Cha ac e is ics o he pipe junc ion wi h 2.4 c oss-sec ion a ea a io o
he case o he low di ision
J. Š igle 1a and O. Špe ka1
1B no Uni e si y o Technology, Facul y o Mechanical Enginee ing, Ene gy Ins i u e, Vic o Kaplan Depa men o Fluid
Enginee ing, Technická 2896/2, 616 69, Czech Republic
Abs ac . The e a e p esen ed cha ac e is ics o he pipe junc ion o he case o he low di ision
in his pape . The pipe junc ion consis s o one s aigh pipe, wi h he diame e 50 mm and one
adjacen pipe wi h diame e 32 mm. The cha ac e is ics ha e been measu ed o i e di e en angles
o he adjacen pipe.
1 In oduc iona
Many pape s, abou he luid low in he pipe junc ion,
ha e been w i en by hese au ho s. I is possible o use i
o he luid low solu ion in he pipe line ne . The e a e
some o he ma hema ical models bu hey a e based on
he un ealis ic assump ions o hei coe icien s ha e no
any physical meaning. [1, 2] The new ma hema ical
model o he 90° pipe junc ion oge he wi h i s
coe icien s is desc ibed in [3, 4]. The imp o ed
ma hema ical model o he a bi a y angle o he
adjacen b anch is in oduced in he [5]. Some discussion
abou ma hema ical model o he uns eady luid low in
pipe junc ion has been p esen ed in [6]. A lo o
nume ical solu ions and expe imen s o he luid low in
he pipe junc ion ha e been done in pas ew yea s. Fo
example nume ical s udy o luid low has been p esen ed
in he [7, 8]. The PIV measu emen o he luid low in
he pipe junc ion has been desc ibed in [9] The
Compa ison o he PIV measu emen wi h nume ical
solu ion o luid low in [10]. The i s new
cha ac e is ics ob ained by measu ing and i s compa ison
wi h CFD calcula ions we e p esen ed a [11]. Many
pape s ha e been w i en abou luid low in pipe junc ion
as i was men ioned abo e and eade can ge mos o
hem on he in e ne . This pape will be he e o e
dedica ed mainly o he p esen a ion o he coe icien s
ob ained by measu emen s. And mo e o e he discussion
abou he esul s will be included in his pape .
2 Ma hema ical model
The pipe junc ion i is deal wi h is d awn in he igu e 1.
a [email p o ec ed]
The ma hema ical model o he luid low in he pipe
junc ion consis s o h ee equa ions which desc ibe he
ela ionship be ween he low a es and p essu es a he
bo de o he pipe junc ion a ea.
Fig. 1. Pipe junc ion
The bo de o he pipe junc ion a ea is delimi ed by he
c oss-sec ions S(a), S(b) and S(c) placed in such dis ance
om he low di ision domain whe e he low is no
dis u bed by he pipe junc ion. The minimum dis ance o
ensu e his assump ion is abou 10 diame e s a each
b anch.
Fi s equa ion o he ma hema ical model o he pipe
junc ion is he powe equa ion. This equa ion ep esen s
he mechanical ene gy conse a ion law in he pipe
junc ion
0
2
1
2
22
2
2
2
2
2
2
2
2

























)mX(
)X(
)mX(
)P()mc()c(
)c(
)mc(
)mb()b(
)b(
)mb(
)ma()a(
)a(
)ma(
Q
S.
Q
Qp
S
Q
Qp
S
Q
Qp
S
Q




.
(1)
The second equa ion is he momen um equa ion. This is a
ec o equa ion bu we will ake in o conside a ion only
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on,
EPJ Web o Con e ences
DOI: 10.1051/
C
Owned by he au ho s, published by EDP Sciences, 2013
,
epjcon 201/
01112 (2013)
45
34501112
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EPJ Web o Con e ences
he componen in he di ec ion o b anch “b” his
di ec ion is ep esen ed by he uni no mal ec o n(b)i.
This is clea om he igu e 1
0
22
2
2

























)X(
)mX(
i)M(i)c()c()c(
)c(
)mc(
i)b()b()b(
)b(
)mb(
i)a()a()a(
)a(
)ma(
S
Q
nSp
S
Q
nSp
S
Q
nSp
S
Q





.
(2)
This equa ion has o be mul iplied by he ec o n(b)i o
ge he componen equa ion in ha di ec ion.
The las hi d equa ion is con inui y equa ion
0



)mc()mb()ma( QQQ . (3)
The summa ion con ec ion is used in he abo e
equa ions. This ma hema ical model is alid unde hese
assump ions
 S eady luid low
 The g a i y ec o is pe pendicula o he pipe
junc ion plain.
 The densi y is cons an
 The Co iolis numbe s and Bousineques
numbe s on he c oss-sec ions S(a), S(b), and S(c)
a e equal o 1.
I is possible o emo e all hese assump ions in he
gene al ma hema ical model [5 ].
The index “X” in he abo e equa ions has o be eplaced
by he index “a”, “b” o “c”. I depends on b anch o
which he coe icien s

(M)i and

(P) will be ela ed o.
3 Pipe junc ion coe icien s
Two coe icien s appea in he ma hema ical model. The
powe coe icien

(P) can be exp essed om he equa ion
(1). This coe icien ep esen s he ene gy losses in he
pipe junc ion. I is p opo ional o he kine ic ene gy pe
ime in one o he b anches o which he coe icien will
be de i ed.
The momen um coe icien

(M)i can be exp essed om
he equa ion (2). This coe icien is p opo ional o he
o ce which he luid impac s on he pipe junc ion. The
o al o ce can be ob ained by mul iplying o his
coe icien wi h he o al momen um o luid in b anch
o which he coe icien was de i ed. In his case we will
ake in o conside a ion he componen o his coe icien
only in he di ec ion o b anch “b”.
The momen um coe icien depends also on he o al
p essu e a all b anches. I all pa ame e s as shape o pipe
junc ion, low con igu a ion, low a es, di e ences o
p essu es be ween he b anches a e he same hen he
coe icien s a ies wi h o al p essu e. The e o e i is
necessa y o ela es all p essu es o he cons an alue
p essu e in some b anch. In case o his pape he
p essu e alue in b anch “a” is aken as ze o. I means
ha he momen um coe icien is de i ed o case ze o
p essu e le el in b anch “a”. I means ha he momen um
coe icien will be exp essed om his modi ied
momen um equa ion
0
22
22

























)X(
)mX(
i)M(i)c()c()ca(
)c(
)mc(
i)b()b()ba(
)b(
)mb(
i)a(
)a(
)ma(
S
Q
nSp
S
Q
nSp
S
Q
n
S
Q





.
(4)
Whe e
)a()b()ba( ppp



(5)
and
)a()c()ca( ppp



. (6)
Now he unknowns a e h ee mass low a es Q(ma), Q(mb)
and Q(mc) and absolu e p essu e p(a) and wo p essu e
di e ences

p(ba) and

p(ca) ins ead o h ee absolu e
p essu es p(a), p(b) and p(c).
Bo h coe icien s can be di ided in o wo pa s. Fi s pa s
will be ela ed o he ic ion losses in he pipe junc ion.
They will be called ic ion coe icien s. Second pa s
will be ela ed o he shape o pipe junc ion. They will be
called geome y coe icien s. I can be exp essed his way
)PG()PF()P(





(7)
and
i)MG(i)MF(i)M(





. (8)
The o al coe icien s can be ob ained by he
measu emen s. The ic ion coe icien s can be ob ained
om he known ic ion losses, ep esen ed by he
p essu e d op, in s aigh pipes. The ic ion powe
coe icien can be exp essed as ollow


)mX(
)mX(
)X(
)c()c()c(
)b()b()b()a()a()a()PF(
Q
Q
S
LQi
LQiLQi
2
2
22




.
(9)
The ic ion momen um coe icien can be exp essed as
ollow


2
)mX(
)X(
i)c()c()c()c(
i)b()b()b()b(
i)a()a()a()a(i)MF(
Q
S
nSLi
nSLi
nSLi





.
(10)
The componen o ic ion momen um coe icien in
di ec ion b anch “b” will be ob ained by mul iplying
p e ious equa ion wi h uni no mal ec o n(b)i.
The quan i ies i(a), i(b), and i(c) a e p essu e d ops in he
b anches “a”, “b” and “c” wi h low a es Q(a), Q(b) and
Q(c) wi hou pipe junc ion in luence.
The geome y coe icien s can be hen e alua ed om he
exp essions (11) and (12)
)PF()P()PG(





(11)
and
01112-p.2
EFM 2012
i)MF(i)M(i)MG(





. (12)
Ad an ages o he geome y coe icien s a e
 They a e independen on he o al low a e in he
pipe junc ion
 They a e he same o he all geome ically simila
pipe junc ions
Bu i is necessa y o emembe ha hese coe icien s
depend on he low con igu a ion in he pipe junc ion and
on he low a e a io be ween wo b anches. In case o
low di isions he h ee low con igu a ions a e possible.
I is appa en om a igu e 2.
“Di a”, (Da) “Di b”, (Db) “Di c”, (Dc)
Fig. 2. Flow con igu a ion o he low di ision.
I is necessa y o know he unc ion o momen um
geome y coe icien and he unc ion o powe
geome y coe icien . I means i is necessa y o know
wo unc ions o cha ac e is ics o each low
con igu a ion.
4 Expe imen desc ip ion
The es ing ci cui was buil in he Vic o Kaplan’s
Depa men o Fluid Enginee ing labo a o y. The pipe
junc ion was d illed in plas ic blocks. I ensu ed he sha p
edges a he di ision domain. Fi e pipe junc ions, each
o he di e en angle o he adjacen b anch, we e
measu ed. The angles o he adjacen b anch we e 30°,
45°, 60°, 75°, 90°. The gauged pa ame e s was as ollows
Flow a es a each b anch gauged by he magne ic low
me e s, absolu e p essu e gauged by he p essu e
ansduce , p essu e di e ences be ween b anches “b”-
“a” and “c”-“a”, p essu e di e ences was gauged bo h by
di e en ial p essu e manome e and also by U- ube
manome e . I is necessa y o say ha o he
cha ac e is ics e alua ion he alues om he U- ube
manome e we e used. The las gauged pa ame e was
wa e empe a u e.
Fig. 3. The diag am o he measu ing ci cui .
The diag am o he es ci cui is d awn in he igu e 3.
The low a e was con olled by pump equipped by he
equency con e e and by he al es placed a each
b anch.
The luid low in some low con igu a ions was uns able
he e o e all pa ame e s we e gauged h ee imes. Ele en
low a e a ios we e measu ed o each low
con igu a ion. Each con igu a ion o all low a e a ios
was measu ed wice o ensu e ha he esul s a e co ec .
The e is a pic u e o es ing ci cle in he igu e 4 and he
pic u e o PIV measu emen s a he igu e 5.
Fig. 4. The es ci cui o he pipe junc ion cha ac e is ics
measu ing.
Fig. 5. The pic u e o he PIV measu ing.
5 Cha ac e is ics o pipe junc ion
The esul an cha ac e is ics will be p esen ed in his
chap e . The cha ac e is ics o nine di e en angles will
be d awn in he cha s. The cha ac e is ics o he angles
bigge han 90° a e ecalcula ed om he cha ac e is ics
o he angles less han 90°.
.
5.1. Flow con igu a ion “Di a”
The coe icien s

(PG) and

(MG)1 a e d awn as a unc ion
o low a e a io q(ca) in he igu e 4 and 5
)a(
)c(
)ca( Q
Q
q
.
(13)
The e a e some in e es ing esul s. I is appa en , om he
igu e 6, ha lowes losses a e o low alues q(ca) o
angle 30°, o he middle alues o q(ca) a e lowes losses
o angle 45° and o high q(ca) a e he lowes losses o
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EPJ Web o Con e ences
angle 60°. The in e es ing esul is ha he losses a e no
lowes o whole ange o q(ca) o pipe junc ion wi h
angle 30°.
In case o momen um coe icien i is in e es ing, ha he
coe icien s o he angles 30° and 150°, 45° and 135°,
60° and 120°, 75° and 105° a e a he close o each o he
and he maximal alues a e o he angle 90°.
Fig. 6. The

(PG) coe icien s o low con igu a ion “Di a” as a
unc ion o low a e a io q
(ca)
.
Fig. 7. The

(MG)i coe icien s o low con igu a ion “Di a” as
a unc ion o low a e a io q
(ca)
.
One will expec ha he bigges momen um coe icien
will be o he angle 150°. Bu his phenomenon can be
explained his way. In case o he angle 150° he p essu e
in b anch “b”, o big alues o q(ca), inc eases a lo . This
p essu e ac s agains o ce caused by he low di ec ion
change. The e o e he esul ing o ce is less han he o ce
o he angle 90°.
5.2. Flow con igu a ion “Di b”
The commen s o his case a e almos iden ical o he he
commen s in case o low con igu a ion Di a. The
a iable o he pipe junc ion coe icien s e alua ion is
low a e a io q(cb) in his case
)b(
)c(
)cb( Q
Q
q
.
(14)
The powe geome y coe icien is iden ical o he one o
he low con igu a ion Di a.
In case o momen um geome y coe icien one will
expec ha he cu es o he low con igu a ion “Di a”
and angle 30° will be symme ical abou ze o alue wi h
he low con igu a ion “Di b” angle 150° and simila ly
o he o he pai s o he angles. Bu his is no ue. I is
caused by he e e ence ze o p essu e in b anch “a”.
I would be symme ical in case when all p essu es will
be ela ed o he ze o p essu e p(b) in case o low
con igu a ion “Di b”. This is he case whe e i is
possible o see he in luence o he absolu e p essu e in
b anches.
Fig. 8. The

(PG) coe icien s o low con igu a ion “Di b” as a
unc ion o low a e a io q
(ca)
.
Fig. 9. The

(MG)i coe icien s o low con igu a ion “Di b” as
a unc ion o low a e a io q
(ca)
.
5.3. Flow con igu a ion “Di c”
The a iable o he pipe junc ion coe icien s e alua ion
is low a e a io q(ac) in his case
)c(
)a(
)ac( Q
Q
q
.
(15)
Fig. 10. The

(PG) coe icien s o low con igu a ion “Di c” as
a unc ion o low a e a io q
(ca)
.
01112-p.4
EFM 2012
The esul s o he powe geome y coe icien look a he
easonable.
Fig. 11. The

(MG)i coe icien s o low con igu a ion “Di c”
as a unc ion o low a e a io q
(ca)
.
This low con igu a ion was e y di icul o measu e,
because he low was e y uns able o he high alues o
q(ac).
The cu es o he pai s o he angles (30°-150°, 45°-135°
and so on) a e no symme ical abou he ze o alue o
momen um coe icien . The eason is he same as in he
case o low con igu a ion “Di a” and “Di b”.
6 Cha ac e is ics o pipe junc ion as a
polynomial unc ions
Cha ac e is ics o he pipe junc ion as a polynomial
unc ions will be p esen ed in his chap e . The
polynomial unc ions we e d awn o app oxima e he
measu ed alues. The gene al o m o he polynomial
unc ion is as ollow


N
i
i
)xy(i qA
0

.
(16)
The quan i y q(xy) is he low a e a io ela ed o he
pa icula low con igu a ion.
Table 1. Polynom coe icien s o he coe icien

(PG) o he
low combina ion “Di a”. Va iable is q
(ca)
Agl. A
(0)
A
(1)
A
(2)
A
(3)
A
(4)
30° 0.03996 0.21542 - 1.98045 5.96808
45° 0.00494 0.69511 - 2.64166 5.30444
60° - 0.03018 0.87583 - 2.18938 4.62091
75° - 0.02644 0.54269 0.22227 2.77669
90° - 0.01502 0.72022 - 0.50585 3.34952
105° - 0.04975 0.13346 2.99545 2.13002
120° - 0.04764 - 0.11965 4.90077 1.62379
135° - 0.05694 0.02440 5.34726 2.24393
150° - 0.04610 0.77084 3.13437 5.61312
Table 2. Polynom coe icien s o he coe icien

(MG)1 o he
low combina ion Di a. Va iable is q
(ca)
Agl. A
(0)
A
(1)
A
(2)
A
(3)
A
(4)
30° 0.06386 - 0.03327 0.50067 -0.30472
45° 0.04349 0.30411 0.01003 -0.17258
60° 0.01982 0.52247 - 0.15851 -0.07909
75° 0.00073 0.67438 0.13419 -0.31179
90° - 0.01433 0.77718 0.18345 -0.27213
105° - 0.04652 0.78045 0.14711 -0.15718
120° - 0.06140 0.90204 - 0.22418 0.08071
135° - 0.07597 0.92531 - 0.75895 0.43393
150° - 0.06522 0.59853 - 0.65970 0.26504
Table 3. Polynom coe icien s o he coe icien

(PG) o he
low combina ion “Di b”. Va iable is q
(cb)
Agl. A
(0)
A
(1)
A
(2)
A
(3)
A
(4)
30° - 0.04610 0.77084 3.13437 5.61312
45° - 0.05694 0.02440 5.34726 2.24393
60° - 0.04764 - 0.11965 4.90077 1.62379
75° - 0.04975 0.13346 2.99545 2.13002
90° - 0.01502 0.72022 - 0.50585 3.34952
105° - 0.02644 0.54269 0.22227 2.77669
120° - 0.03018 0.87583 2.18938 4.62091
135° 0.00494 0.69511 - 2.64166 5.30444
150° 0.03996 0.21542 - 1.98045 5.96808
Table 4. Polynom coe icien s o he coe icien

(MG)1 o he
low combina ion “Di b”. Va iable is q
(cb)
Ang. A
(0)
A
(1)
A
(2)
A
(3)
A
(4)
30° - 0.02744 - 0.15783 0.48191 -0.35726
45° 0.00565 - 0.55647 0.51469 -0.42874
60° 0.01008 - 0.61420 - 0.02193 -0.03182
75° 0.01988 - 0.62134 - 0.28020 0.18022
90° 0.01433 - 0.77718 - 0.18345 0.27213
105° 0.02839 - 0.83435 0.00595 0.28218
120° 0.03744 - 0.83957 0.41874 0.03667
135° 0.04040 - 0.73097 0.34852 0.10066
150° 0.04242 - 0.49984 - 0.07477 0.23415
Table 5. Polynom coe icien s o he coe icien

(PG) o he
low combina ion “Di c”. Va iable is q
(ac)
Ang. A
(0)
A
(1)
A
(2)
A
(3)
A
(4)
30° 6.7239 1.10915 - 19.578 24.26525 -7.34314
45° 7.4166 10.5006 - 81.598 118.1849 -51.384
60° 8.475 3.53416 - 49.05 71.86643 -31.063
75° 4.8848 18.66 - 76.133 108.396 -52.067
90° 4.7174 18.2622 - 57.23 76.319 -37.769
105° 3.7407 16.6857 -63.345 99.87026 -52.067
120° 3.76249 3.21864 - 19.829 52.38555 -31.063
135° 3.12062 3.67499 - 35.345 87.34959 -51.384
150° 2.95850 - 3.1574 9.15858 5.10732 -7.3431
Table 6. Polynom coe icien s o he coe icien

(MG)1 o he
low combina ion “Di c”. Va iable is q
(ac)
Ang A
(0)
A
(1)
A
(2)
A
(3)
A
(4)
30° 0.45341 - 1.165 2.17698 1.82076 -2.9947
45° 0.05468 - 2.706 13.51148 - 17.397 6.70083
60° - 0.184 - 0.857 7.98778 - 12.606 5.75276
75° 0.00097 - 0.2549 0.33661 -0.02042
90° - 0.0566 0.07725 0.03645
105° - 0.1144 0.59046 -0.30261
120° - 0.0394 - 0.6178 2.42304 -1.10727
135° - 0.0125 - 0.1635 - 2.57935 10.44418 -6.9974
150° 0.04895 - 1.9576 7.45342 - 6.42768 1.49592
7 Conclusion
The cha ac e is ics o he pipe junc ion o low di ision wi h
he i e di e en angles o he adjacen b anch we e p esen ed
in his a icle. All cha ac e is ics we e de i ed calcula ed unde
he nex assump ions
 S eady luid low
 The g a i y ec o is pe pendicula o he pipe
junc ion plain.
 The densi y is cons an
 The Co iolis numbe s and Bousineques
numbe s on he c oss-sec ions S(a), S(b), and S(c)
a e equal o 1.
01112-p.5

EPJ Web o Con e ences
 The e e ence p essu e is p(a)=0
The discussion abou he pipe junc ion coe icien s and hei
explana ion was also included in his pape . The in e es ing
esul s was ob ained o he minimum hyd aulic losses o low
con igu a ion “Di a” and “Di b”. The coe icien s o
polynomial unc ions o all measu ed cha ac e is ics we e
lis ed a he end o he pape .
8 Acknowledgemen
The au ho s a e g a e ul o he G an Agency o Czech
Republic (GACR) o unding his esea ch unde p ojec
“Ma hema ical and Nume ical Modeling o Flow in Pipe
Junc ion and i s Compa ison wi h Expe imen ”.
Regis a ion numbe o his p ojec is 101/09/1539.
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