Expanding Grain Model [EGM v1.0] - Illinois Institute of Technology
Contents
1. Time Weight Alpha Maximum E Alpha Range will be from Alpha Minimum to Alpha Maximum with an increment of 1 For example if Alpha Minimum 1 and Alpha Maximum 5 Alpha Range 1 2 3 4 5 Product Layer Diffusivity Range Product Layer Diffusivity cm 2 min Minimum Product Layer Diffusivity cm 2 min Maximum Number of Points Product Layer Diffusivity Range will be the Number of Points evenly spaced between Product Layer Diffusivity Minimum and Product Layer Diffusivity Maximum For example if Minimum 3 Maximum 5 and Number of Points 5 then Range 3 3 5 4 4 5 5 EES EES Figure 44 0 Optimization Parameters Window 4 The Optimization Parameters fields will be active only if an experimental plot is selected first 5 This is what the window looks like after selecting an experimental plot first 37 Expanding Grain Model EGM v1 0 Alpha Range Weights to calculate SSE Alpha Minimum i Weight Alpha Maximum Alpha Range will be from Alpha Minimum to Alpha Maximum with an increment of 1 For example if Alpha Minimum 1 and Alpha Maximum 5 Alpha Range 1 2 3 4 5 Product Layer Diffusivity Range Product Layer Diffusivity cm 2 min Minimum Product Layer Diffusivity cm 2 min Maximum Number of Points Product Layer Diffusivity Range will be the Number of Points evenly spaced between Pr2. 2 Figure 6 0 Toolbar General Tools 1 New Start a new case 2 Open Open a saved case 3 Save Save the current case 4 Close Close the current case Plot Tools 1 Zoom In Zoom in on the plot 2 Zoom Out Zoom out on the plot am 3 Pan Move around the plot A 4 Data Cursor View the values of the selected data point Data Tools 1 z Copy Plot Copies the plot to the memory so that it can be pasted in the desired destina tion E Import and Plot Import the experimental data from MATLAB data file mat or MS Ex cel file xlsx or xis and plot the experimental data 3 P Chemical Properties Shows the list of all the chemical data available in the EGM Expanding Grain Model EGM v1 0 Simulation Tools 1 gt Run Simulation Run the current case after all the parameters have been entered cor rectly 2 Stop Simulation Stop a running simulation or a running optimization 3 Re Run Optimization Run the current case for optimization with respect to a selected ex perimental data set 4 nu Optimization Parameters Enter the parameters for the optimization of the current case 5 e Optimization Results Show the optimization results after successful completion of an optimization run Help Tools 1 L About EGM Displays the information about the EGM version M 2 Disclaimer Displays the disclaimer window 3 User Manual Opens the user manual as a pdf fil
3. 6 41 56 Expanding Grain Model EGM v1 0 Initial value calculations Calculate X ry Cplr Calculate r R Calculate Xp Store Values Increment time Figure 6 4 Flowsheet of the Numerical Solution of the Expanding Grain Model 57 Expanding Grain Model EGM v1 0 Figure 6 5 Schematic of the Grid on Particle Radius The vector of the right hand side b bi ba bn 1 contains the elements bi 0 for i 1 2 N Dn 1 Cb The matrix A is tridiagonal matrix and the set 6 41 is solved by the elimination method of Gauss Having found the solution for equation 6 37 the first order differential equation 6 14 was solved using the Runge Kutta method which was incorporated in the C code The equation for the overall conversion was solved numerically using the trapezoidal rule Details of different methods of the Runge Kutta and the Trapezoidal rule can be found in Chapra and Canale 2002 It must be men tioned here that the quantity dD JOR in the equation 6 29 was determined numerically by using the finite divided difference formulae Chapra and Canale 2002 Of the several parameters or coefficients in the equations of the model describing the process of diffusion and reaction within the particle the only adjustable parameters are Dg and a The model was solved numerically using regression analysis to calculate the parameters Dg and a The analysis involved mini
4. A Initial slope of the reaction 20764 Get Slope Axis TIGHT Axis Limits mu y Min 100 o 50 Gas Conversion 0 Time min Parameters from the file have been loaded Figure 23 0 EGM vvith data loaded 8 The Parameters Display section shows the loaded data 22 Expanding Grain Model EGM v1 0 750 C 1 atm 362 5000 microns 0 5650 136130 cm 2 g 1 0000e 07 cm 2 min 3 2 0764 50 min 0 5000 min Number of Grid Points 150 Over Relaxation Factor 1 8000 Error Tolerance 5 0000e 09 Compound Stoic Coeff CaO so2 02 CaS04 MgO N2 Figure 24 0 Parameters Display Panel showing the loaded data 9 Opening any of the parameters window shows the currently loaded data that can be modi fied if needed Temperature Pressure Figure 25 0 Operating Parameters Window with loaded data 23 Expanding Grain Model EGM v1 0 Running a simulation Once all the parameters are entered correctly into the EGM it is ready for simulation You can see if the case is ready to be simulated or not by looking at the status bar message 1 If the case if ready you will see Ready Not ready Please enter all the parameters 2 Ifthe case is not ready you will see If the case is ready select 1 Simulation gt Run Simulation 2 Or click the Run Simulation tool You will see that the simulation runs whi
5. Import and Plot Import the experimental data from MATLAB data file mat or MS Excel file xlsx or xis and plot the experimental data c Chemical Properties Shows the list of all the chemical data available in the EGM d Convert Excel data to mat Converts the experimental data in excel file to mat file 5 Expanding Grain Model EGM v1 0 3 Simulation Menu File Data Help a Gd Run Simulation p Stop Simulation Run Optimization Optimization Parameters Optimization Results Figure 4 0 Simulation Menu a Run Simulation Run the current case after all the parameters have been entered correctly b Stop Simulation Stop a running simulation or a running optimization c Run Optimization Run the current case for optimization with respect to a selected experimental data set d Optimization Parameters Enter the parameters for the optimization of the current case e Optimization Results Show the optimization results after successful completion of an optimization run 4 Help Menu File Data Simulation DUGRX RR About EGM Disclaimer User Manual Figure 5 0 Help Menu a About EGM Displays the information about the EGM version b Disclaimer Displays the disclaimer vvindovv c User Manual Opens the user manual as a pdf file Note Adobe Reader is required to open the user manual file Expanding Grain Model EGM v1 0 Expanding Grain Model EGM v1 0 Understanding the Toolbar OGax ASO2 AEP awrite i
6. Simulation Time Time Step Number of Grid Points Over Relaxation Factor Error Tolerance OK CANCEL Figure 17 0 Simulation Parameters Window 1 Simulation Time Enter the total time in minutes for the simulation to run Range is from O minute to 1500 minutes 2 Time Step Enter the steps to take between the total simulation time Range is from O mi nute to 5 minutes 3 Number of Grid Points Enter the total number of layers of the sorbent particle to solve for Range is from 1 to 1500 4 Over Relaxation Factor Enter a value between 1 and 2 This will help converge the numeri cal solution faster 5 Error Tolerance Enter the desired tolerance for the error Range is from 1E 8 and 1E 4 18 Expanding Grain Model EGM v1 0 Units The units are fixed in this version There are no other units to select from the units drop down list The dropdown lists may contain multiple units in future versions 19 Expanding Grain Model EGM v1 0 Error Flags 1 Non numeric Input If a proper number is not entered an error text appears in the window as follows Temperature Pressure Figure 18 0 Non numeric input error 2 If a number outside of the range is entered an error text appears in the window as follows Temperature c y Pressure atm y Enter a value between 273 15 and 1500 O conce Figure 19 0 Outside range error
7. 1976 This type of diffusion is commonly known as the solid state diffusion or the intra 54 Expanding Grain Model EGM v1 0 grain diffusion and is expected to depend on the product layer compositions and the reaction tem perature Solution Technique The developed equations of the expanding grain model were solved by finite differences technique using a computer program developed in the C language and compiled using the visual C compiler To obtain the conversion versus time behavior from the expanding grain model equations 6 12 and 6 9 were solved sequentially with their respective boundary condi tions The overall conversion as function of time was calculated from equation 6 14 using r as a function of time obtained by solving the two differential equations The quantity CR is considered as a function of the variable R and the parameter t while the quantity r is taken as the function of the variable f and the parameter R At any given instant t the function ri R t was assumed to be known for values of the parameter Re 0 Ro This assumption is met automatically at t 0 according to the initial condition 6 9a It is possible then from equation 6 12 to determine at a given 1 a profile of the function Cr for RE 0 Ro Having solved the equation 6 12 integration of equation 6 9 from time t to t dt is carried out for all values of the parameter Re 0 Ro The time step dt is so chosen that can be in the inte
8. 2013 2 10 PM 5 27 2013 3 31 PM 6 14 2013 11 06 AM 6 5 2013 11 58 PM 6 14 2013 11 08 AM 6 12 2013 12 32 AM File folder File folder Microsoft Access Microsoft Access Microsoft Access Microsoft Access Microsoft Access File name vl MAT files mat v Figure 21 0 Open dialog box EC 21 5 Selecta file and click Open Expanding Grain Model EGM v1 0 6 If you select an incorrect file the following error message appears Click OK to close it Incorrect File Please select the correct file to load Figure 22 0 File Error message 7 If you select a correct file the data will be loaded to the EGM application and it looks like this File Data Simulation Help DGAUXIN EI AE Pio awe at 2 Operating Parameters Sorbent Parameters Reactions Simulation Parameters Parameters Display Parameter Value Units Plot Controls C Grid ON OFF Temperature Pressure Average Particle Diameter Sorbent initial Porosity BET Surface Area Product Layer Diffusivity Alpha Slope Simulation Time Time Step Number of Grid Points Over Relaxation Factor Error Tolerance Reaction Phase Reactant s Gas Gas Product s Inert s Inert 9 750C 1 atm 362 5000 microns 0 5650 136130 cm 2 g 1 0000e 07 cm 2 min 3 2 0764 50 min 0 5000 min 150 1 8000 5 0000e 09 Compound Stoic Coeff Cao 1 s02 A 02 05 Casos 1 Hgo o N2 o
9. 6 18 in equation 6 15 we get 2 DC 1 9p RG afir art SCR 0 6 19 R R OR P 7 D HE l Y I With boundary conditions 47 Expanding Grain Model EGM v1 0 a di D at R 0 b Cr Cs at R Ro Where Cp Bulk reactant gas concentration seen by the solid reactant mol cc Ro Radius of the reacting solid particle cm Equations 6 19 and 6 14 are solved numerically to obtain the radii of the reacting interface ri Using these values of ri the local X and the overall particle conversions Xp can be determined with the following equations X 56 6 20 Fs 3 R 3 RS pr SE dR 6 21 8 Xp Structural Changes in the Solid during Reaction In the expanding grain model the radius of the grain rg is expected to change due to the differences in the molar volume of the product e g CuSO4 Molar Volume 44 37 cc mol and the reactant e g CuO Molar Volume 12 33 cc mol A schematic representation of the above theory is given in Figure 6 2 r initial grain radius r radius of reacting interface r radius of the expanding grain 48 Expanding Grain Model EGM v1 0 Figure 6 2 Schematic Diagram of a Partially Reacted Grain Based on the unreacted shrinking core mechanism between the gas and the grains of copper oxide a material balance for the solids leads to the following relation 4 i roduc Si ea EE P product Hise JS de r Present 6 22 3 MW oduct MW sactant The ch
10. Alpha 3 Slope 20764 Simulation Time 50 min Time Step 0 5000 min Number of Grid Points 150 Over Relaxation Factor 1 8000 Error Tolerance 5 0000e 09 Reaction Phase Compound Stoic Coeff Reactant s Cao A 502 A Gas 02 05 1 0 N2 0 Gas Conversion i Initial slope of the reaction Get Slope Time min Simulation Complete Figure 27 0 Simulation Complete If you observe the Plot Selection section of the Plot Controls panel you will see that Plot1 has ap peared 25 Expanding Grain Model EGM v1 0 Using Plot Controls In this section you will learn the functions of the Plot Controls and the Plot tools Shown below is the Plot Controls panel Plot Controls Grid ON OFF Axis TIGHT Axis Limits Reset Plot Selection Color Palette Figure 28 0 Plot Controls Panel 1 Grid ON OFF Turns the grids on or off in the plot Gas Conversion 25 Time min Figure 29 0 Plot with Grids ON 26 Expanding Grain Model EGM v1 0 2 Axis TIGHT Tightly fits the axis around the plot a Oy AL a T T T T T T T T T 25 Gas Conversion 0 l l L L 1 f I L L 30 35 40 45 50 25 Time min Figure 30 0 Plot vvith Axis TIGHT Axis Limits The Reset button sets the axis to the default values described below yMin Minimum value of the y axis Default is 0 yMax Ma
11. such varia tions of De have been proposed by VVen 1968 Calvelo and Cunnigham 1970 and Fan et al 1977 For instance the relation used by Fan et al 1977 is of the follovving form D Do a 6 33 exp TE 07 Where Qj empirical constants B Bo Concentrations of the gaseous reactant at a local position in the solid particle and in the bulk respectively In this study the mathematical function to describe t was selected to be of the form exp o Xp The dimensionless parameter a in this study is termed as the tortuosity parameter and is a relative measure of the expected degree of tortuosity that the gas encounters while diffusing through the interior of the sorbent particle A relatively higher value of a indicates that the structural changes 53 Expanding Grain Model EGM v1 0 accompanying the sulfation reaction gives rise to a highly compact pore structure resulting in a com paratively more tortuous path and a higher probability of the occurrence of pore closure A similar functional form also used by Shaaban 1991 and Karnik 2004 while modeling the sulfation of cal cined limestone Thus the variation in the effective diffusivity with the progress of the sulfation reac tion is given as 1 1 D expl aX p 6 34 It is clearly evident from equation 6 34 that at the start of the reaction reaction time 0 the value of the exponential term drops to one and equation 6 34 red
12. 0 DGOAXIR SO RI A Pia mki Operating Parameters Sorbent Parameters Reactions Simulation Parameters Parameters Display Plot Controls Parameter Value Units C Gris oworr Temperature 750C a Pressure tam Average Particle Diameter 362 5000 microns Sorbent initial Porosity 0 5650 BET Surface Area 136130 cm2 g Product Layer Diffusivity 1 0000e 07 cm 2 min lAlpha 3 Slope 2 0764 Simulation Time 50 min Time Step 0 5000 min Number of Grid Points 150 Jover Relaxation Factor 1 8000 Error Tolerance 5 0000e 09 Reaction Phase Compound Stoic Coeff Reactant s Cao A p s02 A Gas 02 05 Gas Conversion Product s 1 inerts Mgo hnert g N2 Select Select lt Reaction Kinetics Initial slope of the reaction Simulation Stopped 25 Time min Figure 35 0 Simulation Stopped A stopped simulation generates the partially completed plot and also displays the plot name in the Plot Selection list box It can be manipulated exactly the same way as a completely simulated plot 30 Expanding Grain Model EGM v1 0 Importing and Plotting Experimental Data Experimental data can be imported from Excel files xlsx xls or MATLAB data files mat 1 Click Data gt Import and Plot 2 Or you can click on the Import and Plot tool 3 The following dialog box will appear 4 B gt JayaSingh SkyDrive Summer 2013 EGM
13. 4 ri Radius of the reaction interface cm Substituting Equation 6 4 in Equation 6 3 we get 1 dN kC 6 5 4a dt For a spherical grain we have N 3 s1 6 6 No r Where Nso Original number of moles of the solid reactant 44 Expanding Grain Model EGM v1 0 Equation 6 6 in its differential form may be written as 2 DES da 6 7 3 1 8 Substituting Equation 6 7 in 6 5 and simplifying we get Adi 6 8 4a dt 3 The advancement of the reaction front is now written as SDA LIA gp 6 9 Where Cso Original Molar Concentration of CuO in the grain mol cc N Cso T 2 47 From the unreacted shrinking core model the concentration profile of the A in the product layer of the grain the following equation can be derived using ld 4 06 6 10 DE di dr 9 l With Dg Effective Diffusivity through the product layer on the grain cm min Cg The concentration of A within product shell of grain mol cc r The radial co ordinate within product shell of grain cm With boundary conditions C Cpatr re 6 11 45 Expanding Grain Model EGM v1 0 de C dr VVith rg The radius of the expanded grain cm The derivation of the expression to calculate the radius of the expanded grain rg will be discussed later When equation 6 10 is integrated with the boundary conditions given in equations 6 11 and 6 12 the following expressi
14. Besides these two common error flags other error flags will also show up in order to help you with the task of correct data entry A lot of them exist in the Reactions window where error flags are shown if for e g an incorrect reaction stoichiometry is entered etc 20 Opening a Case You can open a previously saved case as follows 1 Click File gt Open 2 Alternatively you can click the Open tool Expanding Grain Model EGM v1 0 Note If a case hasn t been opened yet you will see the Open dialog box If you have started a new case or are already working on a case clicking on the Open menu or tool generates the following quest dialog box A file is in progress Would you really like to open a new file WARNING If you select YES all the unsaved data will be lost Figure 20 0 Open quest dialog box 3 Select YES if you want to proceed else select NO 4 The Open dialog box looks like this t b gt Jaya Singh SkyDrive Summer 2013 EGM New vo Organize v New folder E Documents A Name A Music Date modified Search EGM New Type Excel Data Files 6 14 2013 10 46 AM File folder ls Pictures E Videos O images mat Data Files chemicalProp Gi expPlotData GA sample Gi sampleWithOptimResults wSSEdata 28 Homegroup Computer s Local Disk C ca Backup D amp UBUNTU U Es PDGPL 1216 47 Ga Network y 6 18 2013 1 05 AM 6 6
15. EGM i ar 40 DISCIAIMEr TR Rent 41 Expanding Grain Model Theory iii 42 Expanding Grain Model EGM v1 0 Figure Descriptions ze dita temia APN SS aN Sa oa 60 Expanding Grain Model EGM v1 0 About the Expanding Grain Model Expanding Grain Model EGM is a computer program that can estimate the sorbent capacity at dif ferent operating conditions Note See the theoretical section at the end of the manual for complete description and derivation of the model Expanding Grain Model EGM v1 0 Starting the EGM Application 1 Locate the application in its installed location and double click the EGM Application icon 2 The following window appears File Data Simulation Help DGAXIR RO RI AE Pia aww it Figure 1 0 EGM Application Startup Window Expanding Grain Model EGM v1 0 Understanding the Menu bar 1 File Menu Data Simulation Help New E Open Save Save as Excel Report Close Exit Figure 2 0 File Menu a New Start a new case b Open Open a saved case c Save Save the current case d Save as Excel Report Save the current case as Excel File e Close Close the current case f Exit Close the EGM application 2 Data Menu File Simulation Help a G Copy Plot VE P Import and Plot Chemical Properties Convert Excel data to mat Figure 3 0 Open Menu a Copy Plot Copies the plot to the memory so that it can be pasted in the desired des tination b
16. EGM v1 0 7 Once the application reads the experimental data it makes a plot and lists it in the Plot Se lection list box File Data Simulation Help SCIC x AXORB AAP gt mw Mat 2 Operating Parameters Sorbent Parameters Reactions Simulation Parameters Parameters Display Plot Controls T T Parameter it C Grid ON OFF Temperature Pressure Ll Axis TIGHT Average Particle Diameter Sorbent Initial Porosity Axis Limits BET Surface Area Product Layer Diffusivity Alpha Slope Simulation Time Time Step Number of Grid Points Over Relaxation Factor Error Tolerance Reaction Phase Stoic Coeff Select Select Select Select Select Select Select Select lt Gas Conversion Reaction Xinetics Initial slope of the reaction 25 Time min Dol SO2 650C 500ppm mat Figure 38 0 Experimental data plotted using the Import and Plot tool Note If you select an experimental plot in the Plot Selection list box you vvill see the name of the file in the status bar at the bottom Also since it is the experimental data there is nothing to dis play in the Parameters Display panel Hence it is empty as you can see above in Figure 36 0 32 Expanding Grain Model EGM v1 0 Converting Excel data file to mat Experimental data can be read from Excel files It is often useful to convert the Excel data file to mat file for the purpose of l
17. Expanding Grain Model Developed By Process Design and Gas Processing Laboratory PDGPL Department of Chemical and Biological Engineering Illinois Institute of Technology Chicago IL 60616 Website mypages iit edu abbasian AAA Expanding Grain Model EGM v1 0 Table of Contents About the Expanding Grain Model ococncccccncncccconnnnnnnnnnnnnnnnnononnnnnnnnnnnnnnonononnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnos 3 Starting the EGM lA A 4 Understanding the Menu bar iii 5 Understanding the Toolbar iii 8 Starting a NEW CASE nisi AERAZIONE TA ITER NSA TIRI TERI 10 Entering parameters into the EGM iii 11 Operating Parameters ei 12 Sorbent Parameters os iure iene EAE EEEE EEE EEE 13 REACTIONS SORET EE E E T E T A E E E E T A T E EEEE 14 Simulation PaTAMEtOLS urinaria A EAEE a A io 18 UES nui a it 19 EFFOr Fla SS PPP OOO UI O 20 Opening UC A aaa 21 Runningsa simulation asia al mei ti ms 24 Using Plot Controls iia sasisscazeccdenedsdscassseseoccacasedasassesegeteccessdaaausassnntuecsees IRNERIO TRI 26 Stopping simulation ui 30 Importing and Plotting Experimental Data ii 31 Converting Excel data file to mat iii 33 Running OptimizatiOn 3 ARRESE AA LAZ ARA RARA NASA RARA RES RTRT SERA RIA 36 Optimization Parameters iii 37 Optimization Results n 39 ADOUC
18. New v Search EGM New Organize v New folder A h y Fr Favorites Name Date modified Type MI Desktop Excel Data Files 6 20 2013 12 22 AM File folder l Downloads O images 6 18 2013 1 05 AM File folder Recent places mat Data Files 6 6 2013 2 10 PM File folder Google Drive Gi chemicalProp 5 27 2013 3 31PM Microsoft Access di SkyDrive s ChemicalProperties 5 27 2013 10 17PM Microsoft Excel W F Dropbox BA Dol SO2 650C 500ppm 6 20 2013 12 22 AM Microsoft Access al expPlotData 6 14 2013 11 06 AM Microsoft Access A Libraries sample 6 5 2013 11 58PM Microsoft Access Documents BA sampleWithOptimResults 6 14 2013 11 08 AM Microsoft Access a Music Gi wSSEdata 6 12 2013 12 32 AM Microsoft Access Pictures EI Videos v File name Data Files TEA v Figure 36 0 Select a File to Import and Plot dialog box 4 Select the file that contains the experimental data and click Open You can either select an Excel data file or a MATLAB mat file 5 Ifyou select the Excel file it takes a while for the application to read the Excel file Loading mat file is relatively faster 6 Ifyou select an Excel file or mat file other than the one that contains the data you will get the following error message Incorrect File Please select the corect data file to import and plot Figure 37 0 Incorrect File Error for Import and Plot feature 31 Expanding Grain Model
19. all will become more frequent than collisions between gas molecules A common approach taken to model such systems is to assume that the laws of molecular and Knudsen diffusion are obeyed in a porous medium and then to work in terms of an effective diffusivity This effective diffusivity which is smaller than the molecular diffusion coefficient and the Knudsen diffusion coefficient is then selected so as to incorporate the factors mentioned earlier Mason and Malinauskas 1983 Smith 1970 Numerous methods have been proposed for both the estimation of the effective diffusivity and for the representation of pore diffusion through the use of this parameter A detailed explanation and review of these can be found in the references listed above 50 Expanding Grain Model EGM v1 0 Diffusing Species A gt DR MX Concentration Cas Position Figure 6 3 Schematic Diagram of the Diffusion of a Gaseous Reactant into a Porous Solid ial effective diffusivity Deo of the gaseous reactant through the particle is described ini The ion ing equat the follow using 6 27 Where The tortuosity factor T in cm m ion Molecular Diffus Dm In ion cm m Knudsen Diffus Dx tion Szekely et al 1976 ing equa ing the follow t is calculated using the us icien The Knudsen coeff 00 Ey 51 Expanding Grain Model EGM v1 0 Where Rgas universal gas consta
20. ange in the radius of the particle at any time t can then be easily calculated using the following equation TOE 3 3 Y I lr Z de E j 6 23 It may be recalled that the expansion factor Zy vvas defined in chapter 5 as MW _ P react product 5 1 a MW er 1 Es P product The values of Zy were shown to range from 1 5 to 2 1 indicating that the grains expand swell during sulfation The porosity changes in the pellet can be related to changes in the particle size by the following equation 113 r ui 6 24 Ig r From the derivation presented it is clear that the changing porosity of the particle is related to the local conversion of the grain at a radius R by the following relationship 1 1 e 1 Z 1 x 6 25 Thus the maximum possible local conversion of the grain can be determined when e 0 and is given by the following equation Eo Xima Mie 6 26 Based on the above equation it is evident that if o is greater than Zv Di then the complete V conversion of the grain and thus of the particle is possible otherwise the phenomenon of pore 49 Expanding Grain Model EGM v1 0 closure will occur and the conversion of the solid particle will level off below the complete conver sion Modeling of the Varying Gas Diffusivities using EGM The progress of a non catalytic gas solid reaction is governed by the intrinsic chemical reaction and the diffusion of the gaseous reactant into the s
21. e Note Adobe Reader is required to open the user manual file Expanding Grain Model EGM v1 0 Starting a new case 1 Click File gt New 2 Or click the New tool 3 The following window appears File Data Simulation Help Operating Parameters Sorbent Parameters Reactions Simulation Parameters Parameters Display 400 gt Plot Controls Parameter Value Units Grid OWOFF Temperature c A Pressure atm Axis TIGHT Average Particle Diameter microns Sorbent intial Porosity Axis Limits a BET Surface Area i o Product Layer Diffusivity cm 2Imin y Min min min Number of Grid Points Over Relaxation Factor Error Tolerance Reaction Phase Gas Conversion 0 1 25 Time min Welcome Please begin by entering the parameters Figure 7 0 New Case Window 10 Expanding Grain Model EGM v1 0 Entering parameters into the EGM There are four main categories of parameters that must be entered into the EGM in order toruna simulation They are Operating Parameters Sorbent Parameters Reactions Simulation Parameters Figure 8 0 EGM Parameters Buttons More on each of these parameters categories are explained below 11 Expanding Grain Model EGM v1 0 Operating Parameters Clicking on the operating parameters button displays the following window Temperature Pressure EEES Ee Figure 9 0 Operating Parame
22. e 44 0 Figure 45 0 Figure 46 0 Figure 47 0 Expanding Grain Model EGM v1 0 Parameters Display Panel showing the loaded data Operating Parameters Window with loaded data Running Simulation Simulation Complete Plot Controls Panel Plot with Grids ON Plot with Axis TIGHT Plot selection increased line thickness Plot selection displays the relevant data Color Palette Plot color changed from blue to red Simulation Stopped Select a File to Import and Plot dialog box Incorrect File Error for Import and Plot feature Experimental data plotted using the Import and Plot tool Excel Data File Sample Select Excel data file dialog box File Error for Convert Excel data to mat feature Save dialog box for Convert Excel to mat feature Running Optimization Optimization Parameters Window Active Optimization Parameters Window Optimization Results Window About EGM 61 Figure 48 0 Disclaimer Expanding Grain Model EGM v1 0 62
23. ect an Excel data file dialog box T dd JayaSingh SkyDrive Summer 2013 EGM New v GB Search EGM New Organize v Nevy folder La VI Ft Favorites Name Date modified Type MI Desktop o Excel Data Files 20 2013 3 05 PM File folder l Downloads images 18 2013 1 05 AM File folder Recent places di mat Data Files 6 6 2013 2 10 PM File folder Google Drive ChemicalProperties 5 27 2013 10 17PM Microsoft Excel VV d SkyDrive 35 Dropbox Libraries E Documents d Music Pictures E Videos v File name Il v Excel Data File xlsx xls cone Figure 40 0 Select Excel data file dialog box 3 Select the proper Excel data file If you select an incorrect file you will see the following er ror message Incorrect File Please select the correct Excel data file Figure 41 0 File Error for Convert Excel data to mat feature 4 If you select the correct file you will see another dialog box which asks for the filename to save the data The default filename is the same as the Excel data filename 34 Expanding Grain Model EGM v1 0 6 TE JayaSingh SkyDrive Summer 2013 EGM New Y Search EGM New Organize v New folder rares Name Date modified Type EE Desktop a Excel Data Files 6 20 2013 3 18 PM File folder Downloads dl images 6 18 2013 1 05AM File folder Recent places di mat Data Files 6 6 2013 2 10 PM File folder Google Drive c
24. egal liability or responsibility for the accuracy completeness or usefulness of any results generated by the software or represents that its use would not infringe privately owned rights Reference herein to any specific commercial product process or service by trade name mark manufacturer or otherwise does not necessarily constitute or imply its endorsement recommendation or favoring by the United States Government or any agency thereof The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof CLOSE Figure 48 0 Disclaimer Window 41 Expanding Grain Model EGM v1 0 Expanding Grain Model Theory Directly taken from Vasudeo Gavaskar s Thesis The Expanding Grain Model EGM also known as the Particle Pellet model has been com monly used to describe the sulfation reaction of calcined limestone When the solid is porous in na ture the gaseous reactant diffuses into the interior of the particles and reacts with the active solid species at the surface of the interior of the pores This physical picture can be described by the grain model in which the solid pellet is visualized as consisting of a number of small non porous particles which are called grains Surrounding these grains are macro pores through which the gas has to dif fuse to reach the various grains A schematic diagram of such a porous particle along with the assem blage o
25. elect Select Select Select Select Select Load Data OK CANCEL Clear Table Help Figure 11 0 Reactions Window NIC EE EE IE ICE EL 1 Phase Select the phase of the compound that you are about to enter in the Compound col umn Available phases are Reactant s Gas Product s Inert s and Inert g 2 Compound Enter the compound name as a formula For eg CaO for Calcium Oxide Com pound name cannot start with a number or a lower case alphabet 3 Stoic Coeff Enter the stoichiometric coefficient for the compound taking part in the reac tion It must be a negative value for reactants and a positive value for products 4 Mole Frac Enter the mole fraction of the species taking part in the reaction No mole frac tion is required for the Product s species All the solid mole fractions must add up to 1 and all the gas mole fractions must add up to 1 5 MW g mol Enter the molecular weight of the compound in grams mole 6 Density g cm Enter the density for the compound It is only required for solids 7 Diffusion Volume Enter the diffusion volume for the compound It is only required for gases 8 Main Reactant Gas Check the box for one main solid reactant and one main gaseous reac tant Multiple selections are not allowed for each phase Besides the reactions table and the general OK and CANCEL buttons there a
26. f non porous grains is given in Figure 6 1 The reaction then occurs at the surface of each non porous grain according to the un reacted shrinking core model A product layer forms with the pas sage of time in the outer regions of the grains thus offering some diffusional resistance to the reac tion The difference in the molar volumes of the solid product and the reactant also changes the grain size affecting the pore volume and in turn decreasing the diffusion of the gaseous reactant through the sorbent particle a00000 09000204 000000080 000 090009090 00 O Ga 090096 42 Expanding Grain Model EGM v1 0 Figure 6 1 A Schematic Diagram of a Porous Sorbent Particle and the Assemblage of the Non Porous Grains A detailed analysis of the problem is presented below with the following assumptions 1 The sorbent particles are spherical in shape with uniform CuO composition 2 Mass transfer limitations were eliminated using high gas flow rates during TGA runs as was proved using different experiments described in chapter V 3 Temperature is uniform through out the particle This assumption was based on the relatively low heat of reaction of the sulfation of CuO in the range of 300 C to 500 C 697 cal g CuSOa to 692 cal g CuSO and the high gas flow rates used in this study 4 Asingle porous sorbent particle is made up of a large number of non porous spherical grains of uniform radius 5 Individual g
27. ffusivity cm 2 min Optimum 1 12e 05 SSE Optimum 0 02192 CANCEL Figure 46 0 Optimization Results Window Plot Type panel contains various types of plots to choose from Grid ON OFF option turns the grids on or off Optimal Solution panel shows the alpha product layer diffusivity and SSE at the optimal point Load Optimum button load the optimal alpha and product layer diffusivity to the EGM pa rameters which can be used to run the simulation at the optimal condition 39 About EGM Expanding Grain Model EGM v1 0 Expanding Grain Model EGM V1 0 Copyright c Illinois Institute of Technology 2013 Developed By Process Design and Gas Processing Laboratory PDGPL Department of Chemical and Biological Engineering Chicago IL 60616 Advisor Dr Javad Abbasian GTI Associate Professor of Chemical Engineering Department of Chemical and Biological Engineering Illinois Institute of Technology Chicago IL 60616 Phone 312 567 3047 Email abbasian iit edu Website mypages iit edu abbasian Students Worked on EGM Jaya Bahadur Singh PhD Candidate ChBE Armin Hassanzadeh Khayyat PhD ChBE 2007 Vasudeo S Gavaskar PhD ChBE 2006 Figure 47 0 About EGM Window 40 Disclaimer Expanding Grain Model EGM v1 0 This software was developed by Illinois Institute of Technology IIT Neither IIT nor any of its employees makes any warranty expressed or implied or assumes any l
28. gration that within interval t t dt the quantity Cr is independent of time i e dt is chosen to be sufficiently small This procedure is repeated until the final time ty Figure 6 4 shows the flow diagram for the solution of the differential equations involved in the grain model The equation for the concentration profile through the porous particle 6 12 can be simplified and written as lg omen de 1 OD dC y 0 6 36 OR 2e R DR PeR With boundary conditions a di 0 atR 0 R b Cr Co at R Ro where yis the local reaction rate given by equation 6 11 The second order differential equation 6 29 was solved by the finite difference method A net of 150 grid points see figure 6 5 on the particle radius and a time step of 0 05 min provided a numerical solution of sufficient accuracy The following approximation of the finite difference method was employed 9 C _ Ca R AR 2Cx R Cx R AR OR 2 AR 6 37 55 Expanding Grain Model EGM v1 0 dC _ C2 R AR C R AR OR 2 AR IN 6 38 R R N 6 39 where N is the number of grid points and Re R R AR The value of Ca Ro is given by the boundary condition b given above The value of Cr 0 is determined by the approximation of the boundary condition a C R AR C R AR 2 AR 0 6 40 On introducing the approximations 6 29 6 32 the vector C is the solution of the set of linear alge braic equations A C b
29. ha 3 y Max 100 Slope 2 0764 S Slope 11 2070 x Simulation Time Simulation Time 50 min Fr 8 Time Step Time Step 0 5000 min x Max 50 Number of Grid Points Number of Grid Points 150 Over Relaxation Factor Over Relaxation Factor 1 8000 Plot Selection Error Tolerance Error Tolerance 5 0000e 09 A Plott Reaction Reaction Phase Phase Reactant s Reactant s Gas Gas Gas Gas Product s Product s Inert s Inert s Inert 9 Inert g Select Select Select Select in Reaction Kinetics Reaction Kinetics Initial slope of the reaction Initial slope of the reaction Figure 32 0 Plot selection displays the relevant data 9 Delete Plot Deletes the selected plot 10 Color Palette Toggles the color palette 50 28 Expanding Grain Model EGM v1 0 Figure 33 0 Color Palette Select a plot Click the Color Palette button to show the palette Select a color to change the color of the selected plot Plot Controls Grid ON OFF Axis TIGHT Gas Conversion lt 100 T T T T 90 80 70 60 50 40 30 20 l 25 Time min 50 Figure 34 0 Plot color changed from blue to red 29 Stopping simulation A running simulation can be stopped by 1 Click Simulation gt Stop Simulation 2 Or click the Stop Simulation tool File Data Simulation Help Expanding Grain Model EGM v1
30. hemicalProp 5 27 2013 3 31 PM Microsoft Access d SkyDrive Gi Dol SO2 650C 500ppm 6 20 2013 12 22 AM Microsoft Access 45 Dropbox expPlotData 6 14 2013 11 06 AM Microsoft Access sample 6 5 2013 11 58PM Microsoft Access Libraries sampleWithOptimResults 6 14 2013 11 08 AM Microsoft Access Documents 6 12 2013 12 32 AM Microsoft Access a Music lil Pictures he File name Dol Si Save as type MAT files mat a Hide Folders Figure 42 0 Save dialog box for Convert Excel data to mat feature 5 Use the default filename or give a new filename and click Save 6 You will see the Data saved successfully to mat file Message 35 Expanding Grain Model EGM v1 0 Running Optimization The Expanding Grain Model has two unknown parameters They are alpha and product layer diffu sivity These parameters are obtained by performing optimization to closely fit a set of experi mental data Before running the optimization you must meet all of the following requirements 1 All the parameters must be entered or loaded into the EGM using Open tool 2 Experimental data must be imported and plotted Make sure that the slope is specified based on the experimental data 3 The plotted experimental data must be selected from the Plot Selection list box Once you have done all of the above 1 Click Simulation gt Run Optimizaton 2 Alternatively you can click on the Run Optimi
31. le plotting the points File Data Simulation Help DSaX AAOR8 GaP gt awe t Operating Parameters Sorbent Parameters Reactions Simulation Parameters Parameters Display Plot Controls Parameter Vawe Units ears Temperature 750C A Pressure 1 atm Axis TIGHT Average Particle Diameter 362 5000 microns Sorbent initial Porosity 0 5650 Axis Limits BET Surface Area 126130 cm2 g Product Layer Diffusivity 1 0000e 07 cm 2 min Alpha 3 Slope 2 0764 Simulation Time 50 min Time Step 0 5000 min Number of Grid Points 150 Over Relaxation Factor 1 8000 Error Tolerance 5 0000e 09 Reaction Phase Compound Stoic Coeff Reactant s Gas Gas Product s Inert s Inert g Select Select lt Gas Conversion Reaction Kinetics Initial slope of the reaction Time min Running simulation Figure 26 0 Running Simulation Once the simulation has been successfully completed you will see the following 24 Expanding Grain Model EGM v1 0 File Data Simulation Help DEaX A8 OS GaP gt army at Operating Parameters Sorbent Parameters Simulation Parameters Parameters Display Plot Controls Parameter Value Units Grid ONOFF Temperature 750C a Pressure 1 atm Average Particle Diameter 382 5000 microns Sorbent Initial Porosity 0 5650 BET Surface Area 136130 cm 2 g Product Layer Diffusivity 1 0000e 07 cm 2 min
32. me is only applicable to gases Phase Compound Stoic Coeff Mole Frac MW g mol Density g cm 3 Diffusion Volume Main Reactant Gas Reactant s Cao 1 0 67 56 3 35 S02 1 0 0005 64 066 41 1 02 05 0 1495 32 16 6 CaSO4 1 136 MgO 0 0 33 40 N2 0 0 85 28 Product s Inert s Inert 9 Select Select Select DODODODOORKE TC EH lt lt Ene E O CLOSE Figure 14 0 Reactions Help Window A part of the reactions parameter which is actually present in the main window inside the Reaction Kinetics panel is Initial slope of the reaction as shown below Initial slope of the reaction 2 2733 Get Siope Figure 15 0 Reaction Kinetics panel 1 The slope can be directly entered as a value Range is from O to 1000 2 Alternately you can press the Get Slope button to specify the slope with a line as follows a Click Get Slope The mouse arrow turns to a crosshair b Click at a point inside the Plot area While holding the mouse drag it around to spec ify the slope The value of the slope keeps on updating in the slope field c Once you achieve the desired slope release the mouse button 16 Gas Conversion 96 Expanding Grain Model EGM v1 0 5 10 15 20 a Figure 16 0 Specifying slope with Get Slope 17 Expanding Grain Model EGM v1 0 Simulation Parameters Clicking on the simulation parameters button displays the following window
33. mization of the sum of squares of errors residuals by using the steepest descent method Marguardt 1959 Kuester and Mize 1973 o gt e 0 a L 0 a F 6 42 58 Expanding Grain Model EGM v1 0 where fi is the value of the overall conversion given by the model while Fj is the corresponding experimental point 59 Figure Descriptions Figure 1 0 Figure 2 0 Figure 3 0 Figure 4 0 Figure 5 0 Figure 6 0 Figure 7 0 Figure 8 0 Figure 9 0 Figure 10 0 Figure 11 0 Figure 12 0 Figure 13 0 Figure 14 0 Figure 15 0 Figure 16 0 Figure 17 0 Figure 18 0 Figure 19 0 Figure 20 0 Figure 21 0 Figure 22 0 Figure 23 0 EGM Application Startup Window File Menu Open Menu Simulation Menu Help Menu Toolbar New Case Window EGM Parameters Buttons Operating Parameters Window Sorbent Parameters Window Reactions Window Load Chemical Properties Window Clear Reactions Table Prompt Reactions Help Window Reaction Kinetics Panel Specifying slope with Get Slope Simulation Parameters Window Non numeric input error Outside range error Open quest dialog box Open dialog box File Error message EGM with data loaded Expanding Grain Model EGM v1 0 60 Figure 24 0 Figure 25 0 Figure 26 0 Figure 27 0 Figure 28 0 Figure 29 0 Figure 30 0 Figure 31 0 Figure 32 0 Figure 33 0 Figure 34 0 Figure 35 0 Figure 36 0 Figure 37 0 Figure 38 0 Figure 39 0 Figure 40 0 Figure 41 0 Figure 42 0 Figure 43 0 Figur
34. nt T temperature K Ma molecular weight of the reactant gas SO2 and K i 6 29 128 2 7 g i I el ES 1 n 3 co 6 30 4 7 1 8 where Ko Proportionality factor Na number of grains The molecular diffusion coefficient Du also known as the free gas diffusion was calculated using the equation developed by Fuller Schettler and Giddings 1996 100x10 Y si D 6 31 Where Mj molecular weight of the species j Uy specific volumes of the species j Flow and diffusion of the gaseous reactant through the porous solid reactant are therefore charac terized by Dm Dx and t Unfortunately little or no information is available on the experimentally 52 Expanding Grain Model EGM v1 0 measured tortuosities for non catalytic gas solid reaction systems A commonly used approach in modeling such gas solid reactions is to assume a functional form for T which incorporates the initial pore structure and the structural changes occurring in the solid with increasing conversion The changes in the diffusion of the gaseous reactant through the solid reactant correspond ing to the changes in the particle porosity are generally modeled using certain empirical equations The random pore model of Wakao and Smith 1962 uses the following representation of the change in De 2 D es 6 32 Doo Eq The model equations are then solved vvith varying diffusivities Other empirical forms of
35. oading data faster in the future Before we begin to look at the way of converting Excel data file to mat file let s take a look at the Excel data file The Excel data file must be very simple It must contain the data in the first sheet Sheet1 It must contain the data in the first two columns The first column must contain time data with the header label minutes gt PINE The second column must contain the conversion fraction data with the header label conver sion 5 Here is what it should look like HA 9 z Dol_502_650C_500ppm Excel HOME INSERT PAGELAYOUT FORMULAS DATA REVIEW VIEW PAC Jar E enel paste g BIU D A El gt GN Clipboard amp Font Gi Alignment Ta Number Gi F4 y A B E D E F G H minutes conversion 0 0 0 166667 0 002647558 0 333333 0 005295116 1 2 3 4 5 0 5 0 007942674 E 7 8 9 0 666667 0 010590232 0 833333 0 01323779 1 0 015885348 1 166667 0 018532906 10 1 333333 0 021180464 11 1 5 0 023828022 12 1 666667 0 026475581 13 1 833333 0 030230947 14 2 0 034280852 15 2 166667 0 038551662 16 2 333333 0 040613432 17 2 5 0 045399683 18 2 666667 0 048565973 Sheet1 O Figure 39 0 Excel Data File Sample 33 Expanding Grain Model EGM v1 0 Once we have the proper Excel data file we can convert it to the mat file as follows 1 Click Data gt Convert Excel data to mat There is no tool in the toolbar for this task 2 You will see the Sel
36. oduct Layer Diffusivity Minimum and Product Layer Diffusivity Maximum For example if Minimum 3 Maximum 5 and Number of Points 5 then Range 3 3 5 4 4 5 5 pesi EES Figure 45 0 Active Optimization Parameters Window 6 Description of the fields a Alpha Minimum Minimum value of the alpha parameter Default value is 1 b Alpha Maximum Maximum value of the alpha parameter Default value is 5 c Product Layer Diffusivity Minimum Minimum value of the product layer diffusivity Default value is 1E 8 d Product Layer Diffusivity Maximum Maximum value of the product layer diffusiv ity Default value is 1E 4 e Number of Points The number of points between the minimum and maximum prod uct layer diffusivity points Default value is 10 f Weights to calculate SSE The weights of each point used to calculate the sum of the squared error SSE Default value is an array of 1s 38 Expanding Grain Model EGM v1 0 Optimization Results The results of the optimization can be viewed in the optimization results window 1 2 3 o u Click on Simulation gt Optimization Results Or click on the Optimization Results tool The following window will appear 2 AY D2 O Plot Type Surface Plot Mesh Plot O Stem Plot Water Fall Plot Plot Controls Y Grid ON OFF Optimal Solution Alpha Optimum Product Layer Di
37. olid While the intrinsic chemical reaction rate depends solely on the chemistry to the two reacting species the diffusional phenomenon depends on a number of factors characteristic of the solid reactant There are two types of diffusional resistances which are commonly encountered in a gas solid reaction 1 The effective diffusivity De of the gaseous molecule through the porous solid matrix which is also known as pore diffusivity see Equation 6 12 2 The diffusivity of the gaseous reactant through the product layer Dg i e copper sulfate see Equation 6 10 When the reactant solid is porous diffusion through the pores is necessary for the reactant gas to gain access to the solid surface Szekely et al 1976 Pore diffusion is inherently much more complex than diffusion in liquids or gases and as a consequence is not very well understood Figure 6 3 shows a schematic representation of gaseous diffusion in a porous medium Two main factors that have to be addressed while modeling understanding the pore diffusion phenomenon are Sze kely et al 1976 1 The actual diffusion path does not follow a straight line but will be quite tortu ous and the extent of this tortuosity will generally depend on the pore structure of the solids and 2 In majority of the cases Knudsen diffusion becomes more important than molecular diffusion In a physical sense this means that the Knudsen regime collisions between gas molecules and the solid w
38. on is obtained D C C 6 13 D 4 n 1 si 8 Substituting Ci from equation 6 13 into equation 6 9 vve get dr k D C ira BR forr gt 0 6 14 dt En r DO AS 8 The initial condition for the above equation is ri rgatt 0 6 14a a E AN 6 14b dt The equation 6 14b ensures a lower bound on ri and assures the stability of equation 6 14 by maintaining ri gt 0 The material balance for the reactant gas A around the spherical particle leads to the follow ing differential equation describing the diffusion of A between the grains 0 D R 9Cr R R R Jra o7 0 6 15 With De Effective diffusivity through the particle cm min 46 Expanding Grain Model EGM v1 0 R Radial co ordinate of the reacting particle cm Cr The gaseous reactant within the porous particle mol cc e Local Particle porosity dimensionless y The rate of disappearance of gaseous reactant per mole of initial solid reactant mol cc min The local rate of reaction y per volume of the spherical grain can be derived from equation 6 5 which for the stoichiometry of the sulfation reaction can be written as dN Si ee dN dt dt 4m k C 6 16 The rate of disappearance of the gaseous reactant per volume of the spherical grain is then given as dC aN t Where Verain Original Volume of the spherical grain cm 2 D C y 3kE ae 6 18 D kn 1 n 8 Substituting equation
39. rains are sufficiently small for variations in gas concentrations on their surface to be negligible 6 The surroundings of the grains don t interfere with their growth i e the grains can expand uniformly to the size corresponding to their maximum possible conversion 7 Reaction rate is first order with respect to the gas as was determined in chapter V 8 The pseudo steady state approximation can be applied to this reaction system for describing the concentration of the reactant gas within the particle General Equations of the Expanding Grain Model The reaction between the porous solid and gas can be represented by A g S s Products s The grain radius rg of a spherical grain within the unreacted spherical sorbent particle is calculated from material balance and is given by 3 Sa 6 1 S P solid rg 43 Expanding Grain Model EGM v1 0 Where Sg BET surface area cm g Psolid True density of the solid reactant g cc The surface dependent reaction rate at the reaction interface is written as a epee 6 2 S MW CuO dt Where Ci Concentration of gaseous reactant at the reaction interface mol cc 1 dN Upon writing Equation 6 2 in terms of the moles of the solid reactant R N C SMWeo dt 6 3 With N Number of moles of the solid reactant The surface area of the reaction front Sq interface is given by S int erface Fi So MW a0 N 4m 6
40. re three additional but tons in the reactions window They are 14 Expanding Grain Model EGM v1 0 1 Load Data Button This button is inactive most of the time It becomes active only if the Phase is selected first and then the corresponding Compound field is clicked Clicking on this button shows the following window Click on a cell to load the corresponding row data 2 0200 4 20 1800 39 9500 83 8000 131 2900 28 44 0100 44 0130 17 0310 Figure 12 0 Load Chemical Properties Window Select any Compound and click on OK to load that compound s data into the reactions table 2 Clear Table Button Clicking on this button displays the following prompt Clear Reactions Table Figure 13 0 Clear Reactions Table Prompt Clicking on YES clears the reactions table and resets it to the original state 3 Help Button Clicking on the help button displays the following window 15 Expanding Grain Model EGM v1 0 Example Reaction Notes 1 Stoichiometric coefficient is negative for reactants and CaO SO2 1 2 02 gt CaSO4 positive for products 2 Stoichiometric coefficient of Inert is zero MgO is present in the sorbent as Inert 3 Mole fraction of solids must add up to 1 4 Mole fraction of gases must add up to 1 N2 is used as purge gas Inert 5 Mole fraction is not required for the product 6 Density is not required for gases Here is how the reactions table should look like 7 Diffusion volu
41. ters Window 1 Temperature Enter the temperature of the system Temperature range is from 273 15 C to 1500 C 2 Pressure Enter the pressure of the system Pressure range is from O atm to 100 atm 12 Expanding Grain Model EGM v1 0 Sorbent Parameters Clicking on the sorbent parameters button displays the following window Average Particle Diameter Sorbent Porosity BET Surface Area Product Layer Diffusivity Alpha OK CANCEL Figure 10 0 Sorbent Parameters Window 1 Average Particle Diameter Enter the average particle diameter of the sorbent Range is from 0 microns to 1500 microns 2 Sorbent Porosity Enter the initial porosity of the sorbent Range is from O to 1 3 BET Surface Area Enter the surface area of 1 gram of sorbent obtained from the BET meas urement Range is from 0 cm g to 1E 10 cm g 4 Product Layer Diffusivity Enter the product layer diffusivity of the sorbent Range is from 1E 9 cm min to 1E 4 cm2 min More about this parameter in the Model Theory section 5 Alpha Enter the alpha value Range is from 1 to 10 Only accepts integer values More about this parameter in the Model Theory section 13 Expanding Grain Model EGM v1 0 Reactions Clicking on the reactions button displays the following window Phase Compound Stoic Coeff Mole Frac MW g mol Density g cm 3 Diffusion Volume Main Reactant Gas Select Select Select Select S
42. uces to equation 6 27 for the initial effective diffusivity It should also be noted here that when the local conversion of the grain reaches its maximum value effective diffusivity drops to zero as the porosity drops to zero thus preventing further diffusion of the gaseous reactant into the sorbent and describing the pore closure phenom enon It is clear from equations 6 20 and 6 27 that the varying diffusivity can be related to the initial effective diffusivity through the porous sorbent as follows de expl a X p 6 35 Do Since a is the characteristic of the solid reactant and describes the extent of tortuosity ex pected as the reaction proceeds it is expected to be directly proportional to the expansion factor Zy Since Zy was shown to increase with increasing copper contents of the sorbent a is also expected to have a higher value for higher copper loadings in the sorbent formulations However is not ex pected to be a function of the reaction conditions such as reaction temperature pressure and the gas compositions The discussion so far has been limited to describing the diffusion of SO through porous solid reactant However in many cases one has to include the effect of the diffusion of the gas through the product layer formed CuSO This process is usually slow but has been found to control the overall rate of reduction of metal oxides and the oxidation of metals under certain conditions Sze kely et al
43. ximum value of the y axis Default is 100 xMin Minimum value of the x axis Default is 0 xMax Maximum value of the x axis Default is the total simulation time entered in the Sim ulation Parameters Window Plot Selection The list box contains all the plots plotted after the simulation Selecting a plot does the following a Make the plot stand out by increasing the line thickness See Figure 31 below b Display the data related to the selected plot in the Parameters Display panel Note the values of the temperature and the slope for Plot1 and Plot2 in Figure 32 below 27 Piot Plot2 Gas Conversion 96 Expanding Grain Model EGM v1 0 5 10 15 20 25 30 40 45 Time min Figure 31 0 Plot selection increased line thickness Parameters Display Plot Controls Parameters Display Plot Controls Parameter Value Units Grid ON OFF Parameter Value Units Grid ON OFF Temperature 750C Temperature 850C Pressure 1 atm Axis TIGHT Pressure 1 atm Axis TIGHT Average Particle Diameter 362 5000 microns Average Particle Diameter 362 5000 microns Sorbent Initial Porosity 0 5650 Ads Limits St Sorbent Initial Porosity 0 5650 Axis Limits Rawat BET Surface Area 136130 cm 2 g BET Surface Area 136130 cm 2 g Product Layer Diffusivity 1 0000e 07 cm 2 min y Min 0 Product Layer Diffusivity 1 0000e 07 cm 2 min y Min 0 Alpha 3 y Max 100 Alp
44. zation tool on the toolbar 3 You will see the optimization running as follows AG uxXIRR OA GHP gt awit Operating Parameters Sorbent Parameters Reactions Simulation Parameters Parameters Display Plot Controls T T Parameter Value Units Grid ON OFF Temperature 750 C a Pressure 1 atm Axis TIGHT Average Particle Diameter 362 5000 microns Sorbent initial Porosity 0 5650 BET Surface Area 138130 cm2 g Product Layer Diffusivity 1 1120e 05 cm2 min Alpha 1 Slope 2 0636 Simulation Time 50 min Time Step 0 5000 min Number of Grid Points 150 Over Relaxation Factor 1 8000 Error Tolerance 5 0000e 09 Reaction Phase Compound Stoic Coeff Reactant s cao a Gas 502 1 Gas 02 05 Product s 1 Inert s ugo Iinert g Select Select lt Gas Conversion Reaction Kinetics Initial slope of the reaction Time min Running Optimization 2 00 completed Figure 43 0 Running Optimization 4 Wait for the optimization to complete 36 Expanding Grain Model EGM v1 0 Optimization Parameters The parameters for optimization can be adjusted in the optimization parameters window In order to open the optimization parameters window 1 Click the Simulation gt Optimization Parameters 2 Or click the Optimization Parameters tool 3 The following window will appear Alpha Range Weights to calculate SSE Alpha Minimum
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