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 information is proposed. Numerical simulations with the proposed algorithm are performed as well.
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 [
One way to determine the flow profile is to recover it by ultrasonic
multipath 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 [
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: t12 =
L c + vl cos α , t21 =
L c − vl cos α
(1)
tan α,
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 [
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: (3) (4) vl =
Z L 0 v(l)dl, Q =
v(x, y)dS, k =
Q πR2vl
Z
s
,
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 [
Equation (4) is valid for any symmetric flow. However, in the case of
asymmetric 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
general 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 [
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 [
For some types of flow profiles, the analytical expression has been obtained by
Salami [
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.
The model of flow profile, which have been proposed by authors [
In general, the flow rate could be calculated as an integral of velocity distribution over the cross-section area as
Z
s(z) Q = vz(x, y)dS =
Z x2=R "Z y2=√(R2−x2) x1=−R y1=−√(R2−x2)
# vz(x, y)dy dx, Thus
Z R
−R Q = vz(x)dx = c2 tan(α) Z R 2 −R Δt(x)dx, where Δt(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 distribution of nodes locations and corresponding coefficients (the Gauss quadrature (5) (6) where ξ is the normalized coordinate of the measurement plane, λi is the coefficient of weight function (determined by solution of the Gauss quadrature task for number of measurements); Δt(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 (7) (8) multipath flow meter, the flow rate can be calculated in accordance with (6) as Q = k c2R tan(α) Xn λiΔt(ξi) = k X λivz(ξi),
n
2
i
i=1
where the meter factor k can be considered as 1 if there are more than 4 measured
planes [
sin Z
S
−R
sin
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 [
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 [
= y xi
Z +A −A sin
+ α sin π 1 − (x2 + y2)1/2 1/2 × exp(−0.2 arctan 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 = R k X λivz (ξi) − αv0 X λiF2(ξi2) = f −1(m), 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 symmetric 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 position of planes (xi) is supposed to be chosen here in accordance with Table 1 (13) (14) (15) (17) (18) 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))], Pn λiF1(xi, m) = hk Pn λivz(xi) − αv0 Pn λiF2(xi)i/v0 = f −1(m). i=1 i=1 i=1 (19) (20) 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: 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 determined 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
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.
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 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 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:
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 [
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 [
The reached that the model can be considered as reliable. In further developments 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.