Title: Static and dynamic characteristics of a hydraulic Wheatstone bridge

mass flowmeter

Authors: A. Svetea,*, J. Kutina, I. Bajsića

a 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

physicalmathematical 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 mq 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 mq if ρm vrq q or ρm vrq q , 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 physicalmathematical 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 Physicalmathematical steady-state flow model

The main assumptions of the physicalmathematical 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):

1 4m m mq q q , 1 2ρm vr mq q q , 2 3m m mq q q , 4 3 ρm m vrq q q , (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:

2

2

for 0/ ρΔ

for 0/ ρmii mi

imii mi

qK qp

qK q

, (2)

where Δ ip (for the proposed direction of the fluid flow) equals:

1 1 2Δp p p , 2 2 3Δp p p , 3 4 3Δp p p , 4 1 4Δp p p . (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]:

2 2 213

for ρ

Δρ for ρ

2ρ

vr m m vr

m vr m vr

Kq q q q

p Kq q q q

, 2 2 2

24

ρ for ρ2ρΔ

for ρ

m vr m vr

vr m m vr

Kq q q q

p

Kq q q q

. (4)

If the flowmeter uses the differential pressure across the bridge ∆p13 when ρm vrq q and the

differential pressure across the connecting conduit ∆p24 when ρm vrq q , the ideal flowmeter’s

characteristic can be written as:

Δ id vr mp Kq q or Δ id

mvr

pKq

q . (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:

1

ρ

2m vr

m

q qq

, 2

ρ

2m vr

m

q qq

, 3

ρ

2m vr

m

q qq

, 4

ρ

2m vr

m

q qq

. (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, ρm vrq q . In the case when the measured

flow is smaller than the recirculating flow, ρm vrq q , 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:

3,9591ξ 720,31i

miq . (7)

The pressure loss coefficient Ki employed in Section 2 can be related to the dimensionless

pressure loss coefficient ξi by

2 4

8ξ

πi

iKD

, (8)

which is subsequently used as an input parameter for the physicalmathematical 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 computer-

control 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 vrq 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 ρm vrq q the differential

pressure ∆p13 varies linearly with mq , and when ρm vrq q the differential pressure ∆p24 varies

linearly with mq . 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 physicalmathematical

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:

ΔWB

vr m

pK

q q , (9)

which equals the orifice pressure loss coefficient in the case of the ideal flowmeter

configuration, WBK 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 physicalmathematical 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 A WBu K of the slope in the

regression line Δ / vr WB mp q K q and the standard uncertainty B WBu K associated with the

contributions of particular measurands,

2 2 2Δ

ΔB WB vr m

WB vr m

u K u p u q u q

K p q q

, (10)

where each term in Eq. (10) considers the accuracy of the sensor and the repeatability effects.

The relative standard uncertainties /B WB WBu K K 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,

A WBu K and an average of B WBu K , are combined and expanded with the coverage factor k

= 2. The expanded measurement uncertainties of WBK 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

physicalmathematical 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 Δ vq and the pressure time derivative d / dp t in a rigid pipe segment [11]:

dΔ

dv

pq C

t , (11)

where C depends on the bulk modulus 2β ρc (where c is the fluid speed of sound) and the

fluid volume V in the pipe segment:

2β ρ

V VC

c . (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 d / dvq t in the pipe segment [11]:

dΔ

dvq

p Lt

, (13)

where L depends on the length of the pipe section l and its internal cross-sectional area A:

ρlL

A . (14)

Thus, the law of conservation of mass (1) can be rewritten in the modified form:

11 4 1

dρ

dm m m

pq q q C

t , 2

1 2 2

dρ ρ

dvr m m

pq q q C

t ,

32 3 5 3

dρ

dm m m

pq q q C

t , 4

4 3 4

dρ ρ

dm vr m

pq q q C

t , (15)

and the inertia effect of the fluid in the outlet duct can be added to the energy equations (2):

5 53 5

d

ρ dmL q

p pt

. (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 physicalmathematical 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 1p t , 5p t and vq 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 1 1 11 ε sin 2πpp t P ft is assumed, where P1 is the time-

mean value, 1ε p is the pressure pulsation intensity and f is the pulsation frequency. At the

outlet of the hydraulic bridge a constant pressure value (reservoir) 5 5p t P is assumed and

the recirculating pump is considered to be the source of the constant volumetric recirculating

flow vr vrq t Q . 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 iC

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, Δmq p , 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:

Δ Δ

ΔSRE

p t p te

p t

. (17)

The calculated results in Fig. 8 show its dependence on the pressure pulsation intensity Δε p ,

when the differential pressure varies over time according to ΔΔ Δ 1 ε sin 2πpp t P ft

due to pulsating flow. It is evident that with the increase of the intensity of the pulsation Δε p

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 ρm vrq t q . 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 3ρ 750 kg/m and 1300 m/sc ), which has a smaller bulk modulus 2β ρc 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 physicalmathematical 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.

References

1 Fishman B, Ryder F. Mass flowmeter. Patent US 3232104; 1966.

2 Masnik W. Mass flowmeter. Patent US 3662599; 1972.

3 Doi N. Mass flowmeter. Patent US 4031758; 1977.

4 Baker RC. Flow measurement handbook. Cambridge University Press; 2000.

5 Bajsić I. Linearization of the flowmeter. Journal of Mechanical Engineering

1986;32(7/9):118-120 (in Slovene).

6 Cichy M, Bossio RB. A dynamic model of Flo-Tron flowmeters. Proceedings 19th

Intersociety Energy Conversion Engineering Conference 1984;3:1397-1402.

7 Roznowski G, Sieklicki S. The fuel flow measurement in I. C. engines as a

diagnostics method. JMECO Conference 1983.

8 Haney MA. The differential viscometer. I. A new approach to the measurement of

specific viscosities of polymer solutions. Journal of Applied Polymer Science

1985;30(7):3023-3036.

9 Haney MA: The differential viscometer. II. On-line viscosity detector for size-

exclusion chromatography. Journal of Applied Polymer Science 1985;30(7):3037-

3049.

10 Svete A, Kutin J, Bajsić I. Measuring characteristics of the hydraulic Wheatstone

bridge. Ventil 2007;14(6):233-240 (in Slovene).

11 Karnopp DC, Margolis DL, Rosenberg RC. System dynamics: Modeling and

simulation of mechatronic systems. 4th ed. New Jersey: John Wiley & Sons Inc;

2006.

12 Gajan P, Mottram RC, Hebrard P, Andriamihafy H, Platet B. The influence of

pulsating flows on orifice plate flowmeters. Flow Measurement and Instrumentation

1992;3(3):118-129.

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.

p24

p13

Recirculatingpump

Differentialpressure

transmitter

Flowrestrictor K1 K2

qvr

K4 K3

qm1 qm2

qm4 qm3

1 3

2

4

Measured mass flow

qm qm

Figure 1 Schematic view of the hydraulic Wheatstone bridge mass flowmeter.

<

p13

Temperaturesensor

Coriolis massflowmeter

Centrifugal pump

Centrifugalrecirculating pump

Hydraulic Wheatstone bridge

Electromagnetic volumeflowmeter

1

2

3

4

p4

p2

Differential pressuretransmitter

Pressuretransducer

IN

OUT

K1 K2

K4 K3

DAQ Board

Reservoir

PC

Figure 2 Schematic view of the measuring-control system.

0.0 0.1 0.2 0.3 0.4 0.5620

640

660

680

700

720

Ori

fice

pre

ssur

e lo

ss c

oeff

icie

nt

i

Mass flow through the orifice qmi

, kg/s

Experimental results

Approximation (7)

Figure 3 Variation of the dimensionless orifice pressure loss coefficient with the mass flow

rate.

0 5 10 15 20 25 30730

735

740

745

750

755

760

765

770

Vol

umet

ric

reci

rcul

atin

g fl

ow q

vr,

dm

3 /h

Time t, s

Figure 4 Time variation of the regulated volumetric recirculating flow.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Dif

fere

ntia

l pre

ssur

e p

, b

ar

Ratio of the measured mass flow rate to the recirculating flow qm / q

vr

Differential pressure p13

Differential pressure p24

Figure 5 Variation of the differential pressures with the relative measured mass flow rate (qvr

= 750 dm3/h).

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.02

-0.01

0.00

0.01

0.02

Dev

iati

on f

rom

line

ar m

easu

ring

cha

ract

eris

tic

e a , b

ar

Measured mass flow qm, kg/s

Experimental results Theoretical results

Figure 6 Deviation of the measuring characteristic from its linear approximation (qvr = 750

dm3/h).

Obtočnačrpalka

K1 K2

qv0

K4 K3

qm1(t) qm2(t)

qm4(t) qm3(t)

1 3 5

2

4

p1(t) p5(t)

qm(t) qm5(t)

C2

C4

C3 L5C1

Figure 7 Schematic view of the dynamic model of the hydraulic Wheatstone bridge.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

2

4

6

8

10

12

Rel

ativ

e sq

uare

roo

t err

or e

SRE,

%

Pressure pulsation intensity p

Theoretical results

Figure 8 Variation of the relative square root error with the pressure pulsation intensity for

differential pressure devices with a Δmq p form of the measuring characteristic.

0.00 0.25 0.50 0.75 1.00 1.25 1.50-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Nor

mal

ized

mas

s fl

ow r

ate

q m,

kg/

s

Time t, s

Normalized measured mass flow rate Normalized mass flow reading

a

0.000 0.025 0.050 0.075 0.100 0.125 0.150-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0 Normalized measured mass flow rate Normalized mass flow reading

b

Nor

mal

ized

mas

s fl

ow r

ate

q m,

kg/

s

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.

1 10 1000.0

0.5

1.0

1.5

2.0

2.5

3.0

Am

plit

ude

rati

o

Pulsation frequency f, Hz

Water Water (first-order filter with f

c = 1.6 Hz)

Fuel

Figure 10 Amplitude frequency characteristics of the hydraulic Wheatstone bridge.