Chapter172.IntheStreamfield,useListand selectastreamIDThestreammustbeinarecycleloopinthesimulationflowsheet.NoteWhen a Fortran block is in a recycle loop, you can tear variablesdesignatedasWriteVariablesontheFortranSequencesheet.3.Specify any of the remaining optional fields, as you choose.The following parameters are available on the Tear Specifications sheet:FieldDefaultToTolerance0.001Specify Tear convergence toleranceA tear stream isconverged when thefollowing istruefor all streamvariables:-tol s Xealala -Xasmd ol XassimedTraceTolerance/100Specify the trace component thresholdAspen Plus bypasses this convergence test for components that have amole fraction less than the Trace threshold.ComponenAll componentsIdentifytheComponentgroupIDforcomponents tobeconverged in teartGroupstreamsComponent groups are defined on the Components Comp-Group form(seeChapter6).Usea componentgroup when youknow thatsomecomponentshavezeroorconstantflowrates.AComponentGroupmaycauseconvergence problems if the unconvergedcomponents havesignificant flow.Component group specifications are intended primarily for use with thematrix convergencemethods (Broyden,Newton,and SQP)to reducethematrix size and the numberof numerical derivativeperturbationsStatePressure &EnthalpySelect the State variables to be convergedYoucanselectaStateoptionotherthanthedefault(Pressureandenthalpy)when pressure isknown to be constantor enthalpy is notcalculated (mass-balance-only simulations).State specifications are intended primarily for use with the matrixconvergence methods (Broyden, Newton, and SQP) to reduce the matrixsizeandthenumberofnumericalderivativeperturbationsInitialEstimatesforTearStreamsYoucanusetheStreamsheetstoprovideaninitialestimateforthetearstream.Aninitialestimategenerallyaidsrecycleconvergence,andissometimesnecessary,especiallyforrecycleloopsinvolvingdistillationblocks.FormoreinformationonspecifyingstreamsseeChapter917-7Aspen Plus User GuideVersion10.1-0
Aspen Plus User Guide 17-7 Version 10.1-0 Chapter 17 2. In the Stream field, use List and select a stream ID. The stream must be in a recycle loop in the simulation flowsheet. Note When a Fortran block is in a recycle loop, you can tear variables designated as Write Variables on the Fortran Sequence sheet. 3. Specify any of the remaining optional fields, as you choose. The following parameters are available on the Tear Specifications sheet: Field Default To Tolerance 0.001 Specify Tear convergence tolerance A tear stream is converged when the following is true for all stream variables: − ≤ − tol ≤ X X X tol calculated assumed assumed Trace Tolerance/100 Specify the trace component threshold Aspen Plus bypasses this convergence test for components that have a mole fraction less than the Trace threshold. Componen t Group All components Identify the Component group ID for components to be converged in tear streams Component groups are defined on the Components Comp-Group form (see Chapter 6). Use a component group when you know that some components have zero or constant flow rates. A Component Group may cause convergence problems if the unconverged components have significant flow. Component group specifications are intended primarily for use with the matrix convergence methods (Broyden, Newton, and SQP) to reduce the matrix size and the number of numerical derivative perturbations. State Pressure & Enthalpy Select the State variables to be converged You can select a State option other than the default (Pressure and enthalpy) when pressure is known to be constant or enthalpy is not calculated (mass-balance-only simulations). State specifications are intended primarily for use with the matrix convergence methods (Broyden, Newton, and SQP) to reduce the matrix size and the number of numerical derivative perturbations. Initial Estimates for Tear Streams You can use the Stream sheets to provide an initial estimate for the tear stream. An initial estimate generally aids recycle convergence, and is sometimes necessary, especially for recycle loops involving distillation blocks. For more information on specifying streams see Chapter 9
ConverqenceSpecifying User-Defined ConvergenceBlocksUse the Convergence sheets to specify convergence method, tolerance, andconvergence variables for user-defined convergence blocks. System-generatedconvergence blocks generated by Aspen Plus do not use these specifications.To define a convergence block:1.From theData menu,point to Convergence, then Convergence.2.IntheConvergenceObjectManagerclickNew.3.In the CreateNewID dialog box,enter anID or accept thedefault name.4.In the Create NewID dialog box, select the type of convergence block youwant to create.Use this methodTo convergeBROYDEN orTear streams; two ormore design specifications; ortear streams and design specificationsNEWTONsimutaneously.Usewhentherecycleloopsandordesignspecificationsarehighly interrelated.Use Newton when Broyden is unable to converge.COMPLEXOptimization with inequality constraintsDIRECTTearstreamsbysimpledirectsubstitutionConvergencemaybeslow,butsureSECANTSingle design specifications. Recommended for design specification convergence blocksSQPSequentialquadraticprogramming.Optimizationwithanycombinationoftearstreamqualityconstraints, and inequality constraints.WEGSTEINTearstreams.YoucanapplywegsteintoanynumberofstreamssimultaneousyRecommended tear stream convergence method.Formore information on thenumericalmethods, see Convergence Methodsthis chapterClick theTear Streams,DesignSpecifications,FortranTearsorOptimization5.tab to select the elements that you want the convergence block to solve.6. To specify optional parameters, click the Parameters sheet.Convergence MethodsThis section describes the convergence methods available in Aspen Plus.17-8Aspen Plus User GuideVersion 10.1-0
17-8 Aspen Plus User Guide Version 10.1-0 Convergence Specifying User-Defined Convergence Blocks Use the Convergence sheets to specify convergence method, tolerance, and convergence variables for user-defined convergence blocks. System-generated convergence blocks generated by Aspen Plus do not use these specifications. To define a convergence block: 1. From the Data menu, point to Convergence, then Convergence. 2. In the Convergence Object Manager click New. 3. In the Create New ID dialog box, enter an ID or accept the default name. 4. In the Create New ID dialog box, select the type of convergence block you want to create. Use this method To converge BROYDEN or NEWTON Tear streams; two or more design specifications; or tear streams and design specifications simultaneously. Use when the recycle loops and/or design specifications are highly interrelated. Use Newton when Broyden is unable to converge. COMPLEX Optimization with inequality constraints DIRECT Tear streams by simple direct substitution. Convergence may be slow, but sure. SECANT Single design specifications. Recommended for design specification convergence blocks. SQP Sequential quadratic programming. Optimization with any combination of tear streams, equality constraints, and inequality constraints. WEGSTEIN Tear streams. You can apply Wegstein to any number of streams simultaneously. Recommended tear stream convergence method. For more information on the numerical methods, see Convergence Methods, this chapter. 5. Click the Tear Streams, Design Specifications, Fortran Tears or Optimization tab to select the elements that you want the convergence block to solve. 6. To specify optional parameters, click the Parameters sheet. Convergence Methods This section describes the convergence methods available in Aspen Plus
Chapter17The parameters for each method can be found on the Convergence ConvOptionsMethodsform and ontheformfortheConvergenceblock.WEGSTEINMethodThe classical bounded Wegstein method is usually the quickest and most reliablemethodfortearstreamconvergence.Itisanextrapolation of Directsubstitutioniteration. Interactions between variables are ignored; therefore, it does not workwell when variables are strongly coupled.WegsteinmethodcanonlybeusedforTearstreams.ItisthedefaultmethodforAspen Plus tear stream convergence. Apply it to any number of streamssimultaneously.You can control the Wegstein bounds and the frequency ofaccelerationYoucancontrol theWegsteinmethodbyspecifying:FieldDefaultTospecifythe30Maximum FlowsheetMaximumnumberoflowsheetevaluationsEvaluationsWait1Number of direct substitution iterations before the first accelerationiteration0Consecutive DirectNumber of direct substitution iterations between accelerationSubstitution Stepsiterations1ConsecutiveAccelerationNumberofconsecutiveacceleration iterationsSteps-5Lower BoundMinimum valuefor the Wegstein accelerationparameter (g)0Upper BoundMaximum valuefor the Wegstein acceleration parameter (g)WegsteinAccelerationParameterYou can control the Wegstein method by specifying upper and lower limits for:Accelerationparameterq (UpperBound andLowerBound)Numberofdirectsubstitutioniterationsbeforethefirstacceleration(Wait)Numberofdirectsubstitutioniterationsbetweenaccelerationiterations(ConsecutiveDirectSubstitutionSteps)Number of consecutive acceleration iterations (Consecutive AccelerationSteps),Inthebounded Wegstein method,the accelerationparametergis calculatedforeach tear stream variableas follows:AspenPlusUserGuide17-9Version 10.1-0
Aspen Plus User Guide 17-9 Version 10.1-0 Chapter 17 The parameters for each method can be found on the Convergence ConvOptions Methods form and on the form for the Convergence block. WEGSTEIN Method The classical bounded Wegstein method is usually the quickest and most reliable method for tear stream convergence. It is an extrapolation of Direct substitution iteration. Interactions between variables are ignored; therefore, it does not work well when variables are strongly coupled. Wegstein method can only be used for Tear streams. It is the default method for Aspen Plus tear stream convergence. Apply it to any number of streams simultaneously. You can control the Wegstein bounds and the frequency of acceleration. You can control the Wegstein method by specifying: Field Default To specify the Maximum Flowsheet Evaluations 30 Maximum number of flowsheet evaluations Wait 1 Number of direct substitution iterations before the first acceleration iteration Consecutive Direct Substitution Steps 0 Number of direct substitution iterations between acceleration iterations Consecutive Acceleration Steps 1 Number of consecutive acceleration iterations Lower Bound -5 Minimum value for the Wegstein acceleration parameter (q) Upper Bound 0 Maximum value for the Wegstein acceleration parameter (q) Wegstein Acceleration Parameter You can control the Wegstein method by specifying upper and lower limits for: • Acceleration parameter q (Upper Bound and Lower Bound) • Number of direct substitution iterations before the first acceleration (Wait) • Number of direct substitution iterations between acceleration iterations (Consecutive Direct Substitution Steps) • Number of consecutive acceleration iterations (Consecutive Acceleration Steps). In the bounded Wegstein method, the acceleration parameter q is calculated for each tear stream variable as follows:
Convergence-G(X)-G(X-1)Xk Xk-1Where:X=Estimate of the tear stream variableG(X) =Resultingcalculatedvalueofthevariablek=Iteration numberThe new estimate calculated by Wegstein is:Xk+1=qXk+(1-q)G(Xh)= X +(1-q)(G(X)-X)Thefollowing shows theeffect of qon convergence:qConvergenceq<0Accelerationg=0Direct substitution0<q<1DampingBecause oscillation or divergence can occur if q is unbounded, limits are set on q.The default lower and upper bounds on q are -5 and o, respectively.For mostflowsheets, these limits work well and do not need to be changed.Normally,you should use an Upper Bound of the Wegstein accelerationparameterof o.If iterations move the variables slowlytoward convergencesmaller values of the lower bound of the Wegstein acceleration parameter(perhaps-25 or-50)may give better results.If oscillation occurs with directsubstitution, values of the lower and upper bounds between 0 and 1 may help17-10AspenPlusUserGuideVersion 10.1-0
17-10 Aspen Plus User Guide Version 10.1-0 Convergence q s s = − 1 s GX GX X X k k k k = − − − − () ( )1 1 Where: X = Estimate of the tear stream variable G(X) = Resulting calculated value of the variable k = Iteration number The new estimate calculated by Wegstein is: X qX q G X X q GX X kk k k kk + = +− = +− − 1 1 1 ( )( ) ( )( ( ) ) The following shows the effect of q on convergence: q Convergence q < 0 Acceleration q = 0 Direct substitution 0 < q < 1 Damping Because oscillation or divergence can occur if q is unbounded, limits are set on q. The default lower and upper bounds on q are -5 and 0, respectively. For most flowsheets, these limits work well and do not need to be changed. Normally, you should use an Upper Bound of the Wegstein acceleration parameter of 0. If iterations move the variables slowly toward convergence, smaller values of the lower bound of the Wegstein acceleration parameter (perhaps -25 or -50) may give better results. If oscillation occurs with direct substitution, values of the lower and upper bounds between 0 and 1 may help
Chapter17DIRECTMethodFor direct substitution, the new value of the tear stream variable is the valueresultingfromthepreviousflowsheetcalculationpass:Xk+1 = G(XR)Where:X=Estimate oftearstreamvariableG(X)=Resultingcalculatedvalueofthevariablek=Iteration numberWith direct substitution, convergence is slow but sure. It is available for thoserare cases where other methods may be unstable. Direct substitution can alsomake it easy to identify convergence problems,such as component build-up in thesystem. Direct substitution is equivalent to Wegstein with lower bound=upperbound=0.Secant MethodSecant is the secant linear approximation method, with higher orderenhancements.You can select a bracketing/interval halving option. Select thisoption whenever the function is discontinuous, non-monotonic, or flat over aregion.Bracketing will eliminate the flat region and switch back to Secantmethod if possible.You can use Secant for converging single design specifications. Secant is thedefaultmethodfordesign specification convergence,and is recommended foruser-generated convergence blocks.Youcan control theSecantmethod byspecifyingFieldDefaultTo specify30Maximum FlowsheetMaximum number of flowsheet evaluationsEvaluations0.01Step SizeInitial step size, as a fraction of range, for the design specificationmanipulated variable1Maximum Step SizeMaxmumstepsizesafactionofrangeforthedesgnpecificatnmanipulated variableContinuedAspenPlusUserGuide17-11Version10.1-0
Aspen Plus User Guide 17-11 Version 10.1-0 Chapter 17 DIRECT Method For direct substitution, the new value of the tear stream variable is the value resulting from the previous flowsheet calculation pass: X GX k k +1 = ( ) Where: X = Estimate of tear stream variable G(X) = Resulting calculated value of the variable k = Iteration number With direct substitution, convergence is slow but sure. It is available for those rare cases where other methods may be unstable. Direct substitution can also make it easy to identify convergence problems, such as component build-up in the system. Direct substitution is equivalent to Wegstein with lower bound=upper bound=0. Secant Method Secant is the secant linear approximation method, with higher order enhancements. You can select a bracketing/interval halving option. Select this option whenever the function is discontinuous, non-monotonic, or flat over a region. Bracketing will eliminate the flat region and switch back to Secant method if possible. You can use Secant for converging single design specifications. Secant is the default method for design specification convergence, and is recommended for user-generated convergence blocks. You can control the Secant method by specifying: Field Default To specify Maximum Flowsheet Evaluations 30 Maximum number of flowsheet evaluations Step Size 0.01 Initial step size, as a fraction of range, for the design specification manipulated variable Maximum Step Size 1 Maximum step size, as a fraction of range, for the design specification manipulated variable Continued