This section discusses the airflow regimes using k-Îµ turbulence model, then follows into the deposition patterns for micro-size particles for different diameters and densities.
Fluid property, inlet conditions and airway geometry are amongst the three major parameters influencing the modulation of airflow inside the human airways. Simulations for the observation of airflow patterns were conducted on models with steady, k-Îµ turbulence models with air inhaled at breathing rates of 440 ml/s corresponding to mild exercise breathing rates. Figure <> and <> shows Sagittal Cross Sectional view of the air velocity profiles for a single bifurcation section as well as a 3 generation airway model of the human trachea.
In the above two images it can be seen that the overall velocity reached in the 3 generation model is about 33% greater than that in that in the single bifurcation airway model; however the actual local velocity for the tracheal sections of the single bifurcation model are about 1.5 times that of the 3 generation model. The fact that the flow is more developed in the 3 generation model can be accounted for this difference. The 3 generation model faces greater obstruction to the flow downstream from the first bifurcation unlike the single bifurcation model where there are no obstructions downstream the first bifurcation. Please note the scales for the two velocity fields above are different. Also the angled branches in the 3 generation model are not shown as the velocity profile was taken for the plane covering the maximum area and detail. Two recirculation zones can be observed in the two velocity graphs of the airflow models. The first is the recirculation zone located to the left of the bifurcation in the single bifurcation model, which may have been caused due to the compression of the outer layer towards the wall by the faster flowing fluid in the central layers of the model. The second recirculation zone is to the right of the first bifurcation in the 3 generation model.
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As one may notice there exists a slight bulge in the right outer wall of the 3 generation model, the same phenomena of the compression of slower, wall adjacent layers by faster interior layers can be used to account for the recirculation zone here. Hotspots often lie in regions of recirculation as recirculation zones aid particle deposition.
Axial Cross Sectional Airflow
The major factors affecting velocity regimes are the inlet flow rates and the airway geometry (local and global) .The figure above shows the axial cross sections depicting velocity profiles of the 3 generation reconstructed model of the human trachea-bronchial tree. The velocity profiles in the individual sections are discussed below.
Sections 1 displays a cross sectional view of the flow in the glottis. The glottis provides for the laryngeal jet effect forcing the flow towards the rear wall as can be seen in the lower part of the figure above. This laryngeal jet effect is mainly due to the geometry of the larynx and the glottis. The present model replicates the laryngeal jet effect only partially as the interior geometry of the larynx has not been captured, however the flow regimes generated in the present model seem to satisfactorily imitate the effect on particle dispersion and airflow modification. Also recirculation zones are observed at the posterior walls of the section.
Sections 2 and 3 above and Sections 4, 5, 6 below do not show much variation in flow regimes however upon comparison of the sections it can be seen that the turbulent effects induced by the larynx are subsiding as the flow progresses in the straight tracheal tube and also the size of the recirculation zone created due to the bias towards the posterior wall generated by the larynx is reducing.
Section 7 shown above depicts a scene when the flow is approaching the first bifurcation. A secondary vortex to the right of the figure can be observed and an increased velocity flow to the left can also be observed. The secondary vortex in this section is the same as the recirculation zone explained earlier in the generalised explanation of the velocity regimes of the 3 generation model. Secondary vortices are mainly caused as a result of compression of slower layers, geometric features and flow history or the propagation of disturbances in the downstream flow upwards.
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The above sections 8a, 9a, 10a, and 11a belong to the left branch of the airway geometry. Section 8a shows a strong axial field with very slight secondary vortices towards the outer walls. In section 9a some branching can be observed. Strong axial fields characterise the main bronchi however weaker fields are seen in the branching areas. Similar strong axial profiles can be seen in sections 10a and 11a the strength of which increases in section 11a as the flow approaches the outlets. However in realistic scenarios this may not be the case due to further division of flow with progression.
The right branch consists of sections 8b, 9b, 10b, 11b shown above. The secondary axial vortices due to layer compressions can be still seen in section 8b however towards the inner walls a strong axial flow regime can be observed. Section 9b, 10b and 11 b depict strong axial velocity patterns with minimal axial vortices; this may be due to the small area of the branches which do not allow secondary vortices to fully develop. Section 10b depicts branching in which the flow field in the branch is stronger than that in the main stream contrary to the branching in the left bifurcation of the geometry. This can be accounted for on the basis of branching angle and the spatial angle of the branch. It can be seen from the figure<> below that the branching in the left bifurcation is against the direction of the flow thus attracting slower velocity vectors towards itself, however the branching in the right bifurcation is in the direction of flow thus attracting a stronger flow field. The right bifurcation branching is also stronger than the mainstream flow. This may be accounted in terms of the area of the branches and also the obstructions to flow encountered downstream from the bifurcation. It should be noted that in all the sectional velocity graphs there does exist an undercurrent directed towards the outer walls in the plane of the section.
Particle Dispersion studies were conducted for two airway models of the human lungs. The first model was a single bifurcation airway, wherein particles of only 3 micron were inserted through the inlet and the patterns were observed. This was done to verify the correct functioning of the models so as to save time on the final simulations performed on the second, 3 generation geometry of the human airway, for which the computation time is much greater than the Single bifurcation model. The single bifurcation model showed Secondary vortices at regions immediately after the bifurcation, also a major recirculation zone was observed to the left of the bifurcation. Not many studies were conducted on the single bifurcation models as this model served more for the purpose of pilot studies.
The 3 generation models were observed for particle depositions. The inlet conditions were kept steady at 440ml/s and 10000 particles were uniformly distributed throughout the inlet. The uniform distribution was necessary to ensure similarities in the various simulations and helped establish a platform for comparison. Randomised distributions were initially used however dispersion patterns observed were not very stable and varied drastically for the same models in different runs. It is believed that the erratic nature of the distributions may be a result of the short length of the model and hence the low residence time of particles in the model which further suggests low quantities of particles deposited may have be responsible for these erratic patterns. The major focus of the particles was laid on the uniformly distributed injection of particles. The velocity profile as described earlier suggests a bias of particle flow towards the rear end of the tracheal wall in consequence to the laryngeal jet effect. The particle diameters were varied taking values of 3, 5, 7 and 10 Âµm, and also the particle densities were varied taking on values of 1350, 2700, 3800 and 5000 kg/m3Â. Figure <> and <> show the front and back view of the particle flow patterns.
The above two figures very lucidly display the laryngeal jet effect creating the flow bias towards the posterior wall of the geometric model. Further it can be seen that the particle velocities are extremely low at the posterior wall. Figures <>, <>, <> and <> displaying the particle deposition patterns for particles of density 1350 kg/m3 for different particle sizes are shown below.
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The particle transport and deposition of micron size particles is primarily dominated by impaction and sedimentation. The major factors affecting micron particle transport are the Stokes number, the Reynolds numbers at the inlet and the airway geometry. Since air is the medium of transport of micro-sized particles it is important to understand the airflow regimes and patterns globally and locally to account for the particle deposition patterns. The Deposition Fraction (DF) is used to account for the fraction of inlet particles that deposit at any particular site in the human airway model and is the ratio of the number of particles deposited at the site to the number of inlet particles.
Total Deposition Patterns
Density (kg/ cu.m)
Density (kg/ cu.m)
The above Figure <> shows the variation of the Deposition Fraction with an increase in size of the particles for constant densities. It can be seen that for denser particles such as those of 3800 and 5000 kg/m3 there is a clear trend reflecting the increase of total deposition in the airways with an increase in the particle size. As the density reduces this pattern becomes vague such as that for 2700 kg/m3 particles which show a initial reduction in the deposition fractions followed by a regular increase in the deposition fractions with increasing particle size. However for less dense particles such as those of densities less than or equal to 1350 kg/m3 the deposition fraction continuously reduces with an increase in the particle size. These variations in the patterns of lighter particles such as those of density 1350 kg/m3 in sizes 3, 5, 7 and 10 micron and of density 2700 kg/m3 in sizes 3 and 5 micron can be explained on the basis of the influence of sedimentation and impaction effects in the deposition of these particles. For lighter particles the influence of gravity is less adequate to sediment particles by displacing them from their streamline flow, thus with a larger fraction of the particles flow to the lower branches and hence in the present model flow through the outlets. The effect of impaction is greater on lighter particles forcing deposition in the upper airway, unlike the larger particles which flow downstream. This can be seen in figures <>, <>, <> and <> which show greater deposition of lighter particles near the inlet. The general trend for micron size particles is for the Trachea-bronchial deposition fractions to increase with the increase in particle size due to the increase in the deposition due to impaction.
The variation of total deposition fraction with the density for different particle sizes is shown in figure
<>. The deposition fractions as show similar trends as those for the variation with size keeping density constant. The larger particle sizes (7,10 Âµm) show a distinct increase in the deposition fraction with increasing density. However the smaller particles (3,5 Âµm) show an initial decrease and then a regular increase in the deposition fraction patterns. This can also be accounted for on the basis of lower influence on gravitational forces causing less sedimentation of lighter particles which increases as the weight of the particles increase. Also the influence of impaction on lighter particles is much greater causing high deposition of extremely light particles in the upper airway near the model inlet.
Figure <> depicts a linearly increasing trend for the relationship between deposition fraction and stokes numbers. An increase in the Stokes number leads to the increase in the deposition due to fraction. This increase is distinctly characteristic of particles of diameter greater than 5 micron, however for particles with stokes number less than 0.04 the influence of sedimentation on the deposition is greater leading to reducing deposition of particles with increasing stokes number.
The main characteristics of deposition patterns as studied on idealised geometric models includes major particle depositions near carinal ridges and minor particle depositions near walls; however for realistically geometric airways fewer particles deposit at the geometries which is due to the lower values of axial flow velocities, responsible for deposition on carinal ridges and stronger upstream and secondary flows causing large fractions of particles to move outwards thus depositing on the walls. Further particle dispersion patterns for realistic geometries are extremely asymmetric in nature. These points are clearly depicted in the deposition patterns shown in figures <> to <>. Cartilaginous rings aid deposition in proximate areas of the rings <Reference>. Thus depositions in the trachea are greater in models that incorporate cartilaginous ring geometries.
The number of particles entering a particular airway bifurcation depends on the Area of the airway, spatial angle and number of sub-branches or resistance to flow downstream. The knowledge of the variation of deposition patterns with respect to bifurcations for varying diameter and density of the particles is of importance for the targeting of aerosol drugs. The figures<> to <> show the particle deposition fractions for particles of different diameters at the various bifurcations with respect to the Stokes numbers. There has already been established the linear co-dependence between the Stokes number and deposition fraction for the total deposition fractions of particles. A clear reduction in the deposition fraction for distal bifurcations is observed in from figure <> and <> showing charts for 3 and 5 micron particles. However the particles at the outlet are much greater implying that for smaller size particles there is higher penetration of much larger quantities into the airways. Further it may be observed that the deposition at any particular airway does not vary much with an increase in the stokes number or density. However there do exist certain trend breaking points for the smaller size particles which occur at the same locations as those for the total deposition charts in figure <> and <>. These may be attributed to as an effect of the values of sedimentation and impaction deposition that occurs in these particles.
The particle deposition patterns for 7 and 10 micron particles show a constant decrease in the outlet particle flow rate suggesting that with an increase in the stokes number and or the density the larger particles( dp â‰¥ 7Î¼m) do not penetrate to the extremely distal generations of the airways say generations of the respiratory zone. In fact with an increase in the stokes number the particle depositions increase for bifurcation 1 - 3. Though the deposition fractions at bifurcation 1 are always observed to be greater than those at bifurcations 2 and 3 for smaller sized particles an increasing deposition fraction pattern is observed for the second and third bifurcations, so much so that beyond a certain stokes number they over power the outlet particle flow fractions. Cartilaginous rings also aid the deposition of particles in their proximate areas.
The local particle depositions bear consequence to upstream deposition, local airflow and particle motion. Further micron particle depositions are non homogenous and in large cross sectional area airways secondary flows have greater influences on particle depositions. Sedimentation plays an important role in the smaller airways where gravitational force is a major cause of deposition on walls. Presence of sedimentation deposition in lower airways of micro particles reduces the impaction deposition in these airways contrary to the larger airways thus they observe a reduction in DF values for distal bifurcations. The above reasons support the fact that particle depositions in CT scanned models is not limited to carinal ridges and is more wide spread.
CONCLUSIONS AND FUTURE WORK
The velocity profile observes induction of turbulence at the larynx which die in the trachea. The laryngeal jet effect promotes a bias of airflow to the rear wall thus increasing particle depositions at the rear wall.
The recirculation zones and hotspots occur on the carinal ridges and outer walls near bifurcations due to layer compressions and are favourable sites for particle depositions where pathological particle depositions may cause tumours.
The particle depositions tend to linearly increase with an increase in the stokes number, diameter and density of particles.
The increase in stokes number, density and diameter of particles also promotes greater depositions in generations 1-3 of the airways.
For smaller size particles (3, 5Î¼m) the impaction aids the deposition in proximal areas to the inlet and sedimentation accounts for a smaller fraction of deposition of particles. However for larger particles (dp â‰¥7Î¼m) sedimentation aids particle deposition at walls in smaller airways and also aids propagation of particles away from the inlet. However impaction is greatly responsible for particle deposition in the trachea.
Unlike the idealised symmetric models particle deposition patterns are more widespread.
Future work will primarily focus on calculating influence of sedimentation based and impaction based depositions. Also inhalation patterns shall be approximated with more realistic conditions such as cyclic inhalations.
Methodology and Approach
Particle Dispersion inside a human airway has been established to be an area of interest to scientists and doctors for years. However due to the recent familiarisation with simulation techniques and complicated structure of the human trachea-bronchial tree and bearing time constraints in mind, a progressive and explorative approach, removing idealisations step-wise has been applied during this investigation for maximum efficacy.
The methodology for the study of particle dispersions in the human airway may be divided into the Geometric modelling of the human trachea-bronchial tree and, the CFD simulations of Airflow and Particle Dispersion.
Geometric modelling provides for the physical basis upon which experimental studies and/or simulations may be performed. The use of CT scans and MRI data to recreate models of the human trachea and its branches are amongst the most advanced techniques available for this purpose. Geometric modelling for the present investigation was conducted in a similar fashion employing the aforementioned technology. The two approaches that exist for the purpose of reconstruction of geometric models using data obtained from CT scans and MRI scans are, Firstly the use of digital reconstructions and Secondly, the use of inhaled-volume-filling algorithms that establish relationships between any particular airway and its branches. The modelling for the current study is explained below:
CT scans in the form of DICOM files (Digital Imaging and Communications in Medicine) of a 28-year -old Indian male using a single slice helical CT scanner were obtained. The images obtained were in the axial, coronal and the sagittal planes at a resolution of 0.7 x 0.7 mm2, a slice thickness of 0.625mm with a slice separation of 7mm. The slice thickness and separation is of great importance as a lot of detail is lost as the slice separation increases. The present set of images has a higher than desired slice separation resulting in the occurrence of burrs in the re-generated 3D models.
The reconstruction process:
The DICOM files obtained were loaded into the medical image processing software Mimics and ITK-Snap wherein a 2D segmentation process was used to slice-wise extract the airway walls.
The CT Scans obtained represent pictures in the form of voxels. Voxels are analogous to pixels, used for representing data in 2D images. Due to the Three-dimensional nature of our reconstruction, 3D mapping of data in voxels is employed and contain data about the intensities of each sector of the organ mapped under the CT scanner. It is worthwhile noting that each human component such as soft tissue, muscle tissue, bone, cartilage, etc. tends to reflect only a particular range of rays incident on it thus absorbing the others. This reflected range of rays is characteristic of the component type and specific to it and is the property of the organs that makes CT scanning possible and it is this characteristic that we use to identify and create 3D images from the CT scan. DICOM files consequently contain much more information in each particular voxel than regular JPEG and BMP files. The intensities of the frequently used components are usually predefined in medical image reconstruction software, however in the present case, the reconstruction of the trachea-bronchial tree there is a complete absence of tissue, and this void, that is the human airway is surrounded by various soft tissue and cartilage of similar densities and intensities. It is the void surrounded by tissue which was captured in our reconstruction employing a volume filling algorithm.
Segmentation is the process of detection and extraction of the desired sections of the image to be reconstructed. Establishing a Threshold Range for the intensities of interest, the interconnected tubes of the respiratory tracts were iteratively captured by specifying a starting point for the volume filling algorithm that progressively segments neighbouring voxels of intensities falling in the threshold range. The automated segmentation progress was terminated by the user as the area of interest in this particular scenario, was only a part of the entire lungs. Upon having completed automated segmentation a manual segmentation process was conducted wherein the geometric model was finalised by filling in gaps and correcting errors such as the non detection of the certain finer branches. With a good resolution CT scan with finer slice separation the labour involved in manual segmentation may be minimised.
The segmented regions were then transferred into a voxel map creating a 3D geometric model of the human trachea-bronchial tree. Figure <> shows the reconstructed human lung geometry consisting of 1 and 3 generations. Only 3 generations have been considered in the present scenario due to insufficient detail in the CT scans.
The 3D reconstructed geometries were exported in suitable CAD formats to CFD software. For the particular case STL files of the geometries were imported in STAR CCM plus v4.04.011 for fluid flow and structural analysis.
The objective of CFD is to numerically solve the partial differential equations that govern fluid motion. The solution of any fluid flow problem solved using CFD simulations incurs decision making regarding the following few steps; Establishing governing equations for fluid flow problems, Meshing, Solution method and Convergence Criterion. The selection and establishment of the governing equations is explained as part of the Technical background later in this report.
The trachea-bronchial tree once reconstructed using the reconstruction and image processing software Mimics is imported to STARCCM+ v4.04.011 in STL format. Meshing is the division of the entire part (trachea-bronchial tree) into small cells, to quantify the computation of the various parameters and limit their calculation to a specific number of cells or nodes. The variables such as velocity, pressure, density etc. which appear in the fluid flow equations are hence calculated at the centroids of these cells. In this specific case we employ a polyhedral mesh, with 10 prism layers extending to 33% of the base size of 1mm. The prism layers provide extra cells in boundary regions such as those near walls where it is assumed that steeper changes in the values of the variables may occur. Thus we obtain a total of 1100033 Cells, 5763725 Faces and 4185004 vertices over the entire part body for the 3 generation model and 430231 Cells, 2331507 Faces and 1733847 vertices for the single bifurcation model.
Geometry generation is followed by the establishment of governing fluid flow equations and decisions regarding the type of solution methods to be used. The fluid flow fields are known to be transient (Li, 2007); however this study investigates only the uniform steady inhalation effects on turbulent fluid flow in the human respiratory tracts. Typical inlet flow rates for mild exercise of 440ml/s corresponding to the mean Reynolds number of 2797 were considered. It has been previously suggested that the flow in the upper bronchial airways may change from laminar to turbulent flow, going via transitional flow under normal as well as heavy breathing conditions (Zhang, 2003) (Zhang Z. K., 2005). A number of solution methods such as the k-Îµ method and the k-Ï‰ method are known to exist for fluid flow problems. The k-Îµ method used in Reynolds-Averaged Navier-Stokes (RANS) solvers is deemed unsuitable for flow at relatively high Reynolds number, example for flows with Reynolds number over 50,000 (Breuer, 2000) (Travin, 1999) (Tutar, 2001). Further the Low Reynolds Number k-Ï‰ model by Wilcox (1998) (Wilcox, 1998) is suggested to be rather suitable to simulate the laminar-transitional-turbulent flow for lower Reynolds numbers (Ryval, 2004) (Varghese, 2003) (Zhang Z. &., 2003) (Zhang Z. K., 2005). However for the particular scenario at hand the inlet flow suggests that the Reynolds number may not reach values high enough to out rightly disregard the k-Îµ model. Thus simulations for both the models have been run during initial testing for model and conclusions derived suggesting usage of a particular model during particle dispersion studies. The larynx was incorporated in the reconstructed geometric model thus including part of the laryngeal jet effect. The inner more complex geometry of the larynx have not been recreated, thus there is only a partial laryngeal jet effect in existence in the simulated flow.
STAR CCM+ v4.04.011 was used to provide a comprehensive environment for the numerical solution to the study of particle deposition in a human airway. Polyhedral meshes with 1100033 cells were employed to conduct the fluid flow analysis on a single bifurcation and a three-generation model of the respiratory tract.
A Three-dimensional, stationary analysis, in a steady state environment (single time step) was considered in STAR CCM+ to model the flow of air in the geometrically reconstructed model of the human respiratory tracts, wherein all properties of air were considered for air at room temperature. The modelling of the particles was done using the Lagrangian multiphase model, offered by STAR CCM+ with particles modelled to be spherical with constant density and other physical properties approximated to those of a water droplet. Further the interacting particles are of constant dimensions and deposition of particles is considered to have occurred when particles are at one particle radius from surfaces such as walls of the tracts. The study was based on interpretation of data regarding the variation of deposition patterns of particles on the walls of the trachea-bronchial tree with respect to variation in the diameter and the density of inhaled particle droplets. For this purpose 10000 particles of the same diameter and density were injected into the inlet of the model and their deposition patterns studied under the effect of gravity and a drag force suffered as a result of the flow velocity.
The particles were injected into the model at velocities similar to that of the flow with their dimensions taken to be 3,5,7 or 10 Âµm and their density taken to be 1350, 2700, 3800 and 5000 kg/m3. The particle positions were tracked at intervals of 100 iterations and then again for each iteration in the last 10 iterations using 16 parallel processors each simulation running for an average of 128 CPU hours and performing a mean 2510 iterations and were terminated at convergence which was considered to be attained when all residuals attained a value of less than 9x10-3.
The simulations performed were progressive in nature and began with a simple airflow analysis conducted on both, a single bifurcation as well as a 3 generation model of the trachea-bronchial tree, for both laminar and turbulent flows. The turbulent scenarios saw a turbulent intensity of 5% and were explored for both the k-Îµ and k-Ï‰ turbulence models. Upon attainment of convergence exploration was extended to the inclusion of particles, which was again conducted in a progressive manner. The residual plots showed faster convergence of the k-Îµ model than the k-Ï‰ for simulations with particle inclusions however k-Ï‰ turbulence models converged faster for simulations without particle inclusions. The difference in the number of iterations for the two models was far greater than that in the models with particles.
Thus the k-Îµ model was chosen for conducting further simulations on the models with particle inclusions.
The decision regarding the values of the simulation parameters are based upon the theoretical knowledge and understanding of the governing equations. The technical background sets groundwork for these parameters. The governing equations and theory are as follows:
The basic governing equations for any fluid flow problem are the continuity equation or the mass flow equation
The momentum equations:
Where ui is the local velocity, xi is the spatial coordinate, Ï is the density, p is the pressure, t is the time and Tij is the stress tensor. Tij is the matrix formed by expanding the Einstein form of the equation
The airflow inside the human lung airway structures was modelled in STAR CCM+ using the following models:
Coupled Flow and Energy model
The Coupled flow and energy model provides for a solution algorithm to the governing equations of fluid flow and the name is specific to the software. Its advantages lie in the fact that it provides more robust and accurate results for solving incompressible flows with particle inclusions modelled by the Lagrangian Multiphase model (software specific name) are used. Further the convergence rates for the model do not deteriorate with the refinement of the mesh. It provides an algorithm for the solution of the conservation of mass, momentum and energy equations using a time or, in the present scenario, a pseudo-time marching approach.
The Navier-Stokes equations for a control volume V with differential surface area da may be written as:
and Ï, v, E, and p are the density, velocity, total energy per unit mass, and pressure of the fluid, respectively. T is the viscous stress tensor, q'' is the heat flux vector, and vg is the grid velocity vector.
Total energy is related to the total enthalpy H by:
is the vector of body forces. (STAR CCM+ v4.04.011 Tutorials and User Guide)
The usage of the gravity model was necessary as the flow is acting under the effect of gravity. The modelling of gravity for constant density flows is done in STAR CCM+ by means of the Boussinesq model.
where is the gravitational vector and the quantities and are the coefficient of bulk expansion and the operating temperature, respectively. Assuming pressure at all simulation conditions is the same as the reference pressure and is calculated at the same altitude. (STAR CCM+ v4.04.011 Tutorials and User Guide)
The k-Îµ as well as the k-Ï‰ model was used in the simulations performed, however the k-Îµ model proved to converge faster and was preferred for the simulations over the k-Ï‰ turbulence model hence only the k- Îµ model has been discussed here.
Turbulence represents a flow in state of continuous disturbances and instability, exhibiting fluctuations which are irregular, small scale and highly frequent in nature regarding both space and time. However from an engineering standpoint only the mean values of the variables is of interest. The k- Îµ model deals with these mean values and rewrites the fluid flow governing equations for turbulent flows as follows:
The turbulence kinetic energy, k (units m2/s2) is defined as:
and it's rate of dissipation, ï¥ (units m2/s3) as:
The k-ï¥ model is based on the Boussinesq hypothesis links the Reynolds stresses with the mean rates of deformation and is as follows:
where ï¤ij is called the Kronecker delta (equal to 1 if i=j and 0 if iï‚¹j).
The turbulent viscosity, Î¼t is calculated from the following relation:
where CÎ¼ is a constant of the model. This turbulent viscosity is then used to calculate the Reynolds stresses.
The modelled equations for the standard k-Îµ model are:
where Eij is the mean deformation rate ().
In the standard k-e model, the constants CÎ¼=0.09, Ïƒk=1.00, ÏƒÎµ=1.30, C1e=1.44, C2e=1.92.
There exist various versions of the k-Îµ model however STAR CCM+ automatically selects the Reynolds averaged Navier Stokes models with a realizable k-Îµ two layer model and Two layer All y+ wall treatment model for the present simulation.
All models used are explained in detail in the STAR CCM+ user manual and the corresponding pages of relevance are included in Appendix 1.
Particle Flow equations
The particle transport and deposition are both size dependent. Micron size particle deposition s largely influenced by impaction gravity and occurs in the form of sedimentation. Another major influencing factor is the drag force experienced by these particles as a result of the flow.
For micron particle depositions, the Stokes number (St), named after George Gabriel Stokes, an Irish mathematician, bears significant consequence in the governance of deposition particles. It relates the stopping distance to a characteristic dimension of the obstacle, the inlet diameter of the model in this particular case. A Stokes number with a value greater than 1 implies that the particle follows streamline flow through the obstacle; however, a Stokes number of a value less than 1 signifies deposition of the particle as it would break out of the fluid stream and collide with the obstacle walls.
The stopping or deposition distance from the inlet of the model is dependent upon the magnitude of the Stokes number. In the human respiratory system, the local geometries vary rapidly and also extreme variations in the airflow may be observed. Hence a more accurate understanding of deposition physics may only be revealed upon relating the Stokes number measured using the local area changes.
STAR-CCM+ models spherical, non-interacting particles using a Lagrangian multiphase flow, which further employs the Drag Force model as drag forces are amongst a major contributing force in particle depositions.
The generic form of the equation of conservation of momentum for a material particle is
Here Fs represents the forces acting on the surface of the particle, and Fb the body forces. These in turn are decomposed into
Fd is the drag force
Fp is the pressure force
Fvm is the virtual mass force
Fg is the gravity force
Fu is the user-defined body force
The momentum transfer to the particle from the continuous phase is simply Fs. Also employed is the two way coupling model in STAR CCM+ which accumulates the momentum transfer over all the particles and incorporated into the momentum equation of the main flowing fluid.
The drag force is given by
where Cd is the drag coefficient of the particle, Ï the density of the continuous phase, vs the particle slip velocity and Ap the projected area of the particle.
The pressure force is given by
where Vp is the volume of the particle and is the gradient of the static pressure in the continuous phase.
The gravity force is given by
where g is the gravitational acceleration vector.
For the deposition tracking of the particle, the deposition fraction is calculated at each branch or bifurcation of the model using the in-flowing and out-flowing particle flow rates corresponding to the particular bifurcation. The model for the purpose of nomenclature disregarding spatial angles is drawn as a stick figure model as shown below.
The stick model identifies different sections of the trachea-bronchial tree reconstructed model which are used to calculate the depositions as well as observe velocity profiles.