Title:
Static and dynamic characteristics of a hydraulic Wheatstone bridge
mass flowmeter
Authors:
a
A. Svetea,*, J. Kutina, I. Bajsića
Laboratory of Measurements in Process Engineering, Faculty of Mechanical Engineering,
University of Ljubljana, SI-1000 Ljubljana, Slovenia
Andrej Svete, B.Sc., researcher (*Corresponding author):
T: +386-1-4771-131
F: +386-1-4771-118
E: andrej.svete@fs.uni-lj.si
Jože Kutin, Assist. Prof.,Ph.D.:
E: joze.kutin@fs.uni-lj.si
Ivan Bajsić, Assoc. Prof., Ph.D.:
E: ivan.bajsic@fs.uni-lj.si
Abstract
The hydraulic Wheatstone measuring bridge is a potential linear-flowmeter solution for direct
mass flow measurements. In a concrete example of the flowmeter, four matched orifices are
used as the local flow restrictors in the bridge network. This paper presents an analysis of the
nonlinearity of the static measuring characteristics of the hydraulic measuring bridge, which
results from the impact of the flow rate on the local pressure loss coefficient of the orifices.
The results of the experimental analysis were confirmed with solutions of the
physicalmathematical model of the flowmeter. Another objective of this study was to
evaluate the suitability of this flowmeter for pulsating flow measurements. For this reason a
dynamic model of the flowmeter was built by upgrading the steady-state flow model with the
effects of fluid compressibility and inertia. Since the flowmeter's characteristic is linear it was
found to be unaffected by the square root errors, which is the case with standard pressure
differential flowmeters. On the other hand, the flowmeter’s frequency characteristic exhibited
a typical resonance, the value of which depends on the fluid and the dimensional properties of
the bridge.
Keywords: Hydraulic Wheatstone bridge; Mass flowmeter; Orifice pressure loss coefficient;
Linear measuring characteristic; Dynamic error; Pulsating flow
Nomenclature
A
Cross-sectional area of the pipe
C
Fluid capacitance
c
Speed of sound
D
Pipe inner diameter
d
Orifice inner diameter
ea
Absolute error of the measuring characteristic
eSRE
Square root error
f
Pulsation frequency
k
Coverage factor
K
Orifice pressure loss coefficient
KWB
Sensitivity of the flowmeter
l
Length
L
Fluid inertia
qm
Measured mass flow rate
qvr
Volumetric recirculating flow
P1
Time-mean inlet pressure
p
Pressure
∆P
Time-mean differential pressure
∆p
Differential pressure
∆p13
Differential pressure across the bridge
∆p24
Differential pressure across the connecting conduit
u
Standard uncertainty
V
Volume of fluid
Greek Letters

Bulk modulus
ε
Pressure pulsation intensity

Dimensionless orifice pressure loss coefficient
ρ
Fluid density

Time constant
Subscripts
1, 2, 3, 4
Number of branch conduit
A
Uncertainty of type A
B
Uncertainty of type B
id
Ideal
r
Recirculating
c
Cut-off
1 Introduction
The hydraulic Wheatstone bridge is one of the earliest methods used for direct mass flow
measurements (see, e.g., the patents [1, 2, 3]). It has also been successfully developed as a
commercial device, which is primarily used in fuel flow measurement applications. It is
claimed to give a true mass flow rate, unaffected by density, temperature or viscosity changes.
Fig. 1 shows a typical configuration, which consists of four, matched flow restrictors (e.g.,
orifices) arranged in a bridge network and a pump that provides a constant recirculating
volumetric flow qvr through a connecting conduit. The inlet, measured mass flow qm affects
the flow and the pressure distributions in the bridge. The differential pressure across the
flowmeter p13 or across the connecting conduit p24 is found to be proportional to the mass
flow rate qm if qm  ρqvr or qm  ρqvr , respectively [4].
There are not many accessible research studies that investigate the performance of hydraulic,
Wheatstone bridge mass flowmeters. However, the ideal flowmeter’s characteristics, which
are true for flow restrictors with equal and constant pressure loss coefficients, were derived in
[4, 5]. Reference [6] presents a dynamic model of the Wheatstone bridge, which considers the
effects of fluid capacity and inertia. The frequency characteristics of the flowmeter were
experimentally studied in [7]. With a similar configuration to a Wheatstone bridge mass
flowmeter, the differential viscometer, which offers several advantages for measuring the
viscometry of polymer solutions, has also been developed [8, 9].
The purpose of this paper is to introduce new findings on the static and dynamic properties of
the Wheatstone bridge mass flowmeter. In the first part (Sections 2-4) the paper presents
experimental and theoretical analyses of the flowmeter’s nonlinearity due to variations in the
orifice’s pressure loss coefficients. Section 2 presents a physicalmathematical model of the
flowmeter for steady-flow conditions, and Section 3 presents a measuring system for the
experimental investigations of static measuring characteristics. In Section 4 the results of the
theoretical and experimental analyses are discussed and compared. Some initial experimental
work on the static characteristics was previously presented by the authors of this paper in the
Slovene-language journal [10]. The present work introduces an upgraded measuring system
(in a view of stability of the recirculating flow, as discussed in Section 4) and an extended
analysis of the experimental results.
Furthermore, Section 5 presents a discussion of the dynamic characteristics of the hydraulic
Wheatstone bridge mass flowmeter. The dynamic model is built on the basis of reference [6],
but with a (more correct) nonlinear, square-root dependence between the flow rate and the
pressure difference across the flow restrictors. This upgraded model was employed to conduct
an investigation of the effects of pulsating flow on the flowmeter’s performance.
2 Physicalmathematical steady-state flow model
The main assumptions of the physicalmathematical model of the hydraulic Wheatstone
bridge in this section are that the fluid flow in the bridge conduits is steady and the fluid
compressibility can be neglected. In addition, the local pressure losses across the flow
restrictors are considered to be much greater than other local and line pressure losses in the
bridge conduits. The inner diameter of the bridge conduits is assumed to be constant and the
hydraulic bridge is placed in the horizontal plane.
The model is formed from a system of continuity and energy equations. The equations below
are written for the proposed direction of fluid flow, which is indicated in Fig. 1 by the full
arrows. The law of the conservation of mass is written for all four nodes (1, 2, 3 and 4) of the
bridge network (only three of these equations are independent):
qm  qm1  qm 4 , qm1  ρqvr  qm 2 , qm 2  qm3  qm , qm 4  qm3  ρqvr ,
(1)
where qm is the measured mass flow rate, qvr is the volumetric recirculating flow,  is the fluid
density and qmi (i = from 1 to 4) is the mass flow rate in a particular branch conduit. The law
of the conservation of energy is written in the form of non-recoverable pressure losses of the
turbulent flow in a particular branch conduit of the hydraulic bridge, which are defined by the
pressure loss coefficients Ki of the flow restrictors:
 K q 2 / ρ for qmi  0
,
Δpi   i mi2
 K i qmi / ρ for qmi  0
(2)
where Δpi (for the proposed direction of the fluid flow) equals:
Δp1  p1  p2 , Δp2  p2  p3 , Δp3  p4  p3 , Δp4  p1  p4 .
(3)
In general, the pressure loss coefficients can depend on the flow rate through the flow
restrictors, Ki = Ki(qmi).
Using the defined characteristics of the flow restrictors, the fluid density and the boundary
conditions (e.g., the known inlet mass flow qm, the volumetric recirculating flow qvr and the
outlet pressure p3), it is possible to iteratively solve the nonlinear system of Eqs. (1-3) to
calculate the unknown mass flows in the branch conduits and the unknown pressures in the
bridge nodes. The differential pressures across the bridge ∆p13 = p1 – p3 and across the
connecting conduit ∆p24 = p2 – p4, represent the output signals of the hydraulic, Wheatstone
bridge mass flowmeter.
2.1 Analytical results for the ideal flowmeter’s characteristics
In the case of an ideal flowmeter configuration the flow restrictors are considered identical
and their pressure loss coefficients are independent of the flow rate, K = K1 = K2 = K3 = K4.
Following these assumptions the analytical solution for the output differential pressures can
be derived as [4]:
Kqvr qm
for qm  ρqvr


,
Δp13   K 2
2 2
q

ρ
q
for
q

ρ
q


m
vr
m
vr
 2ρ

K 2
2 2
  qm  ρ qvr  for qm  ρqvr
.
Δp24   2ρ

for qm  ρqvr
Kqvr qm

(4)
If the flowmeter uses the differential pressure across the bridge ∆p13 when qm  ρqvr and the
differential pressure across the connecting conduit ∆p24 when qm  ρqvr , the ideal flowmeter’s
characteristic can be written as:
Δpid  Kqvr qm or
Δpid
 Kqm .
qvr
(5)
Eq. (5) shows the linear dependence of the differential output pressure on the measured mass
flow rate if ideal, identical flow restrictors are placed in the hydraulic bridge and if the
volumetric recirculating flow through the connecting conduit remains constant.
Using these assumptions the analytical solution for the mass flow rate in a particular branch
conduit can be written as:
qm1 
qm  ρqvr
q  ρqvr
q  ρqvr
q  ρqvr
, qm 4  m
.
, qm 2  m
, qm3  m
2
2
2
2
(6)
It is evident that the flow conditions, indicated in Fig. 1 with full arrows, occur when the
measured flow is greater than the recirculating flow, qm  ρqvr . In the case when the measured
flow is smaller than the recirculating flow, qm  ρqvr , the flow conditions in the hydraulic
bridge indicated with dashed arrows in Fig. 1 occur. When the measured flow equals the
recirculating flow, qm = qvr, the bridge is balanced and the fluid in the branch conduits 1 and
3 is at rest.
3 Measuring control system
The experimental study of the hydraulic Wheatstone bridge mass flowmeter was carried out in
the water flow rig of the Laboratory of Measurements in Process Engineering at the Faculty of
Mechanical Engineering, University of Ljubljana. The scheme of the measuring control
system is shown in Fig. 2.
The Wheatstone bridge is made of branch conduits with inner diameters of D = 24 mm. Four
precisely matched orifices with inner diameters d = 6 mm are used as the flow restrictors. Fig.
3 shows the variation of the dimensionless pressure loss coefficient ξi of the orifice, which
was determined in preliminary experiments for different combinations of orientations of the
orifice and its fixation elements. The full line in Fig. 3 approximates the experimental results
in the form:
ξ i  720,31 
3,9591
.
qmi
(7)
The pressure loss coefficient Ki employed in Section 2 can be related to the dimensionless
pressure loss coefficient ξi by
Ki 
8ξ i
,
π2 D4
(8)
which is subsequently used as an input parameter for the physicalmathematical model of the
flowmeter.
In this experimental model of the Wheatstone bridge, the centrifugal recirculating pump
(Grundfos, CRN2-110) in connection with the electromagnetic flowmeter (Foxboro, IMT 25,
measuring range 0 to 4560 dm3/h, accuracy 0.3 % of reading or 1 dm3/h) and the computercontrol system was employed to provide a constant recirculating flow. (We should stress that
the experimental system in this study, with the additional flowmeter for measuring the
recirculating flow, does not present a typical configuration of the commercial designs. The
positive displacement pumps are usually employed in the connecting conduit to provide
nearly constant volumetric flow.) Fig. 4 shows an example of the time signal from the
electromagnetic flowmeter when the control system was set to regulate 750 dm3/h. Some
observable noise in the regulated recirculating flow results from the limited resolution of the
pump’s frequency regulator (10 bit) and the noise in the electromagnetic flowmeter output.
During each measurement a 10-seconds averaging was employed.
The pressure difference across the bridge ∆p13 is measured with a differential pressure
transmitter (Endress+Hauser, PMD 70, measuring range 0 to 3 bar, accuracy 0.1 % of
measuring range) and the pressure difference across the connecting conduit ∆p24 is measured
with two static pressure transmitters (Foxboro Eckardt, BIA 408, measuring range 0 to 10 bar
and 0 to 5 bar, accuracy 0.1 % of measuring range). The flow of water through the flowmeter
is provided by a centrifugal pump (Grundfos, CRN4-120). The reference value of the mass
flow rate is measured with a Coriolis mass flowmeter (Foxboro, CFS 10, measuring range 0
to 5400 kg/h, accuracy 0.2 % of reading or 1 kg/h). The water temperature is also measured
and used to calculate the water density. All the meters have electrical output signals, which
are connected to the data-acquisition (DAQ) board (National Instruments, DAQPad-6020E).
The frequencies of both pumps are controlled by the electrical signals from this DAQ board.
The controller of the recirculating flow and the user interface are realized in the LabVIEW
programming environment.
4 Experimental and theoretical results
The experimental study of the presented hydraulic Wheatstone bridge was carried out at four
different volumetric recirculating flows qvr of 750, 1000, 1250 and 1500 dm3/h (approx. 0.21,
0.28, 0.35 and 0.42 dm3/s), in the range of measured flow rates qm of 0 to 0.84 kg/s. Fig. 5
shows the measured values of the differential pressures versus the relative measured mass
flow rate at the recirculating flow of 750 dm3/h. The experimental results are in agreement
with the theoretical findings in Section 2. It is evident that when qm  ρqvr the differential
pressure ∆p13 varies linearly with qm , and when qm  ρqvr the differential pressure ∆p24 varies
linearly with qm . The differential pressures also follow the form of the analytical solutions (4)
in other segments of the measured characteristics.
Fig. 6 gives a closer view of the linear part of the static measuring characteristic for the
recirculating flow of 750 dm3/h. The deviation from the ideal linear measuring characteristic
is the difference between the actual measuring characteristic and its linear approximation. In
Fig. 6 the experimental results are compared with the solution of the physicalmathematical
model in Section 2 (with the considered flow variations of Ki according to Eqs. (7) and (8)).
Both results show similar trends and values of the deviations. The influence of the variation of
the orifice pressure loss coefficient is seen as a certain degree of nonlinearity in the measuring
characteristic. A typical change occurs in the transition through the point when the measured
mass flow equals the recirculating mass flow rate.
The scatter that is evident in the experimental results probably depends on the repeatability of
the regulated recirculating flow, the repeatability of the pressure transmitters and the
repeatability of the pressure losses through the orifices. Although 10-seconds averaging was
applied for each measurement it does not completely remove the repeatability effects. In
comparison with [10] the scatter in Fig. 6 is lower, which is certainly due to the higher
resolution of the recirculating pump’s frequency regulator (now it is 10 bit, before it was 6
bit).
We may define the sensitivity of the Wheatstone bridge as a slope coefficient of the linear
approximation of the dependence of ∆p/qvr on qm:
KWB 
Δp
,
qvr qm
(9)
which equals the orifice pressure loss coefficient in the case of the ideal flowmeter
configuration, KWB  K (see Eq. (5)). Table 1 shows the experimental and theoretical results
for the flowmeter sensitivity. The experimental results are slightly higher and vary more with
the recirculating flow than those obtained with the mathematical modeling. This can be partly
explained by the limitations of the physicalmathematical model, where only the local
pressure losses across the flow restrictors were considered, although there are also other
sources of pressure losses (e.g., the pressure losses in straight conduits and bends).
Furthermore, we should pay regard to the uncertainty of measurement of the flowmeter
constant. It is evaluated by combining the standard uncertainty u A  KWB  of the slope in the
regression line Δp / qvr  KWB qm and the standard uncertainty uB  KWB  associated with the
contributions of particular measurands,
u B  KWB 
 u  Δp    u  q vr    u  q m  
 
 
 
 ,
KWB
 Δp   qvr   qm 
2
2
2
(10)
where each term in Eq. (10) considers the accuracy of the sensor and the repeatability effects.
The relative standard uncertainties uB  KWB  / KWB are, for example, between 0.3 and 1.5 %
(with an average of about 0.5 %) for the recirculating flow of 750 dm3/h, and between 0.2 and
0.4 % (with an average of about 0.25 %) for the recirculating flow of 1500 dm3/h. The relative
uncertainties are higher for smaller differential pressures since the pressure transmitters have
the accuracy specified in % of the measuring range. Finally, the standard uncertainties,
u A  KWB  and an average of uB  KWB  , are combined and expanded with the coverage factor k
= 2. The expanded measurement uncertainties of KWB are about 1.1, 0.75, 0.65 and 0.55 % for
the recirculating flows of 750, 1000, 1250 and 1500 dm3/h, respectively. It is evident that the
differences between the experimental and theoretical results in Table 1 fall within the
estimated expanded uncertainties.
The last column of Table 1 gives the estimated values of the linearity of the flowmeter
characteristic obtained with the mathematical modeling. The linearity is expressed as the
relative value of the maximum deviation from the linear approximation compared to the
measuring range. The results show better linearity at higher recirculating flows.
5 Discussion on the flowmeter’s dynamic performance
In order to examine the dynamic behavior of the hydraulic Wheatstone measuring bridge the
physicalmathematical steady-state flow model from Section 2 was modified. Our upgraded
dynamic model of the flowmeter was derived from a dynamic model proposed in [6]. Instead
of a linear relation between the differential pressure and the flow rate through the flow
restrictor (as it was assumed in [6]), a more correct square-root relation is taken into account
here. As is shown schematically in Fig. 7, the model takes into account the fluid dynamics
modeled by the lumped fluid capacitances Ci added on the nodes 1, 2, 3 and 4 of the bridge
network and the lumped fluid inertia L5 associated with the flow in the bridge’s outlet duct.
The fluid capacitance C is introduced into the model as the relation between the volume flow
rate change Δqv and the pressure time derivative dp / dt in a rigid pipe segment [11]:
Δqv  C
dp
,
dt
(11)
where C depends on the bulk modulus β  c 2ρ (where c is the fluid speed of sound) and the
fluid volume V in the pipe segment:
C
V
V
 2 .
β cρ
(12)
The fluid inertia L is introduced into the model as the relation between the pressure change
Δp and the volume flow rate time derivative dqv / dt in the pipe segment [11]:
Δp  L
dqv
,
dt
(13)
where L depends on the length of the pipe section l and its internal cross-sectional area A:
L
ρl
.
A
(14)
Thus, the law of conservation of mass (1) can be rewritten in the modified form:
qm  qm1  qm 4  ρC1
dp1
dp
, ρqvr  qm1  qm 2  ρC2 2 ,
dt
dt
qm 2  qm3  qm5  ρC3
dp3
dp
, qm 4  ρqvr  qm3  ρC4 4 ,
dt
dt
(15)
and the inertia effect of the fluid in the outlet duct can be added to the energy equations (2):
p3  p5 
L5 dqm 5
.
ρ dt
(16)
In Eq. (15) the mass flow rates can be replaced with the pressure losses across the flow
restrictors by using Eq. (2). So the physicalmathematical model of the flowmeter under
discussion is generally fulfilled with five, first-order, nonlinear differential equations. The
equations are solved with the fourth-order Runge-Kutta fixed-step method for defined (in
general, time dependent) boundary conditions p1  t  , p5  t  and qv  t  , but for the initial
conditions we use solutions from the steady-state flow model in Section 2.
The below presented results of the modeling of the flowmeter's dynamic performance are
obtained on the basis of the following assumptions. At the inlet of the hydraulic bridge a
sinusoidal source of pulsations p1  t   P1 1  ε p1 sin  2πft   is assumed, where P1 is the timemean value, ε p1 is the pressure pulsation intensity and f is the pulsation frequency. At the
outlet of the hydraulic bridge a constant pressure value (reservoir) p5  t   P5 is assumed and
the recirculating pump is considered to be the source of the constant volumetric recirculating
flow qvr  t   Qvr . In addition to the above assumptions the orifice pressure loss coefficient K
is assumed to be independent of the flow rate. Referring to the actual configuration of the
Wheatstone bridge flowmeter discussed in the previous sections, the water capacitances Ci
are set to about 4.7  10-13 m3 Pa-1 (considered equal at all four nodes, approximate volume of
0.6 dm3, water density 1000 kg/m3 and speed of sound 1450 m/s) and the water inertia L5 is
set to about 1.7  107 Pa m-3 s2 (approximate outlet duct length of 10 m).
5.1 Time-mean value of the pulsating flow
In standard differential pressure devices (orifice plates, nozzles etc.), the nonlinear
relationship between the flow rate and the pressure difference, qm  Δp , is the possible
source of the so-called square root error in the indicated time-mean value of the flow rate
under pulsating flow conditions [12]. The square root error occurs when the averaging of the
measured differential pressure is done before calculating the flow rate, which can be, e.g., the
result of a slow frequency response of the differential pressure transmitter. The resulting
relative square root error can be written as:
eSRE 
Δp  t   Δp  t 
Δp  t 
.
(17)
The calculated results in Fig. 8 show its dependence on the pressure pulsation intensity ε Δp ,
when the differential pressure varies over time according to Δp  t   ΔP 1  ε Δp sin  2πft  
due to pulsating flow. It is evident that with the increase of the intensity of the pulsation ε Δp
from 0 to 1, the relative error reaches values up to about 11 %.
As might be expected, the simulations of the hydraulic Wheatstone bridge under pulsating
flow conditions show that the errors in the determination of the time-mean value of the
measured flow from the time-mean pressure differences were found to be practically zero.
Because of the linear measuring characteristic of the hydraulic bridge, the correct values of
time-mean flow rate can even be measured using differential pressure transmitters with slow
frequency responses.
5.2 Frequency characteristics
Next we look at how the reading of the hydraulic Wheatstone bridge follows the fluctuating
component of the measured pulsating flow. Simulations were performed with the flow rates
considered as qm  t   ρqvr . Fig. 9 presents the time variations of the fluctuating components
of the actual measured mass flow rate and the flowmeter reading (normalized on the actual
measured mass flow rate) for two frequencies of the pulsating flow. The flowmeter nearly
ideally measures the flow rate at f = 2 Hz (Fig. 9(a)), but its response at f = 20 Hz (Fig. 9(b))
shows clear amplitude and phase dynamic errors.
Fig. 10 further presents the normalized amplitude-frequency characteristic for pulsation
frequencies from 1 Hz to 100 Hz. The measuring system has a typical resonance that is at
about 30 Hz for water (full line). The simulations were also performed for another fluid (fuel
with ρ  750 kg/m3 and c  1300 m/s ), which has a smaller bulk modulus β  c 2ρ compared
with water and so results in a smaller resonance frequency (dashed line). If the flowmeter
configuration would have a smaller fluid volume, which defines the fluid capacitance and the
resulting compressibility effects, the resonance frequency would be higher.
We should stress that the so-far discussed dynamic characteristics of the hydraulic,
Wheatstone measuring bridge do not consider the dynamics of the pressure-difference
measurement system, which depend (at least) on the connecting tubing between the flow
system and the pressure sensors (see e.g. [13]) as well as on the dynamic properties of the
pressure sensor. The dotted line in Fig. 10 shows the flowmeter's amplitude-frequency
characteristic, when the pressure-measurement system is considered as a first-order dynamic
system with a time constant of 0.1 s (e.g., a low-pass filter in the pressure-sensor electronics
with a low cut-off frequency fc = 1.6 Hz). The trend of the calculated frequency characteristic
with the low-pass filter is similar to the experimental results shown in [7].
6 Conclusions
This paper discusses the static and dynamic characteristics of a hydraulic Wheatstone bridge,
which represents a potential solution for a direct mass flow meter with linear measuring
characteristics. The analytical results of the physicalmathematical model in steady-flow
conditions confirm the linear output of the differential pressure to the measured mass flow
rate if the identical flow restrictors (e.g. orifices) with constant pressure loss coefficients are
arranged in the bridge network and if the recirculating flow remains constant. The
experimental and the theoretical results in this paper show that the dependence of the pressure
loss coefficient on the flow rate through the flow restrictors brings a certain degree of
nonlinearity to the flowmeter’s characteristics. A typical deviation of the actual measuring
characteristic from its linear approximation is found in the transition through the point where
the measured flow equals the recirculating flow in the bridge. The experimental results show
good agreement with the theoretical results.
In order to investigate the flowmeter’s dynamic characteristics a dynamic model was also
developed. The dynamic model takes into account the fluid dynamic characteristics affected
by the lumped fluid capacitances of the volumes of the bridge segments and the inertia of the
fluid in the hydraulic bridge's outlet duct. The simulations of the Wheatstone bridge
flowmeter confirm that the square root errors (present in the measurements with standard
differential pressure devices) do not occur in the measured time-mean value of the pulsating
flow because of the linear measuring characteristic of the flowmeter, even if a differential
pressure transmitter with a slow frequency response is used. The simulation results also
indicate that the flowmeter nearly ideally follows the pulsating flow rate only up to some
defined values of the pulsation frequencies. The calculated frequency characteristic of the
hydraulic bridge shows a typical resonance frequency, which depends on the fluid properties
(the speed of sound and the density) and the internal volumes of the bridge network. The
resonance frequency is expected to become higher for nearly incompressible fluids and
smaller volumes. The dynamics of the differential pressure measurement system can also play
an important role in the flowmeter dynamics.
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13
Bajsić I, Kutin J, Žagar T. Response time of a pressure measurement system with a
connecting tube. Instrumentation Science & Technology 2007;35(4):399-409.
2
Flow
restrictor
K1
K2
qm1
qm2
qvr
Measured mass flow 1
qm
3
Recirculating
pump
Differential
pressure
transmitter
p24
qm
qm4
qm3
K4
K3
4
p13
Figure 1 Schematic view of the hydraulic Wheatstone bridge mass flowmeter.
K1
2
K2
Hydraulic Wheatstone bridge
Electromagnetic volume
flowmeter
p2
3
1
Pressure
transducer
Centrifugal
recirculating pump
4
K4
p4
p13
K3
Differential pressure
transmitter
OUT
PC
IN
DAQ Board
Reservoir
<
Temperature
sensor
Coriolis mass
flowmeter
Centrifugal pump
Figure 2 Schematic view of the measuring-control system.
Orifice pressure loss coefficient i
720
700
680
660
640
Experimental results
Approximation (7)
620
0.0
0.1
0.2
0.3
0.4
0.5
Mass flow through the orifice qmi , kg/s
Figure 3 Variation of the dimensionless orifice pressure loss coefficient with the mass flow
rate.
Volumetric recirculating flow qvr, dm3/h
770
765
760
755
750
745
740
735
730
0
5
10
15
20
25
Time t, s
Figure 4 Time variation of the regulated volumetric recirculating flow.
30
3.5
Differential pressure p13
Differential pressure p24
Differential pressure p, bar
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Ratio of the measured mass flow rate to the recirculating flow qm / qvr
Figure 5 Variation of the differential pressures with the relative measured mass flow rate (qvr
Deviation from linear measuring characteristic ea , bar
= 750 dm3/h).
0.02
Experimental results
Theoretical results
0.01
0.00
-0.01
-0.02
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Measured mass flow qm, kg/s
Figure 6 Deviation of the measuring characteristic from its linear approximation (qvr = 750
dm3/h).
C2
2
C1
p1(t)
K2
K1
qm1(t)
qm2(t)
qv0
1
L5
p5(t) 5
3
Obtočna
črpalka
qm(t)
C3
qm5(t)
qm4(t)
qm3(t)
K4
K3
4
C4
Figure 7 Schematic view of the dynamic model of the hydraulic Wheatstone bridge.
12
Theoretical results
Relative square root error eSRE, %
10
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Pressure pulsation intensity p
Figure 8 Variation of the relative square root error with the pressure pulsation intensity for
differential pressure devices with a qm  Δp form of the measuring characteristic.
2.0
Normalized mass flow rate qm, kg/s
1.5
Normalized measured mass flow rate
Normalized mass flow reading
a
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Time t, s
2.0
Normalized mass flow rate qm, kg/s
1.5
Normalized measured mass flow rate
Normalized mass flow reading
b
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
0.000
0.025
0.050
0.075
0.100
0.125
0.150
Time t, s
Figure 9 Time variations of the normalized fluctuating part of the flowmeter reading for two
different pulsation frequencies: (a) f = 2 Hz; (b) f = 20 Hz.
3.0
Water
Water (first-order filter with fc = 1.6 Hz)
2.5
Fuel
Amplitude ratio
2.0
1.5
1.0
0.5
0.0
1
10
100
Pulsation frequency f, Hz
Figure 10 Amplitude frequency characteristics of the hydraulic Wheatstone bridge.