User's manual for Ecolego Toolbox and the Discretisations Block
Contents
1. Block Parameters Radionuclide Block X RadionuclideBlock mask Radionuclide Block Parameters Radionuclide names Kt C 14 CI 36 Se 79 1 129 Cs 135 Ni 59 Nb 94 Halflife 15730 301 000 1130000 15700000 2300000 76000 20000 Decay chain P0000000000000000000000000000000000000000000 Unit Bq Cancel Help Apply Fig 1 4 Example of Radionuclide block dialog window Radionuclide transport models often have to consider multiple nuclides most often when the ingrowth of daughter nuclides is of interest but also when two or more nuclides might have some relationship other than belonging to the same decay chain for instance when there are similarities in the physico chemical properties which might lead them to compete in adsorption processes or root uptake processes To handle this effectively it was decided to look at the models implemented as being vectorised Since a signal in Simulink can either be a scalar or a vector this raises no big problems Instead of multiplying the signal corresponding to one nuclide a matrix multiplication is performed with the vector signal The matrix contains all information about the decay times and also about decay chains The goal is to calculate the change in each nuclide per time unit either in units of nuclei or activity If Ni Nz and N are the number of nuclei Mole in a decay chain where N is the parent nucl2. Electromagnetic fields SSI is working on the risks associated with electromagnetic fields and adopts countermea sures when risks are identified Emergency preparedness SSI maintains a round the clock emergency response organisation to protect people and the environment from the consequences of nuclear accidents and other radiation related accidents SSI Education is charged with providing a wide range of education in the field of radiation protection Its courses are financed by students fees EDITORS REDAKT RER Robert Broed and Shulan Xu SSI rapport 2008 10 Facilia consulting AB Sweden mars 2008 ISSN 0282 4434 TITLE TITEL User s manual for Ecolego Toolbox and the Discretization Block Anvandarmanual f r Ecolego Toolbox och diskretiseringsblocket DEPARTMENT AVDELNING of Nuclear Facilities and Waste Management Avdel ningen f r karnteknik och avfall SUMMARY The CLIMB modelling team Catchment LInked Models of radiological effects in the Biosphere was instituted in 2004 to provide SSI with an independent mod elling capability when reviewing SKB s assessment of long term safety for a geological repository Modelling in CLIMB covers all aspects of performance assessment PA from near field releases to radiological consequences in the surface environment Software used to implement assessment models has been developed within the project The software comprises a toolbox based on the commercial packages Matlab
3. Gar 3 9 16 Adis ade Ade b lt lt Ee Dia X Am 7 2 down l ott as TR Fig 3 1 Schematic of the compartmental model for description of transport processes in a stream tube concept The transfer rate from mobile to stagnant liquid is given by he _ 2a D 3 10 where d is the length of the first layer compartment in y direction The transfer rate from stagnant liquid rock matrix to mobile liquid is given by 2D 3 11 som R d The transfer rate of diffusion from rock matrix compartment j to j 1 is given by Di 3 12 Aada Rid a dia 2 where d and dj are length of the matrix compartments in the y direction j and j 1 respectively The transfer rate of diffusion from rock matrix compartment j 1 to j is given by Di cil Dt 3 13 a Rd ld d 2 17 3 1 3 Implementation of compartment model in the Block and comparison of the results with the semi analytical solutions We take as an example the pin hole failure case in SR Can assessment of long term safety for a spent nuclear fuel repository SKB 2006 as a calculation example Calculated release from near field due to the leakage from the damaged canister through bentonite buffer to the fracture is shown in Fig 3 2 Details of near field transport calculation are found in SKB 2006 and Maul et al 2003 The near field release flux was used as the boundary condition for the far field transport problem Input da
4. 16 Stockholm Sweden Visiting address Solna strandvag 96 Telephone 46 8 729 71 00 Fax 46 8 729 71 08 WWW SSI SE SSI report
5. 7 in which Q is the water flux m y Mo is the total mass of solute inserted into the fracture moles XA is the Dirac delta function y and Z is the maximum penetration depth m 3 1 2 The Representation of radionuclide transport by compartment model There is similarity between a finite difference approximation of an advection dispersion A D type equation and compartmental models Therefore compartmental models can be used to obtain identical solutions to analytical solutions of A D equations when certain criteria are fulfilled Xu et al 2007 The corresponding transport problem modelled by a compartmental model as a two dimensional array of compartments is schematically shown in Fig 3 1 In the x direction along the stream flow compartments are linked together to represent advective and dispersive fluxes both forwards and backwards In the y direction perpendicular to the stream flow compartments are represent the process of matrix diffusion in a stagnant liquid The transfer rates shown in Fig 3 1 are given below taken from Maul and Robinson 2002 The symbols used in the following expressions have the same definitions as in 3 1 and 3 2 The transfer rate of advection from compartment i to i 1 is given by u 3 8 adv X n where X is length of transport domain and ny is number of compartments in the x direction The transfer rate for dispersion both forward or backward is given by D ais f Rais
6. and Simulink used to solve compartment based differential equation systems but with an added user friendly graphical interface This report documents the new simulation toolbox and a newly de veloped Discretisation Block which is a powerful tool for solving problems involving a network of compartments in two dimensions SAMMANFATTNING Ar 2004 initierade SSI modelleringsgruppen CLIMB Catchment LInked Models of radiological effects in the Biosphere for att bygga upp en oberoende modellerings kompetens f r granskning av SKB s sakerhetsanalyser Modelleringen inom CLIMB t cker alla aspekter av sakerhetsanalysen fran utlackage fran de tekniska barriarerna till radiologiska kon sekvenser i ytmiljon En mjukvara for implementeringen av sakerhetsanalysmodeller utvecklades inom CLIMB projektet Mjukvaran ar en Toolbox baserad pa det kommersiella paketet Mat lab och Simulink f r att l sa differentialekvationssystem kopplade till compartment modeller men med mer anvandarvanligt anvandargranssnitt Den har rapporten dokumenterar den nya Toolbox en och ett nyligen utvecklat diskretiseringsblock som ar ett kraftfullt verktyg f r att l sa problem med en serie av compartments i tva dimensioner N a es aid S Statens str lskyddsinstitut Swedish Radiation Protection Authority Contents 1 1 1 2 1 2 1 1 2 2 1 2 3 1 2 4 1 2 5 1 2 6 1 3 1 4 1 5 2 1 22 2 3 3 1 3 1 1 3 1 2 3 2 3 2 1 3 2 2 3 2 3 ECGOlE SO TOOIDOK sete aa t
7. anywhere within the masked subsystem in question by simply writing kd in the appropriate location this could be in the value field of a constant block in any dialog parameter for any Simulink block or in an expression of a transfer function block Once the parameters are defined in a masked subsystem and the user has pressed apply or ok the mask dialog will open the next time the user left double clicks the subsystem For an example of this see Fig 1 6a and 1 6b In the opened mask dialog the user can specify any values to the listed parameters EJ Mask Editor DS _ O x Icon Parameters Initialization Documentation mDialog parameters Density kg m3 tho edit M fv fv Distribution coefficient m3 kg kd edit M IV v Length m dx edit M IV v le l ble r Options for selected parameter Popups one per line In dialog fy Show parameter Vv Enable parameter Dialog callback Unmask OK Cancel Help Apply Fig 1 6a Example of the editing of a masked subsystems parameters Function Block Parameters DS X Subsystem mask m Parameters Density kg m3 2700 Distribution coefficient m3 ka 1e 3 1e 3 16 2 0 1 1 Length rm eo ooo Cancel Help Apply Fig 1 6b The resulting opened mask dialog of the parameters defined according to fig 1 6a 10 1 3 Interf
8. for each of the discretisations along the second dimension while maintaining the original single of the discretisation along the first dimension Thus if the system is discretised in 10 levels along the first dimension and 5 along the second dimension the total number of Compartments would be 60 10 5 10 Were this to be constructed manually 60 Compartments with 120 Transfer Functions linking them would need to be set up In such modelling it is often the case that the effect of discretisation on system behaviour is part of the study and therefore a manual method is impractical The task is greatly simplified by just changing the values given for the number of required discretisation elements in the two dimensions x and y to get the required size of the system see figure 2 1 Function Block Parameters Discretisator Subsystem mask Parameters nf 60 nly Fig 2 1 An example of the mask dialog for the 2D Discretisation Block 14 3 Application of Discretisation Block 3 1 Radionuclide transport in far field 3 1 1 A dual porosity model for radionuclide transport Transport of radionuclide by groundwater in fractured rock is known as far field transport in assessments of the long term safety of spent nuclear fuel repositories in crystalline bedrock The processes involved in the transport are advection and dispersion along preferential flow paths and diffusion into the rock matrix as well
9. while in the compartmental model no such a distribution function is employed 400 350 n 50 300 I nyiv 100 7 E 250 Nyiy 150 7 200 Nyiv 200 J 5 o 150 lt 4 3 g 8 100 50 0 fi fi L Te i Dre 0 10 20 30 40 50 60 70 Time hours Fig 3 8 Comparison of calculated concentration time distribution for Cr at station D by using different number of compartments in representing stream water in the compartment model 25 Table 3 3 Description of transfer rates in compartmental models definitions and values of the parameters in the descriptions are found in Table 3 4 Transfer rate Description of transfer rate TRaav U L Niv TRais D L N riy y TRwat sed Vz 2h TRsed wat z 2 where R 1 kple R z Med TRapp down V g 2 R Z N sed TRaown upp V g 2 R Z N sed Table 3 4 Parameter values used to evaluate the breakthrough curve of Cr in the Sava Brook experiment 1998 after Johansson et al 2000 W rman et al 2002 Symbol Definitions unit values U The effective flow velocity m s 0 088 D The longitudinal dispersion coefficient m s 0 8 L The length of the river m 3980 amp lt V_ gt 2 The advective velocity into the bed m s 3 96x10 sediment h The hydraulic radius ratio of cross m 0 77 section area and wetted pe
10. SSI Rapport 2008 10 Rapport fran Statens stralskyddsinstitut SSI report tillg nglig i sin helhet via www ssi se User s manual for Ecolego Toolbox and the Discretization Block Robert Broed and Shulan Xu NS Statens stralskyddsinstitut Swedish Radiation Protection Authority amp i lr i Ml LJ SSI s Activity Symbols Ultraviolet solar and optical radiation Ultraviolet radiation from the sun and solariums can result in both long term and short term effects Other types of optical radiation primarily from lasers can also be hazardous SSI provides guidance and information Solariums The risk of tanning in a solarium are probably the same as tanning in natural sunlight Therefore SSI s regulations also provide advice for people tanning in solariums Radon The largest contribution to the total radiation dose to the Swedish population comes from indoor air SSI works with risk assessments measurement techniques and advises other authorities Health care The second largest contribution to the total radiation dose to the Swedish population comes from health care SSI is working to reduce the radiation dose to employees and patients through its regulations and its inspection activities Radiation in industry and research According to the Radiation Protection Act a licence is required to conduct activities involving ionising radiation SSI promulgates regulations and checks compliance with these regulations co
11. a b will be removed from the donor Compartment each time step Thus the value of the Transfer Function Block is not proportional to the value of the donor Compartment On the other hand if the expression was instead a b INVENTORY where INVENTORY is the value of the donor Compartment then the value of the Transfer Function Block will be directly proportional to the value of the donor Compartment Note the variable INVENTORY can be used to automatically access the donor Compartments value IF the expression is written directly in the Transfer Function Block and not using the external rate port option If the external rate port is used this value has to be connected manually like any parameter to the inputs of the Function Block 1 2 5 Conversion Blocks The Conversion Blocks when being connected with an output from another block automatically converts the value between either Mole to Bq or Bq to Mole These blocks also uses the information stored in the Radionuclide Block 1 2 6 Parameter Input Although a parameter in Simulink can be defined in several different ways to simplify the use of Ecolego Toolbox model parameters are by convention defined in so called subsystem masks By following this convention it is possible to import and export model parameters to and from Ecolego Toolbox for example to from MS Excel The user still has the option of suing any valid Simulink method to define parameters however parameters de
12. ace with MS Excel In addition to the block library in Ecolego Toolbox described in section 1 2 two Matlab functions are included in the toolbox simplifying the handling of model parameters The two files are e Simulink xls m e xls simulink m The functions are called from the Matlab prompt and works on the currently active Ecolego Toolbox model The goal of these functions is to assist in editing and viewing the parameters of a model Since a parameter in Simulink can be defined in many different ways these files work under the assumption that the user has defined the model parameters in so called Masked Subsystems in Simulink A Masked Subsystem is basically a set of blocks grouped together hierarchically and having a local workspace associated with it The Masked Subsystems workspace can be edited to include any number of parameters Any parameter defined in a Masked Subsystem is available for all blocks below it in the model hierarchy via references The function simulink_xls m scans through the model for any Masked Subsystems and extracts the data for any defined parameters The data is summarized in a MS Excel file containing the parameter name location in the model and value if specified To use the function the user enters simulink xls at the Matlab prompt after which a dialog asks the user to enter a name of the MS Excel file to be created The function xls_simulink m works the opposite way by reading the
13. as sorption on to the solid matrix A dual porosity model for radionuclide transport along a stream tube is often used for describing radionuclide transport in far field Norman and Kjellbert 1990 i SE ok Bl OC ce p C AC a Di 0 3 1 Ot Ox Oz Ia i ac aC i qimi i l 4i 1 Mil s ar po ER Oak a C 0 e where t time y C an effective stream tube average of the concentration of radionuclide i in the mobile liquid moles m C a surface and stream tube averaged concentration of radionuclide i in the stagnant pore liquid in the impervious rock matrix moles m u velocity of the mobile liquid m y D longitudinal dispersion coefficient m y aw total surface area of the boundary of the flow porosity per unit volume of mobile liquid m x distance along stream direction m Z penetration depth into matrix orthogonal to stream direction m Di effective matrix diffusion coefficient for radionuclide i m y A decay constant for radionuclide i y R retardation factor due to sorption into the rock matrix which is defined by R 0 Kip p bulk density of rock matrix kg m K distribution coefficient for radionuclide i inside rock matrix m kg 0 matrix porosity For a solute pulse travelling in the fracture a delta source the initial and boundary conditions are 15 C x t 0 Ci x t 0 0 3 3 C x 0 1 5 3 4 Q Ci z 0 t C x t 3 5 OG 2 3 6 Oz les C x t 00 0 3
14. ate the release of radionuclide from the near field due to the leakage from a damaged canister through the bentonite buffer to the fracture The reason for choosing this problem is that the results have been verified in a comparison of Ecolego and AMBER Maul et al 2003 12 2 The Discretisation Block 2 1 Introduction In many situations in compartment modelling it is the goal to model the transport of some contaminant through a medium of some sort Since a compartment represents a unit of volume in which the contaminants entering are immediately assumed to be homogenously distributed this gives rise to a problem when the total volume is large This problem can be solved by using a series of connected compartments all together representing the total volume of the medium In this manner the dependency on the spatial variable can be obtained Often the optimal number of compartments required to correctly approximate transport through the medium in question can be large It can also vary depending on radionuclide properties or some other parameter in the system Thus the need to connect the number of compartments via the many transfer function connections can be both time consuming and prone to error since the number of interconnections rapidly becomes large To get around this problem a Discretisation Block was developed for Simulink 2 2 1 D Discretisation Block This block only consists of one underlying integrator i e Compartment whic
15. data in an MS Excel file and then for all matching parameters and Masked Subsystems in an active Ecolego Toolbox model updates the parameters The major benefit of this is that a user created model can have different parameter sets stored in MS Excel files To use the function the user enters xls_simulink at the Matlab prompt after which a dialog asks the user to select an MS Excel file 1 4 Installation and use To install Ecolego Toolbox simply copy the folder containing the harddrive Then in Matlab add this path with subfolders to the Matlab path This is done by selecting in Matlab the following menu items File gt Set Path In the window that appears press the button labeled Add with Subfolders then locate the folder where the files were copied to and press ok After that press the button labeled Save and close 11 Once the Ecolego Toolbox is installed the Simulink blockset will appear in the Simulink Library browser window the next time Simulink is started To use a block from the library simply click and drag it into the model window from the library window 1 5 Verification of the Toolbox The Toolbox was verified by comparing the results with the results obtained from the Ecolego for the same problem In the test the results agreed perfectly The problem used in the verification is taken from assessment of long term safety for a spent nuclear fuel repository Lindgren and Lindstr m 1999 to calcul
16. e blocks constituting a Simulink model created in the tool Ecolego Avila et al 2003 Thus a user of the Ecolego Toolbox will immediately recognize the structure and functionalities in an Ecolego created Simulink model even though some layout and design differences exists between the two tools Also a model created with the Ecolego Toolbox can be imported to Ecolego This allows for the use of Ecolego s data visualization capabilities and parameter handling as well as its probabilistic simulation engine etc 1 2 Blocks in the Ecolego Toolbox Except for those functionalities which are included in Simulink the following Blocks shown in Fig 1 1 are used in the Ecolego Toolbox to facilitate model construction Compartment Block Function Block Radionuclide Block Transfer Function Block Conversion Blocks Bq to Mole Mole to Bq Below follows a description of each block in the Ecolego Toolbox library 1 2 1 Compartment Block The Compartment block represents a section of the system in which the concentration of the radionuclides are homogenously distributed i e a single Compartment has exactly one value of the concentration for any given radionuclide Each Compartment has an unlimited number of inputs and outputs as selected by the user The inputs represent the time dependent flux of radionuclides into the Compartment and the outputs the time dependent outflux of radionuclides over time Mathematically the Compartment integrate
17. e model IS model the water infiltration model WI model and advective storage path model ASP model By using the method of temporal moments of the residence time the relationships between parameters of the different models can be determined W rman 2000 resulting in identical model predictions up to the first three temporal moments Thus selection of any of these models is not critical for predictability We use the ASP model to describe radionuclide transport in streams The governing equations of the ASP model W rman et al 2002 is written as 2 ac 1l a AUC p am 3 14 ot A Ox Ox where C is the activity concentration in stream water Bq m Ar m7 is the cross sectional area of the main stream including side pockets A is the cross sectional area of 22 the main stream excluding side pockets U is the flow velocity in the main stream m s Q UA is the discharge m s and D is the main stream dispersion coefficient m s 1 The effective flow velocity in the main stream channel corrected for side pockets with stagnant water is given by U Q Ar W rman 1998 The net solute mass flux Bq m s in the dissolved phase in the stream water can be written integrating over the distribution of transport pathways Js E OE rT co Vel Mpa ga where gy is solute mass per unit volume of water in the hyporheic zone Bq m V is the infiltration velocity m s into the bed in the direction of the s
18. eet le dena brass Ae rel ne brass sa aaa a a ennt aat 3 TITO UC ATO AREE EE ones S snutetesuetta es aatoh sneeenentaes 3 Blocks in the Ecolego TOOIDOX a essersesserrereereesrereesrerresrrsrnesrrssrernnssrerne nere rr rr ne ennen rerna 3 Compartment eein Tror bero r SNES ess an T dave ches eteduued Seve Ee E T AR Ke chadedbiee 3 F RGUONBIO CK ceca ts cos paazenzebston ead any edad ts ooh A a oe tan lvancengensueled evades ees 5 Radionuclide Manager Block o sssssssssrsssssrssseerresrerersrerresrrsrnesrrsrrernnrsrernrrnrrnr sne ere nr rn nanna 5 Transfer Function BlOCK ccsssssssaccsvciezssasaneedsdedecsenvestece sesutounedunnsed se dtecdeageersengatcentesss 7 Conversion BIOCKS 22 28 0i0 a a oea p a cana AE daha ssa asta vhs dudes LR Sen Nera ade bese SR 9 Parameter Iput arn E E oe cevvateadueddogucecuspesuvelsopasetesgeseones a eE 9 Interface with MS Excel encro raae inse deren oss ense a A E a A a 11 Installation and USE nnanneii n a ssd Lans a E a E 11 Verification of the TOO DOX so me ee bir s n a a a aata aset aoaia 12 The Discretisation Block msossrsrssresesreresresrsreresresreresrerrsrerrsrrsr ere rs rese rr ere ner r ere rn n a 13 TNT OGU CHOI 500630 sets b ssor a sbra a a tn e e bet rr Abele br a bra a a 13 1 D Discretisation Block 0 0 cieececcesseeseesecseeeseeeecsecaeeecaecaeeeseesecaeeaeeeeeaeeaeeeneeaes 13 2 D DiscretisatiOn Block ienien i eiea dul was see brat se TS RN E 13 Application of Discretisation Block s ssessssss
19. er m and equivalent to A P in ASP model A is decay constant for radionuclide s L is the transport length in the x direction m Z is the depth of the sediment m n is number of compartments in the x 23 direction and sea is number of compartments in the z direction R is the sorption capacity of compartment j and can be expressed as R 1 k p 3 18 Where amp is the porosity of the sediment compartment j kaj is the distribution coefficient in the compartment j p is the bulk density of the sediment in compartment j Further the total inventories in stream and sediment compartments are expressed as M CV 3 19 m g lev k 0 3 20 Where C is the dissolved activity concentration in the stream compartment i Bq m V is the volume of compartment i m g is the solute concentration in the sediment pore water in the compartment j Bq m vj is the volume of the sediment compartment j m TRaav TR dis Fig 3 7 Schematic of the compartmental model for conceptual description of transport processes in a stream 24 3 2 3 Model implementation and discretisation Table 3 3 summarises the expressions of those transfer rates derived from the previous section Data obtained from a tracer experiment performed in Sava Brook in Uppland County Johansson et al 2001 was used to verify the compartment model of the stream In the experiment moderately sorbing Cr was used as the tracer The co
20. fall Bjorn Dverstorp och Bo Str mberg 110 SEK 2008 04 E SKI s and SSI s review of SKB s safety report SR Can Avdelningen for karnteknik och avfall Bjorn Dverstorp och Bo Str mberg 110 SEK 2008 05 International Expert Review of Sr Can Safety Assessment Methodology External review contribution in support of SSI s and SKI s review of SR Can Avdelningen for karnteknik och avfall Budhi Sagar et al 110 SEK 2008 06 Review of SKB s Safety Assessment SR Can Contributions in support of SKI s and SSI s review by external consultants Avdelningen for karnteknik och avfall Pierre Glynn et al 110 SEK 2008 07 Modelling of long term geochemical evo lution and study of mechanical perturbation of bentonite buffer of a KBS 3 repository Avdelningen for karnteknik och avfall Marsal F et al 110 SEK 2008 08 SSI s independent consequence calcula tions in support of the regulatory review of the SR Can safety assessmenty Avdelningen for karnteknik och avfall Shulan Xu Anders Worman Bjorn Dverstorp Ryk K os George Shaw och Lars Marklund 110 SEK 2008 09 The Generalised Ecosystem Modelling Ap proach in radiological assessment Avdelningen for karnteknik och avfall Ryk Ktos 110 SEK 2008 10 User s manual for Ecolego Toolbox and the Discretization Block Avdelningen for karnteknik och avfall Robert Broed and Shulan Xu 110 SEK TATENS STR LSKYDDSINSTITUT SSI r en central tillsyns myndighet som verkar f r ett gott
21. fined in this way cannot then be included when interfacing Ecolego Toolbox with MS Excel A subsystem mask basically represents a workspace for a given Simulink subsystem note a mask also has other functionalities that are not presented here in which the user can define the parameters This is similar to the ordinary Matlab workspace the only difference being the manner in which the user defines the parameters and how these are available in the model In contrast to the Matlab workspace parameters defined in a masked subsystem are only available to blocks contained within such a subsystem In this way it is possible to construct hierarchical structures when building a model and even use parameters with the same names but different values depending on their location in the model To define a parameter the user must right click a masked subsystem in the model and then select edit mask This opens a dialog window where the user should select the Parameters tab In this view the user can add remove and edit the order of the list of defined parameters Each parameter has a prompt and a variable which needs to be defined The prompt is the text that will appear when the user left double clicks on the masked subsystem after the parameter has been defined The variable contains a variable name that is used by the model For instance if a parameter has assigned the variable kd the value for kd is available
22. h is being fed the product of its output i e the states with a matrix of size NXN where N is the number of required discretisations The matrix is set up to represent a one dimensional and sequential transport between the discretisation nodes states Both forward and backward transport is allowed as well as specifying initial conditions for any of the states Furthermore the block allows for multiple radionuclides including the calculations for decay and ingrowth in the same manner as is performed in the Compartment Block The transfer coefficients are fed as inputs to the block and can thus describe any required process affecting the overall transport for instance advection dispersion diffusion etc As for the Function Block these inputs the transfer coefficients can be time varying allowing for full time dependency Also inputs can be fed to any of the given discretisation nodes The number of discretisations is changed by entering the required number in the blocks dialog window 2 3 2 D Discretisation Block The original version of the Discretisation Block was for 1 dimensional transport only To be able to model systems with 2 dimensions such as for example water transport in a rock fracture with matrix diffusion the original block had to be extended Due to the fact that only 2D matrix operations are allowed in Simulink a workaround solution had to be 13 devised The solution was to write code that added a Compartment
23. ide and 4 A and 43 are the decay constants we have A N A N en Which in matrix form can be written as 4 0 OT TN AN dN io Pr A Ay 0 x N 3 A N AN 1 2 0 Ay A N ANY 4 N If the unit used in the model is activity 4 AN Bq we can multiply both sides of equation 1 1 with decay constants to obtain dN dA Zh SA AN aA a A AN ee IN AT RAN AN amp Ay dad 1 3 dN dA jy AG A haN AN g aaah Which now is equivalent to dA 4 0 0 A A A ZA 4 0 1x14 44 44 1 4 dt 0 A A A A A A As In the equations above the vectors N and A are the state vectors of any individual compartment i e compartment inventories In the example a decay chain consisting of three nuclides is used but the same principle holds for any number and mix of nuclides and decay chains used in a model 1 2 4 Transfer Function Block The Transfer Function block is what is used to make a connection representing transfer of radionuclides between any two Compartments or to be used as a sink accounting for losses from the system The Transfer Function block is basically a Function Block but with added functionality in that it automatically identifies what block is the so called donor Compartment i e the Compartment from which the radionuclides are to be removed by the rate governed by the Transfer Function Block This is accomplished via the use of a pair of matching Goto and From b
24. il to Bj rn Dverstorp and Bo Str mberg dated 19 3 07 on the subject of input files for SR Can calculations Hollenbeck K J 1998 INVLAP M A Matlab function for numerical inversion of Laplace transforms by De Hoog Algorithm http www isva dtu dk staff karl invlap htm Johansson H Jonsson K Forsman K J W6rman A 2001 Retention of conservative and sorptive solutes in streams simultaneous tracer experiment The Science of the Total Environment 266 1 3 229 238 Lindgren M and Lindstr m F 1999 SR 97 radionuclide transport calculations SKB Report TR 99 23 Svensk Karnbranslehantering AB Maul P R and Robinson P C 2002 Exploration of important issues for the safety of SFR using performance assessment calculations SKI Report 02 62 Statens Karnkraftinspektion SKI Sweden Maul P Robinson P Avila R Broed R Pereira A and Xu S 2003 AMBER and Ecolego intercomparisons using calculations from SR 97 SKI report 2003 28 SSI report 2003 11 28 Norman S and Kjellbert N 1990 FARF31 A far field radionuclide migration code for use with the PROPER package SKB TR 90 01 Svensk K rnbr nslehantering AB SKB 2006 Long term safety for KBS 3 repositories at Forsmark and Laxemar a first evaluation Main report of the SR Can project SKB TR 06 09 Svensk K rnbr nslehantering AB Worman A 1998 Analytical solution and timescale for transport of reactive solutes in r
25. ivers and streams Water Resources Research 34 10 2703 2716 Worman A 2000 Comparison of models for transient storage of solutes in small streams Water Resources Research 36 2 455 468 Worman A Packman A I Johansson H and Jonsson K 2002 Effect of flow induced exchange in hyporheic zones on longitudinal transport of solutes in streams and rivers Water Resources Research Vol 38 NO 1 10 1029 2001WR000769 Xu S W6rman A and Dverstorp B 2007 Criteria for resolution scales and parameterisation of compartmental models of hydrological and ecological mass flow in watersheds Journal of Hydrology 335 364 373 29 SSI rapporter 2008 SSI reports 2008 2008 01 Myndigheternas granskning av SKB s pre limin ra s kerhetsbed mningar f r Forsmark och Laxemar Avdelningen f r k rnteknik och avfall och SKI Maria Nord n ivind Toverud Petra Wallberg Bo Str mberg Anders Wiebert Bj rn Dverstorp Fritz Kaut sky Eva Simic och Shulan Xu 90 SEK 2008 02 Patientstr ldoser vid r ntgendiagnostik i Sverige 1999 och 2006 Avdelningen for personal och patientstralskydd Wolfram Leitz och Anja Alm n 110 SEK 2008 03 Radiologiska unders kningar i Sverige under 2005 Avdelningen for personal och patientstralskydd Anja Alm n Sven Richter och Wolfram Leitz 110 SEK 2008 04 SKI s och SSI s gemensamma granskning av SKB s Sakerhetsrapport SR Can Gransknings rapport Avdelningen for karnteknik och av
26. locks and a callback function that is executed whenever the connection is changed on the Transfer Function Block Several Transfer Function Blocks can be connected to the same donor Compartment Further the block allows for direct input of mathematical expressions describing the transfer rate in either Bq Yr or Mole Yr depending on the selected unit in the Radionuclide Block However these functions all assume that any parameter in the expression is a constant To use time varying parameters the user has to select the Show External Rate Port checkbox in the blocks dialog window Fig 1 5 When this is Function Block Parameters TF X m TransferFunction mask Transfer Function m Parameters Expression a b Connected to Compartment et _ MV Show the expression I Show external rate port Cancel Help Apply Fig 1 5 Example of Transfer Function block dialog window selected a new port is made available on the block where the user can input the output of a Function Block or any other required signal representing the transfer rate Note that the expressions are not automatically assumed to be proportional to the donor Compartments value so the donor Compartments value has to be used in the expression if this is the goal Example The expression a b entered in the Transfer Function block or fed as an input as described above would mean that the amount of
27. m and the relationship of X u and D u t Pe Norman and Kjellbert 1990 Table 3 2 Distribution and diffusivity coefficients used in calculations Hedin 2007 Mo 36C 13508 1297 SN Ka kg m 1x10 0 4 2x107 0 1x10 D m y 8 138x10 1 356107 1 424x10 5 629x10 4 611x107 10 C 14 T Cl 36 Near field release Bo y 10 Time years Fig 3 2 Calculated near field releases from pathway Q1 for the deterministic pin hole failure case which are used as boundary condition for far field release calculations 19 Function Block Parameters TF5 2 Function Block Parameters Example Model 8 1378e 8 1 3563e 7 6 7815e 7 5 6287e 8 1 4241e 6 4 6114e 7 6 7815e 7 0 005 0 005 0 005 0 005 0 005 0 005 0 001 0 0 001 0 0 042 0 01 1 Fig 3 3 Implementation of compartmental model representing far field transport in the discretisation Block a an example of transfer rate Eq 3 10 is filled in Function block window b parameter values are filled in Masked subsystem dialog window c the number of compartments in the x and y directions is filled in Masked subsystem dialog window 20 n 5 n 10 4 10 n 20 4 n 40 il gt n 60 IT i o 10 4 r g N 3 Doa 105 4 5 x i Vy 10 Y J 10 3 a 1 5 10 10 10 Time yea
28. n the expression is added as an input to the Function Block Each such input can be time varying thus allowing for fully time dependent functions El Function Block Parameters TF5 2 X m FunctionBlock mask Function Block m Parameters Expression E D _e porosity R_m Comment Diffusion in rock matrix Old expression 2 D_e porosity R_m Fig 1 3 Example of Function block dialog window mathematical expressions can be typed in the Expression field 1 2 3 Radionuclide Manager Block The Radionuclide Block contains information about the radionuclides selected in the model It also contains information about their respective halflives and about any eventual decay chains The Radionuclide Block is required in all models making use of the Compartment or Conversion blocks in a model All information can be accessed and edited directly in the block itself Fig 1 4 but a special GUI Graphical User Interface has been written that allows for simplified graphical editing of the relevant data The Radionuclide Block creates an object known as the Decaymatrix which is a matrix containing all decay associated data for all selected radionuclides in the model This matrix is used within each Compartment to calculate decay and ingrowth for each radionuclide It is also possible to select the fundamental unit for the quantity of radionuclide whether Bq or Mole in the Radionuclide Block
29. ncentration time distributions were obtained at eight stations along a distance of 30 km The input data used in this application are based on the distance between station C and D The data are shown in Table 3 4 in which some parameter values are obtained from model fitting W rman et al 2002 Following a similar procedure to that described in the previous section the model was implemented in the Discretisation Block First we tuned the number of compartments in the x direction Fig 3 8 shows the breakthrough curve converges when the number of compartments is large than 200 Then we tuned the number of compartments in sediments z direction The number of compartments in z direction is not sensitive either the depth of the sediment on the model predictions in this case The reason for this might be the residence time of radionuclide in the studied domain is much shorter than the residence time of radionuclide in the sediment therefore the discretisation of sediment has no effect on the model response Finally the lumped parameter TRwat sea Was obtained by fitting the simulated breakthrough curve with experimental data when 250 compartments are used Fig 3 8 The calibrated lumped parameter value of TRwat sea 18 0 033 hour which is a factor of 0 55 of the value used for ASP model W6rman et al 2002 The reason for this is that in ASP model mass flux from water into sediment is integrated over the distribution of transport pathways
30. nducts inspections and investigations and can stop hazardous activities Nuclear power SSI requires that nuclear power plants should have adequate radiation protection for the generalpublic employees and the environment SSI also checks compliance with these requirements on a continuous basis Waste SSI works to ensure that all radioactive waste is managed in a manner that is safe from the standpoint of radiation protection Mobile telephony Mobile telephones and base stations emit electromagnetic fields SSI is monitoring developments and research in mobile telephony and associated health risks Transport SSI is involved in work in Sweden and abroad to ensure the safe transportation of radioactive substances used in the health care sector industrial radiation sources and spent nuclear fuel Environment A safe radiation environment is one of the 15 environmental quality objectives that the Swedish parliament has decided must be met in order to achieve an ecologically sustainable development in society SSI is responsible for ensuring that this objective is reached Biofuel Biofuel from trees which contains for example from the Chernobyl accident is an issue where SSI is currently conducting research and formulating regulations Cosmic radiation Airline flight crews can be exposed to high levels of cosmic radiation SSI participates in joint international projects to identify the occupational exposure within this job category
31. rimeter R The retardation factor in bed sediment 20 000 for Cr Z The penetration depth in the bed m 0 4 sediment Niv Number of compartments in stream 250 Nsed Number of compartments in sediments 4 26 600 r r r Station C 500 2 400 ma 4 300 a i Station D 1 200 Concentration cpm g sample 100 0 10 20 30 40 50 60 70 Time hours Fig 3 9 Measured concentration time distribution for Cr at station D marked with 0 in Sava Brook experiment Johansson et al 2001 and predicted curve solid line at station D using compartmental river model with 250 compartments and TRwat sea aS 0 033 hour 27 References Avila R Broed R and Pereira A 2003 Ecolego a Toolbox for radioecological risk assessments International conference on protection of the environment from the effects of ionising radiation 6 10 October 2003 Stockholm Sweden Bencala K E 1983 Simulation of solute transport in a mountain pool and riffle stream with kinetic mass transfer model for sorption Water Resource Research 19 3 732 738 De Hoog F R Knight J H and Stokes A N 1982 An improved method for numerical inversion of Laplace transforms J Sci Stat Compt 3 357 366 Elliott A H and Brooks N H 1997 Transfer of nonsorbing solutes to a streambed with bed forms Theory Water Resource Research 33 1 123 136 Hedin A 2007 E ma
32. rs Fig 3 4 Comparison of calculated far field release by using different number of compartments in the x direction representing mobile liquid in compartment model The number of compartments in the y direction representing rock matrix is set to be constant 6 compartments in this case 10 F n 4 4 n 6 10 F n 8 J n 10 q n 12 cy 3 f 1071 3 oO g g E 2 gt 3 10 A 3 A LL AAA re 1 pf 10 j 3 10 3 i g i Ei 5 10 10 10 Time years Fig 3 5 Comparison of calculated far field release by using different number of compartments in y direction representing rock matrix in compartment model The number of compartments in x direction representing mobile liquid is set to be constant 60 compartments in this case 21 36C 5947 Ni Far field release Boy 3 Time years Fig 3 6 Comparison of the semi analytical solid line and compartment circles models for far field release 3 2 Radionuclide transport in streams 3 2 1 Description of the stream model Radionuclide transport in streams can be described by processes such as advection dispersion and exchange with hyporheic zones as well as adsorption e g Bencala and Walters 1983 Elliott and Brooks 1997 Over the last two decades different models describing transport processes in streams have been developed such as the first order mass transfer model FOT model the impermeabl
33. s the difference between the total influx and total outflux over time In addition to this the radioactive decay including ingrowth of possible daughter nuclides is calculated inside the Compartment The compartment has an Initial Condition setting that by default is zero If the user wishes to have an initial condition other than this it is possible to select Display Initial Condition Port on Compartment window see Fig 1 2 The Compartment window will be displayed by double clicking on the Compartment symbol This adds an additional input port to the Compartment block and whatever value s are input here becomes the new initial conditions for the Compartment Radionuclides Compartmen aa gt transfer b Function Block Fig 1 1 Symbols of various Blocks in Ecolego Toolbox library 7 Function Block Parameters Compartment Fig 1 2 Example of Compartment block dialog window 1 2 2 Function Block The Function Block is a higher level maths expression block compared to the built in maths blocks in Simulink It allows complex mathematical expressions to be entered in ordinary Matlab syntax on the Function Block dialog window shown in Fig 1 3 which is displayed when double clicking the Function Block The expressions then are parsed and built as a set of Simulink low level math operations blocks representing the mathematical expression Each variable and or parameter used i
34. sssrrsssrersrsrerersrernrsrrsrnernrsnrernnrnrernr nen nr nnna 15 Radionuclide transport in far field oo ccc ccccsssceteceteceseceeeceeeceeeceeeeeeseeeaeeeseeeneeesaes 15 A dual porosity model for radionuclide trafnSpOrt sessssrsessresrsrsrrsrererresrsrerrnrenn 15 The Representation of radionuclide transport by compartment model 16 Implementation of compartment model in the Block and comparison of the results with the semi analytical SOlUtiONS sssssssssrsesserrserrerrrsreneernrsrrernrsrernrrnrsnr ennen enn r ena 18 Radionuclide transport in StrealMS ssseeseersssrsrsrererrrerrsreerrsrrernrsrerrrrrrrne rn rs nr enn r ena 22 Description of the stream MOCC smssrsssssrsesserserrerrerrerrrnrernrerrsrrer renen nr rn sne rn nere ennen ena 22 Derivation of compartment model for description of radionuclide transport in STIG ANS sS e 2598 4c E a STADENS SKEN Toncda leu E A EAT A A NER 23 Model implementation and disCretisatiOl sssssessersssrerersrsrrrsrerrsrnrenerrrsnrennrnr ena 25 ii 1 Ecolego Toolbox 1 1 Introduction Ecolego Toolbox is a set of Simulink blocks created to facilitate the creation and modelling of compartment based systems in the Simulink environment Specifically the blocks has been designed to be used in the field of radionuclide transport modelling The name Ecolego Toolbox comes from the fact that the underlying principle for how the blocks function is the same as that of th
35. str lskydd f r m nniskan och milj n nu och I framtiden SSI s tter gr nser f r str ldoser till allm nheten och f r dem som arbetar med str lning utf rdar f reskrifter och kontrollerar att de efterlevs SSI h ller beredskap dygnet runt mot olyckor med str lning Myndigheten informerar utbildar och utf rdar r d och rekom mendationer samt st der och utv rderar forskning SSI bedriver ven internationellt utvecklingssamarbete Myndigheten som sorterar under Milj departementet har I 10 anst llda och r bel gen i Solna THE SWEDISH RADIATION PROTECTION AUTHORITY ssi is a central regulatory authority charged with promoting effective radiation protection for people and the environment today and in the future SSI sets limits on radiation doses to the public and to those that work with radiation SSI has staff on standby round the clock to respond to radiation accidents Other roles include information education issuing advice and recommendations and funding and evaluating research SSI is also involved in international development cooperation SSI with 110 employees located at Solna near Stockholm reports to the Ministry of Environment SN Statens str lskyddsinstitut Swedish Radiation Protection Authority Adress Statens stralskyddsinstitut S 171 16 Stockholm Bes ksadress Solna strandvag 96 Telefon 08 729 71 00 Fax 08 729 71 08 Address Swedish Radiation Protection Authority SE I71
36. ta to the problem are shown in Table 3 1 and 3 2 The same parameter values were used in both the dual porosity model and the compartment model calculations except the numbers of compartments which are only used for the compartment model The dual porosity model is solved by means of Laplace transforming of 3 1 and 3 2 Maul et al 2003 Transformation back to the real domain is performed numerically by means of the series expansion algorithm of De Hoog et al 1982 implemented in a Matlab code developed by Hollenbeck 1998 Implementation of the compartment model in the Discretisation Block simply requires that the expressions for the transfer rates in the Function block dialog windows are filled in see Fig 3 3a together with the parameter values and the number of compartments required in both the x and y directions using the Masked subsystem dialog windows see Fig 3 3b and 3 3c respectively The numbers of compartments for both x and y directions in the compartment model were tuned until the solution convergence This was done in two steps Firstly we kept number of compartments in the y direction constant and tuned number of compartments in the x direction until the solution no longer changed with the number of compartments Secondly we set number of compartments in x direction as obtained in the first step and tuned the number of compartments in y direction until the solution became stable The simulated breakthrough c
37. treamlines denoted by V c T J V r T r JD is the probability density function PDF of T weighted by the velocity _ and exfiltration velocity out of the bed in the direction of the streamlines by component normal to the bed surface V T is the total residence time from inlet to exit of hyporheic flow path s 7 is the exfiltration residence time s 0 lt 7r lt T P is the wetted perimeter m A is the cross sectional area of the stream m and is an area reduction factor equal to V Vz that accounts for the fact that the streamlines are not necessarily always perpendicular to the bed surface 3 2 2 Derivation of compartment model for description of radionuclide transport in streams Similar to the APS model the mass balance for the compartments in the stream and the sediment based on the conceptual description in Fig 3 7 can be written as STENS EN cay fe a Lae ype Le ee Ie dt els L n L n oe 2 Z na R adv TRais Ti Rais TReea _ wat 3 16 dmo g ao Spo oN Ma gv My OV M Am dt 2 Z Nea R Zna Ria 2 Z n R 2h i TRup down TReown up TR ed wai TR at sed 3 17 Where M is the total inventory in stream compartment i Bq or kg m is the total inventory in sediment compartment j Bq or kg U is the advective velocity D is dispersion coefficient Vz is the infiltration velocity 5 is the area reduction factor as described previously for ASP model h is the depth of the riv
38. urve remains unchanged when the number of compartments in the x direction was tuned to be more than 40 while the number of compartments in the y direction was kept constant see Fig 3 4 The length of the compartments in both directions was equally divided We used 60 compartments for the discretisation of rock matrix in the x direction and then tuned the number of compartments in the y direction As can be seen from Fig 3 5 when the number of compartments is more than 12 the solution converges i e the breakthrough curves obtained with 10 and 12 compartments are almost overlapped Fig 3 6 shows the far field release fluxes calculated from both models for five radionuclides in pin hole failure case It can be seen that the agreement between two model solutions is excellent 18 Table 3 1 Parameter values used in far field transport calculation Hedin 2007 Symbols Definitions Units Values xu length of transport domain m 500 ul velocity of the mobile liquid my 12 5 D longitudinal dispersion coefficient my 625 Ay half width of fracture m 1x10 p bulk density of rock matrix kg m 2700 0 matrix porosity 0 001 Z the maximum penetration depth m 0 03 H In Hedin 2007 the values of the transport time t and Peclet Pe number are given as t 40 y and Pe 10 t and Pe have been interpreted into corresponding parameters u and D in our calculation based on assuming X 500
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