1. CHE 185 – PROCESS CONTROL AND DYNAMICS
DYNAMIC MODELING FUNDAMENTALS

2. 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

3. 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

4. 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

5. Lumped Parameter Process Example

6. TYPES OF MODELS DISTRIBUTED PARAMETER MODELS
ALLOW FOR GRADIENTS FOR A VARIABLE WITHIN THE PROCESS UNIT
DYNAMIC MODELS USE PARTIAL DIFFERENTIAL EQUATIONS.

7. Distributed Parameter Process Example

8. 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

9. FUNDAMENTAL AND EMPIRICAL MODELS
CHEMICAL REACTION EQUATIONS
THERMODYNAMIC RELATIONSHIPS, INCLUDING
EQUATIONS OF STATE
PHASE RELATIONSHIPS SUCH AS VLE EQUATIONS

10. 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

11. 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.

12. DYNAMIC MODELS FOR CONTROL SYSTEMS
ACTUATOR MODELS HAVE THE GENERAL FORM: 𝑑𝑉 𝑑𝑡 = 1 𝜏 𝑣 ( 𝑉 𝑆𝑃𝐸𝐶 −𝑉)
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

13. 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

14. 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

15. First Order Dynamic Response of an Actuator

16. 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

17. 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 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.

18. 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

19. EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE
BASIS: F0 = m3/min, VT = 2.1 m3, CAinit = 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

20. 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:

21. 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

22. 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:

23. EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE
THIS EQUATION CAN BE TRANSFORMED INTO A SEPARABLE EQUATION USING AN INTEGRATING FACTOR, IF :

24. EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE
SO THE RESULTING EQUATION BECOMES:

25. 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

26. EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE
THE FINAL EQUATION IN TERMS OF THE DEVIATION BECOMES:

27. EXAMPLE OF A MODEL APPLICATION FOR A PROCESS RESPONSE
RESULTS OF THE CALCULATION:

28. 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:

29. 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:

30. 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

31. 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

32. 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

33. 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

34. NUMERICAL INTEGRATION OF ODE’s
EULER METHOD

35. NUMERICAL INTEGRATION OF ODE’s
RUNGE-KUTTA METHOD

36. 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.