Water Hammer Phenomenon Analysis using the Method of Characteristics and Direct Measurements using a ”stripped”
Electromagnetic Flow Meter
by
Joel Carlsson
August 2016
Division of Nuclear Reactor Technology, Department of Physics Royal Institute of Technology, SE-106 91 Stockholm, Sweden
TRITA-FYS 2016:33
ISSN 0280-316X
ISRN KTH/FYS/–16:33SE
surements using a ”stripped” Electromagnetic Flow Meter
AUTHOR: Joel Carlsson
SUPERVISOR: Professor H. Anglart
PARTNER: Royal Institute of Technology
INSTITUTION: Department of Engineering Physics
INDUSTRIAL PARTNER: Vattenfall AB / YRLF
INDUSTRIAL SUPERVISORS: Dr. L. Facciolo and Dr. K. Angele
NUMBER OF PAGES: viii, 52
This thesis deals with physical and mathematical models in order to simulate, explain and reconstruct real world events of water hammer at Nuclear Power Plants (NPP’s).
The present thesis constitutes a part of Vattenfall’s Thermal Hydraulic Loads program for development and experimental validation of RELAP5/MOD3 models for water hammer transients. The RELAP5/MOD3 code is used at NPPs for load calculations, which is a prerequisite for the license to operate
In the present thesis, the Method of Characteristics (MOC) was implemented as a com- plement to the RELAP5/MOD3 code. As opposed to RELAP5/MOD3, the MOC- code also includes the compressibility of H
2O and the deformation of the pipe. The performance of the MOC code and the RELAP5/MOD3 code was evaluated against water hammer experiments performed in a test rig in ¨ Alvkarleby, Sweden. The results show that the MOC code produces more accurate results and that the RELAP5/MOD3 code underestimate the mass flow gradient during the transient and therefore the loads on the pipe. An extended MOC (taking the FSI-effects into account) is necessary to yield even better agreement with the experimental results.
In connection to the experiments, work has also been performed, within the framework of this thesis, to test a possible direct fast transient mass flow measurement using a
”stripped” conventional electromagnetic flow meter. The initial results are promising and indicate the possibility of measuring the mass flow at fast transients.
i
First and foremost I want to thank my industrial supervisors, Dr. Luca Facciolo and Dr. Kristian Angele for their guidance, support, and most importantly encouragement during my time at Vattenfall AB R&D. I appreciate all their contributions of time, ideas, and knowledge to make this experience productive and stimulating.
I also gratefully acknowledge the support and guidance of the Engineering Physics De- partment of the Royal Institute of Technology and supervisor Professor Henryk Anglart.
This project represents a six-months work under the financial support of Vattenfall AB R&D. To the staff and other master thesis students, many thanks for your friendship and good advice.
I would also like to express gratitude to Dr. Hans Henriksson and Eric Lillberg for helping particularly in the beginning, to start up and guiding me through the course of this thesis.
Lastly, I would like to thank my family and friends for all their love and support. For my parents and older siblings who raised and supported me in all my pursuits.
ii
Abstract i
Acknowledgements ii
List of Figures v
List of Tables vi
Nomenclature vii
1 Introduction 1
2 Background and Theory 4
2.1 Classical Water Hammer Phenomenon and Theory . . . . 4
2.1.1 Conservation of Mass . . . . 6
2.1.2 Conservation of Momentum . . . . 7
2.1.3 The Pressure (Water Hammer) Wave Speed . . . . 8
2.1.4 The Classical Water Hammer Equations . . . 11
2.1.5 Unsteady Friction . . . 12
2.2 Direct Mass Flow Measurement . . . 13
2.2.1 Electromagnetic flow meters . . . 13
2.3 Gibson’s indirect method . . . 14
3 Method of Characteristics 16 3.1 Solution by Method of Characteristics . . . 16
3.1.1 Different MOC schemes for more complex systems . . . 20
3.2 Boundary Conditions . . . 21
4 Experiments 25 4.1 Measuring Devices and System . . . 27
5 Results and Discussion 30 5.1 Results . . . 30
5.1.1 Experimental results . . . 30
5.1.1.1 Direct Mass Flow Measurement . . . 32
5.1.2 Comparison between the RELAP and the MOC codes . . . 34
iii
5.1.3 Validation of the MOC and RELAP codes using experimental data 36 5.2 Discussion . . . 37 5.2.1 Extended MOC . . . 37
6 Conclusions 39
A Supplement on Water Hammer and the Friction Term 41
B Model Description 43
C Calibration protocol of electromagnetic flow meter 48
Bibliography 50
2.1 Water hammer transient . . . . 5
2.2 Control volume diagram of a conduit for the continuity equation . . . . . 7
2.3 Control volume diagram of a conduit for the momentum equation . . . . . 8
2.4 Overview of the important principals of an EMF . . . 14
3.1 Characteristic lines in x-t plane. . . . 18
3.2 Computational mesh/grid with index for MOC . . . 19
3.3 Boundary condition, Ideal Pipe Attachment to Pressure Vessel . . . 22
3.4 Boundary condition, Valve at Discharge End of Pipe . . . 23
4.1 Photograph of the test rig at Vattenfall R&D in ¨ Alvkarleby, Sweden . . . 25
4.2 Overview of the important components of the test rig . . . 26
4.3 The fixed valve in the test-rig. . . 27
4.4 Illustration of a pressure sensor model used . . . 28
4.5 The two electromagnetic flow meters at the DN50 pipe line between the pump and the reservoir in the test-rig. . . 29
5.1 The positive pressure wave propagating upstream from the valve. . . 31
5.2 The negative pressure wave propagating downstream from the valve. . . . 32
5.3 Voltage measurements proportional to the flow velocity together with filtered and magnet field. . . . 32
5.4 Voltage measurement together with the filtered one. . . 33
5.5 Filtered voltage together with time shifted pressure. . . 34
5.6 Pressure normalized by Joukowsky’s equation. . . 35
5.7 Pressure normalized by Joukowsky’s equation with MOC
F SI. . . 36
5.8 Flow rate normalized . . . 36
5.9 Pressure normalized by Joukowsky’s equation with experimental data. . . 37
B.1 Pressure change in the middle of the pipe until steady state was reached . 44 B.2 Mass flow in the middle of the pipe change until steady state was reached 45 B.3 The valve ratio change used in valve boundary condition . . . 45
C.1 Calibration protocol of electromagnetic flow meter at SP, 2015-04-21 . . . 48
C.2 Calibration protocol of electromagnetic flow meter at SP, 2015-04-21 . . . 49
v
4.1 Pressure sensors . . . 28 4.2 Electromagnetic Flow Meters . . . 28 B.1 Specification of parameters used in MOC . . . 43
vi
Roman
A cross-sectional area of the pipe [m
2] a flow area [m
2]
B magnetic flux density [T ]
c pressure (water hammer) wave speed [m/s]
C contraction coefficient
Co Courant number
CS control surface [m
2] CV control volume [m
3]
D diameter [m]
E Young’s elasticity modulus [P a] (or voltage [V ], see context)
f Darcy-Weisbach friction factor (or an adapted friction factor, see context)
F force [N ]
g gravitational acceleration [m/s
2] H piezometric head [m]
k dimensionless constant
K water bulk modulus [P a] (or loss coefficient, see context) L length of pipe [m]
m mass [kg]
˙
m mass flow [kg/s]
M Mach number
n outward normal vector p momentum [kg · m/s]
P piezometric pressure [P a]
t time [s]
u local freestream velocity [m/s]
V cross-sectional area average velocity [m/s] (or volume [m
3], see context) v fluid velocity [m/s]
Z elevation of the pipe center line from a given datum [m]
vii
Greek
α angle between pipe and horizontal direction [rad]
β momentum correction coefficient
ε hoop strain
δ pipe wall thickness [m]
δx short x-directional length [m]
∆x spatial grid size [m]
∆t time step [s]
λ linear multiplier µ viscosity [P a · s]
ψ nondimensional parameter
σ hoop (circumferential) stress [P a]
ρ fluid density [kg/m
3] τ shear stress [P a]
Sub and superscripts
A position, see context B position, see context
e electrodes
end end of pipe
FSI fluid structure interaction G position, see context i space variable j time variable L length of pipe
S surface
Steady steady condition Unsteady unsteady condition
v vena-contracta
w wall
Introduction
Large pipe systems with long pipelines transporting fluids over great distances is a reality in in modern society. The usage of small pipe diameters, high-velocity together with sophisticated fluid control devices, many types of pumps and valves, coupled with electronic sensors have increased the importance of correct design.
Nuclear power plants have systems with large networks of piping both for the production of electricity and to ensure water cooling at all times in a reliable and safe way. The water in nuclear power plants is often under high pressure and at high flow rate generated by pumps. Pump failure, improper operations of valves and accidental events like power losses and pipe ruptures create transient flows, which can lead to pressure waves through the pipe system. A sudden change in flow like that generates a pressure pulse. This phenomenon is called water hammer. Water hammer events rarely lead to pipe rupture in nuclear power plants but due to conservative calculations together with limited models many calculations are being made which costs time and money in the end.
Transient water hammer analysis is essential to verify design and operation of piping systems to prevent damaging equipment and pipes. At this moment, 99 incidents induced by water hammer exists in the IRS (Incident Reporting System), the nuclear safety authorities reporting system, that is forward and collected in a database at the NEA (Nuclear Energy Agency) and the IAEA (International Atomic Energy Agency). Thus, correct water hammer analyses are essential for improving and withstanding a high level of safety and availability of the electricity production from nuclear power plants.
Safer and more efficient reactors would not only be more profitable economically for the nuclear power industry but it could also lead to better acceptance by the public towards the nuclear industry, which could allow more research to improve and explore the potential of nuclear power.
1
The majority owner of Swedish nuclear power, seven of the nine current existing elec- tricity producing reactors in Sweden (four in Ringhals and three at Forsmark) [1], the energy corporation Vattenfall received the request from Forsmark and Ringhals to vali- date RELAP5/MOD3 (Reactor Excursion and Leak Analysis Program) models, a best estimate transient simulation code, for water hammer transients. The code provides a coupled thermal-hydrodynamic and point-kinetics capability with reactivity feedback which makes it possible for example to simulate loss-of-coolant accidents (LOCAs) and allows the study of various anticipated transients without scram (ATWS) [2], [3]. It is mainly used for structural verification at Forsmark and Ringhals nuclear power plants.
The present thesis constitutes a part of Vattenfall’s Thermal Hydraulic Loads program for experimental validation of RELAP5/MOD3 models for water hammer transients.
To succeed in full filling the validation objective, a test rig with a rapidly closing pneu- matically controlled valve has been built at Vattenfall R&D in ¨ Alvkarleby, Sweden. The rig is intended to be used for studies of water hammer transients and to generate val- idation data. This Vattenfall project is large and comprehensive and there are parts which will not be presented in detailed in this report but plays a major role towards the validation. These are:
- The execution of the experiments.
- The RELAP5/MOD3 models of the test rig.
- A Computational Fluid Dynamic (CFD) model using the OpenFOAM software of the test rig.
One drawback in the experiments is that the present electromagnetic flow meters does not have fast enough time resolution to measure the transient flow rate which originates from the water hammer. Therefore the mass flow rate data from RELAP5/MOD3 cannot be validated.
During the thesis studies, effort has been made to:
(1) solve the classic water hammer equations, including a small compressibility effect of water and the deformation of the pipe which are not included in RELAP5/MOD3 simulations.
(2) identify and implement a possible direct method to measure the transient mass flow rate in the test rig and
(3) investigate the usage of Gibsons indirect method for transient flow measurement
via transient pressure measurements provided from the test rig (Vattenfall Hydro power
provided a code using the Gibson method) .
All these are means to improve the reliability of the RELAP5/MOD3 models for water
hammer transients. This report will focus on (1) the classic water hammer equations
simulations.
Background and Theory
This chapter describes the background and the theory which the present work is based upon. It summarizes the classical water hammer equations and a possible approach to directly measure the mass flow during the water hammer transient.
2.1 Classical Water Hammer Phenomenon and Theory
Any disturbances or change, planned or accidental, in a pipe system’s mean flow will initiate pressure waves in the system. As the pressure waves propagate, they create transient pressure and flow conditions. Potentially these transient conditions can have serious consequences if not properly addressed by proper analysis, design and operational considerations. For example, when a valve starts to close, the pressure increases until it is completely closed. Depending on the time it takes for the valve to close, the first pressure increase will have reached a position in space which will give the pressure a wavelength. When the valve is fully closed, the pressure has reached maximum and the wave propagate forward to even out the pressure difference.
Fig. 2.1 represents a water hammer transient of a long horizontal pipe filled with only water. The left boundary is a reservoir with constant pressure and the right boundary is a valve (gate) situated in the end of the figure. (a) A steady water flow is propagating in the right direction of the figure. (b) The valve closes instantaneously (no wavelength) initiating the water hammer phenomena, which is illustrated by the initiation of the pressure increase in fig. 2.1.
(c) The positive water hammer propagates forward to even out the pressure difference.
The negative water hammer travels in the opposite direction of the positive. The spatial extension continues in the flow direction to the right of the valve. (d) When the pressure
4
Figure 2.1: Water hammer transient Source: Micro-hydropower sourcebook[4]
reaches the reservoir it reflects positively back and (e) propagate back towards the valve leaving behind the initiate pressure. (f) The flow velocity is pushed in the opposite direction of the pressure when it propagate back. (g) At the valve, the pressure will reflect negatively (h) and propagate towards the reservoir again. (i) At negative pressure the flow velocity will be zero. (j) When the negative pressure reflects again at the reservoir (k) the pressure will return to initial condition (l) initiating the flow velocity in same direction as the negative pressure propagte back.
In real life, (the valve cannot close instantaneously and) there will be a wavelength of
the pressure wave. When measuring the pressure at a distance from the reservoir within
the wave length of the pressure increase, a superposition of the wave with itself (the first
and the last part of the wave length) will result in lower amplitude than for a position further away from the reservoir.
Joukowsky laid the foundation of the ”fundamental water hammer theory”, which de- scribes the pressure amplitude.
∆P = ±ρc∆V or ∆H = ± c∆V
g (2.1)
where c = the pressure (water hammer) wave speed, ρ = fluid density, g = acceleration caused by gravity, P = ρg(H − Z) = piezometric pressure, Z = elevation of the pipe center line from a given datum, H = piezometric head, V = R
A
udA/A = cross-sectional average velocity, u = local freestream velocity of the fluid and A = cross-sectional area of the pipe. The head (H) is a physical parameter often used in hydraulics calculations but the focus in this thesis will be on the pressure (P). The negative sign in Eq.2.1 describes a water hammer wave moving downstream while the positive sign describes the water hammer wave moving upstream.
Water hammer transients are normally assumed to be axi-symmetric since the axial changes of mass, momentum, and energy are a lot greater than their radial counterparts.
With the axi-symmetric assumption, the one-dimensional water hammer equations in transient pipe flows are derived by applying the conservations of mass and momentum in a control volume.
2.1.1 Conservation of Mass
The mass conservation principle results from the fact that mass cannot be created or destroyed. Thus, the following equation is valid:
Dm
dt = 0 (2.2)
describing that the change of mass is equal to zero in the system. The mass conservation principle related to a control volume CV and its control surface CS can be expressed using the Reynolds transport theorem:
Dm dt =
Z Z Z
CV
ρ dV = ∂
∂t Z Z Z
CV
ρ dV + Z Z
CS
ρv · n v · n v · n dA = 0 (2.3)
where n = outward normal vector to the control surface and v = fluid velocity vector.
Eq. 2.3 scaled down to one-dimension in a pipe with variable cross-section area, referring to Fig. 2.2, can be written as follows:
∂
∂t
Z
x+δx xρA dx + Z Z
CS
ρv · n v · n v · n dA = 0 (2.4)
The local form of Eq. 2.4, obtained by shrinking the length of the control volume to zero (i.e. δx tends to zero), is Eq. 2.5.
∂ (ρA)
∂t + ∂ (ρAV )
∂x = 0 (2.5)
Eq. 2.5, also called the ”continuity equation”, is the area-averaged mass balance equation for one-dimensional unsteady and compressible fluids in a flexible pipe.
Figure 2.2: Control volume diagram of a conduit for the continuity equation Source: A review of water hammer theory and practice[5]
2.1.2 Conservation of Momentum
Newton’s second law of motion states that the change of momentum of a system is equal to the forces exerted on the system by its surrounding, as described in Eq. 2.6
Dp p p Dt = X
F
F F (2.6)
By expanding the vector sum of the forces and applying it to a control volume, Eq. 2.7 yields
X F F F = Dp p p Dt = D
Dt Z Z Z
CV
v v vρ dV = Z Z Z
CV
∂v v vρ
∂t dV + Z Z
CS
v v vρv · n v · n v · n dA (2.7)
Applying Eq. 2.7 to a specific control volume of a conduit, referring to Fig. 2.3; con-
sidering the gravity, wall shear and pressure gradient forces on the fluid and taking the
limit as done previously (i.e. δx tends to zero). The following local form of the axial momentum equation is derived:
∂(ρAV )
∂t + ∂(βρAV
2)
∂x = −A ∂P
∂x − πDτ
w− γAsinα (2.8) where τ
w= shear stress at the pipe wall, γ = ρg which is the unit gravity force, α = angle between the pipe and the horizontal direction and β = R
A
u
2dA which is the momentum correction coefficient. By using the product rule of differentiation to Eq.
2.8, and dividing everything by ρA yields Eq. 2.9:
∂V
∂t + V ∂V
∂x + ∂(β − 1)ρAV
2∂x + 1
ρ
∂P
∂x + gsinα + τ
wπD
ρA = 0 (2.9)
Figure 2.3: Control volume diagram of a conduit for the momentum equation Source: A review of water hammer theory and practice [5]
2.1.3 The Pressure (Water Hammer) Wave Speed
The water hammer wave speed for a flow of slightly compressible fluid of a non-elastic pipe is defined by Eq. 2.10
c = s
K
ρ (2.10)
where K is the bulk modulus of elasticity of the fluid. When pipe walls are elastic, the dynamic pressure waves are propagating both in the fluid and in the conduit walls.
The coupling between fluid and structure material, also known as the fluid structure
interaction (FSI), is typical for the water hammer phenomena in an elastic pipe. FSI
is defined by the interaction of some deformable structure (pipe) with an internal fluid
flow. It is necessary to take FSI into consideration which is done in the classical water
hammer equations by the definition of the water hammer wave speed which will be
derived below.
By expanding the terms in the parentheses and using the definition of total derivatives the continuity equation, Eq. 2.5, becomes Eq. 2.11:
1 ρ
Dρ Dt + 1
A DA
Dt + ∂V
∂x = 0 (2.11)
Since the pressure changes will cause the changes of the cross-sectional area, it will be desirable to express the time derivative of the cross-section area in terms of the pressure.
By using the linear elasticity model for a circular pipe with internal pressure and the assumption that the axial stresses are zero, the strain-stress relationship can be written as follows:
ε = σ
E (2.12)
where ε = hoop strain, σ = hoop stress and E = Young’s elasticity modulus. The hoop stress (or circumferential stress) in the following thin-walled pipe model can be written as:
σ = F
A = P D
2δ (2.13)
where P = internal pressure, D = internal pipe diameter and δ = pipe wall thickness.
The time derivative of Eq. 2.13 can be expressed as:
dσ dt = d
dt
P D 2δ
= P 2δ
dD dt + D
2δ DP
Dt (2.14)
Combining Eqs. 2.12 and 2.14 yields,
E dε dt = P
2δ dD
dt + D 2δ
DP
Dt (2.15)
dD
dt = D dε
dt (2.16)
Since Eq. 2.16 is true, the time derivative of the strain can be expressed as Eq. 2.17.
dε dt = D
2δ DP
Dt
E − P D 2δ
(2.17)
The time derivative of the cross-section area can be expressed in terms of the strain as in Eq. 2.18 which follows:
1 A
DA Dt = 1
πD2 4
d
πD2 4
dt = 1
D
22D dD dt = 2 dε
dt (2.18)
By combining Eq. 2.17 with Eq. 2.18 the following equation is created:
1 A
DA
Dt = 2D
2δE − P D DP
Dt (2.19)
The deformations of the pipe (i.e. the cross-sectional area changes) in Eq. 2.19 are now completely expressed by the pressure changes for a circular pipe with internal pressure.
To continue the rewriting of the continuity equation, the first term in Eq. 2.11 can be expressed in terms of pressure time derivative as:
1 ρ
Dρ Dt = 1
ρ dρ dP
DP Dt = 1
ρ 1
dP dρ
DP Dt = 1
dP dρ/ρ
DP Dt = 1
K DP
Dt (2.20)
when Eqs. 2.11, 2.18 and 2.20 are combined, following Eq. 2.21 yields,
1
K + 2D
2δE − P D
· DP Dt + ∂V
∂x = 0 (2.21)
From Eq. 2.10 the water hammer wave speed in the case of an elastic pipe is defined by Eq. 2.22
c
2= 1
1K
+
2δE−P D2Dρ
= K
1 +
KE2δ−P D/E2Dρ
(2.22)
In normal engineering applications, P D/E 2δ implying that Eq. 2.22 may be written as
c
2=
K ρ
1 +
KDEδ(2.23)
In 1963, Halliwell [6] presented the following general expression for the classical water
hammer wave speed
c =
s K
ρ 1 +
KEψ (2.24)
where ψ is a nondimensional parameter that depends on the elastic properties of the conduit. Expressions for ψ for various conditions important to this study are the fol- lowing:
Rigid Conduit:
Fully rigid conduit that cannot move.
ψ = 0 (2.25)
Two cases for the anchoring of the conduit against longitudinal movement are:
Thin-Walled Elastic Conduit:
(1) Conduit anchored against longitudinal movement throughout its length
ψ = D
δ (1 − ν
2) (2.26)
and (2) the conduit with frequent expansion joints that was derived earlier.
ψ = D
δ (2.27)
2.1.4 The Classical Water Hammer Equations
By combining the new defined wave speed from Eq. 2.23 with Eq. 2.21, following equation yields,
DP
Dt + ρc
2· ∂V
∂x = 0 (2.28)
Note that Eq. 2.28 is for a conduit with expansion joints due to the assumption that the axial stresses are zero.
The usage of a small Mach number (M = V /c) approximation recognized by Allievi [7]
makes it possible for a final simplification of the momentum Eq. 2.9 and the continuity
Eq. 2.28 to the classical water hammer equations represented by Eqs. 2.29 and 2.30:
∂V
∂t + 1 ρ · ∂P
∂x + τ
w(t)πD
ρA = 0 (2.29)
∂P
∂t + ρc
2· ∂V
∂x = 0 (2.30)
The classical water hammer equations are valid for axi-symmetric flow of a slightly compressible fluid in a flexible pipe, where the Mach number is very small and τ
w(t) = quasi-steady wall shear as a function of time [5].
More rigorous derivation of the classical water hammer equations than the one presented here can be found in [8], [9] and [10].
2.1.5 Unsteady Friction
There exists different ways to model the friction between the pipe and the fluid. In the third term of the momentum Eq. 2.29, it is valid to relate the wall shear to cross- sectional average velocity in steady state condition. Weather it is valid under unsteady condition, as for the water hammer transient, is questionable. The attenuation of the water hammer is highly dependent on a correct friction between the pipe and fluid. It has been shown in previous research that the use of steady state wall shear models for the unsteady problem of water hammer is satisfactory for the first pressure pulse and in some cases for the second but the oscillations coming afterwards show very slow attenuation as compared to that measured in experiments. This fact does not cause problems for determining the maximum or minimum pressures in the system. Since the interest for this study lies in the approach to the first maximum pulse peak, it was decided to use the Darcy-Weisbach equation. The Darcy-Weisbach equation is commonly used in water hammer models and is described by Eq. 2.31,
τ
w(t) = ρf (t)|V (t)|V (t)
8 (2.31)
where f (t) = Darcy-Weisbach friction factor. Several unsteady friction models have
been proposed and tested for better agreement of the water hammer attenuation. More
about unsteady friction methods and models can be found in Appendix A.
2.2 Direct Mass Flow Measurement
The flow meters that are commercially available today has a typical time resolution of 0.1 seconds or slower. The thesis included a literature review to identify a possible direct method to measure the transient mass flow rate in the test rig. Already performed RELAP5 simulations revealed an estimation of the time resolution needed, in order to accurately measure the mass flow, to be faster than 0.1 ms, i.e. higher than 10kHz.
Since the interest in this research is the water hammer phenomenon, the measurement method should preferably have no impact on the pressure wave propagation, regarding both the reflection and transmission of the wave and its velocity. The literature review did not reveal a way to measure the mass flow with the required time resolution.
The commercially electromagnetic flow meters (EMF) today have their main focus on the uncertainty of the measurement not on the time resolution. In theory the physical restriction of time resolution governing the EMF, (i.e. the speed of the moving electrons) should not be a limitation.
2.2.1 Electromagnetic flow meters
The electromagnetic flow meter (or the magnetic flow meter) is based on Faraday’s law of electromagnetic induction. Faraday’s law describes the way an electromagnetic field may be generated by a changing magnetic environment. The measurement principal is based on two coils creating a magnetic field perpendicular to the flow direction. The magnetic flux density (B) have to be known. The magnetic field is generated in pulses.
For the magnetic flow meter to work it requires a conducting fluid, for example, in the case of the experiment here, water that contains ions. The ions corresponds to conductors, flowing through the magnetic field changing it and creating a voltage (E
S).
The induced voltage is created over the flow which can be detected by two measuring electrodes. The electrodes read the induced voltage while the magnetic field is held constant. A simplified expression for the voltage, proportional to the flow velocity, is given by Eq. 2.32,
E
S= kBD
ev (2.32)
D
e= the distance between electrodes (inside diameter of the tube), v = mean velocity
of the liquid and k = dimensionless constant [11].
Figure 2.4: Overview of the important principals of an EMF Source: Transactions Volume 4, Flow and Level measurement[11]
The commercially available electromagnetic flow meters do not have a fast enough time resolution to measure the transient flow rate, which is generated by the water hammer.
This is due to an averaging of the original signal in the transmitter in order to reduce the uncertainty. By bypassing the signal processing and taking the original raw signal, a faster time resolution could possibly be achieved. This will increase the uncertainty of the measurement but it is not clear how large the ”noise” level on the flow measurement will be.
2.3 Gibson’s indirect method
The Gibson method, also known as the pressure-time method, for transient flow mea- surement is an indirect measurement method for the mass flow. The Gibson method was studied and considered for of the water hammer validation of the RELAP code. The method calculates the mass flow transient via the time-history of pressure difference measurements (provided from the experiment) by Eq. 2.33.
Q
0= A ρL
Z
t2t1
(∆p(t) + ∆P
f(t))dt + q
l(2.33)
Q
0= the flow rate in the initial steady-state conditions, ∆p = the static pressure differ-
ence between the measuring sections, ∆P
f= the pressure drop between the measuring
sections caused by hydraulic resistance, q
l= the flow rate in final steady-state conditions
(which is the leakage flow through the valve), t
1= the initial time limit of integration
and t
2= the final time limit of integration. Vattenfall Hydro power provided a code
using the Gibson method, which had been used successfully for slower water hammer
transients in hydro power plants applications. In slower water hammer transients, the
fluid compressibility and pipe deformations can be neglected. These assumptions are
made when deriving Eq. 2.33. In the fast water hammer transient considered in this the-
sis, there will be effects from pipe deformations and fluid compressibility on the pressure
oscillations. The simplifications done in the Gibson method are included in the classical
water hammer equations in a special manner through the water hammer wave speed
presented earlier. The Gibson theory and code was studied here but it was decided not
to be used for fast transients due to the above reasons [12], [13].
Method of Characteristics
In this chapter the equations for simulating the transient flow behavior in pipes with the method of characteristics are derived together with the boundary conditions used for producing the results.
Water hammer models are improving and becoming more widely used. Today, several commercially available water hammer software packages exist using different numerical methods. The method of characteristics (MOC) is the most popular numerical method when it comes to one dimensional, hydraulic transient problems, especially if the pressure wave velocity is constant as in the present case. It is desirable due to its attributes of accuracy, simplicity, numerical efficiency, and programming simplicity [5].
3.1 Solution by Method of Characteristics
The continuity Eq.3.1 and momentum Eq.3.2 together with Eq. 2.31 derived in the previous chapter describe transient flows in closed conduits. In these equations, distance x and time t are two independent variables and pressure p and flow velocity V are two dependent variables. The other variables are the system parameters which are assumed not to vary with time.
L
1= ∂V
∂t + 1 ρ · ∂P
∂x + f |V |V
2D = 0 (3.1)
L
2= ∂P
∂t + ρc
2· ∂V
∂x = 0 (3.2)
16
A linear combination of Eqs. 3.1 and 3.2 is L = L
1+ λL
2, where λ is a linear multiplier.
By rearranging the linear combination the following equation yields,
∂V
∂t + ρc
2λ · ∂V
∂x
+ λ ∂P
∂t + 1 ρλ
∂P
∂x
+ f |V |V
2D = 0 (3.3)
The total derivatives for the flow speed (V = V (t, x)) and pressure (P = P (t, x)), if they depend on both the time and the position, as in this case, are represented by the following equations,
dV dt = ∂V
∂t + ∂V
∂x dx
dt (3.4)
and
dP dt = ∂P
∂t + ∂P
∂x dx
dt (3.5)
From Eqs. 3.3, 3.4 and 3.5 the unknown multiplier is solved as follows,
1 ρλ = dx
dt = ρc
2λ (3.6)
which is
λ = ± 1
ρc (3.7)
By combining Eq. 3.3 with the other three Eqs. 3.4, 3.5 and 3.7 following equations are derived,
dV dt + 1
ρc dP
dt + f |V |V
2D = 0 (3.8)
if
dx
dt = c (3.9)
and
dV dt − 1
ρc dP
dt + f |V |V
2D = 0 (3.10)
if
dx
dt = −c (3.11)
What MOC basically does is that it eliminates the independent variable x and the
partial differential equations gets converted into ordinary differential equations (ODE:s)
in the dependent variable t. The ODE:s are Eq.3.8, 3.9, 3.10 and 3.11. They are called
compatibility equations. However, this simplification makes a restriction on where the
new ODE:s are valid. The original momentum and continuity equations were valid everywhere in the x-t plane, but when it comes to the new ODE:s they are only valid along the straight lines (if the wave speed is constant) described by the governing linear equations, Eq.3.9 and Eq.3.11 (with the slopes plus/minus the wave speed) received when the ODE:s were derived. These lines are called characteristic lines.
The derived ODE:s are valid along the pipe length (i.e. for 0 < x < L) and special boundary conditions are required at the ends, i.e. at x = 0 and at x = L. Assume the valve is rapidly closing. This reduces the flow at the valve to zero which increases the pressure at the valve. Since the pressure increase, a positive pressure wave (the pressure is higher behind the wave front than that in front) travels in the upstream direction.
Figure 3.1: Characteristic lines in x-t plane.
To compute the pressures and flow velocities, they need to be known at the initial conditions or calculated during the previous time step. To compute the dependent variables at next time step, which is going to be along a characteristic line, Eqs. 3.8 and 3.10 need to be multiplied by dt and integrated. The integration limit is going to be two points along the positive characteristic line (A-G in Fig.3.1) for Eq.3.8 and two points along the negative characteristic line (B-G in Fig.3.1) for Eq.3.10. The friction losses in the third term are not linear and therefore cause a problem. By a first order approximation, in other words, the flow velocity remains constant on the characteristic line, the evaluation of the integral becomes possible. This first-order approximation usually yields satisfactory results for engineering applications. If the results are not sufficient, one can reduce the time step to a smaller value and improve the contribution from the nonlinear part.
(V
G− V
A) + 1
ρc (P
G− P
A) + f
2D ∆tV
A|V
A| = 0 (3.12)
and
(V
G− V
B) − 1
ρc (P
G− P
B) + f
2D ∆tV
B|V
B| = 0 (3.13) The ∆t in Eq.3.12 is replaced by Eq.3.9 (in finite form) and the ∆t in Eq.3.13 is replaced by Eq.3.11 (in finite form) creating the following equations,
(V
G− V
A) + 1
ρc (P
G− P
A) + f c
2D ∆xV
A|V
A| = 0 (3.14) and
(V
G− V
B) − 1
ρc (P
G− P
B) − f c
2D ∆xV
B|V
B| = 0 (3.15) These equations above can be written in a more programming friendly manner, where index i is the space variable and index j is the time variable as can be seen in Fig. 3.2
Figure 3.2: Computational mesh/grid with index for MOC
V
i,j− V
i−1,j−1+ 1
ρc (P
i,j− P
i−1,j−1) + f c
2D ∆xV
i−1,j−1|V
i−1,j−1| = 0 (3.16) and
V
i,j− V
i+1,j−1− 1
ρc (P
i,j− P
i+1,j−1) − f c
2D ∆xV
i+1,j−1|V
i+1,j−1| = 0 (3.17)
By rearranging Eq.3.16, it will constitute the boundary condition at x = L and by rearranging Eq.3.17, it will constitute the boundary condition at x = 0. By combing Eq.3.16 with Eq.3.17, all the nodes in between the boundaries can be calculated through Eqs.3.18 and 3.19
P
i,j= 1
2 (P
i−1,j−1+ P
i+1,j−1)+
ρc
2 (V
i−1,j−1− V
i+1,j−1) − ρc 2 ( f c
2D ∆xV
i−1,j−1|V
i−1,j−1| + f c
2D ∆xV
i+1,j−1|V
i+1,j−1|) (3.18) and
V
i,j= 1 2
1
ρc (P
i−,j−1− P
i+1,j−1)+
1
2 (V
i−1,j−1+ V
i+1,j−1) + 1 2 ( f c
2D ∆xV
i+1,j−1|V
i+1,j−1| − f c
2D ∆xV
i−1,j−1|V
i−1,j−1|) (3.19) To summarize, the model utilized to produce results uses a fixed-grid MOC. The com- putational grid uses a constant ∆x (here spatial grid size) and constant ∆t (time step) throughout the numerical simulation together with constant pressure wave speeds prop- agating along the characteristic lines [8].
3.1.1 Different MOC schemes for more complex systems
The fixed-grid MOC requires all pipes to satisfy the Courant condition in Eq.3.20. The Courant number (Co), is defined as the ratio of the actual wave speed and the numerical wave speed (∆x/∆t).
Co = c
∆x/∆t = c · ∆t
∆x = {∆t = ∆x
c } = 1 (3.20)
In multiple pipe systems or systems with variable wave speeds, problems occurs. Wave-
speed or pipe-length adjustments and/or interpolations have to be applied to insure
reasonable results. The down side with the adjustments are the introduction of errors
that follows them. A wave speed adjustment can be thought of as an adjustment of one
or more of the physical parameters that are used to calculate the wave speed (Eq.2.24),
for example the wall thickness, the elastic modulus or the density. The adjustment
directly creates an artificially faster or slower wave speed, causing different timing of
interactions and reflections of the pressure waves in the system. In other words, it
misleads the physical characteristics of the problem [14].
There are different interpolation schemes that can be used with advantages and disad- vantages. There exists both possibilities to make interpolations in the space-line and in the time-line. Space-line interpolation approximates the solution to the fixed-grid MOC. By interpolating the dependent variables (the pressures and flow velocities) from adjacent grid locations to the point where the characteristic line intersects the previous time step, the solution is given. The same procedure applies for the time-line.
Several technical articles have been written on the matter of interpolation to deal with adjustable time steps and fluctuations of the wave speed. Different interpolations and methods have been tested to decrease the errors that occur, linearly, implicit methods, Hermite scheme, Holly-Preissmann schemes and spline interpolations [15], [16] and other approaches as an algorithm to derive a flexible class of discretization approaches for interpolating between grid points to obtain values of the dependent variables (V and P) at the start of characteristic curves [17]. One of the articles highlight the adjustment of the wave speed because of the fact that all interpolation methods alter the wave speed of the water-hammer model [14].
3.2 Boundary Conditions
The simplest boundary conditions are specified values of the relevant variables, the pressure and flow velocity in this case. However, many practical boundary conditions are specified as time-dependent pressure, velocity or pressure-velocity functions. The two boundary conditions used to produce the results are going to be derived in this part of the report.
As mentioned before Eqs. 3.18 and 3.19 are valid along the pipe length (i.e., for 0<x<L) and special boundary conditions are required at the ends, i.e., at x = 0 and at x = L.
The first boundary condition expressed in a more programming friendly manner that more easily can be implemented in code, similar to Eqs. 3.16, 3.17, 3.18 and 3.19, is represented by a tank with constant pressure.
The boundary condition is called Ideal Pipe Attachment to Pressure Vessel and is rep- resented in Fig.3.3.
The equation describing the flow for the boundary condition is Eq.3.21
P
0= P
i,j+ V
i,j2ρ
2 (3.21)
By rearranging Eq.3.17, Eq.3.22 yields
Figure 3.3: Boundary condition, Ideal Pipe Attachment to Pressure Vessel
P
i,j= P
i+1,j−1+ ρc(V
i,j− V
i+1,j−1) − ρc f c
2D ∆xV
i+1,j−1|V
i+1,j−1| (3.22) By inserting Eq.3.22 into Eq.3.21 and then rearrange, Eq.3.23 yields
V
i,j2+ 2cV
i,j− 2cV
i+1,j−1+ 2
ρ (P
i+1,j−1− P
0) − f c
2D ∆xV
i+1,j−1|V
i+1,j−1| = 0 (3.23)
By solving the second order equation that is Eq. 3.23, ignoring the negative square root and after rearranging, the following equation yields
V
i,j= −c + c s
1 + 2V
i+1,j−1c − 2
ρc (P
i+1,j−1− P
0) + f
D ∆xV
i+1,j−1|V
i+1,j−1| (3.24)
Eq. 3.24 is used for the boundary at x = 0.
The second boundary condition derived for direct implementation in code, is representing
a valve. The name of the boundary condition is Valve at Discharge End of Pipe and is
illustrated in Fig.3.4
Figure 3.4: Boundary condition, Valve at Discharge End of Pipe
Fig.3.4 is described by Eq.3.25
P
i,j− P
end= K
LV
i,j2ρ
2 (3.25)
Valve boundary conditions are often coupled to opening or closing transients which deter- mine the loss coefficient (K
L) as a function of time. The loss coefficient is approximated by Eq.3.26, which is the flow velocity in the full pipe area A.
K
L(t) ≈
A
C
va
v(t) − 1
2(3.26)
where C
va
v(t) is the vena-contracta area and C
vthe contraction coefficient which is assumed to be equal to 1 for a gently curved entrance.
The same approach is used for this boundary condition as for the previous one. Eq.
3.16, can be written as Eq. 3.27
P
i,j= P
i−1,j−1− ρc(V
i,j− V
i−1,j−1) − ρc f c
2D ∆xV
i−1,j−1|V
i−1,j−1| (3.27)
V
i,j2+ 2c
K
L(t) V
i,j− 2c
K
L(t) V
i−1,j−1− 2
ρK
L(t) (P
i−1,j−1−P
end)+ f c
2DK
L(t) ∆xV
i−1,j−1|V
i−1,j−1| = 0 (3.28) Solving the second order equation above, ignoring the negative square root and after rearranging, the following equation yields
V
i,j= − c
K
L(t) + c K
L(t) s
1 + 2K
L(t)V
i−1,j−1c − 2K
L(t)
ρc
2(P
i+1,j−1− P
0) − f K
L(t)
D ∆xV
i−1,j−1|V
i−1,j−1| (3.29)
Eq. 3.29 is used for the boundary at x = L.
Experiments
As mentioned in the introduction, an experimental facility for the study of water hammer transients has been built at Vattenfall AB, Generation R&D in ¨ Alvkarleby, Sweden. To quantify the water hammer propagation in the pipe accurate measurements of both the pressure and the mass flow are required. Such data will be used for validation of the RELAP and the MOC codes.
Figure 4.1: Photograph of the test rig at Vattenfall R&D in ¨ Alvkarleby, Sweden
25
The test-rig that is displayed in Fig. 4.1 is a closed loop system, which consists of two long approximately 30 m straight pipes (DN100) connected with a 180 degrees bend.
Figure 4.2: Overview of the important components of the test rig
The flow is created by a centrifugal pump, which is rpm-regulated on one of the electro- magnetic flow meters, assuring a constant flow rate. The flow runs counterclockwise in the test rig, as illustrated in Fig. 4.2 by the big arrows on the left hand side.
Before the pump a rapidly closing valve is situated. It is kept open with pressurized air, and closes mechanically with a spring. The generation of the water hammer appears when the valve closes at the same time as there is a constant flow rate through the pipe system. A positive pressure pulse is travelling left in the figure from the valve and a negative to the right. The idea is that the straight pipe sections along with the 180 degree are long enough for the valve to be able to close before the reflection of the wave comes back and interfere with the first pressure pulse. The valve has been shown to have a minimum closing time of about 30ms. Slower closing times can be selected by varying the pressure drop in the pneumatic system. The sonic speed in water calculated from the experiments is about 1250m/s. The wave therefore travels about 37.5m before the valve is fully closed, which implies that the requirement is reached (2 x 30m).
The valve has an asymmetric configuration: it has an inner diameter of 83 mm with two
different expansions that reach a diameter of 100 mm. In between the valve and the
pump there is a conical contraction from DN100 to DN50, illustrated in Fig. 4.2. The
valve can be set to different angles manually.
Figure 4.3: The fixed valve in the test-rig.
To be able to model the characteristics of the valve, the area as a function of the angular position (α) was measured using a potentiometer connected to the valve. The angle α, is the angle which the valve rotates around, and when it reaches approximately 75 degrees, the valve is completely closed.
The pipe surface roughness has been measured and it is commercial standard about 3.5µm±10 %. The test water used in experiments is soften water. It is important for the function of the flow meters that the water has a large enough conductivity. The system pressure is produced and controlled by using Nitrogen gas (N
2). The N
2-tube is coupled to the top of the reservoir. The system pressure in the gas is applied on top of the water level in the reservoir and hence in the entire pipe system.
The air initially contained in the reservoir and in the pipeline will be let out through the degassing valves in the top of the upper pipe line during the filling of the system with water. To achieve an effective degassing of the air dissolved in the water, a vacuum pump, is connected to the top of the tank. A manual valve is used to close the line between the tank and the vacuum pump.
4.1 Measuring Devices and System
The measuring system used is a 2 x 16 channels DAQ system to sample all the signals
from the sensors. Eight channels can sample at 50 kHz and the rest at 300 kHz. A
sampling frequency of 20 kHz (i.e. ∆t =0.05 ms) is sufficient. Then one can resolve a
pressure wave which occurs during 1 ms with 10 points. A LabView program is used to collect all the signals.
The measuring devices consist of flow meters, pressure sensors and strain gauges (in- strument for measuring strain).
The uncertainty of the pressure sensors is ±0.25% of full range and the frequency re- sponse is stated to be 180 kHz according to the manufacturer [18]. The sensor positions and the names of the positions are listed in Table 4.1
Figure 4.4: Illustration of a pressure sensor model used
Position Name of position
Between valve and pump P1
Upstreams from the valve P2
Before 180 degree bend P3
After 180 degree bend P4
Between flow meter and tank P5
Before the S-shaped pipe near the tank P6 Before the S-shaped pipe near the tank P7
In the middle of the above pipe P8
Table 4.1: Pressure sensors
There are two electromagnetic flow meters installed after the pump. One of the flow meter’s purpose is as mentioned to regulate the flow speed generated by the pump. The other one is used for measurements and also initially to study the possibility of measuring the transient behavior of the flow. The flow meters has a measurement uncertainty of about 0.5 % according to the recent calibration at SP, see Appendix C. The velocity in the DN50 flow meter is about 1.2 m/s. At flow rates lower than that, the uncertainty in the flow measurement will increase.
Position Name of position
Between pump and tank Q1
Right after Q1 Q2
Table 4.2: Electromagnetic Flow Meters
Figure 4.5: The two electromagnetic flow meters at the DN50 pipe line between the
pump and the reservoir in the test-rig.
Results and Discussion
This chapter contain analyses of results and associated uncertainties followed by a discus- sion. Since the RELAP and the MOC codes are using different equations for the analyses of water hammer (RELAP is using weighted Navier-Stokes equations and MOC is using the momentum and continuity equations adapted for conduits with a special wave speed term) there will be differences in the results. The differences have been compared and analysed as the results from the two codes are validated against the experimental data.
The general results from the experiment together with the direct mass flow measurement results will also be presented.
The ultimate objective for performing these simulations are to predict the loads on the pipes. The force is proportional to the gradient of the mass flow in each segment. The reason why the focus is only on the first change in mass flow and pressure is that this usually represents the largest force.
5.1 Results
5.1.1 Experimental results
The experimental pressure measurements are displayed relative to the local pressure at stationary flow before the water hammer. The positive water hammer wave, displayed in Fig. 5.1, will travel upstream until it is reflected in the reservoir. The time it takes for the wave to get fully damped out was more than 30 seconds. The noise on top of the first positive pressure peak in Fig. 5.1 is approximately ±10 %. Where the noise comes from will be brought up in the discussion section.
30
A pressure wave (for an instantaneous valve closure) calculated with Joukowsky’s fun- damental equation of water hammer should theoretically produce a pressure amplitude of ρ · ∆V · c = 999 · 0.31 · 1450 = 4.48 bar (Eq. 2.1). The lower amplitude observed in Fig.
5.1 can be explained by the lower speed of sound, i.e. the water hammer wave speed. If the water hammer only affects the water, the wave speed would be the speed of sound in water (1483.8 m/s). However, the elastic pipe expands due to the increased pressure, sending out a wave in the structure as well. The structural wave is approximately four times faster than the one in the fluid. This causes an interaction between the fluid and the structure, resulting in a slower wave speed than the sonic speed in water, see Appendix B. With a wave speed of 1255 m/s (see Fig. 5.1), the amplitude is calculated to 3.8 bar, which corresponds quite well with what is displayed, not taking into account the ”noise”, in Fig. 5.1.
Tid [s]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Pressure [bar]
-6 -4 -2 0 2 4 6
P2
Figure 5.1: The positive pressure wave propagating upstream from the valve.
The results from Fig. 5.1 constitutes the validation data for the MOC simulation.
The negative water hammer wave continuing in the flow direction after the valve clo-
sure through the pump and gets damped faster than the positive wave, Fig. 5.2 Note
also that the negative pressure wave does not display almost any noise and has higher
amplitude, see Fig. 5.2. The distance between the valve and the tank is shorter than
the wavelength of the wave (approximately 9 m). Consequently, the reflected part of
the wave is superimposed with the part of the wave that have not yet been reflected
and the wave amplitude will therefore be higher and more ”sinusoidal” in its shape as
is illustrated in Fig. 5.2
Tid [s]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Pressure [bar]
-6 -4 -2 0 2 4 6
P1
Figure 5.2: The negative pressure wave propagating downstream from the valve.
5.1.1.1 Direct Mass Flow Measurement
The first measurements were taken in the DN50 pipe of the negative wave between the valve and the reservoir (more exact between the pump and the reservoir). The EMF used for the measurement was in position Q2.
The EMF shifted magnet field is to ensure that the parts do not get magnetized. This shift is represented in Fig. 5.3 by the blue peaks. During the magnet field shift the volt- age measurements proportional to the velocity of the fluid will be incorrect. Furthermore, the measured voltage has an offset which will be needed to take into consideration when calculating the corresponding flow velocity and mass flow.
Time [s]
1.9 2 2.1 2.2 2.3 2.4 2.5
Q2 [V]
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Q2 original Q2 filter MagnField
Figure 5.3: Voltage measurements proportional to the flow velocity together with
filtered and magnet field.
Time [s]
1.9 2 2.1 2.2 2.3 2.4 2.5
Q2 [V]
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03
Q2 original Q2 filter
Figure 5.4: Voltage measurement together with the filtered one.
Some embedded MATLAB function for filtering was tried before the filter used in Figs.
5.3 and 5.4 were chosen. The filtered version of the raw voltage signal is using a Savitzky- Golay smoothing filter of the embedded MATLAB function sgolayfilt(x,k=3,f=41), where x = the unfiltered data, k = The polynomial order and f = the frame size. In Fig. 5.4, the magnetic field has been removed and the voltage measurement together with the fil- tered voltage has been zoomed in around the time of the transient. The direct unfiltered voltage is noisy. After filtering it appears less noisy, and the important first gradient induced by the water hammer wave seem to be caught arounf t = 2.08 s. A deeper investigation into weather the voltage measurement may be to rely on was done.
As stated in Table 4.1, P1 is in between the valve and the pump. Hence, there will be a time shift in the pressure measurement compared to the EMF measurement from Q2.
This is why the P1 measurements has been shifted relative to Q2 so that the fist water hammer wave ends up simultaneously for both as displayed in Fig. 5.5.
Fig. 5.5 display the filtered mass flow.
˙
m = V · A · ρ (5.1)
The procedure from the filtered voltage in Fig. 5.4 to the corrected mass flow in Fig. 5.5 is the following: The voltage is firstly corrected for the offset creating the symmetric shift around zero. The negative voltage will then be flipped so that their will be a continuously positive voltage. This will create an amplitude between zero and the positive voltage.
During steady state conditions, the amplitude will be approximately constant. The
direct measured mass flow from the other EMF (Q1) will produce the correct mass
Time [s]
2 2.05 2.1 2.15 2.2 2.25 2.3
-6 -4 -2 0 2 4 6
Q2 filter P1
Figure 5.5: Filtered voltage together with time shifted pressure.
flow and can with the amplitude therefore work as the proportionality for the voltage.
Eq. 2.32 explain the the relationship between the voltage and the flow velocity. Eq.
5.1 describes the relationship between flow velocity and mass flow. With the mean velocity of the fluid (v) equal to the cross-sectional average velocity(V ) Eq. 5.1 can be substituted into Eq. 2.32 and Eq. 5.2 yields.
E
S= kBD
ev = kBD
eAρ m = K ˙ ˙ m (5.2)
Since A and ρ are assumed constant in Eq. 5.1, and B and D likewise are assumed constant in Eq. 2.32 they will together form a constant (K) in Eq. 5.2.
A good correlation between the pressure and the velocity change can be seen. The change in pressure happens almost simultaneously with the change in velocity. These results indicate that it is possible to measure fast mass flow transients using EMF.
5.1.2 Comparison between the RELAP and the MOC codes
A code-to-code comparison was investigated. The same boundary conditions are used for RELAP and MOC. The difference to address the water hammer phenomenon exist within the equations and the numerical methods used by the different models.
The MOC-model that was used to produce results is presented in Appendix B together
with a flowchart for the code. The model was simplified with respect to the test rig
and consists of only a reservoir and a pipe segment connecting the reservoir with a fast
closing valve. Furthermore, to build up the initial values for the transient simulation, a
pressure difference was set so that the reservoir corresponds to a position upstream and the valve correspond to the downstream. Even though differences exist between the two models, the results are expected to be similar.
The MOC model is using the same wave speed as the RELAP code calculated in its simulation (see speed of sound in water in Table B.1). The results of the pressure for the the MOC and the RELAP codes are shown in Fig. 5.6.
Figure 5.6: Pressure normalized by Joukowsky’s equation.
Two MOC models were implemented. The pipe deformations (FSI) and fluid com- pressibility in the classical water hammer equations is implemented in a special manner through the water hammer wave speed derived in Eq. 2.23. The MOC
F SImodel uses the adjusted water hammer wave speed that was measured from the experiment (see Appendix B).
Fig. 5.7 displays a small difference between the two MOC models for the first gradient of the pressure.
The slopes of the flow rate curves in Fig. 5.8 which are all equally proportional to the mass flow will give the force acting on the pipe. The following equation is used for calculating the force on the pipe:
F = Z
L0
