=Paper=
{{Paper
|id=Vol-1452/paper17
|storemode=property
|title=Accuracy Analysis of Estimation of 2-D Flow Profile in Conduits by Results of Multipath Flow Measurements
|pdfUrl=https://ceur-ws.org/Vol-1452/paper17.pdf
|volume=Vol-1452
|dblpUrl=https://dblp.org/rec/conf/aist/RonkinK15
}}
==Accuracy Analysis of Estimation of 2-D Flow Profile in Conduits by Results of Multipath Flow Measurements==
https://ceur-ws.org/Vol-1452/paper17.pdf
Accuracy Analysis of Estimation of 2–D Flow
Profile in Conduits by Results of Multipath
Flow Measurements
Mikhail Ronkin and Aleksey Kalmykov
Ural Federal University, pr. Mira, 19, Yekaterinburg, 620002,
Russian Federation
z sm@mail.ru
Abstract. The paper presents a model of distorted velocity distribution
of a flow in conduit, which passes through its bent section. The proposed
model is properly analyzed. An algorithm for recovering the pipe profile
by multipath ultrasonic measurements in the presence of a priori infor-
mation is proposed. Numerical simulations with the proposed algorithm
are performed as well.
Keywords: flow measurements, flow profile, multipath flowmeter
1 Introduction
In practice, it is important to know the set of values of the stream head of a flow
in each section of the pipeline. The flow head is one of the main characteristics
of the conduit. When the flow passes through a bent section (such as elbow or
valve, etc.), its head gets lost. In practice, losses of the head are determined by
experiments [1]. The loss of head is connected with the flow profile distortion.
There is a symmetric flow profile in a long straight section of pipeline (Fig. 1a,b).
However, when flow passes a bent section (such as elbow, valve, etc.), the flow
profile, i.e., the distribution of velocity in the cross section of the pipe, becomes
distorted (Fig. 1 c,d). Thus, the profile of flow characterizes a bent section of
the pipe [1].
One way to determine the flow profile is to recover it by ultrasonic multi-
path flow measurements. The ultrasonic technology allows measurement of the
velocity distribution in a number of planes between two transducers/receivers of
acoustic waves as it is shown in Fig. 2a. The distortion of a flow profile can be
recovered by approximation of such measured values in a sufficient number of
planes [2].
The task of ultrasonic measurement of the flow rate is the radar task of
measurement of propagation time in an investigated media. The emitted wave
accelerates when moving in the direction of a flow and slows down in the opposite
direction (Fig. 2b). The velocity of a flow is calculated as a difference of measured
times of propagation:
L L
t12 = , t21 = , (1)
c + vl cos α c − vl cos α
143
∆tc2 ∆tc2
vl = , vz = tan α, (2)
2L cos α 2L
where t12 is the time of propagation in the flow direction; t21 is the time of
propagation in the opposite direction; c is the speed of acoustic wave; α is the
angle between direction of the flow and direction of the wave propagation; L is
the length of the wave path in the investigated media; vl is the projection of the
flow velocity on the wave path; vz is the velocity in z –direction of the flow [3].
The velocity profile is always distributed not uniformly through the cross–
section area of a pipe. This means that velocity in the central region is usually
higher than near the wall. The measured value of velocity vl is the average of
this distribution in the direction of the path. Measured value usually differs from
average velocity through the whole cross section vc :
Z L Z
vl = v(l)dl, Q = v(x, y)dS, (3)
0 s
Q
k= , (4)
πR2 vl
where v(l) is the distribution of velocity in the direction of the wave propagation;
Q is the flow rate; v(x, y) is the distribution of velocity through the whole cross–
section of the conduit; R is the radius of the pipe; S is the cross–section area of
the pipe; k is the meter factor, which is connected with the measured and actual
flow rate [3].
Equation (4) is valid for any symmetric flow. However, in the case of asym-
metric propagation of the flow wave patch, the flow rate would be calculated
with error. The type of function v(x, y) for asymmetric flow does not have gen-
eral analytical expression. The function of flow distribution strongly depends on
the conduit configuration, and its characteristics (such as material, roughness,
temperature of the controlled media, etc.). The error of the calculated flow rate
for a single path flow meter can achieve a 10% and be even larger [3].
The meter factor can be calibrated in laboratory conditions. Such calibration
is generally performed on a long straight section of a conduit where a symmetry
flow profile takes place [3]. In the case of multipath flow meter, all asymmetric
features are assumed to be present for calibration of a symmetric flow. In -
practice, it is necessary to have more than 4 paths to consider the meter factor as
1 with error less than 0.1% [4, 5]. However, the solution of the task of recovering
a flow profile with such accurate flow measurements is an open issue.
For some types of flow profiles, the analytical expression has been obtained by
Salami [6]. Based on his works, some authors proposed analysis of configuration
of the multipath measurements and suggested some interpretations of analyti-
cal profiles [4–8]. The solution of the profile recovering problem by ultrasonic
flow measurements was only proposed on the basis of ultrasonic tomography
or by Abel’s integration transform, which doesn’t take into account asymmetry
features of flow [2, 9].
144
In this work, a technique is suggested for recovering the flow profiles using the
multipath measurements and a priori information about discretionary function,
which can be determined for each type of a bent section of the conduit.
Fig. 1. Flow profiles for a) m=2, a=0; b) m=5, a=0; ) m=5, a=0.3; d) m=5, a=0.7
2 Model of flow profile
The model of flow profile, which have been proposed by authors [6, 7] can be
considered as a particular case of a flow profile that has passed through a conduit
section, whose specified characteristic determines the distortion of the velocity
distribution. The proposed model and generalized approach can be analyzed
regarding to the accuracy of the profile recovering.
In general, the flow rate could be calculated as an integral of velocity distri-
bution over the cross-section area as
Z Z "Z √ 2 2
#
x2 =R y2 = (R −x )
Q= vz (x, y)dS = √
vz (x, y)dy dx, (5)
s(z) x1 =−R y1 =− (R2 −x2 )
Thus Z R Z R
c2 tan(α)
Q= vz (x)dx = ∆t(x)dx, (6)
−R 2 −R
where ∆t(x) is the distribution of difference values of measured propagation time
in the flow direction and in the opposite one versus the coordinate. In theory
of numerical integration, it is proposed to substitute the integral by sum with
specified weight, which may be chosen, for instance, by criteria of optimal distri-
bution of nodes locations and corresponding coefficients (the Gauss quadrature
145
Fig. 2. Schematic draw picture a) multipath flowmeter and b) the center plane cut, ηi
is the ith plane
method). The solution is presented in Table 1.
Z Z 1 n
X
x R
ξ= ∆(x)dx → R ∆(ξ)dξ = R λi ∆t(ξ), (7)
R −R −1 i
where ξ is the normalized coordinate of the measurement plane, λi is the coef-
ficient of weight function (determined by solution of the Gauss quadrature task
for number of measurements); ∆t(ξi ), i = 1, 2 . . . n are differences of propagation
times in planes with coordinate ξi ; n is the number of planes. In the case of a
Table 1. Solution of Gauss quadrature task [10]
n=2 n=3 n=4 n=5
λi ± 0,5773 0 ± 0,7746 ± 0.3400 ± 0.8611 0 ± 0.5385 ± 0.9062
ξi 1 0.8889 0.5556 0, 6521 0,3479 0.5689 0.2369 0.4786
multipath flow meter, the flow rate can be calculated in accordance with (6) as
n n
c2 R tan(α) X X
Q=k λi ∆t(ξi ) = k λi vz (ξi ), (8)
2 i i=1
where the meter factor k can be considered as 1 if there are more than 4 measured
planes [4]. There are many profiles proposed by Salami [6], which have been
verified by experiments. For instance, authors [2] noted that profile of flow that
passes through a single elbow could be expressed as follows:
vz (r, ϕ) π
= sin (1 − r)1/m + α sin π(1 − r)1/2 exp(−0.2ϕ) sin(ϕ), (9)
v0 2
146
where vz (r, ϕ) is the velocity distribution in cylindrical coordinates; r is the
radius; ϕ is the angle; v0 is the velocity of flow at the center of cross-section
(vz (0, 0) = v0 ); α is the velocity of flow at the center of cross-section; m is
the coefficient of symmetric flow profile, which characterizes the flow profile
depending on velocity if asymmetric coefficient is equal to zero. In the original
work [6], the author proposed values m = 5, α = 0.3. The first part of the
right side of equation (4) corresponds to the symmetric part of the flow, and the
second part corresponds to asymmetry distortion with coefficient α [6]. Without
asymmetry distortion, the flow rate is calculated according to the equations
(8)-(9) as
Z π Z 2π Z R π
1 1
Q= sin (1 − r) m
dS = sin (1 − r) m rdrdϕ, (10)
S 2 0 −R 2
n
X
Q = kR λi vz (ξi ) . (11)
i=1
For symmetric model of flow (10, 11) calibration m = f (k) = f (Q/v0 ) could
be performed. Normalized profiles are shown in Fig. 1 a,b. It can be noted
that profile with m=2 corresponds to the laminar flow mode, and cannot be
considered in model. Profile with m=5 corresponds to the turbulent mode [2].
It is well known that the profile of a flow changes toward symmetry distribution
with distance from a section of the conduit, which caused distortion [3]. So, it
may be supposed that the influence of the second part of equation (9) decreases as
asymmetric coefficient tends to zero with increasing distance from the distorted
section of the conduit. Based on the condition of constant flow (Q(z) = const)
it should be noted that m is also changing with distance from the section of the
conduit that caused the distortion.
It may be proposed that type of the second part of the right side of equation
(9) depends on the type of section of the conduit, which caused distortion. Such
expressions can be defined theoretically or from experiments. In this work, type
of the profile (9), which corresponds to signal elbow [2], is considered. Profiles of
flow for m = 5, α = 0.3 and α = 0.7 are shown in Fig. 2c,d. It has been mentioned
above that in each measurement plane, the value of velocity is the integral over
the direction of acoustic wave propagation. The normalized measured velocity
in the plane with coordinate xi corresponds to equation (9) can be expressed as
vz (xi )
=
v0
Z +A 1/m 1/2
π 2 2 1/2 2 2 1/2
= sin (1 − (xi + y ) + α sin π 1 − (x + y ) ×
−A 2
y y
× exp(−0.2 arctan sin arctan dy, (12)
xi xi
147
p
where A = R2 − x2i . Denote:
Z +A 1/m
π
F1 (xi , m) = sin (1 − (x2i + y 2 )1/2 dy, (13)
−A 2
1/2
F2 (xi ) = α sin π 1 − (x2 + y 2 )1/2 ×
y y
× exp(−0.2 arctan sin arctan dy. (14)
xi xi
Consequently,
vz (xi )
=F1 (xi , m) +αF2 (xi ) , (15)
v0
where F1 (xi , m) is the function, which characterizes symmetric part of flow,
F2 (xi ) is the function, which characterizes an asymmetric part of flow when it
passes through a bent section. Expression of the flow rate corresponding to (15)
is
Z R Z R n
X
Q= vz (x)dx = v0 [F1 (xi , m) + αF2 (xi )] dx ≈ kR λi vz (ξi ). (16)
−R −R i=1
In accordance with (15), the flow rate can be expressed as
n
X
Q ≈ v0 R λi [F1 (ξi , m) + αF2 (ξi )] . (17)
i
In equation (17), the meter factor is not used because since it is included in the
calibration dependence for F1 (xi , m). From equation (17) in accordance with
(11) and (16), the symmetric flow rate normalized on v0 can be expressed as
" n n
#
Qc X X
=R k λi vz (ξi ) − αv0 λi F2 (ξi 2) = f −1 (m), (18)
v0 i=1 i=1
where f −1 (m) = Qc /v0 denotes the calibration relation in coordinates f (Qc /v0 ).
Such calibration can be implemented because of symmetry of function F1 (xi , m)
and symmetry of profile Qc . Hence, if we know the flow rate under symmetric
conditions and radius of the pipe, we can deduce v0 .
There are three unknown parameters in equation (17): v0 is the value of
velocity at the center of the pipe cross-section; m is the coefficient of symmet-
ric part of flow; α is the asymmetric coefficient. Here, it is assumed that type
of function F2 (xi ) is known as a function of distorted section of the conduit.
Relation m = f (F1 (x, m)) = f (Q/vc ) assumed to be known from preliminary
calibrations on a long straight pipeline. The determination of parameters named
above requires solution of the system of equations for each parameter. The po-
sition of planes (xi ) is supposed to be chosen here in accordance with Table 1
148
and from symmetry of F1 (xi , m). The solution of the system mentioned above
has the following expression:
α = [(vz (x3 ) − vz (x1 )) F1 (x1 , m)]/[F2 (x2 )vz (x1 ) − vz (x3 )F2 (x1 )],
v0 = [F2 (x3 )vz (x1 ) − vz (x3 )F2 (x1 )]/[F1 (x1 , m) (F2 (x1 ) − F2 (x3 ))],
(19)
P
n h Pn Pn i
λi F1 (xi , m) = k λi vz (xi ) − αv0 λi F2 (xi ) /v0 = f −1 (m).
i=1 i=1 i=1
Here, the first and the second equations are calculated depending on F1 and
hence on m. However, the multiplier αv0 may be calculated without the third
equation:
vz (x3 ) − vz (x1 )
αv0 = . (20)
F2 (x1 ) − F2 (x3 )
In the third equation, the meter factor can be considered as 1 if more than 4
measurement planes are used. Consequently, in the third equation m is deter-
mined from the calibration relation. The solution of system (19) allows one to
calculate velocity distribution in accordance with (15) for distorted flow profile
in cross-section of the conduit.
3 Results of Modeling
In the Matlab software, simulation of the algorithm described above is carried
out for function of distorted profile (9). The first stage of modeling requires the
number of measurement planes and their positions. In our study, 4 planes (k ≈ 1
in this case) have been chosen according to the Table 1. For the chosen model
of distortion relation, functions m = f (Q/v0 ) have been calibrated, shown on
Fig. 3.
x1 = 0.34R, x2 = 0.86R, x3 = −0.34R, x4 = −0.86R,
λ1 = 0.652, λ2 = 0.347, λ3 = 0.652, λ4 = 0.347.
In Figure 4 function F2 (x) for the chosen type of distortion section of the conduit
is shown. In the model of profile (9), coefficients were selected as m = 5, α =
0.3, v0 = 1, along with the measured values of velocities:
vz (x1 ) = 1.9; vz (x2) = 1; vz (x3) = 1.4; vz (x4 ) = 0.5.
The calculated value αv0 accordingly to (20) αv0 = 0.3. The calculated value of
the flow rate (considering that k = 1) is Q = 2.7307.
The relation for m (Fig.3) has been found from the third equation of system
(9). For calibration, the value m = 2.83 has been determined. The difference
of the measured and the preliminary chosen m value may be explained by not
sufficient accuracy of the assumption that k ≈ 1 or with influence of F1 on F2 ,
which is not taken into account in the proposed model.
149
Fig. 3. - Relation m = f (Q/v0 )
Fig. 4. Function F2 (x) for the chosen type of distortion section of the conduit
150
The calculated values of F1 (m, xi ), in accordance with (13) and the deter-
mined m, are
F1 (m, x1 ) = F1 (m, x3 ) = 1.51; F1 (m, x2 ) = F1 (m, x4 ) = 1.1.
From the estimated value of m F1 (m, xi ) and by calculation of equations (20)
α = 0.26, v0 = 1.16. The error of determination α, v0 obviously connected with
accuracy of estimation m.
The error of recovering the profile of a flow (9) with the calculated values of
parameters and set values estimated as
Qteor
1− 100% = 6.7%, (21)
Qcalc
where Qteor is the flow rate calculated form the selected values m, α, v0 , and Qcalc
is the flow rate calculated from estimated values of parameters. The relation of
error versus the asymmetry coefficient and m = 5, 7, 10 shown in Fig. 5. The
Fig. 5. Relation of error depending on asymmetry coefficient with m = 5, 7, 10 and
v0 = 1
accuracy analysis shows that the error dependence has a constant part that
corresponds to symmetry flow distribution m = f (Q/v0 ). Such constant value of
the error can be decreased by correction of calibration relationship m = f (Q/v0 ).
The result with such calibration for each value m is shown in Fig. 6. In the whole,
such behavior of the error can be caused by the insufficient accuracy of the model
assumptions. Particularly, it may be supposed that the asymmetry part of F2 (x)
in (13) is influenced by the symmetry part of F1 (x, m).
The general algorithm of flow profile recovering on the basis of multipath
ultrasonic flow rate measurements and a priori information about distorting
function of the conduit section has the following stages:
151
Fig. 6. Relation of error with corrected m = f (Q/v0 ) depending on asymmetry coeffi-
cient with m = 5, 7, 10 and v0 = 1.
1. selection of the function F2 (x, m) type;
2. carrying out calibration for symmetric flow m = F (Q);
3. measurements of flow velocities in planes for each xi and determination the
flow rate by equation (16);
4. calculation value αv0 from (20);
5. determination of m in agree with (18) and calibration of the relationship;
6. evaluation of correction relation m = F (Q) for decreasing symmetric flow
error (α = 0);
7. calculation of values F1 (m, xi ), α and v0 by equations (19);
8. calculation of velocity distribution and flow profile.
4 Conclusion
The new model of behavior of velocity distribution in conduit and the algorithm
for flow profile recovering based on it are proposed in the paper. The model
generalizes analytical expressions for distorted flow profiles in the conduit, which
were proposed by Salami. In contrast to [6], here, some coefficients of the model
has been generalized and their physical interpretation provided.
The algorithm for profile recovering on the basis of multipath ultrasonic
measurements and a priori information about distorting section of conduit is
proposed. The model can be used for any type of profile with theoretical or
experimental description.
The numerical simulation for generalized expression of the function for profile
after single elbow has been carried out [2]. The results of simulation have been
presented in the paper. The flow profile after single elbow, measured by ultrasonic
in 4 planes is considered. The error is estimated as the relation between the flow
152
rate calculated with theoretical set of parameters and parameters that were
determined by the proposed algorithm. The obtained accuracy gives less than
1% error.
The reached that the model can be considered as reliable. In further devel-
opments of the model, the influence of symmetry part of flow on asymmetry
part will be investigated. Moreover, investigation of number of measurements
and orientation of planes influence on accuracy should be done.
References
1. Idelchik, I.E.: Handbook of hydraulic resistance. Mashinostroenie, Moscow (1992)
(in Russian)
2. Rychagov, M., Tereshenko, S.: Multipath flowrate measurements of symmetric and
asymmetric flows. Invers Problems, 16, 495–504 (2000)
3. Kremlevskij, P.P.: Flowmeters and counters of the amount of substance. Handbook.
Politeh, Saint-Peterburg (2004)
4. Brown, G.J., Augenstein, D.R., and Cousins T.: An 8-patch ultrasonic master
meter for oil custody transfer. XVIII IMEKO WORLD CONGRESS, 17–22 (2006)
5. Rychagov, M.N.: Ultrazvukovye izmereniya potokov v mnogoploskostnyh izmeri-
tel’nyh mod-uljah. Akusticheskij zhurnal 44 (6), 829–836 (1998) (in Russian)
6. Salami, L.A.: Application of a computer to asymmetric flow measurement in cir-
cular pipes. Trans. Inst. MC. 6(4), 197–206 (1984)
7. Moore, P., Brown and J., Stimpsom, B.: Ultrasonic transit–time flowmeters mod-
eled with theoretical velocity profiles: methodology. Meas. Sci. Technol., 11, 1802–
1811 (2000)
8. Zheng, D., Zhao, D. and Mei, J.: Improved numerical investigation method for
flowrate of ultrasonic flowmeter based on Gauss quadrature for non-ideal flow
fields. Flow measurement and Instrumentation. 41, 28-25 (2000)
9. Lui, J., Wang, B., Cui, Y. and Wang, H.: Ultrasonic tomographic velocimeter for
visualization of axial flow fields in pipes. Flow measurement and instrumentation,
41, 57-66 (2000)
10. Krylov, V.I.: An approximate calculation of integrals. Nauka, Moscow (1967)
153