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ANSYS POLYSTAT 12.1 User`s Guide

1. gt fluid D the square cavity a closed domain the channel flow an open domain Steady state Time dependent flow a steady flow is a flow that does not change with time The general case of a time dependent flow is a flow that evolves continuously with time our mixing module can study these two kinds of flow However for transient flows the flow domain must not change with time except for flows with moving impellers simulated with the mesh superposition technique If the flow is transient we have to calculate and store the current flow at successive time steps let s note them flow t1 flow t2 However in order to calculate particles path we have to know the velocity field at intermediate times two techniques are implemented In the first case the piecewise steady case we assume the flow is steady between two time steps t1 t2 with t1 lt t2 and is equal to flow t1 This assumption is valid if inertia is neglected and if the boundary conditions change abruptly One example is the oscillating square cavity state 1 DT1 sec state 2 DT2 sec v 0 v 0 fluid v 0 v 0 fluid v 0 November 2009 Polystat User s Guide Example of a piecewise steady flow In this case two velocity fields alternate The first state lasts for DT1 seconds the upper wall moves to the right while the other walls are at rest The second state lasts for DT2 seconds the lower wall moves to the right while the o
2. Polystat x FILE PROPERTIES TRAJECTORIES SLICES STATISTICS HELP SEE MOD DEL properties CREATE anew property NEW disagglomeration properties With this menu you have the ability to define new properties by combination of the existing ones Along each trajectory we have stored a list of values for a small set of parameters also named properties like explained in the next drawing trajectory t2 t3 For each particle for each position stored at time t 0 t1 t2 t5 etc we can store in the mixing files from POLYFLOW several of the following properties 1 the current time the position coordinates the space integration length of the trajectory from the initial position 2 the natural logarithm of stretching In A or In n the direction of stretching m or the rate of stretching J A or if 77 the rate of dissipation D 3 the pressure the velocity the temperature 4 the determinant of the tensor F the divergence of the velocity Both parameters give information about the accuracy of the calculation as the flow is supposed to be incompressible the determinant of F must remain equal to one and the divergence of the velocity equal to Zero 5 the mixing index the first eigenvalue of tensor T the vorticity 1 The words stretching and elongation are equivalent November 2009 78 Polystat User s Guide 4 2 2 SEE PROPERTIES After the reading of the
3. prefix _ trajectory index csv b The other format is the FV format One saves the selected set of trajectories in a single FVP file that can be loaded in FieldView This option is useful if your simulation is steady state and if you want a better graphic treatment of your trajectories for transient simulations see 4 1 7 POLYSTAT will generate one file with the name prefix fvp Enter now the prefix of the files to generate or select it with the browser click on the browser button After the writing the set of trajectories already written appear in the list named Selected set of trajectories of the window Save Set of Trajectories When you have finished click on the OK button in this window to go back to the main window 4 1 7 THE WRITE SLICES OPTION Save Set of Slices x Select a set of slices AY Z gt Selected set of slices In this window you will select the set of slices you want to save in formatted files First you select one set in the upper list Then you click on the gt gt button The next window appears 74 November 2009 Polystat User s Guide Exporting sample The TIME is tine 4j Select the format of the file s sv format aj Enter the prefix of the filets Sanple Browser OK CANCEL HELP You have now the ability to choose the coordinates and time to be used useful in some case where absolute and re
4. KKK KKKKKKKKKKKKKKKEK The current MIXING Task is defined on whole mesh 1 Upper vel menu enter 1 or CR Extension to the whole mesh 1 Removal of subdomain 1 enter 1 2 Removal of subdomain 2 enter 2 3 Removal of subdomain 3 enter 3 4 Removal of subdomain 4 enter 4 5 Removal of subdomain 5 enter 5 6 Removal of subdomain 6 enter 6 Enter your choice With this menu we can remove or add subdomains of the whole mesh in order to define the flow domain as shown in the example below sl s5 s6 The flow domain is the union of the subdomains s2 s3 and s4 We have to remove the subdomains s1 s5 and s6 which are solid parts of the problem November 2009 38 Polystat User s Guide 3 4 DEFINITION OF THE BOUNDARY CONDITIONS KKK KK KKK KKKKKEKKKKKKKKKKKKKK x Boundary conditions K KKEKKKKKKKKKK KKK KKKKKKKKKKKK T Upper level menu enter 1 or CR 1 boundary 1 is non penetrable enter 1 2 boundary 2 is non penetrable enter 2 3 Add Remove or Modify stopping planes enter 3 Select the boundary condition you want to modify With this menu we can make two things Firstly we can specify for each boundary its type wall entry exit Secondly with the last option you can add what we call stopping planes this concept is explained below By selecting each boundary we will modify its type if n
5. November 2009 Polystat User s Guide 4 4 1 SEE SET OF SLICES If you select the option SEE MOD DEL sets of slices in the SLICES menu of the main window you will see the list of the existing sets of slices See Modify Delete existing sets of slices ri PEE ee If you want to modify some data of a set select it in the list and then click on the MODIFY button The window that served for the creation of that set will appear then modify some data If you want to store the modified data click on OK Otherwise click on CANCEL To remove one set from the list select it in the list and then click on the DELETE button To remove all the created sets click directly on the DELETE ALL button In the two cases POLYSTAT asks for a confirmation of your choice 108 Polystat User s Guide 4 4 2 THE AUTOMATIC SLICING OPTION Create automatically a list of slices With this method you generate automatically a list of ordered slices You have first to select a set of trajectories on which the slicing will be done Second you specify the first slice of the list you select a property a position and a direction for the plane Third you enter the number of slices you want and the distance increment between two successive planes Finally you enter the name of the new set This slicing is based on a single property and all the slices are parallel to each other This is not the case for the ma
6. With this method it is possible to combine two sum functions with an arithmetic operator addition multiplication division and subtraction Select the two sum functions the operator and enter the name of the new function The calculation will be done like this first we define a list of x values distributed linearly along the X axis and enclosed between the X minimum and the X maximum of the two functions Second for each x value we search for the y value of the two functions y1 function1 x amp y2 function2 x The y value of the result function corresponding to x will be y y1 lt operator gt y2 You will find an example of use of this method in Addendum B November 2009 125 November 2009 Polystat User s Guide 4 5 2 9 SMOOTHING ON FUNCTIONS Smoothing First select the function to smooth Then select the kind of smoothing you want and specify some parameters we will explain that below You must also enter the number of values to represent the smoothed function a good practice is to use the same number of values that represent the data function You have also to give a name to the new function The type of the result function will be the same than that of the data function It is impossible to smooth a function that is already a smoothed function The method of smoothing is the following to calculate one value of the result function we calculate the mean of values that surround it in the da
7. The minimum and maximum sizes are evaluated as shown in the picture below They corresponds in fact to the 5 th percentile and 95 th percentile of the f function Area 0 05 Area 0 05 a min max b size 98 Polystat User s Guide 4 2 4 3 FRACTION OF AGGLOMERATES OF GIVEN SIZE Fraction of agglomerates of given size With this property the user can evaluate the mass fraction of agglomerates at a given time having a size between a given range Sa Sb Sb FraC sast f S t ds Sa November 2009 99 November 2009 Polystat User s Guide 4 2 4 4 NUMBER OF AGGLOMERATES OF GIVEN SIZE Number of agglomerates of given size With this property the user can evaluate the number of agglomerates at a given time having a size between a given range Sa Sb Sb where FraCsq s5 t f s t ds Sa Sb Sb t f s t s ds J FCD ds Sa Sa Vs is the volume of a sample able to contain a large number of agglomerates of various sizes X is the mass fraction of agglomerates included in the sample Vs and k is a shape factor k 1 6 if we assume that agglomerates are spheres k 1 if they are cubes 100 Polystat User s Guide 4 2 4 5 MASS OF AGGLOMERATES OF GIVEN SIZE Mass of agglomerates of given size With this property the user can evaluate the mass of agglomerates at a given time having a size between a given range Sa Sb Msas V
8. a the All trajectories set contains all the trajectories read in the mixing files b the Interrupted trajectories set contains all the trajectories the calculations of which have been interrupted for different possible numerical problems and read in the mixing files c the set_from_filename sets contain the trajectories stored in the mixing file filename We can not modify or delete those sets If we remove a mixing file in the window READ data the list of existing sets is adjusted automatically If new sets of trajectories have been created it is possible to modify or to remove them from the list If you want to modify some data of a set select it in the list and then click on the MODIFY button The window that served for the creation of that set will appear then modify some data If you want to store the modified data click on OK Otherwise click on CANCEL To remove one set from the list select it in the list and then click on the DELETE button To remove all the created sets click directly on the DELETE ALL button In the two cases POLYSTAT asks for a confirmation of your choice Based on existing sets of trajectories it is easy to define new sets Three possibilities exist a the creation of a new set based on a condition to respect b the combination of two sets to create a new one c the selection of a single trajectory based on a criterion of proximity 103 November 2009 Polystat
9. 44 Polystat User s Guide KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK x Parameters for the stopping planes A KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK Current setup several stopping planes plane id coeff A B Cy D 1 100000E 00 200000E 00 300000E 00 400000E 00 2 500000E 00 600000E 00 700000E 00 800000E 00 WARNING a trajectory of a point is stopped when its position x y z is such that A x B y C z D is negative 1 Upper level menu enter 1 or CR 1 Add a new stopping plane enter 1 2 Modify the parameters of an existing stopping plane enter 2 3 Remove an existing stopping plane enter 3 Enter your choice With the options 2 and 3 you can modify or delete existing stopping planes 3 5 DEFINITION OF THE FLOW FIELD 3 5 1 STEADY STATE FLOW For a steady state flow simulation this menu appears like this KKEKKKKKKKKKKKKKKKKKKKKKKK 7 Flow Definition 5 KKEKKKKKKKKKKKKKKKKKKKKKKK Current choice filename res This file is formatted 1 Upper level menu enter 1 or CR 1 Enter the name of the result file enter 1 Enter the time of use 3 This file is not formatted enter 3 Enter your choice One has just to specify the name of the result file containing the velocity field Polyflow result file format and if this file is formatted or not November 2009 45 Polystat User s Guide 3 5 2
10. TIME DEPENDENT FLOW For time dependant flows the Flow Definition menu is KKKEKKKKKKKKKKKKKKKKKKKK K z Flow definition KKEKKKKKKKKKKKKKKKKKKKK Continuous transient flow Automatic selection of result files time step 1000000E 01 the res files are formatted res files from res 1 to res 10 first res file res 2 1 Upper level menu enter 1 or CR T Piecewise steady flow enter 1 gt 2 Continuous transient flow enter 2 gt 3 Automatic selection of result files enter 3 4 Manual selection of result files enter 4 5 Modify the time step enter 5 6 Modify the prefix of the result files enter 6 7 Modify the number of result files enter 7 8 Modify the first result file id enter 8 9 Modify the format of result files enter 9 Modify the list of result files Enter your choice One has first to specify if the flow is piecewise steady option 1 or continuously transient option 2 the default Next we must specify the list of result files containing the successive flow fields two ways exist e In the automatic selection mode the default the user specifies the time step constant between two successive flows the prefix and the format of the Polyflow result files one particular file for each flow the number id of the first flow and the number of result files to be read For example in the menu above
11. d with d df l dl 30 0 The difference of the standard deviations s t o 0 P with o f d d f 1 dl 31 0 24 Polystat User s Guide t 1 green_surface As for the segregation scale such parameters have limitations as they are global indices we cannot detect local defects 2 3 2 DISTRIBUTION IN ZONES As for the distribution index we want to quantify distributive mixing But with this new method we will be able to detect zones of the mixer where material points are missing and where there is an excess of points As usual we distribute a cluster of particles initially concentrated in a small box see figure below As a function of time the flow will distribute this set of points We define a set of adjacent and non overlapping zones covering all the flow domain November 2009 23 November 2009 Polystat User s Guide In the figure below we have four zones Next we distribute randomly in all the flow domain the same number of points N we assume that such a distribution is the optimal one At time t for each zone for the two distributions we will determine the number of points included in it Based on these numbers we can evaluate a relative error of distribution for each zone Z elz _ nbr Z as 32 where nbr is the number of points of the real distribution included in zone_Z at time t and nbo is the number of points of the optimal distribution
12. the flow is defined in 10 files named res 1 res 2 until res 10 The prefix is res The first one is res 2 The time step is 1 second between successive flows Let s remark that all those files must exist there can not be holes in the numbering of the result files If the lifetime of the material points is greater than time_step number_of_result_files we use the succession of velocity fields in a loop In the manual selection mode the menu changes and is now like this November 2009 46 Polystat User s Guide KKKEKKKKKKKKKKKKKKKKKKKK 7 Flow definition z KKEKKKKKKKKKKKKKKKKKKKK Continuous transient flow Manual selection of result files 1 res file has been defined Upper level menu enter 1 or CR 1 Piecewise steady flow enter 1 gt 2 Continuous transient flow enter 2 3 Automatic selection of result files enter 3 gt 4 Manual selection of result files enter 4 Modify the time step Modify the prefix of the result files Modify the number of result files Modify the first result file id Modify the format of result files 10 Modify the list of result files enter 10 Enter your choice The user must specify one by one the list of Polyflow result files defining the flow option 10 The following menu appears KKKKKKKKKKKKKKKKKKKKKKK Flow definition KKKKKKKKKKKKKKKKKKKKKKK
13. 0000000e 00 0 0000000e 00 0 0000000e 00 With this function we will obtain for each slice for each zone the evolution of the deviation of the real distribution compared to optimal distribution if one wants to save the function the filename containing this curve is built like this prefix zone index zon nbc Z _ nbo Z nbtotal nbtotal zone Z with 6 Z c 1 1 where nbc Z nbo z is the number of points of the real optimal distribution included in zone Z and nbtotal is the total number of points in the real distribution 123 November 2009 Polystat User s Guide If 5 zone Z is zero that means that the right number of points is included in zone Z If zone Z is negative that means that the number of points included in zone Z is smaller than the optimum there is a lack of points in that zone If 5 zone Z is positive that means that the number of points included in zone Z is larger than the optimum there is an excess of points in that zone the evolution of the global deviation of the real distribution compared to optimal distribution if one wants to save the function the filename containing this curve is built like this prefix glo zon nbzones y bo with s c 0 1 Z 1 rie Nie additional properties are also evaluated but only available through the save functions option We will save in csv files one file per slice_index the zones partitioning the deviation from opt
14. 2 radius For dim 2 X Nx n radius A 4 3 For dim 3 o x Nx 3 m radius where Nx is the number of points of the cluster around x at a distance smaller than the radius On the other side at perfect distribution we expect that the points are distributed equally in all the flow domain we should find everywhere the same number of points per unit volume Then we can easily determine the perfect points concentration Dp it corresponds to the number of points divided by the volume of the flow domain For other situations the reader can easily adapt the method of evaluation of the perfect points concentration as explained in 4 5 2 15 27 Polystat User s Guide Eventually the standard deviation p of points concentration at time t is evaluated as follows p 34 where N is the number of points in the cluster at time t the x correspond to the location of points i at time t and 0 is the perfect points concentration If one looks carefully at this definition the user must be aware that we evaluate points concentration only at positions where there are points There is no points concentration evaluation in zones of the mixer empty of material points If distributive mixing improves the points concentration deviation should decrease At perfect distribution we should have same points concentration in any location in the cluster and the deviation 6 should be zero 2 4 Disagglomeratio
15. POLYSTAT we read the files real_0001 to real_000X and the files opti_0001 to opti_000X We define two sets of trajectories the first one named real_set contains all the trajectories from the real_ files the second one named opti_set contains all the trajectories from the opti_ files We define two sets of slices the first one named real_slicing is a slicing on the time N slices every At seconds for the real_set set of trajectories The second one named opti_slicing is one slice defined for time t 0 and on the opti_set set We define two Distance Distribution functions the first one named real_distribution is based on the real_slicing set of slices The second one named opti_distribution is based on the opti_slicing set of slices We define one Deviation function to calculate the deviation of the real distribution real_distribution function from the perfect distribution opti_distribution function 143 Polystat User s Guide ADDENDUM B THE GLOBAL EFFICIENCY OF STRETCHING November 2009 Let s suppose the flow to be 2D steady state in a closed domain We will explain how we can calculate the time evolution of the global efficiency of stretching see chapter 2 for the definition of this parameter For example for the linear stretch at time t this efficiency is t lt lt e gt gt M t In A dQ i D dt dQ 0 Q Qg To calculate such a parameter the follo
16. User s Guide 4 3 2 THE CREATE A NEW SET OF TRAJECTORIES OPTION By clicking on this option in the main window the following window appears Add a new set of trajectories With this method we will select a subset of trajectories that respect a condition Three kinds of condition are possible a a trajectory is selected if along this one the property has always values in the specified interval For example we can select the trajectories that have always the determinant of F between 99 and 1 01 we reject inaccurate trajectories b a trajectory is selected if along this one the property has never values in the specified interval c a trajectory is selected if along this one the property has one time at least a value in the specified interval For example we can select the trajectories that cross a specified box in the flow domain If the selected property is a scalar the interval is defined in the rectangles 1 and 4 the others are set to zero 104 Polystat User s Guide 4 3 3 THE COMBINE SETS OF TRAJECTORIES OPTION If we select this option the following window appears set C set A lt operatom set B With this option we can combine logically two sets of trajectories A and B in order to create a new set C For example in the window above we selected the sets All trajectories and Interrupted_trajectories and the Intersection operator That means that the new set C will
17. and is observed for large particles This process generates two or more agglomerates Erosion and rupture depend on the size of agglomerates the shear rate and the shear stress In the following text we distinguish two types of solid particles the aggregates and the agglomerates The first ones are the smallest particles that can not be eroded or broken anymore The second ones are larger particles formed of a number of aggregates linked together by cohesive forces Rupture Erosion ee asa Cas ED On Gr e Ee CR begs gt 3 Sos Ssa A eS oy Bg a or SBS 02L oO G2 See ool D See ee Erosion The erosion model implemented in Polystat is based on the work of Collin and Peuvrel Disdier 9 on the dispersive mixing of carbon black agglomerates N234 in a SBR matrix Of 29 November 2009 Polystat User s Guide course due to the large variety of models and raw materials it is possible to adapt or modify the implemented model accessible in the CLIPS file disagglomeration clp that can be found in POLYFLOW bin directory the corresponding functions are interpreted by Polystat at run time We consider hereafter the case of low concentration of carbon black pellets meaning that we neglect erosion due to friction between pellets we only consider erosion due to hydrodynamic forces During erosion small particles are removed continuously from the agglomerates that diminish in size This removal can occur if
18. easy to show see ref 3 that A X M t 4M CM 6 while m is given by FM A A good mixing quality requires high values of throughout time and space A local evaluation of the efficiency of mixing see ref 2 is given by the ratio e X M t Mi P 5 J 2 where D 4trD The values of this instantaneous efficiency are always included in the interval 1 1 We can easily show that D e X M t 6 We note that is a local measure along the path of a material point the time averaged efficiency is defined as t e X M t if e X M t dt 7 0 However there exists another way to define a mean efficiency over time t A dt In A e1 X M t 2 _ nl 8 D dt D dt 0 0 13 November 2009 Polystat User s Guide The physical interpretation of 8 is the following for one material point at time t 2 is the ratio of what we get the final stretching obtained at time t over what we put the total mechanical dissipation until time t Eventually we can define a global efficiency over all the material points distributed initially in the flow In A dQ Ker M D 0 D dt dQ 0 Q o This global efficiency is the ratio of the output the mixing obtained the total stretching of the matter until time t over the input the energy we get the total mechanical dissipation until time t 2 1 2 KINEMATIC PARAMETERS FOR 3
19. first eigenvalue of the extra stress tensor T indicates the local stress important parameter to evaluate the capillary number Select only the properties that are necessary You will save time and memory 58 3 10 PARAMETERS FOR THE STORAGE OF THE RESULTS Polystat User s Guide The initial screen looks like this The current PREFIX the NBFILE the The result f KKK KKK KKK KKK KKK KKKKKKKKKKKKKKK x Storage of the results x kkxkxkxkxkxkxkxkkxkxkxkxkxkxkxkxkkkkkkxkxkxkkxkxkxxk values of the parameters are prefix of the result files number of results files iles are formatted NBPTMX the maximum number of trajectories na CPUMAX Pr pr ts Oni single result file he maximum CPU time in hours enerate a single result file We store th at each time successive positions of the material points step of 1 0000000E 00 seconds mixing 9 1 99 00 4 1 Upper level menu enter 1 or CR 1 Modification of PREFIX enter 1 2 Modification of NBFILE enter 2 3 The result files are unformatted enter 3 4 Modification of NBPTMX enter 4 5 Modification of CPUMAX enter 5 6 Storage of all the points of the trajectories enter 6 gt 7 Storage at each time step exactly enter 7 8 Storage after each time step minimum enter 8 9 Storage at each displacemen
20. i Let d nax will calculate nbval values of the correlation function for distances uniformly distributed between 0 and d be the maximum distance between two material points in the flow domain We max To calculate one value of correlation function for a distance d we 21 November 2009 Polystat User s Guide select randomly nbpair pairs of points if their relative distances are in the interval d max d max 2nbval 2 nbval interpolation through the discrete calculated values The segregation scale S t may then be easily calculated by numerical integration on the basis of equation 25 d The correlation coefficient R r t is completed by a linear RG t distance dq d i d nbval dmax There exists a limitation to this method as the number of points is finite the mean size of the pixels is finite too we cannot calculate accurately a segregation scale that is smaller than this characteristic size If the segregation falls below that size that means that the mean thickness of the striations in the flow is smaller than the size of the pixels the concentration field will appear like a random distribution of pixels of the two colors there are no more continuous lines of one color Another problem of the segregation scale is that it cannot detect a local defect in the flow it is a global index of the quality of mixing Finally the segregation scale depends on the size of the flow domain T
21. in time thus each slice will be a volume We assume that each slice as the same perfect points concentration that is equal to the number of points tracked divided by the volume of the flow domain c For a 2D open flow domain we perform a slicing in space in the direction of the flow thus each slice will be a line We assume that each slice as the same perfect points concentration that is equal to the number of points in a slice S divided by the length of the part of the slice S that cuts the flow domain d For a 2D closed flow domain we perform a slicing in time thus each slice will be a surface We assume that each slice as the same perfect points concentration that is equal to the number of points tracked divided by the area of the flow domain 134 November 2009 Polystat User s Guide In order to evaluate the points concentration in each slice around each point one must specify the radius of the sample all points in the neighborhood of point x at a distance smaller than this radius will be taken to evaluate the local points concentration at position x This radius must be chosen carefully if it is too small no points will be found and the local concentration will not be relevant if it is too large we will evaluate a global concentration that will be identical for all points in the slice and that will not change from slice to slice A first estimation is to take as the radius a tenth of a typical distance in the slice diam
22. included in the same zone If is zero for a zone the right number of points is found in that zone If is negative for a zone there is a lack of points in that zone compared to optimum If is positive for a zone there are too many points in that zone compared to optimum Eventually we can define a global index based on all the zones 1 nb zones Eg 5 x ko 33 z 1 The number of points and the zones partitioning can influence dramatically the indices described above When comparing two different mixers it is recommended to keep constant the ratio number of points zone In order to have relevant results this ratio should be higher than 100 26 November 2009 Polystat User s Guide 2 3 3 DEVIATION OF POINTS CONCENTRATION As for the distribution index we want to quantify distributive mixing However compared to that parameter we do not need to compute a perfect points distribution we just need the actual one As usual we distribute a cluster of particles initially concentrated in a small box see figure below As a function of time the flow will distribute this set of points At time t we have N points distributed more or less in the flow domain For each point x we search its neighbors x that stay at a distance smaller than a sample radius Depending on the dimensionality dim of the cluster of points at time t we can evaluate the local points concentration x For dim 1 o x Nx
23. mixing files if you select the option SEE MOD DEL properties in the PROPERTIES menu of the main window you will see the list of the existing properties See Modify Delete existing properties It is possible to modify or to remove from the list one or several properties If you want to modify some data of a property select it in the list and then click on the MODIFY button The window that served for the creation of that property will appear then modify some data If you want to store the modified data click on OK Otherwise click on CANCEL To remove one property from the list select it in the list and then click on the DELETE button To remove all the created properties click directly on the DELETE ALL button In both cases POLYSTAT asks for a confirmation of your choice 1 Only for the properties created in Polystat November 2009 79 Polystat User s Guide 4 2 3 CREATE PROPERTIES Based on existing properties it is easy to define new properties First select the CREATE a new property option in the PROPERTIES menu of the main window The following window appears CREATE a new PROPERTY Then click on the button corresponding to the appropriate operation November 2009 80 November 2009 Polystat User s Guide 4 2 3 1 IAI This method allows you to calculate the amplitude of a property the velocity for example or more precisely the evolution of the a
24. name to the new property the result This property allows you to change if necessary the frame of reference one in rotation with respect to the other The data needed are visualized in the following picture direction of the rotation axis angular velocity rad s one point of the rotation axis The creation window appears like this 84 November 2009 Polystat User s Guide If the property A is a coordinate field click on the corresponding button The resulting property C will be C t Xo Rot a t A t Xo where Xo is one point of the rotation axis and Rot the matrix of rotation at time t However if the property A is not a coordinate field click on the A is another field button The resulting property C is C t Rot a t A t 85 Polystat User s Guide 4 2 3 9 TRANSLATE Translate this method allows you to translate a vectorial property you have to select the property A to translate and the time the data some data specifying the translation direction and amplitude of the translation velocity and to give a name to the new property the result This property allows you to change if necessary the frame of reference one in translation with respect to the other The creation window appears like this C Translate A If the property A is a coordinate field click on the corresponding button The resulting property C will be where Vtrans is the
25. options define the flow where material points will travel we need to know the flow domain not necessarily the complete domain defined by the mesh the type of boundaries of this domain there are several kinds of possible boundary conditions a material point can cross an entry but not a wall the velocity fields necessary to calculate the trajectory of the material points In the option 4 we define the zones of the flow domain where we generate the initial position of the material points This initial position is always determined randomly inside each generation zone The option 5 allows the user to specify numerical parameters necessary for the precision of the trajectory calculation in the flow With the option 6 we enter numerical parameters necessary for the calculation of the mixing parameters evolving along the trajectories Next with the option 7 we select the properties mixing parameters or pressure temperature we want to know along the trajectories With the option 8 we determine how to store the results of the simulation With the created result files we will make a global analysis of the mixing with POLYSTAT Finally with the option 9 we specify the moving parts overlapping the flow domain if the mesh superposition technique has been used previously to calculate the flow field 37 Polystat User s Guide 3 3 DEFINITION OF THE FLOW DOMAIN KKEKEKKKKKKKKKKKKKKKEK amp Flow domain K x
26. points will be done only for distance between 0 and a value specified by the user It is recommended to choose a value greater than the maximum distance evaluated above In all cases it is recommended to have V n identical if one wants to compare different flow domains or set ups on a same flow domain 121 Polystat User s Guide You will obtain a distribution function for each slice of the set If you visualize this function for a given slice you find on the X axis the distance and on the Y axis the distribution function f d A Distance between pairs of points November 2009 122 November 2009 Polystat User s Guide 4 5 2 7 THE DISTRIBUTION IN ZONES FUNCTION Distribution in zones To calculate the evolution of the distribution of material points initially concentrated in a box you need to specify two set of slices the first one contains points coming from the real distribution the second one contains points coming from an optimal distribution Next you must select a coordinate property that will serve to calculate in which zones are included the points Eventually the user must specify in an ascii file the coordinates of the center of the zones The format of this file is as follows on each line there are the three coordinates X Y Z of one center If the flow domain is 2D the Z is set to zero 0 0000000e 00 0 0000000e 00 0 0000000e 00 0 0000000e 00 0 0000000e 00 0 0000000e 00 0
27. possible link between successive slices and successive positions of impellers In the page Slice select first a set of slices or a set of trajectories second a first slice or trajectory index of this set If you want to see successive slices or trajectories enter also a positive increment step Then click on the button First the first slice or trajectory is drawn in the graphic display Now each time you click on the next button the current slice or trajectory is increased by the step so that we can analyze easily the complete set If we want to avoid to click again and again on the next button click on the animate button all will be automatic In that case do not forget to specify a pause in second corresponding to a waiting time between two successive drawings If the mesh superposition technique has been used in POLYFLOW to calculate a flow with moving parts it is possible to see the location of those impellers at different time steps To do so select yes to the question Set into motion Then select the first position of the impellers corresponding to the first slice of your set of slices Enter also an increment step for the successive positions of the impellers you want to look at Then click on the button First the first slice and the first position of the impellers are drawn in the graphic display Now each time you click on the next button the current slice is incr
28. t make any mistake because a wrong answer can unfortunately interrupt definitively your session It is possible to remove from the list one several or all the files To remove one file from the list select it in the list and then click on the DELETE button To remove all the files click directly on the DELETE ALL button In both cases POLYSTAT asks for a confirmation of your choice If there are no file in the list it is impossible to do anything with POLYSTAT of course If the current list seems complete to you click on the OK button and you will go back to the main window 66 November 2009 Polystat User s Guide 4 1 2 THE READ MESH OPTION When you click on this option the following window appears In the upper part of the window just enter the name of the mesh file Polyflow mesh format and its format If the file is not on the current directory click on the Browser button a file browser appears in order to search the location of the file In the lower part of the window you can enter specific data in order to visualize later the motion of moving parts calculated with the mesh superposition technique in a previous POLYFLOW run First answer yes to the question Impellers motion Next enter the prefix of the result files Polyflow result files in which are stored the successive flow fields enter also the number N of those files in the last index input area Finally specify the forma
29. the process of mixing Dankwertz in the 50 ies analyzed the mixing as an homogenization process of a concentration field initially two different fluids are separated in two adjacent zones as the time goes on the local concentration of each fluid evolves everywhere in the fluid and if the mixing is perfect the concentration must tend to the same value everywhere in the flow to quantify this homogenization process Dankwertz defined two parameters firstly the segregation scale is the average thickness of the striations existing in the flow domain Secondly the intensity of segregation is the standard deviation of the concentration around its mean These parameters are used when there are only two fluids to mix and when their proportion in the flow domain is more or less equivalent Later in the 80 ies Ottino defined other parameters based on the Continuum Mechanics Theory He showed that mixing is a process increasing the interface existing between fluids But instead of measuring the surface of the interface that is almost impossible in complex flows he prefers to measure local increases of infinitesimal surfaces distributed everywhere in the flow But these parameters are not very useful if we analyze the distribution of a small amount of pigments tracers and so on in the flow domain small percentage in volume of the total flow 1 Polystat User s Guide domain Based on the work of Manas Zloczower we have defined a new paramet
30. them option3 Entry boundary in a spatially periodic flow Then the following message appears Current value of the number of crosses is 0 Enter its new value CR no modification You enter here the number of loops that the material points can do in the flow domain the calculation of a trajectory is stopped when the material point reaches the exit after this number or if the lifetime of this particle has expired In the next menu you will specify which is the exit boundary connected with the entry KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKEKKKKKKKKKKKK Search of the corresponding boundary A kkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkkxkxkxkxkxkxkxkxkxkkxkkkxkxkxkxkxkkkkkxkkxxk xx k Determine in the following list which boundary is connected to the boundary Boundary der 1 Upper level menu enter 1 or CR boundary 1 is an entry of a spatially periodic flow 2 boundary 2 is non penetrable enter 2 3 boundary 3 is non penetrable enter 3 Select the corresponding boundary November 2009 41 Polystat User s Guide Finally you return to the previous menu that has changed like this KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK KK KKK i Boundary condition along boundary 1 i KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK KK KKK Current choice Entry of a spatially periodic flow Boundary connected to the boundary Boundary 2 exit This boundary
31. translation velocity and t the time However if the property A is not a coordinate field click on the A is another field button The resulting property C is C t A t V trans November 2009 86 Polystat User s Guide 4 2 3 10 INTEGRATE Integrate this method allows you to integrate a property along the trajectories in time you have to select the property A to integrate and the time the data and to give a name to the new property the result C Integrate A 4 2 3 11 DERIVATE Derivate this method allows you to derivate a property along the trajectories you have to select the property A to derivate and the time the data and to give a name to the new property the result C Derivate A November 2009 87 Polystat User s Guide 4 2 3 12 K k this method allows you to create a property that is constant along all the trajectories you have to select the type of the constant scalar vector with 2 components vector with 3 components to specify its value and to give a name to the new property the result 4 2 3 13 A SLICE A slice this method allows you to transport the value of a property along the trajectories you have to select the property to transport to define a slice see 4 6 1 and to give a name to the new property the result C A Slice November 2009 88 Polystat User s Guide The next picture explains the concept def
32. we obtain a Polyflow result file containing the velocity field the shear rate and other fields of interest If the flow is transient it is recommended to save the Polyflow results files at exacts time steps At However if the flow is piecewise steady with N successive boundary conditions N small in loop or not it is sometimes easier to perform N Polydata sessions one for each specific set of boundary conditions We will run POLYFLOW N times one for each data file and we will obtain N result files containing each one a set of fields specific to a particular set of boundary conditions November 2009 Polystat User s Guide Gambit Icem Patran finite element mesh Polydata data file N times if piecewise steady flow Polyflow N successive steady states result file velocity field shear rate e The mixing simulation in a second step we enter back in Polydata to define a MIXING task we specify the mesh the velocity fields to use and the initial position of the material points and other properties to evaluate along trajectories Next we run POLYFLOW with this data file POLYFLOW generates randomly the initial position of a set of material points and calculates their trajectory in the flow domain Along these trajectories POLYFLOW calculates also the evolution of some properties and kinematic parameters temperature viscosity stretching rate of stretching rate of dissipation And finally POLYF
33. 55 143 ADDENDUM B THE GLOBAL EFFICIENCY OF STRETCHING 144 TABLEOFCONIENTS oeecrescncz cestccs scat rassesaacas econ ae TAA iiA tei 145 November 2009 148 CHAPTER 1 Polystat User s Guide INTRODUCTION 1 1 OBJECTIVES In industry the mixing process is widely present it occurs in different kind of machines in a continuous or batch process like in Banburry mixers Kenics mixers extruders stirred vessels and so on The objectives are various too distribution of pigments or other compatibilizers generation of interfaces between different fluids in order to enhance chemical reactions The main objective of this module is to offer the user the ability to quantify the mixing in the process of his interest We will define later a set of objective parameters that are relevant for different situations But we tried to go a step further there is at this time always a certain evolution in the way scientists are quantifying mixing That s why we developed software that uses existing parameters and allows the user to also define new parameters This module simulates mixing for various flows but the situation is in general so complex that numerical simulation cannot take into account all the real phenomena existing in such processes In a next section of this chapter we will explain the needed assumptions and hypotheses 1 2 HOW TO CHARACTERIZE MIXING November 2009 There are various ways to define
34. ANSYS POLYSTAT 12 1 User s Guide November 2009 Copyright 2009 by ANSYS Inc All Rights Reserved No part of this document may be reproduced or otherwise used in any form without express written permission from ANSYS Inc Airpak Mechanical APDL Workbench AUTODYN CFX FIDAP FloWizard FLUENT GAMBIT Iceboard Icechip Icemax Icepak Icepro Icewave MixSim POLYFLOW TGrid and any and all ANSYS Inc brand product service and feature names logos and slogans are registered trademarks or trademarks of ANSYS Inc or its subsidiaries located in the United States or other countries All other brand product service and feature names or trademarks are the property of their respective owners CATIA V5 is a registered trademark of Dassault Systemes CHEMKIN is a registered trademark of Reaction Design Inc Portions of this program include material copyrighted by PathScale Corporation 2003 2004 ANSYS Inc is certified to ISO 9001 2008 See the on line documentation for the complete Legal Notices for ANSYS proprietary software and third party software If you are unable to access the Legal Notice contact ANSYS Inc Polystat User s Guide TABLE OF CONTENTS CHAPTER 1 INTRODUCTION ig sxes sssesssolseuseaxan niobate cls vase ain 1 PA ODICCH VCS es eich ia ieee N estes lovee A Manda eecagsni Neds Dee ONO Neh tone Re ae 1 1 2 How to Characterize Mixing sinrega a a EPE E naan 1 1 3 Classification of flows capabilities of
35. Current setup the velocity fields are included in the files Index Time of use Filenam 1 1 00000E 20 res 1 Upper level menu enter 1 or CR 1 Add a velocity field enter 1 2 Remove a velocity field enter 2 3 Modify the attributes of a velocity field enter 3 4 Change the order of use of the velocity fields enter 4 Enter your choice In this menu one can add remove modify the result files For a particular velocity field its parameters are summarized in a menu like the next one November 2009 47 Polystat User s Guide Enter your choice KKEKKKKKKKKKKKKKKKKKKKKKKK a Velocity Field i a KKEKKKKKKKKKKKKKKKKKKKKKKK Current choice filename res This file is formatted time of use 1 00000E 20 1 Upper level menu enter 1 or CR 1 Enter the name of the result file enter 1 2 Enter the time of use enter 2 3 This file is not formatted enter 3 November 2009 In this menu we define a new velocity field we specify in which result file is stored this field option 1 and if this result file is formatted or not option3 Finally we enter the time of use of the current velocity field option 2 If the lifetime of the material points is greater than the sum of all times of use we use the succession of velocity fields in a loop Note the manual selection mode is not practical if dozen result files must be specified 48 Polystat Us
36. D FLOWS For 3D flows we will calculate the local stretching of infinitesimal surfaces by the mean of the area stretch n In the initial configuration Qg let define an infinitesimal surface dA with a normal direction N With time this surface deforms at time t this surface is noted da with a new normal direction N The area stretch n is the ratio of the deformed surface da at time t over the initial surface dA A da 7 X N t J 10 n n amp N t If the fluid is incompressible we obtain see ref 2 n N C N 11 If the fluid is incompressible the normal direction to the surface da is 1 x a EDN 12 n 14 November 2009 Polystat User s Guide A good mixing quality requires high values of n throughout time and space A local evaluation of the efficiency of mixing see ref 2 is given by the ratio R n n X N t _ en D 13 The values of this instantaneous efficiency are always included in the interval 1 1 After some transformations it is easy to show that o EN EPA 14 We note that ey is a local measure along the path of a material point the time averaged efficiency is defined as t ey X t 1 e X N t de 15 0 However there exists another way to define a mean efficiency over time t UAS aa en X p se aN 16 t t J pw var 0 0 Like for 8 the physical interpretation of 18 is the following for one mat
37. ECT one single trajectory Option s sssssssssssssessisssssesrsssessieressessee 106 4Ath SLICES MENU tvs oeiee dete asr AE EE E welacten ed AATAS REEERE 107 44 1 See Set of SLICES isscatscssssatiteesi ienesis a aa asari iarann rarena 108 4 4 2 the Automatic Slicing Option ss ss sssssssssesiesesstesissesstesssstsrienesesresnesseetessessne 109 4 4 3 the manual Slicing option cccccseesesseecseecseeeseseseseecseecsesesseeseseecssesseeeasens 110 4 4 4 the sub Slicing Option ccccccseseseseseseseecseecseseseseseseecseecsssessssseseeseessssesseess 111 4 5 the STATISTICS Menisi ieira ieo iae SAE EE ie 113 4 5 1 See STATISTICAL functions niini anra a ar ie 113 4 5 2 Create STATISTICAL functions ccccececssecessesesesessseesseessseesssesessesssesseeeseens 114 4 5 2 0 the See property along a trajectory fUNCTION cece 115 4 5 2 1 the SUM Funct On isses ia na e ai rE E alae 116 4 5 2 2 the MEAN amp STANDARD DEVIATION function 00 117 4 5 2 3 the CORRELATION fUnCtiOn cccceecceeseeeteeetesstesteeeetesesenseens 118 4 5 2 4 the PROBABILITY fUnction cccccceecseeseeeseeetessseeseeeeeseneneeens 119 4 5 2 5 the Auto Correlation On Concentration function 120 4 5 2 6 the Distance distribution function s sssssessssssssesrsssessseressessee 121 4 5 2 7 the distribution in Zones function s sssssesssssssresrsssessisressessee 123 4 5 2 8 Arithmetic operation o
38. KKKKKKKKK Current setup index intensity generation zone 1 1 Box in the Flow Domain 2 4 Flow Domain 4 Create a new topo object enter 4 3 Modify the name of a topo object enter 3 2 Delete a topo object enter 2 1 Upper level menu enter 1 or CR 1 Add a new generation zone enter 1 2 Modify the parameters of an existing generation zone enter 2 3 Remove an existing generation zone enter 3 Enter your choice What s index The index is the identification number of the zones the second zone is the Flow Domain for example What s intensity In the above example we have defined two zones every time we generate 4 points in the second zone Flow Domain we generate only 1 point in the first zone Box in the Flow Domain If we select the option Add a new generation zone the following screen appears kkkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkkxkxkxxk 5 Generation Zone i KKKKKKKKKKKKKKKKKKKKKK KKK Current choice NAME Flow Domain TOPO OBJECT All the flow domain INTENSITY 1 i Upper level menu enter 1 or CR 1 Enter the name of this generation zone enter 1 gt 2 This zone is all the flow domain enter 2 3 This zone is a box included in the flow domain enter 3 4 This zone is all the inflows of the flow domain enter 4 5 This zone is a topo object enter 5 6 This zone is one inflow of the flow
39. LOW generates files containing these results Polydata data file result file s velocities shear rate result files trajectories properties kinematic parameters e The statistical post processing finally we use the post processor Polystat to analyze all these trajectories we will calculate the time evolution of global mixing parameters such as the segregation scale or the evolution of the mean stretching and so on Polystat has been built in such a way that we can define new parameters and test them New d Ty w In such a way Polystat can also be used to analyze other processes than mixing parameters are created by combining existing parameters in various ways for example the quality of a molten glass exiting a furnace Eventually we calculate statistical functions of these parameters these functions can be visualized inside Polystat Excel It is also possible to visualize with Polystat the spatial distribution of a kinematic parameter at a given time or in a cutting plane or along a given trajectory November 2009 Polystat User s Guide data files trajectories kinematic parameters Polystat Polycurve result files statistical curves Polystat User s Guide 1 5 THE NUMERICAL TECHNIQUES INVOLVED IN THE MIXING MODULE November 2009 Generally the trajectories are calculated by the time integration of the equation Xx v with an Eul
40. NCTION Let us now define the distribution function F associated with the scalar field The quantity F g B t is defined as follows F B t Pla t lt B 20 where the right hand side is the probability that the field be smaller than at time t 16 Polystat User s Guide A new graph of the distribution function is calculated at every time t see figure below Probability 2 1 3 3 DENSITY OF PROBABILITY FUNCTION Based on the distribution function F of a scalar field a we can define the density of probability function f B t as follows _ OF O00 21 The function f t is the frequency with which we find a value of in the range B Aa B Aa at time t A new graph of the density of probability function is calculated at every time t see figure below A Density of Probability tl November 2009 17 November 2009 Polystat User s Guide 2 1 3 4 PERCENTILES An easier representation of the mixing progress is based on the time dependence of percentiles For the field a let us define 0 t such that F t p 2 a t indicates that at time t p of the material points have a value of o lower than a t as you see on the figure below 4 a ap O time t With the percentiles we can study the evolution of the mixing for specific fractions of the population of material points it s interesting for example to know the value of the length stret
41. Option cccccecseeseseseseeceseresesesesessscessessseessnenseeseeeeeeney 67 ALT 3 the RUN OPbOn eesin i a lo binidces eres svaserees eaves 68 4 1 4 the DRAW results Option ces ctesesesetesescreseneseteeeenenseeneeeeeenenes 69 4 1 5 the DRAW stat Option cccccceceesesesesesceseesesesetessscesstesstenesstenseeneeeneeenenty 72 4 1 6 the WRITE trajectories Option cccceccsescseeseeseescteseteesetetenessnenseeeeeeney 73 4 1 7 the WRITE slices option ccccccccceesesesesescsesctesesesesesescesesesssensseessseeeeeeeeney 74 4 1 8 the WRITE stat Options si ssitstt eres carsteA nEaN E E E EE EEE EEEa 76 419 ANE Saye a OPUON 22 scses svessvssacasasess o anisi iaa EEEE Eia Ea EER ES EE aN 77 4 2 the PROPERTIES MENU seriei iesi oaeiae iaa aaaea E a a a a E aat 78 PBI Tiat O EAEE E ET das fasten adedhicdvaet sccem teed oens tected 78 4X2 See PROPERTIES scsi isin neran aeea iniiis stsncasiea iioesastisiordvasshesb E AEAEE EEEa AnS 79 423 Creat PROPER TIES israiska asnih iiuknin raaith 80 AZSA a ET E E SA ANNE 81 ADB ANR a E E E EE E E E E a E tea 82 AD EE a A E E A A A EA E teres 82 AZSA lO A iias eiee restini anla iaa ties 82 4230 AFB A B priere raer ter Eoaea ES AEEA SLT EUERE testes bataeeivtiadesee 82 EDA A o E A 82 AD BST PE EE E E ES 83 42S ROAU ON See sectors TETEE N T NEE 84 429 Translate eiaa n a e han E e E i iaindttess 86 4 2310 MEETA cc ccccccessenscese sess scesestiasuesestepssesansseuesestaresesesti
42. R 0 Creation of a new moving part enter 0 Enter your choice in a given moving part First we have to create it option 0 The following menu appears KKEKKKKKKKKKKKKKKKKKKKK x moving part 1 KKEKKKKKKKKKKKKKKKKKKKK Enter your choice 1 Upper level menu enter 1 or CR 1 Definition of the moving part domain enter 1 2 Modification of title enter 2 For this moving part we specify its domain option 1 and its name option 2 November 2009 With this menu we specify the moving parts if any overlapping the flow domain Those data are used to remove particles generated randomly in the flow domain and also included Caution you must specify the moving parts in the SAME order as in the previous flow task 61 CHAPTER 4 Polystat User s Guide THE POLYSTAT USER S MANUAL After the calculation of a large set of trajectories performed with POLYFLOW we have to use POLYSTAT in order to treat and or visualize those results to obtain a global overview of the mixing process in the current flow To perform such statistical treatment several steps must be performed in a specific order the main menu of POLYSTAT appears as follows and shows these steps Polystat Ed FILE PROPERTIES TRAJECTORIES SLICES STATISTICS HELP The major steps are the following 1 in the menu FILE we read the results generated by POLYFLOW the data necessa
43. S X Pcg Frac Sb where Fracjs sp t f s t ds Sa Vs is the volume of a sample able to contain a large number of agglomerates of various sizes X is the mass fraction of agglomerates included in the sample Vs and p is the density mass per unit volume of pure agglomerates November 2009 101 Polystat User s Guide 4 3 THE TRAJECTORIES MENU The See Mod Del sets of trajectories option allows the user to manage the list of existing sets of trajectories With the CREATE option the user will define a new set as a subset of an existing one based on a selection criterion With the COMBINE option the user will define a new set by selecting trajectories with boolean operations between two existing sets Eventually with the SELECT option the user can select a single trajectory from an existing set based on a selection criterion 4 3 1 SEE SET OF TRAJECTORIES After the creation of new properties you have the ability to select the set of trajectories on which you will do some statistical treatment Before any creation of such sets if you select the option SEE MOD DEL sets of trajectories in the TRAJECTORIES menu of the main window you will see the list of the existing sets of trajectories See Modify Delete existing sets of trajectories a ox e e Iei In this list you can see three different types of sets of trajectories November 2009 102 November 2009 Polystat User s Guide
44. X axis you have the property associated to the first slice and on the Y axis the property associated to the second slice a Field_2 slice_2 gt Field_1 slice_1 118 November 2009 Polystat User s Guide 4 5 2 4 THE PROBABILITY FUNCTION Probability Function To calculate the evolution of the probability function also named distribution function of a property you need to specify which set of slices will be used and to select a property Don t forget to enter the number of values to represent this function You have also to give a name to the new function If you want to weight the function depending on the local velocity don t forget to select a velocity field see 4 6 3 for more information You will obtain a probability function for each slice of the set If you visualize this function for a given slice you find on the X axis the property and on the Y axis the probability probability for slice S property P 119 Polystat User s Guide 4 5 2 5 THE AUTO CORRELATION ON CONCENTRATION FUNCTION Auto Correlation Function of the Concentration To calculate the evolution of the auto correlation function of the concentration you need to specify which set of slices will be used You must select a coordinate property that will serve to calculate the distance between pairs of points in a slice and you must also select a concentration field Don t forget to specify the numbe
45. ability Function or Distribution Function 16 2 1 3 3 Density of Probability Function ccccccceeseesesescesceseeseecseeeseeees 17 21 BA Percentiles cdaseicsssescen devas situa itorssateta cave sist beslagereterd gtssetoectetstaveneteiows 18 ZLI AFUSTA S ctatsastiat iiss siaiidinioa Manik S 18 2 1 3 6 Correlations isis cade E ona ten rane N ES 19 2 2 HOMOQEMIZAtION cececcccsesesseseseenseeesesesscsescsessesescsessesescsenscsescsensesescsensesescsenscsescnenecseecnenees 19 221 Definition iyii iiaa h Seah a AA ase aes Seats a R 19 2 2 2 Numerical method iire r Eri a A ERE EEA 21 2 3 Disttib tive MINN aae a ra isis ev deacastecedt od ofhcdvtesvevdvibas dec cestvbec uited ctvereet strat ESA ROELS t TAONE E aa 23 AAB E nalaat Tiret e tie EE T E E T E E 23 2 3 2 Distribution i ZONES iiia ka araen anaiari ae s iE airearen ieaiaia 25 2 3 3 Deviation of points concentration sse sssseessisseriessistesiestestenieseesieseestestenens 27 ZA Disa glomiertatiON sisisi nei a EE EE E E O E E a Ea 28 2 5 COMMEN A E E E A A T slain aoa ohne as 33 2 6 Bibliograp hyre naseer i eei e eee harvested bade de abe 34 November 2009 145 Polystat User s Guide CHAPTER 3 MIXING TASK IN POLYDATA cee ceeeceeeeseeseeeeeeees 35 3 1 The Createa new Task Mentiin iania sedeata slates a Eta 36 3 2 General Menu of a Mixing Task cccccsssesseessesssesessseessessssssseseesseecseessssessseeseeesssesesesaeess 37 3 3 Definition of the f
46. ains to draw In this list you can see the set of domains to visualize If you want to change the color of one domain click on its name in the list of domains to draw selection its name now appears in the box Selected domain as well as its current color in the box Color This color can be changed by clicking on the button L Sa If you don t want to see a domain click on its name in the list of the domains to draw selection then click on the button Del If you click on the button Del All all the domains will disappear 2 inthe page Prop you have to specify the property you want to see on the slices or along the trajectories Next in the page Prop select a coordinate field used to position the pixels or the segments in the domain and a property to draw Once a property is chosen you can see the range of values for this property in boxes named min value and max value If it is a vector field the range is based on the norm of the field Then you have to define a list of values that serve to color the pixels or the segments If you want to change the set of values you can delete some of them select one value in the list then click on the button Del or delete all the values click on the button Del All If no values a pair at least are defined no pixels can be seen 3 in the page Slice you have to specify the set of slices or trajectories you want to visualize You can also specify a
47. and B the data and give a name to the new property the result The two properties must have the same type scalar or vectorial 4 2 3 6 A B A B the new property is the division of the property A by the property B you have to select the two properties A and B the data and give a name to the new property the result The property A can be a scalar or a vector but the property B must be a scalar 82 November 2009 Polystat User s Guide 4 2 3 7 A B A B the new property is the multiplication of two properties A and B you have to select the kind of multiplication and the two properties A and B the data and give a name to the new property the result Three cases are possible 1 Scalar product dot product both properties A and B are vectors The result property c is a scalar c a1b a2b2 a3b3 2 Vectorial product cross product both properties A and B are vectors The result property c is a vector cy a2b3 a3b2 c2 a3by a1b3 c3 a b2 a2b1 3 Other product you can multiply one scalar by another scalar the result is a scalar or one vector by a scalar the result is a vector 83 November 2009 Polystat User s Guide 4 2 3 8 ROTATION Rotate this method allows you to rotate a vectorial property you have to select the property A to rotate and the time the data some data specifying the rotation rotation axis angular velocity and to give a
48. at you have done above is called sub slicing you defined a list of slices on another list of slices To define a sub slicing you select first an existing set of slices Next you define your slicing data as for an automatic slicing What differs is the number of sub slices for each slice and if the sub slicing is periodic or not Let s define N the number of sub slices by slice After sub slicing the sub slice indices will be for the slice 1 the sub slices are numbered from 1 to N for the slice 2 the sub slices are numbered from N 1 to 2N for the slice 3 the sub slices are numbered from 2N 1 to 3N etc 111 November 2009 Polystat User s Guide If we select the Periodic option we gather the instants in the following way for a given slice the instants included in the series of intervals named i are gathered in the sub slice i number of sub slices slice 3 time interval of sub slice 2 time If we do not select the Periodic option we gather the instants in the following way for a given slice the instants included in the interval named i are gathered in the sub slice i number of sub slices slice 3 a time interval of sub slice 2 time 112 Polystat User s Guide 4 5 THE STATISTICS MENU November 2009 After the creation of properties set of trajectories and sets of slices you have to define the statistical functions you want to calculate on these objects By
49. but which depends strongly upon the initial distribution of concentration Remark Dankwertz defined another parameter the intensity of segregation 2 I t oro 26 Because of our assumptions the concentration attached to any material point remains constant with time this parameter does not change with time and will not be calculated 2 2 2 NUMERICAL METHOD We distribute uniformly but randomly a set of material points in all the flow domain at time t 0 To each point is associated a concentration depending on its situation in a zone of fluid A c 1 or a zone of fluid B c 0 When time progresses we calculate the new location of each point it is easy to visualize the current state of the concentration field for each point we plot in the domain a small pixel if 2D flow with the appropriate color Once the successive coordinates of material points are stored a minor effort is needed to calculate at time t the concentration corresponding to another set of initial conditions The limitation of the method lies of course in the size of the material point identified by the pixel but the number of points can be increased at will together with the computation time Let us now assume that we have tracked N material points It is easy to calculate the average and the standard deviation of the concentration N cay lei P 27 i x y Oe Wei 28 i l where C is the concentration 0 or 1 of the material point
50. can be crossed 5 times Xexit i A i j Xentry j B i where A i j 0000000E 00 O0000000E 00 O0000000E 00 0000000E 00 O0000000E 00 0000000E 00 0000000E 00 O0000000E 00 0000000E 00 and B i 0000000E 00 0000000E 00 Q0000000E 00 1 Upper level menu enter 1 or CR 0 Non penetrable boundary enter 0 1 Inflow enter 1 2 Outflow enter 2 gt 3 Entry boundary in a spatially periodic flow enter 3 Exit boundary in a spatially periodic flow gt 5 Parameters to connect spatially periodic boundaries enter B5 Enter your choice You have now to specify how the coordinates of the points in the entry section must be transformed to relate them with the corresponding ones in the exit section The general relation between points in the two sections is X A X art B Select now the option 5 to specify the coefficients of the rotation matrix and the translation vector B Current value of the coefficient A 1 1 is 0000000E 00 Enter its new value CR no modification November 2009 42 November 2009 Polystat User s Guide Note When you choose the exit boundary connected to the entry boundary of a spatially periodic flow the boundary type of the exit boundary is automatically updated It is not necessary to define it again B Specification of a stopping plane Why adding stopping planes With this method we
51. can save CPU time if we are interested only in the mixing in a fraction of the domain why do we have to spend time in calculations that are unnecessary By defining stopping planes we have a better control on what we need There exists another situation where it is interesting to define such planes If the flow is 2D 1 2 planar the mesh is 2D but the velocity field has 3 components Vz is perpendicular to the mesh For such flows there are two systems of coordinates the relative coordinates for the trajectory in the plane of the mesh and the absolute coordinates in the real flow domain 3D we continue to calculate the trajectory until the material point reaches in the absolute system the stopping plane stopping plane absolute trajectory relative trajectory Ze plane Z 0 containing the mesh and the flow domain 43 Polystat User s Guide When we select the last option of the menu Boundary Condition this new menu appears KKK KK KKK KKK KKK KKK KKK KKK KKK KKK KKEKKKKKKKKKKK x Parameters for the stopping planes A kkxkkxkxkxkxkxkxkxkxkxkxkxkxkxkxkkkxkxkxkxkxkxkxkxkkxkxkxkxkxkxkxkxkxkxkkxk xx Enter your Current setup no stopping planes WARNING a trajectory of a point is stopped when its position x y z is such that A x B y C z D is negative 1 Upper level menu enter 1 or CR 1 Add a new stopping plane enter 1 Modify the parameters of an existing stop
52. ch reached by the 5 or 10 of the points with the lowest stretching these percentiles can easily show local defects in the stretching 2 1 3 5 HISTOGRAMS Another way to represent the frequency of values of a field amp is to define histograms the user specifies a set of intervals of values of a and he obtains the percentage of the points population that have a value of amp in each interval at time t see figure below al a2 a3 a4 Oy On We see that p of the points population has a value of between a1 and 2 at time t 18 Polystat User s Guide 2 1 3 6 CORRELATIONS Finally once the number of material points is sufficiently large it is possible to examine the correlation between fields either at the same or at different times We have to define two times t1 and t2 and two fields a and p For every material point we plot a t1 in abscissa and B t2 in ordinate an analysis of the graph reveals a possible correlation between the fields B time t2 a time t1 2 2 HOMOGENIZATION November 2009 2 2 1 DEFINITION Suppose we want to mix two fluids A and B both fluids occupy at time t 0 two separated zones of the flow domain see figure below Qt Fluid A Fluid A Let c X t denote the concentration of fluid A throughout the mixing process Since no diffusion occurs between fluids A and B we conclude that c equals either 0 or 1 and remains 19 November 2009 Polysta
53. contain the trajectories that belong to the two sets A and B If we select the Union operator the new set C will contain all the trajectories of the two sets A and B there is no duplication of the trajectories that belong to the two sets If we select the Minus operator the new set C will contain the trajectories of the set A that do not belong to the set B If we select the Difference operator the new set C will contain the trajectories of the set A that do not belong to the set B and the trajectories of the set B that do not belong to the set A We can write this operation as C A Minus B Union B Minus A November 2009 105 November 2009 Polystat User s Guide 4 3 4 THE SELECT ONE SINGLE TRAJECTORY OPTION By selecting this option the following window appears SELECT a single trajectory With this method we will select a single trajectory from a set that is the closest to a specified position We have to specify the following data a aset of trajectories in which we will search one trajectory b a property c a position d an option for the selection if we choose Initially we select the trajectory whom its initial position is the closest to the specified position If we choose Independently we don t specify any peculiar moment The selected property is not necessarily a coordinate property We can for example select the trajectory that has an initial velocity the clo
54. ction P we sort the set of pairs o Oy with 0 lt 0j41 forj 1 to nbi where nbi is the number of instants in slice i Finally the probability to find a value below Q j is nbi P Q SO Zox 2K The value of the weight is a if no weighting oj 1 b if weighting everywhere 0 j e j c if weighting when vj e j positive Oj j e ij if Qj e ij 0 0 otherwise where Qj e ij Jis the dot product of the velocity at instant j by the normal of the slice i 142 Polystat User s Guide ADDENDUM A THE SIMULATION OF THE DISTRIBUTION 1 2 3 4 5 6 7 8 November 2009 Let s suppose the flow to be 3D steady state in a closed domain To perform the analysis of the distribution the following steps are necessary We calculate the flow velocity and pressure We calculate in the mixing task 1 the real distribution we suppose the material points to be initially concentrated in a small volume We have to calculate their trajectory for a given time interval We store these trajectories in files named real_0001 to real_000X for example We calculate in the mixing task 2 the optimal distribution we suppose the material points to be initially distributed in all the flow domain We have to calculate their trajectory for a very short time interval infinitesimal amount of time We store these trajectories in files named opti_0001 to opti_000X In
55. ctly on the instants of slices The second one are functions based on other functions To create a new function click on the corresponding button Let s have a look now to every function in detail November 2009 114 Polystat User s Guide 4 5 2 0 THE SEE PROPERTY ALONG A TRAJECTORY FUNCTION f See property along trajectory With this function the user will see the time evolution of a given property calculated along any trajectory he wants The user has to specify which set of trajectories will be used to select the property to see and the time One has also to give a name to the new function If you visualize this function you find on the X axis the time and on the Y axis the property of interest In this graph you have a new function y f x for each trajectory P Trajectory T1 Trajectory T2 time November 2009 115 Polystat User s Guide 4 5 2 1 THE SUM FUNCTION To calculate the sum function of a property you need to specify which set of slices will be used and to select a property You have also to give a name to the new function If you want to weight the sum in function of the local velocity don t forget to select the velocity field Additional information on weighting is available at the end of Chapter 4 see 4 6 3 You will obtain the evolution of the sum of a property along the slices If you visualize this function you find on the X axis the index of the slice and on the Y a
56. d a the vectorial property A b the component x y or z to extract 1 x 2 y 3 z c the name of the resulting property C Extract A 92 Polystat User s Guide 4 2 3 17 STEP This method allows you to create a new property C defined as follows in H mode C ti Amplitude if property A at time ti 2 threshold value C ti 0 if property A at time ti lt threshold value in 1 H mode C ti 0 if property A at time ti 2 threshold value C ti Amplitude if property A at time ti lt threshold value The following parameters must be defined a the property A b the H or 1 H mode c an amplitude d a threshold value e the name of the resulting property C STEP A November 2009 93 Polystat User s Guide 4 2 3 18 INSTANTANEOUS EFFICIENCY OF MIXING This method allows you to create the instantaneous efficiency of mixing based on the rate of dissipation and the rate of stretching As seen in Chapter 2 this efficiency is defined as uA i in 2D flows e A rate_of _ stretching D rate_of_dissipation yA _ rate_of_stretching D _rate_of_dissipation in 3D flows ey This property is accessible only if the rate of stretching and the rate of dissipation have been calculated along the trajectories In the creation window the default values for the properties are correct This window appears like this Instantaneous Efficiency of Mixing November 2009 94 N
57. default no function exists General ones can be created by selecting CREATE a new function while functions specific to the disagglomeration model can be found in NEW disagglomeration functions 4 5 1 SEE STATISTICAL FUNCTIONS If you select the option SEE MOD DEL functions in the STATISTICS menu of the main window you will see the list of the existing functions See Modify Delete existing functions mi e ma pose If you want to modify some data of a function select it in the list and then click on the MODIFY button The window that served for the creation of that function will appear then modify some data If you want to store the modified data click on OK Otherwise click on CANCEL To remove one function from the list select it in the list and then click on the DELETE button To remove all the created functions click directly on the DELETE ALL button In both cases POLYSTAT asks for a confirmation of your choice 113 Polystat User s Guide 4 5 2 CREATE STATISTICAL FUNCTIONS If you select the option CREATE a new function in the STATISTICS menu of the main window you will see a window showing the list of functions that can be calculated Create a new function If you want to go back to the main window click on the CANCEL button As you can see in the CREATE a new function window there exist two kinds of functions the first one are functions based dire
58. domain enter 6 7 This zone is a CSV slice included in the flow domain enter 7 8 Enter the intensity of the generation of points enter 8 Enter your choice November 2009 50 November 2009 Polystat User s Guide Option 1 you can rename the new zone optional and local to POLYDATA There exist different kinds of generation zones Option 2 the entire flow domain We generate randomly the initial position of material points in the flow domain this option is well suited for flows in a closed domain Option 3 we define a box by its two corners xmin ymin zmin and xmax ymax zmax This box must have a non empty intersection with the flow domain Two modes of generation are available With the first one we generate randomly the initial position of material points in the box With the second one called equidistant distribution the material points are initially distributed at equal distance d specified by the user between neighboring points If the flow domain is 2D the points are distributed at vertex positions of a lattice of identical equilateral triangles see figure below If the flow domain is 3D the points are distributed at vertex positions of a lattice of equilateral tetrahedra The generated points that are outside the flow domain will be rejected Equidistant distribution of points in a 2D box Option 4 in a previous menu we have defined the boundaries that are the inflows of the f
59. e gt gt button Then every function is sent to the internal data structure of Polycurve Let s remark that the first time you click on the gt gt button the Polycurve main window will appear But no graphic is created To do so click on the G button in this window A new graphic appears empty in the graphic window To add a curve to it click on the C button A list appears it contains the set of the curves sent to Polycurve Select the curves you want to add to the current graphic 12 Polystat User s Guide 4 1 6 THE WRITE TRAJECTORIES OPTION Save Set of Trajectories In this window you will select the set of trajectories you want to save in formatted files First you select one set in the upper list Then you click on the gt gt button The next window appears Export You have now the ability to choose the coordinates and time to be used useful in some case where absolute and relative coordinates exist or if you defined your own coordinates or time November 2009 73 November 2009 Polystat User s Guide You can also select the format of writing a The default format is CSV format One saves the selected set of trajectories in a set of CSV files The CSV file format see Polyflow User s Manual is a common format for tabulated data that can be read into spreadsheet programs such as Excel POLYSTAT will generate one file per trajectory their names are built like this
60. e Y axis the value of the deviation Deviation index of the slices To simulate and analyze the distribution of matter a series of steps are necessary to perform Those are explained in detail in addendum A November 2009 133 November 2009 Polystat User s Guide 4 5 2 15 THE POINTS CONCENTRATION DEVIATION FUNCTION Points Concentration Deviation x Select a set of slices new_set_of slices 4 Select a coordinate field coordinates 4j Perfect points concentration Sample radius Minimum number of neighboring points The points in a given slice are distributed ina BurFace 4j Ok CANCEL HELP This function will allow you to calculate the deviation of points concentration in a slice compared to a perfect points concentration This deviation will be evaluated for each slice of the selected set of slices The user must select a set of slices and a coordinate field Moreover the user must introduce the points concentration for a perfect distribution in each slice This perfect distribution may be evaluated as follows a For a 3D open flow domain we perform a slicing in space in the direction of the flow thus each slice will be a surface We assume that each slice as the same perfect points concentration that is equal to the number of points in a slice S divided by the area of the part of the slice S that cuts the flow domain b For a 3D closed flow domain we perform a slicing
61. e rectangle 1 the rectangles 2 and 3 are set to zero The normal direction must be set to 1 0 0 November 2009 140 November 2009 Polystat User s Guide 4 6 2 THE ZONES A zone is an interval of values for a specified property For example for a vector property as the position we have to define the two extreme corners of a box xmax ymax zmax xmin ymin zmin If the property is a scalar we define the minimum and the maximum values of the interval To define a zone the following window appears Define a new zone First you select a property on which we define the zone Second you specify the interval enter the minimum value in the rectangles 1 to 3 and the maximum value in the rectangles 4 to 6 Finally enter the name of the zone If the selected property is a scalar the minimum value is specified in the rectangle 1 and the maximum value is specified in the rectangle 4 the rectangles 2 3 5 and 6 are set to zero 141 November 2009 Polystat User s Guide 4 6 3 WEIGHTING The slicing must been defined on a coordinate field in order to weight a property by the velocity The general formula for the sum the mean and the standard deviation are a slice i X Qj Oj j a slice i aj Der j j 6 slice i where the index j indicates the j th instant of slice i Qj is the value of at instant j and j is the weight at instant j To calculate the probability fun
62. e will use these velocity fields as data for solving a mixing problem When we begin a Polydata session the initial screen looks like this KKEKKKKKKKKKEKKKKKKKKKKKK e POLYDATA 7 KKEKKKKKKKKKKKKKKKKKKKKK Version 3 12 0 Save and exit 1 gt 1 Read a mesh file enter 1 2 Mesh decomposition and optimization enter 2 Combine mesh files 4 Convert a mesh file enter 4 Convert old results files Convert old csv files 6 Filename syntax enter 6 Outputs 8 Read an old data file enter 8 2 gt 9 Create a new task enter 9 Redefine global parameters of a task Enter your choice First enter as usual the mesh file option 1 Next you will create a new task option 9 A new menu will appear November 2009 35 Polystat User s Guide 3 1 THE CREATE A NEW TASK MENU KKEKKKKKKKKKKKKKKKKKKKKKKK Create a new task x KKEKKKKKKKKKKKKKKKKKKKKKKK Current setup MIXING task Steady stat 2D planar geometry 1 Accept the current setup enter 1 or CR 1 F E M task enter 1 gt gt 2 MIXING task enter 2 gt 3 Steady state problem s enter 3 4 Time dependent problem s enter 4 Evolution problem s Optimisation problem s Rigid rotation gt 7 2D planar geometry enter 7 8 2D axisymmetric geometry enter 8 9 2D 1 2 planar geometry e
63. ear rafe 10 fist ao 50 10 0 15 0 20 0 25 0 size microns Effect of erosion on the mass distribution function as a function of time the matrix viscosity is 11000 Pa s and the shear rate is 10 s Rupture For the model of rupture also based on 9 we assume that the rupture into a few fragments occurs if an agglomerate is submitted to a shear stress higher than a critical shear stress during a given amount of time called rupture time This critical shear stress for rupture function of size S can be written rupture B ee S Omin s 37 Indeed we are more interested by the inverse relation we have to know which agglomerates can break for a given shear stress o The equation 37 becomes p Ss 38 O Omin All agglomerates with a size greater than S can break for a shear stress o In our model we assume that the rupture time is a constant the default value is set to 0 1 seconds in the disagglomeration clp file and that only a fraction of all particles of a given size will break if rupture criteria are met Indeed we can understand easily that all particles of a given size do not have same cohesion or same impregnation level infiltration of the matrix inside the agglomerate due to diffusion That is why we can specify the rupture rate the fraction of particles that break when rupture criteria are met however the default value is set to 1 in the disagglomeration clp file meaning t
64. eased by the slice step and the current impeller position is increased by the impeller step The motion of the impellers is now linked to the successive slices of the current set The user has the responsibility to define the correct set of slices corresponding to the motion of the impellers he has to define an automatic slicing on time with an increment corresponding to the time step existing between two successive positions of the 71 November 2009 Polystat User s Guide impellers He has also to choose correctly the first slice the first impeller position step slice and impeller step The default values are not correct Finally in the page Slice you can select the pixel or segment size logarithmic scale Moreover if you answer yes to the question Draw the color set the color scale is displayed in the upper left corner of the Graphic Display window 4 1 5 THE DRAW STAT OPTION Draw Statistical Results In this window you will select the functions you want to visualize with Polycurve Polycurve is a graphic editor linked to POLYSTAT its main goal is to allow the user to create visualize and save graphics in Postscript files see the Polymat Manual for a complete description of Polycurve A graphic can contain several curves We can add and remove a curve a text a line change a title a color the range of the axes First you select the functions in the upper list Then you click on th
65. ecessary A Specification of the boundary type KEKE KK KK KKK KKK KKK KKK KKK KKK KKK KKK KKKKKKKKKKKK X Boundary condition along boundary 1 x kkkxkxkxkxkxkxkxkxkxkxkxkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkx k Current choice Non penetrable boundary 1 Upper level menu enter 1 or CR gt 0 Non penetrable boundary enter 0 1 Inflow enter 1 2 Outflow enter 2 3 Entry boundary in a spatially periodic flow enter 3 4 Exit boundary in a spatially periodic flow enter 4 5 Parameters to connect spatially periodic boundaries enter 5 Enter your choice November 2009 By default the boundary is supposed to be non penetrable This type covers the following cases walls axes of symmetry free surfaces It means that a material point can not cross the 39 November 2009 Polystat User s Guide boundary If it does however for numerical reasons a specific flag attached to each material point is set and the calculation of the trajectory is interrupted the stopping is ABNORMAL But there are other kinds of boundaries Option 1 the inflow it s a part of the boundary where the fluid enters in the flow domain Normally a material point can not cross an inflow boundary a material point can not go against the flow it is not a salmon Option 2 the outflow it s a part of the boundary where the fluid exits the flow domain When a material point reaches an outflow boundary we stop the calculation I
66. er 6 in order to measure this process let us suppose that initially we place a set of particles in a small zone in the flow domain as a function of time these particles move in the flow and distribute Our parameter 5 measures the deviation of the current distribution with respect to a perfect distribution of particles in the flow domain Another technique similar to the previous one is now available we divide the flow domain in a set of adjacent and non overlapping zones Initially we place a set of particles in a small box in the flow domain and they distribute progressively Then for a given time we count the number of points in each zone We get a good distributive mixing if each zone contains a number of points proportional to its surface volume A third option to estimate distributive mixing is to evaluate the local points concentration in various locations in the flow domain and to compare it with a perfect points concentration corresponding to the case where we find the same number of points per unit volume everywhere in the mixer Eventually a new parameter p measures the deviation in points concentration The dispersive mixing is another important aspect of the mixing it concerns the break up of drops into small droplets or the disagglomeration of solid particles in a matrix The stress applied by the matrix on drops or on solid particles is the engine that can lead to dispersion if the stresses are high enough to co
67. er explicit scheme it s enough if we are only interested in the successive positions of material points But if we need to know precisely the deformation accumulated along these trajectories a very accurate numerical technique is required We chose to combine two techniques first we use an explicit Runge Kutta scheme of the fourth order second instead of integrating the motion of a particle in the real space we perform a coordinate transformation we calculate the trajectory in the parent element we integrate with the Runge Kutta method E f v 1 To know the successive positions of the particle in the real space we use x Dxivi The algorithm is the following 1 initialization e find an element E containing the initial position X e find the local coordinates of X in this element E 2 while no problem amp no required stop gt we integrate equation 1 until we cross a boundary of the element E e if a boundary of E is crossed we adapt the time step of integration in such a way that the position is on the boundary e if we are on a boundary of E in x we search the element adjacent to E where to continue the integration let s note this element E find the local coordinates in element E of the current position x goto gt We explained this because some important numerical parameters used by this algorithm must be defined by the user in Polydata see Chapte
68. er s Guide 3 6 PARAMETERS FOR THE GENERATION OF THE MATERIAL POINTS The initial screen looks like this Current setup no generation zones 4 Create a new topo object enter 4 3 Modify the name of a topo object enter 3 KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK x Generation of material points KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK f Upper level menu 1 Add a new generation zone Modify the parameters of an existing generation zone Remove an existing generation zone 2 Delete a topo object enter 2 Enter your choice enter 1 November 2009 In this menu we define zones where the material points are positioned initially By clicking on the option 1 we define an additional zone In order to have a well defined mixing problem we must have one generation zone at least As soon as more than one generation zone is defined a new property is added to the list of properties evaluated along the trajectories and selected in 3 9 This new property named label is constant along each trajectory and indicates the zone id of the material point s origin 49 Polystat User s Guide When we have generated several zones such as shown below we can modify the parameters of zones or delete zones by clicking on the options 2 and 3 KKEKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK K Generation of material points KKEKKKKKKKKKKKKKKKKKKKKKKKKKK
69. erial point at time t en 2 is the ratio of what we get the final stretching obtained at time t over what we put the total mechanical dissipation until time t Like for 2D kinematic parameters we can define a global efficiency over all the material points distributed initially in the flow In n dQ eq p 2 a7 D dt dQ 0 15 November 2009 Polystat User s Guide 2 1 3 STATISTICAL ANALYSIS Let us assign to the N material points an initial orientation M which does not need to be identical for all points While tracking the material points as a function of time we also calculate successive values of e and lt e gt A global representation of A e and lt e gt is again obtained by associating a material point with a small rectangle of size dx X dy the color of the rectangle is associated with the value of the field to be represented The quality of the representation increases with the number of points N When the number of material points is sufficiently large we may proceed with a statistical treatment of the calculated quantities Several statistical tools have been implemented in the new software POLYSTAT 2 1 3 1 MEAN AND STANDARD DEVIATION For any scalar kinematic parameter we can calculate the time evolution of its mean and its standard deviation Sy 260 alt 18 a 0 0W N 2 6g t 19 2 1 3 2 CUMULATED PROBABILITY FUNCTION OR DISTRIBUTION FU
70. esdesestensdesestiestes 87 ADD VAS IIOVIN ALC E O E E E aoe 87 November 2009 146 Polystat User s Guide ADS DA ETN E EES E EEE E E ade tentetebesaee 88 APNI ASCE heei ei creeks e E T E E 88 4 2 3 14 Concentration E 89 4 23 19 Min Max aen a i a a a eia 91 ADS O EXTA a an a AN ENEAN ERATES EANTA PARTES 92 Ae DS NZ Step PESE E EE EEEE 93 4 2 3 18 Instantaneous efficiency Of Mixing ssessssssssessiessssrerisseseesesseene 94 4 2 3 19 Time Averaged efficiency Of Mixing s sssssssiesessrerirsrssresseseene 95 4 2 4 Disagglomeration PROPERTIES s ssessssssesssssertessessestessssrestesesstentensesteseeneea 96 4 2 4 1 Disagglomeration ccccccecseesssesesssecssenseensessecssessesssessseseesseesseeseeeeess 97 4 2 4 2 Typical size of aggloMerates ccceccecsessecseesseesseeeesseesseeseeeeaeseess 98 4 2 4 3 Fraction of agglomerates of given SIZE sceceseseseseseetseseseeeeeeeees 99 4 2 4 4 Number of agglomerates of given siZe s ss ssssssssesrssssrsieresseseee 100 4 2 4 5 Mass of agglomerates of given size ss ssssssssessisssssesrsssessieressessee 101 4 3 the TRAJECTORIES Menuisier ii a ien ii ei ie aieri aeii 102 43 1 See set of TRAJECTORIES ieaie en ier ia aere e ia eaa ai 102 4 3 2 the CREATE a new set of trajectories Option s sssssessssssssesrsssessesresseseee 104 4 3 3 the COMBINE sets of trajectories Option s ss sssssesssessssesriesessesrsssessieressessee 105 4 3 4 the SEL
71. et s explain the use of every button 1 The button Save this option allows you to save the current drawing in a postscript file that can be sent for printing 2 The button Close with this option we close the graphic display and the graphic options windows the current graphic options are saved 3 The button Help some information summarize the meaning of every button page and operation Now let s explain the three pages of the Graphic Options window Graphic Options i a i L om ee Go in the three pages in the following order 1 inthe page mesh you have to specify the parts of the mesh you want to see In the upper part of the page Mesh the list of existing domains is shown A domain is a topological object based on the mesh The following objects are built automatically a the domains b an intersection of a domain with another domain for example the intersection between the domains 1 and 2 is written Domain1 Domain2 c an intersection of a domain and a boundary for example the intersection between the domain 1 and the boundary 3 is written Domain1 Boundary3 d the perimeter of the previous objects the perimeter of Domain1 Boundary3 is written Domain1 Boundary3 70 November 2009 Polystat User s Guide If you want to visualize a domain of this list click on its name It will disappear from this list and appear in the list named Dom
72. eter of a rotor or of a screw for example The user can also specify the minimum number of points needed to evaluate the points concentration at a position x the default is set to 3 If the number of neighboring points is smaller than this minimum we increase progressively the radius for that position x only until we have that right number of neighbors Next the user must not forget to specify the dimensionality of the samples If the samples are straight line segments dim 1 surfaces dim 2 or volumes dim 3 This is needed to evaluate the local points concentration x as explained in 2 3 3 The deviation 6 p in a slice S is evaluated as follows p S where N is the number of points in the slice S the x correspond to the location of points i in the slice S and us is the perfect points concentration Eventually the user can give a name to the function 135 Polystat User s Guide 4 5 3 NEW DISAGGLOMERATION FUNCTIONS If you select the option New Disagglomeration Functions in the STATISTICS menu of the main window you will see a window showing the list of functions that can be calculated on properties derived from properties linked to disagglomeration model NEW disagglomeration functions The first three functions are specific to disagglomeration model and are explained in the next pages while the last three functions are identical to general statistics functions For them see explanatio
73. gin to generate a new result file without erasing the result files from an old session Options 6 to 10 we have to select one of these five ways to store trajectories With the option 6 we store every calculated position very expensive on a memory space point of view Option 7 allows the user to store the successive positions of a material point every DT seconds exactly With option 8 two successive stored positions have a minimal time step of DT seconds The time step DT is imposed in option 11 With option 9 we store a position every time the length of the trajectory has increased of a distance DL With option 10 we store a new position if the increase in the trajectory length is greater than DL The displacement DL is imposed in option 11 If you select option 9 or 10 option 11 becomes 11 Modification of the displacement enter 11 and you can modify the exact or minimum displacement to move before we store the actual position of a point The increase in that case is the difference between the trajectory length for the current stored position and the trajectory length for the previous stored position November 2009 60 Polystat User s Guide 3 11 DEFINITION OF MOVING PARTS The initial screen looks like this KKK KKKKKKKKKKKKKKKKKKKKKKKK x Define moving parts KKEKKKKKKKKKKKKKKKKKKKKKKKKK 2 Deletion of a moving part enter 2 1 Upper level menu enter 1 or C
74. hat all agglomerates break 31 November 2009 Polystat User s Guide Eventually we have to model how a set of agglomerates of size S will break in numerous fragments of various sizes We assume here that the volumes of the fragments follow a Gaussian distribution between 0 and the parent agglomerate volume In average the parent agglomerates of size S are cut into two fragments of equal volume leading to a mean size of 0 85 Of course once again this behavior can be modified by changing the corresponding function in the disagglomeration clp file The transfer function for rupture can be seen in the following figure with a rupture rate of 1 Sirel before rupture s afer rupture o 08S s Size Transfer function for rupture with a rupture rate of 1 and a mean size of fragments equal to 0 8 times the parent size S Let us briefly enter in the details of implementation of the rupture model We associate to each discretized agglomerate size S an induction time T initialized to zero As we progress along the trajectory we evaluate the shear stress 6 based on equation 38 we get immediately all the classes where rupture can occur For classes S lt S their induction time T is reset to zero For the other classes their associated induction time is increased by the local time step At For all classes where the induction time is above the rupture time the rupture occurs and the corresponding induc
75. he parameters of this method are e The initial distribution of the zones of fluid A and B dmax the maximum possible distance between two points in the flow domain e nbpair the number of pairs of points necessary to calculate one value of the correlation function nbval the number of values of the correlation function to calculate 22 Polystat User s Guide 2 3 DISTRIBUTIVE MIXING November 2009 2 3 1 DISTRIBUTION INDEX Suppose we want to distribute a cluster of particles initially concentrated in a small box see figure below We suppose that the particles do not affect the flow field and that there is no interaction between them Their number is supposed to be large As a function of time the flow will distribute this set of points It s important to define a distribution index 6 to quantify this process Its definition is based on the work performed by Manas Zloczower and her colleagues see refs 5 6 At time t we have N points distributed more or less in the flow domain Option 1 These points can form N N 1 2 pairs of points For each pair of point x and Xj we calculate their inter distance dij E x Xj The maximum inter distance will be of the order of the diameter of the mixer j and we store their inter distance dij Xj Xj We have thus only N distances With this method of calculation Option 2 For each point X we search its closest neighbor X we are able t
76. here exists a list of probability functions one for each slice of a sorted list you will obtain a density of probability function for each slice S 04 8 PS M 5 If you visualize this function for a given slice you find on the X axis the property values the property has been chosen earlier when the probability function has been defined and on the Y axis the density of probability density of probability for slice S property Q In order to avoid wiggles it is a good practice to smooth the probability function before the calculation of the density of probability And in general it is also necessary to smooth the density of probability function before visualisation 128 November 2009 Polystat User s Guide 4 5 2 11 THE PERCENTILES FUNCTION Percentiles To calculate the evolution of percentiles you need to specify which probability function to use You must define a list of percentiles and also give a name to the new function To add a percentile to the list ranged from 0 to 100 enter a value in the box a and then hit lt CR gt To delete one percentile select it in the list and then click on the Del button As there exists a list of probability functions one for each slice of a sorted list you will obtain the evolution of a percentile along the slices one curve for each percentile If you visualize all the percentiles on the X axis you have the index of the slice and on the Y axis
77. how if the calculations were accurate if det F remains 1 along the trajectories In the other case unfortunately if we constraint F such that det F remains 1 along trajectories we cannot be sure that this will improve the accuracy of the results That is a future analysis of the results in The initial screen looks like this x Selection of properties List of selected properties Time always Coordinates always i Upper level menu en 1 Enable Space integration en 2 Enable Rate of stretching en 3 Enable Rate of dissipation en 4 Enable Direction of stretching en 5 Enable Logarithm of stretching en 6 Enable Pressure en 7 Enable Velocity en 8 Enable Temperatur en 9 Enable Divergence of velocity en 10 Enable Determinant of F en 11 Enable Viscosity en 12 Enable Mixing index en 13 Enable First eigenvalue of T en Enter your choice KKK KK KKK KKK KKK KKKKKKEKKEKKKKKKKKEK KKK KK KKK KKK KKK KE KKK KKKKEKKKKKKKKEK Ler Cer Ler Cer Cer Cer Cer Cer Cer Cer cer Cer Cer Cer bh POODIANAHOARWNE PRR WN With this menu we select in a list the properties we want to store in the result mixing files The time and the coordinates are always saved November 2009 57 November 2009 Polystat User s Guide The first property the space integratio
78. imal distribution and the concentration of points in each area of the flow domain If you want to see those fields enter Polydata read the mesh and select convert old csv files enter the name of csv file and ask to save them in files readable by your graphic post processor The concentration of real points will be evaluated like this for each point X of the optimal distribution we determine the number of points of the real and optimal distributions included in a sphere of radius R and center X The concentration at point X will be nbr c X _ withc X c 0 1 nbr nbo where nbr number of points of the real distribution C sphere X R and nbo number of points of the optimal distribution C sphere X R The radius of the sphere is evaluated as follows First we determine the smallest box that surrounds all the points of the optimal distribution Next we calculate its volume V A typical distance between points is d IV nbi where nbi is the number of instants Eventually the radius of the sphere is set to 3d However the user can impose its own value To do so one has to create a p3rc file my_p3rc and add a line containing RADIUS user_value Eventually one runs POLYSTAT like this polystat s my_p3rc No modification of the p3rc file done during a session of POLYSTAT can influence it 124 Polystat User s Guide 4 5 2 8 ARITHMETIC OPERATION ON SUM FUNCTIONS Sum A lt operator gt Sum B
79. ined above a slice one trajectory A i intersection i of the A i trajectory and the slice value of the property A at transport of A i to all the intersection i the other instants of the trajectory In general we want to visualize the current value of a property for example the temperature attached to a material point at the current position of this point But for the study of mixing it may be useful to see results differently For example if we study the stretching of a set of particles in a Kenics mixer it may be useful to see at the initial position of the particles their final stretching By this way we can detect zones of the inlet from which the stretching is bad or good 4 2 3 14 CONCENTRATION Concentration this method allows you to define any concentration field in an initial configuration As the concentration field is constant for a material point no diffusion no chemical reactions we transport the value of the concentration along the trajectories without making any change you have to define the initial configuration and give a name to the new property the result The initial configuration is composed of two things first you have to specify a slice see 4 6 1 Second you specify a list of zones see 4 6 2 In this slice the instants included in a zone have a concentration value of 1 The instants external to all the zones have a concentration of 0 slice c 1 traject
80. int i c slice idx A p p i 0 For a given slice f size Point 1 Point 2 Point 3 ai agglomerates size size J agglomerates size 138 November 2009 Polystat User s Guide 4 5 3 2 THE PROBABILITY FUNCTION Probability Function The user has to select a density of probability function and to give a name to the new function With this function one evaluates for each slice idx a probability function P as the integral of a density of probability function f on a property a Ot P a idx f a idx da oco 139 Polystat User s Guide 4 6 ADDITIONAL DEFINITIONS 4 6 1 THE SLICES A slice is the set of instants of the trajectories that respect some specified condition A condition is for example the time t must be equal to 30 seconds or the position x must be included in the plane 10x 2y 3z 6 0 etc The general way to define a condition is the following for a property P the value of P must be included in a plane this plane is defined by a position and a normal direction To define a slice the following window appears define a slice First you select a property on which the condition will apply Second you specify the plane to respect To define it enter the position in the rectangles 1 to 3 and the normal direction in the rectangles 4 to 6 Finally enter the name of the slice If the selected property is a scalar the value to respect is specified in th
81. ix slice index hst percentiles prefix percentile index pct distribution in zones global deviation prefix glo zon deviation for zone i prefix zone index zon November 2009 76 Polystat User s Guide zones partitioning deviation for zone i points concentration prefix csv slice index The files are written in the crv file format I the optional header is the first 5 lines line 1 lt empty line gt line 2 title or name of the current curve line 3 lt empty line gt line 4 lt empty line gt line 5 lt empty line gt II the next lines are the set of points defining the curve On each line there is one pair of x y values the format is 14 characters per value with 7 digits after the dot A white space separates the two numbers next lines 0 0000000E 00 0 0000000E 00 0 0000000E 00 0 0000000E 00 4 1 9 THE SAVE OPTION This option allows you to save in a file with a sav suffix your current POLYSTAT session In the session file are saved all information regarding properties those existing in the mixing files but also the created ones set of trajectories set of slices and of course statistical functions Let us note that this session file does not contain information regarding mesh and result files used only for graphical purpose November 2009 TT Polystat User s Guide 4 2 THE PROPERTIES MENU 4 2 1 DEFINITION
82. k for Description of Mechanical Mixing of Fluids AIChE Journal Vol 27 n 4 pp 565 577 3 OTTINO J M 1989 The Kinematics of Mixing Stretching Chaos and Transport Cambridge University Press 4 TADMOR amp GOGOS 1979 Principles of Polymer Processing John Wiley and Sons Chapters 7 and 11 5 WONG T H MANAS ZLOCZOWER I 1994 Two Dimensional Dynamic Study of the Distributive Mixing in an Internal Mixer Intern Polymer Proc Vol IX n 1 pp 3 10 6 YANG H H MANAS ZLOCZOWER I 1994 Analysis of Mixing Performance in a VIC Mixer Intern Polymer Processing Vol IX n 4 pp 291 302 7 B Alsteens V Legat A Model for the Disagglomeration of Carbon Black in Rubber Matrix Proceedings of the 6th Conference Eurheo Erlangen Germany September 2002 8 B Alsteens Mathematical Modelling and Simulation of Dispersive Mixing PhD thesis Universit Catholique de Louvain Louvain le Neuve Belgium 2005 9 V Collin E Peuvrel Disdier Dispersion Mechanisms of Carbon Black in Elastomers Conference on European Rubber Research Practical Improvement of the Mixing Process Paderborn Germany 2005 34 Polystat User s Guide CHAPTER 3 MIXING TASK IN POLYDATA In order to create a mixing problem we suppose that in a previous session you defined and solved the Navier Stokes equations on the flow domain We have thus one or several result files containing the velocity field In this session w
83. lative coordinates exist or if you defined your own coordinates or time The selected property for the COORDINATES will be named COORDINATES and will appear as the first property in the file Useful if you need to read a CSV file as a generation zone see 3 6 for additional information You have now the ability to save slices in two different formats a The default format is CSV format One saves the selected set of slices in a set of CSV files The CSV file format see Polyflow User s Manual is a common format for tabulated data that can be read into spreadsheet programs such as Excel POLYSTAT will generate one file per slice their names are built like this prefix _ slice index csv b The other format is the FV format One saves the selected set of slices in a single FVP file that can be loaded in FieldView This option is useful if your simulation is transient and if you want a better graphic treatment of your trajectories for steady state simulations see 4 1 6 POLYSTAT will generate one file with the name prefix fvp After the writing the set of slices already written appear in the list named Selected set of slices of the window Save Set of Slices When you have finished click on the OK button in this window to go back to the main window 75 4 1 8 THE WRITE STAT OPTION Polystat User s Guide Save Statistical Results In this window you will select the f
84. le flow domain Option 8 You have also the ability to generate with a higher frequency points in one zone in comparison with others to do that you modify the intensity factor of the created zones higher the intensity higher the frequency of point generation in that zone Note 1 If among the generation zones some of them are of the CSV file type it is mandatory to impose the intensity of each zone equal to the number of material points starting from that zone Then in the Storage of results menu 3 10 ensure that NBPTMX NBFILE the number of material points per file times the number of files is equal to the sum of the intensities Note 2 If among the generation zones some of them are of the BOX with equidistant distribution type it is difficult to estimate the exact number of points that will eventually be generated in that zone That is why an approximation of the number of points upper bound that will be generated is provided to the user when defining the box generation zone This number can be used to define appropriately the intensity of generation nbptmx and nbfile as for previous note 1 52 Polystat User s Guide Note on the definition of topological objects The management of topo objects is done in the menu Generation of the material points To create a new topo object select Create a new topo object option in the menu Generation of the material points The foll
85. lected object 53 Polystat User s Guide 3 7 PARAMETERS FOR THE TRACKING In order to calculate the trajectory of a material point we have to specify a set of numerical parameters These parameters have different purposes to decide when to stop the calculation when there is a problem The initial screen looks like this KKEEKK KK KKK KKK KKK KKK KKK KKEKKKKKKKKKKK a Parameters for the tracking a kkkxkxkxkxkxkxkxkxkkxkxkxkxkxkxkxkxkxkkkkxkxkxkxkxkxkkxkkxxk xx Current setup TIMAX the lifetime of the material points VNORMX the maximum velocity in the flow field 1 0000000 1 0000000 04 00 Gl FI 1 Upper level menu enter 1 or CR 1 Modification of TIMAX enter 1 2 Modification of VNORMX enter 2 3 Options enter 3 Enter your choice You have to modify absolutely the two parameters TIMAX and VNORMX 1 TIMAX is the lifetime in seconds of the material points we will calculate their trajectory until that time is reached In order to lower the CPU time of your simulation don t put a huge value for this time if it is not necessary 2 VNORMX is the maximum velocity magnitude in the flow field It will serve to detect stagnation points in the flow a stagnation point is a point that has a negligible velocity in comparison with the maximum velocity By default negligible means one millionth of the maximum velocity By selecting opti
86. low problem With this option we generate randomly the initial position of material points in these entries this option is well suited for flows in an open domain Option 5 with this option we can define a zone by selecting a topo object in a list If some topo object is missing it is possible to add new ones in the list see note on next page Option 6 in a previous menu we have defined the boundaries that are the inflows of the flow problem With this option we select one inflow section among the boundaries tagged as inflow in the boundary conditions menu We generate randomly the initial position of material points in the selected entry this option is well suited for flows in an open domain Option 7 with this option the user can specify its own initial distribution of points saved in a CSV file Comma Separated Variables format After selecting this option the user introduces the name of the CSV file Then a question appears to the screen All fields are reinitialized to default values Do you agree with that 51 November 2009 Polystat User s Guide if you enter yes that means that we just read the coordinates of the points in the CSV file the other properties will be reinitialized to their default values time 0 logarithm_of_stretching 0 space_integration 0 if you enter no that means that POLYFLOW will read not only the coordinates of the points in the CSV file but also so
87. lowdomaif neissa a ie a a a 38 3 4 Definition of the boundary conditions cecesesesesssee sees sseeeseseesseecsesesnssseseesseesssesssesatens 39 3 9 Definition of the flow field ccc icccssecsescesbasssbsestuaneasshasusotaesveestaradarenpessausentarsherabsesdacseaterss 45 3 5 1 Steady State FLOW sisissicsessvehvssreesivexacgshcnnsssnsnas babes caeheabasdpanseandasascaedbos sphaincpuavety 45 3 5 2 Time Dependent FloW ierra EEE E EE NE 46 3 6 Parameters for the Generation of the material points ss sesssssssssiessssesesssesiessssteseseese 49 3 7 Parameters for the Tracking c ccccecceseseseseseeseseseseseseecssscseessesesesescsesssensseseseneseeeseeeneeesenty 54 3 8 Parameters for the Kinematic Mixing Properties cccccseecsescseesesetetesetessteeteeeeenes 56 3 9 Selection Of PrOperties io sx dices fh R Aastha etic ees Mitral AE aE seeded EIE RES 57 3 10 Parameters for the Storage of the Results cccccecsescseeseseseescnessteneseeneessneeeeeenenes 59 3 11 Definition Of MOVING Parts sei sieciectarstencrarstecohsescsvatendasheahead isos ida aisanana pi ai teats 61 CHAPTER 4 THE POLYSTAT USER S MANUAL 62 ANT the FILE Mens iscsi isexsoatsixcshentbaioedtansaacetasitsisiaidasses tabi aaa aariaa aa Bavestorsagp Leeasaesiabanareelanay 64 4 1 0 the Opens Optioniives Jin ta E dail ania AE ender annie 65 4 11 the READ data Options cii0au cies cninanimnaaitiinimiadvebaaniiiaaenesd 65 4 1 2 the READ mesh
88. luated Keywords mesh superposition technique batch mixer transient flow problem forces and torque dispersive mixing mixing index eigen values of the stress tensor shear rate vorticity Polystat statistical analysis 1 6 6 EXAMPLE 91 DISPERSION In this sixth example we present the models of erosion and rupture in a simple shear flow By this way we analyze the effect of various functions and parameters of these models Keywords dispersion disagglomeration erosion rupture Polystat 10 CHAPTER 2 Polystat User s Guide THE MIXING THEORY 2 0 INTRODUCTION November 2009 In polymer blending a minor component is generally present as drops or filaments in a continuous phase of a major component Mixing is a process of deformation and rupture of the drops but also a process of distribution of those drops in the whole flow domain A good mixing is characterized by small and identical drops distributed uniformly in the all flow domain Deformation of drops is promoted by the viscous stress T exerted on the drops by the flow field and counteracted by the interfacial stress O R where is the interfacial tension and R the local radius The capillary number Ca is useful to characterize mixing Ca R 0 oj For a given pair of polymers a critical Capillary number may be found It corresponds to the situation where the viscous stress competes with the interfacial stress the drop is extended and finall
89. m initial direction of stretching enter 1 2 Imposed initial direction of stretching enter 2 gt 3 No constraint over the tensor F enter 3 4 Constraint over the tensor F detF 1 enter 4 choice November 2009 As a material point moves in the flow a small volume of matter attached to it will deform To calculate its stretching and its rate of stretching we have to specify a direction where to measure this stretching as explained in the theoretical background We have two possibilities options 1 and 2 1 We don t specify a direction the computer will generate randomly an initial direction of stretching This direction will be different from material point to material point 2 We specify a direction dx dy dz every material point has the same initial direction The two possibilities are in fact equivalent after a while in a statistical point of view The second parameter to define is about the way we calculate the tensor F the gradient of deformation tensor For 2D simulations no problems the method guarantees that the determinant of F will remain 1 incompressibility it has no influence if you select option 3 or 4 56 why we prefer not to constraint F default 3 9 SELECTION OF PROPERTIES Polystat User s Guide But for 3D flows we cannot be sure that the tensor F will remain 1 along a trajectory of a material point If we do not constraint the tensor F POLYSTAT will s
90. me other properties If they are found we initialize the property with the value found in the file otherwise we initialize with the default values as usual The properties that are read in the CSV file are the coordinates must be the first property written in the file COORDINATES in upper case with always 3 components time logarithm_of_stretching space_integration direction_of_stretching abel All the points defined in the CSV file that are included in the flow domain will be tracked the other points are rejected Another use of this option is the following let us assume you cut your flow simulation in several parts to simplify and reduce the size of the problem the flow on each part is evaluated separately Then you want to make some tracking of material points across all the parts You define a first tracking through the first flow part Next with the generated mixing files in POLYSTAT you perform a slicing at the exit of the first flow domain save the slice in a CSV file Next you can define a second mixing task in POLYDATA on the second flow part the generation zone will be the CSV file you created in POLYSTAT By this way you will pursue the tracking in the second flow part In this case it is useful to answer no to the question All fields are reinitialized to default values Do you agree with that in order to keep the history of deformation and stretching throughout the who
91. mpete with surface tension of drops or with internal mechanic resistance of solid particles dispersion occurs Dispersion will be better if some elongational effect exists in the flow This information is available by adding some post processors while defining the set up for the flow calculation Next it will be possible to evaluate them along trajectories of material points Moreover a model has been included in Polystat to calculate the disagglomeration process along trajectories We have found a general and accurate method to calculate all these parameters in a single simulation The main steps of this method are firstly we calculate the flow as usual secondly we compute the trajectories of a large set of material points initially concentrated in the whole flow domain or not with in complement the calculation along these trajectories of the local deformation of the matter and other relevant properties Finally we analyze these results with statistical tools in order to obtain a global objective and quantitative overview of the mixing evolution 1 3 CLASSIFICATION OF FLOWS CAPABILITIES OF THE MIXING MODULE November 2009 With the mixing module all the kind of flows can not be studied There exist limitations But first let us define some concepts November 2009 Polystat User s Guide Open Closed domain a closed domain is a domain where there is no entry and no exit of fluid An open domain is the opposite
92. mplitude of a property along all the trajectories After clicking on the button A l the following window appears C Amplitude A The following data are necessary in this case 1 you select a vectorial or scalar property the data 2 you give a name to the new property the result There can not be two properties with the same name 3 if you agree with the data click on OK the new property will be stored in the list of existing properties If you click on CANCEL there is no storage of this property 81 November 2009 Polystat User s Guide For the other methods the process is always the same What changes is the number and the type of the data needed to calculate the new property Now let s define each method 4 2 3 2 AAX A x the new property is the property A exponent x you have to select the property A the data to enter the exponent x and to give a name to the new property the result 4 2 3 3 EXP A exp A the new property is the exponential of the property A you have to select the property A the data and give a name to the new property the result 4 2 3 4 LOG A log A the new property is the natural logarithm of the property A you have to select the property A the data and give a name to the new property the result 4 2 3 5 A B A B A B A B the new property is the addition or the subtraction of the properties A and B you have to select the two properties A
93. n We wish to evaluate the dispersive mixing of solid particles in a fluid matrix in studying the evolution of the size of the agglomerates 7 8 Let us consider a set of agglomerates of different sizes at the start of the mixing in an internal mixer In each point of the volume of the mixer we define a little volume Vx called representative volume that contains agglomerates of different sizes as illustrated here below o Vx time If the number of agglomerates is large enough in volume Vx this distribution of agglomerates sizes can be summarized in a mass density function f s t where t is the time and s the size mean diameter of the agglomerate Its unit is 1 um November 2009 28 November 2009 Polystat User s Guide It is discretized by a piecewise linear curve T s 0 00 f s t ds 1 f fy S gt So Si S2 Sm Sb The mass fraction of agglomerates of size in interval Sa Sb is f f s ds Sa Of course this mass fraction distribution will change with time as the volume Vx attached to the point X moves in the mixer because of erosion and rupture taking place during mixing We assume there is no transfer of agglomerates between adjacent volumes due to the high viscosity of the matrix Erosion is a slow and continuous process observed for all admissible sizes of agglomerates This process generates a lot of small particles Rupture occurs when a critical stress is reached
94. n S t is the length of the trajectory up to the current time t dt S t v t The properties 2 to 5 are the kinematics parameters defined in Chapter 2 a for 2D flows i the rate of stretching is equal to 1 2 ii the rate of dissipation is equal to D iii the direction of stretching is the vector m iv the logarithm of stretching is the natural logarithm of i b for 3D flows i the rate of stretching is equal to 1 11 ii the rate of dissipation is equal to D iii the direction of stretching is the vector n iv the logarithm of stretching is the natural logarithm of n The properties 6 to 8 are based on the fields stored in the RES files needed for the tracking pressure velocity and temperature are coming from the flow calculation If those fields are not in the RES files they are initialized to zero The properties 9 and 10 are useful to evaluate the accuracy of the calculation The properties 11 to 13 are based on fields stored in the RES files needed for the tracking viscosity mixing index and first eigenvalue of tensor T are post processors defined as additional sub tasks to the flow calculation If those fields are not in the RES files they are initialized to zero Those three fields are especially useful to the analysis of the dispersive mixing The mixing index M indicates if locally the flow is rigid M 0 or is a shear flow M 0 5 or is an extensional flow M 1 Moreover the
95. n SUM functions cece neeeteeetetetetees 125 4 5 2 9 Smoothing on functions 00 ceeceecseeeses ete eeeseteeeeeseeneteneeesesesenseens 126 4 5 2 10 the DENSITY of Probability function ccccecesseeeereeneeens 128 4 5 2 11 the Percentiles function 0 0 cece ete cteeteeteeseeseeneeesesenenenens 129 4 5 2 12 the Histograms fUNCtiONn ccccecsesesesseeseseeeetessseneteteeeeseenseens 130 4 5 2 13 the Segregation Scale function ss ssssssssessisssssessssessieressessee 131 4 5 2 14 the Deviation function 0 ccceceeste ete eteeteteteeseeneeneeesesesenseens 132 4 5 2 15 the Points Concentration Deviation function sss s1sss2s 134 4 5 3 New Disagglomeration Functions ccccceecseeesesesesescseecseseseseseseecssseseeeseess 136 4 5 3 0 the See disagglomeration along a single trajectory function 137 4 5 3 1 the Density of Probability function ssssssssssssssesssssssssssessesses 138 November 2009 147 Polystat User s Guide 4 5 3 2 the Probability fUNCtION ccceceesessseseecseesseesseeesteeeseeesseeeeeeeens 139 4 6 Additional definitions cccccccccccsssscecesseccesscscsesesseeseusescessesessecsesessssccseuseceessesesssesenses 140 A Gil Meslier ese aE AE ovaveadtves davies Nive EN NEE OER 140 a OAA U EAE N AEE EE EAEE EOE E E AS O EAE 141 A P AIT d Qi nT 91E TEE E AAE A EEEE E EEA EAEE 142 ADDENDUM A THE SIMULATION OF THE DISTRIBUTION
96. n order to quantify the progressive deformations of the interface in the channel Keywords 2D 1 2 planar flow viscoelasticity coextrusion secondary motion Polystat concentration field segregation scale 1 6 3 EXAMPLE 52 FLAT DIE In this third example we will analyze a 3D steady state non isothermal flow through the die section of an extruder We are specially interested by the residence time distribution and the melting characteristics of the matter leaving the die Keywords 3D steady state flow non isothermal flat die Polystat statistical analysis residence time distribution melting index November 2009 Polystat User s Guide 1 6 4 EXAMPLE 46 PERIODIC FLOW THROUGH A KENICS MIXER In this fourth example we analyze the distributive mixing generated by a Kenics mixer As the complete flow domain is too large to be used we reduce the flow calculation to a single mixing element of the mixer We assume the flow field to be spatially periodic We analyze the generation of striations through successive mixing sections and the efficiency of the process Keywords 3D steady state flow periodic boundary conditions static mixer Polystat distributive mixing 1 6 5 EXAMPLE 37 MIXER 2 D In this fifth example we simulate the 2 D transient flow produced by the rigid rotation of two cams in a batch mixer Moreover we evaluate the dispersive mixing capability of the mixer Forces and torque along the cams are also eva
97. n this case the stopping is NORMAL in the sense regular Mok SN A e e n D A D D De a n A a D S e R a A e S e S E e S The options 3 to 5 are specific to spatially periodic flows Remember that a problem is spatially periodic if there exists an elementary module on which we can calculate the flow field and where the velocity field in the inflow section is equal exactly to the velocity field in the outflow section The flow field is repeated infinitely in space With this kind of flow when a material point reaches the outflow we can continue the calculation by injecting this particle back into the inflow the particle will travel several times the same module we have thus two systems of coordinates The first is relative to the coordinates in the module The second is absolute and is attached to the real trajectory in the real infinitely repeated domain The next picture illustrates such concepts Flow domain module Absolute trajectory Relative trajectory 40 Polystat User s Guide In order to calculate trajectories in spatially periodic flows we have to connect the inflow with the outflow A limitation of the program exists the geometrical dimensions and the mesh distribution in the inflow and the outflow must be equal In order to define the spatially connected boundaries first we have to select the boundary which is the entry of the flow domain and to select the correct boundary condition for
98. ns in 4 5 2 9 4 5 2 11 and 4 5 2 12 November 2009 136 November 2009 Polystat User s Guide 4 5 3 0 THE SEE DISAGGLOMERATION ALONG A SINGLE TRAJECTORY FUNCTION See disagglomeration along a single trajectory With this function the user will see the time or displacement evolution of the disagglomeration property calculated along any trajectory he wants The user has to specify which single trajectory will be used to select the disagglomeration property to see and the time or displacement The user must also specify the time or space interval where the disagglomeration property will be stored Eventually the use gives a name to the new function If you visualize this function you find on the X axis the size of agglomerates and on the Y axis the density of probability to find agglomerates of a given size In this graph you have a new function y f x for each time or displacement step f size t2 gt tl gt agglomerates size 137 November 2009 Polystat User s Guide 4 5 3 1 THE DENSITY OF PROBABILITY FUNCTION Density of Probability The user has to select a set of slices and a disagglomeration property Eventually the use gives a name to the new function With this function one evaluates for each slice a mean disagglomeration function based on the disagglomeration property known at each point included in the slice n f s idx 1 Y f s point i Y po
99. nter 9 10 2D 1 2 axisymmetric geometry enter 10 2D shell geometry Enter your choice Select option 2 to create anew MIXING TASK By choosing between options 3 or 4 you specify if the flow field previously calculated is steady option 3 or time dependent option 4 Finally by choosing between options 7 to 10 you specify if the flow is 2D planar or 2D 1 2 axisymmetric Let us note that the mixing task is not available for 2D or 3D blow molding If you accept the current definition of the problem select the option 1 a new menu will appear as you can see on the next page 1 If the domain is 3D you don t have to choose between the options 7 to 10 November 2009 36 Polystat User s Guide 3 2 GENERAL MENU OF A MIXING TASK KKEKKKKKKKKKKKKKKKKKKKK x MIXING Task 1 x KKEKKKKKKKKEKKKKKKKKKKKK 1 Upper level menu enter 1 or CR 1 Definition of the flow domain enter 1 2 Definition of the boundary conditions enter 2 3 Definition of the velocity fields enter 3 4 Generation of the material points enter 4 5 Parameters for the tracking enter 5 6 Parameters for the kinematic variables enter 6 7 Selection of properties enter 7 8 Storage of the results enter 8 9 Definition of moving parts enter 9 November 2009 This general menu decomposes a mixing problem in several parts The first three
100. nual method November 2009 109 Polystat User s Guide 4 4 3 THE MANUAL SLICING OPTION Create manually a list of slices With this method you generate manually one by one a list of ordered slices You have first to select a set of trajectories on which the slicing will be done Second you specify each slice one by one Each slice must have a different name You can modify or delete existing slices You can also modify the order of the slices it is important to notice because the statistical functions are also ordered in function of the slices on which they are based Finally you enter the name of the new set This slicing is not based on a single property each slice can be defined on a different property The slices are not necessarily parallel to each other This method is more general but is more time consuming for the user November 2009 110 November 2009 Polystat User s Guide 4 4 4 THE SUB SLICING OPTION Generate a subslicing on a set of slices Suppose that you have a 3D unsteady flow and that you want to visualize the spatial repartition of the stretching in a plane cutting your flow domain Suppose that you define a slice whom the selected property is the coordinates the instants in this slice can have various time What you want to do is to distribute those instants among a list of time intervals and look at the spatial repartition of the stretching for one time interval Wh
101. o better discriminate distributive capacities of similar mixers The maximum inter distance will be of the order of 3 V N where V is the volume of the mixer With this set of distances we can calculate the density of probability function on the distance f d the probability to find a pair of points chosen randomly such that their inter distance is included in range d d Ad at time t is f d Ad 23 November 2009 Polystat User s Guide density of i probability at time t Area probability distance 0 d dmax Suppose on the other hand that you have distributed randomly a same number of points in all the flow domain we can assume that such distribution is ideal With the same tools we can calculate the function f d for this optimal distribution It is noted d The distribution index is defined as the deviation of the function f d real distribution from opt the function f d optimal distribution 00 1 opt B FOF Dal Se 0 1 29 0 As the distribution improves the index 6 decreases This index is dimensionless it is independent of the size of the flow domain The evolution of 5 depends of course on the initial position of the box Another important parameter not to forget is the number or material points to distribute a careful analysis must be done to measure its influence Two other parameters can also be evaluated The difference of the means dm t d
102. on 3 the user can modify some numerical parameters acting on the accuracy of the integration scheme November 2009 54 Polystat User s Guide KKEK KKK KKK KKK KKK KKK KKK KKEKKKKKKKKKKK x Parameters for the tracking E kkkxkxkxkxkxkxkxkxkkxkxkxkxkxkxkxkxkxkkkxkxkxkxkxkxkxkkkkxkxx Current setup NBELEM the number of steps of integration to cross an element of the mesh 3 EPSPNT the tolerance criterion on a distance 1 0000000E 06 EPSVIT the tolerance criterion on a velocity 1 0000000E 06 EPSTIM the tolerance criterion on a time 1 0000000E 06 1 Upper level menu enter 1 or CR 1 Modification of NBELEM enter 1 2 Modification of EPSPNT enter 2 3 Modification of EPSVI enter 3 4 Modification of EPSTIM enter 4 Enter your choice 1 NBELEM This parameter indicates how many integration steps we want to cross one element of the finite element mesh the default 3 asks that in the mean any material point crosses one finite element in 3 steps This parameter is important for the accuracy of the calculation of kinematic parameters higher is NBELEM better is the accuracy It s not necessary to have NBELEM higher than 3 expensive in CPU time and intrinsic accuracy of the method reached 2 3 and 4 EPSPNT EPSVIT and EPSTIM are respectively tolerances on a distance on a velocity and ona
103. on I ratio see equation 40 2 5 COMMENT November 2009 In the presentation of the mixing parameters that we calculate we always define them as evolving with time This kind of representation is well suited if the flow occurs in a closed domain in that case the mixing evolves with time But what if the flow occurs in an open domain such as in a single screw extruder or in a Kenics mixer In such a case the mixing quality evolves from the entry of the machine to the exit To analyze this process we generate a set of points in the plane section of the entry then 33 Polystat User s Guide we calculate their trajectory through the machine until they reach the exit For the statistical analysis we will generate a set of slicing planes uniformly distributed from the entry to the exit For each slice we determine the intersections with the trajectories Then at these intersections we interpolate the values of the kinematic parameters For each slice we can then calculate the mean value of a field a or the distribution function of a field B and so on As the slices are sorted from the entry to the exit we can analyze the evolution of the mixing slice by slice 2 6 BIBLIOGRAPHY November 2009 1 DANKWERTZ P V 1952 The Definition and Measurement of some Characteristics of Mixtures Applied Science Research Section A Vol 3 pp 279 296 2 OTTINO J M RANZ W E MACOSKO C W July 1981 A Framewor
104. ory 1 instant trajectory 2 November 2009 89 November 2009 Polystat User s Guide To define such property the following window appears C Concentration Every zone must have a different name To define a new zone click on the Add button If you want to modify an existing zone select it in the list and then click on the Mod button If you want to delete an existing zone select it in the list and then click on the Del button 90 November 2009 Polystat User s Guide 4 2 3 15 MIN MAX This method allows you to calculate the minimum or the maximum of a property A along trajectories The following parameters must be defined a the property A of interest b if we want the minimum or the maximum of the property c the name of the resulting property d if we want the absolute extreme value of property P along the whole trajectory option 0 gt t_end or a time evolving extreme value of property P option 0 gt t option 0 gt t_end for a given trajectory T min max A at any time min max A ti for ti t_o to t_end option 0 gt t for a given trajectory T min max A at time t min max A ti for ti t_o tot C min or max A 91 November 2009 Polystat User s Guide 4 2 3 16 EXTRACT This method allows you to create a new scalar property by extracting a component of a vectorial property A The following parameters must be define
105. ovember 2009 Polystat User s Guide 4 2 3 19 TIME AVERAGED EFFICIENCY OF MIXING This method allows you to create the time averaged efficiency of mixing based on the time the rate of dissipation and the rate of stretching As seen in Chapter 2 this efficiency is defined as t in 2D flows lt ey gt yea dt t in 3D flows lt en gt 4 yen dt This property is accessible only if the time the rate of stretching and the rate of dissipation have been calculated along the trajectories In the creation window the default values for the properties are correct This window appears like this fi Time Averaged Efficiency of Mixing 95 November 2009 Polystat User s Guide 4 2 4 DISAGGLOMERATION PROPERTIES In this specialized window it will be possible to create properties directly related to the model of disagglomeration of solid particles presented in chapter 2 NEW disagglomeration properties Of course it is not possible to evaluate size fractions number of agglomerates without having first define a property of type disagglomeration 96 November 2009 Polystat User s Guide 4 2 4 1 DISAGGLOMERATION Disagglomeration With this property Disagglomeration we will know the time evolution of the mass fraction distribution for a set of agglomerates of various sizes As explained in chapter 2 the mechanisms of erosion and rupture depends on time shear rate and viscosit
106. ow must always be incompressible There is no void formation in the flow The flow domain is completely filled with the same fluid if we want to mix two or several fluids they must have the same rheological behavior no diffusion nor chemical reactions between them and no interfacial tension It is thus clear that despite the fact that we calculate a mixing problem the flow calculation is identical to that of a single homogeneous fluid We will examine the time dependence of a set of mixing parameters without making any distinction between the fluids we want to mix However there is no limitation on the model of fluid generalized Newtonian or visco elastic models are available the dimensional complexity of the problem 2D planar 2D axisymmetric 2D 1 2 planar 3 components for the velocity field 2D 1 2 axisymmetric swirling flows 3D the thermal complexity of the problem isothermal or non isothermal simulations are possible 1 4 GENERAL EXPLANATION ON THE WAY TO SOLVE A MIXING TASK November 2009 Three major steps must be performed in order to solve a mixing task i we calculate the flow ii we calculate a set of trajectories iii we perform statistics on this set The flow simulation we have to define a finite element mesh via Gambit Icem Patran next we enter in Polydata where a F E M task is defined in order to calculate the flow ONLY With the data file we run POLYFLOW and finally
107. owing menu appears KKEKKKKKKKKKKKKKKKKKKK KKK x Add topo objects KKEKKKKKKKKKKKKKKKKKKK KKK object 1 S1 B5 object 2 S2 B4 operator INTERSECTION 2 Reset enter 2 1 Upper level menu enter 1 or CR 1 Select object 1 enter 1 2 Select object 2 enter 2 3 Select UNION operator enter 3 Create new object Enter your choice November 2009 The user must select two existing objects in a list and one operator UNION or INTERSECTION When this is done he selects option Create a new object and specifies the name of the new topo object see menu above Let us mention that object like S1 B5 means intersection of subdomain S1 and boundary B5 Be cautious if one uses INTERSECTION operator the resulting object may be empty There is no check to avoid that situation running POLYFLOW with this kind of degenerated generation zone will lead to a fatal error In order to modify the name of a topo object select Modify the name of a topo object option in the menu Generation of the material points Then select one existing object in the list Eventually select option Modify object name and enters a new name for this object In order to delete a topo object select Delete a topo object option in the menu Generation of the material points Then select one existing object in the list Eventually select option Delete object to delete effectively the se
108. perty for a specific slice or for a given trajectory By clicking the DRAW results option three windows will appear the graphic display the graphic options window and the view options window as shown on next figure Graphic Options Graphic windows of POLYSTAT With the View Options window we can modify the visualization point of view by rotating translating or zooming the domain In order to do this first select the type of operation translate rotate zoom Second choose an axis direction of the translation axis of rotation If you have chosen the zooming operation you have to select the type all Finally you can enter the amplitude of translation rotation or zooming If the flow domain disappears don t worry you can undo the last operation by clicking on the button Undo or return to the initial configuration by clicking on the button Initialize Don t forget to click on the button Draw to see the new position of the domain Be careful the axes are attached to the domain The origin of the three axes is the mass center of the finite element mesh Note that you can change the background color of the graphic display window by selecting another color in list In the Graphic Options window there exist two zones in the first one three buttons Save Close and Help are placed in the second one we can see three pages Slice Prop and Mesh 69 November 2009 Polystat User s Guide First l
109. ping plane Remove an existing stopping plane choice If we add a new stopping plane option 1 we have to enter the coefficients defining the plane and the final status of the trajectory if the crossing of the cutting plane is valid then select normal Current val ond T n T Pes Enter y es Enter its new v Current value o Enter its new v Current value o Enter its new v Current value o lue of the coefficient A of the new plane is 0000000E 00 alue CR no modification f the coefficient B of the new plane is 0000000E 00 alue CR no modification fF the coefficient C of the new plane is 0000000E 00 alue CR no modification the coefficient D of the new plane is 0000000E 00 Enter its new value CR no modification When a point crosses this plane the crossing is normal y or not n I or n o CR yes November 2009 You must be careful when defining a stopping plane in the algorithm of trajectory calculation after every time step of integration we enter the current position of the material point in each plane equation if the results are all positive then there is no crossing and we continue the calculation Otherwise we stop to calculate this trajectory After the definition of several stopping planes the menu has changed and appears like this
110. r 3 Parameters for the tracking for example we must define the coefficient NBELEM that indicates the mean number of integration steps necessary to cross an element Polystat User s Guide 1 6 EXAMPLES November 2009 Here below one can find a short description of the POLYFLOW examples devoted to mixing Refer to the documents corresponding to those examples for their full description on the POLYFLOW Documentation CD 1 6 1 EXAMPLE 50 THE RECTANGULAR CAVITY This example is the tutorial of the mixing module In this first example we will compare the mixing efficiency of a steady state flow with a piecewise steady flow It is 2D planar and isothermal flow problem We explain in detail how to use Polystat a how to create new properties b how to define a slicing on time and c how to define the statistical functions needed for the comparison of the two cases Moreover we explain how to extract useful information from those statistical curves Keywords piecewise steady flow distributive mixing reorientation process mixing efficiency Polystat statistical analysis 1 6 2 EXAMPLE 51 COEXTRUSION OF POLYMERS IN A SQUARE CHANNEL In this second example two viscoelastic fluids are injected in a channel They have identical rheological properties but different colors We analyze the axial evolution of the interface between those fluids in the square channel we calculate the axial evolution of the segregation scale i
111. r of values necessary to represent the auto correlation function You have also to give a name to the new function You will obtain an auto correlation function for each slice of the set If you visualize this function for a given slice you find on the X axis the distance and on the Y axis the auto correlation function of the concentration for slice S distance November 2009 120 November 2009 Polystat User s Guide 4 5 2 6 THE DISTANCE DISTRIBUTION FUNCTION Distance distribution To calculate the evolution of the distribution function of distances between material points you need to specify which set of slices will be used You must select a coordinate property that will serve to calculate the distance between pairs of points in a slice Don t forget to enter the number of values to represent the distribution function You have also to give a name to the new function Finally you have the possibility to choose between two methods a the first determines the distance distribution between all pairs of points The maximum distance measured will be about the size of the flow domain if the flow domain is closed b the other determines the distance distribution only for pairs of points that are close neighbors The maximum distance measured for a closed flow domain will be Max distance 2 3 V n where V is the volume of the flow domain and n the number of material points The search of neighboring
112. retching of these vectors are interesting properties that vary from place to place in the flow domain and that evolve with time Finally we perform a statistical analysis of the set of results in order to have a global overview of the process With such a method we can have an objective and quantitative evaluation of the mixing of any process we can for example find areas in the domain where the mixing is poor low stretching instead of exponential increase For 3D flows we generalize the concept the interface is now a surface and we will calculate the stretching of infinitesimal surfaces attached to material points Let Q and Q denote the domain occupied by the homogeneous fluid at time 0 and t respectively The motion of the fluid is described by the relationship x 4 X t 1 where X denotes the position of a material point P in Q and x in Q The symbols F and C denote the deformation gradient and the right Cauchy Green strain tensor between both configurations The velocity gradient and the rate of deformation tensor at time t are denoted by Land D respectively For later use we note that F LF 2 where a dot denotes the material time derivative 12 November 2009 Polystat User s Guide 2 1 1 KINEMATIC PARAMETERS FOR 2D FLOWS Consider in Q a material fiber dX with a unit orientation M which deforms into a material fiber dx with a unit orientation m at time t Let denote the length stretch ldx ldX It is
113. rty in a slice through the flow domain By clicking the RUN option you ask for the calculation of all the objects defined earlier new properties new sets of trajectories new sets of slices new statistical functions With the DRAW results option you can visualize one selected slice trajectory in the flow domain you will see the spatial repartition of any property in this slice trajectory The drawings can be saved in Postscript files With the DRAW stat option you can visualize the calculated statistical functions and save them in Postscript files With the WRITE trajectories option you select a set of trajectories to save on files in the csv file format see Polyflow User s Manual for more details With the WRITE slices option you select a set of slices to save on files in the csv file format see Polyflow User s Manual for more details With the WRITE stat option you select statistical functions to save on files in the crv file format see Polymat User s Manual for more details 64 Polystat User s Guide The Save option allows you to save your current POLYSTAT session Finally to quit the program click on the QUIT option and confirm your choice 4 1 0 THE OPEN OPTION This option allows you to read a file with a sav suffix containing a previous POLYSTAT session By this way it is possible to pursue an interrupted session on the same set of mixing files or to apply
114. ry to POLYSTAT 2 in the menu PROPERTIES we can ask the program to calculate new parameters evolving along the trajectories For example we can define any concentration field or a new mixing index These new parameters are always a combination of existing parameters those calculated in POLYFLOW and stored in the mixing result files We will see later the different possibilities accessible to the user 3 in the menu TRAJECTORIES we have the ability to select a subset of trajectories on which we will perform the statistical treatment For example we can eliminate all the trajectories that terminated abnormally on a wall for example We will see later the different possibilities accessible to the user D The Help option summarizes this analysis process and gives information on the way to contact us if necessary telephone Email and fax number November 2009 62 Polystat User s Guide 4 in the menu SLICES we determine the way to slice the selected trajectories For example let s suppose that we analyze the flow through a cylinder like shown on the next picture wall Inflow Outflow Cutting plane To analyze this flow we place a set of material points in the inflow section and we calculate their trajectory until they reach the outflow section We will cut the trajectories with planes disposed regularly from the entry to the exit this is the slicing step In each plane we will calculate statistical function
115. s Those functions will evolve from entry to exit and show the way the mixing changes Remark if the flow occurs in a closed domain we want to know the time evolution of statistical functions and the slicing will be done on the time 5 in the menu STATISTICS we define the set of statistical functions we want to calculate on the defined set of slices 6 finally we go back to the FILE menu and click on the RUN option By clicking on this option we order POLYSTAT to calculate actually all our desiderata new properties the subset of trajectories the set of slices the set of statistical functions This calculation can last for a while When the calculation is over we can analyze visualize and store our results After this brief description of the way to use POLYSTAT we will now explain in detail every menu option and window November 2009 63 Polystat User s Guide 4 1 THE FILE MENU November 2009 The Open option allows you to read an old POLYSTAT session file The READ data option allows you to read the files containing the trajectories calculated by POLYFLOW All the trajectories and the kinematic parameters calculated along those trajectories are stored in POLYSTAT The READ mesh option allows you to read the file containing the finite element mesh Polyflow format used to calculate the flow and the trajectories This mesh is used only by the option DRAW results to visualize a prope
116. same statistical treatment on a new set of mixing files in this case the mixing files must have same name same type ascii or binary and with the same number of mixing files See section 4 1 9 for additional information on the list of objects saved or not saved in the POLYSTAT session file 4 1 1 THE READ DATA OPTION Read Mixing files created with Polyflow This window shows the list of mixing files already read If you want to add several mixing files to the list in a single command click on the ADD all button the following window will appear November 2009 65 November 2009 Polystat User s Guide Read mixing files In this window the user has to specify the prefix of the mixing files their first and last indexes and their format If the mixing files are not in the current directory click on the browser button to search their location with a specific file browser The names of the mixing files are built like this prefix 000i where i is the current index Once those data are entered click on the OK button to close the window and actually read the mixing files Click on Cancel to close the window without any reading If you want to add just one file click on the ADD button a file browser will appear in which you have to select the file It is not possible to read the same file twice When a file has been selected a new window appears asking if this file is formatted or not don
117. sest to the value 1 2 3 If the selected property is a scalar enter the position in the rectangle 1 the rectangles 2 and 3 are set to zero 106 Polystat User s Guide 4 4 THE SLICES MENU November 2009 After the creation of new sets of trajectories you must select one set on which you will perform some statistical treatment Before this step you have first to define a list of slices that cut the trajectories This list is necessary to analyze the evolution of the mixing from one slice to the next from the beginning of the process to its end By default no set of slices exists Based on an existing set of trajectories it is easy to define a set of slices Three possibilities exist a the automatic creation of a list of slices b the manual definition of each slice one by one c the sub slicing which is the generation of a new list of slices based on each slice of an other list Remember that a trajectory is a set of instants ordered in time Each slice will contain a set of instants that are the intersections of the slice and the trajectories If the intersection of a trajectory and a slice is not a stored instant we create a new instant by interpolation with the previous and the next instants that surround the intersection as explained below slice previous trajectory O stored instants of the trajectory W intersection of the trajectory and the slice new instant created by interpolation 107
118. t The result file names will be built like this prefix id where id ranges from 1 first index to n last index The OK button is used to close the window and read the files The Cancel button closes the window without any reading The Reset button reinitializes the internal data structure containing the mesh and the successive positions of the impellers Note 1 Don t make any mistake in the format of the files because a wrong answer can interrupt definitively your session Note 2 It is absolutely necessary to read a mesh before any visualization of your results 67 November 2009 Polystat User s Guide 4 1 3 THE RUN OPTION By selecting the RUN option you want to actually calculate all the objects you defined elsewhere properties trajectories But it takes time That is why firstly you will have to confirm your choice If you confirm your order POLYSTAT will calculate successively the new properties the new sets of trajectories the sets of slices and the statistical functions Upon the time a message informs the user about the current calculations being done When those calculations are finished we can visualize our results it s the purpose of the next paragraph 68 November 2009 Polystat User s Guide 4 1 4 THE DRAW RESULTS OPTION With this option we can see the results of our calculations we visualize in the flow domain the spatial distribution of a pro
119. t exactly enter 9 10 Storage after each displacement minimum enter 10 11 Modification of the time step enter 11 With this menu we will determine the way we store the calculated trajectories and the evolution of different properties These results can be stored in several files in order to be analyzed by POLYSTAT Options 1 to 3 the prefix of the files is by default mixing We will create files with the names prefix 000i where i varies from 1 to NBFILE which is the number of files to generate With the option 3 you can choose if the files are formatted ASCII mode readable by human beings and all machines or not Despite the fact that the unformatted files are smaller in size economy of memory space the default option is formatted November 2009 59 Polystat User s Guide Options 4 and 5 NBPTMX and CPUMAX are parameters that allows the user to specify the way POLYFLOW will generate the files NBPTMX is the maximum number of trajectories that are stored in a single result file CPUMAX is the maximum CPU time in hours that is spent before we close the current file and we begin to store results in the next file By this way we generate continuously files we can begin the statistical analysis on a short population of material points and in the same time POLYFLOW continues to work If we are happy of our results we can interrupt POLYFLOW If we want we can run again POLYFLOW with the same data file it will be
120. t User s Guide constant along the trajectory of a material point Its evolution is governed by the transport equation c 0 23 The concept of concentration allows us to introduce the notion of segregation scale see ref 1 and 4 At time t consider a set of M pairs of material points separated by a distance r For the j th pair and time t let c and c7 denote the concentrations at both points of the pair moreover let denote the average concentration of all points and the standard deviation At time t the correlation coefficient R r t for the concentration is defined as follows D 2 e7 lt R r t 24 2 M The function R r t gives the probability of finding a pair of random points with a relative distance r and with the same concentration R r t 4 distance The Figure above shows a typical graph of R as a function of r Let be such that R t 0 when r we cannot predict whether the members of the pair have the same concentration or not The segregation scale S t is defined as a S t R t t dr 25 0 It is easy to understand that S t is a measure of the size of the regions of homogeneous concentration S t decreases when mixing improves 20 November 2009 Polystat User s Guide While quantities such as e and lt e gt are proper to the flow irrespective of the initial concentration the segregation scale S t is a quantity affected by the flow
121. ta function This process can be iterated several times The calculation of the mean can be weighted in different ways type of smoothing With an equal smoothing the current Y value and every neighbor have the same weight cur 126 November 2009 Polystat User s Guide With a centered smoothing the current Y value has the highest weight and the weight decreases linearly as the distance to the current X value increases X cur With the upwind smoothing we look first at the Y value of the farthest neighbor We weight more the size to the left or the right size of the current X value with the highest Y value With the downwind smoothing we look first at the Y value of the farthest neighbor We weight more the size to the left or the right size of the current X value with the lowest Y value In general we obtain a good result with a centered smoothing and with the following parameters 2 neighbors at left 2 at right and 2 iterations 127 November 2009 Polystat User s Guide 4 5 2 10 THE DENSITY OF PROBABILITY FUNCTION Density of Probability Function To calculate the evolution of a density of probability function you need to specify which probability function to use You must enter the number of values to represent such a function a good practice is to use half the number of values that represent the probability function You have also to give a name to the new function As t
122. the mixing module cece eee teeeeeeeeeeeeees 2 1 4 General explanation on the way to solve a Mixing taSk ccccceessesssesee esse sseeeeeeeeeeees 5 1 5 The numerical techniques involved in the mixing MOdUIe cece tee teeeteeeeeeeeees 8 16 EXAMI ES oes svcstescessts sies estes casasein siscovststassavien eoar aPN aA ctu eSEE EES REE SASEA EEE REENEN ESEESE NiE 9 1 6 1 EXAMPLE 50 The rectangular cavity sssssssssstessssrestessestertessssresiesessteee 9 1 6 2 EXAMPLE 51 Coextrusion of polymers in a square channel 00 9 1 6 3 EXAMPLE S25 flat diS nenni sonics ea aE Ea E a ai E 9 1 6 4 EXAMPLE 46 Periodic flow through a Kenics Mixet cccceeeeeeeee 10 1 6 5 EXAMPLE 37 2 Mixet 2D naeio enea i a atea ANR TEE 10 1 6 6 EXAMPLE 91 Dispersion eorr iai iineoa AE AENEA E AAEE 10 CHAPTER 2 THE MIXING THEORY seiisistsssssesscndssissseossuosntoaiinsasecien 11 20 troduction eeren EAEE EEEE E EEEREN NEE EEAS EEEE ER SEEE 11 2 1 Kinematic ParametetSi ianen neneke irainik aair ana sa Eaa a niea Ea Tsaa as Ea 12 2 1 1 Kinematic Parameters for 2D flOWS cccccceceseesseesseseecseesseeseeeseseeseessseeeeneeess 13 2 1 2 Kinematic Parameters for 3D flOWS ccscceceseessessseseesseeseeeseesseseeseeeeeeseeeeess 14 2 1 3 Statistical analy Sis sic cccssss usnertesasreds veut Reacrsiasia aa E Eae anaes 16 2 1 3 1 Mean and Standard Deviation ccccccccssesessseseseecescseseeseeseeeseeees 16 2 1 3 2 Cumulated Prob
123. the shear stress o is above a given erosion crit variation of size As of the agglomerate after At seconds can be described as follows threshold In this case if we assume that agglomerates are roughly spherical the As __ Salo oge YY n At 3S2 where amp is a coefficient of proportionality y the local shear rate and S the size of the agglomerate at time t and S AS will be the new size of the agglomerate at time t At Based on the assumption of mass conservation the mass distribution function becomes s As 5 f s As t At f s t 36 Moreover as the total mass of solid particles is constant in the control volume Vx the mass fraction of aggregates increases has the mass fraction of agglomerates decreases because of their reduction in size their number staying constant This can be seen in the next figure where we see the effect of erosion on an initial set of agglomerates with sizes ranged between 15 and 25 microns They are mixed in a matrix with a viscosity of 11000 Pa s We applied a constant shear rate of 10 s and we plot the mass distribution function every 25 seconds We observe the shift to left the widening and flattening of the Gaussian curve centered initially at 20 microns as erosion develops But we observe also an increasing peak at extreme left of the graph in the small sizes corresponding to the generation of aggregates 30 November 2009 Polystat User s Guide ay Ffsize sh
124. the value of the property chosen earlier when the probability function has been defined ty P a percentile p1 percentile p2 percentile p3 index of the slices 129 Polystat User s Guide 4 5 2 12 THE HISTOGRAMS FUNCTION Histograms To calculate the evolution of histograms you need to specify which probability function to use You must define a list of intervals by introducing a set of values You have also to give a name to the new function To complete the list enter a value in the box a and then hit lt CR gt To delete one value select it in the list and then click on the Del button As there exists a list of probability functions one for each slice of a sorted list you will obtain a histogram function for each slice If you visualize this function for a given slice on the X axis you have the specified intervals of values the corresponding property has been chosen earlier when the probability function has been defined and on the Y axis the percentage of instants of the slice having a property value in each interval for slice S pl p2 p3 p4 p5 property P November 2009 130 Polystat User s Guide 4 5 2 13 THE SEGREGATION SCALE FUNCTION Segregation Scale To calculate the evolution of the segregation scale you need to specify which auto correlation on concentration function to use You have also to give a name to the new function As there exists a list of a
125. ther walls are at rest With such a flow we can obtain a far better mixing than with a steady state flow In the second case the more general the flow changes continuously between t1 and t2 the flow at time t will be a linear combination of flow t1 and flow t2 flow t 1 a flow t1 flow t2 t tl with Oy and t1 lt t lt t2 t2 4 tl Spatially periodic flows the flow is spatially periodic if there exists an elementary module on which we can calculate the flow field and where the velocity field in the inflow section is equal exactly to the velocity field in the outflow section A spatially periodic flow is necessarily a flow through an open domain flow domain module Example of a spatially periodic flow The flow field is repeated infinitely in space the flow field in the next module is the same as in the current module and in the previous module and so on The limitations of our module are the following Geometrical limitations the domain must not change with time we have to find a frame of reference where the domain occupied by the flow does not vary For example if one wants to analyze mixing in a single screw extruder we assume the screw to be fixed and the barrel to be rotating If there are moving internal parts the mesh superposition technique must be used to simulate their motion Polystat User s Guide Fora piecewise steady flow there must be no inertia The fl
126. time Two points are identical if the distance between these points is smaller than EPSPNT EPSVIT is used to determine if a point is a stagnation point if the local velocity v X t is lower than EPSVIT VNORMX then the material point X at time t is a stagnation point A time step smaller than EPSTIM is considered as zero it will be used to stop the iterative Newton Raphson procedure that finds the time step needed to reach the border of the current finite element containing the point element 1 element 2 Successive positions q of a material point ha EAS November 2009 55 Polystat User s Guide Remember that we calculate the trajectory of a material point piece by piece we integrate the velocity in a finite element until we reach the border of this element Then we find the adjacent element where to continue the calculation and so on It is thus important to determine precisely the time step needed to reach the border of the current element containing the material point 3 8 PARAMETERS FOR THE KINEMATIC MIXING PROPERTIES The initial screen looks like this KKK KK KKK KKK KKK KKK KKK KKK KKK KKK KKKKKKKEKKKKKKKKKK Parameters for the kinematic variables koa KKEKEK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKEKKKKKKKKKK Enter your Current setup Random initial direction of stretching No constraint over the tensor F 1 Accept the current setup enter 1 or CR gt 1 Rando
127. tion of the mixing index and the main stress along the trajectories of material points With those data we can evaluate the fraction of the matter experiencing a given stress value and then evaluate the efficiency of the dispersive mixing 11 Polystat User s Guide We call also dispersive mixing the process where solid particles are broken by erosion or rupture in smaller parts due to stresses applied on them by the matrix carbon black or silica in a rubber matrix for example A new model has been developed by B Alsteens and V Legat see ref 7 to simulate this process of disagglomeration thanks to them this model is already available in Polystat The description of the model can be found in 2 4 2 1 KINEMATIC PARAMETERS November 2009 A first way to measure mixing is to quantify the capacity of the flow to deform matter and to generate interface In the theory presented below we neglect interfacial forces the interface is passive and no break up into droplets can occur For 2D flows the interface between fluids is a line in order to avoid to calculate the evolution of this interface a very complex and impossible task to perform because of the exponential growth of the interface we prefer to calculate the stretching of infinitesimal vectors attached to a large number of material points distributed in all the flow domain As the points move in the flow the vectors are stretched The stretching and the rate of st
128. tion time is reset to zero also for agglomerates that do not break phenomenon occurring when the rupture rate is not 100 Regarding now classes withS 2S but with an induction time below the rupture time they can receive fragments but does not break Their mass frequency f increases but their induction time Ti must be modified because we assume that incoming fragments come with their own zero induction time 32 Polystat User s Guide We apply the following rule Ti t At T t At 85 39 where T is a function of the ratio of mass frequency f of class S at previous time t and current time t At By default we define I as T ratio ratio 40 List of functions used in the erosion and rupture models and available in the disagglomeration clp file DNSPRB to define the initial mass density distribution function default Gaussian distribution between 15 and 25 um AS EROSION_MODEL to define the function ae of erosion model see equation 35 t TRANSFER_RUPTURE to define the transfer function for rupture model CRITICAL SIZE to define the minimum size of agglomerates that can break for a given shear stress see equation 38 RUPTURE_TIME to define the amount of time during which the stress must be above required threshold to get rupture RUPTURE_RATE to define the fraction of agglomerates that will actually break if rupture criteria are met MODIFY_INDUCTION_TIME to define the functi
129. unctions you want to save in formatted files First you select one function in the upper list Then you click on the gt gt button A file browser appears enter now the filename or the prefix of the filename to generate After the writing the functions already written appear in the lower list When you have finished click on the OK button in order to go back to the main window Depending on the type of function to write POLYSTAT can generate one or several files for example if you select a probability function there exists a function of the type y f x for each slice and POLYSTAT will generate a file for each one The name of these files are built like this the user specifies the prefix of the files and POLYSTAT add an index if necessary and a suffix depending on the kind of result you want to save see a property along a trajectory prefix trajectory index see correlation between two fields prefix corr kin segregation scale prefix evol seg deviation from an ideal distribution prefix evol dev sum of a property prefixJevol sum mean of a property prefix evol men standard deviation of a property prefixJevol std deviation of points concentration prefix evol pcd operator on functions prefix evol opr auto correlation on concentration field prefix slice index cre probability prefix slice index prb density of probability prefix slice index dns distance distribution prefix slice index dsp histograms pref
130. uto correlation functions one for each slice of a sorted list you will obtain the evolution of the segregation scale along the slices If you visualize this function on the X axis you have the index of the slice and on the Y axis the value of the segregation scale Segregation Scale index of the slices November 2009 131 Polystat User s Guide 4 5 2 14 THE DEVIATION FUNCTION Deviation This function will allow you to calculate the gap existing between two density of probability functions The second one is supposed to be an optimal function Three different methods exist to evaluate this deviation 8 a integral abs f f 5 2 00 S idx Al F s idx t 6 idas b difference of the means S idx 5 5 c difference of the standard deviations S idx 5 5 November 2009 132 Polystat User s Guide where 00 s idx f s idx s ds FOD and 00 5 idx f s idx s 5 ds This distribution index defined in chapter 2 corresponds to the result of the first method In all cases you have to specify the density of probability function of the real distribution and the density of probability function of the optimal one Next you select a method and eventually you give a name to the new function You will obtain the evolution of the deviation along the slices If you visualize this function on the X axis you have the index of the slice and on th
131. wing steps are necessary a In the mixing files are calculated the stretching In A and the dissipation rate D along trajectories b First we have to define in POLYSTAT a new property the cumulated dissipation which is the time integration of the dissipation rate t D D X t D X t dt 0 This new parameter depends on the material point and on time c We perform a slicing on the time N slices every At seconds d We calculate the sum function of the stretching In A S nh M t In ACX M t dQ Qo e We calculate the sum function of the cumulated dissipation lt D gt ane t lt D gt X t dQ Qo f Finally we divide the two sum functions to obtain the global efficiency of stretching lt lt e gt gt M S nha Lo S lt p gt t 144
132. xis the property a index of the slices November 2009 116 Polystat User s Guide 4 5 2 2 THE MEAN amp STANDARD DEVIATION FUNCTION Mean Standard Deviation To calculate the mean and the standard deviation of a property you need to specify which set of slices will be used and to select a property You have also to give a name to the new function If you want to weight the mean and the standard deviation in function of the local velocity don t forget to select the velocity field Additional information on weighting is available at the end of Chapter 4 see 4 6 3 You will obtain two curves one for the evolution of the mean of a property along the slices and the second for the evolution of the standard deviation If you visualize these functions you find on the X axis the index of the slice and on the Y axis the property a mean g deviation ti index of the slices November 2009 117 November 2009 Polystat User s Guide 4 5 2 3 THE CORRELATION FUNCTION Correlation Function To analyze a possible correlation between two properties in two slices you have first to select exceptionnally a set of trajectories Second you define two slices by clicking successively on the a first slice and a second slice buttons Third you specify the properties associated to each slices Finally you give a name to this new function If you want to visualize this function on the
133. y breaks up into smaller droplets We name this process dispersive mixing Let us note that an extensional flow field is more efficient to break up drops into droplets than a shear flow If the capillary number is much higher than the critical capillary number then the viscous stress overrules the interfacial stress and the drop is extended but does not break up this process is called distributive mixing On the contrary if the capillary number is much lower than the critical capillary number then the interfacial stress dominates and the drop is only slightly deformed In general mixing begins with a distributive step drops are deformed passively followed by a dispersive one drops break up into droplets and finally by the distribution of the droplets in the flow In the paragraphs below we concentrate mainly on distributive mixing 2 1 and 2 2 and on distribution of material points into the flow domain 2 3 However dispersive mixing can also be analyzed the user can add post processors to the flow calculation a the mixing index or flow number indicates if the flow is locally a rigid motion mixing index 0 a shear flow mixing index 0 5 or in extension mixing index 1 b the eigen values of the extra stress tensor T with this field we have access to the main component of the stress which stretches and breaks the drops Once the flow and those post processors are determined we can calculate the evolu
134. y of the matrix Those properties must be defined first Let us note that POLYFLOW solver evaluates the rate of dissipation D instead of the shear rate which is in fact equal to J2D Next the user must specify the number of classes of agglomerates he wants to evaluate He must specify the size of the largest agglomerates and the maximum size of the aggregates particles that can not be broken in smallest pieces anymore Next the user must specify in which CLIPS file are defined transfer function for erosion and rupture mechanisms the kinetics of erosion and rupture function of shear rate viscosity size of agglomerates and of course the initial distribution function of agglomerates size As this file is interpreted during the calculation it is very easy to modify those functions as the understanding of the disagglomeration improves or to test new ideas 97 November 2009 Polystat User s Guide 4 2 4 2 TYPICAL SIZE OF AGGLOMERATES With this function it is possible to evaluate the minimum mean or maximum size of agglomerates without taking into account the aggregates To do that we evaluate a new distribution function ranged between the maximum size of aggregates a and the maximum size of agglomerates b b f s t f s t 7 JE s t ds in range a b and f s t 0 otherwise a The mean size of agglomerates is thus b b_ b_ Mean size t F s 0 sds f s t ds where f s t ds 1 a a a

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