180 H. Raszillier and V. Raszillier - Dimensional and symmetry analysis of Coriolis mass flowmeters

Dimensional and symmetry analysis of Coriolis mass flowmeters H. RASZlLLIER* and V. RASZlLLIER*

For Coriolis mass flowmeters working with straight pipe segments a model equation described in detail and solved approximately in previous work is considered here from the viewpoint of combined dimensional and symmetry analysis. This analysis shows, without solving the equation, that only for low (mean) fluid velocities o (compared to a characteristic elastic velocity c) are measurements of density and flowrate reciprocally unbiased, i.e. independent of each other. In addition, i f the detection points for flowmetering are situated symmetrically about the middle of the pipe segment, the (relative) errors in measurement of density and mass flowrate are of the same order, ~2/c2.

Keywords: Coriolis flowmeters, modelling, dimensional analysis

Introduction

Coriolis flowmeters essentially consist of fluid-convey- ing vibrating pipe segments. Their working principle is based on the fact that the fluid flow influences the vibration modes of the pipe, which enables informa- tion about the flow to be drawn from appropriate observations of the pipe vibration. In detail, the pipe-fluid interaction in this type of measuring device is rather complicated; in particular, it depends on the geometry of the pipe segment. Even after certain reasonable simplifications, analysis has to be made either numerically ~ or within analytical approximation schemes 2.

In the present paper it will be shown that impor- tant insight into the way Coriolis flowmeters work can be gained even before solving, either numerically 1 or analytically 2, the corresponding (simplified) equations of pipe-fluid interactions. By combining dimensional analysis with the consequences of additional (reflec- tion) symmetry properties of these equations, an un- derstanding of the sensitivity of the measurement of mass flowrate and of the expected order of magnitude of errors in both mass flowrate and density measure- ment can be obtained.

In the first section of the paper the model equa- tion describing the flowmeter will be shortly reviewed. It considers the pipe as a thin elastic beam and the fluid as a thin, perfectly flexible, moving threadline or string. The string is coupled in its transverse motion to the beam, its longitudinal motion 'through' the beam is frictionless. The derivation and also the solution of this equation has been presented in detail in Reference 2. In the second section the symmetry properties of the equation are reviewed. It is essentially due to the fact that if the pipe segment of the instrument has a

*Lehrstuhl ffir Str6mungsmechanik, Universit~t Erlangen-N~rnberg, Cauerstr. 4, D-8520 Erlangen, Germany

geometric reflection symmetry that maps one half (say that of inflow) on the other (of outflow) and vice versa, then with the reversal of flow the flow conditions in these two halves will interchange. This statement goes beyond this specific model equation; it is, in fact, general.

The symmetry described has implications on the behaviour of the frequencies and (local) amplitudes and phases of transverse vibration modes under flow reversal. The model equation allows the dimensional analysis 3 of the Coriolis flowmeter to be put into a precise frame. This flame is described in the third section of the paper in terms of the dimensional parameters and observables relevant to flowmetering. In this frame the w-theorem is then applied; it ex- presses the observables as functions of the appropriate number of dimensionless parameters. The form of this dependence, given by dimensional analysis, is re- stricted by the reflection symmetry mentioned above. This restriction expresses the additional physical information supplied by this type of symmetry.

There is additional information in the model equation. For small flowrates centrifugal forces, which are of second order in flowrate, can be neglected in the equation. The equation thereby gets an approxi- mate form, which is valid to first order in flowrate. This form gives, in addition to dimensional analysis and reflection symmetry, a more precise insight into the behaviour of mode frequencies and phases to first order in flowrate.

In the last section the results of the paper are summarized and commented upon.

Model equation for a Coriolis flowmeter

In the frame of the approximations discussed in the Introduction, one can derive a simple equation for the amplitude u(x, t) of free transverse vibrations of a fluid-conveying pipe segment (of length /)2:

0955-5986/91/030180-05 © 1991 Butterworth-Heinemann Ltd

F low Meas. Instrum. Vo l 2 Ju ly 1991 181

~2u ~2u a2u (mp + mf) ~t ~ + 2mr1) ~ + mfv 2 ~x 2

D4u + E I - - = O (1)

Dx 4

with the boundary conditions

u(O, t) = u(I, t) = 0 (2) ~u ~u a-~ (0, t) = ~ (/, t) = 0 (3)

In these equations rap(mr) is the pipe (fluid) mass per unit length, v is the mean fluid velocity, E is the Young modulus of the pipe, and / is the area moment of inertia of the pipe. With these parameters the mass flowrate of fluid is

Q M = mr1) (4)

It appears 2 that for fluid velocities below a certain critical value

= (5) C ---~ ~mfl

the free vibration modes of the pipe-f luid system are undamped. In complex form they may be represented as

u(x, t ) = u ( x ) e TM (6)

the frequency o) and the amplitude u(x) have to be determined from the nonlinear eigenvalue problem:

d 4 u(x) d 2 u(x) du(x) El - - + m f o 2 - - + 2 i m f o L o - -

dx 4 ( ix 2 dx

-- (.o2(mf + mp)U(X) = 0 (7)

u(0) = u(/) = 0 (8)

u'(0) = u'( I ) = 0 (9)

In Reference 2 the modes have been considered as small perturbations caused by the flow, owing to the smallness of the ratio

1) 2 - - << 1 (10) C 2

of the modes of the equation

d 4 uo(x ) El dx 4 (o02(mf + mp)Uo(X) = 0 (11)

u0(0) = Uo(I) = 0 (12)

u6(O) = u;( I ) = 0 (13)

which describes the vibration modes in the presence of the fluid in the pipe, but without flow (v = 0). Thereby a rather detailed analysis of the modes of the fluid-conveying pipe has been given to first order in the parameter v/c. In particular, it has been shown how the mass flowrate can be determined from the measurement of the time delay between two suitably chosen points of the pipe segment in a chosen mode of free vibration. Here it will be investigated how this kind of prediction can be traced back to the general symmetry properties of Equation (1), without solving the equation explicitly.

Symmetry of the equation for a Coriolis f lowmeter

The basic symmetry of the equation follows from dimensional analysis, which essentially states that any of its physical consequences can be formulated in terms of dimensionless quantities. The instrument for its precise use is the 1r-theorem. There is an additional symmetry that possibly has an impact on the conse- quences of the rr-theorem. Namely, Equation (1) and the boundary conditions (2) and (3) are left unchanged by the symmetry operation (reflection)

t--~ t

x ~ l - x

(14)

(15)

(16)

UR(Xl, to(X/; V); V) = 0 (20)

Because of

UR(Xu t; V) = + U R ( I - Xl, t; --V) (21)

these have to be the same as the zero crossings in the image point after reflection, xr = / - xu when the flow is reversed:

t0(x/; v) = to(I - xl; - v ) (22)

The object of measurement in Coriolis instruments is the time difference between the zero crossings in two points situated symmetrically to the middle, i.e. the difference between t0(x/; v) and t o ( I - x l ; v). Assume that x / < I /2. Then intuition suggests that t0(x/; v) > to(I - xl; v). The difference

T(x/; v) = t0(x/; v) - to(I - x/; v) (23)

is accordingly expected to be positive; it will be called the t ime de lay of (the vibration of the point) x/ with respect to (the vibration of) its image, / - x/. Because of Equation (22) r(x; v) is an odd function of v:

r(x; v) = t0(x; v) - to(I - x; v)

= t o ( I - x; - v ) - t0(x; - v ) (24)

= - T(x; - v)

Equations (18) and (24) are very important for flow- metering. They show for the (expected) case, that the flow produces only a (small) perturbation of the free

1) "--'> --O

which interchanges inflow and outflow halves (x < I /2 and x > I /2, respectively) of the pipe segment by reflection around the midpoint x = I /2 ( x - I / 2 - - . - ( x - •/2)) with simultaneous change of flow direc- tion (v ---> -v) .

Therefore the relations

u(x; v) = + u ( I - x; - v ) (17)

w(v) = w ( - v ) (18)

have to be obeyed by the frequency and the am- plitude of any eigenmode. Consider now the zero crossings of the (real) physical amplitude

UR(X, t; V) = ~(U(X; v)e i~v)t) (19)

= UR(X; v ) c o s w ( v ) t - Ul(X; v ) s i n w ( v ) t

in a g i ven point xl, i .e. the m o m e n t s to(xl; o), fo r which

182 H. Raszillier and V. Raszillier - Dimensional and symmetry analysis o f Coriolis mass flowmeters

vibration modes and co(v), v(x; v) can be expanded in a (convergent) power series of v (precisely: v/c), that

co(v) = co(O) + O(v 2) (25) r(x; v) = vv'(x; O) + O(v 3) (26)

The mode frequency changes only to second order with the flow velocity v, whereas the time delay is l inear in v with a relative error of the same order V2/C 2 as the frequency co. Without the symmetry conditions obeyed these errors would be larger; i.e. of order of v/c. The frequency co and time delay T are the observable magnitudes for the determination of the fluid density and velocity, respectively. In order to unveil, at least in part, their relation with these sought quantities, in the next section dimensional analysis will be exploited.

D i m e n s i o n a l a n a l y s i s

If a dimensional analysis of a Coriolis flowmeter is made by leaning on Equations (7)-(9), the following parameters must be taken into account:

/ length of pipe segment in flowmeter v fluid velocity mf fluid mass per unit (pipe) length mp + mf total mass (pipe + fluid) per unit length El product of the (pipe) Young modulus E and

area moment of inertia I x coordinate of an observation point n (integer) number identifying a (given) free

vibration mode

and as observables

co eigenfrequency of the vibration mode r(x; v) time delay of the vibration in x with re-

spect to / - x.

The dimension matrix of these quantities in the units of length (L), time (T) and mass (M) is (see, for example, Reference 3)

(L) (T) (M)

(/) (mp + mf) (El) (mr) (x) (v) (n) (7:) (co)

B =

1 0 -1 0

3 - 2 -1 0

1 0 1 - 1 0 0 0 1 0 -1

0\ 1 1 1 0 0 0 0 0

(27)

It permits transfer to /, mp-F mf and El as funda- mental quantities, because their submatrix

1 0 O) Bf = - 1 0 1

3 - 2 1 (28)

BF 1 = 1/2 -1 1 (29)

it is easy to compute the matrix that gives the dimen- sionless quantities according to the 7r-theorem:

t 0 1 O) 11 0 0 A = - - 1 / 2 1 / 2

i 0 0 (30) 1/2 -1/2

- -1/2 1/2

The dimensionless quantities are

mf T[ 1 - -

mp + mf

x 1r2-- I

/r 3 ---- v/ (mp

1r4= n

+ m01/2 El

1 ( E/ ) 1/2 /r5 = z ~ - mp + mf

/T6 = COl2 ( mp-I-E/ m!) I/2

(31)

(32)

(33)

(34)

(35)

(36)

Since co and v are observables resulting from the solution of Equations (7)-(9), 7rs and 7r6 will be functions of Irl, rQ, ~r3 and 7r4. In addition, since co is a global quantity, it will be independent of ;r2. The symmetry properties (18) and (24) show that ~r6 will depend only on 7r 2, whereas ~rs will have, as far as rr3 is concerned, the form: rr3 times a function of 7r 2. Explicitly:

( ) F , v2/2 (rap + mf) co = ~ mp + mft mp + mf El , n

(37)

v= v/3 mp + mf ( mf , /22/2 mp + mf El G mp + mf El '

x / n )

(38)

The functions F and G can be redefined as

F (-mp mf , vZ l 2 mp + mf ) + mf El , n =

mf , v212 m p + m() fn mp + mf El

G ( mf , v212 mp + mf x ) mp + mf El ' I ' n =

mf gn( mf v212 mP + mf mp + mf mp + mf El I

(39)

has non-vanishing determinant: det Bf = 2. With (40)

Flow Meas. Instrum. Vol 2 July 1991 183

the dependence of the new functions on n being expressed now as an index, and then get the very convenient form

13 ( mf ,v212 mp + mf x) T(X; V) = mft) -~ gn mp + mf El ' -I

(41) for the time delay.

Consider now Equation (1) to first order in v/c. Then notice that the only appearance of mf is in the combination

QM- - mfv (42)

This implies, on the one side, that the function fn(mf/(mp + mr), 0) is a constant

fn mp + mr' 0 = Yn (43)

it is the mode constant of vibration without flow. So

(.0(12) = "-~ mp + mfl + 0 (44)

and the frequency co(v)~ co(O) can be used to good approximation for measuring the fluid density pf as

mf = pfS (45)

S being the (inner) pipe cross section. On the other side, for the same reason,

g,(mf/(mp + mf), 0, x/I) is only a (mode-dependent) function of x/ I

gn mp + mr' 0, = hn (46)

Therefore ,

l'(x; v) = QM ~-~ hn + O (47)

i.e. to lowest order the time delay T(x; v).may serve as a means of measuring the mass flowrate QM of the fluid: there are no other fluid-dependent quantities in Equ- ation (47). As can be seen from Equations (37) and (41) this measuring principle, as it stands, in fact relies on the accuracy of the lowest order of perturbation. With the higher orders the nature of the fluid comes into play and the whole procedure has to be thought over anew.

Comments

In this paper three elements have been combined for an analysis of Coriolis flowmetering: a (reflection) symmetry of the fluid-conveying pipe segment of the instrument; dimensional analysis based on a model equation incorporating this symmetry; and the form of this equation at low flowrates. The main result is expressed by the pair of formulae for the frequency co(v) and the time delay r(x; v):

,( ( CO(V) = ~--mp + mr/ fn mp

[3 ( mf r(x; v) = OM -~ g, mp + mf

mf 1) 2 / 2 mp + mr) - - ! + mf El

(48) mp + mf x /

' v= 12 El ' -I ] (49)

supplemented by the information that the two limits

( mf m p + mr) limfn - - , v 2/2 = Yn (50) v-~o mp + mf El

( m f , D212 mP + mf x ) ( - ~ ) l imgn , v - , o ,mp + mf El -I = hn

(51)

are independent of the mass ratio mf/mp, i.e. of the nature of the fluid.

These formulae together show that to lowest order in v the observables ~v ) and ~(x; v) are given by

( EI I d2 ~/ n mp CO(0) = -F + mr/ (52)

and

, (.) lr(X; V) = QM El hn (53)

respectively: up to pipe constants the mode frequency is determined entirely by the fluid density (pf! and the time delay by the fluid mass flowrate (QM). The relative error of both relations is of the order v2/c 2. The precise magnitude of error of Equation (53) as a device for measuring the mass flowrate may thereby depend very sensitively on the choice of the vibration mode (n) and on the positions (x, / - x ) of the symmetric pair of detection points.

All these facts are, of course, reflected by the detailed computations of Reference 2. The present analysis allows rather deep qualitative insight into this type of flowmeter before hard computational work has to be started. It also gives a hint as to what degree computations are worthwhile pursuing. From Equa- tions (48) and (49) it is expected that the corrections to the lowest order formulae (52) and (53) will mix density and flowrate measurements and destroy their independence. This means that if the latter equations appear not to be precise enough, the whole principle is expected to lose its simplicity, i.e. the measuring principle as it stands is a first-order principle only. Its complete quantitative formulation requires only the knowledge of the constants 7n of vibration modes without flow and that of the mode function hn(x/I). Whereas the constants 7n are rather easy to obtain, the hn(x/I) require considerable computational effort 2. With detection point x, / - x and mode n fixed, 7n and hn(x/I) can, on the other side, also be viewed as constants to be determined directly by experiment. Although this procedure may seem convenient, it misses the physics behind the function h~(x/I) (Refer- ence 2).

This type of analysis is, as already mentioned in the introduction, not restricted to Coriolis flowmeters working with a straight pipe segment: essential to it is the existence of a (geometric) reflection symmetry that interchanges inflow and outflow parts of the device. For the sake of illustration, the U-tube Coriolis instru- ment analysed in Reference 1 will now be considered. This instrument has a reflection symmetry with respect to the middle plane ( y = 0, i.e. with respect to t - -> - t , s - - ~ 2 b + r r a - s (s is the arc length coordin- ate in the notation of Reference 1, which will be followed)). As one can easily derive from Reference 1, its equations (11)-(14) remain unchanged under this

184 H. Raszillier and V. Raszillier - Dimensional and symmetry analysis of Coriolis mass flowmeters

symmetry, if also the flow is reversed (U ~ - U ) . The observables in this instrument are again co and A~, but two dimensionless parameters have to be added to #1-#4 because of the appearance of the radius a and of the product GJ between the shear modulus G and the polar moment of area J in these equations. These additional parameters can be taken as a/b and (GJ)/(EI). For this instrument the results (48)-(53) are

~ ( U ) =

1

b 2 ̧(M, El 11/2 Mf , U2b2 Mt 4- Mf a 4-Mfl fn Mt 4-~ f El ' b ' ~

(54)

~(s; U) =

b3 ( Mf U2b2 Mt + Mf a G] b) MfU ~ gn Mt 4- Mf' El ' b' El'

(55)

(M, 2b2Mf+Mta ,)(aO,) lim f~ b ' = Y ~ b ' - ~ - U--~0 M t + Mr ' El -~

(56)

( Mf U262 Mf + Mt a GJ b ) lim gn , , , - u~o Mt + Mr' El b El

(a hn u' El' b

and as

oXO) = ~ ~,~ ,

(57)

(58)

b 3 (a GJ s) t-(s; U)--- MfU--~-h n b ' E~' -b (59)

respectively. The conclusions of this paper are seen to remain true for this example.

Refe rences

1 Sultan, G. and Hemp, J. Modelling of the Coriolis mass flowmeter J Sound Vibration 132 (1989) 473-489

2 Raszillier, H. and Durst, F. Coriolis effect in mass flow metering Arch Appl Mech 61 (1991) 192-214

3 G~rtler, H. Dimensionsanalyse Springer, Berlin (1975)