Seismic Part 2

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Nuclear Engineering and Design 238 (2008) 2084–2093

Experimental evaluation of ability of Relap5, Drako®, Flowmaster2TM

and program using unsteady wall friction model to calculatewater hammer loadings on pipelines

Jerzy Marcinkiewicz a,∗, Adam Adamowski b, Mariusz Lewandowski b

a Inspecta Nuclear AB, SE-104 25 Stockholm, Swedenb The Szewalski Institute of Fluid Flow Machinery, Gdansk, Poland

Received 21 March 2007; received in revised form 28 October 2007; accepted 28 October 2007

Abstract

Mechanical loadings on pipe systems caused by water hammer (hydraulic transients) belong to the most important and most difficult to calculate

design loadings in nuclear power plants. The most common procedure in Sweden is to calculate the water hammer loadings on pipe segments,

according to the classical one-dimensional (1D) theory of liquid transient flow in a pipeline, and then transfer the results to strength analyses of

pipeline structure. This procedure assumes that there is quasi-steady respond of the pipeline structure to pressure surges—no dynamic interaction

between the fluid and the pipeline construction. The hydraulic loadings are calculated with 1D so-called “network” programs. Commonly used

in Sweden are Relap5, Drako and Flowmaster2—all using quasi-steady wall friction model. As a third party accredited inspection body Inspecta

Nuclear AB reviews calculations of water hammer loadings. The presented work shall be seen as an attempt to illustrate ability of Relap5,

Flowmaster2 and Drako programs to calculate the water hammer loadings. A special attention was paid to using of Relap5 for calculation of water

hammer pressure surges and forces (including some aspects of influence of Courant number on the calculation results) and also the importance

of considering the dynamic (or unsteady) friction models. The calculations are compared with experimental results. The experiments have been

conducted at a test rig designed and constructed at the Szewalski Institute of Fluid Flow Machinery of the Polish Academy of Sciences (IMP

PAN) in Gdansk, Poland. The analyses show quite small differences between pressures and forces calculated with Relap5, Flowmaster2 and Drako(the differences regard mainly damping of pressure waves). The comparison of calculated and measured pressures and also a force acting on

a pre-defined pipe segment shows significant differences. It is shown that the differences can be reduced by using unsteady friction models in

calculations. Recently, such models have been subjects of works of several researches in the world.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

According to rules stated by Swedish regulatory body (SKI)

the calculated loadings must be reviewed and accepted by a

notified third party inspection body. Inspection Nuclear AB per-

forms such work. The purpose of the review is to confirm that

the loadings are not underestimated. Moreover, a number of keytechnical requirements which influence the review process are

stated in the SKIFS-series (2005) as follows:

• well-tried, verified and recognized methods/codes with good

safety margins shall be used;

∗ Corresponding author. Tel.: +46 8 5011 30 65; fax: +46 8 5011 30 01.

E-mail addresses: [email protected] (J. Marcinkiewicz),

[email protected] (A. Adamowski), [email protected] (M. Lewandowski).

• well-founded input data and prudent conservative assump-

tions shall be used, and also the latest knowledge/experience

shall be considered when calculating the loadings.

Several computer programs are used in Sweden for the cal-

culation of dynamic loadings on pipe systems caused by fluid

transients. These kinds of calculations were at the beginning

dominatedby large companies: ASEA/ABBAtom, Westighouse

Electric,General Electric,Siemens and Areva. Today, because of

availability of software (Relap5, Drako and Flowmaster2) small

consultant companies can also perform such analyses. When the

calculation can be performed with several computer programs

an important question for the reviewer (and also for the users)

is knowledge about their ability to calculate the dynamic load-

ings and eventual differences. This situation created the idea to

perform the following analyses:

0029-5493/$ – see front matter © 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.nucengdes.2007.10.027

mailto:[email protected]



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J. Marcinkiewicz et a l. / Nuclear Engineering and Design 238 (2008 ) 2084–209 3 2085

Nomenclature

a speed of pressure wave (m/s)

A*, C *, κ coefficients in Eq. (17)

Cu Courant number

D pipe diameter (m)

e pipe-wall thickness (m)F force (N)

g acceleration of gravity (m/s2)

H pressure head (m H2O), H = p /(ρg)

J head loss per pipe length unit

k 3 coefficient in Eq. (18)

K equivalent sand roughness height (m)

L pipe length (m)

m mass flow rate (kg/s)

p pressure (N/m2)

r pipe radius (m)

Re Reynolds number Re = VD /

S pipe cross-flow area (m2)

t time (s)

u variable from integral in Eq. (16) related to time

(s)

V average velocity (m/s)

W weighting function

x length co-ordinate (m)

f friction coefficient

Greek symbols

ν kinematic viscosity coefficient (m2 /s)

νt eddy viscosity coefficient (m2 /s)

ρ fluid density (kg/m3)

τ dimensionless time

Subscripts

app approximation

c-s/c-u critical value under steady/unsteady conditions

f friction

l/t laminar/turbulent flow

0 initial conditions

q quasi-steady hypothesis

r radial co-ordinate direction

u unsteady flow conditions

x axial co-ordinate direction

1. For a quite elementary case (one phase water hammer in a

long pipeline caused by rapid valve closure) perform calcu-

lations with three programs Relap5, Drako and Flowmaster2

(quasi-steady friction model) and compare the results (pres-

sures) from these three programs with experimental data

(Adamkowski and Lewandowski, 2006). Calculate also

dynamic force acting on a pre-defined pipe segment and

compare results from the three programs. Also draw up some

recommendations (based on reviewers experience) regarding

use of Relap5 for calculation of water hammer loadings.

2. Perform additional calculations of pressures and dynamic

loadings using a program with dynamic friction capabilities

and compare to Relap5, Drako, Flowmaster2 and experi-

mental results—mainly to illustrate the impact of dynamic

friction on the pressures and loadings.

2. Experiments

Experiments were executed at a test stand specially erected

at the laboratory of IMP PAN in Gdansk (Poland)—Fig. 1. Its

main component is the long copper pipe with internal diameter

D = 0.016 m and wall thickness e = 0.001 m. The pipe is spi-

rally coiled on a steel cylinder with diameter of about 1.7 m

and is rigidly mounted to the cylinder coating in order to min-

imize its vibrations. The inclination angle of the pipe is not

larger than 0.5◦. A quick-closing ball valve is installed at one

end of the pipe. The special spring drive of this valve results

in an almost stepwise complete flow cut-off. The valve closure

duration does not exceed 0.003 s, which is only about 1% of

the period of pressure wave propagation along the pipe (4 L / a

where L is the measuring segment of the pipeline between the

quick-closing valve and the outlet from the high-pressure reser-voir).

Four absolute pressure semi-conductor transducers are

mounted at equidistant (every L /4) sections along the pipeline.

The measuring range of these transducers is 0–4 MPa with trans-

mitted frequency band 0–2 kHz and precision class 0.2%. The

transmitted frequency band of the transducers indicates that

these transducers, within the measuring range (up to 4 MPa),

respond to stepwise pressure changes quicker than 0.0005 s

(1/2000 Hz). This time is shorter than about 0.17% of the pres-

sure wave period (4 L / a =∼0.3 s) or, in other words, transmitted

frequency band for each transducer is at least 600 times wider

then pressure wave frequency equal to 3.3 Hz (a/ (4 L)). A tur-bine flowmeter with the range of 1.5 m3 /h (∼4.2× 10−4 m3 /s)

and precision class of 1% is used for indirect measurement of

flow velocity in the pipeline. The flowmeter was checked using

volumetric method before and after the measurements.

In order to reduce the influence of undissolved air bubbles on

the analysed phenomenon, before starting the experiments, the

hydraulic system of the test rig was filled with water and left for

a few days to remove undissolved air from the liquid as much

as possible.

The experimental test consisted in several runs of unsteady

flow after sudden closing of the valve from different initial flow

conditions—velocity V 0 ( Re0). During the test a constant head

waterlevel waskeptin thehigh-pressure reservoir ( H 0 ∼= 125 m).Based on the water temperatures (22.6 ◦C), the kinematic vis-

cosity coefficients were specified (ν = 9.432× 10−7 m2 /s).

Every test run was started with cutting-off the flow in the test

pipeline. The measurements were preceded by adjustment of

required stable initial conditions (initial velocity V 0 and steady

head water level H 0). After adjusting the initial conditions and

some delay needed for stabilization of the flow conditions (not

less then 180 s), the flow rate measurement was conducted. A

few seconds before fast cutting-off the flow (water hammer phe-

nomenon), recording of the pressure variations and the valve

closure course was started. Flow cut-off was executed by releas-

ing the spring in the driving mechanism of the quick-closing



2086 J. Marcinkiewicz et al. / Nuclear Engineering and Design 238 (2008) 2084–2093

Fig. 1. Layout of the test rig.

valve. After finishing each record, the test rig was prepared to

conduct the next experimental run.

Two chosen examples of measured pressure variations

obtained for different initial Reynolds number Re0 (5700 and

15,800) are presented in Fig. 2.

Fig. 2. Pressure changes measured at the valve and in a cross-section located at

a middleof thepipe—casesof suddenflow cut-off with initial Reynolds number

Re0 = 5700 and 15,800.

3. Calculations with Relap5, Drako and Flowmaster2

and comparison with experimental data

Relap5 code (1999) has been developed for best-estimate

transient simulations in light water reactors coolant systems

during postulated accidents but is also used for thermal

hydraulic analyses in nuclear power plants for processes which

have time scale of magnitude about 1 s. Code uses include

analyses required to support rulemaking, licensing audit calcu-

lations, evaluation of accident mitigation strategies, evaluation

of operator guidelines and experiment planning analysis. The

Relap5Mod3 code is based on a non-homogeneous and non-

equilibrium model for two-phase system that is solved by a fast,

partially implicit numerical scheme. Relap5Mod3.2.2g is used

in the presented analyses.

A special attention regarding choice of time step is recom-

mended by Relap5 developing group when very rapid processes

like steam and water hammer are to be calculated (see chap-

ter 2.1.2.2 in NUREG/CR-5535, 1998). Correct use of Relap5

for analysing very rapid processes need application of proper

cell size to resolve the wave front and time step small enoughto meet the Courant limit (based on sonic speed) and to min-

imize numerical diffusion. It should be observed that using

too small time step can bring numerical oscillations and in

some cases lead to failure of the simulation (Tiselj and Cerne,

2000).

It is also important to point out that Relap5 is not the most

appropriate or user-friendly code for solving water hammer

problems but it is well documented, easy to access and able

to produce reliable results, and therefore is commonly used for

such purposes in Sweden.

Drako code (Kollman, 1999, www.kae-gmbh.de) has been

specially developed for calculating unsteady flow processes in

http://www.kae-gmbh.de/

http://www.kae-gmbh.de/




complex pipe systems and is commonly used for calculating of

water and steam hammer loadings in nuclear power plants.

Three procedures can be used by Drako to solve the differ-

ential equation describing the propagation of pressure waves:

• the method of characteristics,

• the differential procedure based on McCormack—this pro-cedure is a finite differential method of the second order

(non-centered difference scheme for nonlinear hyperbolic

equations),

• Taylor first order gives quite large numerical diffusion and is

not recommended for analyses with steep gradients.

In addition to three conservation equations for mass, momen-

tum and energy Drako uses a material law to describe the

thermodynamic state variables in form of density as a function of

pressure and entropy and enthalpy as a function of pressure and

entropy. The material model for water and steam is based on the

“IAPWS Industrial Formulation 1997 for the Thermodynamic

Properties of Water and Steam (IAPWS-IF97)”. Drako is basedon homogeneous equilibrium model for two-phase systems.

Flowmaster2 (www.flowmaster.com) is a general code for

solving of fluid flow problems in complex pipe systems by the

method of characteristics. An important disadvantage is that the

program cannot consider phase changes which strongly reduce

its availability in nuclear applications.

3.1. About calculating dynamic forces

The dynamic force F (t ) as a function of time t acting during

a fluid transient on a pipe segment of length L between two 90◦

bends A and B can be calculated using one of the followingequations (Lee et al., 1982):

F (t ) = − ∂

∂t

L

0m(t )dx (1)

F (t ) = [pB(t ) − pA(t )]S + [mB(t )V B(t ) − mA(t )V A(t )] + F f (t )

(2)

where m(t ) is mass flow rate, p is the pressure, V (t ) the average

velocity, m(t ) the mass flow rate in respective point (index A or

B), S is the constant cross-section area of the pipe and F f is the

wall friction force.A pipe segment with length 11.56 m starting at high-pressure

tank has been chosen for calculations of dynamic force called

from now F . This force will be used for comparisons between

different calculations.

Eq. (1) has been implemented in Relap5 input file in order to

calculate force F (t ).

Calculation of dynamic forces is a standard option in Drako.

Forces in Flowmaster2 cannot be calculated automatically.

Therefore a calculation outside the program has been made

using Eq. (2). In order to simplify the equations the friction

term F f and momentum terms m(t )V (t ) are not considered (are

small).

3.2. Conditions and assumptions valid for Relap5, Drako

and Flowmaster2 models

The calculation models are similar to the test stand. The case

corresponding to Re0 = 15,800 is analysed, i.e. initial velocity

is 0.94 m/s and water temperature 22.6◦C. The small increase

in wall friction caused by curvature of the pipeline is generally

not considered. In order to make possible direct comparisons of

calculated pressures, the following small modifications of input

data for Relap5 and Drako have been made:

• Relap 5. Because modifying of pressure wave speed is not

possible in the program the pipe length and initial velocities

have been changed (multiplied respectively divided by factor

1500/1297, where 1500 m/s is the velocity of pressure wave

in the water at 23 ◦C and 1297 m/s is the velocity of pressure

wave measured in the test rig). The change of Reynolds num-

ber caused by this modification is judged be negligible for the

results.

• Drako calculates the wave speed automatically consideringthe elasticity of pipe wall. The measured wave speed could

not be exactly reached in the calculations because of sim-

plified formula used by Drako (not considering Poisson’s

coefficient). To reach the wave speed of 1297 m/s the Young’s

modulus of pipe-wall material has been slightly changed.

• No special measures were needed regarding modelling in

Flowmaster2. The wave speed of 1297 m/s was implemented

as an input data.

3.3. Pressures and forces calculated with Relap5

A special attention has been paid to Relap5 calculationsbecause of sensitivityof theprogram to numericaldamping.Sev-

eraltest calculations have beenperformed with differentCourant

numbers Cu (see the following equation):

Cu = (a + V )dt

dx(3)

where a (m/s) is the wave speed, V (m/s) is the average velocity,

dt (s) is the time step and d x (m) is the element length.

Representative calculation results are shown below: pres-

sures in Figs. 3 and 4 and force F (as defined in Section 3.1)

Fig. 3. Pressures at the valve (at x = L) calculated with Relap5 and different

Courant numbers.

http://www.flowmaster.com/

http://www.flowmaster.com/




Fig. 4. Last two pressure peaks from Fig. 3.

Fig. 5. Force F calculated with Relap5 and different Courant numbers.

in Figs. 5 and 6. The experience from Relap5 calculations is that

to avoid a strong numerical damping of force amplitudes the

calculations should be performed with Courant numbers approx-

imately between 0.02 and 0.005. Performing calculations with

Courant numbers lower than 0.005 results not only in increased

calculation time and larger amount of data but also in genera-

tion of high frequency pressure pulsations of purely numerical

nature (Tiselj and Cerne, 2000).

3.4. Comparison of calculated and measured pressures

Fig. 7 shows normalized pressures at the valve (at x = L) mea-

sured and calculated with Relap5, Drako and Flowmaster2 for

Fig. 6. Last four peaks of F from Fig. 5.

Re0 = 15,800. The differences between the pressures calculated

with different programs are negligible for an engineer. The dif-

ferences between the calculated and measured pressures are

caused by the fact that the dynamic friction is not considered

in the calculations.

3.5. Force F calculated with Relap5, Drako and

Flowmaster2

Figs. 8 and 9 show force F calculated with the three pro-

grams. Relap5 and Drako calculate forces according to Eq. (1).

The Flowmaster2 force is calculated using Eq. (2) without fric-

tion and momentum terms. This is the reason for slightly larger

damping of this force.

Fig. 9 shows the last four peaks from Fig. 8. Again the dif-

ferences between the calculated values are small. Some force

fluctuations given by Drako are difficult to avoid when using

McCormack solver (as in this case). As a result the forces will

be slightly overestimated.

4. Calculation of pressures and forces considering

dynamic friction

4.1. Main assumptions

The following equations are used for mathematical descrip-

tion of unsteady liquid flows in closed conduits (Wylie and

Streeter, 1993):

Fig. 7. (a) Measured and calculated normalized pressures at the valve ( x = L) and (b) first, second and third peaks from figure (a).




Fig. 8. Force F calculated with Relap5, Drako and Flowmaster2.

• Continuity equation:

∂H

∂t +

V ∂H

∂x

+ a2

g

∂V

∂x= 0 (4)

• Momentum equation:

∂V

∂t +

V ∂V

∂x

+ g

∂H

∂x+ gJ = 0 (5)

Quantities contained in curly brackets may be neglected

due to their small influence on the results of solution of these

equations for liquid. According to the quasi-steady hypothe-

sis, commonly accepted in the engineering practice until now,

the hydraulic losses represented in Eq. (5) as quantity J , in an

unsteady flow are calculated using formulas valid for steady

flows. Such an approach is considered the main reason of dif-

ferences between experimental and computational results that

can be observed in the form of substantially different rates of pressure wave damping effects. The discrepancies between cal-

culated and measured pressure courses concern mainly wave

damping effect.

An alternative approach is the inclusion of unsteady fric-

tion losses in mathematical description of unsteady fluid flow

in closed conduits. Different unsteady friction models can be

found in the relevant literature.

In general, the models of unsteady friction losses assume that

quantity J is a sum of the quasi-steady flow pipeline resistance

J q (active resistance resulting from viscous friction at the pipe

wall) and pipeline inertance J u (reactance accounting for liquid

Fig. 9. Last four peaks of force F from Fig. 8.

inertia):

J = J q + J u (6)

Modelling quantity J q is based on the quasi-steady flow

hypothesis as was said earlier. It is usually described basing

on the Darcy–Weisbach equation:

J q = f q

gDV |V |

2 (7)

Calculation of the quasi-steady friction coefficient (Darcy

friction coefficient):

• for laminar flows ( Re≤ Rec-s where Rec-s = 2320) depends

only on flow characteristics ( Re) according to the

Hagen–Poiseuille law:

f q–l =64

Re(8)

• for turbulent flows ( Re > Rec-s) depends on flow characteris-

tics ( Re) and absolute pipe-wall roughness (K / D) accordingto the Colebrook–White formula:

1 f q–t

= −2 lg

2.51

Re

f q–t+ K/D

3.71

(9)

The models of unsteady friction losses focus on modelling

quantity J u. These models can be classified into three groups

accordingly to the method of calculating friction factor in

the one-dimensional momentum equation (Vardy and Brown,

2003):

• friction term is related to the instantaneous velocity deriva-

tives in the pipe cross-section (Daily et al., 1956; Cartens and

Roller, 1959; Brunone et al., 1991);

• friction term is related to the past velocity changes in a

given cross-section (Zielke, 1968; Vardy and Brown, 2003;

Zarzycki, 2000; Zarzycki and Kudzma, 2005);

• friction term is calculated according to the irreversible ther-

modynamic theory (Axworthy et al., 2000; Bilicki, 2002).

The first group of models enables quite easy numerical cal-

culations of friction losses. That is why these models are more

frequently used for simulating water hammer courses. The mod-

els belonging to the second group demand much more effort

when preparing algorithms and require more computer stor-age. The requirement of flow history knowledge in a given

pipe cross-section, essential for enabling calculations of flow

parameters in the next time step of simulation, results in the

need to store very large data sets in the computer memory. This

slows down considerably the calculations and makes the mod-

els from this group less attractive and is not so commonly used

in practice. However, it has to be stressed that these models

have been developed using strong theoretical bases. Such strong

bases are also the feature of models belonging to the third group

mentioned. However, these models are still in the preliminary

state of development and require many empirically determined

coefficients.




4.2. Method of solution

A special calculation method has been developed in the

Szewalski Institute of Fluid Flow Machinery in Gdansk

(Lewandowski, 2002). It is based on the Method of Charac-

teristics whereby the system of Eqs. (4) and (5) is transformed

into the following two pairs of ordinary differential equations

(for positive and negative characteristics C+ and C−):

for C+ : dx

dt = +a → dV + g

adH + gJ dt = 0 (10)

for C− : dx

dt = −a → dV + g

adH + gJ dt = 0 (11)

The first-order finite-difference approximation is used in

order to integrate Eqs. (10) and (11) along the characteristic lines

relevant to each of them, respectively. This procedure leads to

the following form of these equations:

C+ : (V P

−V A)

+

g

a

(H P

−H A)

+(gJ )At

=0 (12)

C− : (V P − V B) − g

a(H P − H B) + (gJ )Bt = 0 (13)

In all selected unsteady friction models the values of V and

H in node P (Figs. 10 and 11) are calculated using term (gJ )

related to the velocity partial derivatives, which are determined

at time levels t j−1 and t j. Two types of numerical grids may be

used in MOC—diamond (Fig. 10) and rectangular grid of char-

acteristics (Fig. 11). From among all differences between them

a way of calculating the velocity partial derivatives necessary to

determine quantity J is the most important. It follows from the

fact that the rectangular grid is two times denser than diamond

grid. In both grids longitudinal step x is related to time stept as follows (stability condition):

x

t = a (14)

Increasing density of the grid (of the same structure) has

no significant effect on the computational results. This fact is

Fig. 10. Diamond grid of characteristics with notation used in the calculation

method.

Fig. 11. Rectangular grid of characteristics with notation usedin the calculation

method.

already confirmed by many numerical tests published in the lit-

erature concerning this issue (e.g. in Bergant et al., 2001) thenumerical grid with 8, 16 and 32 computational reaches were

tested. However there are much more differences between cal-

culations obtained basing on different grid structures, diamond

and rectangular. It follows directly from basic feature of rectan-

gular grid which, in fact, consists of two diamond grids shifted

in relation to each other. These two diamond grids are coupled

only through the partial velocity derivatives in quantity J which

are calculated using finite differences. For comparisons between

experimental results and calculations the diamond grid was used.

4.3. Friction models used in calculation method

4.3.1. Vardy and Brown model

One of the main groups of unsteady friction models for

turbulent unsteady flows (the multi-layer models) bases its

mathematical description on Reynolds-Averaged Navier–Stokes

Equations applied to axisymmetric flow. In these equations the

Reynolds averaging procedure is used for averaging the flow

parameters. Adoption of the Boussinesq hypothesis that relates

the turbulent Reynolds stress and averaged velocity field V x, as

in Eq. (15), the eddy viscosity coefficient is introduced into the

mathematical description of the analysed phenomenon νt:

ρV xV r = −ρνt

∂V x

∂r (15)

In models based on such approach, experimental data con-

cerning cross-sectional eddy viscosity distribution are used and

the pipeline inertance can be related to the local acceleration of

flow by means of the weighting function W (t ):

J u =16ν

gD2

t

0

∂V

∂t (u)W (t − u)du (16)

Eq. (16) describing the inertance presented was proposed by

Zielke (1968) f or axisymmetric unsteady laminar flows. He also

proposed a form of weighting function W (t ) for these kind of

flows using inverse transformation of Laplace transform of the




equation of motion for laminar axisymmetric flow of an incom-

pressible fluid. Vardy and Brown (2003, 2004) have utilized

similar methodology for deriving their unsteady friction models.

In this model approximation of real eddy viscosity distribution

is used and flow is divided into two regions—the outer annu-

lar region (near the wall) where the eddy viscosity is assumed

to vary linearly with distance from the pipe wall and the inner

core region (in the vicinity of the pipe axis) in which the eddy

viscosity is assumed to be constant. Using such distribution of

eddyviscosity and applying Laplace transformation to equations

describing pipe unsteady flow, Vardy and Brown obtained a Re-

dependent weighting function W . There are two different forms

of weighting function for smooth and rough pipes and relation

(13) refers to smooth pipes unsteady flow:

W app =A∗e−τ/C∗√

τ with A∗ = 1

2√

π= 0.2821,

C∗ = 12.86

Reκ , κ = log10

15.29

Re0.0567 (17)

In caseof laminar flow ( Re≤ Rec-s) the value of the Reynolds

number in Eq. (17) is replaced by the critical value of Re

( Rec-s =2320) that separates laminar and turbulent flows (the

critical value of Re for transient flows in this model remain the

same as in case of steady flows). According to authors this model

is valid for initial Reynolds numbers Re < 108 and for smooth

pipes only.

4.3.2. Brunone et al. model

This model (Brunone et al., 1991) is related to the second

trend in hydraulic resistance modelling.

The authors of this model had carefully analysed the influ-ence of thequasi-steady hypothesis on theresults of calculations.

On this basis they stated that observed discrepancies between

calculations and experimental data are caused mainly by inap-

propriate mathematical description of inertial forces and wall

friction stress. It results directly from assuming the uniform

velocity profile in calculations. Analyses based on experimental

data concerning velocity profiles in different pipe cross-sections

and at wide range of Re revealed clear correlation between the

pipeline inertance and the acceleration (local and convective

velocity derivative) defined as

J u =k3

g∂V

∂t − a

∂V

∂x

(18)

Originally the k 3 coefficient assumed constant and fitted so as

to match computational and experimental results on a sufficient

level of conformity. Vardy and Brown proposed the follow-

ing empirical relationship to derive this coefficient analytically

(Vardy and Brown, 2003):

k3 =√

C∗

2 (19)

The coefficient C * is calculated according to Eq. (17) and it

cause that coefficient k 3 is variable, depending on the instanta-

neous Re values during unsteady flow.

Fig. 12. Comparison between calculated and measured pressure traces at the

valve—test run with initial Reynolds number Re0 = 5700.

4.4. Comparison between calculated and measured

pressures

Results of calculation (relative pressure head variations at

the valve for initial Reynolds number equal to 5700) are shown

together with experimental ones in Fig. 12. Calculations werecarried out using diamond grid of characteristics.

In contrary to the results of calculations obtained using

Relap5, Drako and Flowmaster2, which are based on the quasi-

steady hypothesis, the results calculated using the unsteady

friction models give good conformity with measured pressure

variations. Similar results have been published also by other

researchers as the authors of this paper (Adamkowski and

Lewandowski, 2004, 2006; Vitkovsky et al., 2000; Bergant et al.,

2001; Bughazem and Anderson, 2000; Ghidaoui et al., 2002).

It is worth emphasizing that the best conformity was obtained

using the Vardy and Brown model. This model predicts very

well not only the rate of pressure wave attenuation but it alsopreserves the wave front shape and the wave frequency. The last

feature is not characteristic for the Brunone et al. model.

4.5. Comparison of force F calculated with and without

dynamic friction

The good agreement between the measured and calculated

pressures implies that forces calculated using models of Vardy

andBrown or Brunone shall be more realisticthan corresponding

forces calculated with the quasi-steady friction model.

Force F has been calculated using Eq. (2) (momentum term

containing m(t )V (t ) was not considered) and dynamic friction




Fig. 13. Re0 = 5700. Force F calculated with Relap5 and using dynamic friction

models.

Fig. 14. Re0 = 5700, (a) first four peaks of force F from Fig. 11 and (b) last six

peaks of force F from Fig. 13.

Fig. 15. Re0 = 15,800. Last six peaks of force F (compare with Fig. 14).

(models of Vardy and Brown and Brunone). Calculations were

conducted for Reynolds numbers Re0 = 5700 and 15,800. The

results for Re0 = 5700 are shown in Figs. 13 and 14.

Theresults for Re0 = 15,800 are very similar, only amplitudes

are larger. Fig.15 shows last sixpeaksof force F for Re0 = 15,800

(compare with Fig. 14). Figs. 13–15 show clearly that consider-

ing the dynamic friction results in a very strong damping of force

F , i.e. forces calculated with Relap5, Drako and Flowmaster2

are conservative.

Figs. 13–15 show also that the Brunone et al. model gives a

small shifting of force frequency (it becomes slightly lower).

5. Conclusions

(a) The results of calculations made by means of the Relap5,

Flowmaster2 and Drako software packages have been com-

pared. Quite small differences between pressure variations

(waves) calculated by means of these packages have been

noticed. The main differences arewith regard only to numer-

ical damping. However, the differences between the forcescalculated with Relap5 and Drako/Flowmaster2 can be

significant because of sensitivity of Relap5 to numerical

damping. In order to reduce numerical damping in Relap5

use of Courant number (based on wave speed) between 0.02

and 0.005 is recommended. However further reducing the

Courant number can imply generating of purely numerical

high frequency oscillations of pressures and forces.

(b) The comparison of calculated (Relap5, Flowmaster2 and

Drako) and measured pressure variations show significant

differences. These differences regard mainly the damping

of pressure waves. The measured pressure oscillations are

more damped than the calculated ones. Use of quasi-steadywallfriction model in tested software packages is considered

as the main reason of observed differences.

(c) The discrepancies between calculated and measured pres-

sure waves canbe reduced by using unsteady friction models

in calculations. It is shown on the base of calculations per-

formed by means of special computer code which includes

chosen unsteady friction models.

(d) The dynamic friction has a very strong damping effect on

dynamic forces acting on pipe segments during the unsteady

flow conditions. Calculation of forces performed with the

quasi-steady friction model is therefore conservative.

(e) Experimental verification of calculations presented in this

paper concerns relatively small Reynolds numbers. It resultsmainlyfrom small diameter of pipe used at thetest rig. There

is indeed a strong need for conducting experimental verifi-

cation of calculations for higher Re. It is worth emphasizing

that experimental results confirming correctnessof unsteady

friction models at high values of Re are not known to the

authors of this paper.

Acknowledgment

The results of investigations published in this paper were

presented on the14th International Conference on Nuclear Engi-

neering ICONE 14, 17–20 July, Miami, FL, USA.




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