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ZESZYT NAUKOWY POLITECHNIKI RZESZOWSKIEJ Nr 266
Budownictwo i inżynieria środowiska z. 54 2009
Volodymyr V. CHERNYUK1
Vadym I. OREL2
EXPERIMENTAL VERIFICATION
OF A NEW METHOD OF CALCULATION
FOR PRESSURE DISTRIBUTIVE PIPELINES
ABSTRACT
In comparison with all the known methods of calculation for pressure distributive
pipelines (PDP), those developed by Chernyuk,V.V. proved to most exactly agree with results
of experiments. Calculated by this technique values of flow rate and of heads of fluid inside
PDP practically coincide with experimental data.
KEYWORDS: pressure distributive pipelines, variable mass fluid flow.
1. INTRODUCTION
Pressure pipelines with discrete fluid dispensation along the path are used in different
branches of economic activity of the human: irrigation (drip, subsurface, surface); ventilation
(discharge systems); metallurgic industry (cooling systems); water transport (distributive
lock-feed piping systems and those of large dry docks); water supply and water drainage
(distributive pipe systems of purification works, dispersed discharge of sewage) and others.
There are different techniques of calculation for pressure distributive pipelines (PDP).
The most perfect of them are based on differential equation of variable mass fluid flow
(DEVMFF) [1]. The creator of the theory of motion of variable mass bodies is prof.
Meschersky,I.V. (1897). In 1928, prof. Makkaveev,I.V. for the first time deduced the general
DEVMFF. In 1937, prof. Nen'ko,Ya.T. obtained the DEVMFF for total stream of fluid and
applied it to problems of calculating perforated PDP [2]. For cylindrical PDP DEVMFF is of
the form [3]:
0dhdxsin
g
dp
g
dVV2cosv
x
0
, (1)
where V is the average velocity of the main stream;
v
is the same for the flow of an outlet jet;
p
is the pressure inside PDP; dzdxsin
is the geometric head; dxidh fx is the loss of
head along PDP;
is the angle between the vectors v and V
;
is the angle of the
inclination to horizon (Fig. 1).
1 Doc., Cand. Sc., Lviv Polytechnic National University, Lviv, UKRAINE
2 Cand. Sc., Lviv Polytechnic National University, Lviv, UKRAINE
28 V. V. Chernyuk, V. I. Orel
In the existing methods, some variables of DEVMFF are expressed in terms of the
main flow )x(
Q, and the angle )x(
of inclination to horizon and is the angle )X(
between
the axis of PDP and the direction of outlet jets are neglected. Besides, in the known methods,
the magnitude of the friction factor )x(
of PDP and that of the coefficient of the flow rate
)x(
of outlets (nozzles) are assumed to be constant along the PDP. The non-complete taking
into account of design variables of a perforated pipeline and that of hydrodynamic
peculiarities of variable mass flow lead to considerable miscalculations [4].
2. TECHNIQUE OF PDP CALCULATION SUGGESTED BY CHERNYUK,V.V. [3]
Doc. Chernyuk,V.V. has suggested a new approach to solving DEVMFF for PDP [3].
It consists in expressing all the variables of DEVMFF in terms of the total operation head
)x(
H in the PDP. The calculation of PDP made by means of the relations obtained from the
solution of DEVMFF practically coincides with the experimental data. The influence of
constant or variable magnitudes of all the geometric parameters of PDP, those of kinematic
and dynamic characteristics of the main stream and of outlet jets, including the angle )x(
of
jet outlets, the angle )x(
of inclination, and the change in modes of flow and in laws of
resistance along the PDP are taken into account. The deduced relations are good for designing
long, intermediate, short, horizontal and inclined PDP. According to the technique suggested
by Chernyuk,V.V. [3], the flow rate dxHb
k
x
i
x
2/1
)x(
of the fluid which is dispensed from PDP in
its segment ki
whose length ki
x is calculated by means of Equation (2);
.sinx2
g2
V
D
x
H4
1
cosvVcos2
g4
xb
HxbdxHb
)
i
x()ki(
2
)
i
x(
)
i
x(
)ki(
)
i
x(
2/1
)
i
x(
)
i
x()
i
x()
i
x()
i
x(
)
i
x(
)ki()ki(
2/1
)
i
x()ki()ki(
k
x
i
x
2/1
)x()ki(
(2)
The calculation of PDP is made against the stream. The lengths of the segments are
taken to be equal to the distances between outlets (nozzles) hole
l. The values of total heads are
determined according to Formula (3);
,sinx2
g2
V
D
x
H2
cosvVcos2
g2
xb
HH
2
)
i
x(ki
2
)
i
x(
)
i
x(
ki)
i
x(
2/1
)
i
x(
)
i
x()
i
x()
i
x()
i
x(
)
i
x(
kiki
2/1
)
i
x()
k
x(
(3)
where )
i
x(
V is the average velocity of the main stream in the cross-section i
x of PDP (Fig. 1);
)
i
x(
v is the velocity of outlet jet; constg2nb o
, sm 5.1 ; 4d 2
is the area of the
outlet hole (nozzle); d is its diameter;
n
is the number of holes per unit length of PDP, 1
m;
D
is the inner diameter of PDP.
Friction factor )x(
for PDP is calculated according to the formulae for 2320Re )
i
x(
(laminar flow):
)
i
x(
)
i
x( Re
64
; (4)
Experimental verification of a new method of calculation for pressure distributive pipelines 29
Fig.1. Schematic diagram of PDP calculation against the stream; 1 – curve of piezometric
head; 2 – curve of total head; 3 – profile of average velocity of running out jets; x – axis of
PDP
for 10
D
Re
)
i
x(
)
i
x(
)
i
x(
(“smooth-pipe” turbulent flow):
25.0
)x(
)x(
i
iRe
3164.0
; (5)
for 500
D
Re10
)
i
x(
)
i
x(
)
i
x(
(transitional turbulent flow):
25.0
)x()x(
)x(
)x(
ii
i
iRe
68
D
11.0
; (6)
for
)
i
x(
)
i
x(
)
i
x( D
Re
>500 (“rough-pipe” turbulent flow):
25.0
)x(
)x(
)x(
i
i
iD
11.0
; (7)
and the value of Reynolds' number for the main stream in PDP is determined according to the
formula
)x()x(
)x()x(
)x(
ii
ii
i
DQ
Re
; (8)
where )
i
x(
is the kinematic viscosity; )
i
x(
v is the velocity of jet;
is the coefficient of
velocity; )
i
x(
and )
i
x(
are the angles, the reference is made counterclockwise as it shown in
Fig.1.
)
0
)
30 V. V. Chernyuk, V. I. Orel
Coefficients of flow rate outlet-hole or of outlet-nozzle )/,(Re )(
)( dlf i
xi holex
where
l is the thickness of PDP wall or the length of outlet nozzle; d is the diameter of outlet-hole
or of outlet nozzle; )(
Re i
x
hole is the Reynolds’ number for the jet which flows through outlet-
hole or through outlet-nozzle in the cross-section i
x of PDP, )(Re )(
)( i
i
xxholer Hf. For
example, for a cylindrical outlet-nozzle at )
i
x(
Fr >10, )
i
x(
We >200, for perfect total compression
and sharp inlet edges the value of the coefficient )
i
x(
can be calculated by means of empiric
formulae obtained by formulae from [5, pages 68-71]. One of these relations for the ratios:
dl 1…1.5, )(
Re i
x
theor =53 10...10 or dl 2…5, )(
Re i
x
theor =4
1015...50 or dl 10…50,
)(
Re i
x
theor =4
1015...80 is of the form [5, page 69];
d
l
i
x
i
theor
x
)(
Re
58
23.1
1
)(
; (9)
where
/2Re )(
)( dgH i
i
xxtheor is the Reynolds’ number for a jet at a “theoretical velocity of
running out” [5, page 61].
At the estuary of the PDP in the cross-section 0x
(Fig. 1) the flow rate equals the
transitive one tr)0( QQ , and the operating head )0()x( HH . The latter is calculated by the
formula )0()0( gH2q
; the value of the flow rate )0(
q through the last outlet-hole which is
to be realized should be substituted into this formula.
3. AIM OF THE PAPER
To experimental test the technique of calculation for PDP developed by
Chernyuk,V.V. [3] on the basis of the new approach to solving DEVMFF for PDP are
determined according to Formula (1)
4. EXPERIMENTAL PROCEDURE
The investigations were carried out on an experimental PDP whose diameter
D = 8.21 mm, the water was supplied by gravitation [6]. The material of the pipes was
stainless steel. The pipes were joined by flanges.
In the network of experimental PDP, holes with diameter of 3.2 mm were drilled along
a generatrix; coaxially to them, water outlets whose lengths was 25 mm and the inner
diameter d = 3.2 mm were welded to the wall. They were situated with the interval multiple of
10d. Depending on the purpose, these outlets were used for dispensation along the path or
they served as unions to which rubber pulse tubes where connected to join with piezometers
(Fig. 2). For convince in reading the schematic diagrams, unions in the diagram (Fig. 2) a
directed upward, and water outlets are oriented downward, as it really was. The inner
diameter of rubber pulse tubes is 8 mm. Heads were measured by piezometers correct to
0.5 mm. The operating head in the experimental PDP was 3740 mm when the valve 11 at its
end was closed (Fig. 2). Head tank 2 which has an overflow wall ensured constant head in the
experimental PDP, constant flow rate; and it prevented pulsations.
The nozzle to pipe cross-section ratio of PDP was calculated according to the formula
[7, page 30]:
Experimental verification of a new method of calculation for pressure distributive pipelines 31
Fig. 2. Schematic diagram of experimental setup: 1 – tank; 2 – head tank; 3 – overflow tank;
4 – overflow pipe; 5 – supply pipe; 6 – experimental PDP; 7 – water outlets; 8 – unions;
9 – rubber pulse tubes; 10 – board of piezometers; 11 – valve; 12 – measuring vessels;
13 – movable trough; 14 – handle; 15 – rolling bearings; 16 – measuring tank; 17 – hinge;
18 – receiving tank; 19 – pump; 20 – water collecting tank; 1`–12` – numeration of unions
(dimensions are given in mm)
n
f , (8)
where ω is the area of the cross-section of water inlet nozzle,
4
d2
; n is the number of
water inlet nozzles in the whole PDP; Ω is the area of the cross-section of the experimental
PDP,
4
D2
.
The non-homogeneity of the water dispensation along the path from the PDP is
calculated like this [7, page 32]:
end
beginning
q
q
, (9)
where beginning
q, end
q are flow rates through the first and the last water inlet nozzles of PDP
respectively.
The flow rates
q
of water through the nozzles where determined in terms of volume
with a help of the measuring vessels 12 (Fig.2).
The relative change of non-homogeneity in the dispensation of water along the path is
caused by the inclination of PDP compared to its zero inclination under other analogical
conditions is
910
14
15 13
6
7
3740
3'
2'
1'
12
8
6'4' 5'
5
90
40
80
1220
15 20 19
10'
7' 8' 9'
12 7
11' 12'
8
9
10
18
16
17
511
4
100
200
240
280
3
1
2
to tank 18
to tank 1
32 V. V. Chernyuk, V. I. Orel
%1001
o
, (10)
where subscripts ψ and 0 denote the water flow in PDP with inclination to horizon at the
angle of ψ ≠ 0 and ψ = 0 respectively.
The flow rate at the end of PDP (Fig. 1)
Qbeginning = Q(xN) = Σq + Qtr , (11)
where Qtr is the transitional flow rate at the end of PDP which was also determined in terms of
volume with a help of the measuring tank 16 (Fig.2) according to Fig. 1, Qtr = Q(о).
5. COMPARISON OF PDP CALCULATION TECHNIQUE [3] WITH
EXPERIMENTAL DATA
PDP of intermediate lengths were investigated with eleven (Fig. 3a) and with eight
(Fig.3b) water inlet nozzles whose nozzle to pipe cross-section ratio f = 1.469 and 1.215 and
whose operating length L = 2644 mm and 1276 mm respectively.
2644
158
160
323
644
D=8,21
317
1'
2
240
117477
640
2'
1 4
159
211
5'4'3'
165
3
7'6'
161
5 6
161158
79
161
11
10'
234
234
9'8'
162
7 8
158
12'11'
163160
9 10
163
82
Qbeg
20,95
20,95
a)
158
160
1
159
1' 3'2'
161
2 3
161158 161
8
234
234
5'4'
162
4 5
158
163
160
6 7
163
82
6' 7' 8'
1197 79
81 1195
Qbeg Qtr
165
82
b)
Fig. 3. Schematic diagram of experimental PDP whose f = 1.469 (a) and 1.215 (b):
1-11 – water inlet nozzles; 1`–12` – unions for connecting rubber pulse tubes; Qbeginning – flow
rate at beginning of PDP; Qtr – transitional flow rate (dimensions are given in mm)
During investigation on PDP with f = 1.469, the transitional flow rate in the cross-
section 0 (Fig. 1) was absence, and the head H(0) = 0.104 m. With this, the non-homogeneity
of the water dispensation along the path from PDP was η = 2.77 (Fig. 4).
The investigation on PDP with f = 1.469 (Fig. 6) were carried out under its different
inclinations according to the schematic diagram given in Fig.5, at the absence (Qtr ≠ 0) and
presence (Qtr =Q(о) = 0) of transitional flow rate in the cross-section 0 (Fig. 1).
According to Fig. 6, the non-homogeneity η of water dispensation along the path from
PDP is shown in Table 1.
Table 1. Non-homogeneity of water dispensation from PDP along the path
Qtr ≠ 0 Qtr = 0 Angle of inclination to horizon
)x(
, agree η Δη/η, % η Δη/η, %
0 1.671 — 1.867 —
5.3 1.945 –16.4 2.291 –22.7
354.7 1.622 2.9 1.692 9.4
Experimental verification of a new method of calculation for pressure distributive pipelines 33
Fig. 4. Comparison of results for PDP whose f = 1.469:
a – piezometric head inside PDP; b – flow rate inside PDP;
1 – experimental data; 2 – calculation according to the formulae (2)–(7);
x-axis is directed against the stream [9]
Fig. 5. Schematic diagram of inclined PDP:
1 – zero inclination (ψ = 0о); 2 – descending of pipe along the flow (ψ = 5.3о); 3 – ascending of
pipe along the flow (ψ = 354.7о)
Fig. 6. Relative variation of water dispensation along the stream for PDP with f = 1.215
for Qtr =Q(о) = 0 (а) and Qtr ≠ 0 (b): 1-3 – experimental data; 4-6 – calculation according to the
formulae (2)–(7); 1, 4 – ψ = 0о; 2, 5 – ψ = 5.3о; 3, 6 – ψ = 354.7о;
L – operating length of PDP; x-axis is directed against the stream
0
0,4
0,8
1,2
1,6
2
0 1 2 3
х, м
p/(
g),
m1
2
0
0,00004
0,00008
0,00012
0,00016
0,0002
0 1 2 3х, м
Q, m 3/s 1
2
a) b)
0,75
1,00
1,25
1,50
1,75
2,00
0,0 0,2 0,4 0,6 0,8 1,0
x/L
qi/q 81 2 3
4 5 6
0,75
1,00
1,25
1,50
1,75
2,00
2,25
0,0 0,2 0,4 0,6 0,8 1,0
x/L
qi/q 81 2 3
4 5 6
a) b)
34 V. V. Chernyuk, V. I. Orel
Thus, the least non-homogeneity of water dispensation from PDP along the path is
observed under ascending of pipe along the flow (ψ = 354.7о), and the greatest one under
descending of pipe along the flow (ψ = 5.3о). The presence of transitional flow rate (Qtr ≠ 0) lessens
the non-homogeneity of water dispensation from PDP along the path. This can be seen from
our calculation and is confirmed by experiments (Fig. 6, Table 1).
6. CONCLUSIONS
The method of calculation for pressure distributive pipelines (PDP) [3] is good for
calculations horizontal, ascending, and descending of PDP; this is confirmed by experiments. The
values of heads, of flow rates of water inside PDP and of the water dispensation along the
path which are calculated according to the formulae (2)–(7) practically coincide with the
experimental data.
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