Measurement Techniques, Vol. 40, No. 5, 1997

EFFECT OF SYMMETRIC OF FLUID VELOCITY ULTRASONIC

DISTORTION OF THE FIELD

ON THE READINGS OF AN

FLOWMETER

F. Grunert, Z. Kabza, and Ya. Pospolita (Poland)

UDC 681.121:532

A numerical method is proposed f o r determining a discharge coefficient that relates mean velocity to the velocity measured with an ultrasonic flowmeter.

Ultrasonic flowmeters (UFMs) are being used with increasing frequency in measurement technology. These instruments offer relatively good accuracy while avoiding creating additional resistance to the flow of the medium being measured. The most common application of UFMs is for the determination of flow velocity by measuring the difference in the times of passage of the acoustic signal in the direction of the flow and counter to the flow. Here, the UFM is used mainly for measurements in liquids, since intensive acoustic vibrations develop in gases due to their low acoustic resistance. One of the factors that affects the measurement accuracy of UFMs is the distortion of the profile of flow velocity. The quantity measured by the flowmeter (the average velocity of the flow over the path traveled by the acoustic signal) is related to the mean velocity in the cross-section of the pipe by a suitable coefficient. The dependence of the coefficient K on the Reynolds number Re is used for single-channel UFMs, which determine flow velocity in a diametral plane [1, 2]. As was confirmed in [3], existing formulas give the correct values of K only for smooth pipes. In [4], the value of K depended on the linear drag coefficient X. The formulas just referred to lose their universality for UFMs that measure the velocity of a liquid flow in nondiametral planes. In these cases, the coefficient K is determined separately for each measurement system [5]. An independent problem is studying the effect of the elements that disturb the flow on the value of K, as well as determining the conditions of UFM installation that will minimize this effect. The fact that the velocity field of a liquid can be disturbed by many different factors makes it very difficult to perform exhaustive experimental investigations (particularly with the appropriate flow regimes) and makes such studies costly in terms of time and materials. Thus, it is preferable to substitute numerical studies for physical experiments when the accuracy of the results is not seriously compromised. Numerical studies first require the construction of a mathematical model of the flow and the solution of the equations of the model by means of a suitable algorithm. In this article, we examine the feasibility of using mathematical modeling to analyze the effect of the distortion of the velocity field of a liquid on the readings of an ultrasonic flowmeter. First, we worked with specialists at a technical university (in Dresden, Germany) and an advanced school of engineering (in Opole, Poland) to study symmetric perturbations of the velocity field of a liquid in a pipe. These perturbations corresponded to points in the pipe with an abruptly changing cross-section (such as the junction of the pipe with a tank). In such cases, the axial symmetry of the flow and, thus, its two-dimensional character appreciably simplified the mathematical modeling. A symmetric perturbation was obtained by placing a constrictor (CR) with certain moduli in the pipe. Then preliminary experiments were conducted in the turbine lab at the university on a measurement stand. The main element of the stand (Fig. l) consists of replaceable straight sections of pipe 4 with diameters of 285 and 355 mm. The sections make it possible to position the flowmeter 6 different distances from the constrictor 5. A combination diffuser and rectifier 3 was installed behind radial fan 2. A nozzle 1 for measurement of flow rate is located in the intake pipe. The distance l had values equal to 0.7, 6, and 13 pipe diameters during the experiments. The UFM was operated in one-, two-, and three-channel modes, which improved measurement accuracy. The computer 8 on the measurement stand determined liquid velocity for the given signal path, as well as mean flow velocity. The average velocity over the radius of Translated from Izmeritel'naya Tekhnika, No. 5, pp. 29-32, May, 1997. 0543-1972/97/4005-0445518.00

9

Plenum Publishing Corporation

445

I

/ y / - ,/2

3

4

,

-'-

5

6

,iT-

8

~

Fig. 1. Measurement system: 1) measurement nozzle; 2) radial fan; 3) flow rectifier; 4) straight segment of pipe; 5) constrictor; 6) ultrasonic flowmeter; 7) printer; 8) computer.

;

,

l e

~,lar

f ff''fFf

C'lf r F v ~ q / ~ Z F f F l !

[

t n ll~l

i ] i

!

I IIIIlI

,i

,

II11111

! , i

I II1~! I ! !

,

ii -h

,

..........

IHI4;:_'

!

i

;

.... ~-

- ~, T -

i

:__L_, ,

t-'-

_

-.' . . . . "-~""

Y2

Fig. 2. System of flow analyzed with a marked differential grid: "/1 and 3'3) inlet and outlet sections; 3'2) restricted axis of symmetry; 3"4) surface of pipe and constrictor.

the pipe U R was determined as the mean of ten measurements of the time of signal transmission with a constant flow. We simultaneously determined mean velocity over the cross-section of the pipe U A from values of flow rate measured by the nozzle 1 and from indirect measurements performed with the UFM in the two-channel mode. The resulting values of U R and U A were used to calculate the coefficient K = UA/U R for each distance I with different values of Re. We studied the flow regime schematized in Fig. 2 in order to construct the mathematical model from the experimental data. The section of the pipe ahead of the CR and a long section after it were examined. Here, it was assumed that fully developed turbulent flow was created in the inlet section of the pipe and that the effect of the CR on the velocity profile in the outlet section was negligible. Average parameters of turbulent flow are described by the Reynolds equations, supplemented by the equations of a turbulence model [6-8]. In the case of two-dimensional axisymmetric flows, the equation of the mathematical model can be written in generalized form [9]:

d(pUO)dz + _1 + O(prVO) = __d l dO) r Or az F~ -~Z + + -lr

fl".

~

+ S.

(1)

,

where the variable ~, denotes (in succession) the axial U and radial V components of the velocity vector, eddy kinetic energy k. and the rate of dissipation of this energy s. Table 1 gives the coefficients F~, and S~, for the individual variables, where 446

G = ~t t

I

2

2 i vl2

~

+ . Tr

~,=P+p,"

+

t ~r

la, = C

v121

§ c)---i

'

p MI E

The quantities C~,, C 1, C 2, %, and or, are constants of the turbulence model, p is density, and # is absolute viscosity. Equation (1) becomes the continuity equation of the flow for the case ~I, = 1 and r' o = S o = o. The prescribed axial symmetry makes it possible to examine the region bounded by the axis of symmetry "/2, the surface of the pipe and the constrictor 3'4, and the inlet 3'1 and outlet 3'3 sections (see Fig. 2). The boundary-layer regime of flow, examined wifla allowance tVorthe notation 3'1-3'4, is represented as follows. The Dirichlet condition ~13'i = Co(r), in which r successively determines U, V, k and e, is assigned in the inlet section 3'1 for all variables. On the symmetry axis V = 0, and the Neumann condition ao/arl3"2 = 0 is assigned for the variables U, k, and e. The following conditions are assigned in the outlet section

f

u~

=

f

u,,A.vJ,,

=o

For q~ = k, e(0q~/0z) l.y4 = 0. The boundary-layer regime is based on the assumptions that the walls of the pipe are impermeable and that no slip occurs on them, i.e., UI~ 3 = 0 and VI,r3 = 0. The conditions for k and e at points immediately adjacent to the wail were formulated on the basis of relations obtained in a model of developed turbulence [7]. To discretize differential equations (1), the continuous region in which they were obtained was replaced by a reticular region (see Fig. 2). Figure 3 shows a fragment of the differential grid and the designations of some of its nodes. Multiplying both sides of (1) by r, we reduce the equation to a form more suited for differential approximation:

a~

,O,,,'V*I -

+ e'I-'7~

+ ~c)

ez

( r r o ~,0. 1 +

(2)

irF,r, O,l, dr )/ + r S o

The differential form of the overall transport equation can be obtained by integrating (2) on the superimposed grid:

{rU.-rF

+

rVO-rF

0 ~zi~ e Ae~-!rUq,-rFq,.

a,z,i o ~ - J n An* - !l ~ v . - r r , , ,

0--~-~'1' j,. ~'w A ~ +

~i'.

where the indices e, w, n, and s determine points equidistant from the central point P and, accordingly, equidistant from the points E, W, N, and S. The value of A ~ determines the field of the surface formed by rotation of the corresponding edge of the control surface through a unit angle, while A~ = AzAr. 9

9

The components Sp and S U are determined by tinearizing the integrand. Approximating the expressions in the braces, we arrive at the general form of the differential equation 9

ap tl,p = a ~ oE .

ar

q'

9

'l'w* a N O N - a~ 9 s - r] S~ ,

447

in which a~ = a t + a w + a N + a s -- rj Sp. The coefficients a ~ for the individual variables 9 are equal to the following, ,I~

,I~

el,

,I,

respectively: a E = AE; a w = Aw; a N = AN; a S = A S. Since the actual values of pressure p are unknown when the differential equations are solved for U and V, the calculated velocity field does not observe the continuity conditions. The actual values of velocity are calculated from the equations:

u~ = u~

'~ u (p~, - ;;

(3)

p A z i ap r. - 1,'2

Vp = V;

p~rjaV

,

(4)

I p s - pp

in which V* and U* denote approximate values of the components of the velocity vector and p' is a pressure correction. The differential equation for p' follows from the continuity equation and has the form: (5)

The corrections p' obtained in solving (5) are used to calculate exact values of the components U and V in accordance with (3) and (4). The criterion for convergence of the iteration is the condition

max {Res (q~)}< X,

(6)

where

Res (qb)-i. -max ! ap ~p i

~. aj,I,-S~} .

.

N.S.E.W

Residual criterion (6) requires the solution of differential equations at all grid points with a specified accuracy for each variable ~. The value of X was determined experimentally. The quantities UR and UA and the coefficient K were calculated after determination of the velocity field. Due to the perturbation of the field, velocity averaged over the radius was calculated by iteration along the entire path of the acoustic signal R

uR =

fu,J .

-R Velocity U on the signal path (for individual radii) was determined by iteration with allowance for its values at each pair of adjacent points. Figures 4 and 5 show the results of experimental measurements and numerical modeling. The coefficient K was determined numerically within the range L/D = 0.8-30 (where D is the diameter of the pipe), i.e. it was determined on almost the entire pipe section on which the constrictor might have affected the velocity field and produced significant perturbations. The calculations were performed primarily for the conditions under which the experiments were conducted. Values of K obtained from experimental measurements in a pipe without a constrictor were used in the pipe section at L/D = 30, where the effect of the constrictor on the velocity profile was negligible. An analysis of the results confirms the appreciable effect of velocity-field perturbations caused by the CR on the value of K. Within the range of L/D corresponding to the recirculation zone, K ranges from 0.4 to 0.7, depending on the modulus m of the CR. The coefficient K reaches a maximum at L/D = I0. It then decreases and at L/D = 30 takes the value characteristic of free flow through the pipe at the given Reynolds number. Within the investigated range of Re, the effect of the constrictor on the value of K is negligible and ranges from 1 to 2%. We should point out the good agreement between the experimental and theoretical results. The differences in the values of K for the determinations of U A made by measuring flow rate with the nozzle are no greater than 2 %. For a more detailed analysis of the accuracy of the results, Table 2 shows values of K that we obtained experimentally and by numerical modeling. Table 2 also presents literature data for flows in pipes with different values of Re. The differences between the experimental results and the data in [10] are no greater than 1.5%, while there is nearly complete agreement between our numerical data and the experimental results in [2, t0]. 448

TABLE 1 Designat]on of the variable

Values of the coefficients of the equation

r,i,

S.

(p~t

o

U

ou)+ 1 o

ov

on

V P'ef

k

G ~ ps

c~k

(C, G-C2 p E)

Pet

~z

i x

)Pi.j+I ~,

I

V

"

l

n

i

:b

1•.1.1

Z;,.,.,

f

Zi

Fig. 3. Fragment of the differential grid and designations of the nodes.

0.9-._

LID

0,5

1

2

3

4

5

10

20

30

Fig. 4. Measured and calculated values of the coefficient K as a function of the distance of the flowmeter from a constrictor with a modulus of 0.45 [solid line) calculations for Re = 0.48.105, dashed line) calculations for Re = 1.9"105 and 3.3.105]: -, o ) Re = 0.48-105; I , F-I) Re = 1.9.105; A, ,,) Re = 3.3.105 (the clear symbols represent flow-rate measurements made with the nozzle).

449

TABLE 2 Reynolds Measure- i Numerical number Re ment: calculations I 0.48 1,3 1,8 1.9 2,8 3,3

9 10 t 9 10 ~ 9 10 s 9 10 s 9 10 s 9 10 s

0,9284 0.929 0,92 0,9554 0,9652 0,9622

Data from [2]

Data from [31

0,9407 0,9446 0.946 0,947 0,948 0,9496

0,9396 0,9451 0,9469 0,9469 0,948 0,9495

1 0

0,9367 0.9409 0.9423 0,9423 0.9442 0.9449

w

=

0,8-

0.6-

I I

LID

0.4 I I

89

3

4

"C

5

20

30

Fig. 5. Measured and calculated values of the coefficient K as a function of the distance of the flowmeter from a constrictor with a modulus of 0.29: o, o ) Re = 1.3-105; I I , rq) Re = 1.8.105; A,, a) Re = 2.8"105 (the clear symbols represent flow-rate measurements made with the nozzle).

q2 ~,--f

I"'" h

,-"

~ ' -

~ . D ""2

o,8 r

/"

//~

V i .Lt.

"---'CI

~-~, i

*.

/

/

0,6

~ " \\

i

;

0.4

LID

,

,

,

.

,

1

2

3

,~

5

,

, ,

,

'

, 1'0

20

30

Fig. 6. Numerically determined values of the coefficient K in the region behind the constrictor and the sudden change in channel crosssection as a function of the ratio (d/D) 2 and the distance of the obstacle from the flowmeter [1) m = 0.7; 2) m = 0.5; 3) m = 0.29; Re = 104; D = 355 mm].

The agreement between the results demonstrates the expediency of using numerical modeling to study the metrological properties of ultrasonic flowmeters. Figure 6 shows the results of numerical studies illustrating the effect of CR modulus on the coefficient K. Figure 6 also shows similar results for the flow with a sudden expansion. The data confirms the substantial effect of the ratio of the diameters d and D on K in both cases, especially in the recirculation zone. The ratio d/D also affects the distance L beyond which the effect of the constrictor on K is negligible. 450

The results from our investigations permit the following conclusions. Symmetric distortion of the velocity profile (such as due to sudden broadening of the pipe cross-section, the presence of an accelerating constrictor, or the placement of a tank in front of the flowmeter) has a substantial effect on the range of variation of the coefficient K and. thus, on the metrological characteristics of the UFM. The value of K depends on the type of local resistance and the distance between the resistance and the flowmeter. In the case of the use of a constrictor and pipes with a sudden expansion, the value of K depends appreciably on the ratio of the diameters of the pipe sections. The effect of the Reynolds number on K is no greater than 2 % throughout the entire range of locations of the UFM. A comparison of the experimental and theoretical results shows their complete agreement. The results obtained by numerical modeling make it possible to avoid time-consuming and costly experimental studies. This is particularly true in cases involving multiple perturbations of the velocity field, which may be too complex to realize experimentally. However, numerical studies of asymmetric distortions of the velocity field require the development of a mathematical model and a modeling program that can solve three-dimensional flow problems.

REFERENCES 1.

2. 3. 4. .

6. 7. 8. 9. 10.

G. I. Birger, Izmer. Tekh., No. 10, 53 (1962). S. S. Kivilis and V. A. Reshetnikov, Izmer. Tekh., No. 3, 77 (1965). P. P. Kremlevskii, Flowmeters and Counters [in Russian], Mashinostroenie, Leningrad (1989). V. I. Myasnikov, Tr. VODGEO (Transactions of the All-Union Scientific Research Institute of Water Supply, Sewer Systems, Hydraulic Engineering Structures, and Engineering Hydrogeology), 10 (1983). S. Walus, Zesz. Nauk. Politeeh. Slask., Ser. Automat., 99 (1990). Z. Kabza and Ya. Pospolita, Izmer. Tekh., No. 3, 30 (1993). B. E. Launder and D. P. Spadling, Comp. Meth. Appl. Mech. Eng., 3, 269 (1974). W. Rodi, Proc. Ecole d'Ete d'Analyse Numerique (1982). S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publ. Corp., N. Y. (1980). T. Kritz, Instrum. Autom., 28, 1912 (1955).

451