the smart choice of fluid control systems

7/28/2019 The smart choice of Fluid Control Systems

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C o n t e n t s

1. O p e n - lo o p a n d c l o s e d - lo o p c o n t r o l P a g e 4

1.1. Func tion a nd seq uence of a n open-loop control s ys tem P a ge 4

1.2. Function and seq uence of a closed-loop control system Pa ge 5

2 . T h e c o n t r o l - lo o p P a g e 6

2.1. The elements of the control loop P a ge 7

2.2. The controlled s ys tem P a ge 82.3. The controller Pa ge 11

2.3.1. On/off controllers P a ge 12

2.3.2. Continuous-a c tion controllers P a ge 14

3 . A d a p t in g t h e c o n t r o l le r t o t h e c o n t r o l le d s y s t e m Pa ge 18

3.1. S elec ting the s uita ble controller P a ge 19

3.2. Determining controller pa ra meters P a ge 19

3.2.1. S etting guidelines in line w ith Zieg ler a nd Nichols P a ge 19

3.2.2. S etting guidelines in line w ith Chien, Hrones a nd Resw ick P a ge 21

4 . R a t in g a n d s e le c t io n o f c o n t r o l v a lv e s Pa ge 22

4.1. Introduction a nd definition of terms P a ge 22

4.2. Ra ting a nd s elec tion of pilot va lves P a ge 22

4.3. Ra ting a nd s elec tion of control va lves P a ge 23

4.3.1. Fluidics funda menta ls P a ge 23

4.3.2. Cha ra cteris tic curves P a ge 24

4.3.2.1. Va lve cha ra c teris tic P a ge 24

4.3.2.2. Flow cha ra cteris tic a nd ra ngea bility P a ge 24

4.3.2.3. Operating cha racteristic and press ure ratio P ag e 25

4.3.3. Ra ting a nd s elec tion P a ge 26


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The terms open-loop c ontrol a nd

closed-loop control are closely inter-

linked.

1.1 .F u n c t io n a n ds e q u e n c e o f a n o p e n -lo o p c o n t r o l s y s te m

An open-loop co ntrol sys tem is cha r-

ac terized b y the fact tha t one or more

input variab les of a sys tem influence

its output variables in accordance with

the systems own interrelations.

One everyday example of an open-

loop control system:

The inside tempe rature of a room is to

be maintained at a constant value as a

function o f the outside tem perature.

The roo m te mpe rature P V (outp ut va ri-

able of the open-loop control system)

is to b e maintained a t a c onsta nt value

by a djusting the electrica lly ope rated

mixer valve and thus , the temp erature

of the hea ting s upply line o r radiato r.

If the outside temperature changes,

the room temperature cha nges a s a

co nseq uence of this. The outside tem-

perature is referred to a s the disturb-

anc e va ria ble and identified w ith the

letter z (z1 in the example).

The ta sk of the o pen-loop co ntrol sys-

tem is to counteract the influence of

the o utside temperature disturbance

variable.

For this purpose, the outside tempera-

ture is measured via the outside temp-

erature sens or.

The mixer valve is a djusted or the

tempera ture o f the rad iato r varied via

the control unit.

The interrela tionship bet w een the o ut-

side temperature and the heating out-

put req uired for maintaining a cons tant

room temperature must be stored in

the control unit (e.g. in the form of

characteristic curves). Use of such a

control unit allows the influence of the

outside temperature on the room

temperature to b e elimina ted.

Bes ides the outside temperature, the

example shown in Figure 1 also con-

tains other disturba nce va riab les w hich

also affect the room temperature:

op en in g a w in d ow o r a d oor

chang ing wind cond it ions

presence of persons in the room.

Since it is not detec ted by the control

unit, the effect of these disturbance

variab les on the room tempera ture is

not compens ated for by the open-loop

control system.

The use o f such a n open-loop co ntrol

system is practical only if it can be as-

sumed that there is a low influence of

(secondary) disturbance variables.

The b loc k diagra m in Figure 2 s how s

the open action sequence cha racteris-

tic of an open-loop control system.

1 . O p e n - l o o p a n d c l o s e d - l o o p c o n t r o l

F igure 1 : O pe n- loop co ntro l o f the ins ide

t e m p e r a t u r e o f a r o o m

Outsidetemperaturesensor

Radiator

Electricallyoperatedmixer valve

He

ating

ret

urnline

Controlunit

Heatingboiler

PVRoomtemperature

Window

Disturbanc e variable z2Opening the window

Disturbance variable z1Change in outsidetemperature

M

Heating

supplyline

Disturbanc e variable z 4Number of persons in the room

Disturbanc e variable z 3Wind conditions

Disturbanc e variable z 2Opening the window

Outside

temperature

sensor

Control unit Mixer va lve Ra dia tor Room

Disturbanc e variable z1

Change in outside

temperature

Room

temperature

P V

F ig u r e 2 : B lo c k d ia g r a m o f t h e o p e n - lo o p c o n t r o l s y s t e m f o r r o o m t e m p e r a t u r e


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5

Disturbance variable z 2Opening the window

Disturbance variable z 1Change in outside temperature

Disturbance variable z 3Wind conditions

Disturbance variable z 4Number of persons in the room

Contro ller Mixer va lve Rad ia tor Room

RoomtemperatureP V+

Set-pointprogramm.

Set-pointvalue

Actual value

Outsidetemperaturesensor

Open-loop control Closed-loop control

Application If no or only one essential If several essential d isturb-

and meas urab le disturbance ance variab les are present.

variab le is present. If disturbance variab les

cannot be detected or can

be only poorly de tec ted with

meas uring sys tems.

If unforeseeab le d is turbance

variables may occur.

Advantages Low im ple me nt a tion e ffo rt . D is t urb a n c e v a ria b le s a re

No stabi lity problems detected and compensated

due to the open a ction for.

s eq uence. The preset va lue (set-point SP )

is more precisely complied.

Disadvantages Disturbance variables The equipment effort and

that occ ur are not detected complexity are greater than

auto ma tica lly. with open-loop co ntrol.

Me a sure me nt is re q uire d A m ea s ure me nt is a lw a ys

for ea ch disturbanc e varia ble required.

to be co mpensa ted for.

All interrelationships of the

system to be controlled must

be known in order to design

the control unit optimally.

Howe ver, if the effect of the o ther dis-

turba nce va riab les is so strong that it

also needs to be c ompensa ted for, it

becomes necessary to control the room

temperature on the ba sis of a closed-

loop control system. The block diagram

in Figure 3 shows the closed action

seq uence that is typical of a c losed -

loop control system.

1. 2 .F u n c t io n a n ds e q u e n c e o f a c lo s e d -lo o p c o n t r o l s y s t e m

The funda ment a l differenc e with res -

pect to an open-loop control system is

that the output variable of the system

(the actual value) is constantly meas-

ured and compa red w ith ano ther vari-

able (the set-point value or the refer-

ence varia ble). If the a ctua l value is

not equal to the set-point value, the

controller responds to this. It changes

the actual value by adjusting it to the

set-point va lue.

One everyday example of a closed-

loop control system:

The inside tempe rature of a room is to

be maintained at a preset temperature.

In this c as e, the effects of d isturba nce

variables on the room temperature

sho uld b e eliminate d.

The effects of a ll disturba nce variables

influencing the room temperature:

change in outside temperature

op en in g a w in d ow

chang ing wind cond it ions

are registered by measurement of the

room temperature and comparing this

with the set-point value. On the basis

of the co mparison between set-point

and actual value, the controller adjusts

the temperature of the heating supply

line or radiator via the mixer valve until

the req uired room tempera ture is

reached.

The ta ble below c onta ins a pplica tion

recommenda tions for open-loop and

closed-loop control.

F igure 4 : C lose d- loop co ntro l o f the

in s id e t e m p e r a t u r e o f a r o o m

Roomtemperaturesensor

Radiator

Electricallyoperatedmixer valve

Heating

returnline

Controlunit

Heatingboiler

PVRoomtemperature

Window

Disturbanc e variable z 2Opening the window

Disturbanc e variable z1

Change in outsidetemperature

Setpointadjuster

Actualvalue

Set-pointvalue

M

Heating

supplyline

Tab le 1 : App l icat ion , advantage s and d isadvanta ges o f open- loop and

closed- loop contro l

F ig u r e 3 : B lo c k d ia g r a m o f t h e c l o s e d - lo o p c o n t r o l s y s t e m f o r r o o m t e m p e r a t u r e


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Closed-loop c ontrol is a proces s tha t

is used in more than just technical

applica tions. Closed-loop control

sys tems run virtually everywhere a nd

always . The process of se tting the

required water temperature when

showering or complying with a speed

limit when driving a car involves

closed -loop c ontrol. These tw o exa m-

ples demo nstrate the task of a closed-

loop control system: adjusting, a

specific variable such as temperature,

speed , f low rate or pressure to a re-

q uired va lue.

In principle, a closed -loop co ntrol pro-

ces s a ppears to be a very simple one.

How ever, w hen implementing tec hni-

ca l closed -loop control systems , prob-

lems are very q uickly enco untered.

The preco ndition for correct function-

ing of a closed-loop control system is

the interplay of the individua l compo -

nents involved in a closed -loop co ntrol

sys tem. The tota lity of comp onents o f

a c losed-loop control system is referred

to a s the co ntrol loop . In the follow ing,

the control loop is explained in further

detail.

A control loop consists of the following

components :

the measuring instrument or sensor

for detecting the variable to be

controlled the controller, the core of the

closed-loop control system

the system to be controlled (this

pa rt is referred to a s c ontrolled

system).

One example:

The fluid level in a ta nk is t o b e ma in-

tained at or a djusted to a preset va lue.

In order to implement a close d-loo p

control system, it is nec ess ary to co n-

tinually me a sure t he filling level in the

tank (proc ess value).

This is d one he re, for example, by a n

ultrasonic level transmitter. The process

value is constantly compared with the

preset target filling level (set-point va -

lue), w hich is s et e.g . on a co ntrol unit

via buttons or se lector sw itches.

The comparison between process value

and set-point value is performed by the

co ntroller. If a d eviation o cc urs be-

tween the proces s a nd set-point value

(control deviation), the controller must

respond to it. The co ntroller has to a d-

just a suitable final co ntrol element or

actuator (a continuous-action control

valve in the exa mple show n in

Figure 5) so that the process value ad-

justs to the set-point value, i.e. so that

the control deviation becomes zero. If

the process value is higher than the

set-point value, the co ntrol valve must

be clos ed further. If the filling level is

too low, the valve must be o pened

wider.

Co ntrol deviations in a co ntrol loop

are caused by two fac tors :

d is turbance variab les

changes in the set-point value.

In our example, the follow ing tw o d is-

turba nce va riab les may o ccur:

outflow from the tank, occurring

abruptly due to opening of one or

mo re ON/OFF va lves

slow filling-level chang e due to

eva poration o f the fluid from the

tank.

2 . T h e c o n t r o l l o o p

Continuous-ac tion controlvalve

Continuous-ac tion leveltransmitter

S e n s o r

Conto l l er

S e n s o r

ON/OFFvalves

Contro l led s ystem

ControllerSe t-point value = req uired va lue

Process value = measured value

Ta nk

F igure 5 : Hardw are re pres entat ion o f a c losed- loop f ill ing - l eve l

cont ro l system


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7

2 . 1.T h e e le m e n t s o f t h ec o n t r o l lo o p

Block diag rams a re used to represent

co ntrol loop s. This mod e o f represen-

tation affords the ad vantag e that it

concentrates on the control-engineer-

ing prob lem. The interplay o f the in-

dividual components of the control

loop is represented grap hica lly. For

the exa mple of a closed -loop filling-

level control sys tem, the block diagram

looks a s follow s:

SP: S et-point value o r reference

variable (required filling level)

PV: Process value or controlled

va ria ble (meas ured filling level)

PVd: Co ntrol deviation

(a ctua l value s et-point value)

CO: Manipulated variable or con-

trol output (output value of

the controller)

z1: Disturbance variable 1

(outflow from the tank)

z2: Disturbance variable 2

(evaporation of fluid from the

tank)

The ba sic s tructure o f this b loc k dia-

gram corresponds to tha t of the c losed-

loop room temperature control struc-

ture. Thus, the follow ing gene ral block

diag ram (sho wn in Figure 7) ca n be

used to s ummarize the closed-loop

co ntrol engineering.

In this c as e, it is a ss umed that the ac -

tion po int of the disturba nce va ria bles

does not a lwa ys need to be at the out-

put of the controlled system, but that

the action of the disturbance variables

ca n be converted to this point.

Usually however, the simplified block

diagram shown in Figure 8 is used.

The disturbanc e variables a re combi-

ned and their action point is at the

output of the controlled system.

Block diag rams a re used to create a

closed-loop control engineering model

of a real syste m. The ma in compo -

nents of the control loop are represen-

ted by function blocks, frequently also

referred to a s transfer elements. The

functiona l relations hip b etw een the in-

dividual blocks and in regards to the

environment is s how n by a ction lines.

Each function block is characterized

by the dependenc e of its o utput signal

on the input signa l. This depe ndenc e

is d esc ribed by the respo nse. There

are numerous possible ways of repre-

senting the res pons e. The mos t con-

ventional wa y is sta ting the step re-

spo nse o r trans fer function. It is plot-ted as a simple timing diagram in the

relevant function block.

The s tep respons e is the c harac teristic

of the output signal, which occurs

when the input signal change s a brupt-

ly a s a function o f time. The trans fer

function (designated h(t)) is the step

respons e sta ndardized w ith respect to

the ma gnitude of the input step o r in-

put s igna l (h(t) = P Vo(t)/P Vi0).

Controller Continuous-ac tion position-ing valve

Measuredvalue p ick-upplus

transmitter

Referencepoint

S P P Vd C O P VTa nk

z 1 z 2

+

F igure 6 : B lock d iagram o f the c losed- loop f i lli ng - l eve l cont ro l system

z 1

z 2

Controller Fina l c ontrolelement/actuator

Measuredvalue p ick-upplustransmitter

Referencepoint

S P P Vd C O P V++Controlledsystem

+

+

+

P V

Disturbancevar. sys tem 2

Disturbancevar. system 1

F igure 7 : G enera l b lock d iagram o f the co nt ro l l oop

Controller Fina l c ontrolelement/actuator

Measured

value p ick-upplustransducer

Referencepoint

S P P Vd C O P V++Controlledsystem

+

P V

Contro l system , cont ro l ler

z

F igure 8 : S im p l if ied gene ra l b lock d iagram o f the c ont ro l loop


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If, in the b loc k diag ram, w e replac e the

individua l control loop elements o r

function b loc ks by the form of repre-

sentation shown in Figure 9, we obtain

the signal flow diagram of the control

loo p. Figure 10 show s the signa l flow

diag ram of the close d-loo p filling-level

control system.

The signa l flow diag ram or the b loc k

diagram is a n important aid to design-

ing control loops and for adapting the

co ntroller to the co ntrolled sys tem. In

many ca ses , ada ptation of the control-

ler is one of the most demanding tasks,

and one that requires basic knowledge

of controlled systems and controllers.

This is c ove red in the follow ing se c-

tions.

2 . 2 .T h e c o n t r o l le d s y s t e m

In order to s elect a suitable co ntroller

and b e ab le to a da pt it to the control-

led system (the system or equipmentto b e co ntrolled), it is ne ces sa ry to

have precise information on the b ehav-

ior of the controlled system. Factors

that must be known include to w hat

extent and in what timeframe the out-

put signal of the controlled system re-

sponds to cha nges o f the input signal.

Rea l trans fer elements differ from idea l

ones by virtue of the fact that they al-

most a lwa ys feature a time-delayed

respons e. This mea ns tha t a ce rtain

time elaps es until the output signa l re-

sponds to a chang ing input signa l.

Controlled s ystems ca n be subdivided

into two categories in terms of their

time respons e or stead y sta te condi-

tion:

Controlled systems with compen-

sation:

In the ca se of controlled s ystems with

compensa tion, the o utput variab le o f

the system reassumes steady-state

co ndition w ithin a spe cific p eriod. One

example of a controlled system with

compensa tion is the flow rate in a pipe.

If the degree of opening of a continu-

ous-action control valve is changed, a

constant f low rate is established af ter

a s pecific period a ssuming c onsta nt

press ure co nditions. The trans fer ele-

ment shown in Figure 11 symbolizes a

controlled system with compensa tion.

P V iInput signal

P V oOutput signal

Time

P Vi

Time

P VoTime

P Vo

P Vi

F ig u r e 9 : S te p r e s p o n s e o f a t r a n s f e r e le m e n t

Controller Fina l c ontrol element/actuator

Controlled sys tem

Measured value pick-up+ transmitter

S P P V d C O P V

z 2 z 1

+

P V

Time

xo

Time

xo

Time

xo

Time

co

?

F igure 10: S ignal fl ow d iagram o f the c losed- loop f ill ing - l eve l cont ro l system

P V iInput signal

P V oOutput signal

Time

P Vi

Time

P VoTime

P Vo

P Vi

F igure 11: Transfer e lem ent w ith p lo t ted s tep re sponse o f a co nt ro l led s ystem w i th

c o m p e n a s t a i o n


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9

Controlled systems without com-

pensation:

In the case of controlled systems with-

out compensa tion, there is no stea dy-

sta te condition. Even with a c onsta nt

input variable (greater than 0), the out-put signal changes at a constant rate

or acceleration. In the example of

clos ed -loo p filling-level co ntrol in a

tank, this relate s to a co ntrolled sys tem

without compensation. If the valve for

filling the ta nk is ope n a nd the ON/OFF

valves a re closed, the filling level in-

creases continuously without as suming

a s teady-sta te condition. In the cas e

of realc ontrolled systems without com-

pens ation, there is ge nerally a limita -

tion: in the examp le of c los ed-loop

filling-level c ontrol, t his res ults from

overflow ing o f the ta nk.

The tra nsfe r eleme nt s how n in Figure

12 symbolizes a controlled system

without c ompensa tion.

When selecting and setting the con-

troller, the aspect of whether the con-

trolled system is a controlled system

with or without compensation is of

crucial importa nce.

The mos t freq uently occ urring cont rol-

led systems and their transfer func-

tions are des cribed below in greater

de ta il. Ta ble 2 provides a n initial over-

view.

P V iInput signal

P V oOutput signal

Time

P Vi

Time

P VoTime

P Vo

P Vi

P V iInput signal

P V oOutput signal

Time

P Vo

P V iInput signal

P V oOutput signal

Time

P Vo

P V iInput signal

P V oOutput signal

Time

P Vo

P V iInput signal

P V oOutput signal

Time

P Vo

P V iInput signal

P V oOutput signal

Time

P Vo

P V iInput signal

P V oOutput signal

Time

P Vo

P V iInput signal

P V oOutput signal

Time

P Vo

D e s ig na t io n Tra ns fe r e le m e nt Ap p lic a t io n

P - s y s t e m Closed- loop pressurecontro l with f luidsClosed- loop f low-ratecontro l with f luids

1st -o rder t im e-delayeds y s t e m

Closed- loop pressurecontro l with f luidsand gasesClosed- loop f low-ratecont rol wi th gasesClosed-loop rotational-speed cont rol

2 n d - o r d e r t im e - d e l a y e ds y s t e m

Closed- looptemperature cont rol

3 r d - o r d e r t im e - d e l a y e ds y s t e m

Closed- looptemperature cont rol(steam via heatexchanger)

T t- s y s t e m Closed- loop conveyingquanti ty control onconveyor b el ts

T1 1st -o rder t im e-delayed

s y s t e m

Closed- loop pH cont rolClosed-loop conductivi tycont rolMixing two f luid streamsin one pipe

I -system Closed-loop f i ll ing levelcont rolClosed-loop conductivi tycont rol

F igure 12: Transfer e lem e nt w i th p lo t ted s te p respo nse o f a co nt ro ll ed system w ithout

c o m p e n s a t i o n

Tab le 2 : O verv iew o f typ ica l cont ro l led s ystem s


9/25

The P-element:

On the P -element o r proportiona l ele-

ment, the o utput signal follow s the in-

put signal directly, with no time delay.

Input and output signal are propor-

tiona l to e a ch o ther. There is no timedelay. Figure 13 shows the behavior or

step response o f a P -element.

The 1st-order time-delay element:

On the 1s t-order time-dela y element,

the output s igna l follow s the input sig-

nal with a time delay. In this case, the

output signal changes immediately,

but increa ses co ntinuously to the full

scale value with a time delay.

An ana log respons e is show n by the

voltag e through a ca pac itor when

cha rging via a series resistor.

The 2nd-order time-delay element:

The 2nd -order time-dela y element is a

controlled system with two d elays

(two 1st-order time-delay elements

co nnected in s eries). 2nd-order time-

delay systems a re charac terized by

three pa rameters, the system ga in Ks

and the tw o time cons tants Tu and Ta.

Unlike the 1st-order time-delay ele-

ment, the ste p respo nse is initially

charac terized b y a horizontal tangent,

features a flex point and then runs

asymptotically towards the full scale

value.

Rea l 2nd-order time-delay elements

are controlled systems with two (ener-

gy) stores, s uch as those o ccurringwhen tempering a tank via a heat ex-

changer.

What follows is a consideration of the

co ntrolla bility o f c ontrolled sys tems

with co mpens a tion (with time

de lays /de a d time). Thes e co ntrolled

systems can be approximated by the

approximate model shown in Figure 16.

0

P V iInput signal

I nput s tep

Time

P V oOutput signal

S t e p r e s p o n s e

Time

P Vo = Ks P ViP Vi

F ig u r e 1 3 : S t e p r e s p o n s e o f a P - e l e m e n t

P V iInput signal

I nput s tep

Time

P V oOutput signal


Time

P Vo = Ks P Vi

T1

P Vi

F igure 14: Step response o f a 1st -o rder t im e-delay e lem ent

P V iInput signal

I nput s tep

Time

P Vi

P V oOutput signal


Time

P Vo = Ks P Vi

TuTt Ta

Flex ta ngent

Flex point

F ig u r e 1 5 : S t e p r e s p o n s e o f a 2 n d - o r d e r t im e - d e l a y e le m e n t

P V iInput signal

I nput s tep

Time

P Vi

P V oOutput signal


Time

P Vo = Ks P Vi

TuTt TaTte

F igure 16: Approxim ate m ode l fo r cont ro l led system s w i th com p ensa t ion and

d e a d t i m e


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11

2 . 3 .T h e c o n t r o l le r

A closed-loop control system must

ensure that the process value is equa l

to the set-point value or is adjusted tothe set-point value under all circum-

stances, even under the influence of

disturbance variables, i.e. the control

deviation must be 0.

In addition, the closed-loop control

system must operate sta bly and the

proces s va lue ma y neither drift from

the required opera ting p oint nor osc il-

late a round it as a c onseq uence of a

chang e of d isturba nce va riab le or se t-

point va lue.

In order to me et thes e requirements

when d esigning a closed-loop control

system or a control loop, the appro-

priate controller must be selected for

the given controlled system and must

be ma tched to the controlled s ystem.In addition to knowledge of the dy-

namic and sta tic b ehavior of the con-

trolled system, this also necess itates

knowledge of the cha racteristics of

the individual controller versions or

co ntroller types .

In the following, the individual control-

ler types are described in greater de-

tail.

Controllers can initially be subdivided

into two main groups:

continuous-action co ntrollers and

on/off controllers.

On the basis of practical experience, it

is po ss ible to make a n a pproximate

sta tement o n the co ntrollab ility of a

controlled system with compensa tion

and equivalent dead time via the ratio

Tte/Ta .

The I-element:

On the I-element (integra l element),

the o utput variable is proportiona l to

the time integral of the input variable.

In the ca se o f a c onsta nt input varia-

ble, the output variable increases con-

tinuously.

The lag element:

On the lag element, there is a similar

behavior to that on the P-element with

sys tem g ain 1 (Ks = 1). How ever, the

lag element does not respond immedi-

ately to changes of the input value. In

the case of a s tepped change of the

input variable P Vi, the same s tepped

chang e of the o utput variab le oc curs

upon expiration o f time Tt.

P V iInput signal

I nput s tep

Time

P V oOutput signal


Time

P Vo = Ks PVi t

P Vi

F igure 17: Step response o f an I -e l em ent

P V iInput signal

I nput s tep

Time

P Vi

P V oOutput signal


Time

P Vo = P Vi

Tt

F ig u r e 1 8 : S t e p r e s p o n s e o f a l a g e le m e n t

Tte/Ta Controllability Control engineering

effort

< 0.1 Very w ell contro lla b le Low

0.1 0.2 Well c ontrolla b le Modera te

0.2 0.4 (S till) contro lla b le High

0.4 0.8 P oorly controlla b le Very hig h

> 0.8 B arely contro lla ble S pec ia l mea sures or c ontroller

structures required

Tab le 3 : Est im a t ion o f the c ont ro l lab i lity o f a syste m w i th com pens at ion


11/25

2 . 3 . 1. O n / o f f c o n t r o l le r s

On/off co ntrollers a re freq uently used

in temperature, filling level, pressure,

pH value and conductivity control

loo ps. On/off co ntrollers a re also usedin day-to-day applications or appli-

a n c es s uc h a s a ut om a tic c o ffe e m a -

chines, irons, refrigera tors or building

heating systems.

Two-point controller:

A two -point c ontroller ope rates in the

same w ay a s a switch. Its output can

ass ume only two s tates : sw itched on

or sw itched off. This me ans that the

co ntrolled final co ntrol element or a c-

tuator is either switched on or opened

or is switched off or closed. A two-

point co ntroller can be s een a s a

P co ntroller (co ntinuous-ac tion c on-

troller) w ith very high g a in.

A two -point c ontroller ca n only be

used in c onjunction with time-delayed

sys tems (1st-order or 2nd-order time-

delay systems) or controlled systems

without co mpens a tion (I-syste ms).

Figure 19 illustrates the principle of

opera tion o f a two -point c ontroller.

Ideally, the switch-on point and switch-

off point of the two -point c ontroller

would coincide. In practice however,the switch-on point and switch-off

point a re rec iproc a lly offse t. The inter-

val between the tw o s witching points

is referred to as (sw itching) hysteresis

and is identified w ith P Vh. If the pro-

cess value P V drops below the preset

set-point value SP minus half the hys-

teresis, the output of the c ontroller is

sw itched on (CO = 100 %).

If the process value rises above the

set-point value plus the hys teresis, the

output is sw itched off again (CO = 0 %).

Half of the hysteresis prevents a con-

sta nt sw itching on a nd sw itching off at

the same point owing to very minor

disturbances.

The de sc ribed two -point controller ca n

be used , e.g., for tempering a room.

The c ontrol resp ons e of the two -point

co ntroller is illustrate d below on the

ba sis of this e xample.

The d iag ram in Figure 21 sho ws the

principle c hara cteristic o f the proces s

value (temperature in the room).

The uppe r diag ram in Figure 21 sho ws

the proces s value (temperature) a nd

the s et-point value (des ired tempera -

ture) as a function of t ime and the lower

diag ram s hows the ma nipulated vari-

ab le (co ntrol output). At instant t = 0

(switch-on instant of the closed-loop

control system), the c ontroller sw itches

its output on and thus opens the heat-

ing va lve. For the period of the d ea d

time (Tt), the a ctua l value initially d oe s

n ot c h a ng e . Aft er t he d e a d t im e e x-

pires, it increa ses . The c hara cteristic

of the proces s va lue corresponds to

the step response of the controlled

system (characteristic of the process

value shown with a dashed line).

2

Control output CO

Process value PV

P VhHysteresis

CO = 100Outputswitched on

CO = 0Outputswitched off S P

Set-point value

F igure 19 : Pr inc ip le o f opera t ion , chara cte r is t ic o f a tw o -po in t

contro l l er

Heatingsystem

Process value

Set-point value

Radiator

Controloutput

Heating valve

Temp era turesensor

Room

F ig u r e 2 0 : H a r d w a r e r e p r e s e n t a t io n o f a c l o s e d - l o o p

t e m p e r a t u r e c o n t r o l s ys t e m

Process value

Set-point value

Controloutput CO

Set-point

Set-point+ 0,5* Hysteresis

Set-point 0,5 * H ys te re sis

100%On

0%Off

Time

Time

Ts

Tt Tt

Peak-to-peak

displacem

entof

the

processvalue

TtTt Tt

Tt

F igure 2 1: Proc ess va lue and m an ipu lated va r iab le o r

contro l l er ou tput o f the c losed- loop tem perature cont ro l

system as a funct ion o f t im e

Tt : Dea d t im e

Ts: System t im e constan t


12/25

13

If the proces s value reaches the set-

point value plus ha lf the hyste resis,

the co ntroller sw itches its o utput off

and thus closes the hea ting valve. For

the duration of the dead time, the pro-

ce ss va lue initia lly s till rises . After thedead time elapses, it drops. If the pro-

cess value drops below the s et-point

value minus ha lf the hyste resis, the

controller switches its output back on

and the heating valve opens. In the

sa me wa y as with the rise in tempera-

ture, it initially s till drops be fore it rise s

ag ain, due to the time delay or dea d

time of the system.

The co ntrol cyc le then sta rts ag a in.

The de ad time o f the co ntrolled sys tem

and the hysteresis of the controller

result in periodic fluctuat ion of the pro-

cess value a bout the s et-point value.

In order to assess the controllability of

a controlled s ystem with compensa tion

and d ead time using a two-point con-

troller, it is possible to use the ratio of

equivalent dead time to system time

co ns ta nt (Tt/Ts ). Figure 21 sh ow s ho w

the two times are determined from the

step response of a controlled s ystem.

The pea k-to-pea k displac ement o f the

actual value is chiefly dependent on

two as pec ts :

the controller hysteresis (this ca n

genera lly be set on the co ntroller).

The p ea k-to-pea k displace ment in-creases with increasing hysteresis

the system time constant (this ge-

nerally ca nnot be c hanged and is

determined by the s tructure of the

co ntrolled sys tem).

The low er the sys tem time co nsta nt,

the greater will be the peak-to-peak

displacement.

The lower the time constant of the con-

trolled system (system time constant)

and the hysteresis of the two-point

co ntroller a re, the m ore frequently the

co ntroller will sw itch. Depend ing on

the d esign o f the c ontroller, the final

co ntrol element/ac tuato r and the sys -

tem, increased wea r on the control

loop c omponents w ill oc cur in the ca se

of frequent switching. Consequently, a

two-point controller cannot be used

on controlled systems without time

delay (P-systems).

The ab ove-des cribed resp ons e of the

two -point c ontroller is referred to as

the heating function. Besides this

heating function, two-point control-

lers are also used for the cooling

function . The principle o f ope ration is

similar in this ca se but the co ntroller

output is s witched on w hen the set-

point value is excee ded . Figure 22

sho ws this p rinciple of op eration.

On mos t mod ern two -point co ntrollers,

it is pos sible to s et the c ircuit function

so that they can be used for both ap-

plications, depending on the setting.

3-point controller:

A 3-point controller is a switch, like a

two -point co ntroller. In contras t to the

2-point c ontroller, its o utput ma y a s-

sume three switch positions. 3-point

co ntrollers a re used, for example, in

the follow ing a pplica tions :

Closed-loop temperature control

Closed-loop temperature control of

a room o n which the disturba nce

variab les ca n be counteracted by

heating and cooling.

Closed-loop pH control systems

For neutralization o f media . The

required pH value can be set by

ad ding a cid or lye.

Control of motorized actuators

Motorized ac tuators operate valves

such as butterfly valves, ball valves

or ga te valves via me cha nisms . The

valves can be opened, closed or

stopped at a ny position.

Figure 23 d emons trates the principle

of o peration o f a 3-point c ontroller.Tte/Ts Controllability

< 0.1 Well co ntrollab le

0.1 0.3 C ontrolla b le

> 0.3 P oorly contro lla b le

Table 4 : Estim ation of the c ontro l lab i li ty

o f a s y s t e m w i t h c o m p e n s a t i o n a n d d e a d

t im e us ing a tw o-po in t cont ro l ler

Control output CO

Process value PV

P VhHysteresis

CO = 100Outputswitched on

CO = 0Outputswitched off S P

Set-point value

F igure 2 2 : Pr inc ip le o f ope rat ion , chara cter i s t ic curve o f

a tw o-po in t co nt ro ll er, co o l ing func t ion

Control output CO

Process value PV

CO = 100%Outputswitched on

CO = 0%Outputswitched off

P VhHysteresis

P VhHysteresis

CO = 100%Outputswitched onS P

Set-point value

Dead band

F igure 2 3 : Pr inc ip le o f ope rat ion , chara cte r is t i c curve o f

a th ree -po in t cont ro l ler


13/25

If the proce ss value PV rises ab ove the

preset set-point value SP plus half the

dea d ba nd plus ha lf the hysteresis,

the output o f the controller is s witched

on (CO = + 100 %). If the proces s va lue

drops below the set-point value plushalf the dea d b and minus half the hys-

teresis, the output is s witched ba ck

off a ga in (CO = 0 %). The d ifference

betwee n the sw itch-on and s witch-off

point is referred to a s the hysteresis

(as on the 2-point controller).

The co ntroller displays the s a me prin-

ciple of opera tion in the other direction.

If the process value PV drops below

the preset set-point value SP minus

half the dea d b and minus half the hys-

teresis, the output o f the c ontroller is

sw itched on (CO = 100 %). If the pro-

ces s value rises ag ain above the set-

point value minus half the dead band

plus ha lf the hyste resis, the output is

sw itched b a ck off a ga in (CO = 0 %).

A 3-point controller is shown by the

follow ing symb ol:

In principle, the 3-po int c ontroller co m-

prises two 2-point controllers whose

set-po int values a re mutually offset.

One of the controllers must be oper-

ated in cooling mo de a nd the other

must be operated in hea ting mo de.

2 . 3 . 2 . C o n t in u o u s - a c t i o n

c o n t r o l l e r s

Co ntinuous-action c ontrollers a re used

for dema nding c ontrol engineering

ta sks . Unlike on/off co ntrollers, t hemanipulated variable of these control-

lers may assume any value within the

range of the ma nipulated variab le/con-

trol output (i.e. the ra nge b etw een the

maximum and minimum possible value

of the c ontrol output, e.g . ge nerally

between the pos itions OPEN and

CLOSED on a control valve; this

then generally corresponds to a range

of 0 ... 100 %). Thes e co ntroller types

are charac terized b y the fact tha t they

respond to any change in control devi-

ation (P Vd = set-point value proces s

value) at the output.

There are different types of co ntinuous-

ac tion co ntroller:

P c ont ro lle r

P I c ont ro lle r

P D c ont ro lle r

PID controlle r

Thes e c ontroller type s d iffer by virtue

of their dynamics , i.e. by virtue of their

time respo nse of their co ntrol output

as a function of the control deviation.

The va rious co ntrollers a re cha racte ri-

zed by their step response, i.e. by the

time c hara cteristic o f their co ntrol out-

put after an abrupt change in input va-

riab le, the c ontrol deviation P Vd .

The individua l types of c ontroller are

explained in grea ter deta il below.

P controller

The P co ntroller is a purely propo r-

tional-action controller. Its control out-put is directly proportional to its input

variable, the control deviation PVd , in

sta tiona ry state . The co ntrol output of

the P controller is calculated as fol-

lows:

Depending on Kp, the control output

may drop b elow (Kp < 1) or increas e

above (Kp > 1) the control deviation.

Kp is referred to a s proportiona l gain

fac tor or p roportiona l coefficient.

As c an be seen from the ab ove ca lcu-

lation formula for the c ontrol output,

the P co ntroller req uires a co ntrol de-

viation (PVd = P V SP ) for forming a

control output (CO = Kp PVd). Fo r

this reas on, co ntrol loops with P c on-

trollers fea ture a permanent c ontrol

deviation which decreases with in-

c reas ing Kp. On the bas is of dynamic

aspec ts however, it is not pos sible to

achieve Kp of any arbitrary magnitude.

This ma y lead to insta bilities of the

co ntrol loo p.

Figure 26 show s the s tep response of

the P controller.

4

S P C O

P V

P V d

F ig u r e 2 4 : S y m b o lic r e p r e s e n t a t i o n o f

a 3 -po in t cont ro ller

S P P V d

C O 1

C O 2

P V

F ig u r e 2 5 : S y m b o lic r e p r e s e n t a t i o n o f

a 3 -po in t cont ro ller co m p r is ing 2-po in t

contro l l ers

C O = K p P VdC O = K p ( Pr o c e ss va lu e S e t -p o in t v a lu e )

P V dInput signal

I nput s tep

Time

C O


Time

CO = Kp P Vd

F ig u r e 2 6 : S t e p r e s p o n s e o f t h e P c o n t r o lle r


14/25

portiona l gain factor Kp to b e se t high-

er than on the pure P co ntroller.

Figure 28 show s the s tep response of

the P D c ontroller. On rea l PD c ontrol-

lers, the D-co mponent is time-dela yed

(time co nst a nt T1), w hich is a llow ed

for in the trans fer function sho w n. The

time co nsta nt T1 ca n, how ever, not be

set d irectly on mo st c ontrollers.

The P D co ntroller is repres ente d b y

the follow ing s ymbo l:

Characteristics of a PD controller: Like the P controller, the P D con-

troller operates without delay and

responds immediately to c hanges

in the co ntrol deviation.

The PD c ontroller responds to the

rate of cha nge o f the control devia-

tion and thus counteracts the build-

up of a higher c ontrol devia tion.

Control loops w ith PD controller

have a permanent co ntrol deviation.

The D-component of the controller

may lead to a situation in which

minor fluctua tions of the proce ss

value, and thus minor fluctua tions

of the control deviation, as ca used,

15

The P c ont roller is repres ente d by the

follow ing s ymbo l:

Characteristics of the P controller:

The P controller operates without

delay a nd very q uickly; it resp onds

immediately to changes in the

co ntrol de viation.

Control loops w ith P controller have

a permanent control deviation.

Sett ing pa rameter: Kp (proportional

ga in fac tor).

PD controller

On the P D co ntroller, not only the co n-

t ro ldevia t ion , but a lso it s ra te of change

is use d to form a co ntrol output. The

controller thus already responds when

a c ontrol deviation occurs and counter-

ac ts the occurrence of a higher control

deviation. The c ontrol output increas es

all the fa ste r the co ntrol deviation

chang es. The control output of the PD

controller is calculated as follows:

As c an b e seen from the ab ove ca lcu-

lation fo rmula for the c ontrol output,

the influence of the D-component is de-

termined via p a ram eter Td. The higher

Td bec omes , the higher the D-compo -

nent becomes when ca lculating the

co ntrol output.

As is also the case on the P controller,

co ntrol loop s w ith P D co ntroller have

a permanent control deviation which

decreas es with increas ing Kp. How-

ever, the D-component produces a

sta bilizing effect which a llow s the pro-

for example, by disturbances in

electrical transfer of the process

value (e.g. by standardized signals),

lead to c onsta nt f luctuations of the

co ntrol output.

S e t tin g p a ra m e te rs :

Kp (proportiona l ga in fa cto r)

Td (de riva tive-ac tion time)

PI controller

The P I controller cons ists of a propor-

tional component and an integral com-

ponent. The integral compo nent ca lcu-

la tes its s hare of the c ontrol output via

the time integra l of the co ntrol devia-

tion. If there is a co ntrol deviation, the

integral component increases the co n-

trol output. This a voids a permanent

control deviation as o ccurs on P con-

trollers a nd P D c ontrollers. The c ontrol

output of a P I controller is c a lculated

as follows :

As ca n be seen from the above ca lcu-

lation formula for the control output,

the influence of the I-component is de-

termined b y pa rameter Tr. The low er Tr

beco mes, the greater the I-component

beco mes w hen ca lculating the control

outp ut. Res et time Tr is t he time w hich

the controller requires to g enerate a

control output of the same magnitude

as that which occurs immediately as

the result of the P-component by

means of the I-component.

S P C O

P V

Kp

P V d

F igure 27: Sym bol i c representat ion o f

a P c ont ro l ler

O = Kp (Td +P Vd (t))d (P Vd (t))

d t

P V d = PV SP: Co ntrol de viat ion

Kp: Propor t ional ga in facto r

Td: De r ivat i ve-act ion t im e

P V d

I nput s tep

Time

C O


Time

CO = Kp P Vd

CO = Kp Tv/T1 P Vd

T1

F ig u r e 2 8 : S t e p r e s p o n s e o f t h e P D c o n t r o lle r

S P C O

P V

Kp, Td

P V d

F igure 29: Sym bol i c representat ion o f

a PD c ontro l ler

C O = K p ( (P Vd ( t )dt)+PVd ( t ) )1

Tr

P V d = PV SP: Co ntrol de viat ion

Kp: Propor t ional ga in factor o r

propor t ional coe f f ic i en t

Tr : Re set tim e


15/25

Figure 30 shows the step response o f

the PI controller.

The P I co ntroller is repres ente d b y the

follow ing symb ol:

Characteristics of the PI controller:

The PI controller is advantageous

in tha t it res pond s q uickly due to its

P-component and eliminates per-

manent control deviations owing to

the I-component.

S ince two parameters can be se t

on the P I controller (Kp a nd Tr), it is

poss ible to better ada pt it to the

dynamics o f the co ntrolled system.

S e t tin g p a ra m e te rs :

Kp (proportiona l ga in fa cto r)

Tr (res et time )

PID controller

The c ontrol output of the P ID control-

ler is ca lculated from the propo rtiona l,

integra l and differentia l compo nent.

The c ontrol output of the P ID control-

ler is calculated as follows:

6

P V dInput signal

I nput s tep

Time

C O


Time

CO = Kp P Vd

CO = 2 Kp P Vd

Tr

F ig u r e 3 0 : S t e p r e s p o n s e o f t h e P I c o n t r o lle r

S P C O

P V

Kp, Tr

P V d

F igure 3 1: Sym bol ic re prese ntat ion o f

the P I cont ro l ler

C O = Kp ( (P Vd ( t )d t )+Td +PVd ( t ) )1

Tr P V d = PV SP: Co ntrol de viat ion

Kp: Propor t ional ga in factor o r

propor t ional coe f f ic i en t

Tr : Re set t im e

Td: De r ivat i ve-act ion t im e

d (P V d ( t ) )

d t


16/25


17/25

There are two requirements ma de o n a

controller or control loop.

Variable command control:

In the cas e of variab le comma nd con-

trol, the set-point value is not constant

but chang es o ver the course of time.

The proces s va lue must be co rrecte d

to the set-po int value. The be havior of

a closed-loop control system in the

ca se o f changing s et-point value is

referred to as response to set-point

changes.

Fixed command control:

In the ca se of f ixed co mmand control,

the s et-point va lue is c ons tant. In this

ca se, the c losed -loop c ontrol system

has the task of maintaining the proces s

value at the value of the s et-point. Dis-

turba nce variab les a cting on the c on-

trolled system should be compensated

for in this c as e. The beha vior of a c los -

ed-loop control system with changing

disturbance variables is referred to as

disturba nce response.

In ad dition to ha ving a goo d response

to set-point changes , a c losed -loop

control system s hould, in most c as es,

feature a g ood d isturba nce response.

If a disturbance o ccurs in a c ontrol

loop, this leads to a control deviation

which the controller must compensate

for. When planning a closed -loop co n-trol system, disturbance variables are

of sp ecial significa nce. If several dis-

turba nce va riab les ac t on a controlled

sys tem, the individual disturba nce va r-

iab les gene rally ha ve a different time

respons e. Many disturba nce variab les

occ ur abruptly and others less ab rupt-

ly. Even the ma gnitude o f the influence

on the process value differs with the

individua l disturba nce variables.

When planning a close d-loo p co ntrol

system, there is the risk that the con-

trol loop beco mes unstable due to the

selected combination of controller and

controlled s ystem or ow ing to the se-

lected para meters of the controller. The

following behaviors may occur, e.g.

after occurrence o f a set-point chang e

or disturba nce va riab le c hange.

The c ont rol loo p is a t the st a bility limit;

the process value oscilla tes at constant

amplitude and frequency.

The c ontrol loop is uns ta ble. The pro-

cess value oscillates with increasing

amplitude.

The co ntrol loo p is sta ble; the proces s

value is corrected to the new set-point

value.

The b eha vior of a w ell-tuned o r well-

set c ontrol loop after a disturbance

variab le c hang e is simila r.

8

3 . A d a p t i n g t h e c o n t r o l l e r t o t h e

c o n t r o l l e d s y s t e m

Time

S et -poi nt va l ue S P

P r o c e s s v a l u e P V

F ig u r e 3 4 : P r o c e s s v a lu e c h a r a c t e r i s tic ,

co ntrol loop at th e s tab i li ty l im i t

Time



F ig u r e 3 5 : P r o c e s s v a lu e c h a r a c t e r i s tic ,

cont ro l l oop unstab le



Time

Tolerance band

Permanent controldeviation PVb

Maximum overshoot PVm

Rise timeTrise

Settling time Tset

F ig u r e 3 6 : P r o c e s s v a lu e c h a r a c t e r i s t ic

a f t e r a s e t - p o in t c h a n g e i n t h e c a s e o f a

stab ly ope rat ing cont ro l loop

Time

Di sturbanceVariable z

Tolerance band

Permanent controldeviation PVb

Maximum overshoot PVm

Trise

Tset



F igure 37: Process va lue character i s t i c

af ter a d is turbance var i ab le change in

t h e c a s e o f a s t a b ly o p e r a t i ng c o n t r o l

loop


18/25

19

The q uality of a close d-loo p c ontrol or

a c ontrol loop is as sess ed on the ba sis

of the following parameters.

Permanent control deviation PVb

The perma nent c ontrol devia tion o c-curring after the adjustment process

has decayed .

Overshoot PVm

Maximum value of the process value

or of the co ntrolled varia ble minus the

proces s value in stead y state.

Rise time Trise

The time w hich elaps es a fter a set-

point or disturbance variable change

until the proces s va lue oc curs for the

first time in an agreed tolerance band

(e.g . 2 % or 5 %) ab out its s ta tiona ry

end value.

Settling time Tset

The time w hich elaps es a fter a set-

point or disturbance variable change

until the process value occurs and

permanently remains in an agreed

toleranc e ba nd (e.g. 2 % or 5 %) a b-

out its stationary end value.

On the basis of these parameters, it is

pos sible to formula te the req uirements

mad e of an optimally tuned c ontrol

loop:

permanent control deviat ion PVb =

0 wherever poss ible

maximum overshoot PVm as low as

possible

se t tling time Tde t as low as possible

ris e tim e Trise as low as possible .

3 . 1.S e le c t in g t h e s u i t a b lec o n t r o l l e r

The co ntroller must be ma tched to the

controlled sys tem in order for a co ntrolloop to o perate optimally.

Suitab le co mbinations o f controllers

and controlled s ystems o n which a

stab le co ntrol response ca n be ac hiev-

ed b y a ppropriate setting o f the con-

troller pa rameters:

Kp (proportional ga in fac tor)

Tr (reset time)

Td (derivative-ac tion time)

can be establ ished on the bas ed on

the dynamics and sta bility of c ontrol

loops and allowing for empirica l values.

There are, of course, also control loops

necess itating other combinations of

co ntrolled sys tem /co ntroller.

Ta ble 6 provides a n overview o f suit-

able combinations of controllers and

controlled systems.

3 . 2 .D e t e r m in in g t h ec o n t r o lle r p a r a m e t e r s

After a s uita ble controller has bee n se-

lected, a seco nd step is to match the

parameters of the controller to the

controlled system.

A number of setting guidelines with

which a fa vorable s etting o f the con-

troller parameters can be determined

experimentally are cited in control-

engineering literature.

In order to a void incorrect se ttings ,

the co nditions under which the rele-vant setting guidelines were establish-

ed must always be followe d. Bes ides

the characteristics of the controlled

system and of the controller them-

selves, other important factors include

whether a disturbance variable change

or a reference variable change is to be

compensated for optimally.

3 . 2 . 1. S e t t in g g u id e lin e s

in l in e w it h Z i e g l e r a n d

N i c h o l s

( o s c il la t i o n m e t h o d )

With this metho d, the c ontroller pa ra-meters are set on the ba sis of the be-

havior of the co ntrol loop at the s tab ili-

ty limit. The c ontroller pa ram eters a re

initially s et s o that the c ontrol loop

starts to oscillate. Critical characteris-

tic values then occur which allow con-

clusions to be drawn in terms of the

co ntroller pa rameters. The p recondi-

tion for using this me thod is tha t the

control loop can be caused to osci lla te .

Procedure:

Set c ontroller as P controller (i.e.

Tr = 9999, Td = 0), initia lly s elec t a

low value for Kp.

Set the required set-point value.

Increase Kp until the process value

executes an undamped

sus ta ined osc illat ion (se e Figure 38).

The proportiona l gain facto r set a t the

sta bility limit is des igna ted Kcrit. The

resultant period of osc illation is des ig-nated Tcrit.

P rocess va l ue

Time

Tcri t

F ig u r e 3 8 : P r o c e s s v a lu e c h a r a c t e r i s t ic

of the co ntrol loop at th e s tab i li ty limi t in

o r d e r t o d e t e r m i ne t h e c o n t r o l p a r a -

m e ters in l ine w ith Z ieg ler and N icho ls


19/25

0

Controlled system Continuous-action controllers On/off controllers

P PI PD PID 2-point 3-point

P-element Uns uita b le Uns uita b le Uns uita b le Uns uita b le Uns uita b le

PTt- Uns uita b le Uns uita b le Uns uita b le Uns uita b le Uns uita b le

element

1st-order Res po ns e Dis turb anc e Uns uita ble Uns uita ble S uita ble S uita ble

time-delay to se t-poin t response

element c ha ng es w ell-s uited

well-suited

1st-order Uns uita b le Uns uita b le C ondition- C ondition-

time-delay a lly a lly

element s uit a ble if s uit a ble if

with dead hy st ere s is h ys te re s is

time is low is low

2nd-order Uns uita b le Res pons e Dis turba nc e S uita b le S uita ble

time-delay to se t-poin t response

element c ha ng es w ell-s uited

well-suited

2nd-order Uns uita b le Uns uita b le Uns uita b le Uns uita b le

time-delay

element

with dead

time

3rd-order Uns uita b le Uns uita b le Uns uita b le Uns uita b le

time-delay

element

I-element Res po ns e Dis turb anc e Res po ns e Dis turb anc e S uita ble S uita ble

to se t-poin t response to se t-poin t response

cha nges w ell-s uited c ha ng es s uita ble

w ell-s uited s uita b le

I-element Uns uita ble Uns uita ble Res po ns e Dis turb anc e S uita ble S uita ble

with to se t-poin t response

1st-order changes well-suited

delay well-suited

Tab le 6 : Su itab i lity o f cont inuous-ac t ion a nd on /o f f c ont ro l lers fo r com binat ion w i th var ious types o f cont ro l led system

Time

P Vo

Time

P Vo

Time

P Vo

Time

P Vo

Time

P Vo

Time

P Vo

Time

P Vo

Time

P Vo

Time

P Vo

Responseto set-point

changeswell-suitedDisturbanceresponsewell-suited

Responseto set-pointchangessuitableDisturbanceresponse s.

Responseto set-pointchangessuitableDisturbanceresponsesuitable

Responseto set-pointchangeswell-suitedDisturbanceresponsewell-suited





Responseto set-pointchangessuitable

Disturbanceresponse s.


20/25

2 1

The co ntroller para meters c a n then be

ca lcula ted in acc orda nce w ith Tab le 7

from Kcrit and Tcrit.

The Ziegler and Nichols s etting w ere

determined for P systems with 1st-

order time de la y a nd d ea d time. Theyap ply only to c ontrol loop s w ith dis-

turba nce response.

3 . 2 . 2 . S e t t in g g u id e l in e s

in l in e w it h C h i e n , H r o n e s

a n d R e s w ic k ( c o n tr o l o u t -

p u t s t e p m e t ho d ) :

With this me thod , the co ntroller pa ra-

meters are set on the basis of the tran-

sient respons e or step response of the

co ntrolled sys tem. The c ontroller emits

a co ntrol output s tep. The times Tu

a nd Tg a re rea d o ff (se e Figure 38)

from the characteristic of the process

va lue or the co ntrolled va ria ble. The

control output step must be selected

so that a n proces s va lue is produced

that lies within the range of the subse-

quent operating point of the c ontrolled

sys tem. Ks is the propo rtiona l coeffi-

cient of the c ontrolled sys tem. It is

ca lculated as follows :

Procedure for determining the step

response of the controlled system:

Sw itch the controller to manual

operating mod e.

Emit a control putput s tep and

record the process value with a

recorder.

In the ca se of critica l controlled sys-

tems (e.g. in the case of the risk of

overheating), switch off in good time.

It must be no ted that the proces s va lue

may increase a ga in a fter sw itch-off on

thermally sluggish systems.

Ta ble 8 lists t he co ntroller pa rame ter

settings a s a function o f Tu, Tg and Ks

for response to s et-point changes and

disturba nce response and for an ape -

riodic co ntrol proces s and a control

proce ss with 20% overs hoo t. The figu-

res a pply to sys tems w ith P-respons e,

with dead time and with 1st-order de-

lay time.

Setting the parameters in line with Ziegler and Nichols:

Controller type Setting parameters

P controller Kp = 0.5 Kcrit

PI controller Kp = 0.45 Kcrit Tr = 0.85 Tcrit

PID controller Kp = 0.6 Kcrit Tr = 0.5 Tcrit Td = 0.12 Tcrit

Tab le 7 : Contro l ler p aram ete rs in li ne w ith Z ieg ler and N icho ls

Tab le 8 : Contro ll er param ete rs in li ne w ith Ch ien , Hrone s and Re sw ick

Cont rol output CO

Time


TimeTu

C O

P V

Tg

F ig u r e 3 8 : S t e p r e s p o n s e o f a c o n t r o lle d

system for determ in ing cont ro l para-

m e ters in li ne w ith Ch ien , Hrones a nd

R e s w i c k

K s = P V

C O

PV: M agn i tude o f the p roce ss va lue s tep

C O : M a g n it u d e o f t h e c o n tr o l o u t p u t s t e p

Controller type Parameters settings

Aperiodic control process Control process with overshoot

(approx. 20% overshoot)

Resp. to set-p. changes Disturbance response Resp. to set-p. changes Disturbance responseP controller

PI controller

PID controller

Kp = 0.3Tg

Tu KsKp = 0.3

Tg

Tu KsKp = 0.7

Tg

Tu KsKp = 0.7

Tg

Tu Ks

Kp = 0.35 Tg

Tu KsKp = 0.6

Tg

Tu KsKp = 0.6

Tg

Tu KsKp = 0.7

Tr = 1.2 Tg Tr = 4.0 Tu Tr = Tg Tr = 2.3 Tu

Tg

Tu Ks

Kp = 0.6 Tg

Tu KsKp = 0.95

Tg

Tu KsKp = 0.95

Tg

Tu KsKp = 1.2

Tr = Tg Tr = 2.4 Tu Tr = 1.35 Tg Tr = 2.0 Tu

Td = 0.5 Tu Td = 0.42 Tu Td = 0.47 Tu Td = 0.42 Tu

Tg

Tu Ks

Setting the parameters in line with Chien, Hrones and Reswick:


21/25

In addition to controllers and sensors,

ac tuato rs or final co ntrol elements

which intervene in the process to be

controlled as a function of the signals

preset by the controller and which

change the proces s variable to be con-

trolled are required for constructing

closed-loop control systems.

4 . 1.In t r o d u c t i o n a n dd e f in it io n o f t e r m s

Va lves are fina l co ntrol elements or ac -

tuators for influencing fluid streams in

pipe s ystems. In a cco rda nce w ith DIN

IEC 534, a posit ioning valve is a device ,

opera ted with a uxiliary ene rgy, w hich

varies the flow rate in a proces s. It con-

sists of a valve fitting, connec ted to the

actuator, which is ab le to change the

position of the restrictor in the valve a s

a function o f the c ontroller signa l (co n-

trol output). G enerally, a control sys tem

is required b etween the a ctuator a nd

controller to ac t a s a signal transd ucer

a nd/or amp lifier. In the ca se of ma ny

pos itioning valves, the c ontrol sys tem

is integrated as far as a f ield bus inter-

face in the ac tuator. In accordance with

DIN IEC 534, positioning valves are sub-

divided on the b as is o f the follow ing

types:

Valves ca n also be classified in ac cord-

ance with the distinction between the

main functions of final control elements/

actuators in compliance with DIN

19226, d ividing them into C ONTROL-

fina l co ntrol elements and ON/OFF-final

co ntrol elements .

ON/OFF valves ha ving only tw o o r a

few circuit states are used for open-

loop control tas ks. Control valves which

are able to continuouslys et the fluid

stream are used for closed-loop pro-

ces s co ntrol tas ks. ON/OFF valves

and co ntrol valves ha ve extremely dif-

ferent tasks in some ca ses , so tha t the

rating a nd selection of both valve types

necessitate greatly different procedu-

res.

4 . 2 .R a t in g a n d s e le c t io no f O N / O F F v a lv e s

This kind of va lves ca n either open o r

clos e a line (ON/OFF va lve) or s witc h

over a m ate rial stream from o ne line to

another.

An important criterion for the valve to

be selected is initially that the required

fluid q uantity be a ble to flow through

the va lve a t a given pressure differen-

tial, i.e. the valve cross-section must

be a deq uately large. The follow ing rule

of thumb frequently applies: the line

cross-section is equal to valve (fluidic

connec tion) cross-s ection. The next

requirement is tha t the va lve be ab le to

switch ag ainst the maximum press ure

differential, i.e. that the valve ac tuato r

be ad eq uately pow erful. The ma ximum

sw itcha ble press ure d ifferential is spe -

cified in the data sheet. If the type of

auxiliary energy has been defined and

the ma teria l suitability has bee n

checked, it is already possible to def ine

a specific valve type and to select the

specific valve.

2

4 . R a t i n g a n d s e l e c t i o n o f

c o n t r o l v a l v e s

Valve type Restrictor

Lift-type valve The restricto r is g enerally des igned as a c one.

Through-w ay valve It moves perpendicular to the sea t plane.

3-way valve

Angle valve

Gate valve The restricto r is a fla t or wed ge-s hap ed plate.

Diaphragm valve A flexible restrictor performs the function of

of isolation and sealing.

Ball valve The res trict or is a ba ll w ith a c ylindrica l bo re

or a seg ment of a ba ll.

Butterfly valve A disc mounted in such a w ay a s to allow it to rotate.

Plug valve The restricto r may be a cylindrica l, c onical oreccentrically mounted ball segment.

Tab le 8 : C lass i f icat ion o f posi t ion ing va lves in acc ordanc e w ith D IN IEC 5 3 4


22/25

u m e an f low ve loc i ty

D h hydrau l ic d iam ete r ,

Dh = kA /U

v kinem a t ic v iscos i ty

2 3

4 . 3 .R a t in g a n d s e l e c t io no f c o n t r o l v a l ve s

Control valves a re a ble to c onsta ntly

chang e their opening cross-sectionand thus continuously influence fluid

stream s. They thus represe nt variable

flow resisto rs.

4 . 3 . 1. F lu i d ic s

f u n d a m e n t a l s

Flow resistances occur in process in-

sta llations in va rious forms:

as resistances in capillaries , gaps,

nozzles, diaphragms and

valves

as line resistances in pipes, hoses

and ducts

as leakage res is tances in gaps and

porous components.

In gene ral, the ratio o f press ure drop

p to fluid flow Q can be defined as

the flow resistance R of a component.

Ba sically, a distinction must be mad e

between two types of resistance on

the bas is of the physica l ca uses:

fric t ional resistances due to f low

involving friction

cross-sectional resistances owing

to variations in the flow cross-

section.

The follow ing distinction betw een

cas es for dependence between p

and Q must be made for frictional re-sista nces in non-comp ressible fluids

as a function of the Reynolds number

From this, we ca n conc lude that the

flow resistance R is constant only in

the ca se o f laminar flow owing to

p Q. Otherwise, a non-linea r rela-

tionship a lwa ys a pplies betwe en pres-

sure drop p a nd fluid flow Q.

The fo llow ing a pplies to fluid resist-

ances in the case of cross-sectional

variation in non-compressible fluids

and with turbulent flow :

The p ermanent pressure loss p loss is

taken as the ba sis for the flow resist-

anc e R. The flow resista nce c oefficient

is introduced as a non-dimensional

pressure los s by referring the p erma-

nent pressure loss to the dynamic

pressure.

The follow ing a pplies to the mo del

case of flow through an orifice plate:

a nd

and, after introduction of a flow coef-

ficient a, w e o btain the flow -rate equa -

tion

In the ca se o f a high Reynolds number,

i.e. in turbulent flow, the following ap-

plies to cross-sectional resistances in

non-comp ressible fluids : p Q2.

The flow-ra te va ria ble kv which is de-

fined a s follow s is us ed to identify

valves a s o rifice-type fluidic co mpo-

nents:

the kv value (in m 3/h) is the vo lume

flow of wa ter at + 5 to + 30 C pas sing

through the va lve a t the releva nt

stroke s with a pressure loss

p0valve = 100 kP a. (1 ba r; 14.5 psi)

Analogous to this, the flow-rate coef-

ficient cv is des cribed in the America n

literature, defined a s fo llow s:

the cv va lue (in US ga l/min) is the vol-

ume flow of wa ter at 60 F which pas-

ses through at a press ure loss of 1 psi

with the relevant s troke s .

Moreover, the QNn-Wert (in l/min) is

specified as the flow characteristic for

compressed air under stand ardized

conditions for pneumatic valves.

The follow ing c onversion fa ctors ap ply:

Kv cv: kv = 0.86 cvKv : kv = 4 d 2/()1/2

Kv QNn: kv = 1 078 QNn

R =p

Q

Re =uD h

v

Ra ng e of the Flow form Interrela tions hip

Reynolds number betw een p and QRe low La mina r p Q

Re hig h Turbulent p Q7/4

Re Re critical Tra ns itio na l fo rm To b e d ete rm. e xpe rime nta l.

=ploss

u 2

2

Q = Aorifice2p

= 1Apipe

Aorifice

((

ploss = (1 m b ) pB

(( [

[2

contract ion coef f i c i en t

f low coe f f ic i en t de nsi ty of the f lu id

p B ef fect . p ress . th rough the or i fice p late

m B ope n ing ra t ioAorif ice

Apipe


23/25

4 . 3 . 2 . C h a r a c t e r is t ic

c u r v e s

4.3.2.1. Valve characteristic

The va lve c hara cteristic represe nts the

dependence of the aperture cross -sec tion A on the s troke s of the va lve

sp indle: A = f(s).

In the simplest c as e, the valve charac -

teristic is linea r, i.e. A = K1 s :

Note: Des pite the linea rity of the valve

cha racte ristic, the re is no linea r inter-

rela tions hip betw een the vo lumetric

flow rate Q through the valve and the

valve stroke s due to non-consta nt

pressure drops through the valve.

The equa l-percenta ge va lve chara c-

teristic is d esc ribed by a consta nt per-

centage increase in the aperture cross-

section A with stroke s (referred to the

releva nt a perture c ross-s ection A pre-

sent).

The equa l-percenta ge valve chara cte-

ristic a pproxima tes practica l req uire-

ments to a greater extent than the li-

near characteristic, since

low variat ions in stroke s cause

low A, i.e. fine infeed movements high variat ions in stroke s cause

high A, i.e. coarse infeed move-

ments.

In the ca se o f s = 0, a minimum aper-

ture cross-section A0 is pres ent. The

valve closes only with an additional

sea ling edg e.

Va rious valve cha rac terist ics a re imple-

mented by the contour of the closure

elements, the valve cones. Convention-

al designs include, for example, par-

ab olic cone, lantern cone, perforated

co ne, V-port cone and ma ny others.

4.3.2.2. Flow characteristic and

rangeability

The mos t co nventiona l charac teristic

curve required for valve selection is

the flow cha racte ristic. The flow cha r-

ac teristic represents the depe ndence

of the standardized flow rate kv on the

st roke s: kv = f(s).

ON/OFF flow characteristic

Due to t heir plate co ne, ON/OFF valves

initially have a linear flow characteristic

in the rang e of s ma ll stroke (up to a p-

prox. 30 % stroke). At an opening angle

of 30 to 40 % stroke (s), such valves

alrea dy a chieve app rox. 90 % flow

rate (kv). As the aperture opens even

further, the flow rate rises only very

slowly through to the ma ximum, a t full

stroke. Since the total stroke of the

ON/OFF va lves is low in rela tion to the

stroke of control valves, the actualtas k of producing a high c hange in

flow rate w ith low s troke is performed.

Linear flow characteristic

In the simplest c as e, the flow charac -

teristic is linea r, i.e. kv = K1 s :

Note: Unlike the linear valve character-

istic, the interrelations hip b etw een the

volumetric flow rate Q through the

valve and the va lve stroke s is linea r in

the case of the linear flow characteris-

tic.

Equal-percentage flow characteristic

The eq ual-percenta ge flow cha racte r-

istic is des cribed b y a consta nt per-

centage increase in the flow rate kv

with the stroke s (referred to the rele-

vant flow rate kv present).

At s 0 a m inimum a perture cros s-

section A0 is present, causing a mini-

mum flow kv0. The va lve clos es o nly

with an a dditional sea ling ed ge.

4

S t r o k e s

Aperture

area

A

F igure 3 9 : L inear va lve cha racte r is t ic o f

a c ont ro l va lve

S t r o k e s

Aperture

area

A

F ig u r e 4 0 : E q u a l - p e r c e n t a g e v a lv e

chara cter i s t ic o f a c ont ro l va lve

S t r o k e s

Flow

rate

kv

F ig u r e 4 3 : E q u a l - p e r c e n t a g e f lo w

character i s t i c

S t r o k e s

Flow

rate

kv

F igure 4 1: O PEN/C LO SE f low

character i s t i c

A = A0 e K 2 s

dA(s )

d s= K 2

1

A(s)

S t r o k e s

Flow

rate

kv

F igure 4 2 : Linear f low chara cter i s t ic

kv = kv 0 e K 2 s

d k v (s )

d s= K 2

1

kv (s )


24/25

The o perating c hara cteristic c urve

(V = f (s )) thus differs from th e flow

cha racte ristic c urve kv = f (s) of the

valve co nsidered in iso lation.

The ma gnitude o f the differenc e (thedeg ree of c hara cteristic distortion) is

represented by the pressure ratio .

The p ress ure ratio is s tated for the

fully open valve:

The beha vior of the system a s a n

interplay betw een s ource (pump) cha -

racteristic a nd loa d (valve) cha racteris-

tic can be shown in the characteristic

ma p (se e Figure 45):

The follow ing s ta nda rdized eq uation

applies to the operating characteristic

i.e. the operating characteristic de-

pends on the pressure ratio and the

flow characteristic

2 5

Examples of operating

c ha ra c t eris t ic s (f. ra n g ea b ilit y =

and various ):

Control valve with linear flow

characteristic:

Control valve with equal-percentage

flow characteristic

An a pproxima tion of a linea r opera ting

charac teristic can b e a chieved

in the case of linear flow c haracter-

istic with high pressure ratio

in the case o f equa l-percentage

flow characteristic with low pres-

sure ratio .

The no n-linea rities for b oth valve typ es

have a pproximately the same mag ni-tude at 0.3.

The ma ximum a perture c ross-s ection

Ama x is reac hed a t maximum stroke s.

The related kv value is referred to a s

kvs value. The rang e be twe en kv0 a nd

kvs is the total range of the manipulat-

ed variable of the valve.

The ratio of kv0 to kvs is referred to a s

the rangeability and defines a valve

characteristic value:

Conventional values a re a s follow s:

4.3.2.3. Operating characteristic

and pressure ratio

The ope rating cha rac teristic ide ntifies

the flow behavior of the valve under

opera ting c ond itions in the insta lla-

tion. It represents the dependence of

the volume flow V on the s troke s of

the va lve s pindle.

The following ma in eleme nts of the in-

sta llation influence the opera ting b e-

havior:

the pump; the pressure generated

by the pump drops a s p over the

entire installation

the tubes with the pressure drops

pLi

and other res is tances p' i in the in-

sta llation (shut-off valves , hea t ex-

changers, pipe elbows, branches,chang es in cross-section and other

insta lled fittings ).

1

k v0

k vs=

1

1

2 5

1

3 0

1

5 0= ; ;

V = f(s )

VentilPumpTube Tube

pL1 pv pL2

V

pL1 + pL2 = pL

p

Va lve

F igure 44: Pressure losses in

an insta l lat ion V

Rang e of the co ntrol output

pL pL

pv pvpv

pL0

pL0

Positioningvalve Operating po ints

Closes

Opens p = pvmin + pL

pvmin

pL

pL0

p

p0

= =p vop o

p vop vo+p Lo+p 'o

F ig u r e 4 5 : C h a r a c t e r is t ic m a p , s o u r c e

character i s t i c and load character i s t i cs

k vs

k v

1

1 + - 1

VV m a x=

k vs

k v

s

s m a x=

2

0 .2

0 .4

0 .6

0 .8

1.0

0 .2 0 .4 0 .6 0 .8 1.0

= 0 .1 0 .3 1.0

1

13 0

=

ss m ax

VV ma x

F igure 4 6 : Contro l va lve w i th l inea r f low

character i s t i c

0 .2

0 .4

0 .6

0 .8

1.0

0 .2 0 .4 0 .6 0 .8 1.0

= 0 .1 0 .3 1.0

1

13 0

=

s

s m ax

VV m ax

F igure 4 7: Contro l va lve w i th equal -per -

c e n t a g e f lo w c h a r a c t e r i s t ic

1

130

(( [[

((

p o : P r e s s u r e d r o p o ve r e n t ir e in s t a lla -tion

P vo : Pressure drop a t fu lly opene d va lve(m ax . flow )

p LO : Pressure drop at tube s, f it t ings. . .p ' o: P r e s s u r e lo s s a t p u m p ( a t m a x .

f low ra te )


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differential press ure a cross the va lve

and at s troke s. The kvs value, a nalog-

ous ly, is the quant ity a t s t roke

s = 100 %.

Analogous to the kv value the flow ratecoefficient cv is described in the Ame-

rica n literat ure. The follow ing c onver-

sion facto r applies: kv = 0.86 cv.

Se e a lso cha pter 4.3.1. Fluidics fun-

damentals.

The kv value must be ca lculated for the

current op erating da ta . A distinction

must be ma de b etween maximum load

(ma ximum q uantity Qma x, minimum

pmin kvma x) a nd minimum loa d (mi-

nimum quantity Qmin

, ma ximum pma x

kvmin) Both load ca ses must be ca l-

culated individua lly and be a djusted

on the basis of the valve rangeability.

The following applies to cold water:

The following applies in general to

fluids (sub-critical):

The following applies to fluids in

general (super-critical):

The kv value is c a lcula ted here in tw o

steps: the kv value for the evaporating

steam quantity kvD and the kv value for

the fluid kvF are ca lculated sepa rately

6

A linea r ope rating c hara cteristic is

achieved only if the valve features an

optimum flow characteristic as the

result of a spe cial valve co ntour.

Flow and operating c haracteristics forvalves with 0.3 and = :

4 . 3 . 3 . R a t in g a n d

s e l e c t i o n

Control valves must b e rated a nd se-

lected with a view to their spec ific ta sk

in order to be able to ensure a fault-

less co ntrol function.

Initially, the connection nominal dia-

meter must b e defined in acco rda ncewith the medium and the rela ted, e ffi-

cient flow velocity.

The following g uide line va lues a pply:

2 m/s for f luids

20 m /s f o r g a s e s

45 m/s for s team.

The fo llow ing formulae a re helpful for

practical application:

Fluids:

Gases:

Steam:

General:

In the case of simple control valves on

which a connection nominal diameter

is assigned directly to a kvs value, the

anticipated flow veloc ity sho uld a t mi-

nimum be checked.

The nom inal press ure s tag e results

from know ledg e of the va lve ma teria l,the opera t ing tempera ture and the max.

operating pressure, e.g. from DIN 2401,

or from a valve da ta s heet.

The a ctua l clos ed-loop c ontrol func-

tion, i.e. s etting the fluid flow rate o f a

given temperature and given pressure

while s imultane ously producing a de-

fined pressure loss, is determined by

the flow cha racte ristic, the kv value

The kv value is a reference variab le and

is d efined a s follows : kv value = qua n-

tity in m3/h of cold wa ter (+ 5 + 30 C)

which flows through the valve at 1 bar.

0 .2

0 .4

0 .6

0 .8

1.0

0 .2 0 .4 0 .6 0 .8 1.0

k vk vs

lin . o p t . e p .

Flowcharacteristics

kv0kvs

= 0.33 ss m ax

F igure 4 8 : F low c harac ter i s t ics:l i near , op t im um , equal -percentage

0 .2

0 .4

0 .6

0 .8

1.0

0 .2 0 .4 0 .6 0 .8 1.0

lin . o p t . e p .

Operatingcharacteristics

0.061 ss ma x

VV m ax

F igure 4 9 : O pe rat ing chara cte r is t ics:

l i near , op t im um , equal -percentage

N W = 0 .4 2 QN W : C o n n e c t io n n o m i n a l d ia m e t e r

Q : Vo lum e tr ic f low rate in l / h

N W = 4 .2 Q Np 1

Q N : Vo lum e tr ic f low rate in Nm 3 / h

p 1: Pres sure upst rea m o f the va lve in

bar abso lu te

G : M a s s f lo w r a t e in k g / h

v": Spe ci f ic volume in m 3 / k g

N W = 2 .8 G v "

Q B : Vo lum e tr ic f low rate in m 3 / h

c: F low ve loc i ty in m /s

N W = 18 ,8 Q Bc

Q : Vo lum e tr ic f low rate in m 3 / h

p: Pres sure d i fferen t i a l a t the va lve inb a r

k v = Q 1

p

k v = Q 0 .0 3 2 1p

p s 2 : S a t u r a t e d s t e a m p r e s s . , in b a r a b s . ,

r e la t e d t o t h e t e m p e r a t u r e d o w n s t r e a m

of the va lve

1: De nsi ty o f the m e d ium in k g /m 3

p : P r e s s u r e d if f e r e n t ia l a t t h e v a lv e inb a r

k v = G 0 .0 3 2 1

1 pG : M a s s f lo w r a t e in k g / h

p : P r e s s u r e d if f e r e n t ia l a t t h e v a lv e inb a r

p 2 < p s 2

p 2 > p s 2

p s 2 : S a t u ra t e d s t e a m p r e s s u r e , in b a r

a b s o l ut e , r e l a t e d t o t h e t e m p e r a t u r e

d o w n s t r e a m o f t h e v a lv e

1

130

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