Calculation of the sensitivity of a straight tube Coriolis mass flowmeter with free ends

Flow Measurement and Instrumentation 12 (2002) 411–420 www.elsevier.com/locate/flowmeasinst

Calculation of the sensitivity of a straight tube Coriolis mass flowmeter with free ends J. Hemp

*

Department of Process and Systems Engineering, School of Engineering, Cranfield University, Cranfield MK43 OAL, UK Received 26 July 2001; received in revised form 4 September 2001; accepted 20 September 2001

Abstract The weight vector theory for Coriolis mass flowmeters is applied to a simple theoretical meter configuration consisting of a single unsupported straight tube unattached to adjacent piping. The tube has free ends and vibrates in the fundamental mode. It is shown how the sensitivity of this meter depends in part on the interaction of the flow velocity profile with fluid vibrations occurring near the tube ends. This end effect is negative i.e. the meter reads lower than would be expected if end effects were ignored. On account of the end effect there is a predicted variation in sensitivity with Reynolds number (of the order 1% in a tube 25 diameters long) and this can be minimised by a certain choice of the sensor positions.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Coriolis mass flowmeter; Flowmeter sensitivity; Velocity profile effects in flowmeters; Flowmeter weight vector; Flowmeter end effects; Flowmeter optimisation

1. Introduction The configuration studied is illustrated in Fig. 1. It consists of a straight flowmeter tube section 0⬍z⬍l of elastic material infinitely close to adjoining rigid pipes but unattached to them. The internal radius b of the flowmeter tube is the same as that of the adjoining pipes. The flowmeter tube vibrates transversely in the x-direction in its fundamental mode with small amplitude vibrations. The ends of the flowmeter tube are not fixed or constrained in any way. Steady incompressible fully developed flow takes place through the adjoining pipes and the flowmeter tube. A flow measurement is obtained

Fig. 1.

from the small phase difference observed in the vibration l of points P and P⬘ at z= ±z on the flowmeter tube. 2 This configuration models a design of Coriolis mass meter which employs a straight tube only lightly supported (for example by rubber seals) rather than clamped or fixed to a supporting structure. Such a design has the possible advantage of less coupling with adjacent piping through the connecting flanges of the flowmeter. Coupling with adjacent pipes is known to be a cause of zero offset [1]. The usual method of mathematical analysis of Coriolis mass flowmeter configurations is by way of beam theory

The Coriolis mass flowmeter configuration studied.

* Tel.: +44-1234-750-111; fax: +44-1234-751-875. E-mail address: [email protected] (J. Hemp). 0955-5986/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 5 - 5 9 8 6 ( 0 1 ) 0 0 0 3 1 - 0

412

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

with flow causing a lumped Coriolis reaction force per unit length of the flowmeter tube proportional to the mass flow rate and the local angular velocity of the tube (see for example [2,3]). This does not allow study of end effects or velocity profile effects. However the weight vector theory [4,5] does allow a study of end effects and velocity profile effects. According to this theory the phase difference ⌬f between the vibrational velocity at points P and P⬘ in Fig. 1 is given by the following integral over the entire volume of the fluid [4].

冕

⌬f⫽ v·WfdV

⌬⬇

5v u∗

where the friction velocity u* is related to the pipe friction factor f and mean flow velocity v¯ by

(1)

冪8

u∗⫽v¯

where v is the flow velocity (time averaged in the case of turbulent flow) and Wf is a weight vector given by

冉 冊

Wf⫽Im

W , (ux)P

W⫽⫺r((u(2).ⵜ)u(1)⫺(u(1).ⵜ)u(2))

冪w

2v

f

f being approximately related to the Reynolds number (in smooth pipes) by [7]

(2)

f⫽0.316Re−1/4 4000⬍Re⬍105.

(3)

Therefore in turbulent flow

where u(1) and u(2) are the vibrational velocities (in the complex notation) set up in the fluid (with no steady flow present) when [in case (1)] the flowmeter tube is driven as in normal operation by a central force [Fig. 2(a)] and when [in case (2)] the flowmeter tube is driven by equal and opposite unit forces applied at the sensing points P and P⬘ [Fig. 2(b)]. In Eq. (2) (ux)P is the xcomponent of the tube velocity at the sensing point P. As explained in [4,5] Eqs. (1)–(3) hold in the case when viscosity can be neglected in the vibrational velocity fields u(1) and u(2) and the condition for this is that the boundary layer d in the vibrational flows is small compared to the thickness ⌬ of the laminar sublayer in the steady flow (i.e. the thickness of the layer near the tube wall in which the axial steady flow velocity changes rapidly with position). The order of magnitude of d is given by [6] d⬇

where v is the kinematic viscosity of the fluid and w the angular frequency of operation of the flowmeter. In the present case of fully developed (possibly turbulent) flow, the order of magnitude of ⌬ is (in the case of laminar flow) simply the tube radius b and in the case of turbulent flow is [7]

(4)

⌬ ⬇50.3Re−7/8 4000⬍Re⬍105 b

(5)

It was explained previously ([5, p. 263]) that if the steady flow velocity is linearly increasing with distance from the wall throughout the thickness of the vibrational velocity boundary layer then there is no error due to the assumption of no viscosity in the vibrational flows. To find the condition of validity of this assumption it is therefore necessary to find the order of magnitude of the first nonlinear term in the expansion of steady flow velocity as a function of distance from the wall. To do this use is made of Spalding’s formula for the fully developed velocity profile [8]. In terms of non-dimensional velocity u+=v/u∗ and distance from the wall yu∗ this ‘back to front’ formula is y+= v y+⫽u+⫹0.1108

冉

冊

(0.4u+)5 (0.4u+)6 ⫹ ⫹% . 5! 6!

Inverting the series we have u+⫽y+⫺0.1108

(0.4y+)5 ⫹%. 5!

The condition for neglect of viscosity in the vibrational flow is (eq. 42 in [5]) that R1 where

||

R⫽

Fig. 2.

Vibrational velocity fields (a) in case (1), (b) in case (2).

vd v¯

and vd is the value at y=d of the first non-linear term in the expansion of v(y) in powers of y. This gives for the condition of validity of Eqs. (1)– (3) the inequality

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

(0.4y+)3 5! R⫽u∗0.1108 u∗

|

+

y =d

冪f

8

u∗ v

⫽

冉冊 d ⌬

In the cylindrical coordinates of Fig. 1 these are

5

冪

33.8

1. 8 f

(6)

r2 v(r)⬀1⫺ 2 (laminar flow) b

冉 冊

1 n

(turbulent flow)

∂p iwrur⫽⫺ , ∂r

∂p ∂p iwruq⫽⫺ , iwruz⫽⫺ r∂q ∂z

1 ∂(rur) 1 ∂uq ∂uz ⫹ ⫹ ⫽0 r ∂r r ∂q ∂z

Take a specific example. For a flow of water at mean speed v¯ =1 m/s through a smooth walled flowmeter tube of inner diameter 10 mm, Re=10,000, f=0.0316 and ⌬=0.079 mm. If the flowmeter tube is of stainless steel 250 mm long with a wall thickness of 1 mm its fundamental bending mode frequency (with free ends) is 920 Hz. This means d=0.0186 mm and R=1.34×10−6. So the condition is fulfilled in this example. (If v¯ is increased to 5 m/s the condition is still fulfilled with R=0.0012.) In this paper Eqs. (1)–(3) are applied to predict the sensitivity (phase shift divided by flow rate) of the flowmeter configuration in Fig. 1 to fully developed flow. Away from the tube ends the vibrational velocities u(1) and u(2) are calculated using a ‘locally rigid tube’ approximation. That is, inside most of the flowmeter tube, u(1) and u(2) depend essentially on the local tube wall motion (its translation and rotation). Near the ends of the flowmeter tube the effect of adjacent pipes is taken into account by treating them as providing rigid boundaries to the fluid vibrational velocities. Having obtained expressions for u(1) and u(2) inside the flowmeter tube and beyond it (inside the adjacent pipes), the weight vector Wf and the axisymmetric flow weight function W(r) are calculated. The effects of fully developed flow are then computed using the following formulae for the velocity profiles:

r v(r)⬀ 1⫺ b

413

(7)

where n depends on Reynolds number [9] (e.g. n=6 for Re=4000, n=7 for Re=110,000).

2. Calculation of the vibrational flow fields u(1) and u(2) and the weight vector away from the ends

ur⫽Ucosq on r⫽b… where U(=U(z)) is the local linear velocity of the tube as shown in Fig. 3. Eq. (9) have the approximate (locally rigid tube) solution ur⫽Ucosq uq⫽⫺Usinq

(10)

uz⫽⍀rcosq p⫽⫺iwrUrcosq where ⍀(=∂U/∂z) is the local angular velocity of the tube as shown in Fig. 3. On substitution of Eq. (10) into Eq. (9) all is satisfied except for the continuity equation. The continuity equation is approximately satisfied when r

∂2 U U ∂⍀ U  or when r 2  , ∂z r ∂z r

that is when b2 1. l2

(11)

On substituting the solution (10) into the expression (3) for the weight vector it is found that (to the same approximation) W⫽r(U(1⍀(2)⫺U(2)⍀(1))k.

(12)

where superscripts refer as usual to cases (1) and (2) defined in Section 1. So the weight vector away from the ends of the tube is independent of r and q and depends only on z. This implies that the flowmeter averages the flow profiles perfectly except near the ends of the tube. It is assumed that Eq. (12) is an adequate expression for the weight vector for 0⬍z⬍l when end effects are neglected. So the weight vector without end effects (denoted W0) is W0⫽

With neglect of viscosity and compressibility of the fluid the equations for u(1) or u(2) are

(9)

再

r(U(1)⍀(2)−U(2)⍀(1))k 0⬍z⬍l 0

z⬍0,z⬎l

(13)

iwru⫽⫺ⵜp ⵜ.u⫽0

(8)

un=unt on the tube wall. Where r is the fluid density, p the pressure and suffices ‘n’ and ‘t’ denote ‘normal component’ and ‘belonging to the tube’ respectively.

Fig. 3. Local linear and rotational velocities (U and ⍀) of the tube.

414

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

The expected fractional error in Eq. (13) (away from the ends) is, by Eq. (11), of the order of b2/l2 whereas the end effects to be calculated will be of relative order b/l.

u2r=zcosq

u2r=0

u2q=−zsinq for z⬎0, u2q=0 for z⬍0, u2z=rcosq

(18)

u2z=0

and 3. Calculation of the vibrational flow fields u(1) and u(2) near the end of the tube (at z=0) With reference to Fig. 4, in the first approximation the vibrational velocity field u(1) or u(2) in the vicinity of the end at z=0 is [by Eq. (10)] u0r=(U+z⍀)cosq

u0r=0

u0q=−(U+z⍀)sinq for z⬎0, u0q=0 for z⬍0, u0z=⍀rcosq

(14)

u0z=0

where (thoughout this section of the paper) U and ⍀ denote the linear velocity and angular velocity of the tube at z=0. Eq. (14) gives u(1) or u(2), U and ⍀ taking on the values U(1) and ⍀(1) in case (1) or U(2) and ⍀(2) in case (2). Eq. (14) gives u(1) or u(2), U and ⍀ taking on the values U(1) and ⍀(1) in case (1) or U(2) and ⍀(2) in case (2). Solution (14) has discontinuities at z=0 and therefore needs correcting. Let the complete solution be u⫽u0⫹u⬘

(15)

where u⬘ is a solution of Eq. (9) with boundary condition ur⬘=0 on r=b, with equal and opposite discontinuity at z=0 and with a zero limit for z→±⬁. The fields u0 and u⬘ are usefully divided into two parts. A part proportional to U and a part proportional to ⍀. Thus we put u0=Uu1+⍀u2

(16)

u⬘=Uu⬘1+⍀u⬘2

where, in order to correct the discontinuities of u(1) and u(2) respectively, f1 and f2 satisfy f1(⫺z)⫽⫺f1(z),f2(⫺z)⫽f2(z)

(20)

and (for z⬎0) the equations

冦

ⵜ2f1=0

ⵜ2f2=0

冦

∂f1 =0 on r=b ∂r

∂f2 =0 on r=b ∂r

(21)

1 ∂f2 1 f1= rcosq on z=0+ = rcosq on z=0+ 2 ∂z 2

Now Eq. (21) implies ∂f2 f 1⫽ ∂z

(22)

and that f2⫽ycosq, y(⫺z)⫽y(z)

(23)

where y is defined (for z⬎0) by

冉 冊

∂2y 1 ∂ ∂y 1 r ⫺ 2y⫹ 2 ⫽0 r ∂r ∂r r ∂z ∂y ⫽0 on r⫽b ∂r ∂y 1 ⫽ r on z⫽0⫹ ∂z 2

(24)

The solution of the system of Eq. (24) is of the form

u1r=0

u1q=−sinq for z⬎0, u1q=0 for z⬍0, u1z=0

(19)

y→0 for z→⬁

where u1r=cosq

u1⬘⫽⫺ⵜf1,u2⬘⫽⫺ⵜf2

(17)

u1z=0

冘 冉 冊 ⬁

y⫽

anJ1 j⬘1,n

n⫽1

Fig. 4.

Geometry near the tube end at z=0.

r −j ⬘ z e 1,n b b

(25)

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

where Jm is the Bessel function of order m, j1,n⬘ the nth positive root of J1⬘(z)=0. The values of the coefficients are found to be J2(j1,n⬘) 1 an⫽⫺b . 2 ⬘2 (J1(j1,n⬘)) j 1,n−1 2

(26)

So the vibrational velocity fields u(1) and u(2) in the vicinity of the end at z=0 can be directly expressed using the relevant equations above.

4. The axisymmetric weight function In this paper only the effects of fully developed flow profiles are examined. Therefore it is necessary only to calculate the phase shift ⌬f in Eq. (1) for an axisymmetric velocity profile. With v=v(r)k in Eq. (1)

415

5. Calculation of the end effect axisymmetric weight function W⬘(r) In Eq. (31) Wz⬘ is the z-component of the end effect weight vector W⬘ given by W⬘=W⫺ (2) .ⵜ)u ⫺ W0=r((u(1).ⵜ)u(2)⫺(u(2).ⵜ)u(1))⫺r((u(1) 0 0 (1) (u(2) 0 .ⵜ)u0 ). Let this be calculated in the vicinity of the end at z=0. Substituting u=u0+u⬘ it is evident that 1 (2) (1) (2) (1) (2) W⬘=((u(1) 0 .ⵜ)u⬘ +(u⬘ .ⵜ)u0 +(u .ⵜ)u⬘ )⫺((1)↔(2)) r where((1)↔(2)) denotes the previous bracket with superscripts (1) and (2) interchanged. Substituting (from Eq. (16)) (1) (1) u(1) 0 =U u1+⍀ u2,

(2) (2) u(2) 0 =U u1+⍀ u2

u⬘(1)=U(1)u1⬘+⍀(1)u2⬘, u(2)⬘=U(2)⬘u1⬘+⍀(2)u2⬘ one finds

冕 b

⌬f⫽ v(r)Wf(r)2prdr

(27)

W⬘⫽r(U(1)⍀(2)⫺U(2)⍀(1))((u1·ⵜ)u2⬘⫺(u2·ⵜ)u1⬘

(33)

⫹(u1⬘·ⵜ)u2⫺(u2⬘·ⵜ)u1⫹(u1⬘·ⵜ)u2⬘⫺(u2⬘·ⵜ)u1⬘)

0

where Wf(r)⫽Im

冉 冊 W(r) (ux)P

(28)

冕冕

2p ⬁

1 2p

Wzdzdq,

(29)

0 ⫺⬁

Wf(r) may be called the ‘axisymmetric weight function for phase shift’ and W(r) may be called the ‘fundamental axisymmetric weight function’ or simply ‘the axisymmetric weight function’. Neglecting end effects, the weight vector is as in Eq. (13) and the axisymmetric weight function is

冕冕

2pl l

1 W0(r)⫽ 2p

r(U ⍀ ⫺U ⍀ )dzdq (1)

(2)

(2)

(1)

(30)

⫽r (U(1)⍀(2)⫺U(2)⍀(1))dz.

⫹uq

冊 冉

∂vq ∂vq ∂vz ∂vz ∂vz ⫹zˆ ur ⫹uq ⫹uz ⫹uz r∂q ∂z ∂r r∂q ∂z

This is applied to find the z-components of each term in Eq. (33). Making use of Eqs. (17) and (18) and other equations in Section 3 the following results are obtained for z⬎0. ∂2 ⌿ ∂y ⫺sin2q ((u1.ⵜ)u2⬘)z⫽⫺cos2q ∂r∂z r∂z ∂3y ∂2 y ∂3y 2 2 ((u2.ⵜ)u2⬘)z⫽⫺zcos2q 2⫺zsin q 2⫺rcos q ∂r∂z r∂z ∂z3

((u2⬘.ⵜ)u1)z⫽0 ((u1⬘.ⵜ)u2⬘)z⫽cos2q

冉 冊

((u2⬘.ⵜ)u1⬘)z⫽cos2q

∂y ∂3y y ∂2y 2 ⫹sin q ∂r ∂r∂z2 r r∂z2

0

The total axisymmetric weight function is W(r)⫽W0(r)⫹W⬘(r)

(31)

where

冕冕

2p ⬁

1 W⬘(r)⫽ 2p

冊 冉 冊

∂2y ∂y ⫺sin2q ((u1⬘.ⵜ)u2)z⫽⫺cos2q ∂r∂z r∂z

0 0

冕

冉

∂vr ∂vr 1 ∂vr ˆ ∂vq 1 ⫹q ur ⫹uqvr (u.ⵜ)v⫽rˆ ur ⫹uq ⫺uqvq ⫹uz ∂r r∂q r ∂z ∂r r

in which W(r)⫽

Next note the general expression of (u.ⵜ)v in cylindrical coordinates. For any vector fields u and v,

Wz⬘dzdq, Wz⬘⫽Wz⫺W0z

(32)

0 ⫺⬁

The correction W⬘(r) to the axisymmetric weight function is calculated in the next section.

⫹cos2q

冉 冊

冉 冊

∂2y 2 ∂2y 2 ∂y ⫹sin2q ⫹cos2q 2 ∂r∂z r∂z ∂z

2

∂y ∂3y ∂z ∂z3

For z⬍0 the first four terms are zero (since u1=u2=0 for z⬍0) and the last two terms remain the same (and are symmetric in z).

416

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

Substituting these results into the z-component of Eq. (33) an expression for Wz⬘ near the end at z=0 is obtained. By symmetry (with reference to Fig. 1)

˜ (r) in Eq. (36) was obtained using the first 20 function W terms in the series (25) and is estimated to be accurate to 1%.

l l Wz⬘(r,q, ⫺z)⫽Wz⬘(r,q, ⫹z). 2 2

6. Calculation of the end effect on meter sensitivity

Therefore putting the expression for Wz⬘ near the end at z=0 into Eq. (32) and performing the integration over q the following result is obtained.

Of interest is the fractional increase E in meter sensitivity due to end effects. For axisymmetric flows this is (from Section 4) given by

冕冉 ⬁

W⬘(r)⫽2r(U(1)⍀(2)⫺U(2)⍀(1))

∂ y ∂y ⫺ ⫺ ∂r∂z r∂r

冉 冊 冉 冊 冊

1 ∂y 1 ∂y 1∂y ∂y ∂y ⫹ z ⫹ z ⫹ r ⫹ ⫹ 2 ∂r∂z2 2 r∂z2 2 ∂z3 ∂r∂z r∂z 2

冉 冊

3

2

2

2

(34)

v(r)W0f(r)2prdr

0

where

the factor 2 accounting for effects at both ends of the tube. In Eq. (34) U and ⍀ are constants (independent of r) and, as previously explained, take on the values [in case (1) and case (2)] of the linear and angular velocities of the tube at z=0. The integral in Eq. (34) is a real function of r and can be computed using the series expansion (25) with Eq. (26) for the coefficients. The analysis was carried out using algebraic computational software and a polynomial fit to the function was made. As a result it was found from Eq. (34) that (35)

where ˜ (r)⫽⫺0.941⫹0.816(r/b)2⫺0.214(r/b)4 W

(37)

E⫽ b

∂2y 2 ∂y ∂3y y ∂2y ∂y ∂3y ⫹ ⫺ ⫺ ⫺ dz ∂z2 ∂r ∂r∂z2 r r∂z2 ∂z ∂z3

˜ (r) W⬘(r)⫽2r(U(1)⍀(2)⫺U(2)⍀(1))|z=0bW

v(r)Wf⬘(r)2prdr

0

0

3

冕 冕 b

2

(36)

⫹0.551(r/b) . 6

This end effect weight function is plotted in Fig. 5. The

冉 冊

Wf⬘(r)⫽Im

冉 冊

W⬘(r) W0(r) , W0f(r)⫽Im (ux)P (ux)P

(38)

are respectively the end effect weight function for phase difference and the weight function for phase difference assuming no end effects. Expression (37) can be rewritten in terms of W⬘(r) and W0(r) given by Eqs. (30) and (35). Let it be supposed that the tube motion (without flow) is frictionless (no energy losses) and that the tube is driven [Fig. 2(a)] by a real valued force close to the fundamental frequency. With no flow the tube motion is therefore all of one phase and the linear and angular tube velocities U(1) and ⍀(1) are pure imaginary. It follows that W⬘(r) in Eq. (35) and W0(r) in Eq. (30) are real. Therefore from Eq. (38) Wf⬘(r)⫽W⬘(r)Im

冉 冊

冉 冊

1 1 , W0f(r)⫽W0(r)Im (ux)P (ux)P

and substitution in Eq. (37) gives

冕 冕 b

v(r)W⬘(r)2prdr

0

(39)

E⫽ b

v(r)W0(r)2prdr

0

or by Eqs. (30) and (35) E⫽E1E2

(40)

with b(U(1)⍀(2)−U(2)⍀(1))|z=0 E1⫽2 l

冕

(U(1)⍀(2)−U(2)⍀(1))dz

˜ (r). Fig. 5. The end-effect axisymmetric weight function W

0

(41)

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

Table 1 Values of E2 for various fully developed profiles

[10] according to which the differential equation for sinusoidal tube motion at angular frequency w is [10]

Profile

E2

Parabolic n=6 (Re=4000) n=7 (Re=110,000) n=10 (Re=2E06) Flat

⫺0.649 ⫺0.518 ⫺0.511 ⫺0.498 ⫺0.466

∂4y 4 ⫺b y⫽0 ∂x4

b⫽

冪a, w

冪rA

a⫽

EI

(44)

where EI is the tube rigidity and rA the tube mass per unit length (which must include the mass of the fluid). The boundary conditions for the free ends are

b

E2⫽

(43)

with

and

冕

417

l ∂2y ∂3y ⫽ ⫽0 at x⫽⫾ ∂x2 ∂x3 2

˜ (r)v(r)rdr W

0

冕 b

.

(42)

v(r)rdr

0

The factor E1 of E depends only on the position of the sensing points P and P⬘ in Fig. 1 (i.e. only on z/l). Its value will be derived in Section 7. The factor E2 of E depends only on the velocity profile. Computed values [found using Eqs. (7) and (36)) are given in Table 1.

7. Calculation of the factor E1 in the expression for end effect on meter sensitivity To find E1 in Eq. (40) it is necessary to calculate U and ⍀ in [cases (1) and (2)] all along the tube. To do this take new coordinates (Fig. 6) with origin at the tube centre and x-axis along the tube axis. Let displacement occur in the y-direction with amplitude y. It has been assumed that (b/l)21 [see Eq. (11)]. This is also the condition for the use of (Bernoulli–Euler) beam theory

(45)

(The effect of the reaction couple due to fluid pressure over the end boundaries of the fluid contained in the tube is of order b2/l2 and can therefore be neglected both here and in the calculation of the sensitivity in Section 8.) As is well known the natural fundamental mode of vibration is given by the symmetric solution y⫽A¯ cosbx⫹B¯ coshbx

(46)

of Eq. (43) where A¯ and B¯ are constants. By Eq. (45) bl cosh 2 A¯ ¯B⫽ bl cos 2

(47)

provided bl bl tan ⫽⫺tanh 2 2

(48)

which is the characteristic equation with solution bl⫽4.730

(49)

determining [in Eq. (44)] the frequency w0 of the fundamental mode. It is supposed that the flowmeter tube is driven close to the fundamental mode frequency and that it is adequate to take the limit as the driving frequency tends to w0 and the force itself tends to zero. In this way the above natural vibration mode applies and so it is possible to say that U(1)⫽y˙ ⫽iw(cosbx⫹B¯ coshbx)

(50)

∂y˙ ⍀(1)⫽ ⫽iw(⫺bsinbx⫹B¯ bsinhbx) ∂x

Fig. 6. New coordinates for calculation of tube vibrations (a) case (1), (b) case (2).

where A¯ has been put equal to 1 since the absolute value of result (50) is of no consequence in evaluating the RHS of Eq. (41). (For the same reason the common factor iw in Eq. (50) can be dropped.) To find U(2) and ⍀(2) it is necessary to solve Eq. (43)

418

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

when unit forces are applied at P and P⬘ [as shown in Fig. 6(b)]. The boundary conditions (45) still apply but in addition there are the boundary conditions y|x⫽z−⫽y|x⫽z⫹ ∂y ∂x

|

⫽

x⫽z−

∂2 y ∂x2

∂y ∂x

x⫽z−

⫺EI

冉 | ∂3y ∂x3

b(Acosbx⫹Bcoshbx)

0⬍x⬍z

b(Ccosbx⫹Dcoshbx−Esinbx⫹Fsinhbx) z⬍x⬍l/2

Substituting Eqs. (50) and (55) into Eq. (41) and introducing the non-dimensional variable

|

(51)

x⫽bx

(56)

x⫽z⫹

∂2 y ⫽ 2 ∂x

|

再

⍀(2)⫽

|

x⫽z−

x⫽z⫹

the following expression for E1 is obtained.

| 冊

∂3y ⫺ 3 ∂x

⫽1

(57)

A1−A2 z ⫽2a l I1+I2

(58)

where

x⫽z⫹

at the point P⬘ and similar boundary conditions at P. Since the motion of the tube is now anti-symmetric in x the following solution to Eq. (43) is appropriate. y⫽Asinbx⫹Bsinhbx,

冉冊

b z E1⫽ f l l

|x|⬍z

f

冉冊

in which A1⫽(cosa⫹B¯ cosha)(Ccosa⫹Dcosa⫺Esina⫹Fsinha)

z⬍x⬍

y⫽Csinbx⫹Dsinhbx⫹Ecosbx⫹Fcoshbxm,

l (52) 2

l ⫺ ⬍x⬍⫺z 2

y⫽Csinbx⫹Dsinhbx⫺Ecosbx⫺Fcoshbx,

A2⫽(Csina⫹Dsinha⫹Ecosa⫹Fcosha)(⫺sina⫹B¯ sinha)

冕

((cosx⫹B¯ coshx)(Acosx⫹Bcoshx)⫺(Asinx⫹Bsinhx)(⫺sinx⫹B¯ sinhx))dx

冕

((cosx⫹B¯ coshx)(Ccosx⫹Dcoshx⫺Esinz⫹Fsinhz)

2az/l

I1⫽

(59)

0

a

Applying the boundary conditions (45) and (51) to Eq. (52) a system of six equations for the six constants A, B,…F is obtained. To within a constant common factor [which is of no significance when finally substituting in Eq. (41)] the equations for A, B,…F are

冢

0

0

−sina sinha

0

0

−cosa cosha sina

−cosg coshg cosg

−cosa cosha

−coshg −sing

sing

sinhg −sing

cosg

coshg −cosg −coshg sing

−sing sinhg sing

sinha −sinhg

冣冢 冣

−sinhg −cosg −coshg −sinhg cosg

−sinhg −coshg

B

D

2ax/l

⫺(Csinx⫹Dsinhx⫹Ecosx⫹Fcoshx)(⫺sinx⫹B¯ sinhx))dx

In Eq. (53) the value of a is fixed by Eq. (49). Therefore the A, B,…F depend only on z/l. Also in Eq. (59) B¯ is a constant given by Eq. (47), i.e.

A

C

I2⫽

(53)

E

cosa B¯ ⫽ . cosha

(60)

Therefore in Eq. (57) E1 is the product of b/l and a function of z/l only. A plot of the function f(z/l) is given in Fig. 7.

F

冢冣 0 0

⫽

1 0 0 0

where z bl a⫽ , and g⫽bz⫽2a . 2 l

(54)

The solution to Eq. (53) can be obtained using computer algebra software and then a suitable expression for U(2) and ⍀(2) or x⬎0 is as follows.

再

U(2)⫽

Asinbx⫹Bsinhbx

0⬍x⬍z

Csinbx⫹Dsinhbx⫹Ecosbx⫹Fcoshbx z⬍x⬍l/2

(55)

Fig. 7.

The function f(z/l).

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

8. Calculation of the sensitivity without end effect

where, by Eqs. (54) and (60),

Finally the sensitivity to fully developed flow without end effect is calculated. For this purpose use is made of the weight vector Eq. (13) or to the axisymmetric weight function Eq. (30) derived from it. Substituting Eq. (30) into Eqs. (27) and (28) gives

b 3⫽

˙ Im ⌬f⫽M

冢

冕 l

(U(1)⍀(2)−U(2)⍀(1))dz

0

U

冉 冊

(1)

l −z 2

冣

where (ux)P has been replaced by U(1)

B¯ ⫽

cosa cosha

(66)

and where I1 and I2 are as in Eq. (59). Collecting the Eqs. (61)–(66) the phase difference (due to fully developed flow) between the sensing points P and P⬘ in Fig. 1 (neglecting end effects) is

冉冊

(61)

冉 冊

l ⫺z and where 2

冕

(62)

0

is the mass flow rate. The (large) bracketed term in Eq. (61) can be evaluated using results in Section 7. In terms of the axial coordinate x used in Section 7

冕

(67)

where F

˙ ⫽r v(r)2prdr M

冕

2a 3 8a3 ⫽ 3 , l l

l3w z ¯ F ⌬f⫽M EI l

h

l

冉 冊

419

冉冊

z ⫽ l

−2(I1+I2) . z cosa z 3 8a cos 2a + cosh 2a l cosha l

冉 冉 冊

冉 冊冊

The function F(z/l) (with a=bl/2=4.730/2) is plotted in Fig. 8. The sensitivity of the meter is seen to diverge for z/l=0.2758. For this particular value of z/l the sensing points are at the nodes of the tube vibration with no flow. This divergence (or strong dependence of sensitivity on sensor position) near to nodes has been noted before [3]. In practice one avoids placing the sensors near the nodes and then Fig. 8 gives the sensitivity (neglecting end effects) as a function of sensor position.

l/2

9. Conclusions

(U(1)⍀(2)−U(2)⍀(1))dz 2 (U(1)⍀(2)−U(2)⍀(1))dx

0

U(1)

冉 冊 l −z 2

⫽

0

U(1)(−z)

.

(63)

For U(1)(x) and ⍀(1)(x) on the RHS of Eq. (63), Eq. (50) can be used without the factor iw [the absolute value of U(1)(x) and ⍀(1)(x) being of no importance in Eq. (63)]. For U(2)(x) and ⍀(2)(x) on the RHS of Eq. (63) the absolute values are needed. The expressions Eq. (55) for U(2)(x) and ⍀(2)(x) must therefore be multiplied by iw [since this was omitted in passing from Eqs. (52)–(55)]. Expressions (55) must also be multiplied by the factor ⫺

1 1 EI b3

When neglecting end effects the sensitivity of the freeended configuration (Fig. 1) studied in this paper is given by Eq. (67), i.e. the expected phase difference ⌬f between motion at sensing points P and P⬘ is

冉冊

3 ˙ l wF z ⌬f⫽M EI l

(69)

˙ is the mass flow rate, l the tube length, w the where M angular frequency of operation and EI the tube rigidity. The non-dimensional sensor position function F(z/l) is

(64)

since the constants A…F [found from Eq. (53)] are normalised by replacing the factor (64) by 1 [on the RHS of Eq. (53)] as is evident from the last boundary condition in Eq. (51) and the expression for y in Eq. (52). As a result

冕

(68)

l/2

(U(1)⍀(2)−U(2)⍀(1))dx

0

U(1)(−z)

冉 冊

1 1 iw EI b3 ⫽ cosbz+B¯ coshbz (I1+I2) −

(65)

Fig. 8.

The function F(z/l).

420

J. Hemp / Flow Measurement and Instrumentation 12 (2002) 411–420

plotted in Fig. 8 (for the fundamental mode of excitation). Eq. (69) can be compared with the phase difference equation in eq. (5.14) of [3] calculated for a straight tube flowmeter with rigidly clamped ends. Dimensionally the equations are the same but the sensor position factor in Eq. (69) is generally much larger than (typically 10 times larger) than in eq. (5.14) of [3]. This is presumably due physically to the fact that, for the same amplitude of vibration with no flow, the Coriolis forces that arise when flow occurs are able to cause a greater secondary motion of the tube in the free-ended case than they are in the fixed-ended case. In the free-ended case only the inertia of the tube (to turning about its centre) has to be overcome. The fractional increase E in sensitivity due to end effects (for the meter configuration illustrated in Fig. 1) is by Eqs. (40) and (57)

冉冊

b z E⫽ f E2 l l

(70)

where the function f(z/l) is plotted in Fig. 7 and the factor E2 is tabulated in Table 1. Firstly, it is seen that the order of magnitude of E is b/l. This is perhaps to be expected because b/l is the ratio of the volume of fluid near the ends (where the fluid vibrations are substantially different) to the total volume of fluid inside the tube. Secondly, it can be observed that the function f(x/l) has a minimum at z/l=0.369. This shows there may be some advantage in placing the sensors at this relative distance from the tube centre. Thirdly, with this ‘best’ choice of sensor positions, f(z/l)=2.50 so the value of E becomes b E⫽2.50 E2 l

(71)

For example, in a tube 25 diameters long the end effect increase in sensitivity is then (by Table 1) 2.50(1/50) (⫺0.649)=⫺0.0324=⫺3.24% for laminar flow or ⫺2.59% for turbulent flow at a Reynolds number of 4000. This represents an increase in sensitivity of about 0.65% upon transition from laminar to turbulent flow. This paper demonstrates the power of the weight vector approach to calculating flowmeter sensitivities. The particular case considered (straight tube meter with free ends and fully developed flow) allows an quasi-analytic

solution. It is useful to summarise the assumptions made in the analysis. These are as follows: 1. The viscous boundary layer in the vibrational fluid flow is small compared to the laminar sublayer thickness in the steady turbulent flow (or to the tube radius in laminar flow). 2. The amplitude of vibration is small compared to the laminar sublayer thickness in the steady turbulent flow (or the tube radius in laminar flow). 3. In the case of turbulent flow the signals from the sensors are narrow band filtered to remove effects due to turbulence fluctuations (see [4, p. 252]). 4. It is assumed that (b/l)21 so that simple beam theory (with neglect of rotational inertia and shear stress) and the ‘local rigid tube’ approximation can be made in calculating the vibrational flow of the fluid. 5. Energy losses are neglected. Note that if the same (weight vector) approach with the same assumptions 1–5 is applied to the case of a straight tube rigidly clamped at its ends the result is a zero end effect and a zero profile effect. This is because the fluid vibrational motion has then no discontinuity at the ends z=0 and z=l, both U and ⍀ being zero there. The sensitivity in this (fixed-ended) case is (neglecting b2/l2 compared to 1) the same as that predicted by the lumped Coriolis force model. References [1] N. M. Keita, The zero drift effect in Coriolis mass flowmeters, in: International Conference of Mass Flow Measurement Direct and Indirect, London, February 1989. [2] G. Sultan, J. Hemp, Modelling of the Coriolis mass flowmeter, Journal of Sound and Vibration 132 (3) (1989) 473–489. [3] H. Raszillier, F. Durst, Coriolis effect in mass flow metering, Archive of Applied Mechanics 61 (1991) 192–214. [4] J. Hemp, The weight vector theory of Coriolis mass flowmeters, Flow Measurement and Instrumentation 5 (4) (1994) 247–253. [5] J. Hemp, L.A. Hendry, The weight vector theory of Coriolis mass flowmeters. Part 2. Boundary source of secondary vibration, Flow Measurement and Instrumentation 6 (4) (1995) 259–264. [6] L. Landau, E.M. Lifshitz, Fluid Mechanics, 2nd ed., Pergamon Press, Oxford, 1987. [7] F.M. White, Fluid Mechanics, McGraw-Hill, New York, 1986. [8] A.J. Ward-Smith, Internal Fluid Flow, Oxford University Press, Oxford, 1980. [9] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1955. [10] K.F. Graff, Wave Motion in Elastic Solids, Dover, New York, 1975.