CHE 185 – PROCESS
CONTROL AND DYNAMICS
DYNAMICS OF SPECIFIC
TYPES OF PROCESSES
INTEGRATING PROCESSES
• A FIRST ORDER PROCESS WITH
CAPACITANCE
• TYPICAL PROCESS IS THE LEVEL IN A
TANK
INTEGRATING PROCESSES
• THE HOLDUP TIME IN THIS UNIT IS
DEFINED AS τH
– THE GENERAL RELATIONSHIP BETWEEN
CHANGES IN L AND F IS
– WHERE THERE IS NO RELATIONSHIP
BETWEEN F AND L, SUCH AS IN THE
GRAVITY DRAINED TANK EXAMPLE.
INTEGRATING PROCESSES
• GENERAL FORM FOR THE TRANSFER
EQUATION IS THEN:
• FOR THE SPECIFIC CASE WHEN FIN IS
CONSTANT, THE FORM OF THE
EQUATION IS:
INTEGRATING PROCESSES
• THE INTEGRATOR FORM FOR THE
FIRST ORDER DIFFERENTIAL
EQUATION
𝑑(𝑌−𝑌𝑠𝑠 )
– INCLUDES A DERIVATIVE TERM
𝑑𝑡
THAT IS INDEPENDENT OF 𝑌 − 𝑌𝑠𝑠
– HAS A POLE = 0 IN THE TRANSFER
FUNCTION
INTEGRATING PROCESSES
• ANALYTICAL FORM FOR THE
INTEGRATOR IS:
AND IT CAN ACCUMULATE VALUE AS
LONG AS INPUT IS NOT ZERO
INTEGRATING PROCESSES
• FOR A STEP INPUT, THE RESPONSE IS:
• FOR AN IMPULSE OF MAGNITUDE A:
THE RESPONSE IS:
DEADTIME (TRANSPORT DELAY)
• THIS IS THE RESULT OF AN INHERENT
DELAY IN THE SYSTEM TO CHANGES
– EXAMPLE 1 - CONSIDER THE TIME DELAY
WHEN A LARGE TANKER IS BEING
MANEUVERED
– THE CAPTAIN SENDS A NOTIFICATION TO
THE CREW TO CHANGE SPEED AND OR
DIRECTION
– THE CREW RESPONDS BY MAKING THE
CHANGES
– THE TANKER STARTS TO RESPOND TO
THE CHANGES
DEADTIME (TRANSPORT DELAY)
• EXAMPLE 2 - CONSIDER A HEATING
SYSTEM WITH THE FOLLOWING
CONFIGURATION:
TC
FUEL
COMB
AIR
HX
HEAT
TRANSFER
FLUID
HX
PROCESS
FLOW
T
DEADTIME (TRANSPORT DELAY)
• THERE IS A TIME DELAY BETWEEN
WHEN THE FUEL FLOW CHANGES AND
THE HOTTER/COOLER HEAT
TRANSFER FLUID ARRIVES AT THE
PROCESS HEAT EXCHANGER, θP.
• THE BLOCK DIAGRAM FOR THE FOPDT
SYSTEM MAY BE APPROXIMATED BY:
DEADTIME (TRANSPORT DELAY)
• SENSING THE SIGNAL FROM THE
PROCESS FLOW AND ADJUSTING THE
SIGNIFICANCE IS RELATED TO THE
OVERALL PROCESS RESPONSE TIME
• GENERAL FORM OF THE EQUATION
FOR DEADTIME IS: Y(t)=X(t−θ )
• GENERAL FORM OF THE TRANSFER
FUNCTION IS:
DEADTIME (TRANSPORT DELAY)
• FIRST ORDER PLUS DEAD TIME
(FOPDT) RESULTS FROM COMBINING
THE FIRST ORDER TRANSFER
FUNCTION WITH THE DEADTIME
TRANSFER FUNCTION:
INVERSE ACTING PROCESSES
• PROCESSES WHERE THERE ARE TWO
COMPETING ACTIONS OCCURRING
SIMULTANEOUSLY
• TYPICAL PROCESS MIGHT INCLUDE A
BYPASS SYSTEM FOR A HEAT
EXCHANGE
INVERSE ACTING PROCESSES
• THE BLOCK FLOW DIAGRAM FOR THIS
PROCESS IS
G1(s)
Y1(s)
+ Y(s)
X(s)
G2(s)
Y2(s)
+
• THE OVERALL EQUATIONS FOR THIS
SYSTEM ARE:
Y1 ( s )  G1 ( s ) X ( s )
Y2 ( s )  G2 ( s ) X ( s )
Y1 ( s )  Y2 ( s )  Y ( s )
INVERSE ACTING PROCESSES
• THE OVERALL EQUATIONS FOR THIS
SYSTEM CAN BE COMBINED TO YIELD
• IF BOTH PROCESSES ARE FIRST ORDER,
THEN SUBSTITUTION WILL PRODUCE:
INVERSE ACTING PROCESSES
• WHICH CAN BE REARRANGED TO YIELD:
INVERSE ACTING PROCESSES
• WHEN THESE PROCESSES ARE OF
OPPOSITE SIGN, THEN EQUATION
(6.9.1) RESULTS
– .THIS EQUATION HAS A ZERO TERM THAT
CAN BE POSITIVE, THEN INVERSE ACTION
CAN OCCUR
– OTHER RESULTS ARE SHOWN IN THE
FOLLOWING GRAPH AND FIGURE 6.9.1,
WHERE τ3 IS VARIED.
INVERSE ACTING PROCESSES
• THE GENERAL FORM OF THE
EQUATION OBTAINED FROM INVERSE
LaPLACE TRANSFORMS IS:
• USING VALUES OF 1 FOR KP AND ΔX,
τ1 = 2, AND τ2 = 1:
INVERSE ACTING PROCESSES
LEAD-LAG COMPENSATION
• LEAD COMPENSATOR CAN INCREASE
THE STABILITY OR SPEED OF
RESPONSE OF A SYSTEM
• A LAG COMPENSATOR CAN REDUCE
(BUT NOT ELIMINATE) THE STEADYSTATE ERROR
• LEAD-LAG COMPENSATORS CAN BE
USED FOR THE SAME OPTIMIZATION
AS PID, PI, PD, I, D CONTROLLERS.
LEAD-LAG COMPENSATION
• LEAD COMPENSATORS AND LAG COMPENSATORS
INTRODUCE A POLE–ZERO PAIR INTO THE OPEN
LOOP TRANSFER FUNCTION:
• LEAD-LAG COMPENSATOR IS A LEAD COMPENSATOR
CASCADED WITH A LAG COMPENSATOR. TRANSFER
FUNCTION IS:
• z1 AND p1 ARE THE ZERO AND POLE OF THE LEAD
COMPENSATOR AND z2 AND p2 ARE THE ZERO AND
POLE OF THE LAG COMPENSATOR
LEAD-LAG COMPENSATION
FOR A FIRST ORDER PROCESS WHERE THERE IS A
CONSIDERATION OF THE TIME DERIVATIVE OF THE
𝑑𝑢(𝑡)
INPUT:
𝑏1 𝑢 𝑡 + 𝑏2
𝑑𝑡
RESULTS IN A TRANSFER FUNCTION:
𝐺 𝑠 =
𝑌(𝑠)
𝑈 𝑠
=
𝐾𝑝 (𝑏1 +𝑏2 (𝑠)
𝜏𝑝 𝑠+1
OR THE GENERAL LaPLACE FORM FOR LEAD-LAG:
𝜏𝑙𝑒𝑎𝑑 𝑠 + 1
𝐺𝑙𝑒𝑎𝑑−𝑙𝑎𝑔 𝑠 = 𝐾𝑙𝑒𝑎𝑑−𝑙𝑎𝑔
𝜏𝑙𝑎𝑔 𝑠 + 1
AND THE TIME VERSION IS (EQN. 6.10.1):
−1
𝜏𝑙𝑒𝑎𝑑
𝑦 𝑡 = 𝐾𝑙𝑒𝑎𝑑−𝑙𝑎𝑔
− 1 𝑒 𝜏𝑙𝑎𝑔 + 1
𝜏𝑙𝑎𝑔
RECYCLE (PROCESS
INTEGRATION) SYSTEMS
• GENERAL FLOWSHEET FOR RECYCLE
OF ENERGY IS SHOWN FOR EXAMPLE
6.12
RECYCLE (PROCESS
INTEGRATION) SYSTEMS
• GENERAL FLOWSHEET FOR RECYCLE
OF ENERGY IS SHOWN FOR EXAMPLE
6.12
• THE TRANSFER FUNCTIONS FOR THIS
SYSTEM ARE:
RECYCLE (PROCESS
INTEGRATION) SYSTEMS
• THE BLOCK DIAGRAM FOR THIS
SYSTEM IS:
T0(s)
T1(s)
GH1(s)
Tf(s)
+
+
Tr(s)
GR(s)
T2(s)
GH2(s)
• WHERE T2 REPRESENTS ENERGY
RECYCLED TO THE REACTOR VIA THE
MIXED FEED STREAM
RECYCLE (PROCESS
INTEGRATION) SYSTEMS
• THE BLOCK DIAGRAM FOR THIS THE
OVERALL TRANSFER FUNCTION FOR
THIS SYSTEM IS:
Tr ( s )
GR ( s )GH1( s )
Go ( s ) 

T0 ( s ) 1  GR ( s )GH 2 ( s )