Process Dynamics and Control - Appunti ed esempi

PROCESS DYNAMICS

and CONTROL

SUMMARY

CONTROL LOOP (CLOSED).................................................................................................. 3

DYNAMIC MODELS of CHEMICAL PROCESSES................................................................ 7

FIRST PRINCIPLES MODELS.......................................................................................... 7

LAPLACE TRANSFORMS.................................................................................................... 12

REPRESENTATIVE FUNCTIONS....................................................................................13

TRANSFER FUNCTION MODELS........................................................................................16

DYNAMIC BEHAVIOUR of FIRST ORDER and SECOND ORDER PROCESSES............. 24

GENERIC DYNAMIC RESPONSE........................................................................................ 33

SYSTEMS with NUMERATOR DYNAMICS..................................................................... 34

SYSTEMS with DEAD TIME............................................................................................ 37

EMPIRICAL MODELS from PROCESS DATA..................................................................... 40

FEEDBACK CONTROLLERS............................................................................................... 43

ON - OFF CONTROL....................................................................................................... 45

PROPORTIONAL CONTROL.......................................................................................... 45

INTEGRAL CONTROL.....................................................................................................47

PROPORTIONAL INTEGRAL CONTROL....................................................................... 48

DERIVATIVE CONTROL.................................................................................................. 49

PID CONTROLLERS....................................................................................................... 50

DIGITAL CONTROLLERS..........................................................................................54

FLOW CONTROL (FC).................................................................................................... 57

LIQUID-LEVEL CONTROL (LC)...................................................................................... 59

TEMPERATURE CONTROL............................................................................................59

CONDENSATION CONTROL.......................................................................................... 60

PRESSURE CONTROL (PC)...........................................................................................61

COMPOSITION CONTROL (CC).....................................................................................62

INSTRUMENTATION............................................................................................................. 63

SENSOR-TRANSMITTER............................................................................................... 63

CONTROL VALVES......................................................................................................... 64

GUIDELINES for SELECTION of VALVE for LIQUIDS.............................................. 75

STABILITY of CLOSED-LOOP CONTROL SYSTEMS........................................................ 76

PID CONTROLLER DESIGN................................................................................................ 95

DIRECT SYNTHESIS.......................................................................................................96

INTERNAL MODEL CONTROL....................................................................................... 98

FEEDFORWARD CONTROL.............................................................................................. 109

RATIO CONTROL................................................................................................................117

CASCADE CONTROL......................................................................................................... 118

MULTI-LEVEL CASCADE.............................................................................................. 124

INFERENTIAL CONTROL...................................................................................................125

DISTILLATION............................................................................................................... 125

REACTOR......................................................................................................................126

SELECTIVE CONTROL...................................................................................................... 127

OVERRIDE CONTROL.................................................................................................. 127

1

AUCTIONEERING CONTROL.......................................................................................129

SPLIT-RANGE CONTROL.................................................................................................. 131

VALVE POSITION CONTROL.............................................................................................133

MULTIPLE INPUT MULTIPLE OUTPUT (MIMO) SYSTEMS..............................................137

DECENTRALIZED CONTROL.......................................................................................138

PAIRING................................................................................................................... 139

CENTRALIZED (MULTIVARIABLE) CONTROL.............................................................144

DECOUPLERS.........................................................................................................144

CONTROL of DISTILLATION COLUMNS.......................................................................... 146

DISTURBANCE..............................................................................................................146

PRESSURE CONTROL................................................................................................. 147

LEVEL CONTROL..........................................................................................................149

COMPOSITION CONTROL........................................................................................... 149

COMMON CONFIGURATIONS..................................................................................... 151

2

The process represents the set of transformations converting raw materials into products

(and byproducts).

The term «process» is used to indicate both the processing operation and the processing

equipment, and usually it is equal to the word «system».

We need process control:

●​ for safety reasons → a process must be operated far from potentially harmful

conditions (for human lives, environment and equipment)

●​ for profit reasons:

○​ reaching final product specifications

○​ minimizing waste and environmental impact

○​ minimizing energy and use of materials

○​ maximizing production rate

A process control system is made for:

●​ monitoring the state of certain variables (controlled variables) that can indicate the

state of the system

●​ introducing changes in appropriate variables (manipulated variables) to keep

controlled variables at their target

CONTROL LOOP (CLOSED)

The basic elements of a control loop are:

●​ a sensor (or transmitter) that monitors a certain variable

●​ a controller that decides which corrective action must be taken based on the signal of

the sensor

●​ a final control element on which the action is made

For example, in this process, where a sensor monitors the outlet temperature of the process

fluid, the controller will partially open (or close) the steam valve as this outlet temperature

changes, to obtain again the nominal T. There are many reasons for the change of the

temperature:

●​ leak of steam

●​ fouling, so the global heat exchange coefficient decreases ( )

= ∆ 3

●​ steam at different pressure

●​ input process fluid at a different temperature

In a process we can identify:

●​ inputs, which are those variables that can induce changes in the internal process

conditions. They could be:

○​ manipulated variables that can be adjusted by the system

○​ disturbances which are impossible to modify

●​ outputs, which are measurements to obtain the internal condition of the system

●​ states are a minimum set of variables that are necessary to completely describe the

internal condition of the system

TYPICAL CONTROL PROBLEMS

●​ regulatory control (most common), where the control system tries to cancel the

effect of disturbances to maintain the output to a constant target (called set point)

●​ servo [asservimento] control, where the control system tries to make the output track

a set-point trajectory (for example in a batch reactor the temperature naturally

change its temperature)

EXAMPLE: MIXING PROCESS

A tank is used to mix two inlet streams to obtain an output stream with an A composition of

x . We can manipulate the flow w , which is pure component A to reach that objective with a

SP 2

control valve. The controlled variable is the composition of the outlet.

Assumptions:

●​ perfect mixing

●​ composition of stream 2 doesn’t change

●​ composition of stream 1 is x and can change over time

1

●​ volume is constant (thanks to an overflow line / weir) → the volume will have an

impact on the mixing

●​ no reaction

The flow needed to obtain the desired concentration is obtained from the global material

balances: overall balance → + =

1 2

species A balance → →

+ = + =

1 1 2 2 1 1 2 4

−

design equation of the process → 1

= 1−

2 1

Now let’s assume that the composition x could change over time. We have two possibilities

1

to control the specified composition x :

SP

●​ feedback control

○​ the sensor detects a variation in the outlet composition (for example x

increases), the information is sent to the controller that acts on the control

valve, decreasing the flowrate w 2

○​ we should change w proportionally to the distance of the outlet

2

composition from the specified composition (error)

[ ]

○​ In this case the control law is where K is the

() = + − () C

2 2

controller gain (the higher the K the more sensitive is the controller)

C

●​ feedforward control

○​ the sensor detects the variation in the inlet composition (for example x 1

increases), the information is sent to the controller that acts on the control

valve, decreasing the flowrate w 2 5

− ()

○​ In this case the control law is that remarks the steady

1

() = 1−

2 1

state condition

●​ feedforward + feedback that consists in measuring both x and x to manipulate w

1 2

●​ design change: for example using a much larger tank to avoid large variation of

x if the composition of the input changes over time

outlet

In P&I diagrams we would see several instruments that are indicated with some letters and

numbers:

●​ the first letter indicates the measured or the controlled variable

●​ the consequent letters indicates the function

●​ the number indicates the loop number

We can distinguish between:

●​ SISO systems that manipulates a single variable and controls a single variable

●​ MIMO systems that manipulates and controls more than one variable 6

DYNAMIC MODELS of

CHEMICAL PROCESSES

Dynamic models are used:

●​ to improve process understanding on how it will behave in response to disturbances

●​ to develop the control strategy for a new process or improve an existing one by

tuning the controllers

●​ for operator training, in order to face up normal or abnormal situations

Dynamic models could be:

●​ first principles models that uses principles (knowledge) of chemical engineering,

chemistry, physics ecc to describe the system

○​ the used parameters have physical meaning

○​ they may be difficult to derive and computationally demanding

●​ empirical models that are obtained by fitting experimental data using an assigned

model structure

○​ can be used in a limited range of operating conditions since usually

extrapolation is not a good technique

○​ easy to derive and fast to compute

●​ semi-empirical (hybrid) models which are a combination of the first-principles

models and empirical models

FIRST PRINCIPLES MODELS

They can be derived by using two kinds of equations:

●​ conservation laws

●​ constitutive equations

First-principles models represent an abstraction of reality where:

●​ the level of details to be included in the model depends on its required use

●​ a compromise between accuracy and complexity is needed

The model must not be more complicated than just needed for the particular application

it is developed for → we must not create a model which is useful for all purposes.

When we use first principles models, differential equations arise, and we must integrate

them over time to understand how the system changes.

Considering a generic variable as output y we could have:

●​ inputs that have a direct effect on y because they directly appear in the differential

equation of y → y changes instantaneously

●​ inputs that have an indirect effect on y because they affect other variables that are

present in the differential equation of y → we must consider a chain of changes

○​ y does not change instantaneously

DEGREES of FREEDOM

1.​ List all quantities in the model that are known constants 7

2.​ Determine the number N of independent equations and the number N of process

E V

variables

3.​ Calculate the number of degrees of freedom as = −

4.​ Identify the N output variables that will be calculated using the process model

E

5.​ identify the N input variables that must be assigned in order to saturate the N

F F

degrees

a.​ If N = 0 it has a solution

F

b.​ If N > 0 the equations have an infinite number of solutions so we need to fix

F

some variables arbitrarily (the variables that the engineer can rely on to

manipulate the system)

c.​ If N < 0 the set of equations has no solution

F

EXAMPLE: SIMPLE HEATER

The simple heater is made by a stirred tank that contains a liquid of volume V and with a heat

capacity which is independent of temperature (reasonable for small ). We assume perfect

∆

mixing.

The inlet is at temperature T and the mass flowrate is w while the outlet stream is w(t) and

i i

has a temperature T(t).

An electrical resistance dissipates an electric power Q instantaneously to the liquid.

el

We assume that the inlet flowrate is equal to the outlet flowrate w = w and the density is

i

independent of temperature (reasonable for small ). There are not heat losses to the

∆

surroundings.

Build the dynamic model of the system, i.e. find w(t) and T(t).

Due to the assumption of perfect mixing, concentration and temperature are

homogeneous inside the tank, and the outlet temperature T(t) is equal at any time to the

temperature inside the tank.

Since w = w, writing the mass balance around the whole system: →

= − +

i () = − = 0

meaning that mass inside the tank is constant. The density does not depend on

temperature thus we also have a constant volume of the liquid ( ).

= ρ 8

In order to find a dynamic model for temperature, we need to write the energy balance:

[ ]

ρ = + + −

ℎ

Assuming that W is smaller than the other contribution, we can neglect it:

shaft [ ]

ρ = + −

Density and volume are constant with respect to time while w = w so:

i

( )

ρ = + −

The variation of enthalpy is given by so the enthalpy at a certain temperature is

=

( )

.

() = + ∫ = + −

( ) ( )

We choose and therefore the difference .

− = −

The balance eventually becomes ( )

ρ = + −

Therefore we have N = 1 equations and N = 4 variables (T, w , Q and T ) since c , and V

ρ

E V i liq i P

are parameters → .

= − = 4 − 1 = 3

Among the 4 variables:

●​ T , Q and w are inputs and function of time

i liq i

●​ T is an output and is a function of time

By changing one of the inputs, the time derivative of T will change and therefore

temperature will change instantaneously.

If the assumption of w = w doesn’t hold anymore, what will be the dynamic model of the

i

system?

In this case the volume is no longer constant in time. The material balance results

ρ

() .

= = ρ = −

The difference from the previous example is that we need to add an equation to the model:

−

=

ρ

The energy balance is

[ ] [ ]

→ .

ρ = + − ρ = + −

Knowing that :

=

[ ( ) ] ( ) ( )

ρ − = + − − −

[ ( ) ] ( ) ( )

ρ − = + − − −

[ ( ) ]

The time derivative is carried out:

−

( )

−

[ ( ) ] ( )

− = + −

T is a constant while can be substituted from the mass balance:

ref 9

( )

− −

( ) ( )

.

+ − = + −

ρ

The energy balance is therefore: −

( ) ( ) ( )

⎡⎢ ⎤⎥

ρ + − = + − − −

ρ

⎣ ⎦

By adding some algebra passages, we obtain the dynamic model of the system:

−

=

ρ

( )

= + −

ρ ρ

Even if the energy balance is basically the same in mathematical appearance with respect to

( )

the previous case ( ), now volume is no longer a parameter,

ρ = + −

but it is an input.

Now we consider again that w = w (constant volume), but the electric power Q dissipated

i el

by the resistance is not equal to the heat that is absorbed by the liquid Q because it has a

liq

certain mass and therefore a heat capacity c .

P

Since the resistance has its own c it will heat up and the heat will be delivered to the liquid

P

because of a difference in temperature by convection.

We assume that the temperature of the resistance T is equal everywhere (resistance is very

e

thin or the resistance of conduction is negligible).

The heat flux absorbed by the liquid is: ( )

= ℎ −

where h is the heat transfer coefficient and S the area of the resistance.

e e

The only dynamic equation is the energy balance (the same as the first example)

( ) ( )

, but now Q is so the balance becomes

ρ = + − ℎ −

liq

( ) ( )

ρ = ℎ − + −

T is a function of time so we need an energy balance for the resistance (control volume is

e ⎡⎢ ⎤⎥

the resistance): ρ = + +

⎣ ⎦

ℎ

where:

●​ Q is the rate at which heat is transferred to the resistance = −

res

●​ W is negligible

shaft

●​ there are no flow terms

●​ ρ =

●​ dH = c dT

e P,e e

[ ] ( )

() = − ℎ −

,

The equation must be solved together with the balance on the liquid volume:

[ ] ( )

() = − ℎ −

,

( ) ( )

ρ = ℎ − + −

10

If we step change the temperature of the inlet T , the temperature of the outlet T will be

i

directly affected by this variation (because T appears in the differential equation).

If we step change the heat flux coming to the resistance Q el

1.​ it will change directly T (apparently doesn’t change T)

e

2.​ T than will directly affec