The smart choice of Fluid Control Systems
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Transcript of 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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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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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
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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
S t e p r e s p o n s e
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
S t e p r e s p o n s e
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
S t e p r e s p o n s e
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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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
S t e p r e s p o n s e
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
S t e p r e s p o n s e
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
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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
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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
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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
S t e p r e s p o n s e
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
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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
S t e p r e s p o n s e
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
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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
S t e p r e s p o n s e
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
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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
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 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
S et -poi nt va l ue S P
P r o c e s s v a l u e P V
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
S et -poi nt va l ue S P
P r o c e s s v a l u e P V
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
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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
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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-pointchangeswell-suitedDisturbanceresponsewell-suited
Responseto set-pointchangeswell-suitedDisturbanceresponsewell-suited
Responseto set-pointchangessuitableDisturbanceresponsesuitable
Responseto set-pointchangessuitableDisturbanceresponsesuitable
Responseto set-pointchangessuitable
Disturbanceresponse s.
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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
P r o c e s s v a l u e P V
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:
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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
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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
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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 )
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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
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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