Numerical solutions of reeks hall equation for particulate concentration in recirculating turbulent fluid flow

Comparison between results from Vainshtein model and solutions to Equation 18 see reference [4] with Vainshtein resuspension rate constant.. Comparison between numerical solution from dy

NUMERICAL SOLUTIONS OF REEKS-HALL EQUATION

FOR PARTICULATE CONCENTRATION

IN RECIRCULATING TURBULENT FLUID FLOW

A Dissertation Presented to the Faculty of the Graduate School

at the University of Missouri

In Partial Fulfillment

of the Requirements for the Degree Doctor of Philosophy

by GIANG N NGUYEN

Dr Sudarshan K Loyalka, Dissertation Supervisor

MAY 2014

The undersigned, appointed by the dean of the Graduate School,

have examined the dissertation entitled

NUMERICAL SOLUTIONS OF REEKS.HALL EQUATION

FOR PARTICULATE C ONCENTRATION

IN RECIRCULATING TURBULENT FLUID FLOW

presented by Giang N Nguyen

a candidate for the degree ofDoctor of Philosophy

and hereby certify that, in their opinion, it is worthy of acceptance.

DEDICATION

This work is dedicated to my family and my parents I would like to express my deeply thanks to my father, Nguyễn Văn Thuỷ, and my mother, Nguyễn Thị Phương Dung, for all of their support to my education and professional career As time goes by, I further realize the importance of their decision to let me take English class and computer class when these were not popular at that time in our area These initial bricks laid down have helped expand my horizon to a much wider scale, make me go further than many other friends I also would like to thank my younger brother, Nguyễn Nam Hiếu, and his family for taking care of our parents while I am abroad pursuing my academic degrees for quite

a long time

ACKNOWLEDGEMENTS

Firstly, I would like to express my deeply sincere thanks to Professor Sudarshan

K Loyalka for his supervision, guidance, and his kind support in my study and research, his limitless effort in reviewing and correcting my manuscripts, and fruitful suggestions when I thought I had reached to a dead end of ideas for improvement and solution in research Working with you is my great honor and makes me become more and more a real researcher and scientist

I would like to express my special thanks to Professor Tushar K Ghosh as being my academic advisor, with his helpful support I have efficiently taken necessary classes to fulfill PhD program requirements and achieve a firm background for doing research, at present and also in the future I would like to thank other faculty members, Professor Dabir

S Viswanath, Professor Mark A Prelas, Professor Robert V Tompson, for your contributions, review, comments, and questions that help me have a better research project

I would like to thank my fellows who have assisted me in my research, Dr Mathew

P Simones, Michael L Reinig for their fruitful help on using computer tools, installing computer hardware and softwares

I would like to thank Latricia J Vaughn, department secretary, for her wonderful support on paper work, administration procedures, and many other issues since the first time I got admission letter till the day I graduate; James C Bennett for his assistance on financial issues

And last but not least, I would like to thank Vietnam Education Foundation for providing me a scholarship so I have a great opportunity to pursue a PhD degree in Nuclear

Engineering in the United States, a dream of my life, and I also thank Professor Sudarshan

K Loyalka for his funding resource from Department of Energy NERI-C grant

DE-FG07-14892, 08-043 in supporting my research

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

LIST OF FIGURES vii

LIST OF TABLES ix

NOMENCLATURE x

ABSTRACT xv

I INTRODUCTION I.1 Significance of the work 1

I.2 Aerosol resuspension phenomenon and impact 3

I.3 Objectives and outline 4

II LITERATURE REVIEW II.1 Resuspension models 5

II.1.1 Reeks, Reed, and Hall model (RRH model) 5

II.1.2 Vainshtein model 7

II.1.3 Rock  ‘n  Roll model 10

II.1.4 Williams’  exact  solution  to  Reeks  and  Hall  equation 13

II.1.5 Multilayer aerosol deposits resuspension 17

II.1.6 Validation status 19

II.1.7 Resuspension models in applications 25

II.2 Resuspension experiments 26

II.2.1 Wells et al’s experiment on deposition of aerosol particles to surfaces in the coolant of a commercial carbon dioxide cooled reactor 26

II.2.2 Wen and Kasper’s  experiment  on  particle  reentrainment  from  surfaces 27

II.2.3 STORM experiment 28

III METHODOLOGY III.1 Overview 30

III.2 Numerical and exact solutions III.2.1 Description of numerical technique 30

III.2.2 Exact solution 38

III.3 Experimental data 39

III.4 Improvement 42

III.4.1 Investigation with resuspension rate p as function dependent on time 42

III.4.2 Sensitivity analysis with coefficients a and b of Reeks and Hall equation 46

III.4.3 An attempt on finite difference numerical method to solve Reeks and Hall equation 46

IV RESULTS IV.1 Experimental data adaptation 49

IV.2 RRH numerical validation 52

IV.2.1 Numerical solution 52

IV.2.2 Benchmarking against Williams exact solution 55

IV.2.3 Comparison with experimental data 58

IV.3 Vainshtein model numerical validation 62

IV.4 Rock  ‘n  Roll  model numerical validation 68

IV.5 Modification and improvement 74

IV.5.1 Investigation with resuspension rate p as function dependent on time 74

IV.5.2 Sensitivity analysis with coefficients a and b of Reeks and Hall equation 77

IV.5.3 An attempt on finite difference numerical method to solve Reeks and Hall equation 80

V CONCLUSIONS 84

REFERENCES 87 VITA 93

LIST OF FIGURES

1 Particle-surface  geometry  for  Rock  ‘n  Roll  model 10

2 Fractional resuspension rate Λ(𝑡) as a function of time for 𝑟̅ = 0 15

3 Particulate concentration for 𝑟̅   =  4.636 ×10-3 from exact solution and from Reeks and Hall approximation 16

4 Dimensionless resuspension flux Φ  versus  dimensionless time 𝑡 19

5 The injection experiment results for  2  μm  iron  oxide  particles  at full reactor coolant flow 20

6 Comparison of resuspension measurements with model predictions for the nominal 20 μm  alumina spheres 21

7 Comparison  of  MELCOR  Rock  n’  Roll  resuspension  model  with  STORM  Test   SR11 resuspension data 22

8 Comparison between results from Vainshtein model and solutions to Equation (18) (see reference [4]) with Vainshtein resuspension rate constant 23

9 Comparison between results obtained by various resuspension models against Reeks and Hall experimental data 24

10 Main results from Wells et al’s experiment: Variation of normalized concentration during the first 18 minutes (full coolant flow) 27

11 Reentrained particle concentration versus time for a section of thin Teflon tubing at 𝑄 = 2.7  𝑚 /ℎ Particles are unspecified 28

12 Fractional resuspension rate for 5 𝜇𝑚 iron oxide particle 31

13 Schematic diagram of experiment 40

14 Experimental result 42

15 Fluid flow velocity in the first scenario 44

16 Fluid flow velocity in the second scenario 45

17 Related  functions  of  5  μm  iron  oxide  particle 53

18 Particulate concentration by numerical method 55

19 Comparison with exact solution 57

20 Comparison with experimental data 59

21 Comparison between numerical solution, exact solution, and experimental data with the new set of coefficients: 𝑎 = 0.035, 𝑏 = 0.022 61

22 Log-normal distribution function 𝜑 5  μm  iron  oxide  particle  case 65

23 Resuspension rate constant 𝑝(𝑟) in Vainshtein model 65

24 Resuspension rate constant Λ(𝑡)  5  μm  iron  oxide  particle  case 66

25 Comparison between numerical solution from Vainshtein model and experimental data 68

26 Lognormal distribution function 𝜑(𝑓 ) 5  μm  iron  oxide  particle  case 72

27 Resuspension rate constant function 𝑝(𝑓 )  5  μm  iron  oxide  particle  case 72

28 Fractional resuspension rate function Λ(𝑡)  5  μm  iron  oxide  particle  case 72

29 Comparison  between  numerical  solution  from  dynamic  Rock  ‘n  Roll  model  and   experimental data Coefficients 𝑎 = 0.035, 𝑏 = 0.022 74

30 Comparison between time dependent in sine function form and time independent cases  for  5  μm  particle 75

31 Comparison between time dependent and independent cases of fluid flow rate for 5 μm  particle  in  a  reactor  shutdown  procedure 77

32 Numerical solutions with various coefficients a and b combination (for  5  μm  iron oxide particle) 79

33 Sensitivity analysis with time step ∆ℎ (for  5  μm  iron  oxide  particle) 82

34 Numerical solutions with time step ∆ℎ = 0.5  𝑠 for 0.6, 2, and 17 μm  particles 83

LIST OF TABLES

1 Gas properties 50

2 Input parameters 50

3 Calculated parameters as obtained from input parameters in Table 2 52

4 Calculated values of tangential pull-off force and drag force 64

5 Calculated values of several  parameters  used  for  dynamic  Rock  ‘n  Roll  model 71

NOMENCLATURE

𝐴(𝜉) Probability density function of intrinsic particle parameter 𝜉

𝑎

Distance between two contact points

A coefficient in Reeks and Hall equation

𝑚

𝑠

𝛽 Damping constant

𝛽 Fluid damping constant

𝛽 Mechanical damping constant

𝐶 Normalized particulate concentration

𝜂, 𝜂 Resonant coefficient

𝐸 Normalized energy spectrum of fluctuating lift-force

𝐸 Universal energy spectrum of lift force

〈𝐹〉 Average of aerodynamic force 𝑁

𝑓 Fluctuating component of aerodynamic force 𝐹(𝑡)

〈𝑓̇ 〉 Covariance of time derivative of aerodynamic force fluctuations 𝑁 /𝑠

𝑓 ln 𝑓 is mean of ln 𝑓 in its normal distribution representation

𝑓 Normalized adhesive force

〈𝑓 〉, 𝑓̅ Average normalized adhesive force

𝑓 𝑓 = ∫ 𝜑(𝑟 )𝑑𝑟̅

𝐹 Adhesive force of a sphere particle on a perfectly smooth surface 𝑁

𝛾 Free surface energy of contact material (𝑖 = 1, 2) 𝐽/𝑚

𝐾 Elastic constant

𝑘 Numerical constant depending on form of potential well

𝜈 Poisson’s  ratio  of  particle

𝜈 Poisson’s  ratio  of  wall  surface

𝜔 Natural frequency of particle in surface potential well 𝑟𝑎𝑑

𝜔 Natural frequency of oscillation

𝑝̂ Probability of resuspension at a specific time point 𝑠

𝑃 Hertzian force 𝑁

𝜑 Log-normal distribution function of normalized adhesive radii or

normalized adhesive force

Φ Dimensionless resuspension flux

𝑃𝑟 Prandtl number

𝜉 Intrinsic particle parameter

Ξ Domain where 𝐴(𝜉) and 𝑝(𝜉) are defined

𝑟 Normalized adhesive radius

𝑟̅ Mean of 𝑟 in log-normal distribution

𝑟̂ ln𝑟̂ is mean of ln 𝑟 in its normal distribution

𝑡 Duration time for initial resuspension 𝑠

ABSTRACT

Source term is an important issue in safety assessment of nuclear power plants Therefore, modeling of particulate concentration in reactor coolant systems during normal operation and hypothesized or real accidents is of continuing interest We report here on exploration

of a numerical solution of the Reeks and Hall equation with the use of fractional resuspension rate in its original integral form The numerical results for particulate concentration are compared with those obtained from the exact expression given by

Williams and experimental data provided by Wells et al The numerical results agree very well with exact results and also agree well with the data of Wells et al Applications of

numerical method to problems with time dependent resuspension rate (for which exact solutions are not available), are explored and some typical results are reported Research is carried out for three related resuspension models: Reeks, Reed, and Hall (1988), Vainshtein

et al (1997), and   Rock   ‘n   Roll   (2001)   Results   from   Rock   ‘n   Roll   model   show   some  

advantages over the other two models Since the advanced numerical technique we used may not be entirely suitable for use in large integrated computed codes, we have also explored use of a first order finite difference scheme for solving the Reeks-Hall equation

This first order scheme is sensitive to time-step size, but can work in some cases

CHAPTER I INTRODUCTION

I.1 Significance of the work

Currently, nuclear power plants are well designed with defense-in-depth principle and multiple barriers to prevent radioactive materials from escaping and releasing into atmosphere and surrounding environment The three basic barriers used in nuclear power plant include fuel pellet, reactor cooling system, and reactor containment building which help to confine and contain radioactive materials within nuclear power plant during normal operation and accident However, in some initiating events or transients, there are probabilities for decay products to escape from fuel rods to coolant system and further due

to cracking, swelling in fuel pellet and fuel rod Although filtration systems are available

to collect them later but a good estimation of how much material has been released will help a response

With new reactor designs in generation IV, especially Very High Temperature Reactor (VHTR), fuel elements and structural materials must operate at high temperature conditions, of about 900°C to 1,100°C Again, proper safety features are needed in case of accidents

The source term plays an important role in safety assessment and analysis for nuclear

power plants Technically it is based on evaluation of radioactive species and particulate

concentration in primary coolant system Among various contributing factors and phenomena, it has been thought that resuspension of particulates can be important

To count for contribution from resuspension phenomenon, several calculation models have been developed and verified Among these, basic model given by Reeks, Reed, and Hall

(1988) [1] is a more analytical approach in which an integro-differential equation was established It is briefly mentioned as RRH model hereafter Development of RRH model

is based on theory of particle movement in a potential well, also called as energy balance When particle gains enough kinetic energy to overcome the potential wall, it becomes free and resuspended

Vainshtein et al (1997) [2] further developed the RRH model based on force balance

approach by utilizing tangential pull-off force for separating particles from adhesive surface This resuspension mechanism requires less energy to detach particles, therefore, it gives higher resuspension rate Further development of the RRH model, Reeks and Hall (2001) [3] introduced   the   Rock   ‘n   Roll   model   in   which   resuspension   is   facilitated   by  particle’s  vibration  and  rolling  movement  on  wall  surface

Currently,  the  Vainshtein  and  Rock  ‘n  Roll  models  have  been apparently integrated into MELCOR code, a fully integrated code that is capable of modeling severe accident progression in light-water reactor nuclear power plants However, there still exist considerable limitations in both modeling and verification/validation or some details of comparison and validation presented in Idaho National Laboratory Report on Resuspension Model for MELCOR for Fusion and Very High Temperature Reactor Applications (2011) [4]

Since the integro-differential equation that was established in the same paper with RRH model [1], there have been several publications dealing with solving and validating this equation, analytically and numerically Reeks and Hall (1988) [5] presented comparison

between numerical solution and experimental data provided by Wells et al [6] Numerical

solution was achieved by using an implicit technique and based on an approximation of

fractional   resuspension   rate   Λ(t)   Williams (1992) [7] presented an exact solution by Laplace transformation

I.2 Aerosol resuspension phenomenon and impact

Radioactive aerosol in turbulent fluid flow is formed by radioactive material attaching to various types of particles This process yields some concentration of radioactive material

in fluid flow and may cause harmful effects to human health As time goes by, aerosol gradually deposits onto wall surfaces of cooling system due to adhesive force and its concentration in fluid flow decreases This deposition process will limit the dispersion of radioactive material in the reactor primary systems and amount escaping to containment building, or even further, if there is leakage or explosion

However, under the impact of lift force and drag force from turbulent flow, deposited aerosol can resuspend and join into fluid flow again Mechanisms for resuspension process are presented in several models, including energy accumulation, force balance Resuspension process causes particulate concentration in fluid flow to increase Therefore, higher level of radioactive material exists in coolant system and it is available for escaping into surrounding environment Radioactive material escaping from nuclear reactor will cause radiation exposure to workers, surrounding population and food chains

Therefore, a good calculation of how much radioactive material available in primary coolant system will help achieve improved estimations of impacts to human health and surrounding environment Based on that one can take actions to protect human health and environment, and minimize harmful outcomes

I.3 Objectives and outline

In an effort to contribute more on the application of resuspension models, this research focuses on finding numerical method to solve Reeks and Hall equation with the use of fractional   resuspension   rate   Λ(t)   in   its   original   integration   Numerical   solution   is   then  benchmarked against exact solution by Williams (1992) [7] and validated by comparison

with experimental data provided by Wells et al (1984) [6] Following validation, time

dependent resuspension rate is explored, and some sample results are reported

Investigation is initially carried out with the basic RRH model It is then extended to Vainshtein   model   and   Rock   ‘n   Roll   model   As   all   of three models have the same foundation, it will be easier to make comparison and justification

CHAPTER II LITERATURE REVIEW

II.1 Resuspension models

II.1.1 Reeks, Reed, and Hall model (RRH model)

To determine contribution from resuspension to particulate concentration in a turbulent flow, Reeks, Reed, and Hall established a resuspension model in 1988, we named it RRH model In this model, particles are considered vibrating in potential well This potential well is formed by adhesive force Under impact from turbulent flow, vibrating movement

is enhanced and when particle achieves its energy high enough to overcome the wall of potential well, it becomes resuspended Energy from turbulent flow is transferred to particle by lift force By this approach, probability for particle to resuspend is higher than the case where only force balance of adhesive force and lift force is applied

For particle in potential well, its resuspension rate constant is represented by:

𝑝 =𝜔2𝜋exp −

where 𝑟 is normalized adhesive radii; 𝑝(𝑟 ) is resuspension rate constant as a function of

𝑟 ;  𝜑(𝑟 ) is log-normal distribution function of 𝑟

𝑝(𝑟 ) is expressed in more detail as:

where 𝑟 is adhesive radius corresponding to adhesive force 𝑓 of wall surface 𝑅 is radius

of a sphere particle on a perfectly smooth surface that has adhesive force equal to 𝐹

𝑟 follows log-normal distribution:

(2𝜋) /

1𝑟

Another important parameter normally used in resuspension calculation is fractional resuspension rate, which is defined from remaining fraction 𝑓 (𝑡) as follows:

II.1.2 Vainshtein model

In 1997, Vainshtein, Ziskind, Fichman, and Gutfinger established their own resuspension model, hereafter called as Vainshtein model This model also uses potential well and particle fluctuates inside this potential well However, different from RRH model, energy transfer from turbulent flow to particle is by tangential pull-off force, 𝐹 Particle is whether resuspended or not depending on balance between adhesive force moment and tangential pull-off force moment Investigation from research shows that resuspension rate caused by tangential pull-off force is higher than the case by lift force in RRH model

Similar to RRH model, important parameters of Vainshtein model are determined as follows

Resuspension rate constant is defined as:

where 𝑢 is friction velocity, 𝜈 is fluid kinematic viscosity, ∆𝛾 is adhesive surface energy,

𝑟 is particle radius, 𝐾 is elastic constant, 𝜇 is fluid dynamic viscosity, and 𝛾̇ is shear rate Average value of shear rate is determined by

〈𝛾̇〉 = 0.3𝑢

Fraction of particle remaining and distribution of adhesive radii are the same as with RRH model:

𝑓 (𝑡) = exp[−𝑝(𝑟 )𝑡] 𝜑(𝑟 )𝑑𝑟 (16)

(2𝜋) /

1𝑟

1

ln 𝜎 exp −

(ln 𝑟 − ln 𝑟̂)

As we can see from Eqs (12) and (14), Vainshtein et al indirectly define 𝑝 as a function of

particle radius 𝑟, meanwhile, they define lognormal distribution function 𝜑 as a function

of normalized adhesive radii, 𝑟 Therefore, we need a step further to interpret 𝑝 as a function of 𝑟 or 𝜑 as a function of 𝑟

From Eq (4) and its note on definition of adhesive radius, we have:

𝑓

𝐹 𝑑𝑟 =

1

II.1.3 Rock  ‘n  Roll model

Rock  ‘n  Roll  model  was  established  by  Reeks  and  Hall  in   2001,  named  as  RnR  model  hereafter It is an advance of RRH model In this model, particle is considered as being in

a potential well, oscillating about a point of contact Energy transfer to particle is not only from lift force but also from drag force Particle removal is based on balance between moments from lift force and drag force, named together as aerodynamic force moment, versus moment from adhesive force

Figure 1 Particle-surface  geometry  for  Rock  ‘n  Roll  model

There are two Rock  ‘n  Roll  models The first RnR model takes into account resonant energy transfer (when typical frequency of aerodynamic forces is close to natural frequency of

particle deformation) and it is named as Dynamic  Rock  ‘n  Roll  model The second RnR

model does not take into account resonant energy transfer (based on investigation result that contribution from resonant energy transfer is rather small and insignificant) and it is

named as Quasi-static  Rock  ‘n  Roll  model

Dynamic  Rock  ‘n  Roll  model

Basic formula for resuspension rate constant is the same as in RRH model:

𝑝 = 𝑛 exp − 𝑄

However, bursting frequency 𝑛,  height of potential wall 𝑄, and average potential energy

〈𝑃𝐸〉 of particle are determined in a different way In this model approach, resuspension rate constant is rewritten as follows:

𝐹 and 𝐹 are lift force and drag force, respectively

Hence,  𝑝 in Eq (27) can be expanded in these force terms as follows:

𝑝 = 𝑛 exp −𝑘 (𝑓 + 1/2𝑚𝑔 − (𝑟/𝑎)𝑚𝑔 − 〈𝐹〉)

Bursting frequency 𝑛 is defined and determined by:

𝑎 is distance from adhesive point Q and pivot point P (see Figure 1); 〈𝐹〉 is average of

aerodynamic force; 〈𝑓 〉 is covariance of aerodynamic force fluctuations; 〈𝑓̇ 〉 is covariance of time derivative of aerodynamic force fluctuations; 𝜂 is resonant coefficient;

𝜔 is natural frequency of oscillation

When 𝜂 = 0 (non resonance), Eq (30) becomes:

Λ(𝑡) =   −𝑓̇ (𝑡) = 𝑝(𝑓 ) exp[−𝑝(𝑓 )𝑡] 𝜑(𝑓 )𝑑𝑓 (33)

which is then substituted into Reeks and Hall equation (Eq (10)) for finding particulate concentration in turbulent fluid flow

Quasi-static  Rock  ‘n  Roll  model

In this model, particle is considered impacted by aerodynamic force with frequency off from natural frequency of particle-surface deformation Therefore, there is not resonant energy transfer Based on result from investigation, resuspension rate constant for quasi-static RnR model is written as follows

〈𝑓 〉 exp −(𝑓 − 〈𝐹〉)2〈𝑓 〉1

II.1.4 Williams’  exact  solution  to  Reeks  and  Hall  equation

Williams (1992) [7] solved Reeks and Hall equation analytically by Laplace transformation and its inversion Solution achieved shows similar behavior as what predicted by Reeks,

Reed, and Hall [1], but more than that, it helps recognize and analyze characteristic features

of the model

Williams used Laplace transformation to solve Reeks and Hall equation and found exact solution for particulate concentration in turbulent fluid flow, taking into account both resuspension and deposition phenomena He assumed fractional resuspension rate Λ(𝑡) used in its original form by definition in Reference [1]:

where 𝑟 is normalized adhesive radius, 𝑓 = ∫ 𝜑(𝑟 )𝑑𝑟̅

Fractional resuspension rate was also investigated and result achieved is as in Figure 2 which shows fractional resuspension rate in time range from 10-6 second to 106 seconds In logarithmic scale system, fractional resuspension rate is rather flat up to about 10-3 s and then from 0.1 s it has a form asymptotically with 𝜉/𝑡 It confirms the behavior noted by Reeks and Hall with 𝛾 = 1.0417 and 𝜉 = 0.009693

Figure 2 Fractional resuspension rate Λ(𝑡) as a function of time for 𝑟̅ = 0

By applying 𝑟̅ = 0, it is simpler to investigate characteristics of the exact solution which is then rewritten in the form

𝐶(𝑡) = 𝑏 𝑝(𝑟 )𝜑(𝑟 )𝑒 𝑑𝑟

𝑝(𝑟 ) − 𝑎 + 𝑏 ∫ 𝑝(𝑟 )𝜑(𝑟 )𝑑𝑟𝑝(𝑟 ) − 𝑝(𝑟 ) + 𝑏𝜋𝑝(𝑟 )𝜑(𝑟 )𝑝 (𝑟 ) (41) Comparison between exact solution with Reeks and Hall results is also presented in Figure

3 and it shows very good agreement

Figure 3 Particulate concentration for 𝑟̅ = 4.636×10-3 from exact solution and from

Reeks and Hall approximation

From results achieved, it confirms that an exact solution of Reeks and Hall equation can

be obtained by Laplace transformation Laplace inversion is possible and leads to an analytical exact solution that highlights important parameters and explains nature of solution

II.1.5 Multilayer aerosol deposits resuspension

RRH model initially applies for single deposits layer However, in reality this assumption may not be true where particles remaining on wall surface have multilayer structure This multilayer structure has variant resuspension properties because not only interaction between neighbor layers is dominant against interaction with wall surface due to distance but also fluid flow can interact with more than one particle layer at the same time Several researches have taken care about this issue Fromentin (1989) [8] set up experiment PARESS (Particle RESuspension Study) to investigate resuspension flux and found that variation of resuspension flux with time has a power law form as follows:

where c and d are parameters dependent on nature of deposit and flow velocity

Friess and Yadigaroglu (2001) [9] systematically reviewed relevant researches, including results from Fromentin, and approached to a multilayer aerosol deposits resuspension model In their model, particles from different layers can involve in resuspension process, depending on surface coverage fraction and  particles’  exposure  to  fluid  flow

For single layer deposits, normalized resuspension flux is determined by

where 𝐴(𝜉) is probability density function of intrinsic particle parameter 𝜉; 𝑝(𝜉) is resuspension rate constant; Ξ is domain where 𝐴(𝜉) and 𝑝(𝜉) are defined This resuspension flux is equivalent to fractional resuspension rate Λ(𝑡)

In consideration of multilayer deposits, there is an extreme case of infinitely thick deposits For this scenario, resuspension flux is

In the case of finite multilayer aerosol deposits, resuspension flux is defined as

Φ (𝑡) = Φ (𝑡) + Φ (𝑡 − 𝑡 )Φ (𝑡 )𝑑𝑡 ,      𝐿 ≥ 2 (46)

where 𝐿 is number of initial layers

Based on knowledge from multilayer aerosol deposits resuspension, Friess and Yadigaroglu (2001) adapted and added multilayer topology to monolayer models, such as RRH model By this way, it helps monolayer models work better with deposit configurations of different thicknesses

Figure 4 Dimensionless resuspension flux Φ versus dimensionless time 𝑡

Φ = 𝑘 Φ ;  𝑡 = 𝑘 𝑡

II.1.6 Validation status

Currently, although the aforementioned models of particulate resuspension have been developed for quite a long time but not much experimental data is available for a full set

of validation to particulate concentration achieved from Reeks and Hall equation At the beginning, measured data were sparse, a few are given by Reeks and Hall [1] for resuspension factor of silt on grass, gas-born concentration of iron oxide particles Most of initial validation for RRH model was carried out by Reeks and Hall [5] where they used

experimental data from Wells et al (1984) [6] They compared experimental data with calculation results and investigated solution behavior Wells et al had done an experiment

by injection of aerosols into recirculating coolant system of a Commercial Advanced Cooled Reactor (CAGR) Aerosols injected have  four  diameter  sizes:  0.6  μm,  2  μm,  5  μm,  and  17  μm  The  first  three  sizes  are  of  iron  oxide,  this  oxide  is  typically  found  in  nuclear  reactors  The  17  μm  aerosol  is  aloxite  where  its  use  was  based  on  material  availability  2  

Gas-μm   and   5   Gas-μm   are   the   most   representatives for particulate dimension present in reactor coolant

In their first effort solving Reeks and Hall equation, they assumed an approximation of fractional resuspension rate Λ(𝑡) ≈ with 𝜀 chosen in the range from 1.01 to 1.1, and used

a numerical implicit technique to solve Eq (10) Comparison with experimental data from

Wells et al (1984) shows rather good agreement Figure 5 presents comparison between

numerical solution and experimental  data  for  2  μm  iron  oxide

Figure 5 The injection experiment results for  2  μm  iron  oxide  particles  at full reactor

coolant flow

Best fit of numerical solution with experimental data is gained by varying 𝜀 values However, these values are in a small range and they can be corresponding to error in accuracy of experimental data

As time allowed for improvement and advancement, Reeks and Hall provided much more useful  experimental  data  at  the  time  they  established  the  Rock  ‘n  Roll model [3] in 2001 However, these data are mostly in the form of particulate remaining fraction on surface versus friction velocity Experimental data used for validation are measurements for 10

μm,  20  μm  alumina  particles,  and  graphite  particle Comparison is made for both RRH and Rock  ‘n  Roll  models and it shows that remaining fraction  provided  by  Rock  ‘n  Roll  model  gives better approach to experimental data than RRH model

Figure 6 Comparison of resuspension measurements with model predictions for the

nominal 20 μm  alumina spheres

In the technical report from Idaho National Laboratory (INL, 2011) on aerosol

measured data from STORM (Simplified Test of Resuspension Mechanism) experiment were used for validating resuspension module in MELCOR code Its validation summary shows expression on using mass deposited parameter, not the particulate concentration in fluid flow Results achieved show rather good agreement between experimental data and MELCOR solution An example is shown in Figure 7 below

Figure 7 Comparison  of  MELCOR  Rock  n’  Roll  resuspension  model  with  STORM  Test  

SR11 resuspension data

Vainshtein model was established in its original paper in 1997 [2] In this paper, Vainshtein

et al used a set of data for calculating remaining fraction parameter based on their newly

established model Authors in INL Technical Report (2011) benchmarked their numerical

calculation  of  remaining  fraction  against  “exact”  solution  given  above  by  Vainshtein  et al

Comparison result is shown in Figure 8 below There is a good agreement between results from two numerical algorithms and Vainshtein model

Figure 8 Comparison between results from Vainshtein model and solutions to Equation

(18) (see reference [4]) with Vainshtein resuspension rate constant

In a more comprehensive investigation and validation work done by Stempniezwicz and Komen (2010) [10], they have made comparison for results from several resuspension models against Reeks and Hall experimental data [3], for both aluminum particle sizes 10

μm    and  20  μm  Models  used  include  Rock  ‘n  Roll,  Vainshtein,  NRG3,  NRG4 The two resuspension models NRG3 and NRG4 are “On/Off”   type, resuspension occurs when aerodynamic force is large enough in comparison with adhesive force, meaning drag force becomes greater than tangential pull-off force (NRG3); or drag moment becomes greater than adhesive moment (NRG4) All of these models are constructed and investigated in thermal-hydraulic code SPECTRA Similar to most of other validation, work was done only with remaining fraction parameter as shown in Figure 9