coefficient of

thermal expansion. Most of the tube materials are similar.

Youngs modulus becomes different by the composition

and heat treatment of the flow tubes. This condition

suggests that rigorous quality management of flow tubes

on their composition and heat treatment is needed to

reduce the measurement error and difference among man-

ufactured transducers.9) The effect of process pressure, i.e., the inner pressure of

the flow tube, is small for the measurement. Transduc-

ers with small flow tubes, i.e., smaller than 3/4 in size

(13.36 mm in diameter and 1.65 mm in wall thickness),

do not need pressure compensation of up to 10 MPa

for 0.05% of the reading. The larger flow tubes needsome consideration in process pressure for accuracy. If

the flowmeter specification is 0.1%, no pressure com-pensation is needed in most of the pressure ranges.

10) CMFs with twin U-shaped flow tubes are suitable for ac-

curate flow measurement and are less sensitive to external

vibration.CMFs with straight-flow tubes are also accurate

flowmeters. They may be more sensitive to external vi-

bration in some extent.

APPENDIXA

CAPACITIVES ENSOR

First, we show the structure of a capacitive sensor in Fig. 5

as an example of displacement sensors. We use two of them for

sensors S1 and S2 in Figs. 1 and 2. Each sensor consists of the

electrodes F, G, and H, which are electrically separated from

each other: Electrodes F and H are mechanically connected

with the bracket FH. Therefore, electrodes F and H movetogether. We connect the electrodes F and H of sensor S1 to

Fig. 5. Capacitive sensor.

point P of flow tube 1 and electrode G to point Q of flow tube 2.

We denote 2 g to the distance between F and H. The actual

thickness (F to H) of the sensor is several millimeters or less to

detect the small movements of the flow tubes, although Fig. 5 is

expanded for illustration. Electrode G is at the center betweenF and H when the sensor is in neutral.

Next, observe the movement of the electrodes. The points P

and Q move in opposite directions, i.e., P byd11and Q byd12.We write the displacements d11and d12in (11) and (12) in vec-

tor notation. Their signs, i and i, indicate that P and Q movein the opposite directions. Because the mechanical connections

of the sensor agree with the directions, we consider the scalar

components ofd11and d12hereafter. When F moves to F and

G moves to G, the resulting distances between FG and GH

are[g (d11+ d12)]and[g + (d11+ d12)], respectively.The electric connections are given as follows. We apply

a positive dc voltage U3

in the reference to the ground to

electrode F and a negative voltage U3to electrode H throughflexible wires. We take the output signal v1 between electrodeG and the ground.

Then, we calculate the output voltage. We denote C11 theelectric capacitance between the electrodes F and G andC12between the electrodes G and H for S1. The voltage across C11isU11and that acrossC12is U12. We assume each capacitanceholds electric charge Q. (The following equations apply to S2

in the same way.)

Then

U11= Q/C11 (33)

U12= Q/C12 (34)U11+ U12= 2U3. (35)

Based on (33)(35), we obtain

U11= 2 C12

C11+ C12U3. (36)

On the other hand, we have

C11= A

g (d11+ d12) (37)

C12= A

g+ (d11

+ d12

) (38)

whereAis the area of the electrode.

8/10/2019 Kazahaya 2011

12/12


By substituting (37) and (38) to (36), we obtain

U11= U32(d11+ d12)

g U3.

Because we measure the sensor output in reference to the

ground

v1= U3 U11= 2U3

g (d11+ d12). (39)

Thus, the sensor output v1 is proportional to the total of theinput displacementd11+ d12.

In an actual construction, the voltage at electrode G may have

some dc voltage. We takev1through a capacitor to eliminate it.Consequently, v1(andv2) is an ac signal in this paper.

Although we use the capacitive sensors for modeling in this

paper, other displacement sensors may be also used, because

the type of the sensors do not change the meter factor.

APPENDIXB

DENSITYA LGORITHM

We record here the algorithm that may be used to obtain the

density of process fluid for a reference purpose, although we

do not need to measure fluid density to measure mass flow rate

with a CMF.

To obtain the density data, the CMF must operate at the

resonance frequency of the transducer, as shown in Fig. 1,

because density measurement uses the physics of mechanical

resonance.

The fluid density may be expressed by

= 998.2f2wf2

f2e f2

f2e f2w

in kg/m3 (40)

where 998.2 kg/m3 is the density of water at 20 C,fe is theresonance frequency when the flow tube is empty, and f wis theresonance frequency when the flow tube is filled with water at

20 C, both without flow. The values offeand fware measuredby wet calibration and are stored in the electronics.

A CMF always measuresffor obtaining the mass flow rate.The microprocessor calculates the fluid density with (40) by

using f. Thus, no additional hardware is needed for densitymeasurement.

ACKNOWLEDGMENT

The author would like to thank J. Spaihts, a Quality Man-

agement Specialist with the Fischer & Porter Company for

his suggestions to develop a general theory of all CMFs to

clarify what are happening in them; Dr. Y. Shibutani, Professor

with the Department of Engineering, (National) University of

Osaka, Japan, for his advice; Dr. R. Zoughi, the Editor-in-

Chief; Associate Editor; the reviewers of this paper for their

constructive critique; and C. Ingelin of the IEEE T RANSAC-

TIONS ON INSTRUMENTATION ANDMEASUREMENT for her

help in communication with the IEEE, including uploading theoriginal and revised manuscripts.

REFERENCES

[1] R. A. Serway and J. W. Jewett, Physics for Scientists and Engineers.Pacific Grove, CA: Brooks/Cole, 2009.

[2] F. P. Beer and E. R. Johnston, Jr., Vector Mechanics for Engineering:Statics and Dynamics. New York: McGraw-Hill, 2009.

[3] J. P. Vanyom,Rotating Fluid in Engineering and Science . New York:Dover, 2001.

[4] G. Sultan and J. Hemp, Modeling of Coriolis mass flowmeters,J. SoundVib., vol. 132, no. 3, pp. 473489, Aug. 1989.

[5] G. Sultan, Single straight-tube Coriolis mass flow meter, Flow Meas.Instrum., vol. 3, no. 4, pp. 241246, Oct. 1992.

[6] H. Raszillier and F. Durst, Coriolis effect in mass flow metering,Arch.Appl. Mech., vol. 61, no. 3, pp. 192214, Mar. 1991.

[7] H. Raszillier and V. Raszillier, Dimensional and symmetry analysisof Coriolis mass flow meters, Flow Meas. Instrum., vol. 2,no. 3,pp. 180184, Jul. 1991.

[8] H. Raszillier, N. Alleborn, and F. Durst, Mode mixing in Coriolis flowmeters,Arch. Appl. Mech., vol. 63, no. 4/5, pp. 219227, Apr. 1993.

[9] Y. Basrrawi, Coriolis force mass flow measurement devices, presentedat the 49th Int. Instrumentation Symp. (ISA), Orlando, FL, May 48,2003, Paper TP3AERO022.

[10] M. Anklin, W. Drahm, and A. Rieder, Coriolis mass flowmeters:Overview of the current state of the art and latest research, Flow Meas.

Instrum., vol. 17, no. 6, pp. 317323, Dec. 2006.

[11] Compatibility of Analog Signals for Electronic Industrial Process Instru-ments, ANSI/ISA-S50.1-1982.

[12] HART Communication Specification HCF_SPEC-13.[13] Standards Digital data communications for measurement and

controlFieldbus for use in industrial control systems, IEC 61158,1999.

[14] F. P. Beer and E. R. Johnston, Jr., Vector Mechanics for Engineering:Statics and Dynamics, 5th ed. New York: McGraw-Hill, 1988, p. 744.


[16] W. A. Nash, Strength of Materials, 2th ed. New York: McGraw-Hill,1972, Solved problem 10.6 in p186.

[17] J. E. Smith, Method and structure for flow measurement, U.S. PatentRE31 450, Feb. 1982.

[18] U. Lange, A. Levien, T. Pankratz, and H. Raszillier, Effect of detectorsmasses on calibration of Coriolis flow meters, Flow Meas. Instrum.,vol. 5, no. 4, pp. 255262, Oct. 1994.


[20] D. Hartog, Strength of Material. New York: McGraw-Hill, 1977,pp. 136137.

[21] J. Yard and W. O. Strohmeier, Coriolis-type flowmeter, U.S. Patent4 957005, Sep. 18, 1990.

Masahiro Matt Kazahaya (M76LSM06) wasborn in Japan in 1932. He received the B.S. degreein physics and the Ph.D. degree in technology fromOkayama (National) University, Okayama, Japan,and the MBA degree from the Wharton GraduateSchool, University of Pennsylvania, Philadelphia.

For 12 years, he was a Visiting Instructor withthe Wharton Graduate School, the University ofPennsylvania. He was a Vice President of Marketing,Technology, and Corporate Planning with the Fischer& Porter Company (now a part of the ABB Group),

Warminster, PA. He is currently the Owner and President of MKK International,Inc., Southampton, PA, a business and technology management consultingfirm. He published two Japanese books: SeminarIntroduction to Business

ManagementandProblems in Japanese Employment System.Dr. Kazahaya is a Lifetime Senior Member of the International Society ofAutomation (ISA).

kazahaya 2011

Documents