CHE 185 – PROCESS CONTROL AND DYNAMICS DYNAMIC MODELING FUNDAMENTALS DYNAMIC MODELING • PROCESSES ARE DESIGNED FOR STEADY STATE, BUT ALL EXPERIENCE SOME DYNAMIC BEHAVIOR. – THE REASON FOR MODELING THIS BEHAVIOR IS TO DETERMINE HOW THE SYSTEM WILL RESPOND TO CHANGES. • DEFINES THE DYNAMIC PATH • PREDICTS THE SUBSEQUENT STATE USES FOR DYNAMIC MODELS • EVALUATION OF PROCESS CONTROL SCHEMES – – – – – SINGLE LOOPS INTEGRATED LOOPS STARTUP/SHUTDOWN PROCEDURES SAFETY PROCEDURES BATCH AND SEMI-BATCH OPERATIONS • TRAINING • PROCESS OPTIMIZATION TYPES OF MODELS • LUMPED PARAMETER MODELS – ASSUME UNIFORM CONDITIONS WITHIN A PROCESS OPERATION – STEADY STATE MODELS USE ALGEBRAIC EQUATIONS FOR SOLUTIONS – DYNAMIC MODELS EMPLOY ORDINARY DIFFERENTIAL EQUATIONS LUMPED PARAMETER PROCESS EXAMPLE TYPES OF MODELS • DISTRIBUTED PARAMETER MODELS – ALLOW FOR GRADIENTS FOR A VARIABLE WITHIN THE PROCESS UNIT – DYNAMIC MODELS USE PARTIAL DIFFERENTIAL EQUATIONS. DISTRIBUTED PARAMETER PROCESS EXAMPLE FUNDAMENTAL AND EMPIRICAL MODELS • PROVIDE ANOTHER SET OF CONSTRAINTS • MASS AND ENERGY CONSERVATION RELATIONSHIPS – ACCUMULATION = IN - OUT + GENERATIONS – MASS IN - MASS OUT = ACCUMULATION – {U + KE + PE}IN - {U + KE + PE}OUT + Q - W = {U + KE +PE}ACCUMULATION FUNDAMENTAL AND EMPIRICAL MODELS • CHEMICAL REACTION EQUATIONS • THERMODYNAMIC RELATIONSHIPS, INCLUDING – EQUATIONS OF STATE – PHASE RELATIONSHIPS SUCH AS VLE EQUATIONS DEGREE OF FREEDOM ANALYSIS • AS IN THE PREVIOUS COURSES, UNIQUE SOLUTIONS TO MODELS REQUIRE n-EQUATIONS AND nUNKNOWNS • DEGREES OF FREEDOM, (UNKNOWNS EQUATIONS) IS – ZERO FOR AN EXACT SPECIFICATION – >ZERO FOR AN UNDERSPECIFIED SYSTEM WHERE THE NUMBER OF SOLUTIONS IS INFINITE – <ZERO FOR AN OVERSPECIFIED SYSTEM – WHERE THERE IS NO SOLUTION VARIABLE TYPES • DEPENDENT VARIABLES - CALCULATED FROM THE SOLUTION TO THE MODELS • INDEPENDENT VARIABLES - REQUIRE SOME FORM OF SPECIFICATION TO OBTAIN THE SOLUTION AND REPRESENT ADDITIONAL DEGREES OF FREEDOM • PARAMETERS - ARE SYSTEM PROPERTIES OR EQUATION CONSTANTS USED IN THE MODELS. DYNAMIC MODELS FOR CONTROL SYSTEMS • ACTUATOR MODELS HAVE THE ππ 1 GENERAL FORM: = (ππππΈπΆ − π) ππ‘ ππ£ – THE CHANGE IN THE VARIABLE WITH RESPECT TO TIME IS A FUNCTION OF • THE DEVIATION FROM THE SET POINT (VSPEC - V) • AND THE ACTUATOR DYNAMIC TIME CONSTANT τv • THE SYSTEM RESPONSE IS MEASURED BY THE SENSOR SYSTEM THAT HAS INHERENT DYNAMICS GENERAL MODELING PROCEDURE • FORMULATE THE MODEL – ASSUME THE ACTUATOR BEHAVES AS A FIRST ORDER PROCESS – THE GAIN FOR THE SYSTEM • IS THE RATIO OF THE SIGNAL SENT TO THE ACTUATOR TO THE DEVIATION FROM THE SET POINT • ASSUMED TO BE UNITY SO THE TIME CONSTANT REPRESENTS THE SYSTEM DYNAMIC RESPONSE EXAMPLE OF DYNAMIC MODEL FOR ACTUATORS • EQUATIONS ASSUME THAT THE ACTUATOR BEHAVES AS A FIRST ORDER PROCESS • DYNAMIC BEHAVIOR OF THE ACTUATOR IS DESCRIBED BY THE TIME CONSTANT SINCE THE GAIN IS UNITY FIRST ORDER DYNAMIC RESPONSE OF AN ACTUATOR EXAMPLE OF DYNAMIC MODEL FOR SENSORS – EQUATIONS ASSUME THAT THE ACTUATOR BEHAVES AS A FIRST ORDER PROCESS – DYNAMIC BEHAVIOR OF THE ACTUATOR IS DESCRIBED BY THE TIME CONSTANT SINCE THE GAIN IS UNITY – T AND L ARE THE ACTUAL TEMPERATURE AND LEVEL RESULTS FOR SIMPLE SYSTEM MODEL • SEE EXAMPLE 3.1 – THE PROCESS MODEL FOR A CST THERMAL MIXING TANK WHICH ASSUMES UNIFORM MIXING – RESULTS IN A LINEAR FIRST ORDER DIFFERENTIAL EQUATION FOR THE ENERGY BALANCE – SEE FIGURE 3.5.6 FOR THE COMPARISON OF THE MODEL BASED ON THE PROCESSONLY RESPONSE AND THE MODEL WHICH INCLUDES THE SENSOR AND THE ACTUATOR WITH THE PROCESS. EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • STEP INCREASE IN A CONCENTRATION FOR A STREAM FLOWING INTO A MIXING TANK – GIVEN: A MIX TANK WITH A STEP CHANGE IN THE FEED LINE CONCENTRATION – WANTED: DETERMINE THE TIME REQUIRED FOR THE PROCESS OUTPUT TO REACH 90% OF THE NEW OUTPUT CONCENTRATION, CA EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • BASIS: F0 = 0.085 m3/min, VT = 2.1 m3, CAinit = 0.925 mole A/m3. AT t = 0. CA0 = 1.85 mole A/m3 AFTER THE STEP CHANGE. – ASSUME CONSTANT DENSITY, CONSTANT FLOW IN, AND A WELL-MIXED VESSEL • SOLUTION (USING THE TANK LIQUID AS THE SYSTEM): – USE OVERALL AND COMPONENT BALANCES – MASS BALANCE OVER Δt: – F0ρΔt - F01ρΔt = (ρV)(t + )t) - (ρV)t EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • DIVIDING BY Δt AND TAKING THE LIMIT AS Δt → 0 • FOR A CONSTANT TANK LEVEL AND CONSTANT DENSITY, THIS SIMPLIFIES TO: EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • SIMILARLY, USING A COMPONENT BALANCE ON A: • MWAFCA0Δt - MWAFCAΔt = (MWAVCA)(t + Δt) - (MWAVCA)t • DIVIDING BY Δt AND TAKING THE LIMIT AS Δt → 0 EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • DOF ANALYSIS SHOWS THE INDEPENDENT VARIABLES ARE F0 AND CA0 AND THE TWO PREVIOUS EQUATIONS SO THERE IS AN UNIQUE SOLUTION • SOLUTION FOR THE NON-ZERO EQUATION: LET τ = V/F AND REARRANGE: EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • THIS EQUATION CAN BE TRANSFORMED INTO A SEPARABLE EQUATION USING AN INTEGRATING FACTOR, IF : EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • SO THE RESULTING EQUATION BECOMES: EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • EVALUATION • THE INTEGRATING CONSTANT IS EVALUATED USING THE INITIAL CONDITION CA(t) = CAinit AT t = 0. • FOR THE TIME CONSTANT EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • THE FINAL EQUATION IN TERMS OF THE DEVIATION BECOMES: EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • RESULTS OF THE CALCULATION: EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • CONSIDERING THE ORIGINAL OBJECTIVE, THE DATA CAN BE ANALYZED TO DETERMINE THE TIME REQUIRED TO REACH 90% OF THE CHANGE BY CALCULATING THE CHANGE IN TERMS OF TIME CONSTANTS: EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • ANALYSIS INDICATES THE TIME WAS BETWEEN 2τ AND 3τ.ALTERNATELY, THE EQUATION COULD BE REARRANGED ANDS OLVED FOR t AT 90% CHANGE: • CA = CAinit + 0.9(CA0 - CAinit) OR: EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE • OTHER FACTORS THAT COULD AFFECT THE RESULTS OF THIS TYPE OF ANALYSIS ARE: – THE ACCURACY OF THE CONTROL ON THE FLOWS AND VOLUME OF THE TANK – THE ACCURACY OF THE CONCENTRATION MEASUREMENTS – THE ACTUAL RATE OF THE STEP CHANGE SENSOR NOISE • THE VARIATION IN A MEASUREMENT RESULTING FROM THE SENSOR AND NOT FROM THE ACTUAL CHANGES – CAUSED BY MANY MECHANICAL OR ELECTRICAL FLUCTUATIONS – IS INCLUDED IN THE MODEL FOR ACCURATE DYNAMICS PROCEDURE TO EVALUATE NOISE • (SECTION 3.6) DETERMINE REPEATABILITY σ = STD. DEV. • GENERATE A RANDOM NUMBER (APPENDIX C) • USE THE RANDOM NUMBER TO REPRESENT THE NOISE IN THE MEASUREMENT • ADD THIS TO THE NOISE-FREE MEASUREMENT TO GET AN APPROXIMATION OF THE ACTUAL RANGE NUMERICAL INTEGRATION OF ODE’s • METHODS CAN BE USED WHEN CONVENIENT ANALYTICAL SOLUTIONS DO NOT EXIST – ACCURACY AND STABILITY OF SOLUTIONS – REDUCING STEP SIZE FOR NUMERICAL – INTEGRATION CAN IMPROVE ACCURACY AND STABILITY – INCREASING THE NUMBER OF TERMS IN EIGENFUNCTIONS CAN INCREASE ACCURACY – EXPLICIT METHODS APPLIED ARE NORMALLY THE EULER METHOD OR THE RUNGE-KUTTA METHOD NUMERICAL INTEGRATION OF ODE’s • EULER METHOD NUMERICAL INTEGRATION OF ODE’s • RUNGE-KUTTA METHOD NUMERICAL INTEGRATION OF ODE’s • IMPLICIT METHODS OVERCOME STABILITYU LIMITS ON Δt BUT ARE USUALLY MORE DIFFICULT TO APPLY • IMPLICIT TECHNIQUES INCLUDE THE TRAPEZOIDAL METHOD IS THE MOST FLEXIBLE AND IS EFFECTIVE • THERE ARE MANY MORE METHODS AVAILABLE, BUT THESE WILL COVER A LARGE NUMBER OF CASES.