Chapter : Process Modelling Flashcards
what makes process modelling different from process analysis
the task involves process analysis with dynamics so there may be accumulation,generation and consumption terms so the models may comprise of differential equations
derive the mass balance
notes
derive volumetric balance
notes
how do we find a differential equation for the level in the tank from the mass balance
from the mass balance we can relate mass to volume through density formula. following that we can relate volume to level through the volume formula. therefore M= Aph
what is constant in the mass balance
flowrates, mass , level are not constant because they are functions of time. so density and area is the only constants
how is mass related to volume
density formula M= p x V
how is volume related to level
V = A x h
how is mass related
M = A x p x h
what is the control system based on for slide 7 and 8
the differential level equation because level is the controlled variable
what is the rate of change of stored heat
heat inflow - heat outflow
derive the heat balance
in notes
why does Fin=Fout
because volume is constant
why is hcond negligible compared to hst
the specific enthalpy of the condensate is negligible compared to the specific enthalpy of steam because the value is much smaller
what does the final heat balance equation relate
it relates the temperature inside the tank with the flow rate of steak and our disturbances
what are the potential disturbances for the heat equation
the temperature of steam may be subject to disturbances
the flowrate of steam may be subject to disturbances
the model may be wrong because steam can create foulingwhich leads to suboptimal performance heat exchange
why does the component balance using mass fraction and mass flowrate not have independent equations
because the mass fractions both add to unity
xA+xB=1
why does the component balance using concentrations and volumetric flow rate not have independent equations
becuase in a binary mixture CA+CB = p if the concentrations are in kg/m^3
how do you convert between mass flowrate and volumetric flowrate
Fm=Fv * p
how do you convert between mass and volume
M= V * p
how do you convert from concentration to mass fraction
CA= p * xA
what is the concentration dependence of a reaction with product inhibition
r = k * CA/K+CB
what is the time delay
Td= L/v = L x A/ Fv
where the time delay is the time taken to travel distance l with velocity v
what are deviation variables used for
continuous processes operate at steady state, deviation variables measure deviations from the steady state
what are examples of deviation variables
T = Tss+∆T
L = Lss+∆L
∆T or ∆L may be positive or negative
what are the steps for linearization
The steps are:
* Step 1: Formulate the appropriate balance equations
* Step 2: Find the steady state
* Step 3:
» 3.1 Define deviation variables
» 3.2 Substitute into the balance equation
» 3.3 Make linearizing approximations
* Step 4: Use the steady state relationship from the second step to simplify
the expression
End result is a linear differential equation describing the dynamics of small
deviations from the operating point
what are examples of control system signals?
unit step decay, exponential decay, damped unit step, damped cosine with decaying amplitude, underdamped second order step
why do do the graphs start from the origin
because we’re assuming steady state
what happens in the unit step
the unit step usually represents a change to the set point e.g a control engineer changing the set point for a flow rate
what do are control system signals functions of
they are all sums of exponential basis functions
what is an example of a unit step function
e^0t for t>0
what is an example of exponential decay
e^(-t/2) for t>0
what is an example of damped unit step
1-e^(-t/2) for t>0
what is euler’s theorem and when is it needed
on slide 9
it is needed for signals containing sin or cos
what does the real and imaginary part do for the graphed function
the imaginary part give the oscillation
the negative real part gives the decay
what would happen if the real part was positive
if the real part was positive the graph would increase to infinity with every osicalltion which would be completely unstable
what is si
si values show which exponential and are called poles. complex values of si indicate the signal is oscillating. they always ome in complex conjugate pairs
what is ai
ai values show how strong the signal is. the ai values are valled residues for complex and amplitudes for real.
how do you visualize si on an s-plane
si values or numerical values which may be real or complex
si values show which exponential basis signals are present in f(t)
how does si affect the signal if we’re visualizing on the s-plane
if we’re on the positive real axis then we have growth
if we’re on the negative real axis then we will have decay, the further along the negative axus, the faster the decay will e
if we’re on the imaginary axis then we will have some sort of oscillation
if we have a mixture of real and imaginary parts then we could have a decaying oscillation or growing oscillation which would be an unstable system
what is the downside of the s-plane
we can only see si so we can;t see the amplitudes of the signals
if the real part is less negative then what do we know
the decay will be slower because the larger period of oscillation means there will be fewer crossings of the x-axis so the decay will slower
what is the period of oscillation based on
the frequency of oscillation/ angular frequency
what determines the time for the signal to decay
the real part of the exponent
if the imaginary part is greater what do we know about the signal
the greater the imaginary part, the higher the frequency of the sinusodial wave
