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Il processo a due serbatoi
Rτ A 2 2 1+ +R R dt R Rp 1 1 1 1 2 1 2{( ) +h́ s τ R s+ R R R1 p 2 1 1 2= 2=K´ 2 ( ) p 2+ + +τ τ s τ τ A R s+1 +R R( )V́ s p 1 p 2 p 1 p 2 1 2 1 2i R R1 2=τ Ap 2 2 +R R1 2( )h́ s R1 2= +τ R s+ R R( )h́ s p 2 1 1 22R 2( )=G ( ) ( )=G s s ∙ G s1 2 2 ( )+ +τ + +1τ τ s τ A R sp1 p 2 p 1 p 2 1 22( )+ + −4 >∆= τ τ A R τ τ 0p1 p 2 1 2 p 1 p 2{ +τ +τ A Rp 1 p2 1 2ζ= K√2 τ τ p( )=G ( ) ( )=→G s s ∙G s Overdampedp1 p 2 1 2 2 2 + +1τ s 2ζτs2 =ττ τp 1 p 2=RK p 2The dynamic response of the two-tank process in never underdamped.Non-Interacting Systems Interacting SystemsNon-interacting systems will The time constants of interacting always result in an overdamped or processes may no longer becritically damped second-order directly associated with the timeresponse. constants of individual capacities.The poles of the overall system are Interacting
capacities are more equal to the individual poles and "sluggish" than the non-equal the inverse of the individual interacting.time constants. The transfer function of the first system is a second order with a negative zero. If individual time constants are equal, then the poles are equal. For systems in series, increasing the number of systems increases the sluggishness of the response. 718.3. Dead Time A definition of time constant is given by the following general law: Capacity = Time constant * Time Rate Mass Mass = τ * Mass Flow Rate Out Volume = τ * Flow Rate Out Length = τ * Velocity Out The "dead time" is defined as: "the time elapsed between a variation of the input variable and the beginning of the observed effect on the output variable. f(t) = y(t) Whenever material or energy is transported in a process unit or plant component, there is a time delay associated with it. The transportation time between points 1 and 2.in pipe flow is:{ ( )=TL T t−t= L 0 dt → ex :d v ( )=cc t−tL 0 d{ 0 t< 0( )=y t t ≥0( )f t−t d
The main condition to be valid this sentence is the "plug flow" in the pipe: ℜ≫2100 .8.3.1. Examples of Distant Measuring Point 72
The following are assumed:
The pipe between the tank and the measuring point is well insulated;
The flow of the liquid through the pipe is ideal plug flow. L( )( )=T =T t t−t → tL d d v 73S O
Transport and weighing of a basic sorbent to neutralize in combustion off-gas: 28.3.2. Transfer Function[ ] −s t( )( )=f ( )=L ( ) ( )=ey t t−t → y s y t f́ sdd
It satisfies the property of linearity: ( )( ) ( )=cfcf t → y t t−t d ( ) ( )( ) ( ) ( )=c+c +cc f t f t → y t f t−t f t−t1 1 2 2 1 1 d 2 2 d( )ý s −s t( )= =eG s → TF non rationald( )f́ s 749. First-Order Plus Dead Time Model (FOPDT)The best way to understand process data is through modelling. Modellingmeansfitting a first-order plus dead time dynamic process model to the data set.
<dy>(</dy>) = K + τ <y t u t−tp p ddt>
Where:
- <y t> is the measured process variable;
- <u t> is an input variable;
- K is the steady state process gain;
- τ is the overall process time constant;
- pt is the apparent dead time.
The FODTP model is low order and linear, so it can ONLY APPROXIMATE the behavior of real processes.
K −tp se dτ s+ 1p9.1. Method of Process Reaction Curve (Step Test)
Two different cases:
1. Self-regulating process
- Process starts at steady state;
- An input variable is stepped to new value;
- The measured process variable is recorded and allowed to complete response.
2. Non-self-regulating process
- The dynamic response to the input step change is a straight line;
- The slope of the response is K/p.
Example of dynamic response to a step input change for a tank with an output flow rate withdrawn by a pump:
9.2. Integrating FOPDT Model
It is applied when thedynamic response doesn’t reach a new value at steadystate. This model has two parameters:¿Steady state process gain determined as the angular coefficient of the K presponse line; tApparent dead time , determined as the time elapsed between the instant dof the step and the first successive instant in which the dynamic responseoccurs. 76¿Kdy t s¿ p( ) ( )==K f t−t →G s e d−Ip d FOPDTdt s9.3. Evaluation Criterion for FOPDT Model GoodnessTo estimates the goodness of the FOPDT model, we can use the method of thesum of squared errors (SSE).N∑ 2( )−SSE= y yi model ,ii=1{ =measured∨experimentaly valuei=y predicted value at the same time with FOPDTmodel, i9.4. Procedure for the Calculation of the ParametersWe can find different kind of procedure:Software Procedure It consists of an “automatic” approximation to a first-order system andparameters estimation with the software LOOP-PRO. It requires a file to recorddata
of the response curve. It is available for response to the step test of:- Self-regulating systems;- Non-self-regulating systems;- Non-inflected responses;- Inflected responses.
Graphical Procedure
- K ( )describes how much the measured process variable changes iny tp ( )response to changes in the controller output . A step test starts and endsu tKat steady state, so can be computed from the plot data:
pSteady state change∈measured process variable ∆ y= =K p Stady state change∈controller output ∆u
Usually a large process gain means the process will show a big response to eachcontrol action.
Let’s have an example for “gravity-drained tanks”: 77∆ y 2.9 m−1.9 m 0.1 m= = =K p ∆ u 60 %−50 % %
Time constant and dead time are computed from the plot of the measuredprocess variable. The calculation procedure may depend on:- If the measured process variable plot presents or doesn’t present aninflection point;- If a software is used to minimize
The model fitting error:
- If calculation is graphically carried out directly on the plot
- If calculation is carried out analytically using some data points on the plot
Case 1 – The curve doesn’t present an inflection point
Case 2 – The curve presents an inflection point
We can also define two other parameters:
td=Controllability Ratio τp /τΔ y
Kp = Maximum Velocity Response = RR Δu /τp
78Analytical Procedure
- Locate where the measured process variable first shows a clear initial response to the step change. Call this time tstart.
- Locate where the measured process variable reaches 63.2% of its total final change. Name the corresponding time tmin.
- From the example, the time constant is the time difference between tstart and tmin. It must be positive and present the units of time.
For this example: ystart = 9.6 m, yt = 1.93 m, Δy = 0.95 m
From the example, tstart = 2.53 min, tmin = 1.93 min
example:63.2 =t −t =1.6τ miny( ) passes through aty t p 63.2 ystart63.2=11.2t min63.2 =t −tt d ystart Ustep 79“Manual” approximation to the first order lag – Analytical procedure using areas Hypotheses:- Monotonical sigmoidalresponse;- Initial and final equilibriumconditions.If we approximate theresponse curve to a real firstorder plus dead time, forwhich:( )= ( )y t y 0 ; 0 ≤ t<t d( )−t−t dτ( )=∆y t y 1−e ; t>tp0 dA A0 1+τ = =t τ ed p p∆ y ∆ y0 0 8010. Feedback ControlFor a correct management of a chemical process, it is necessary to checkoperations in according to the operating conditions of the project. For this reason,“control systems” play an important role in process management.They are physical systems that establish a correspondence relation between an inputquantity called “reference” and an output quantity which consists of the “controlledentity”, also inpresence of other inputs acting as "disturbances".
The main roles of the control system are:
- Suppression of external disturbances on the process;
- Assurance of the operational stability of the process;
- Process performance optimization.
Let's have an example of control system for house heating:
| Measurement | Calculation | Action |
|---|---|---|
| House Temperature | House is colder than the set point temperature? | Open fuel valve |
| House Temperature | House is hotter than the set point temperature? | Close fuel valve |
| House Temperature | House temperature is at set point temperature? | No actions |
10.1. Feedback Loop 81
Example of "feedback loop" for house heating:
Example of "feedback loop" for general case: 821
10.2. Characteristic Components of the Control
An automatic control and regulation system is usually made of the following components of the control loop:
"Measuring
An instrument is a proper device for the measurement of a physical quantity of the process such as temperature or pressure.
A transmitter consists of a device which transforms the measured physical property into a signal transmitted to a receiver. The signal can be pneumatic or electric.
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