CHE 185 – PROCESS CONTROL AND DYNAMICS DYNAMIC MODELING FUNDAMENTALS

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 n-UNKNOWNS

• 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 GENERAL FORM: – 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 PROCESS-ONLY 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.