Transcript P in - XAMK Moodle

Dynamic processes
Causes of delay in process
systems
Whenever material or energy flows into or out of certain system, it takes
time. Thus, the level of a liquid, the temperature of the vessel or the
position of the solid mass cannot change suddenly. They are subjects to
delays which are dependent on the magnitude of the capacity and the
resistance to the inflow of material or energy.
Heater process:
Electric hot water boiler has electric power Pin (w).
We assume that the metal parts of the systems (resistor and casing) are Picture 1
small compared to mass of water.
Thermal power Q = c * m * ΔT
c = heat capacity of water
m = mass
ΔT = Twater - Tambient = temperature difference
(1) ΔT
Tambient
Twater
Pin
First let's assume that insulation of the boiler is very good. There are no
heat transfer from boiler.
Heater resistor is switched on with constant power.
Next block diagram describes the case
Pin
WATER
ΔTout
Temperature
difference
Power/w
STEP
Picture 2
Equation (1) gives ΔT = Q/(c * m)
(2)
Energy of vessel = Q = W = ∫Pdt
(3)
Equations (2) and (3) gives:
ΔT = ∫Pdt/(c * m)
(4)
We have got temperature difference to the process, if input power is step
function.
With constant power equation (4) gives:
ΔT = P * t/(c * m)
Time-domain plot is growing strait line.
Laplace transform:
Laplace transforms are used to solve dynamic systems.
Time-domain signals are transferred to s-domain signals.
s = jω, ω = angular velocity
Computer based transient analyses are using L-transform.
Integral change to multiplication and gives easy handling for
equations.
Matlab simulink can utilize L-transforms.
Time-domain
Time-domain
Time-domain
Pin
s-domain
∫Pdt = W
1/S
1/c*m
STEP
integrator
ΔT
GAIN
Picture 3
P
ΔT
Transient Analyse
ΔT
P
Time
Pout
Pout
Let´s assume that isolation of the boiler is not perfect.
ΔT
Heat is transferred through the walls of vessel.
Q = U * A * ΔT
Twater
Tambient
(5)
Q = Transferred heat w.
U = Conductance w/m2K
ΔT = Tout - Tin = Temperature difference
A = Area of casing m2
Pin
Pin
+
STEP
s-domain
∫Pdt = W
1/S
-
GAIN
integrator
U * A * ΔT
1/c*m
ΔT
U * A = 0.2
GAIN
Picture 4
ΔT
Temperature difference of the boiler is not any more
strait line. It is exponential and it reaches certain
maximum level after a long time.
In this example power loses are huge and
temperature inside of the vessel doesn't get over 100
degrees.
ΔT
Pin
Transient analyse
ΔT
Time
We can edit the block diagram and substitute the
feedback:
Block diagram in picture 4 can be more simple.
Feed-back block diagram with integrator is substituted to
one transfer function.
Step response will remain the same figure and
magnitude.
Pin
STEP
5/(5s+1)
ΔT
Transfer function
Picture 5
Transfer function formula can be written: A/(τs +
1)
A = Gain of the transfer function = ΔT/Pin ,
K/w
τ = Time constant , second
Next graph describes common step response. Any process with
one "energy store" is indentical. The process has 1st order delay.
Step function 0...100
A
Output = A * (1 - e-t/τ)
Exponent function of
1 st order delay
63,2
τ = Time constant
Time/s
Next processes have similar
responses:
in
R
+
out
C
RC-circuit, τ = R * C
+Force
ΔT =Tmass - Tambient
out
mass
in
Heat power
Velocity
out
in
- Force, example wind force
SECOND ORDER DELAYS
Always we cannot describe the process with one
delay.
Example of heat process:
water
Cattle
Electric heater
plate
Heat resistor
Power/w
τ1
Resistor
Tresist
Tplate
τ2
Heater plate
τ3
Cattle
Tcattle
τ4
Water
Twater
Step response 1...4 delays, τ1 = τ2 = τ3 = τ4 = 1
s
out
in
input step = 100
time
TRANSPORT DELAY
Delays are also caused by the time taken for material to travel along pipes or
conveyors from one place to another. This type delay, shown in next figure.
Temperature mixed water
Hot water
velocity = v
Distance to measuring point =
L
Time delay = L/v
Could water
Step response of the prosess
Transport delay = 2 s, Capacitive delay =
3s
RELATIVE DIFFICULTY OF THE PROCESS
Step response of the process is run to the open-loop
system:
Step
B
Relative
change
in output signal
Step response
Tt
A
Controllability
S = Tr/Tt
S < 4 difficult to control
S > 4 easy to control
Tr
Sketch deepest tangent to the response and mark the points A
and B.
A = minimum level of response.
B = maximum level of response (after a long period)
Dynamic Process Simulation Information
What is the difference between mathematical models and simulations?
On this site we will use the term "mathematical models" to represent sets of equations
that mathematically describe the process. The term "simulator" refers to a computer program or
a digital system running a computer program that implements the mathematical model.
The simulator may be connected to the control system or may be embedded within the control system.
What types of simulations used in process engineering?
Static
Static simulations, typically used in process design, simulate the process at steady state conditions,
usually at the design operating conditions. Time is not a variable.
Dynamic
Dynamic models consider time as a variable and simulate the process over a period of time.
A dynamic simulation can be used to estimate or illustrate the response, over time, to a change in the process. This
primary concern of this site is dynamic models.