### DYNAMIC MODELING FUNDAMENTALS

```CHE 185 – PROCESS
CONTROL AND DYNAMICS
DYNAMIC MODELING
FUNDAMENTALS
DYNAMIC MODELING
• PROCESSES ARE DESIGNED FOR
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
FREEDOM
• PARAMETERS - ARE SYSTEM
PROPERTIES OR EQUATION
CONSTANTS USED IN THE MODELS.
DYNAMIC MODELS FOR
CONTROL SYSTEMS
• ACTUATOR MODELS HAVE THE

1
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 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.
```