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Fouling due to Particulate flow across

tube bundles

Seshagopal Narayanan Report Number WET 2007.21

Supervisors: Dr. Ir. C.C.M. Rindt Prof. Dr. M.V. Krishnamurthy Eindhoven University of Technology Vellore Institute of Technology Department of Mechanical Engineering Division Thermo Fluids Engineering Section Energy Technology

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Abstract

In this project a numerical model for fouling of tube bundles is developed. A three step sequential methodology involving turbulent flow, particle tracking and particle adhesion is adopted to model the phenomenon. In this model, greater emphasis is laid on the turbulent flow field and particle transport problems as reliable models already exist for particle adhesion. Firstly a robust model for turbulent flow is developed and this model is used as a base for development of particle tracking model. Both these models are used finally with an existing particle adhesion model to calculate fouling rates. The turbulent flow field is solved by utilizing a combination of k-ω model and Reynolds stress model to close the set of RANS equations. Data for parameters like velocity, fluctuations and Reynolds stresses are extracted and compared with existing experimental results from literature. The predicted values give good agreement with measured values. Additionally, the vortex shedding phenomenon is also studied and the Strouhal number is determined. A Lagrangian particle tracking model is developed and is used to determine particle trajectories. Finally all of this data is coupled with an existing particle adhesion model to calculate deposition rates on cylinder surface. The data trend is compared with fouling experiments to validate the model.

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Nomenclature

a Local speed of sound m/s CD Drag Coefficient - Cµ Viscosity coefficient in k-ε model - Dtube Diameter of the tube m FD Drag Force kgm/s2 Gk Production of turbulent kinetic energy m2/s2 Gω Production of specific dissipation s-1

k Turbulent kinetic energy m2/s2 Mt Mach number - R Universal gas constant kgm2/Ts2 Re Reynolds number - S Mean rate of strain tensor s-1 Str Strouhal Number - T Temperature K u Velocity m/s uffff Mean velocity m/s

u . Velocity fluctuation m/s U∞ Free stream velocity m/s u. i u. j

fffffffffffffffffffff Reynolds stress tensor kg/ms2

Yk Dissipation of kinetic energy m2/s2 Greek Letters α angle of incidence deg/rad α* Damping coefficient - Γk effective diffusivity of k m2/s Γω effective diffusivity of ω m2/s γ Ratio of specific heats - δ ij Kronecker Delta - ε Scalar dissipation rate m2/s3

κ Von Karman constant - ρ Density kg/m2

σk Turbulent Prandtl number for k - σω Turbulent Prandtl number for ω - θ angle on cylinder deg µ Dynamic viscosity kg/ms µt Turbulent viscosity kg/ms ω specific diffusion rate s-1 ζ Normally distributed random number -

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Contents 1. Introduction 1 1.1 Introduction………………………………………………………………........1 1.2 Problem Overview…………………………………………………………….2 2. The Flow Model 4 2.1 Introduction……………………………………………………………………4 2.2 The Closure Problem…………………………………………………….........4 2.3 Turbulence Model Selection…………………………………………………..5 2.3.1 Eddy Viscosity Model……………………………………………….5 2.3.2 Reynolds Stress Model……………………………………………...6 2.3.3 A Quick Comparison of the Turbulence Models……………………6 2.4 Problem Definition and Solution Approach…………………………………..7 2.4.1 Boundary Conditions………………………………………………..7 2.4.2 Mesh Generation…………………………………………………….8 2.5 Solution Scheme………………………………………………………………9 2.5.1 The Reynolds Stress Model Simulations……………………………9 2.6 Results and Discussions……………………………………………………...10 2.6.1 X Velocity Data………………………………………………........11 2.6.2 Vortex Shedding……………………………………………….......12 2.6.3 Reynolds Stress Data………………………………………………13 2.6.4 Turbulent Fluctuations……………………………………………..15 2.6.5 Some Comments on the results.........................................................16 2.7 Conclusions......................................................................................................17 3. Particle Tracking Model 18 3.1 Introduction......................................................................................................18 3.2 The Discrete Phase Model...............................................................................18 3.2.1 The Random Walk Model.................................................................19 3.2.2 Particle Forces..................................................................................20 3.3 Flow Geometry................................................................................................21 3.4 Solution and Results........................................................................................22 3.5 Conclusions.....................................................................................................25 4. Particle Deposition 26 4.1 Introduction.....................................................................................................26 4.2 Particle Deposition Model...............................................................................26 4.3 Flow Geometry and Solution Scheme.............................................................27 4.4 Results and Discussion....................................................................................27 5. Conclusions and Recommendations 29 5.1 Conclusions......................................................................................................29 5.2 Recommendations for further research............................................................30 Bibliography 31

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Appendices A. The k-ω model equations 32 B. The Reynolds Stress Model Equations 33

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CHAPTER 1

1.1 Introduction

Cylindrical tube bundles are used universally across all major heat exchanger systems. In particular, these tube bundles are used in cross flow of heat recovery boilers. Such heat recovery systems form the bulk of the power generating units not only in India but in most parts of the world. They have remained the lifeblood of advanced industrialized nations and developing countries alike. But all tube bundles face a critical problem, one that reduces their efficiency over time and greatly increases their maintenance costs. That problem is one of fouling. Fouling can be defined as the formation of a particulate layer over the heat transfer surface. The particulate layer generally manifests itself in the form of scales and thick powdery layers. Wherever they occur they serve to vastly increase the resistance to heat transfer. This increased resistance leads to lesser net heat transfer and overall poorer performance of the heat exchanger. In extreme cases, heat transfer rates are reduced by almost 40% [2] making the heat exchanger economically unviable. This phenomenon leads to increased expenditure for periodic maintenance activities like ‘flushing’. Not only are these measures expensive, they also require that the heat exchanger be shut down. This leads to further loss of revenue for the power generating units. With the size and complexity of modern power stations, shutting down and switching back on is in itself an arduous task. The problem is quite acute in heat recovery boilers. The hot flue gases flowing over the tube bundles of heat recovery boilers carry with them flyash and other solid particles. These particles adhere to the outside surfaces of the (generally) cylindrical tube bundles. Observations of actual heat recovery surfaces [2] have shown that a thick layer of particles is formed over the outer surface within hours. Such a layer seriously affects the performance critical heat recovery systems like economisers and superheaters. These problems have also been reported in refuse waste incinerators and biomass gasifiers. Their occurrence in biomass gasifiers seriously undermines their viability as a clean, renewable source of energy.

The modeling of these phenomena has to be carried out over 3 steps: modeling turbulent flow, particle transport and particle-wall interaction. Each of these steps is inherently complex and the problem is compounded by the inter-relationships between them. In high temperature applications and in liquid flows, chemical reactions add an extra dimension to the problem. A general schematic of gas side fouling is presented in Figure 1.1. It shows in particular particle transport in a flue gas stream.

Figure 1.1 Particulation process for tube bundles placed in cross flow with the flue gas stream

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Additionally photographs in Fig 1.2 show a typically fouled tube bundle. The photographs are taken from a superheater and economizer of a refuse waste incinerator in Moerdijk, Netherlands. Data from this installation was used in the fouling study of Van Beek [2], upon which this work has been based.

Figure 1.2 Deposits observed on the tubes in the final superheater (left) and economiser (right) of the boiler in

Moerdijk Armed with the equations for fouling and careful experimental measurements, engineers have established fouling factors for most of the common materials in use today. But it’s only in recent years that a concerted effort has been made to mathematically model the problem. Fouling is still quite poorly understood. Once the phenomenon is fully understood, computational models can be made which will improve the predictive capabilities of heat exchanger design engineers and help them design heat exchangers which will minimize fouling. Predictions can be made of critical values like flow rates, mass loading and temperature etc. at which fouling starts. Ultimately the goal would be a comprehensive fouling model which encompasses all the three distinct phenomena in the fouling mechanism and can be applied universally over most heat exchanger geometries. 1.2 Problem overview This work is based on the work of Van Beek [2]. In this project, focus has been laid on the turbulent flow and the particle transport. The work of Van Beek focused instead on the particle adhesion problem. As is conceded by Beek himself [2], the flow and particle transport models used by him were crude and not very effective. It is this aspect of the problem that this author has decided to pursue. Firstly, the flow is modeled. Cross flow in tube bundles in inherently complex with a turbulent main stream and laminar boundary layers. It is also characterized by vortex shedding at the trailing edges of the cylinders. The problem is further compounded by the occurrence of transition as the flow is low turbulent at the inlet and becomes high turbulent between the tube bundles. It is to be noted that however high the turbulence intensity may be between the tube bundles, the boundary layer remains laminar as observed by Konstandinis [9] for the flow in a staggered tube bundle. A combination of well-established turbulence models as implemented in commercial software FLUENT™ was used to model the flow and

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the results were compared with the most reliable experimental data available to validate the flow model. Once the flow model is validated, particles are introduced into the flow. The particle properties (diameter, density etc) used is taken from available literature concerning experimental work in commercial heat exchangers. The particles are, in general, flyash of different sizes, which get entrained in the flue gas stream as is shown in Fig 1.1. A Lagrangian particle tracking technique is employed to compute particle trajectories. At the beginning perfect sticking is assumed and particle deposition rates are computed and compared with available experimental data. After this, existing particle wall interaction models are used to determine actual particle deposition rates. Different flow rates and flow configurations were simulated to determine the exact location where fouling begins. Commercial software FLUENT™ was used to perform the calculations and simulations. The data from these simulations are compared with experimental data available in literature. It is to be noted however that such experimental data is quite rare and isn’t readily available in literature.

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CHAPTER 2

The Flow Model 2.1 Introduction All heat recovery boilers utilize a particular configuration of tube bundles. The most common configurations used are the inline and staggered configurations as shown below in the figure.

Figure 2.1 Common tube bundle configurations

SL and ST are the streamwise and transverse pitches respectively. Whatever the tube bundle configuration, the flow through them is turbulent at flow rates normally encountered commercially. This presents a problem as turbulence is a complicated phenomenon and difficult to model. Nevertheless, over the years the problem has been studied by numerous scientists and several mathematical models are available for turbulence modeling. Unfortunately, there is no single model which can account for all the different situations in which turbulence occurs. So, the first challenge was to select a suitable turbulence model for the situation at hand. It was decided to use a staggered tube bundle configuration as the test case for validation and selection of a suitable turbulence model. The reasons for this are that the staggered tube bundle arrangement has historically been the most challenging turbulence modeling problem. In fact, flow through staggered tube bundle flows was used as the test case for validating turbulence models in the 1980-81 AFOSR-HTTM Stanford Conference on Complex Turbulence Flows where the best turbulence models of the day were tested against the best experimental data of the day. Other than this, highly reliable and accurate experimental data are available for flow turbulent flow through staggered tube bundles. The following 2 sections give a brief overview of the physics of turbulence and available turbulence modeling approaches. 2.2 The Closure Problem Since turbulence is a continuum phenomenon, it is completely described by the Navier-Stokes equations given below. Though this equation describes turbulence fully, it is not very useful in this form as turbulence is inherently unsteady and is characterized with fluctuations. Hence the following Reynolds averaging is usually performed.

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(2.2)

where ūi and úi are the mean and fluctuating velocity components (i = 1, 2, 3). The same holds true for other scalars in the transport equation. Hence we get Reynolds Averaged Navier Stokes Equations (RANS). They are of the following form:-

(2.3)

(2.4)

As can be seen, a new term been added to the equation. This term is known as the Reynolds Stress component. Now to solve any differential equation the number of unknowns should be equal to the number of equations. In the above equation, considering 3 dimensional incompressible case, there are 10 unknowns (1 pressure, 3 velocity components and 6 Reynolds stress components) while there are only 4 equations available (1 continuity and 3 momentum equations). This is the classical closure problem where the number of unknowns exceeds the number of available equations. Hence over the years several researchers have established turbulence models each with varying degrees of success. The next section describes in brief some of the turbulence models considered solving the problem and closing the equations. 2.3 Turbulence Model Selection The entire problem of turbulence modeling basically boils down to modeling the unknown Reynolds stresses. There are in general three distinct approaches to modeling these stresses each with their attendant assumptions. They are listed in the following sub sections. 2.3.1 Eddy viscosity Models These are the earliest and simplest classes of turbulence models. They all generally base themselves on the Boussinesq hypothesis [7] which is given below.

(2.5)

Here µt is the turbulent viscosity, a term introduced by Boussinesq. Offshoots of this model are the one equation Spalart-Allmaras model and 2 equation k-ε and k-ω models. There are several variations of these models and for complete details of these models one if referred to Wilcox [14]. One of the most important assumptions of these models was that of isotropy of velocity fluctuations. This assumption is not strictly true and is flawed which probably accounts for their failure in modeling advanced turbulent flows. Nevertheless some of these models, in particular the k-ε and k-ω models are still widely used

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and are considered as industrial standard techniques. They are easy to implement and computationally less intensive. 2.3.2 Reynolds Stress Models The Reynolds Stress Model (RSM) is the most robust RANS model available till date. It achieves closure by solving transport equations for the Reynolds stresses together with an equation for the dissipation rate. It abandons the isotropic eddy viscosity hypothesis and thus is expected to provide better results, especially near walls. The RSM accounts for the effects of streamline curvature, swirl, rotation and rapid changes in strain rates. All of these are relevant to our chosen problem. However all of this is at a higher computational expense as the RSM solves 5 additional transport equations in the 2D case and 7 additional transport equations in the 3D case. It is also very hard to converge. It is not guaranteed apriori that RSM gives better results than eddy viscosity models for all flows but for flows involving strong streamline curvature and vortex shedding the RSM assumption of anisotropy becomes very important. 2.3.3 A Quick Comparison of the turbulence models The following table gives at a glance the various turbulence models and their strengths, weaknesses and application ranges.

Model Behavior and Usage Spalart-Allmaras

• Economical for large meshes • performs poorly for 3D flows, free shear flows and flows with

strong separation • suitable for mildly complex(quasi 2D flows)

Standard k-ε • Robust, but performs poorly for complex flows • Suitable for initial set up testing, parametric studies etc.

RNG k-ε • Suitable for locally transitional complex shear flows with separation, strong swirl and vortex shedding

Realizable k-ε • Similar benefits to RNG • Easier to converge and more accurate

Standard k-ω • Suitable for wall-bounded, free shear and low Re number flows • Can predict transition

SST k-ω • Similar benefits to standard model, but less sensitive to outer disturbances

• Suitable for wall bounded flows RSM • The most physically sound RANS model

• Complex and hard to converge • Can handle anisotropy, so suited for flows with strong swirl and

shear.

Table 2.1 Comparison of the various turbulence models considered

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2.4 Problem definition and solution approach Flow through a staggered tube bundle is considered. The flow geometry contains 5 rows of staggered tubes with constant streamwise and transverse pitches of 45 mm. It was decided to use a 2D domain as the flow essentially was observed to be 2 dimensional in character [9]. The experimental set up of Simonin and Barcouda[11] was used as the template for geometry creation in these simulations. This set up was also modeled by Hassan and Barsamian [7] and Kuerten [10] in their simulations. In the experiments, water was used as the working fluid. Flow data was extracted at the positions shown in Fig 2.2. The flow parameters used are listed in Table 2.2.

Table 2.2 Properties

Values

Density(ρ) 999.73 kg/m3

Kinematic viscosity(ν) 1.28 x 10-6 m/s2

Inlet flow velocity(U∞) 1.06 m/s Reynolds Number(Re) 18000

Inlet turbulent kinetic energy(k) 0.01 m2/s2

Inlet specific dissipation rate(ω) 111.11 s-1

Figure 2.2 The tube bundle geometry

2.4.1 Boundary Conditions At inlet, the transverse component of velocity is considered zero. The streamwise component equals the free stream velocity U∞. At outlet, an outflow boundary condition as defined by FLUENT is used. It means that a fully developed flow condition is assumed and the only constraint is a diffusion flux of zero for all quantities. All other boundaries are wall boundaries with zero normal and tangential velocities at their surfaces (the no slip condition). Though it has been observed in experiments and reported in literature that the flow is periodic between the tube bundles periodic boundary conditions have not been set.

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Instead it is decided to let the flow become periodic on its own. Such an approach is physically more realistic.

2.4.2 Mesh Generation The mesh was generated on modeling software GAMBIT™. After creation of the geometry shown above, the entire setup was meshed using a triangular paved meshing scheme with interval size = 0.3. Thus a fine 2d mesh was created with 352056 nodes and 639920 mesh faces. Also, a boundary layer was set up along the tube walls with 26 rows. The 1st grid point was located 0.01 mm from the tube wall and the growth rate was set at 1.1.

Figure 2.3 The mesh created in GAMBIT

Figure 2.4 Magnified view of the near wall mesh and boundary layer

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The grid is unstructured and uses a combination of triangular and rectangular grid faces. The near wall mesh is refined and made denser than the mesh away from the wall. This is done to improve the predictive capability of the model near the wall as this is where the gradients are largest. By making the mesh away from the wall coarser, it also saves on computational expense and time. 2.5 Solution Scheme FLUENT uses a control volume approach. The basic steps involved in this approach are: ● Division of the domain into discrete control volumes using the computational grid. ● Integration of the governing equations on the individual control volumes to construct

algebraic equations for the discrete dependent variables (unknowns) such as velocities, pressure, temperature, and conserved scalars. ● Linearization of the discrete variables and solution of the set of linear equations to obtain

updated values of the dependent variables.

Initially a steady simulation was performed. The standard k-ω model with correction for transitional flows was implemented as the turbulence model. A pressure based solver is used. This solver utilizes algorithms belonging to a general class of algorithms called the projection method. In this method, the continuity of the velocity field is achieved by means of a pressure correction equation. The pressure correction is derived from the momentum equations in such a way that the continuity equation is satisfied. The particular algorithm used is the SIMPLE algorithm. For initial calculations 1st order upwind discretization schemes were used for all quantities and steady iterations were performed till convergence was achieved. Adaptive grid refinement was performed with velocity gradients as the refinement factor. The discretization schemes were changed to a more accurate 2nd order upwind method and under relaxation factors were reduced to encourage stable solutions. The iterations were again carried out till convergence. This took much longer as the grid had been further refined. It took a large number of iterations and approximately 30 minutes for convergence to be achieved. The residuals were kept at 10-4 throughout the simulations.

2.5.1 The Reynolds Stress Model simulations Once the steady simulations of the k-ω model were performed, attention is turned to the unsteady part. Since turbulence is primarily an unsteady phenomenon, only unsteady simulations can effectively capture all of the behavior. For the unsteady simulations, the most advanced RANS model, the RSM was used. This was used only for the unsteady simulations as the RSM tends to produce unstable results when steady state simulations are performed with it. This behavior is well documented in literature and is also mentioned in the FLUENT™ User’s Manual [5]. In fact, it goes on to recommend the usage of either k-ε or k-ω model for initial steady simulations and then using these results as initial conditions for unsteady simulations with RSM. This author’s simulations are pursuant to those suggestions.

The solver schemes were left unchanged from the previous k-ω simulations. The results from these calculations were used as initial condition for the unsteady RSM simulations. The time step size was 1x 10-3 secs and the simulation was carried out for 100 time steps. The boundary conditions were left unchanged from the previous simulation.

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2.6 Results and Discussions After the simulations were over, results were extracted in the form of contour plots, vector plots and XY plots. For the XY plots, the data sets were extracted along the line as shown in the figure. It corresponds to a surface 16.5 mm from the centre of the 3rd

row of cylinders and in between the 3rd and 4th rows. The data range is also as shown in Figure 2.2. This was chosen as the experimental data of Simonin and Barcouda[11] were also extracted from here. 2.6.1 X velocity data X velocity data is extracted along the line shown in Fig 2.2. In the contour plots the dark red color regions indicates higher velocities and dark blue color regions indicates reversed flow. Yellow and green indicate intermediate velocities. Light blue represents very low velocities.

Figure 2.5 X velocity contours

Figure 2.6 Magnified velocity contours

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Figure 2.7 Periodicity of velocity Figure 2.8 Comparison with experimental data Figures 2.5, 2.6 and 2.7 show variation of x velocity in the domain of data extraction. Figure 2.7 shows the periodicity of the velocity between the tube bundles. It validates the earlier observations made of flow periodicity. The gap between the graph lines are due presence of cylinders. One can see from the contour plots in Figure 2.5 and 2.6 that all the turbulence is generated due to the tubes alone. The tops and bottoms of the cylinders have the highest velocities while both the leeward and windward side of the cylinders show a very low velocity. But the reasons are this observation is different in both the cases. For the windward side the low velocities are due to flow separation occurring there. As the fluid approaches the face of the cylinder it is forced to separate and go around the cylinder face. This flow separation causes low velocities at the windward side. At the top and bottom of the cylinder the low velocity streams are met by the higher velocity free stream. This increases the flow velocities here. Fig 2.8 shows the x velocity plot for the line of data extraction shown in Fig 2.2. The predicted values give a good agreement with experimental data. The velocity is initially low just behind the cylinder. It rises as the y coordinate increases thus taking the domain closer to the high velocity top region of the cylinder. It then falls again as it enters the vicinity of the next cylinder.

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2.6.2 Vortex Shedding An important aspect of turbulent flow is vortex shedding. Vortex shedding is the process where turbulent eddies are transported from the trailing edges of cylinders in the form of vortices. It is an unsteady process with a more or less regular frequency.

Figure 2.9 Vortex shedding frequency Fig 2.9 shows the variation of x velocity at a point just behind the bottommost

cylinder of the 4th row. This periodic change with time is due to the periodic shedding of vortices from behind the cylinder. It is observed that the time period for this vortex shedding is approximately 20 time steps. Since the time step chosen was .001 seconds, this gives a time period of 2 x 10-2 seconds. Thus the dimensionless Strouhal number can be calculated. It is a measure of the vortex shedding frequency. The Strouhal number can be expressed as:-

Str = L2πTffffffffffffff (2.6)

L is the characteristic length scale, in this case, Dtube, and T is the time period of vortex shedding. Thus using the equation, a Strouhal number of 0.125. This value is of the same order as the Strouhal Number for a single cylinder, which is 0.18 [4].

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Figures 2.10 (i) to (iv) show the build up and shedding of vortices from the trailing edges of the cylinder graphically. Each figure is a snapshot at particular time intervals. The red color indicates maximum vorticity magnitude with the magnitude decreasing from yellow to blue.

(i) (ii)

(iii) (iv)

Figure 2.10 Snapshots of vorticity magnitudes at times T/4(i), T/2(ii), 3T/2(iii) and T (iv) where T is time period of vortex shedding

The figures show the advance of vortices graphically. At 5th time step, the vortices are still building up. They get stronger with each passing moment until they are finally shed from the surface at the end of 20 time steps. This pulsing behavior is a major characteristic of turbulent flows and is the main cause of flow recirculation.

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2.6.3 Reynolds stress data The presence of the Reynolds stress terms in the RANS is what causes the closure problem in turbulence as has been discussed earlier. This stress term is very difficult to model and remains one of the focus areas for turbulence modeling. Among the RANS models, the RSM comes closest to modeling these stresses with reasonable level of accuracy.

(a) (b)

Figure 2.11 Normal Reynolds stresses UU and VV compared with experimental data

Figure 2.12 UV Reynolds stresses compared with experimental data

Figures 2.11 (a), (b) and 2.12 give a comparison between the predicted and measured values of Reynolds stresses. The stress values show a high as expected near the top of the cylinder. This is the spot from where vortex shedding also begins. Hence it can be concluded that the Reynolds stresses play a very important role in the formation of vortices. In fact, it can be surmised that vortex shedding is actually due to the transport of the Reynolds Stresses in the wake of the tubes. The agreement between the predicted and measured values is to a reasonable extent accurate. Errors do creep in due to phenomena like jet flapping and the 3D nature of vortex shedding.

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2.6.4 .Turbulent Fluctuations

Turbulent flow is characterized by random velocity fluctuations. The fluctuation behavior was also studied here.

Figure 2.17 Transverse fluctuations Figure 2.17 shows contours of transverse fluctuations v’ over the tube bundle. The fluctuations are most pronounced at the trailing edges of the cylinders. This is as expected as this is origin for the vortices. Another interesting point to note is that the fluctuations are high in between the tubes and not in the free stream. This indicates that all of the turbulence is generated by the tubes alone. The yellow and red spots indicate higher magnitudes of transverse fluctuations. Another important parameter is the comparison between streamwise and transverse fluctuations. For the purpose, data is extracted from the 2nd cylinder from the at the bottom row along the line shown in Fig. 2.18.

Figure 2.18 Cross section for fluctuation data extraction

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Figure 2.19 Comparison between u' and v'

Both u’ and v’ are 0 at the wall. u’ increases more rapidly as we move away from the wall. At y = 16 mm and 28 mm, both the fluctuations are equal. Around the mid point between the 2 cylinders the transverse fluctuation is much higher than the streamwise one. This behavior is not yet fully understood at this moment. However it does graphically depict the anisotropy of the fluctuations. 2.6.5 Comments on the results Several researchers have, over the years, tried to model turbulent cross flow across tube bundles. A few of the prominent ones are by Benhamadouche et. al [3], Bouris et. al [4], Hassan and Barsamian [7] and Kuerten [10]. The methods used have been as varied as the degree of their successes. The common thread among these works has been the use of Simonin and Barcode’s experimental data (except Achenbach’s work as he used his own experimental results). Another common thread among them is that, to the best of the author’s knowledge, none of the simulations yielded completely accurate agreement with the experimental data over all the quantities measured. In general, the velocity is predicted very well but the stresses have been both under and over predicted. For example, in the LES simulations of Rollet-Miet et.al, the UU stresses are over-predicted by a factor of 2 while the agreement is good for the rest of the quantities involved. They compared their results with DNS studies as well and got the same results. The work of Benhamadouche et.al is even more revealing of the vagaries of turbulence modeling. In their work, they used the same set of parameters as Rollet-Miet et.al but used a different modeling technique and code. They also modeled the turbulent flow using a specially developed RSM code and T-RANS code developed by Hanjalic et.al [3]. The simulation results were compared with DNS results of Moulinec et al [3]. This time they found the VV stresses to be over predicted and UV stresses to be underpredicted. Even the advanced DNS results did not yield completely accurate correlations. In their discussions, the authors concluded that the errors were due to the limited accuracy of the experimental results themselves and the presence of complex phenomena like vortex shedding and jet flapping. They have also concluded that RSM calculations did not yield good results. But the comparative study between LES and RSM by Kuerten concluded that the RSM (as implemented in FLUENT) was just as good as the more advanced LES.

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With this information at hand, this author concludes that the errors are also due to the implementation of the numerical code itself. It has been documented by Zevenhoven [25] that different CFD techniques and softwares yield different results even when the simulation parameters are kept the same. Hence some of the errors in the prediction of stresses are not physically significant. It can be safely assumed that they are numerical oddities, a product of the numerical schemes and the models used. In that sense, the results obtained here are, to the best of the author’s knowledge, the most reliable 2D RANS predictions yet. This is mostly due to the transitional flow corrections used in the k-ω model for initial steady simulation. The k-ω model also does away with empirical wall functions with would skew otherwise stable results. The k-ε model used by Van Beek and Kuerten cannot account for transitional effects and uses wall functions. Hence when used as a starting point for unsteady calculations they result in slightly skewed results. With the k-ω model here, the transition is well predicted and the near wall behavior is more accurately modeled. This serves as a better starting point for unsteady RSM simulations. Due to this combination the use of LES was deemed unnecessary. The 3D nature of LES and subsequent computational costs were a detriment. This conclusion was also reached by Kuerten who found that the slight increase in accuracy by LES did not justify its increased costs. Hence, these flow simulation results can be used as a reliable starting point for particle tracking. 2.7 Conclusions A 2D geometry was created and meshed using GAMBIT. The mesh was imported to FLUENT and the k-ω model was used to simulate the steady initial flow field. The results of the simulations served as a starting point for unsteady RSM simulations. The x velocity, vorticity, Reynolds stress and turbulent fluctuations results of the simulations were extracted at pre determined points and compared with experimental data . The phenomenon of vortex shedding was graphically captured and the Strouhal number was determined for this flow. The number was found to be in the range of Strouhal numbers calculated for this type of flow. The calculations showed a good agreement with experimental data and followed the trend of the other calculations. The stress results were closely analysed and errors were attributed to vortex shedding, jet flapping and inherent numerical scheme errors. In particular the close relationship between vortex shedding, stress transport and turbulent fluctuations was observed and analysed. Overall the results are satisfactory enough for purposes of particle tracking, which will be dealt with in the next chapter.

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CHAPTER 3 Particle Tracking Model 3.1 Introduction The flows in any commercially operated tube bundle are always particle laden. Particles add an extra dynamic to the problem. In the previous chapter flow through a staggered tube bundle with water as the fluid was simulated. This was done only the validate the flow model. In this chapter that same model is used to simulate particulate flow across cylinders with air as the fluid. The simulations are carried out across a single cylinder first and then across an inline tube bundle. Particle trajectories are plotted in both cases to determine the effectiveness of the particle tracking scheme. An effective model for particle tracking will be the precursor for calculating particle deposition rates and visualizing fouling. This aspect is dealt with in the next chapter. 3.2 The Discrete Phase Model

A particle tracking model as implemented in FLUENT is used. It is known as the Discrete Phase Model. The underlying assumptions of this model are: ● The particles in the flow are inert particles i.e. they don't react with the continuous phase,

namely air. ● The volume fraction of the particulate phase (also called the discrete phase) is very small,

less than 1% of the total domain volume. This assumption is reasonable as this is the case commercially as well.

A Lagrangian particle tracking scheme is used. The Lagrangian approach focuses on particle tracks. The continuous phase is solved via an Eulerian control volume approach. The difference between the two approaches is shown in Fig 3.1. In this sequential approach, the flow iterations are performed and these serve as inputs to the discrete phase equations. Thus the discrete phase trajectories are updated after every set of flow calculations.

Figure 3.1 Eulerian vs Lagrangian approach

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The effects of turbulence are also taken into account using specialized sub models available inside the DPM (Discrete Phase Model). There are 2 models available: - the particle cloud model and Discrete Random Walk Model [5]. In this study, the Random Walk Model is used. 3.2.1 Random Walk Model In this model, the interaction of particles with a succession of fluid phase eddies is simulated. Each eddy is characterized by:- ● A Gaussian distributed random velocity fluctuation u', v', w'. ● An eddy time scale The values of u', v' and w' are given by:-

u . =ζ1 u. 2ffffffffff

qwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

; v . =ζ2 v. 2ffffffffff

qwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

; w . =ζ3 w. 2ffffffffffff

qwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

(3.1)

ζi is a normally distributed random number and the remainder of the right hand side is the r.m.s of the velocity fluctuation. In the k-e and k-w models where isotropy is assumed, all the components are equal. In the RSM, each component of the velocity fluctuation has a different randomly associated number. The expressions for the fluctuations are given by:

u. 2ffffffffff

qwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

= v. 2ffffffffff

qwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

= w. 2ffffffffffff

qwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

=2k3ffffffffswwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

(3.2)

The characteristic eddy life time is given as:

(3.3)

r is a random number between 0 and 1. TL is the integral time given by:-

(3.4)

The value of 0.30 is chosen specifically for the RSM. It is 0.15 for k-ε and k-ω models.

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3.2.2 Particle Forces The discrete phase equations are simply Newton's 2nd law. A force balance is carried out on all the particles in a Lagrangian reference frame. The force balance in Cartesian coordinates for the x direction is given by:

(3.5)

up is the particle velocity(x direction), FD is the drag force, gx is the gravitational term and Fx is any additional force. The drag force FD is given by :-

(3.6) µ is the dynamic viscosity, ρp is the particle density, dp is the particle diameter, CD is the drag coefficient and Re is the relative Reynolds number given by :

(3.7) The other miscellaneous forces which may be present are virtual mass force, Saffman’s Lift Force and thermophoretic force. These forces have been studied in literature and been deemed negligible for the present purposes[5]. The drag coefficient CD is correlated as :

(3.8)

This formulation is for smooth spherical particles. This assumption is used in the present particle tracking schemes as well. a1, a2 and a3 are constants that apply over several ranges of Re numbers and are given in FLUENT User’s Manual[5].

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3.3 Flow geometry The flow geometry used is the one given in fig. 3.2. It is a single cylinder with particles released as shown.

Figure 3.2 Single cylinder flow

The values are : Dtube = 50 mm; U∞ = 4.56 m/s; xinj = -3Dtube; Hinj = 0.6Dtube

Figure 3.3 The mesh

The particle tracking is carried out over a single cylinder first. There is a symmetry line at the bottom and top. The particles and the flow are injected from the left boundary. The boundary conditions for the inlet, outlet and wall for the fluid phase is kept the same as was done for the geometry considered in the flow model. For the particle tracking however, the following boundary conditions are maintained :

• Any particle reaching the inlet or outlet, is deemed to have “escaped”, thus having left the computational domain and any further trajectory calculations are stopped for that particle.

• At the cylinder wall, a restitution coefficient of 0.25 for both radial and tangential components was set. This value was arrived at by Van Beek et.al [2] in their experiments.

• At the symmetry boundaries, the particles reflect back with coefficient =1.

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The particles are staggered during their injection i.e. all of the particles are not injected at the same time. They are injected at random intervals with different number of particles at each injection.

Table 3.1 Fluid properties The flow properties used are given in Table 3.1. The particles in the flow can be

characterized by the Stokes number given by :-

(3.9) τp :- Particle relaxation time which is given by :-

(3.10)

The simulations were performed with 2 sets of particles, one with average diameters of 7.3 µm and another with average diameter of 23.1µm. These give Stokes numbers of 0.05 and 0.5 respectively. The density of the particles was kept at a uniform 2670 kg/m3. 3.4 Solution and Results Firstly, the flow was solved using the models and techniques discussed in the previous chapter. After this the discrete phase model was turned on. Approximately 30000 particles were injected into the domain and were tracked through 200 time steps. The time step size was kept at 0.001 seconds throughout. The impact velocities and incident angles were recorded as the particles hit the cylinder wall.

Figure 3.4 Particle tracks at time .05 seconds

Property Values Fluid Air

Density 1.225 kg/m3

Dynamic Viscosity 5.4 x 10-5 kg/m-s Reynolds Number 5000

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Fig. 3.4 show the initial particle tracks in the complete domain by mirroring data across the bottom symmetry line as well. The darker spots depict areas of higher particle concentration. As can be seen, most of the particles hit the cylinder on at the windward side. There are very few particles that impact on the leeward side. One can however notice some particles at the trailing edges of the cylinder. As can be recalled from the previous chapter this is the spot where vortices start and it is likely that these particles are there as they are trapped by the vortices. Figure 3.5 shows the particle tracks after completion after .1 seconds.

Figure 3.5 Particle tracks after .1 seconds

The flow is more developed now and some particles are escaping as can be seen. There are now more particles impacting at the leeward side while others are trapped in the wake of the flow. Still the windward stagnation region has the highest particle concentration.

Figure 3.6 Radial particle velocities impacting the surface of the cylinder

Fig. 3.6 shows the radial particle velocities as they impact the wall versus the x coordinate. The data ranges from the windward stagnation point( x = -0.025 m) to the leeward side( x = 0.025 m). The cylinder center is the origin. Interestingly though the particle concentrations are highest at the stagnation region, the radial velocities there are quite low. This can be attributed to the fact that particles impact the wall and then as they bounce off they get slowed down by the flow . At the top the radial velocities are highest mostly because of the high mean flow rates and also because of the stress production which tends to accelerate the flow. This is the area of high shear as well.

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Fig 3.7 show individual particle tracks of 10 particles colored by their velocity magnitudes.

Figure 3.7 Individual particle tracks

All of the particles are assumed to originate from the same point at the inlet boundary. They undergo multiple reflections. It is to be noted however that all reflections from the upper and lower boundaries are perfect reflections as these are assumed to be symmetry boundaries. As can be seen, particles hitting the stagnation region bounce back and are immediately slowed down by the flow and get carried away by it to the top of the cylinder. There they are accelerated by the mean flow and then undergo multiple collisions with the cylinder top. Some of the particles have enough velocity to escape but some of the slower particles get trapped near the wall. Figs 3.8 a and b show histogram plots of the percentage of particles colliding with the cylinder wall with respect to the x and y coordinates. It is to be noted that the centre of the cylinder is the origin(0,0). The x and y coordinates on the graph are the (x,y) coordinates on the cylinder surface. As can be seen, majority, almost 60 % of the particles collides with the surface at the stagnation point. There are almost no collisions at the leeward side of the cylinder. The number of particles colliding with the wall keeps on reducing as the cylinder top surface is reached.

Figure 3.9 Radial velocity and α vs. θ along the cylinder surface. α and θ are defined in Fig. 3.2 These values are time averaged over a time period of 1 second. The radial velocities are high in the stagnation region in the opposite direction to the flow. These are due to high the cylinder( θ = 0) are very low since here the tangential free streams dominate. The radial velocities rise once again at the leeward side as the particles get caught up in the wake vortices. A combination of particles with sizes ranging from 7.3 µm to 23.1 µm was used.

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3.5 Conclusions A Lagrangian particle tracking model called the Discrete Phase model was chosen and implemented in FLUENT software. The simulations were performed on a single cylinder to validate and test the effectiveness of the particle tracking model. A single cylinder geometry was constructed and meshed. The flow field was solved using the turbulence models elucidated in the previous chapter. Particles were then, introduced into the flow and were tracked through time. Their collisions with the cylinder surface were recorded and incident velocities and angles were calculated. A statistical sampling of the particles hitting the wall was also performed. It was found that the majority of the particles strike the front stagnation area of the cylinder. They bounce off and are slowed down by the flow. Their paths get skewed by the flow and some of them strike the upper surface of the cylinder with much lower velocities. At the leeward side of the cylinder almost none of the particles struck the cylinder wall. It can be concluded that they get trapped in the vortex recirculation region of the flow. This particle tracking model serves as a base for calculating particle deposition rates on the tube bundles. This is dealt with in the next chapter.

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CHAPTER 4 Particle deposition 4.1 Introduction The next step in the analysis of fouling is to calculate particle deposition rates. For this purpose, an existing model from literature is employed and coupled with the particle tracking model employed. As explained in Chapter 1 itself, this study concentrates more on flow and particle tracking rather than on particle wall interaction models. Hence the use of existing models for particle adhesion is used without any modification. In the next section, the model used is described. 4.2 Particle deposition model The model used is the Haugen model developed by Haugen et.al[8]. It was originally developed to model sand erosion of wear-resistant materials, but with the use of appropriate coefficients, it has been extended for use with a variety of combination of commonly used materials. The basic equation used is given by :

E = M P KF α` a

V pn (4.1)

E is the accretion or deposition rate, Mp is the particle mass, α is the particle impact angle, Vp is the particle velocity, K and n are constants that depend on the type of materials being used and F(α) is a functional relationship with α. The functional relationship is given by :

F α` a

=Xi = 1

8

@ 1` ai + 1 Ai α

π180fffffffffffd ei

(4.2)

The coefficients Ai are given in Table 4.1.

A1 A2 A3 A4 A5 A6 A7 A8 9.370 42.295 110.864 175.804 170.137 98.298 31.211 4.170

Table 4.1 Constants to be used in Eq. 4.2

The tube bundle was assumed to be built of steel and the particles were assumed to be calcium carbonate. For these materials, the values of K and n are documented as 2.0 x 10-9 and 2.6 respectively. In FLEUNT, the erosion-accretion phenomenon is modeled by a standard equation :

E =mpA C d p

b c

f α` a

v b v` a

A face

ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff (4.3)

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C(dp) is the diameter function and b(v) is the velocity exponent function. Hence by simply correlating each term of Eq 4.1 with that of Eq 4.3, the Haugen model is implemented in FLUENT. Thus, C(dp) = K and b(v) = n.

4.3 Flow geometry and solution scheme

The flow geometry is given in Fig 4.1. This geometry mimics the geometry used by Abd El-Hady et.al[1] in their experimental studies on particle deposition on tube bundles.

Figure 4.1 Flow geometry

The inlet is from the left boundary and outflow is from the right boundary. At the inlet, the air free stream velocity is 1.25 m/s and particle mass flow rate is .0004 kg/s. The cylinders are numbered as shown. The particle sizes are 40 µm with a std deviation of 16 µm. The Reynolds number of the flow is 32000 and the particle density is 2670 kg/m3. All other boundary conditions are the same as for the previous simulations. The flow field is solved first and then the particle tracking is carried out as per the methods determined in the previous chapters. At the cylinder walls, the deposition model of Haugen[8] is applied and the calculations were carried out. The simulations were carried out for 10 minutes with time step size of 10 seconds.

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4.4 Results and Discussions Data was extracted for cylinder 5 and cylinder 8. In both these cases, the deposition rate was calculated and plotted as functions of cylinder angle θ. The negative angles in Fig 4.1 refer to the windward side while the positive angles refer to the leeward side. θ = 0 deg implies the top of the cylinder.

(a) (b)

Figure 4.2 Deposition rates on cylinder 8 (a) and on cylinder 5 (b) after 10 minute

These plots reveal interesting details about particle deposition and ultimately fouling. As can be seen, the particle deposition on cylinder 8 is greater than that on cylinder 5. In fact, in general, particle deposition on lower cylinders is higher than the particle deposition on cylinders in the middle and the top. This is due to the influence of gravity. Particles on the lower surface of the upper cylinder fall down on the top surface of the lower cylinders. This observation has been made by Abd-ElHady et.al also[1]. Focusing on particle deposition across cylinder surface, one finds that the maximum deposition occurs at the top of the cylinder for the flow regime under consideration. This is probably due to the transport of the slower particles from the stagnation point towards the top of the cylinder by the flow. Particles undergoing collisions at those low speeds are most likely to stick to the surface. There is very little particle deposition on the leeward side of the tubes. This is consistent with the findings of Abd-Elhady et.al[1]. It shows that gravity also has a role to play in the fouling of tubes.

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CHAPTER 5 Conclusions and Recommendations 5.1 Conclusions The problem of fouling in heat recovery boilers was studied from the point of view of optimizing the flow model. The flow through the tube bundles was observed to be highly turbulent but with laminar boundary layers. This observation was verified through mathematical simulations performed using commercial software FLUENT. The critical aspect in the flow was the choice of turbulence models used. A host of turbulence models available were studied. The k-ω and RSM models were chosen as the most likely candidates to solve the complicated turbulent flow field. The k-ε turbulence models and its variants used by Beek et.al[2] were found to be unsatisfactory according to the author’s own admissions and through the work of this author and several other researchers. Staggered tube bundles were used as the computational domain. The steady flow was solved using the k-ω model and then the results from this simulations were used as the starting point for unsteady simulations using the RSM. Using the experiments of Simonin and Barcouda[11] as benchmarks, data was extracted at various points in the form of contour plots and x-y plots. In particular, the x velocity, vorticity, Reynolds stress and fluctuation data was presented and discussed. The x velocity and Reynolds stress data was compared with the experimental data and the agreement was deemed to be satisfactory. The vortex shedding frequency and Strouhal number for the flow was also calculated and its value was found to lie in the expected range. Thus, it was concluded that the combination of the steady k-ω and unsteady Reynolds stress model serves as a reliable base for particle tracking calculations. The particle tracking was performed with the use of the Discrete Phase Model implemented in FLUENT. The use of the RSM coupled with the DPM ensured physically more realistic results as isotropy was not assumed apriori. The particle tracking calculations was performed for flow over a single cylinder. Particle trajectories were determined by the Lagrangian particle tracking methods and particle collisions with the wall was recorded. Key factors like incident velocity and impact angle were determined by the use of specially written user defined functions. It was found that the particles collided a lot more with the windward stagnation region of the cylinder than the leeward side of the cylinder. This observation was entirely along expected lines. Statistical samples of the particle tracks across the cylinder surface also revealed similar behavior. Thus the particle tracking methodology was deemed to be reliable. Finally, particle deposition rates were calculated. For this case, a tube bundle was used again with the experimental set up of Abd – ElHady et.al[1] serving as template. This was an inline tube bundle as opposed to the staggered tube bundles used for the flow model calculations. A particle adhesion model was not developed from scratch. Instead, an available particle adhesion model was used and implemented in FLUENT. The particle deposition rates were calculated and plotted for certain characteristic cylinders. It was found that gravity also has a major role to play in the fouling of tube bundles. The experimental findings of Abd-Elhady et.al[1] are used to validate the comprehensive model. The findings of that study, which had concluded that for horizontal flow across tube bundles, fouling begins from the stagnation region and from the top of the cylinder, was verified in the

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simulations. Thus, the objective of this study which was to develop a fouling model by focusing on optimizing the flow model is met. 5.2 Recommendations for further research The modeling of turbulence is still a very complicated affair. In this study, the 2 models used were essentially RANS models. The newer and presumably more accurate LES model was not used as it is still in its infancy. With further development of the LES, it can be used as a reliable and probably more accurate turbulence model. From the particle tracking point of view, several forces like the Saffman Lift Force, thermophoretic force, Brownian force and virtual mass force were neglected from the calculations as they were found to be negligible compared to the inertial force. But in several real situations, these forces, especially the thermophoretic force can play a major role in particle transport. A complete model incorporating these forces as well is still awaited. Finally from the particle adhesion point of view, all calculations in this and several other studies have been almost exclusively performed using physical forces and energies as starting points. Chemical reactions have not been explicitly modeled. In reality, in many high temperature applications, chemical reactions on the surface play a major role in particle adhesion. A truly complete model of fouling will have to include the effects of surface reactions as well.

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BIBLIOGRAPHY [1] Abd-Elhady M.S., Rindt C.C.M., Wijers J.G., Steenhoven A.A. van, “ Optimization of flow direction to minimize particulate fouling of heat exchangers” (In press) (2007) [2] Beek M.C, Gas side Fouling in Heat-Recovery Boilers, PhD Thesis, Eindhoven University of Technology, 2001 [3] Benhamadouche S., Laurence D., “LES, coarse LES, and transient RANS comparisons on the flow across a tube bundle”, Int. J. of Heat and Fluid Flow 24(2003) pp 470-479 [4] Bouris D., Papadakis G., Bergeles G., “Numerical evaluation of alternate tube configurations for particle deposition rate reduction in heat exchanger tube bundles”, Int. J of Heat and Fluid Flow 22(2001), pp 525-536 [5] FLUENT User’s Guide, version 6.3.26 [6] Graham D.I; James P.W., Turbulent Dispersion of particles using eddy interaction models, Int. J. Multiphase Flow, vol. 22, No. 1, pp 157-175, 1996 [7] Hassan Y.A., Barsamian H.R., “Tube Bundle Flows with the large Eddy simulation technique in curvilinear coordinates”, Int. J. of Heat and Mass Transfer 47(2004) pp 3057-3071 [8] Haugen K., Krenvold O., Ronold A., Sandberg R., “ Sand Erosion of wear resistant materials: Erosion in choke valves “, Wear 186-187(1995) [9] Konstantinidis E., Balabani S., Yianneskis M., “ A Study of Vortex Shedding in a Staggered Tube Array for Steady and Pulsating Cross Flow “, J. of Fluids Engg. Vol. 124. pp 737-74. [10] Kuerten J.G.M. “ Numerical Simulation of flow through tube bundles “ Report for EU Project BIO ASH, July 2006. [11] Simonin O., Barcouda M., “ Measurements and prediction of turbulent flow entering a staggered tube bundle”, Fourth International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal(1988) [12] Prithiviraj, M; Andrews, Malcolm J; “Comparison of a three dimensional Numerical model worth existing methods for prediction of Flow in Shell and Tube Heat Exchangers”, Heat Transfer Engineering vol.20, no.2, 1999. [13] Watterson J.K., Dawes W.N., Savill A.M., White A.J., “Predicting turbulent flow in a staggered tube bundle “, Int. J. of Heat and Fluid Flow 20 (1999), pp 581-591. [14] Wilcox, D. Reassessment of the scale-determining equation. AIAA Journal 26, 11 (1988), pp 1299-1310

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APPENDIX A The k-ω model equations The mathematical form of the equation is explained below:

and (A.1)

(A.2) In these equations, Gk represents the generation of turbulence kinetic energy due to mean velocity gradients; Gω represents the generation of ω. Γk and Γω, represents the effective diffusivity of k and ω. respectively. Yk and Yω represent the dissipation of k and ω due to turbulence. All of the above terms are calculated as described below. Sk and Sω are user-defined source terms. Modeling the Effective Diffusivity The effective diffusivities for the k-ω model are given by:

(A.3) where σk and σω are the turbulent Prandtl numbers for k and ω. respectively. The turbulent viscosity µt is computed by combining k and ω as follows

(A.4) Low-Reynolds-Number Correction The coefficient α+ damps the turbulent viscosity causing a low-Reynolds-number correction. It is given by:

(A.5) Where

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Modeling the Turbulence Production Production of k The term Gk represents the production of turbulence kinetic energy. From the exact equation for the transport of k, this term may be defined as:

(A.6) To evaluate Gk in a manner consistent with the Boussinesq hypothesis:

(A.7) where S is the modulus of the mean rate-of-strain tensor, defined the same way as for the k-ε model. Production of ω The production of ω is given by:

(A.8) The coefficient α is given by

(A.9) where Rω= 2.95. and Ret is given by previous equations. Modeling the Turbulence Dissipation Dissipation of k The dissipation of k is given by

(A.10)

where

(A.11)

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where

(A.12) and

Dissipation of ω The dissipation of ω is given by

(A.13) where

(A.14)

(A.15) Also,

(A.16) Compressibility Correction The compressibility function. F (Mt), is given by

(A.17) where

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Model Constants

Wall Boundary Conditions The wall boundary conditions for the k equation in the k-ω models are treated in the same way as the k equation is treated when enhanced wall treatments are used with the k-ε models. This means that all boundary conditions for wall-function meshes will correspond to the wall function approach, while for the fine meshes, the appropriate low-Reynolds-number boundary conditions will he applied. In FLUENT the value of ω at the wall is specified as

(A.18) The asymptotic value of ω+ in the laminar sublayer is given by

(A.19) where

(A.20) where

(A.21) and ks is the roughness height.

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APPENDIX B The Reynolds Stress Model Equations The exact transport equations for the transport of the Reynolds stresses, ρ, may be written as follows:

(B.1) Modeling Gradient diffusive transport DTi, j ca be modeled by the generalized gradient-diffusion model of Daly and Harlow:

(B.2)

However, this equation can result in numerical instabilities, so it has been simplified in FLUENT to use a scalar turbulent diffusivity as follows:

(B.3)

(B.4) Cµ = 0.09 and σk = 0.82

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To model the pressure strain term, the linear pressure strain model available in FLUENT was used. The set of equations are given below:

(B.5) where Φij, 1 is the slow pressure strain term, also known as the return to isotropy term, Φij,2 is the rapid pressure-strain term and Φij,w is the wall reflection term. The slow pressure strain term is modeled as

(B.6) with C1 = 1.8 The rapid pressure strain term is modeled as

(B.7) where C2 = .62, P = ½ Pkk, G= ½ Gkk and C=½ Ckk The wall-reflection term, is responsible for the redistribution of normal stresses near the wall. It tends to damp the normal stress perpendicular to the wall, while enhancing the stresses parallel to the wall. This term is modeled as

(B.8) where Ć1= 0.5, Ć2= 0.3. nk is the xk component of the unit normal to the wall. d is the normal distance to the wall and Cl=Cµ

3/4/κ, where Cµ= 0.09 and κ is the von Kàrmàn constant = 0.4187. Since enhanced wall treatments were used, the standard model was slightly modified to specify values of C1,C2,Ć1,Ć2 as functions of the Reynolds stress invariants and turbulent Reynolds number.

(B.9)

xxxviii38

where

(B.10) aij is the Reynolds-stress anisotropy tensor, defined as

(B.11) Effects of Buoyancy on Turbulence The production of turbulence due to buoyancy is modeled as

(B.12) where Prt is the turbulent Prandtl number with default value of 0.85. Modeling the Turbulence Kinetic Energy The following transport equation is used for modeling production of turbulent kinetic energy :

(B.13) Modeling the Dissipation Rate The turbulent dissipation rate is modeled as

(B.14) where YM = 2ρεMT

2, is an additional dilation dissipation term. The Mach number Mt is defined as

(B.15)

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where a =√γRT is the speed of sound. This compressibility modification always takes effect when the compressible form of the ideal gas law is used. The scalar dissipation rate, ε, is computed with a model transport equation similar to that used in the standard k-ε model:

(B.16) where σε=1.0, Cε1=1.44, Cε2=1.92. Cε3 is evaluated as a function of the local flow direction relative to the gravitational vector and S is a user-defined source term.

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.