1. Figure 1 - Fed-Batch Reactor with PID Controllers and Fixed Profile
1-3 PT
1-1 PC
1-1
fixed
profile
set point FC
1-1 TT
1-3
Vent
feed
ratio FY
1-1 FT
1-1
Feed A
Coolant
set point FC
1-2 FT
1-2
Feed B TC
1-2
set point TC
1-1 TT
1-2
Coolant
Makeup TT
1-1
Reactor
Figure 1 - Fed-Batch Reactor with
PID Controllers and Fixed Profile
Discharge
Coolant
Return

2. For Self-Regulating Processes:to
Tu = 2 1 + - td (1a)
to + td
[1 + (to*2*p/Tu)2 ]0.5
Ku = (1b) Ko

3. TC
1-3 PT
1-1 PC
1-1
optimum
profile
set point FC
1-1 TT
1-3
Vent
MPC
feed
ratio FY
1-1 FT
1-1
overhead
constraints
Feed A
jacket
constraint
Coolant
set point FC
1-2 FT
1-2
Feed B TC
1-2
set point TC
1-1 TT
1-2
Coolant
Makeup TT
1-1
Reactor
Figure 2 - Fed-Batch Reactor with Model Predictive Control (MPC) for Optimum Profile Layered on Top of PID Controllers
Discharge
Coolant
Return

4. td to to’ Process Variable (%) Time (seconds)
0.72Eo 1
Change in
Manual
Controller
Output (%) 2
Open Loop
Error Eo (%)
0.63Eo
curve 1 = Self-Regulating
curve 2 = Integrating
curve 3 = Runaway 3
Time
(seconds) td to
to’
Eo is the open loop error (e.u.)
Ko is steady state overall open loop gain (%/%)
TCp is the process self-regulating (negative feedback) time constant (minutes)
TCp’ is the process runaway (positive feedback) time constant (minutes)
TDo is the observed open loop dead time or time delay (minutes)
The three major types of process are self-regulating where the process bends to a final value, integrating where the the process ramps to a physical limit, and runaway where the process accelerates to a physical limit. True integrators and runaway process rarely exist, but most level loops and some exothermic loops exhibit degrees of non self-regulation for various operating zones. Also, low range pressure and strong acid or base pH loops appear to the controller to behave as integrating and runaway processes, respectively. Whether real or not, the controller only knows what it sees and the loop behaves accordingly.
Positive Feedback
Time Constant
Dead Time (Time Delay)
Open Loop Time Constant (Time Lag)
Figure 3 – Open Loop Response for Three Major Types of Dynamic Process Response

5. For Integrating Processes:to
Tu = 4 1 + td (2a) td
(2*p/Tu)*[1 + (to*2*p/Tu)2 ]0.5
Ku = (2b) Ko

6. For Runaway Processes:
N N = (to’ + to)  to’ to
Tu = 4 1 + td (3a)
D D = (to’ - to)  (to’ - td) td
[1 + (to*2*p/Tu)2 ]1/2 *[1 + (to’*2*p/Tu)2 ]0.5
Ku = (3b) Ko

7. Where:
to = open loop negative feedback time constant (process time lag) (seconds)
td = total loop time delay (dead time) (seconds)
to’ = runaway process positive feed back time constant (seconds)
Ko = open loop process gain (1/seconds for integrating process otherwise dimensionless)
Ku = ultimate gain (dimensionless)
Tu = ultimate period (seconds)