in
s
tal
.
pc
z
.
pl8/202661
An
a
l
ys
i
s
of
t
h
e
t
r
a
n
s
i
e
n
t st
a
t
e
d
u
ri
n
g
t
h
e
o
pe
r
a
t
i
on
of
a
c
e
n
t
ri
fu
ga
l
p
u
mp
i
n
c
on
j
un
ct
i
on
w
i
t
h
a
n
a
ir
v
e
ss
e
l
An
a
li
z
a
st
a
nu
n
i
e
u
st
a
l
on
eg
o
p
o
d
c
z
a
s ws
p
ó
ł
p
r
a
cy
p
o
mp
y w
ir
o
w
e
j
z
e
z
b
i
o
r
n
i
k
i
em
p
o
w
i
e
tr
z
a
K
R
Z
Y
S
Z
T
O
F
K
ARAŚ
K
I
E
W
IC
Z
D
O
I
:
10
.
17512/
I
NST
AL.
2026
.
08
.
04
The paper pre
s
ent
s
an anal
y
s
i
s
o
f tran
s
ient
o
perating c
o
nditi
o
n
s
during the c
oo
perati
o
n
o
f a centrifugal pump
w
ith
a c
o
mpre
ss
ed air ve
ss
el
.
The c
o
n
s
idered
s
y
s
tem c
o
n
s
i
s
t
s
o
f a pump delivering liquid fr
o
m a
s
ucti
o
n re
s
erv
o
ir int
o
a cl
o
s
ed tan
k
c
o
ntaining an air cu
s
hi
o
n
s
ubjected t
o
c
o
mpre
ss
i
o
n
.
Unli
k
e c
o
nventi
o
nal pumping
s
y
s
tem
s
o
perating
cl
o
s
e t
o
a
s
tead
y
o
perating p
o
int, the anal
y
s
ed
s
y
s
tem i
s
characteri
z
ed b
y
c
o
ntinu
o
u
s
l
y
changing h
y
draulic c
o
nditi
o
n
s
cau
s
ed b
y
the increa
s
ing pre
ss
ure
o
f the c
o
mpre
ss
ed ga
s
and the
w
ater c
o
lumn height
.
Thi
s
mean
s
that the pump
o
pe-
rate
s
under d
y
namic c
o
nditi
o
n
s
, and it
s
o
perating p
o
int i
s
n
o
t a
s
tead
y
s
tate
.
Thi
s
t
y
pe
o
f
o
perati
o
n in a centrifugal
pump
–
air tan
k
s
y
s
tem ha
s
n
o
t been anal
y
s
ed
s
o
far
.
Thi
s
article pre
s
ent
s
a meth
o
d
o
l
o
g
y
f
o
r c
o
mputing the parameter
s
o
f
s
uch a
s
y
s
tem in a tran
s
ient
s
tate
.
I
t pr
o
vide
s
equati
o
n
s
de
s
cribing the d
y
namic
s
o
f pump
s
tartup, change
s
in the
r
o
tati
o
nal
s
peed
o
f the r
o
tating a
ss
embl
y
, and the accelerati
o
n
o
f the liquid c
o
lumn in the
s
y
s
tem
.
The influence
o
f
r
o
tati
o
nal inertia, mechanical l
o
ss
e
s
, and variati
o
n
o
f the
s
tatic head cau
s
ed b
y
air c
o
mpre
ss
i
o
n i
s
di
s
cu
ss
ed
.
I
t
w
a
s
empha
s
i
z
ed that the
s
y
s
tem’
s
characteri
s
tic
s
change much m
o
re
s
l
ow
l
y
than th
o
s
e
o
f the pump during
s
tartup,
w
hich
re
s
ult
s
in a peri
o
dic e
x
ce
ss
o
f energ
y
tran
s
ferred fr
o
m the pump t
o
the
s
y
s
tem c
o
mpared t
o
the energ
y
l
o
s
t in the
s
y
s
tem
.
The
s
tud
y
di
s
cu
ss
e
s
the applicabilit
y
o
f
s
imilarit
y
la
w
s
w
hich have n
o
t been anal
y
z
ed in the available literature f
o
r the
ca
s
e
o
f d
y
namic pr
o
ce
ss
e
s
and Suter characteri
s
tic
s
f
o
r determining in
s
tantane
o
u
s
pump
o
perating parameter
s
under
tran
s
ient c
o
nditi
o
n
s
.
A
ttenti
o
n i
s
dra
w
n t
o
the limitati
o
n
s
a
ss
o
ciated
w
ith inc
o
mplete fl
ow
s
imilarit
y
during
o
ff--de
s
ign
o
perati
o
n
.
The anal
y
z
ed
s
y
s
tem ma
y
f
o
rm the ba
s
i
s
f
o
r b
o
th c
o
nventi
o
nal h
y
dr
o
ph
o
re in
s
tallati
o
n
s
and c
o
m-pre
ss
ed-
air energ
y
s
t
o
rage
s
y
s
tem
s
empl
oy
ing pump
s
o
perating rever
s
ibl
y
a
s
turbine
s
(P
A
T
s
)
.
Ke
ywo
rd
s
:
centrifugal pump, air ve
ss
el, energ
y
s
t
o
rage, tran
s
ient
s
tate
W art
yk
ule pr
z
ed
s
ta
w
i
o
n
o
anali
z
ę
s
tanu nieu
s
tal
o
neg
o
p
o
dc
z
a
s
w
s
p
ó
łprac
y
p
o
mp
y
w
ir
ow
ej
z
e
z
bi
o
rni
k
iem
z
a
w
ierając
y
m
s
prę
ż
o
ne p
ow
ietr
z
e
.
Ro
z
w
a
ż
an
y
u
k
ład
o
bejmuje p
o
mpę tł
o
c
z
ącą ciec
z
z
e
z
bi
o
rni
k
a
ss
a
w
neg
o
d
o
z
am
k
nięteg
o
z
bi
o
rni
k
a,
w
k
t
ó
r
y
m na
s
tępuje
s
prę
ż
anie p
o
du
s
z
k
i p
ow
ietr
z
nej
.
W pr
z
eci
w
ień
s
t
w
ie d
o
k
la
s
y
c
z
n
y
ch
u
k
ład
ów
p
o
mp
owy
ch pracując
y
ch
w
p
o
bli
ż
u u
s
tal
o
neg
o
pun
k
tu prac
y
, anali
z
ow
an
y
u
k
ład chara
k
ter
y
z
uje
s
ię ciągłą
z
mianą chara
k
ter
y
s
t
yk
i h
y
draulic
z
nej
wy
ni
k
ającą
z
e
w
z
r
o
s
tu ci
ś
nienia
s
prę
ż
aneg
o
ga
z
u i
wy
s
oko
ś
ci
s
łupa
wo
d
y.
O
z
nac
z
a t
o
,
ż
e p
o
mpa pracuje
w
w
arun
k
ach d
y
namic
z
n
y
ch, a jej pun
k
t prac
y
nie je
s
t
s
tanem u
s
tal
o
n
y
m
.
Ten r
o
d
z
aj
prac
y
w
u
k
ład
z
ie p
o
mpa
w
ir
ow
a
–
z
bi
o
rni
k
p
ow
ietr
z
a nie b
y
ł d
o
t
y
chc
z
a
s
anali
z
ow
an
y.
A
rt
yk
uł pr
z
ed
s
ta
w
ia met
o
d
yk
ę
o
blic
z
ania parametr
ów
ta
k
ieg
o
u
k
ładu
w
s
tanie nieu
s
tal
o
n
y
m
.
P
o
dane
s
ą r
ów
nania
o
pi
s
ujące d
y
nami
k
ę r
o
z
ruchu
p
o
mp
y
,
z
mian
y
pręd
ko
ś
ci
o
br
o
t
ow
ej
z
e
s
p
o
łu
w
irująceg
o
o
ra
z
pr
z
y
s
pie
s
z
anie
s
łupa ciec
z
y
w
u
k
ład
z
ie
.
O
m
ów
i
o
n
o
w
pł
yw
m
o
mentu be
z
w
ładn
o
ś
ci,
s
trat mechanic
z
n
y
ch
o
ra
z
z
mian
s
tat
y
c
z
nej
wy
s
oko
ś
ci p
o
dn
o
s
z
enia
z
w
ią
z
anej
z
e
s
prę
ż
aniem p
ow
ietr
z
a
.
P
o
d
k
re
ś
l
o
n
o
,
ż
e chara
k
ter
y
s
t
yk
a u
k
ładu
z
mienia
s
ię
z
nac
z
nie
wo
lniej ni
ż
chara
k
ter
y
s
t
yk
a
p
o
mp
y
p
o
dc
z
a
s
r
o
z
ruchu, c
o
p
owo
duje
ok
re
s
ow
ą pr
z
e
w
agę energii
o
dda
w
anej pr
z
e
z
p
o
mpę d
o
u
k
ładu
o
d tej tra-
c
o
nej
w
u
k
ład
z
ie
.
W prac
y
o
m
ów
i
o
n
o
m
o
ż
li
wo
ś
ć
wyko
r
z
y
s
tania pra
w
p
o
d
o
bień
s
t
w
a
–
nie anali
z
ow
an
y
ch
w
t
y
m
ko
nte
k
ś
cie
w
d
o
s
tępn
y
ch publi
k
acjach
o
ra
z
chara
k
ter
y
s
t
yk
Sutera d
o
wy
z
nac
z
ania ch
w
il
owy
ch parametr
ów
prac
y
p
o
mp
y
w
s
tanach pr
z
ej
ś
ci
owy
ch
.
Z
w
r
ó
c
o
n
o
u
w
agę na
o
granic
z
enia
wy
ni
k
ające
z
niepełneg
o
p
o
d
o
bień
s
t
w
a
pr
z
epł
yw
u p
o
dc
z
a
s
prac
y
p
o
z
a pun
k
tem pr
o
je
k
t
owy
m
.
A
nali
z
ow
an
y
u
k
ład m
o
ż
e
s
tan
ow
ić p
o
d
s
ta
w
ę
z
ar
ów
n
o
k
la
s
y
-
c
z
n
y
ch in
s
talacji h
y
dr
o
f
o
r
owy
ch, ja
k
i
s
y
s
tem
ów
maga
z
y
n
ow
ania energii
wyko
r
z
y
s
tując
y
ch
s
prę
ż
o
ne p
ow
ietr
z
e
o
ra
z
p
o
mp
y
pracujące re
w
er
s
y
jnie ja
ko
turbin
y
(P
A
T)
.
Sł
ow
a
k
luc
z
ow
e
:
p
o
mpa
o
d
ś
r
o
d
kow
a,
z
bi
o
rni
k
p
ow
ietr
z
a, maga
z
y
n
ow
anie energii,
s
tan pr
z
ej
ś
ci
owy
Kr
z
y
s
z
t
o
f Kara
ś
k
ie
w
ic
z
O
R
C
I
D:
0000-0003-4709-2355, P
o
litechni
k
a War
s
z
a
w
s
k
a, W
y
d
z
iał
M
echanic
z
n
y
E
nerget
yk
i i L
o
tnict
w
a,
ul
.
N
owow
iej
s
k
a 24, 00-665 War
s
z
a
w
a, P
o
l
s
k
a, e-mail
:
k
r
z
y
s
z
t
o
f
.k
ara
s
k
ie
w
ic
z
@
p
w.
edu
.
pl
I
n
st
a
l
a
c
j
e
p
r
z
em
ysł
o
w
e
/I
n
d
u
st
ri
a
l
i
n
st
a
ll
a
t
i
on
s
D
ata zgło
s
zenia: 19.05.2026.
D
ata p
r
z
y
jęcia: 22.07.2026.
D
ata publi
k
acji: 26.08.2026.
I
n
t
r
o
d
u
ct
i
on
F
o
r man
y
decade
s
, c
o
mpre
ss
ed air ha
s
been
o
ne
o
f the m
o
s
t
w
idel
y
u
s
ed energ
y
media in techn
o
l
o
g
y.
I
t i
s
u
s
ed in indu
s
trial
aut
o
mati
o
n, pneumatic drive
s
, energ
y
s
t
o
-
rage, and
s
afet
y
s
y
s
tem
s
.
A
t the
s
ame time,
pump
s
–
and centrifugal pump
s
in particu-
lar
–
are am
o
ng the ba
s
ic device
s
u
s
ed t
o
tran
s
p
o
rt liquid
s
and build pre
ss
ure in
h
y
draulic
s
y
s
tem
s
.
The c
o
mbinati
o
n
o
f the
s
e
t
wo
field
s
–
h
y
draulic
s
and pneumatic
s
–
o
pen
s
up intere
s
ting p
o
ss
ibilitie
s
in
w
hich
c
o
mpre
ss
ed air i
s
n
o
t merel
y
an au
x
iliar
y
medium but an active element in the pr
o
ce
ss
o
f energ
y
s
t
o
rage
o
r rec
o
ver
y.
I
n engineering applicati
o
n
s
, c
o
mpre
s
-
s
ed air can interact
w
ith a pump in man
y
w
a
y
s
.
I
t can
s
tabili
z
e pre
ss
ure in a pre
ss
ure
b
oo
s
ter
s
y
s
tem b
y
limiting the number
o
f
m
o
t
o
r
s
tart
s
and c
o
mpen
s
ating f
o
r temp
o
ra-
r
y
fluctuati
o
n
s
in fl
ow
demand
.
I
n m
o
re
advanced applicati
o
n
s
, c
o
mpre
ss
ed air
s
erve
s
a
s
an energ
y
s
t
o
rage medium
:
the
pump f
o
rce
s
w
ater int
o
an air ve
ss
el, c
o
m-
pre
ss
ing the ga
s
in
s
ide, and the
s
t
o
red ener-
g
y
can later be u
s
ed t
o
drive the fluid bac
k
thr
o
ugh a turbine, a pump
o
perating in
rever
s
e,
o
r an
o
ther p
ow
er rec
o
ver
y
s
y
s
tem
.
A
di
s
tinctive feature
o
f thi
s
t
y
pe
o
f
s
y
s
-
tem i
s
that the pump
o
perate
s
in a tran
s
ient
s
tate
.
A
s
the
s
t
o
rage tan
k
fill
s
, the v
o
lume
o
f
the air cu
s
hi
o
n change
s
, and
w
ith it the bac
k
pre
ss
ure acting
o
n the pump
.
Thi
s
re
s
ult
s
in
a c
o
ntinu
o
u
s
change in the
s
y
s
tem’
s
s
tatic
head and, c
o
n
s
equentl
y
, it
s
o
verall perf
o
r-
mance characteri
s
tic
s
.
I
n c
o
ntra
s
t t
o
c
o
nven-ti
o
nal
o
perati
o
n under relativel
y
c
o
n
s
tant parameter
s
, a pump
wo
r
k
ing
w
ith a c
o
m-pre
ss
ed air ve
ss
el
o
perate
s
under d
y
namic c
o
nditi
o
n
s
, requiring the
anal
y
s
i
s
o
f tran
s
ient re
s
p
o
n
s
e
s
.
T
h
e
d
y
n
am
i
cs
of
t
h
e
a
i
r
c
o
mp
r
e
ss
i
on
p
r
o
c
e
ss
b
y t
h
e
p
u
mp
R
egardle
ss
o
f
w
hether the air ve
ss
el i
s
a t
y
pical pre
ss
ure ve
ss
el in b
oo
s
ter pump
s
et
o
r a ve
ss
el f
o
r accumulating p
o
tential ener-
g
y
rec
o
vered in the P
A
T (Pump
A
s
Turbine),
the centrifugal pump
s
tart
s
o
perati
o
n fr
o
m
a
s
t
o
pped
s
tate, i
.
e
.
,
z
er
o
r
o
tati
o
nal
s
peed,
and increa
s
e
s
thi
s
s
peed until it reache
s
the
rated value
.
I
t
s
perf
o
rmance curve ri
s
e
s
fr
o
m
the
s
tarting p
o
int t
o
the
Q
-H curve c
o
rre-
s
p
o
nding t
o
rated c
o
nditi
o
n
s
(
F
ig
.
1)
.
Change
s
in the angular m
o
mentum
o
f
the pump’
s
r
o
tating a
ss
embl
y
are given b
y
the d
y
namic equati
o
n (1)
:
d
M
ω
s
(
ω
)
=
I
dt
+
M
m
+
M
p
(1)
w
here
:
I
–
the t
o
tal m
o
ment
o
f inertia
o
f all
r
o
tating ma
ss
e
s
in the pump a
ss
embl
y
m
s
P
d
ω
ω
η
M
(
ω
)
=
I
d
t
+
ω
+
ρ
g
QH
(4)
s
p
mm
t
y
o
f the r
o
tating a
ss
embl
y
, P
mmpm
s
=
P
+
P
–
dtA
d
=
,
w
here
A
i
s
the average cr
o
ss
-
(i
.
e
.
, the pump’
s
r
o
tating a
ss
embl
y
,
o
f fracti
o
n
s
o
f a
s
ec
o
nd (auth
o
r’
s
ow
n re
s
e-
c
o
upling
s
, gearb
o
x
(if an
y
), and the electric arch)
.
m
o
t
o
r’
s
r
o
t
o
r),
M
(
ω
)
–
the m
o
t
o
r’
s
t
o
rque
D
uring
o
perati
o
n, the pump mu
s
t
acce-(a
s
s
pecified b
y
the manufacturer), lerate a ma
ss
o
f
w
ater m
=
ρ
λ
A
in
the
s
ec-
M
=
P/
ω
–
the pump t
o
rque,
w
hich varie
s
ti
o
n
o
f the
s
y
s
tem e
x
tending fr
o
m the l
ow
er
during pump
o
perati
o
n and act
s
a
s
a bra- re
s
erv
o
ir (
w
ell
o
r
s
ucti
o
n tan
k
) t
o
the c
o
m-
k
ing t
o
rque
o
n the drive,
M
=
P /
ω
–
the pre
ss
ed air ve
ss
el
.
The initial fl
ow
rate i
s
z
er
o
s
um
o
f the fricti
o
n t
o
rque
s
o
ccurring in the and graduall
y
increa
s
e
s
w
ith an accelerati
o
n
pump gland and bearing
s
, a
s
w
ell a
s
in the
d
c
1
d
Q
m
o
t
o
r bearing
s
and fan
ω
–
angular vel
o
ci-
s
ecti
o
nal
t
area
o
f the pipe and l i
s
it
s
mechanical p
ow
er l
o
s
t due t
o
fricti
o
n in the
length
.
The d
y
namic
s
o
f the
s
e change
s
are
pump and m
o
t
o
r
.
e
x
pre
ss
ed b
y
the equati
o
n f
o
r pump head
:
()
,
H
Q
H
a
Q
F
o
r a pump, the mechanical l
o
ss
p
ow
er
2
l
d
Q
can be determined u
s
ing f
o
rmula
[
2
]:
s
tat
gA
dt
ω
=++
(5)
P
Q
n
P
Q
n
s
tat
0,40,3
Hi
s
the
s
tatic head
o
f the pumping
mp
=
0,0045
r
e
f
r
e
f
(2)
s
y
s
tem, and a i
s
the l
o
ss
c
o
efficient f
o
r that
opt
s
y
s
tem
.
F
ig
.
1 Change
s
in pump characteri
s
tic
s
due t
o
an increa
s
e in r
o
tati
o
nal
s
peed
Ry
s
.
1
.
Z
mian
y
chara
k
ter
y
s
t
yk
p
o
mp
y
na
s
k
ute
k
w
z
r
o
s
tu pręd
ko
ś
ci
o
br
o
t
ow
ej
refref
ss
()
P
P
η
−
1
−
1
(3)
1
w
here
:
Q
=
1 m
3
/
s
, n
=
1500 1/min
I
n
t
e
r
a
ct
i
on
be
tw
ee
n
t
h
e
p
u
mp
a
n
d
t
h
e
v
e
ss
e
l
M
echanical l
o
ss
e
s
in inducti
o
n m
o
t
o
r
s
t
y
picall
y
acc
o
unt f
o
r ab
o
ut 2-10
%
[
1
]
o
fThe pumping
s
y
s
tem c
o
n
s
i
s
t
s
o
f a l
ow
er
t
o
tal rated p
ow
er l
o
ss
e
s
; theref
o
re, a
ss
u- (
s
ucti
o
n) tan
k
, piping, fitting
s
(including
ming P i
s
the m
o
t
o
r’
s
s
haft p
ow
er and
η
i
s
a chec
k
valve), and an upper tan
k.
A
cen-
the m
o
t
o
r’
s
efficienc
y:
trifugal pump deliver
s
w
ater fr
o
m the
o
pen
s
ucti
o
n tan
k
t
o
the cl
o
s
ed di
s
charge ve
ss
el
m
s
=
0,02
÷
0,1
filled
w
ith air (
F
ig
.
2)
.
The ri
s
ing
w
ater
ss
c
o
lumn H
w
c
o
mpre
ss
e
s
the air
.
The initial air pre
ss
ure in the upper tan
k
When the engine
s
tart
s
,
s
tatic fricti
o
n i
s
i
s
p , and the final pre
ss
ure, at
w
hich the
pre
s
ent,
w
hich then change
s
t
o
k
inetic fricti
o
n
.
A
ss
uming the availabilit
y
o
f the pump’
s
efficienc
y
area
η
(
Q
,H) and appl
y
ing the
relati
o
n
s
hip f
o
r
s
haft p
ow
er
ρ
g
Q
H/
η
–
equati
o
n (1) can be
w
ritten a
s
:
Studie
s
s
h
ow
that the time required t
o
reach rated
s
peed f
o
r a large pump
w
ith
a m
o
t
o
r p
ow
er
o
f 1
.
1
M
W and a m
o
ment
o
f
inertia
o
f r
o
tating ma
ss
e
s
o
f
I
=
1050
k
gm²
i
s
le
ss
than 3
s
ec
o
nd
s
[
3, 4
].
F
o
r much
s
mal-
F
ig
.
2
.
Pump and l
ow
er and upper tan
k
diagram
ler pump
s
, the
s
e time
s
are
o
n the
o
rder
Ry
s
.
2
.
U
k
ład p
o
mp
y
i
z
bi
o
rni
ków
d
o
lneg
o
i g
ó
rneg
o
628/2026in
s
tal
.
pc
z
.
pl
I
in
s
tal
.
pc
z
.
pl8/202663
2
pump i
s
s
t
o
pped b
y
the aut
o
matic c
o
ntr
o
l
s
y
s
tem, i
s
p
.
D
ue t
o
the
s
l
ow
pr
o
ce
ss
, the air c
o
m-
pre
ss
i
o
n
o
perati
o
n i
s
i
s
o
thermal
.
A
s
c
o
mpre
ss
i
o
n pr
o
ceed
s
and the
w
ater
c
o
lumn in the air tan
k
ri
s
e
s
, the
s
tatic part
o
f
the pumping
s
y
s
tem’
s
head increa
s
e
s
.
1
pV
p
−
H
s
tat
=
ρ
g
+
H
w
=
ρ
gA
(
H
v
1
H
w
)
+
H
w
(6)
2
Change
s
in the
s
y
s
tem characteri
s
tic
s
o
ccur much m
o
re
s
l
ow
l
y
than change
s
in the
pump characteri
s
tic
s
due t
o
the rapid incre-
a
s
e in it
s
r
o
tati
o
nal
s
peed
.
Thi
s
re
s
ult
s
in the
pump gaining an energ
y
advantage
o
ver
the
s
y
s
tem, leading t
o
an accelerati
o
n
o
f the
fl
ow
(
E
quati
o
n 5)
.
The pump never reache
s
a
s
tead
y
-
s
tate
o
perating p
o
int in the
s
y
s
tem (
F
ig
.
3)
.
D
uring
s
tartup, the
s
y
s
tem curve
s
ta
y
s
ver
y
l
ow
due
t
o
the
s
y
s
tem’
s
l
ow
s
tatic head and remain
s
flat due t
o
l
ow
fl
ow
re
s
i
s
tance
.
A
fter the air
i
s
c
o
mpre
ss
ed t
o
the required value p , the
pump i
s
s
t
o
pped b
y
the aut
o
matic c
o
ntr
o
l
s
y
s
tem
.
The fir
s
t term in equati
o
n (6), re
s
ul-
ting fr
o
m the air pre
ss
ure, can be man
y
time
s
greater than the
s
ec
o
nd term, re
s
ulting
fr
o
m the
w
ater c
o
lumn height H
w
.
D
e
t
e
r
m
i
n
i
n
g
t
h
e
h
ead
a
n
d
t
o
r
q
u
e
C
h
a
r
a
ct
e
ri
st
i
cs
of
a
p
u
mp
d
u
r
i
n
g
a
ir
c
o
mp
r
e
ss
i
on
p
T
o
determine the pump’
s
perf
o
rmance,
it i
s
nece
ss
ar
y
t
o
k
n
ow
the pump head func-
ti
o
n H(
Q
,
ω
) and the pump t
o
rque
M
(
Q
,
ω
)
.
The functi
o
n H(
Q
,
ω
) can be determined
u
s
ing the la
w
s
o
f
s
imilarit
y.
A
cc
o
rding t
o
the-
s
e la
w
s
, the head varie
s
w
ith the
s
quare
o
f
the r
o
tati
o
nal
s
peed,
w
hile the fl
ow
rate
varie
s
linearl
y
; hence, the relati
o
n
s
hip f
o
r
H(
Q
,
ω
) ta
k
e
s
the f
o
rm
:
()
,
H Q
ω
=
f
ω
2
Q
ω
r
(
ω
/
ω
r
)
(7)
r
0
F
o
r
ω
=
ω
the ba
s
ic pump characteri
s
tic
H
=
f(
Q
) i
s
o
btained
.
The fl
ow
curve can
o
ften be appr
o
x
ima-
ted b
y
the parab
o
la H
=
H c
Q
2
.
I
n that
ca
s
e, the
s
urface H(
Q
,
ω
) ta
k
e
s
the f
o
rm
:
()
,
H
Q
ω
H
c
=−
ω
2
Q
2
ω
r
0
(
ω
/
ω
r
)
(8)
p
The t
o
rque curve
o
f a pump
M
i
s
n
o
t
t
y
picall
y
mea
s
ured b
y
the manufacturer;
in
s
tead, the
s
haft p
ow
er curve i
s
pr
o
vided
.
G
iven that the pump’
s
p
ow
er i
s
pr
o
p
o
r-
ti
o
nal t
o
the cubic
o
f the r
o
tati
o
nal
s
peed,
and thu
s
the t
o
rque i
s
pr
o
p
o
rti
o
nal t
o
the
s
quare
o
f the r
o
tati
o
nal
s
peed, it can be
determined,
s
imilarl
y
t
o
the head, u
s
ing the
f
o
ll
ow
ing equati
o
n
:
()
,
M
Q
ω
=
g
ω
2
Q
p
ω
r
(
ω
/
ω
r
)
(9)
r
pp
A
s
w
ith the head, f
o
r
ω
=
ω
, the ba
s
ic
characteri
s
tic
s
o
f the pump’
s
t
o
rque
M
=
g(
Q
)
o
r p
ow
er P
=
ω
g(
Q
) are
o
btained
.
Similarit
y
la
w
s
are ba
s
ed
o
n the
a
ss
umpti
o
n
o
f
s
imilarit
y
o
f vel
o
cit
y
field
s
at
different r
o
tati
o
nal
s
peed
s
.
H
ow
ever, thi
s
a
ss
umpti
o
n i
s
n
o
t accurate, becau
s
e the
s
i
z
e
s
o
f
s
eparati
o
n
z
o
ne
s
and v
o
rtice
s
f
o
r-
med
w
hen the pump
o
perate
s
o
ut
s
ide the
de
s
ign p
o
int d
o
n
o
t
s
cale
.
Similarl
y
, the
thic
k
ne
ss
e
s
o
f b
o
undar
y
la
y
er
s
d
o
n
o
t
s
ati
s
-
f
y
the
s
imilarit
y
la
w
s
.
S
o
metime
s
, the efficienc
y
η
(
Q
,H)
mea
s
ured b
y
the manufacturer i
s
available
.
T
o
c
o
mpute the d
y
namic
s
o
f the
s
y
s
tem
during air c
o
mpre
ss
i
o
n b
y
the pump, it
s
anal
y
tical f
o
rm i
s
required
w
hich i
s
difficult t
o
o
btain
.
There are c
o
nver
s
i
o
n f
o
rmula
s
f
o
r
ma
x
imum efficienc
y
a
s
a functi
o
n
o
f r
o
tati
o
-
nal
s
peed, e
.
g
.
,
[
2
]
, but
s
ince the
y
appl
y
t
o
o
nl
y
a
s
ingle p
o
int, the
y
cann
o
t be u
s
ed
.
A
n
o
ther meth
o
d i
s
t
o
u
s
e Suter’
s
charac-
teri
s
tic
s
[
8
]
,
w
hich pre
s
ent the full
o
perating
range
o
f the pump in the f
o
rm
o
f dimen
s
i
o
n-
le
ss
c
o
efficient
s
o
f head WH and t
o
rque
W
M
, repre
s
enting the d
o
uble-reduced
head and t
o
rque
.
22
(
h
α
+
q
WM
X
)
=
α
2
+
q
2
W
H
(
X
)
=
h
(10)
w
here
:
·
2
3
π
α
0
f
o
r
0
≤
X
≤
X
=
i
π
+
a
r
c
tg
q
,
i
=
1
f
o
r
π
≤
X
≤
3
π
2
f
o
r
2
π
≤
X
≤
2
π
(11)
nnnn
H
QM
ω
h
=
H
,
q
=
Q
,
m
=
M
,
α
=
ω
A
full de
s
cripti
o
n
o
f the appr
o
x
imati
o
n
o
f
the
s
e c
o
efficient
s
i
s
pr
o
vided, am
o
ng
o
ther place
s
, in
[
5
].
F
ig
.
3
.
Ex
ample perf
o
rmance curve
s
o
f the pump
and
s
y
s
tem
Ry
s
.
3
.
Pr
z
yk
ład
ow
e chara
k
ter
y
s
t
yk
i prac
y
p
o
mp
y
i u
k
ładu
F
ig
.
4
.
Ex
ample pump
o
perating
z
o
ne
s
Ry
s
.
4
.
Pr
z
yk
ład
ow
e
s
tref
y
prac
y
p
o
mp
y
I
n
st
a
l
a
c
j
e
p
r
z
em
ysł
ow
e
F
igure 4
s
h
ow
s
e
x
ample characteri
s
tic
s
o
fWH andW
M
a
s
a functi
o
n
o
f the dimen-
s
i
o
nle
ss
parameter
X.
R
egi
o
n
D
repre
s
ent
s
n
o
rmal pump
o
perati
o
n during air c
o
mpre
s
-
s
i
o
n,
w
hen H
>
0,
Q
≥
0, n
≥
0, and
M
>
0
f
o
r
π
≤
X
≤
1
.
38
π
.
WH and W
M
characteri
s
tic
s
are ava-
ilable f
o
r
s
elected
s
pecific
s
peed
s
[
3,4,5
].
I
f the pump being anal
y
s
ed ha
s
a different
s
pecific
s
peed, the WH and W
M
s
h
o
uld be
interp
o
lated, f
o
r e
x
ample, linearl
y
, ba
s
ed
o
n the available mea
s
ured value
s
value
s
.
Publicati
o
n
[
6
]
pre
s
ent
s
an anal
y
s
i
s
o
f
change
s
in the h
y
draulic parameter
s
o
f
a nuclear p
ow
er plant
s
y
s
tem re
s
ulting fr
o
m
a failure, u
s
ing Suter characteri
s
tic
s
.
C
on
c
l
u
s
i
on
s
The meth
o
d pre
s
ented in thi
s
article
all
ow
s
f
o
r the calculati
o
n
o
f the parameter
s
o
f a pump-air ve
ss
el
s
y
s
tem during the c
o
m-
pre
ss
i
o
n
o
f air b
y
pumping in
w
ater
.
Thi
s
pr
o
ce
ss
i
s
tran
s
ient in nature due t
o
the
changing pump characteri
s
tic
s
re
s
ulting
fr
o
m an increa
s
e in r
o
tati
o
nal
s
peed, a
s
w
ell
a
s
the changing
s
y
s
tem characteri
s
tic
s
re
s
ul-
ting fr
o
m an increa
s
e in air pre
ss
ure and the
w
ater c
o
lumn height in the tan
k.
D
uring
o
perati
o
n, the pump’
s
perf
o
r-
mance characteri
s
tic
s
change much
m
o
re
p
rapidl
y
than th
o
s
e
o
f the pumping
s
y
s
tem,
w
hich lead
s
t
o
the pump’
s
parameter
s
e
x
ce-
eding th
o
s
e
o
f the
s
y
s
tem and t
o
an increa
s
e
in the ma
ss
o
f the pumped
w
ater
.
The pump
and the
s
y
s
tem never reach a
s
tate
o
f equili-
brium
.
T
o
determine the pump head H(
Q
,
ω
)
and pump t
o
rque
M
(
Q
,
ω
)
w
ithin the range
o
f it
s
p
o
s
itive parameter
s
(the
s
o
-called fir
s
t
quadrant),
s
imilarit
y
la
w
s
can be u
s
ed,
w
hich all
ow
the determinati
o
n
o
f the pump’
s
in
s
tantane
o
u
s
o
perating p
o
int
.
A
n alternati-
ve i
s
t
o
u
s
e the Suter characteri
s
tic
.
The accurac
y
o
f a m
o
del ba
s
ed
o
n
s
imi-
larit
y
la
w
s
decrea
s
e
s
during
o
perati
o
n far
fr
o
m the de
s
ign p
o
int,
w
hen
s
eparati
o
n
s
and n
o
n-
s
calable v
o
rte
x
s
tructure
s
o
ccur
.
A
dditi
o
nall
y
, the thic
k
ne
ss
e
s
o
f the b
o
unda-
r
y
la
y
er
s
are n
o
t
s
ubject t
o
s
imilarit
y.
D
e
s
pite
the
s
e limitati
o
n
s
, the
y
are c
o
mm
o
nl
y
u
s
ed t
o
determine pump parameter
s
a
s
a functi
o
n
o
f
vel
o
cit
y
[
6
].
The anal
y
s
ed
s
y
s
tem can be applied n
o
t
o
nl
y
in cla
ss
ic pre
ss
ure b
oo
s
ter
s
y
s
tem
s
but
al
s
o
in c
o
mpre
ss
ed air energ
y
s
t
o
rage
s
y
s
-
tem
s
and energ
y
rec
o
ver
y
s
y
s
tem
s
w
ith
a pump
o
perating in rever
s
e a
s
a turbine
.
F
urther
wo
r
k
s
h
o
uld include a d
y
namic
and energ
y
anal
y
s
i
s
o
f the entire c
o
mpre
s
-
s
i
o
n and energ
y
rec
o
ver
y
pr
o
ce
ss
in pump--
a
s
-turbine (P
A
T) m
o
de
.
L
i
t
e
r
a
t
u
r
e
[
1
]
D
ąbała K
.
Pr
o
p
o
z
y
cje n
owy
ch met
o
d
wy
z
na-
c
z
ania
s
pra
w
n
o
ś
ci
s
ilni
ków
indu
k
c
y
jn
y
ch
k
lat
ko
-
wy
ch -
Z
e
s
z
y
t
y
Pr
o
blem
ow
e
–
M
a
s
z
y
n
y
E
le
k
-
tr
y
c
z
ne Nr 75/2006, p
.
107
–
112
[
2
]
G
ülich J
.
F
.
- Centrifugal Pump
s
- 4th editi
o
n, Sprin-
ger Nature S
w
it
z
erland
A
G
2020,
D
O
I
:
http
s
:
//
d
o
i
.o
rg/10
.
1007/978-3-642-12824-0
[
3
]
Jędral W
.
, Kara
ś
k
ie
w
ic
z
K
.
-
O
blic
z
enia
wy
bra-
n
y
ch pr
z
y
pad
ków
r
o
z
ruchu p
o
mp
y
ś
migł
ow
ej
w
u
k
ład
z
ie
wo
d
y
chł
o
d
z
ącej - Prace Nau
kow
e
P
o
litechni
k
i Ślą
s
k
ie,j
M
o
n
o
gr
.
K
o
nfer
.
, 2003,
z
.
11, p
.
103-108
[
4
]
Jędral W
.
, Kara
ś
k
ie
w
ic
z
K
.
A
nali
z
a
wy
bran
y
ch
pr
z
y
pad
ków
r
o
z
ruchu d
w
u
s
trumieni
ow
ej p
o
mp
y
o
d
ś
r
o
d
kow
ej -
I
X
M
ięd
z
y
n
.
K
o
nf
.
Pr
z
epł
ywow
e
M
a
s
z
y
n
y
Wirni
kow
e, 2003, mat
.ko
nf
.
, p
.
279-
286
[
5
]
Kara
ś
k
ie
w
ic
z
K
.
, S
z
y
mc
z
yk
J
.
-
E
valuati
o
n
o
f the
accurac
y
o
f centrifugal pump
s
c
o
mplete charac-
teri
s
tic
s
and their appr
o
x
imati
o
n u
s
ing
F
o
urier
s
erie
s
-
I
nternati
o
nal J
o
urnal
o
f
F
luid P
ow
er,
2025, t
.
26,
s
.
1-18, d
o
i
:
10
.
13052/ijfp1439-
9776
.
2612
[
6
]
Kara
ś
k
ie
w
ic
z
K
.
,
M
arcin
k
ie
w
ic
z
J
.
, J
o
heman C
.:
The
I
nfluence
o
f Centrifugal Pump Characteri
s
tic
s
o
n
Dy
namic L
o
ading
s
o
n Pipeline
s
after P
ow
er
F
ailure, 26th
I
nternati
o
nal C
o
nference
o
n Nucle-
ar
E
ngineering 2018 1-7
s
.
, 2018
[
7
]
PN-
E
N
I
S
O
9906
:
2012 P
o
mp
y
w
ir
ow
e
–
B
ada-
nia
o
dbi
o
rc
z
e parametr
ów
h
y
draulic
z
n
y
ch
–
k
la
s
y
d
ok
ładn
o
ś
ci 1,2 i 3
[
8
]
Suter P
.
,
R
epre
s
entati
o
n
o
f pump characteri
s
tic
s
f
o
r calculati
o
n
o
f
w
ater hammer,
”
Sul
z
er Techni-
cal
R
evie
w
R
e
s
earch, 1966, p
.
45- 48
648/2026in
s
tal
.
pc
z
.
pl
I