Nuclear Engineering and Design 238 (2008) 2546–2553

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Nuclear Engineering and Design

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Implementation of a strainer model for calculating the pressuredrop across beds of compressible, fibrous materials

A. Grahna,∗, E. Kreppera, S. Altb, W. Kastnerb

a Institut fur Sicherheitsforschung, Forschungszentrum Dresden-Rossendorf, PF 510119, D-01314 Dresden, Germanyb Institut fur Prozessautomatisierung, Hochschule Zittau/Gorlitz, PF 1455, D-02754 Zittau, Germany

a r t i c l e i n f o

Article history:Received 17 April 2008Accepted 24 April 2008

a b s t r a c t

Mineral wool insulation debris, which is generated during a loss-of-coolant-accident (LOCA), has thepotential to undermine the long-term recirculation capability of the emergency core coolant system (ECCS)in a nuclear power plant. Most importantly, ECCS pumps are faced with an increasing pressure drop whileinsulation debris accumulates at the pump suction strainers. The presented study aims at modelling thepressure drop of flows across growing cakes of compressible, fibrous materials and at the implementationof the model into a general-purpose three-dimensional (3D) computational fluid dynamics (CFD) code.

Computed pressure drops are compared with experimentally found values. The ability of the CFD imple-mentation to simulate 3D flows with a non-uniformly distributed particle phase is exemplified using a

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. Introduction

The investigation of insulation debris generation and transporturing loss-of-coolant-accident (LOCA) events as well as the shortnd long-term behaviour of the emergency core coolant systemECCS) must be considered with regard to the safety of pressurend boiling water reactors under such conditions (NRC, 2003; OECDEA, 1995, 2004). The mineral wool blankets that are used to insu-

ate the components of nuclear reactors can be destroyed by jettingteam during LOCA. A portion of the mineral wool fibre debris canhen be transported into the containment sump, which collects theooling water for use in the ECCS in the late phase of LOCA. Mineralool fibres that accumulate at the ECCS pump suction strainers

ead to increased pressure drops which could reduce the pumpsapability to recirculate the cooling water. Hazards associated withuch an incident were emphasized by an incident at the Barseback-2uclear power plant in Sweden in 1992 when a steam valve inad-ertently opened (ENS, 1992). The debris quickly blocked the ECCSump strainers, resulting in a potential compromise in the defense-

n-depth concept for the reactor.

The present paper reports on our efforts in modelling the pres-

ure drop buildup at strainers obstructed by fibrous materialsnd the implementation of the strainer model into the commer-ial, general-purpose computational fluid dynamics (CFD) code

∗ Corresponding author.E-mail address: a.grahn@fzd.de (A. Grahn).

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029-5493/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2008.04.010

h a horizontally embedded strainer plate.© 2008 Elsevier B.V. All rights reserved.

NSYS-CFX (ANSYS Inc., 2008). Special attention is drawn to theompressibility of the fibrous cake.

The work is embedded into a research project on genericxperimental investigation and CFD modelling of separate phe-omena related to the transport of fibrous insulation material inuclear reactor sumps, including sedimentation, re-suspension andtrainer clogging. While experiments are conducted at Universityf Zittau, theoretical work is concentrated at Forschungszentrumresden-Rossendorf.

. Theoretical model

Cakes of fibrous material which form on the upstream side ofhe suction strainers have two particular features that lead to diffi-ulties in the understanding of the flow mechanisms: (1) they aref very high porosity, and (2) due to the deformability of the fibres,uch cakes can be easily compressed under the action of fluid dragorces or an external compacting pressure. Equations describing theow in porous media as well as their compressibility are presented

n the following.

.1. Equation of flow through fibrous media

A standard approach in the investigation of fluid flow in macro-copically homogeneous porous media is to characterize the system

A. Grahn et al. / Nuclear Engineering an

Nomenclature

a, a0, b empirical parameters in the pressure drop equation,Eq. (7)

As solid phase mass specific surfaceC, D empirical parameters of compressibility function,

Eq. (12)d streamwise extension of strainer subdomain, Fig. 7L streamwise fibre bed lengthn streamwise strainer surface normal vectorNs strainer mass load�p pressure droppk mechanical compaction pressure in the fibre bedt, �t time, simulation time stepU continuous phase superficial velocityu continuous phase velocityus solid phase velocityx streamwise space coordinate

Greek lettersε fibre bed porosityεs solid phase volume fraction� continuous phase dynamic viscosity

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ε

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sare opposite in sign. For dpk/dx it follows from (7)

� continuous phase density�s solid phase material density

n terms of Darcy’s Law

= − k

�

�p

L. (1)

It linearly relates the superficial flow velocity U to the hydraulicressure difference �p that is applied to a layer of porous mate-ial of streamwise thickness L and permeability k; � is the dynamiciscosity of the fluid. However, as evidenced by numerous exper-ments, this relationship only holds for very low flow velocities

here viscous forces predominate. Unlike pipe flow, which is char-cterized by a sudden passage from the viscous to the inertialegime at a critical Reynolds number, the departure from the lin-ar U ∼ �p relationship proves to be gradual for flow in porousedia. Consequently, the contribution of inertia to the flow in the

ore space should also be examined in the framework of the lam-nar flow regime before assuming that fully developed turbulenceffects are present and relevant to momentum transport (Andradet al., 1999).

A more general relationship between flow velocity and pressurerop is given by the Forchheimer equation (Dullien, 1979)

�p

L= −(˛�U + ˇ�U2). (2)

It regards the flow resistance of a porous layer as being madep of two parts. The first one, which results from viscous forces,epends linearly on velocity, while the second one, resulting from

nertial effects, is proportional to density � of the liquid phaseimes the square of velocity. The relative importance of both parts iseighted by empirical coefficients ˛ and ˇ. Note that Eq. (2) is noturely empirical, since it can be derived by an appropriate averagef the Navier–Stokes equation for one-dimensional, steady incom-ressible laminar flow of a Newtonian fluid in an incompressible

orous medium (Dullien, 1979).

Initial efforts, dating back to the 1930s, focussed on the determi-ation of the coefficient ˛ of Eq. (2) for purely viscous flow withinorous media. They lead to the well-known Carman–Kozeny equa-

d Design 238 (2008) 2546–2553 2547

ion (Carman, 1937)

�p

L= −k(As�s)2(1 − ε)2

ε3�U, (3)

here As is the surface area per unit mass of the particle phase, �s

ts material density and k, the Kozeny constant, a to-be-determinedmpirical coefficient; porosity ε is defined as

= Vv

Vtot, (4)

nd expresses the ratio between the void (= pore) volume Vv andhe total volume Vtot of the porous bed. Eq. (3) has been exten-ively used in connexion with granular media; it has shown toield bad results for fibrous media, though. Analytical reasoningor this is given in Kyan et al. (1970). For fibre structures Davies1952) proposed the equation

�p

L= −a(As�s)2(1 − ε)1.5[1 + a0(1 − ε)3]�U, (5)

hich showed better agreement with measured pressure dropscross fibrous beds at laminar flow conditions. Based on a largemount of experimental data, Ingmanson et al. (1959) found uni-ersal values of 3.5 and 57 for the empirical coefficients a and a0.o date these constants have been widely used for laminar flowhrough fibrous porous media.

However, Davis’ equation still neglects the contribution of iner-ia to the over-all pressure drop of flow in the pore space. Thiseficiency was remedied by Ergun (1952) who suggested the rela-ion

�p

L= −b

As�s(1 − ε)ε3

�U2, (6)

or the turbulent flow regime in granular media. Nevertheless,xperimental studies (Kyan et al., 1970) indicated that the func-ional relationship in Eq. (6) can as well be applied to turbulentow in fibrous media. The empirical constant b was found to belose to 0.66.

Eqs. (5) and (6) can be combined to give a relation that encom-asses the whole range of flow regimes from laminar to turbulent.

t reads

�p

L=

−{

a(As�s)2(1−ε)1.5 [1+a0(1−ε)3]�U+b

As�s(1 − ε)ε3

�U2}(7

A simple force balance shows that the mechanical pressure,hich acts on the fibres and which results from the fluid drag,

ncreases in streamwise direction along the cake. As fibre cakesre compressible, this leads to a porosity distribution with a maxi-um at the upstream and a minimum at the downstream end. Fig. 1

llustrates this decrease of porosity by the shading getting darkern streamwise direction. Therefore, Eq. (7) can only be used to cal-ulate the differential change of the pressure drop d(�p)/dx fromocal porosity values ε(x). Hence, integration of Eq. (7) in stream-

ise direction is required to obtain the total pressure drop �p overhe fibre cake length L.

The local change in compacting pressure dpk/dx and the pres-ure drop d(�p)/dx of the flow have the same absolute value but

dpk

dx= −d (�p)

dx= a(As�s)2(1 − ε)1.5 [

1 + a0(1 − ε)3]�U

+ bAs�s(1 − ε)

ε3�U2. (8)

2548 A. Grahn et al. / Nuclear Engineering an

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Fig. 1. Fibre cake at a strainer.

As indicated above, the cake porosity depends on the local com-acting pressure. Hence, the complete description of the flow stillequires a compressibility function that relates porosity ε to theompacting pressure pk.

.2. Compressibility function

The volume reduction of a porous bed subject to a compact-ng pressure results from deformation of the solids. Generally,he volume reduction is, in parts, irreversible, because portions ofhe particles that constitute the bed may disintegrate or changeheir mutual orientation within the solid matrix. To the author’snowledge, a complete, theoretical foundation of the irreversibleompression of porous media has not yet been published. Oneossible method to workaround this problem is to use differentxpressions for the first and the subsequent compressions of theorous bed as shown in Jonsson and Jonsson (1992), where theolid is regarded as mechanically conditioned after the first com-ression, that is, having a constant compressibility independentrom the compacting pressure.

The present study considers the flow through a growing bed

f fibres which are deposited from a dilute fibre suspension duringOCA. Thus, the change of material properties due to repeated com-ression and release plays a secondary role and shall be neglectedere. It should be pointed out, however, that the latter assumptionoes not imply a restriction of the applicability of the flow equation

d

pot

Fig. 2. (a) Compaction measurement principle; (b) compaction m

d Design 238 (2008) 2546–2553

resented in the foregoing section as it can be combined with anyorking compressibility function.

For a given fibrous material the compressibility function muste determined experimentally. The measurement principle and thexperimental apparatus are depicted in Fig. 2 a and b. A knownuantity ms of insulation material, placed into a vertical cylinderith cross sectional area A, is subject to a uniform compactingressure pk, resulting from an externally applied force Fk. Then,k amounts to

k = Fk

A, (9)

nd porosity ε can be calculated from height h as

= 1 − ms

�sAh. (10)

The glass cylinder of the experimental apparatus had an inneriameter of 110 mm. The quantity of rock wool placed into theylinder varied from 19.4 to 97 g and compaction pressures rangingrom 0 to 16 kPa were applied.

Most compressibility functions for fibrous beds, relating bedorosity ε to mechanical compacting pressure pk, have the formMeyer, 1962)

= 1 − qprk, (11)

ith empirical parameters q and r. This expression, however, suf-ers from the fact that is does not give adequate estimates for theimiting cases of zero and infinite compaction pressures. Therefore,he four-parameter equation

(pk) = ε∞ + (ε0 − ε∞) e−CpDk , (12)

s suggested, which does not have this shortcoming. It has proveno reproduce measured relationships ε(pk) especially well. Fig. 2illustrates the expected curve as well as two of the parameters,

he porosities ε0 and ε∞ at zero and infinite compacting pressures.etermination of the parameters in Eq. (12) requires a non-linear

east-squares fitting method, such as the Marquardt–Levenberglgorithm, which is, for example, implemented in the open-source

ata plotting software Gnuplot (Williams et al., 2008).

Fig. 3 shows compaction measurements of a mineral wool sam-le. It can be seen that there is no significant effect of temperaturen the compaction properties in the applied range. Therefore,he measurements were fitted by a single set of parameters of

easurement apparatus; (c) typical compaction curve ε(pk).

A. Grahn et al. / Nuclear Engineering and Design 238 (2008) 2546–2553 2549

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ig. 3. Measured relationship of compaction pressure pk and porosity ε (compactionurve) for a given sample of mineral wool.

q. (12) to yield the solid line of the plot. Based on an esti-ated material density of �s = 2500 kg m−3 of the mineral wool

he following values were assigned: ε0 = 0.9833, ε∞ = 0.9147, C =.00712467 Pa−0.5197 and D = 0.5197.

. Application of the model equations to the 1D case

Flow Eq. (8) and the compressibility function (12) constitute annitial value problem for calculating the streamwise compactionressure and porosity profiles along the fibre cake. It has to beolved by integration with respect to x starting at the upstreamnd of the cake towards the strainer plate with initial conditionsk(0) = 0 and ε(0) = ε0, which correspond to zero compaction pres-ure and standard porosity of the mineral wool.

A stopping condition for the integration needs to be formulatedet. Integration should stop, as soon as the amount of fibrous phaseontained in the integration interval corresponds to a previouslyiven strainer mass load Ns,given. Strainer mass load Ns is defined asass of fibres per unit area of strainer plate. The contribution dNs

f an infinitesimal slice dx of the cake to the total strainer mass loads calculated from local porosity ε as

Ns = �s(1 − ε)dx. (13)

Thus, differential equation

dNs

dx= �s(1 − ε), (14)

ust be solved together with the differential equation of flow (8),ubject to the initial condition Ns = 0 at the upstream end of theake. Integration stops on fulfilling the condition Ns(x) = Ns,given,ielding the total length (streamwise thickness) L of the com-ressed fibre cake. The pressure difference over the entire fibreed follows directly from the compacting pressure at the strainerosition as

p = −pk(L). (15)

The system of Eqs. (8), (12) and (14) was implemented andolved numerically by means of GNU-Octave (Eaton et al., 2008).omputed compacting pressure and porosity profiles along beds ofineral wool at different superficial fluid velocities are displayed

n Fig. 4. The mass specific surface As of the fibrous material wasoughly estimated to be 160 m2 kg−1 according to equation

s = 4ds�s

, (16)

ased on a typical fibre diameter of ds = 10�m, visually determinedy microscopy, and the material density of 2500 kg m−3. Water at

tatwt

ig. 4. Computed porosity (a) and compaction pressure (b) profiles along a bedf mineral wool at different superficial water velocities; given strainer mass load:s = 10 kg m−2, strainer position at x = 0.

oom temperature was taken as the liquid phase. The left end xositions of the profiles mark the total length L of the compressedbre bed, while x = 0 marks the strainer position. It can be seen,hat higher flow velocities lead to a stronger non-linearity of theomputed profiles. As expected, the compaction pressure increasesrogressively in the streamwise direction due to decreasingorosity.

The total pressure drop �p was determined, both numericallynd experimentally, as a function of the superficial velocity. Achematic view of the experimental set-up is given in Fig. 5. The ver-ical test section consisted of two circular perspex tubes of 220 mmnner diameter where the first one, mounted on the upstream sidef the strainer plate, had a length of 1.4 m while the second one,erving as the downstream section, had a length of 1 m. The strainerlate was made of pre-fabricated perforated steel plate of 2.5 mmhickness with wholes of 4 mm diameter and 7 mm distance. Theolumetric flow rate was determined using a magneto-inductiveowmeter (Krohne, IFC 090) and the pressure difference usingn electric differential pressure transmitter (ABB, 2010 TD HART).ater was thermostatted by electric heating of the deionized water

ank; the temperature was measured at three locations (water tank,nlet and outlet of the test section) by means of thermocouplesType K, DIN EN 60584-2). After starting the pump, clean wateras circulated through the test rig until stationary flow rate and

emperature established. The rock wool suspension, stored in theolding tank, was then added to the flow at once. Again, the sys-em was given time to adjust flow rate and temperature as well

s to clear the circulating water before measuring the differen-ial pressure. Experimental runs started at the lowest flow rate,hich was stepwise increased for subsequent measurements. After

he final measurement of a run, the test section was dismounted

2550 A. Grahn et al. / Nuclear Engineering and Design 238 (2008) 2546–2553

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Table 1Experimental conditions

Run Ns (kg m−2) T (◦C) � (kg m−3) � (mPa s)

1 6.01 44.6 990.44 0.6012 6.01 59.0 983.69 0.474345

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Fig. 5. Test rig for pressure drop measurements.

n order to remove the accumulated fibre cake which was thenried and weighed for determination of the strainer mass loads.

Results of the differential pressure measurements are shownn Fig. 6. For comparison the computed curves and measured dataoints have been plotted into one diagram. The curves representifferent experimental conditions, i.e. strainer mass loads and fluidroperties, which are summarized in Table 1. It can be seen thatalculated and measured pressure drops are of the same order mag-itude and that the experimentally found non-linear relationshipetween pressure drop and flow velocity, which is characteristicor compressible fibrous media, could be reproduced in a qualita-ive manner. It must be noted that estimated values of solid density

nd fibre surface have been used for calculations. Better quantita-ive agreement could be obtained using exact values which wereot available to the authors.

ig. 6. Pressure drop across compressed fibre bed vs. superficial velocity: experi-ents (points) and computed profiles (lines).

t

∂

wsm

∂

3.87 58.9 983.77 0.4751.96 59.7 983.35 0.4690.32 59.7 983.35 0.469

. Implementation into CFD code ANSYS-CFX

The implementation of the model into a general purpose codeuch as ANSYS-CFX requires the transition from the zero to a two-imensional representation of the strainer plate. Moreover, theodel should be applicable to the simulation of transient flows.

he former task is addressed by placing a subdomain of fixed thick-ess d into the flow geometry. It represents the filter cake and thetrainer plate and separates the upstream from the downstreamegion, as illustrated in Fig. 7.

The cross-stream distribution of the strainer resistance is madep by a parallel connexion of multiple resistances, the magnitudef each depending on the local particle mass load and superficialelocity values. The resistance of the clean strainer plate is to beeglected in this study. Experiments have shown that its contribu-ion to the overall pressure drop is small in comparison to that ofhe fibre cake. Nevertheless, it can easily be accounted for by andditional loss coefficient.

In order to make allowance for transient flows, the strainer massoad distribution at time t has to be calculated by integrating thearticle phase mass flow passing through the strainer subdomainith respect to time according to

s(t) = �s

∫ t

0

εsus,⊥ d�, (17)

here εs represents the local particle phase volume fraction of theow, reaching the strainer subdomain at a velocity us,⊥ perpendicu-

ar to the strainer. For this to work, the particle phase must be able toreely penetrate the strainer subdomain. The particle phase can beemoved from the flow only after passing the strainer subdomain,sing an appropriately formulated sink term in the particle phaseontinuity equation which is applied to the downstream region.ccording to the CFX-Manual on solver theory the continuity equa-

ion, as implemented in the code, reads

t(εs�s) + ∇ · (εs�sus) = Smasss , (18)

here Smasss is a user specified mass source. Initially, the down-

tream region is free of particle phase. Since this state is to beaintained the instationary term must be zero:

t(εs�s) = 0. (19)

Fig. 7. Strainer represented as CFX subdomain.

ing and Design 238 (2008) 2546–2553 2551

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A. Grahn et al. / Nuclear Engineer

Thus, the mass source can be written as

masss = ∇ · (εs�sus). (20)

Eq. (20) even allows for possible changes in the solid materialensity. It is implemented as a User-Fortran routine into the CFXolver. Since the CFX solver does not allow to calculate the diver-ence term on the right hand side directly it had to be expandedrst to give

masss = �s (εs∇ · us + us · ∇εs) + εsus · ∇�s. (21)

Solution fields are available at discrete time steps only. There-ore, the integrand in Eq. (17) has to be evaluated numerically givinghe current strainer mass load Ns, cur. During the coefficient loops,istributions of the volume fraction εs and the associated particlehase velocity us are still intermediate. Hence, strainer mass loads, cur is calculated by adding the approximate mass load incrementf the current time step to the converged mass load of the previousime step:

s, cur = Ns, prev + εs�s�t max(us · n, 0). (22)

is the streamwise surface normal vector of the strainer plate.s, cur is updated at every coefficient loop and overwrites Ns, prev

nly at the end of the current time step, when the solution fieldsre taken as converged. Consequently, Ns, cur and Ns, prev must betored in two different user-defined solver variables.

Now that the mass load distribution Ns,cur is at one’s disposal,he compacting pressure pk acting onto the strainer can be com-uted from the local superficial velocity by solving Eqs. (8), (12)nd (14). Superficial velocity U in Eq. (8) is obtained from liquidhase velocity u by

= (1 − εs)u · n. (23)

The actual task of integrating the system of differential Eqs. (8)nd (14) is passed on to the differential equation solver ‘lsodar’hich is part of the open-source library ‘Odepack’ by Hindmarsh

t al. (2008). ‘lsodar’ is a solver with root finding capabilities as ittops the integration when it finds the root of at least one of a setf constraint functions. In the present case a constraint function isiven by

= Ns(x) − Ns, cur, (24)

here Ns(x) is the particle mass load at the current position x on thentegration path, cf. Fig. 1. The wanted pressure drop �p is readilybtained from the compaction pressure pk at the stopping positionccording to Eq. (15).

Finally, the flow resistance the liquid phase experiences withinhe strainer subdomain must be determined. It is modelled asource Smom in the momentum transport equation using the ‘Direc-ional Loss Model’ of CFX:

momS = −

(˛S�uS + ˇS�u2

S

), (25)

momT = −

(˛T�uT + ˇT�u2

T

). (26)

With this model, linear and quadratic loss coefficients ˛ and ˇay be provided for the streamwise (subscript ‘S’) and transver-

al (subscript ‘T’) directions of flow. The computed pressure dropp already contains the viscous (∝ u) and inertial (∝ u2) contri-

utions to the momentum loss, cf. Eq. (7). Therefore, only one ofhe coefficients ˛ or ˇ needs to be determined, while the other can

e set to zero. Here, the linear coefficient has been chosen. For thetreamwise direction of the liquid phase it reads

S =∣∣∣ �p

�d u · n

∣∣∣ . (27)

tstbo

Fig. 8. Flow geometry with computational grid.

For simplicity the same coefficient is used for the transversalow direction. If the homogeneous multiphase model is used inhe set-up of a particular flow problem, coefficient ˛S has yet toe divided by the continuous phase volume fraction. In the homo-eneous model, all phases share a single velocity field and theomentum source (25) is weighted by the volume fraction of each

hase. Dividing ˛S by the continuous phase volume fraction ensureshat the momentum source be applied entirely to the continuoushase.

Momentum source (25) can now be set for the strainer subdo-ain. On solving the momentum equations in a CFX solver run,

ntegration of Eq. (25) establishes the previously determined pres-ure drop between the up and downstream ends of the strainerubdomain.

. Three-dimensional (3D) simulations with ANSYS-CFX

As shown in Section 3, the strainer pressure drop model yieldsealistic values of �p in the one-dimensional case. CFD codes allowhe simulation of three-dimensional flows in arbitrarily shapedeometries. Hence, for testing the implementation of the strainerressure drop model it would be natural to consider flows with aon-uniformly distributed particle phase, that would lead to par-ially or at least unevenly loaded strainers. For this purpose, a stepike channel geometry with a horizontally embedded strainer wasonstructed.

.1. Problem set-up

The flow geometry that has been used for simulation is shownn Fig. 8. It consists of two straight channel segments with quadraticross sections representing the up and downstream parts. Bothhannels are connected by a strainer domain of size 20 cm × 20m and 1 cm thickness. The channel segments are discretizedy cubic elements of 1 cm edge length, whereas the strainers discretized into three layers of elements of size 1 cm × 1 cm1/3 cm.

The flow is simulated based on the homogeneous Euler–Eulerodel where a common flow field is shared by the continuous and

he solid particle phase. The SST turbulence model was chosen as ithowed best convergence. Buoyancy was not considered. Except forhe quadratic faces at both channel ends, the no-slip condition haseen assigned to all channel bounds. A plug-like, constant velocityf u = 4 cm s−1 was set at the inlet boundary on the left end of the

2552 A. Grahn et al. / Nuclear Engineering and Design 238 (2008) 2546–2553

F

co(p

wspBse

tlamz

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TwrctsfiswoevTdpsPdpat

Fig. 10. Distributions of (a) flow velocity and (b) pressure drop along strainer at thechannel mid plane.

Fig. 11. Fibre mass load distribution along strainer at the channel mid plane.

ig. 9. Flow field (streamlines) in the channel mid plane: (a) t = 0 s and (b) t = 40 s.

hannel, cf. Fig. 8, and the ‘Averaged Static Pressure’ condition at theutlet on the right end. As with the stand-alone 1D implementationSection 3), water at room temperature was used as continuoushase and rock wool as particle phase.

First, a stationary solution of the velocity and pressure fieldsas computed for clear water flowing through the domain. This

olution was then used as initial condition for the transient two-hase flow simulation. Transient runs made use of the 2-stepackward-Euler time discretization scheme and high-resolutionchemes were applied on the spatial derivatives of the transportquations.

A two-phase simulation starts with the particle phase enteringhe flow domain at the inlet face. Throughout the transient run, ainear volume fraction profile of the particle phase was maintainedt the inlet boundary in order to achieve a non-uniform impinge-ent onto the strainer. The particle volume fraction εs was set to

ero at the top and to 0.015 at the bottom of the channel inlet.

.2. Simulation results

Flow fields at different simulation times are shown in Fig. 9.he solution at t = 0 s corresponds to the stationary solution thatas found for clear water filling the computational domain. A large

ecirculation area forms behind the back facing step in the lowerhannel section and two smaller ones at the upper right corner ofhe upstream section as well as beneath the upper wall of the down-tream section. The most noticeable difference between the flowelds at start and end of the simulation is the flow direction in thetrainer subdomain. The particle laden strainer acts like a rectifierhich forces the flow into the vertical direction while smoothing

ut velocity differences, such as the high velocity at the right stepdge at x = 0.7 m. This becomes clearer in Fig. 10 a which plots flowelocity profiles along the strainer for different simulation times.he velocity maximum at the right end of the strainer decreasesuring the simulation. This rectifying effect is caused by the highressure drop across the clogged strainer, Fig. 10 b, leading to a pres-ure gradient whose maximum is in the strainer normal direction.

rofiles of mass load Ns are shown in Fig. 11. Initially, the mass ofeposited fibrous particles is unevenly distributed on the strainerlate due to the orientation of the latter within the flow geometrys well as the prescribed profile of the particle phase volume frac-ion at the channel inlet. While minimum and maximum values of

Fig. 12. Average strainer mass load, compaction pressure and pressure drop atstrainer vs. time.

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A. Grahn et al. / Nuclear Engineer

s differ by about two orders of magnitude at t = 15 s, differencesradually smooth out with time.

Fig. 12 shows how strainer mass load, compaction pressure andressure drop evolve with time. All three quantities have been aver-ged about the cross stream area of the strainer plate. The particlehase reaches the strainer after approximately 13 s of simulationime. At the same time the fibrous filter cake begins to form givingise to the pressure drop increase. As expected, it can be observedhat the pressure drop resulting from the momentum source, Eq.25), follows exactly the compaction pressure at the strainer sur-ace.

. Conclusion

The linear relationship between superficial flow velocity andressure drop, as suggested by Darcy’s law, fails in the case of com-ressible, fibrous media. In the present article a combination of aemi-empirical flow equation and a material equation is proposedhat allows to calculate the pressure drop in beds composed of thislass of materials. The system of model equations constitutes annitial value problem which is solved numerically for given strainer

ass load and flow velocity. Solid density, mass specific surface andhe static compaction properties of the fibrous material need to benown.

Computed porosity and compaction pressure profiles aretrongly non-linear along the fibre bed due to the compactionressure which increases in the streamwise direction. The experi-entally found non-linear relationship between flow velocity and

ressure drop could be reproduced using a one-dimensional imple-entation of the model equations. The model can be used for

mplementation into system codes for nuclear reactor and contain-

ent simulation. Thus, existing system codes might be enabled for

se in the risk assessment of loss-of-coolant accidents.The model has been successfully implemented as an exten-

ion to the general-purpose CFD code ANSYS-CFX. Its capabilityo simulate the transient pressure drop build-up at non-uniformly

O

O

W

d Design 238 (2008) 2546–2553 2553

oaded strainers in arbitrary three-dimensional geometries haseen demonstrated using a step-like flow geometry with a hori-ontally embedded strainer plate. It was shown that the increasingressure drop at the strainer has a rectifying effect on the flow andressure fields.

cknowledgment

This study was funded by the German Federal Ministry of Econ-my and Technology under the grant Nos. 1501270 and 1501307.

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