CC BY
12
Section 4. Mechanical engineering
DOI: http://dx.doi.org/10.20534/AJT-17-1.2-38-50
Vasenin Valery Ivanovich, Perm National Research Polytechnic University, Associate professor of the Department of materials, technologies and machinery design, Candidate of technical sciences
E-mail: vaseninvaleriy@mail.ru Bogomyagkov Aleksey Vasilievitch, Senior teacher
Investigation of the work of the P-shaped gating system
Abstract: The description of laboratory P-shaped gating system is provided. The results of theoretical and experimental determination of speeds and flow of the fluid were stated, depending on the quantity of feeders that work simultaneously. Not only ramification of some part of the stream from collector to collector (or to feeder) takes place in the system, but also the confluence of fluid streams from two collectors, and between the rings with feeders there are working feeders. It was demonstrated that Bernoulli's equation is suitable for calculation of gating systems with variable flow (mass), which varies a lot in collector in the course of stream distribution to feeders. For calculation the gating system is divided into two half-rings. Calculation is performed, using the method of successive approximations until the given value of pressure loss divergence in half-rings reaches the zero point. Pressure losses, speed and fluid flow in the feeder are calculated and compared with each other during flow of the fluid in two parallel hydraulic lines. Four types of pressure losses are taken into account: for friction along the length, in local resistances, for split of the stream into parts, for confluence of streams.
Keywords: pouring basin, down gate, collector, feeder, pressure, resistance coefficient, flow coefficient, speed of the stream, fluid flow.
Introduction
L-shaped, ramified, combined, crosspiece, stepped gating systems (GS), one- and two-ring GS [1; 2], vertical ring GS [3], ring system with feeders of different cross sections [4], GS with 2 down gates [5], GS with collector of variable section were studied previously. The difference between calculated and experimental values of speed and flow amounted to 1-6 %, although Bernoulli's equation (BE) was used in the calculation for stream of the fluid with variable flow (mass). And it was derived for a particular case — for the stream with a constant flow (mass), in the absence of distribution of fluid to feeders [5, 205; 6, 10], that is, for the simplest gating system — the system with one feeder. Therefore, BE is applicable for the stream with variable flow, although it is unclear why it works. The possibility of using BE in the calculation of GS with flow in the collector (scum riser) varying from maximum to zero in the split and confluence of
fluid streams is theoretically not proved. Therefore, it seems reasonable to research through experiments and calculations apparently the most complex GS — ring system, which has 2 rings with feeders and between the rings there are also working feeders (fig. 1). This system is widely used in the founding of magnesium alloys.
Fig. 1. Ring gating system: 1 — down gate; 2 — feeder
Research methods
In the fig. 1 in GS in sections A-A and B-B, the fluid speed is equal to 0. So, only half of GS can be investigated. The fig. 2 shows half of the ring GS — laboratory P-shaped GS. The system consists of pouring basin, down gate, collector and 5 feeders I-V. Internal diameter of pouring basin is 272 mm, height of water in the pouring basin is 103.5 mm. Longitudinal axes of collectors and feeders are located in the same horizontal plane. The level of liquid H — vertical distance from the section 1-1 in the pouring basin to the longitudinal axes of the collector and feeders — was kept constant by continuous filling-up the water in the pouring basin and discharge of its surplus through a special slit in the pouring basin: H = 0.3615 m. The fluid is poured from the top out of the feeders into the moulds. In the sections of the collector 5-5, ..., 16-16 piezometers are installed in order to measure the pressure — glass tubes with a length of 370 mm. and internal diameter of 4.5 mm. 90° curved piezometers were placed in the sections of the down gate 2-2, 3-3 and 4-4 (not shown in the fig. 2). The time of liquid outflow from each feeder was 50-150 s. — depending on the number of simultaneously operating feeders, and the volume of water discharged from each of the feeder was about 8 L. These time and weight restrictions provided the deviation from the average speed — no more than ± 0.0005 m/s. The fluid flow in each feeder was determined at least 6 times.
Major part
First we calculate the characteristics of GS for the cases, when the hydraulic system is open-circuited in the section 16-16 (there is no ring). We work out Bernoulli's equation (BE) for the sections 1-1 and 17-17 of GS (let's assume that only feeder I is in operation):
H = a — + h 2g
1-17 >
(1)
where: a — the coefficient of speed distribution irregularity in a section of the stream (Coriolis coefficient); assume that a = 1,1 [7, 108]; v17 — speed of metal in the section 17-17, m/s; g — free fall acceleration; g = 9.81 m/s 2; h1-17 — pressure loss when fluid flows from the section 1-1 to the section 17-17, m. These pressure losses are as follows:
i , \ 2
h „ =
l
z +x—
~ cm
\
ZK +^-cmT + 2Z
d..
cm J i
v
a+ 2g
vK
a — + 2g
d
(2)
a-
n J
2g
Fig. 2. P-shaped gating system
In equation (2): Zm, ZK and Zn — local resistance coefficients of entering of metal from the pouring basin into the down gate, turn from the down gate to the collector and turn from the collector to the feeder I; Z — local resistance coefficient of 90°-turn from the section 6-6 to the section 7-7 and from the section 7-7 to the section 8-8 (without change of collector sectional areas); X — friction loss coefficient; lcm — length (height) of the down gate, m; dcm, dK and dn — hydraulic diameters of the down gate, collector and feeder, m; vm and vK — fluid speed in the down gate and collector, m/s; lcm — distance from the down gate to the feeder I, m; ln — length of the feeder I, m. The flow in GS during water discharge from the top is determined by the metal speed v17 in the exit section 17-17 of the feeder I and by its cross sectional area: Q = v17Sn. Other fluid speeds in gates of GS are determined by using the flow continuity equation:
Q = v S = v S = v 17S , (3)
cm cm k k 17 nJ \ /
where: Scm, SK — cross sectional areas of the down gate and collector, m 2. Let's express all speeds in the
2
2
v
7
formula (2) through the speed v17, using the flow continuity equation (3):
h = a
1-17(17) "
2^
l
Z +A —
^ cm j
d j
cm y
\
V S !
\ cm y
c
l
ZK + A — + 2Z
V K dK !
+Zn
d
^ V
V s ,
V k j
We denote the formula in square brackets as ^i-i7(i7) — system resistance coefficient from the section 1-1 to the section 17-17, adduced to the fluid speed in the section 17-17:
Z
■17(17)
l
z + A—
~ cm
i „ \
\ S 1
cm
z + A — + 2Z
\
^S ^2
n
J\. Sk
l
+z .
~ n
[11]. Calculation results according to the formulas (5), (7), (8) and (3): Zi-i707) = 0.818621, ^1_1707) = 0.741530,
v17 = 1.882939 m/s, Q17 = 120.587295 -10-6 m 3/s.
If the hydraulic system is open circuited in the section 5-5, then for calculation of characteristics of the feeder I in the formula (5) it is necessary to replace lcmA by • (4) lm-i(i6-8) — distance from the down gate to the feeder I on the way through the sections 16-16, 14-14, 12-12, 8-8; l = 0.992 m. We have: Z. ...... = 0.912471,
1 cm-I(16-8) ~ 1-17(17) '
^j_17(17) = 0.723107, v17 = 1.836157 m/s,
Q17 = 117.591311 • 10-6 m 3/s.
The work of feeders II and III is calculated in the same way. The results of calculations and experiments (in the denominator) are given in the table 1.
N = 100 (Q - Q_)/ Q_, %. Let us calculate a simultaneous work of the feeders I and V. This is a ramified GS. v 9 = v10 = v11 = v 12 = 0. Work out Bernoulli's equation for the sections 1-1 and 17-17 (for the way through the sections 2-2, 5-5, 7-7, 8-8):
(5)
Now the formula (1) may be presented as:
H = av2„(l + Z,-,7(i7))/2g . (6)
The system flow coefficient from the section 1-1 to the section 17-17, adduced to the speed v17, is as follows:
^1-17(17) -17(17)
) • (7)
Speed:
17 _ r*1-17(17)
yJ^gH / a . (8)
The flow Q is calculated using the formula (3). Length of the down gate lm = 0.2675 m, length of the feeder ln = 0.0495 m, distance from the down gate to the feeder I l . = 0.494 m. Diameters of feeder,
cm—I '
collector and down gate: dn = 0.00903 m,
d = d5 = ... = d16 = 0.01603 m, d = 0.02408 m. We ask 5 16 'cm
sume, the same as in the works [8; 9], that the friction loss coefficient 1 = 0.03. The local resistance coefficient of the entering from the pouring basin into the down gate, depending on the radius of rounding of the entering lip, is determined using the reference data [10, 126]: Z = 0.12 . The local resistance coefficient of the 90°-
~ cm
turn from the down gate to the collector and change of flow areas ZK = 0.396 [11]. The local resistance coefficient of the 90°-turn in the collector from the section 6-6 to the section 7-7 (without change of the flow areas before and after the turn) Z = 0.885 [11].
Z = Z6-1 = Z7-8 = Z1-9 = Z14-12 _ Z14-13 _ Z15-16 . lo-
cal resistance coefficient ofthe 90°-turn from the collector to the feeder I (with change of the flow areas) Zn = 0.334
H =
I
\
d
cm y
I
a — + 2g
C5(5) +A — + 2Z
a — + 2g
(9)
z + A— +1
~ n
a
2g
and for the sections 1-1 and 21-21 (for the way through the sections 2-2, 16-16, 14-14, 13-13):
H =
r ^ .2
Z +l—
~ cm j
v d ,
cm
i
zti«(i6) + 2z
v
cm +
2g
\ 2 v
2g
(10)
Z + +1
~ n
a-
2g
where v5, v16, v21 — fluid speeds in the sections 5-5, 16-16 and 21-21, m/s; v. = v.,, v._ = v,,; l V — dis-
J ' ' 5 16 7 17 21' cm-V
tance from the down gate to the feeder V; l , = l ,, = 0.494 m; Z* ^^ — the resistance coefficient
cm-I cm-V ' ~ 4-5(5)
for split of the stream in the down gate in the section 4-4 between the sections 5-5 and 16-16, adduced to the speed of metal in the section 5-5; Z4-16o6) — the resistance coefficient for split of the stream in the down gate in the section 4-4 between the sections 5-5 and 16-16, adduced to the fluid speed in the section 16-16. These coefficients are calculated by the following formula [10, 277]:
2
2
v
7
2
n
+
cm
K
n
2
V
K
2
V
17
n
K
2
v
2i
n
^ =
1 +
/ v )21 / ( / v )2, (11)
where 0 — coefficient, which depends on the rounding of the edges of the place of stream split; in the case ofbig
rounding radius $ = 0.3 ; in the case of zero rounding radius Q = 1.5; for our GS Q = 1.5; v — fluid speed before split of the stream, m/s; vd — fluid speed in one of the gates after split of the stream, m/s.
Table 1. - Characteristics of gating system in the case of work of the feeder I
Indicators Operating feeders
I* I** I II* II III* III
z 0.819 0.912 0.693 0.819 0.639 0.844 0.637
0.742 0.723 0.769 0.742 0.781 0.736 0.782
v 1.883 1.89 1.836 1.83 1.952 1.98 1.883 1.88 1.983 1.99 1.870 1.86 1.986 2.01
Q -10-6 120.58 121.04 117.59 117.20 124.99 126.80 120.58 120.40 127.01 127.44 119.74 119.12 127.19 128.72
N, % -0.4 0.3 -1.4 0.2 -0.3 0.5 -1.2
Note: * — the hydraulic system is open circuited in the section 16-16; ** — the hydraulic system is open circuited in the section 5-5.
The fluid flow:
Q = v S = v.S + = 2v.S = + = 2v17S .
^ cm cm 5 k 16 k 5 k 17 n 21 n 17 n
v, / v = S /2S = 1.128277 — this is a ratio v, / v in
5 cm cm k d
the formula (11). We arrive at the following conclusion:
£-5(5) =£4-16(16) = 2285540 .
The formula (9) will be written as follows:
H = a^L 2g
l
Z + A —
~ cm
d
/ „ V 2S„
cm J V cm J
Z
4-5(5) + 2Z
l
S2
\ K J
+Zn +A — +1 d
The function in square brackets (except for "1") is Z1-17(17), the system resistance coefficient from the section 1-1 to the section 17-17, adduced to the speed v17 and considering the work of both feeders (I and V). We formulate:
Z1-17(17) = Z1-21(21) _ l.°35783 , ^1-17(17) = ^1-21(21) = 0.700865,
v 17 = v21 = 1.779679 m/s, Q17 = Q21 = 113.974307 • 10-6 , Q = 2 Q17 = 227.948615 • 10-6 m 3/s.
If the I and II feeders work and the hydraulic system is open circuited in the section 16-16, then this is a ramified GS. v 10 = v 12 = v 16 = 0. Let us work out Bernoulli's equation for the sections 1-1 and 17-17:
H =
z +A-
~ cm
v
a-cm +
cm J
2g
\
Z4-5(5) — + Z
Z7 -8(8) +A
k j
v8 a — + 2g
a — + y 2g
n,
l
Z„ + A — +1 d„
(12)
a
2
2g:
and for the sections 1-1 and 18-18:
H =
l
Z + a—
~ cm j
d j
cm y
l
v
a-cm +
2g
\
Z4-5(5) + A -cm-A + Z
j
a — + 2g
(13)
Z 7-9(9) +A
/
j
v9
a — + 2g
Zn +A — +1 d
a-
2g
Here v8, v9 — fluid speeds in the sections 8-8 and 9-9,
m/s; v 8 = v 9; lcm _A — distance from the down gate to the point A on the way through the sections 5-5, 6-6, 7-7; lm-A = 0.378 m: lA-I, lA-II — distance from the
point A to the feeders I and II; lA _I = lA _M = 0.116 m; Z7-8(8) — the resistance coefficient for split of the stream in the collector in the section 7-7 between the sections 8-8 and 9-9, adduced to the metal speed in the section 8-8; Z7—— the resistance coefficient for split of the stream in the collector in the section 7-7 between the sections 8-8 and 9-9, adduced to the fluid speed in the section 9-9. The fluid flow:
2
k
2
K
2
Q = v S = v.S = v7S = v„S + v0S =
^ cm cm 5 k 7 k 8 k 9 k = 2v8sk = v17Sn + V18Sn = 2v 17Sn .
vslv7 = S k/2SK = 0.5, z7-8(8) = z7-9(9) = 55 — according
to the formula (11).
Now the formula (12) may be written as follows:
H = a^L 2g
l
Z + A —
~ cm -,
d
„ V 2S,
cm J \ cm J \ /
Z4-5(5) + X
cm-A
+ Z
2S
/
v
zd + ; lA-L S7-8(8V d
S2
K J
V K J
J\ SK J
+ Z +x— +1
The ratio in square brackets (except for "1") -Z1-1707), the system resistance coefficient from the section 1-1 to the section 17-17, adduced to the speed v17 and considering the work of both feeders (I and II). Formulate: Z =Z = 1 910914
S 1-17(17) S 1-18(18) L-y-Vyi-t, ^1-17(17) = ^1-18(18) = 0 586118, V17 = V18 = 1.488307 m/s,
Q17 = Q18 = 95.314284 • 10-6 m 3/s, Q = 2 Q17 = 190.628568 • 10-6 m 3/s.
The work of the feeders I, II, III and the rupture of the hydraulic system in the section 16-16 lead to the formation of combined GS. v5 = v6 = v7, v11 = v14 = v16 = 0 , v8 * v9 * v10 . Bernoulli's equation for the sections 1-1 and 17-17 is already formulated — this is a formula (12). The formula (13) is valid for the sections 1-1 and 18-18; however, it is necessary to replace in it the local resistance coefficient of the feeder Z„ by the resistance coefficient Z18 (for ramification of some part of the stream from the section 9-9 of the collector to the feeder II). Bernoulli's equation for the feeder III for the sections 1-1 and 19-19 is as follows:
f
H =
l
\
Z +i—
~ cm j
a
cm y
v
a-cmm +
(
+ Z4-5(5)
(
+ Z" h 7-
V
JL )a
a /
+ À-
l
■+z
2g
a — + 2g
(14)
k J
r
V10 2g
a — + 2g
l
\
Zn + 1
a
V n J
a-
2g
Here Z10 — the resistance coefficient for the passage of the stream in the collector from the section 9-9 to the section 10-10 in the case of ramification of some part of the stream from the section 9-9 to the feeder II.
z =
~ oms
Resistance coefficients, resulting from ramification of the stream from the collector to the feeder and for the passage of the stream, will be calculated using the formulas for the circulating T-joint [6, 112-115]. Resistance coefficient for the passage in the collector in the case of ramification of some part of the stream to the feeder:
Znp = 0,4 (1 - vni I vK )(vnp I vK )2, (15) and resistance coefficient for ramification of some part of the stream to the feeder:
1 + t(v„ /vK)2]/( /vKf, (16) where v and v „ — metal speeds in the collector before
K np -T
and after ramification of some part of the stream to the feeder, m/s; vn — fluid speed in the feeder, m/s; t — coefficient. For our case at Sn / SK = 0.317, t = 0.15 [12]. The coefficient Znp turns out adduced to the speed of the
passing stream v „, and Z — to the speed in the feed-TO np ' ^ oms -T
er v . As we can see, the coefficients Z „ and Z den ' ~ np ^ oms
pend on the unknown speed ratios vnp / vK and vn / vK, namely, on v10 / v9 and v18 / v9.
Let us introduce the following notations: x 1 = v18 / v19, x2 = v 17 / v19. Fluid flow in the system:
Q = VcmScm = V 5sk = V17Sn + V18Sn + V19S„ =
= Sn (v17 + x1 ■ v17 / x2 + v17 / x2) =
= V17Sn (x2 +X1 +l)/ x2 = V17Snp(17)>
where S 07) = Sn (x2 + Xj +1) / x2 — the area of feeders, adduced to the speed v17. Similarly we write down:
Q = Sn (v„ + vi8 +vi9 ) = Sn (x2 * vi8 / xi + vi8 + vi8 / x, ) = = Vi8Sn (X2 +Xi +i)/ Xi = Vi8Snp(i8).
Here Snp(18) = Sn (x2 + xl +1) / Xj — the area of feeders, adduced to the speed v18. For the feeder III the area of feeders, adducedto the speed v19 — Snp(19) = Sn (x2 + xl + i).
Vcm = Vl7Snp(l7) ! Scm = Vl8Snp(18) ^ Scm = Vl9Snp(l9) ^ Scm , V5 = Vl7Snp(l7)! SK = V18Snp(18) 1 SK = V 19Snp(19)! SK •
Ratios xj, x2, v8 / v7, v9 / v7, v18 / v9, v10 / v9 are unknown. Let us assume, that x l = x 2 = 1.
V^ = QL = Q17 = =
V 7 Q7 Q17 +Q18 +Qi9 Vi7S„ +Vi8S„ +Vi9S„
X2Vi9 x 2
x 2v19+x1v19+1 x 2 +x1 +1
v9 xl +1
Similarly we determine:
v 7 x 2 +x1 +1
. At
^ = x2 = 1, v8 / v7 = 1/3 , v9/ v7 = 2/3 . Z7-8(8) = !0.5 , Z 7-9(9) = 3.75 — according to the formula (11). v8 = v17S / S .
8 17 n k
2
+
K
k
2
v
2
(V18 + V19 )S„ ( +V1S/X1 )S„ = (l+l/Xi )S„
V1 c
SK SK 18 SK
(V18 +V19 )S„ (X1V19 +V19 )S„ = (!+X1 )S„
V1 (
SK
V10 = V 19Sn 1 SK .
S
Now Bernoulli's equation for the feeders I, II and III may be formulated as follows:
H = a^L 2g
Z +A-
~ cm
I,
rs ^
■>(17)
cm J \ cm J
Z 4-5(5) +1 — + Z
Snp(17)
\ f
Z d + lA-I h 7-8(8)"'" ]
k j V k j
V
K J\ K J
+ Zn +A — +1 d„
H = a
2g
Z + A —
~ cm j
d
V
■>(18)
cm J \ cm J
Z 4-5(5) + Z
Snp(18)
Zd +
Z7-9(9) + d
K J \ K J
(l+1/x1 )Sn
K
S
K
+Z18 + 1
H = a^ 2g
Z + A—
~ cm j
d
v
np(19)
cm cm
Z 4-5(5) +XJCmA + Z
Snp(19)
,7-9(9) d .
K J \
j \ ^ k j 2
+
(1+X1 )S„
V SK J
Z10+A
I
+ Z„ + A — +1 d„
The ratios in square brackets (except for "1") are resistance coefficients ZX-mi) , Z1-18(18) and Z1-19(19) . At
xj = x2 = 1, Q10 = 0,5Q9, v10 / v9 = 0,5 . Using the formu-la(l5)we find out, that£10 = 0.4. — = = 1 - Q10 / Q9,
Q9 v9sk
Ll = hA = (1-Q10/QS } . v18/ v9 = 1.575657 , and
Zls = 0.552788 — according to the formula (16). Calculation results:
Z ,-17(17) = 3.460359, C1_18(18) = 4.197866,
Z™ = 4.04667, ^_17(17) = 0.473495, ^_1808) = 0.43 86 1 9, ^1_19(19) = 0.445273, v17 = 1.202326 m/s, v18 = 1.113768 m/s, v19 = 1.130665 m/s. xj = v 18 / v19 = 0.985055, x2 = v17 / v19 = 1.063380. And we prescribed the values x1 = x 2 = 1. We assume thatx1 = 0.985055 , x2 = 1.063380 , perform calculation once again and receive the results: x1 = 0.974609 , x2 = 1.102053. After successive approximations we determine that x1 = 0.946411, x2 = 1.150917. At the same time:
Z1-17(17) = 2915719 ,Z1-18(18) = 4.790815 ,
Z1-19(19) = 4.186801. The results of calculations and experiments are presented in the table 2. Similarly we calculate the work of GS with the feeders I-IV and I-V. Note that for the feeders I-III: v 17 / v 19 = 1.151, for the feeders I-IV: v 17 / v20 = 1.342, for the feeders I-V: v 17 / v2l = 1.553.
Bernoulli's equation in the case of work of the feeder I for the sections 1-1 and 17-17 of GS on the way through the sections 2-2, 5-5, 7-7, 8-8:
H = av17 llg + h1-17(5_8). (17)
Bernoulli's equation for the sections 1-1 and 17-17 of GS on the way through the sections 2-2, 16-16,14-14,12-12, 8-8:
H = ay\i! 2g + Vi7(i6-8). (18)
Pressure losses when the fluid flows from the section 1-1 to 17-17 on the way through the sections 2-2,5-5, 7-7, 8-8:
h
1-17(5-8 /
l
z + A—
~ cm
d
v
a-cm +
Z 4-5(5) +Z+^-
l
cm J \
V
/
h
k J
2g
2 v5
a — +
2g
(19)
Z6 | a-I V dK y
v8 a — + 2g
Z„ +A-
l
a-
n J
2g
Pressure losses when the fluid flows from 1-1 to 17-17 on the way through the sections 2-2, 16-16, 14-14,12-12, 8-8:
h
1-17(16-
l
z +A —
~ cm
v
a-cm +
cm J
2g
Z4-16(16) + 2Z+
cm-A(16-9)
a
Znp +1
h 9-8(8)T/L
A -I
K J
v8 a — + 2g
K J
Z„ +A-
v 2 16 +
2g
2
v127
a
2g
(20)
9 =
9 =
2
n
2
2
Table 2. - Characteristics of gating system
Indicators Operating feeders
I, II* I, II I-III* I-III I-IV* I-IV I-V* I-V
Z 1-17(17) 1.911 1.258 2.916 1.960 3.588 2.682 3.956 3.696
^1-17(17) 0.586 0.665 0.505 0.581 0.467 0.521 0.449 0.461
v y 17 1.488 1.50 1.690 1.74 1.283 1.25 1.476 1.50 1.185 1.12 1.323 1.32 1.141 1.09 1.172 1.17
Z 1-18(18) 1.911 1.025 4.791 1.746 11.029 2.638 22.357 4.421
^1-18(18) 0.586 0.703 0.416 0.603 0.288 0.524 0.207 0.429
v 18 1.488 1.48 1.784 1.83 1.055 1.09 1.532 1.56 0.732 0.76 1.331 1.32 0.525 0.55 1.091 1.12
Z 1-19(19) 4.187 1.713 8.226 2.564 16.147 4.435
№ 1-19(19) 0.439 0.607 0.329 0.530 0.241 0.429
v 19 1.115 1.14 1.542 1.51 0.836 0.85 1.345 1.33 0.613 0.64 1.089 1.17
Z1-20(20) 7.264 3.272 12.157 4.421
№ 1-20(20) 0.348 0.484 0.276 0.429
v y 20 0.883 0.92 1.229 1.25 0.700 0.72 1.091 1.14
Z1-21(21) 10.957 3.696
^1-21(21) 0.289 0.461
v 21 0.734 0.74 1.172 1.15
Q -10-6 190.63 190.85 222.48 228.63 221.16 222.87 291.38 292.67 232.92 233.75 334.82 334.30 237.83 239.52 359.53 368.24
N, % -0.1 -2.7 -0.8 -0.4 -0.4 0.2 -0.7 -2.4
Note: * — the hydraulic system is open circuited in the section 16-16.
Here lmm _A(16_9) — distance from the down gate to the point A on the way through the sections 16-16,14-14, 12-12, 9-9; lcm_a(1M) = 0.876 m.
In the formula (19) — the resistance coeffi-
cient in a sideward ramification in the confluence of the stream from the section 7-7 with the stream from the section 9-9 in the section 8-8. And in the equation (20) Zn-nm — the resistance coefficient for the passage in the confluence of the stream from the section 9-9 with the stream from the section 7-7 in the section 8-8. The coefficients of resistances, which are conditioned by confluence of the streams in the collector, will be calculated using the formulas for the circulating T-joint [6, 114-115]. Resistance coefficient for the passage in the collector in the case of confluence of the steams:
Z M = 1.55V, /vK/VK)2, (21)
and resistance coefficient in a sideward ramification (at V, > 0AVK ):
(„) = 0.75[l + ( / vK )2 - 2(1 - vtf / vK )2 ], (22)
where: VK — metal speed in the collector after the confluence of streams, m/s; v6 — fluid speed in a sideward
ramification, m/s. Coefficients L, , and Z , , are ad' ' ~ np(CRj ~ oms (Π)
duced to the stream speed after the confluence of streams.
Let us introduce the following notation: z = v5 / V16. Fluid flow in the system:
Q = v S = v17S = (v5 + v16)S =
cm cm 17 n \ 5 16/ k
= (z ■ v 16 + V16)SK = v 16 (z + 1)Sk.
Let us assume that v5 = 1,1v 16, i. e. z = 1,1. Then:
v16 / vcm = scm/(z + 1)SK = 1 074550 . Using the formula (11) we find out that£4_16(16) = 2.366058. Similarly we determine the following:
V cmScm = (V5 + V 16 K = (V5 + v5 / ZK =
= v 5 (1 + 1/Z K ,
v5 / v = S /(1 +1/z)S = 1.182005,
5 cm cm \ / k '
= 2.215750. v = v17S /S , v8 = v17S /S,
^ 4-5(5) cm 17 n cm' 8 17 n k'
v5 = v S /(1 +1/z)S = v S /(1 +1/z)S .
5 cm cm \ / k 17 n \ / k
Now the formula (19) may be written as follows:
h = a
1-17(5-8) ^
2£
Z + A
~ cm
d
/ „ \
+
cm J V cm J
Z 4-5(5) +Z+A
d
k /
S.
V
(1 +1/z )S
+
Z7-8(8) +A a
k j f „ \
+
k j\ k j
+Zn +A
. (23)
The function in square brackets will be denoted as Z1-17(5-8) — the system resistance coefficient when the fluid flows from the section 1-1 to the section 17-17 through the sections 2-2, 5-5, 7-7, 8-8. The coefficient is adduced to the fluid speed in the section 17-17. K v v 1
v8 v 5 + v 16 v5 + v5 / z 1 +1 / z
At z = 1.1 using the formulas (21) and (22) we determine the following: Z7-8(8) = 0.537528, z7%8) = 0.247392 . Using the formulas (23), (7), (8) and (3) we find out: Z1-17(17) = 0.659403, ^1-17(17) = 0.776290, v17 = 1.971203 m/s,
Q17 = 126.239922 • 10-6 m 3/s.
The value z was taken randomly. It must be determined through calculations. The ring consists of 2 halfrings: the first half-ring — from the down gate through the sections 5-5, 6-6 and 7-7 to the point A, the second half-ring — from the down gate through the sections 16-16, 14-14, 12-12 and 9-9 to the point A. Pressure losses in these half-rings — parallel pipelines — must be
equal. Pressure losses in the first half-ring:
f . A 2
h
cm—A(5—7)
*=> 4—;
4-5(5)
+ Z+A-
l
' + Z 7—;
7—8(7)
a-
2g
Pressure losses in the second half-ring:
cm-A(16-9)
.-)/-. o 1cm-A(16-9) <Z 4-16(16) + 2Z ---+ z
np
9-8(9)
a
2g
and must be adduced to
Coefficients Z7-8m and ^9_S(S) the speeds before confluence of the streams, using the following equations:
(v 7 / v J , Z:-8(9) =Z:P-8(8) (v,/ V j2.
f 6 _ f 6
T> 7-8(7) T> 7-8(8) 1
Z*) = 0.901653, Z
' np
5 9-8(9) ^ 9-8(8)
_ 'np = 2 3705
'7-8(7) ) >3 9_8(9)
v5 = 0.327652 m/s, v16 = 0.297866 m/s,
v 8 = 0.625518 m/s.
After calculations we obtain the following results:
h
'cm - A (5-7)
= 0.028348 m, hcm -
cm - A (16-9)
0.040521 m,
h5-16 = hcm-A(5-7) - hcm-A(16-9) = -0.012173 m.
Losses h ... .. are less than the losses h a,. In or-
cm-A (5-7) cm-A (16-9)
der to make them equal, it is necessary to increase the speed v5 and decrease the speed v16.
Changing z, we alter the speeds and pressure losses in the half-rings of the hydraulic system. At z = v5 / v16 = 1.384552615, h5_16 = 1.99 • 10-10 m. The difference of pressure in 10-10 m. is surely meaningless. It was necessary to verify efficiency of the proposed calculation method.
Bernoulli's equation in the form (17), (18) and pressure losses in the form (19), (20) are questionable. The characteristics of the feeder I may be calculated using the formulas (18) and (20). Let us compare the characteristics of the feeder I, using the formulas (17), (19) and (18), (20):
ZW7(5_8) = 0.6926712053, ^6-8) = 0.6926712043. The difference is in the ninth sign after the coma. But the calculation of pressure losses for the same feeder is conducted in different hydraulic lines. Apparently, the equations in the form (17)- (20) are valid.
The results of calculations and experiments are presented in the table 1.
During work of the feeders I and II two different streams exist in the ring: one is counterclockwise (16-1412-10), and the other one is clockwise (5-6-7), moreover, the flow is divided into two parts at the point A. In the feeder II the confluence of the streams from sections 9-9 and 10-10 takes place. The streams meet on entering into the feeder II at the point C, which is called the water-parting point or the zero point [7, 240-241]. In the mind's eye we cut our ring along the designated water-parting line and get the circuit, which is depicted in the figure 3. Then using common formulas, we calculate pressure losses for the line 16-14-12-10 h,, , „ and for the line
16-10
5-7-9 h5-9. Then we compare two obtained values of the
2
K
2
cm-A
x
2
17
x
2
n
n
v
5
K
pressure losses. If h16-10 = h5-9, we conclude that pressure in the points C' and C" will be identical, as it must be, since the points C' and C" physically constitute one point C. Therefore, after obtaining the specified equality, we may state that we designated right values ofliquid flow Q5, Q9 and Q16. Ifthe specified equality fails, then we have to change the liquid flow values. We make the 2nd, 3rd and subsequent attempts to ensure that the above mentioned equality is achieved with a given accuracy.
Fig. 3. Calculation scheme for the ring system during the work of the feeders I and II
Bernoulli's equation for the sections 1-1 and 17-17 of the feeder I (for the way through the sections 2-2, 5-5, 7-7, 8-8):
H =
Z +A-
~ cm
v
a-cm +
cm J
zL(5)+Z
"-K J
/
2g
2 V 5
a — + 2g
(24)
Z7 -8(8) +A
k J
v8 a — + 2g
l
\
Zn +A — +1
a
n
2g
Bernoulli's equation for the feeder II (for the way through the sections 2-2, 16-16,14-14,12-12,10-10):
H =
Z +x-
~ cm
v
a-^ +
cm J
2g
¡-'à . cm-11(16-10) 0 ~
z 4-i6(i6) —:— + 2z
\ 2 aVl6 +
+ (25)
2g
z„ +a—+1
a
n J
2g
Here I
— distance from the down gate to the feeder II on the way through the sections 16-16,14-14,
cm-ll(16-10)
12-12, 10-10, m. l
cm—ll(16-10)
= 0.760 m.
For the feeder II (on the way through the sections 2-2, 5-5, 7-7, 9-9) Bernoulli's equation may be formulated as follows:
H =
l
z + A —
~ cm j
d
cm J
v
a-cm +
2g
z;-s(5)+z
/
i
\
k j
/
\ 2 V 5
a — + 2g
(26)
Z à I ; 'A-l V dK
a--+
2g
l
z„ +1 d
2
v
a-
2g
Let us introduce the following notations: x = vl7 / vl8, z = v5 / v16, w = v8 / v9. Fluid flow in the system:
Q = VcmScm = (v 17 +V18 ) Sn = (xV18 +V18 )Sn = = V18 (X + 1)S„ = V 18Snp(18)'
where Snp (18) = (x + l)si8 — the area of the feeders, adduced to the speed v18. Similarly we write down:
Q = (V17 +V18 )S„ =(V17 +V17/X )S„ = = V A ( + X ) = V 17S„p(17).
Here Snp (17) = (l +1 / x )Sn — the area of the feeders, adduced to the speed v17. Also we have:
Q = v S = (v. + v.,)S =
^ cm cm \ 5 16 / k
= (v5 + v5 / z)sk = v5 ( + 1/Z)sk ,
Qs
v 5S
5 k
v 5sk
Q vcmScm v 5 {l+1/z )Sk 1 +1/ z'
v S
v 5 =■
V 17Snp(17)
zS
=v
,p(17)
(1+1/z)SK (1+1/z)SK 17 (1 + z)SK' Now the formula (24) may be written as follows:
H = a^ 2g
Z + a —
~ cm
np(1T)
cm cm
\ /
zLs)+a -cm-A+Z
zS
\2
np(1T)
(1 + z )S
K J
g , i 1a-i
ZT -8(8) +A
S2
J \ J
l
+Z„ + a—+1 d„
The ratio in square brackets (except for "1") -Z1-1l(1l), the system resistance coefficient from the section 1-1 to the section 17-17, adduced to the speed v17 and considering the work of both feeders (I and II).
ÏL = Q± = V 8SK = v 8 = 1 v7 Q7 v8SK + v9SK v8 + v8 / w 1+1/W ' We assume (randomly) that: x = 1, z = 1,1, w = 1. In this case Snp(l7) = 2Sn, Snp(18) = 2Sn. At w = 1, v8 / v7 = 0.5, Z7-8(8) = 5.5 — according to the formula (11). Coefficients
2
2
1
2
2
+
K
2
K
2
v
18
Z4-5(5) and Z4-16(16) were already determined. Calculation results:
Z1-17(17) = 1.530863, U-1707) = 0.628587, v17 = 1.596147 in m/s, Q17 = 102.220587 • 10-6 m 3/s.
For the feeder II on the way through the sections 2-2, 16-16, 14-14, 12-12, 10-10 the following ratios are valid:
Q = V cmS cm = (V5 + V16 ) = (ZV16 + V16 ) = V16 (Z + l)^
( \ v -S
v., / v = S / (1 + z )S , v., = v,„ =■
16 cm cm \ / K 16 10
V18Snp(18)
(l+z)SK (l+z)SK'
i6 vi6SK
vA
Q
Q v S v.6 (l+z)S 1 + z'
^ cm cm 16 V / k
The formula (25) will be formulated as follows:
H = a^
2g
Z + A —
~ cm j
d
cm J
V
">(18)
cm
-16(16)
-i cm-II(16-10) „ „
+ X---- + 2Z
S
np(18)
l
+ Z„ — +1 d
.(1 + z )sk y
In square brackets (except for "1") — the system resistance coefficient Z1-18(18) from the section 1-1 to the section 18-18, adduced to the speed v18 and considering a simultaneous work of the feeders I and II. We find: Z 1-18(18) = 1.113125 , ^1-18(18) = 0.687919 , v18 = 1.746806 m/s, Q18 = 111.869076 • 10-6 m 3/s.
The flow in the system: Q = Q17 + Q18 = 214.089664 • 10-6 m 3/s. v 5 = v6 = v7 = 0.555664 m/s, v10 = v 14 = v16 = 0.505149 in m/s, x = v 17 / v18 = 0.913752, w = v8 / v9 = Q17/ (Q16 - Q18) = 10.302815.
In the ring hydraulic system the pressure losses hcm-II(5-9) from the down gate to the feeder II on the way through the sections 5-5, 7-7 and 9-9 should be equal to the pressure losses hcm II(16-10) from the down gate to the feeder II on the way through the sections 16-16, 14-14 and 10-10. These pressure losses may be calculated using the following formulas:
h
t-II(5-9)
C-5(5) + Z
\ 2 V 5
a — + 2g
/
I
\
zd i ^ "a
S8-9(9) 71 J V dK
a
2g'
r
''cm—II(16-10)
fb ,1 cm-II(16-10) . t y
Z 4-16(16) -j-+ 2S
\ d /
V K J
\ 2 V,,
a
2g
(27)
. (28)
All values in the formulas (27) and (28) are known. We find out that:
h II(5 9) = 0.090665 m, h II(16 10) = 0.079521 m, the difference between them hII = 0.011144 m. The losses
Km—II(5—9) are greater, than Km-II(16-10) , it is necessary to decrease the fluid speed on the way through the sections 5-5, 6-6 and 7-7. At x = 0.913752, w = 10.302815 and z = 1.1 as per calculation hII = 0.004235 m, x = 0.938313, w = 12.186871. Using such approximations with further changes of x, w and z we bring hII to the value, which is less than 10-6 m. The table 2 presents calculation results for the feeders I and II at x = 0.947055, z = 1.070017 and w = 15.943246. v9 / v7 = 0.059021, i. e. only 5.9 % of the fluid flow comes to the feeder II in the section 7-7, the rest — to the feeder I.
Characteristics of the feeder II may be also calculated using the formula (26). Let us compare the characteristics of the feeder II using the formulas (26) and (25): Z 1-18(5-9) = 1.025445, Z 1-18(16-10) = 1.025438. The difference is in the sixth sign after the comma. But the calculation of pressure losses for the same feeder was conducted in different hydraulic lines. Apparently, the formulas (24)-(26) can be regarded as proven — considering their experimental verification.
During work of the feeders I, II and III, the zero point is apparently in the feeder II. Bernoulli's equation for the feeders I and II is already formulated — these are the formulas (24) and (26). For the feeder III for the sections 1-1 and 19-19 (on the way through the sections 2-2, 16-16,14-14,12-12, 11-11) Bernoulli's equation is as follows:
H =
z + A-
~ cm
L
\
cm J
a — + 2g
^4 -1
16(16)
* cm-III(16-11) -
+ A-+ z
a+ 2g
Z19 +A — + 1
a
2g
Here l
cm—III (16—11)
— distance from the down gate to the feeder III on the way through the sections 16-16,14-14, 12-12,11-11, m. Zcm-IIIa6_n) = 0.631 m. Z19 — the resistance coefficient for ramification of some part of the stream from the section 11-11 of the collector to the feeder III.
Let us introduce the following notations: x 1 = v 18 / v19,
X2 = vl7 / ^ ,
2 = v / V , W = V / V
5 ' 16 > 1 8 9
w = v / v
yv2 y 10 ' Ml
The adduced areas of the feeders will be as follows:
np(17)
= (
X, + x, +
1)/
X2 , Snp(l8)
= S, (
X, + x, +
l)
l
2
X
K
2
X
2
v
2
v
K
2
v
19
n
2
9
S 09) = Sn (x 2 + x 1 + i) .Let us write down the evident equations: v8 = v7w1 / (l+w1) , v9 = v7 / (l+w1) , v8 = vl7Sn /SK, v 11 = w2vn, Q5 = Qz/(1 + z), Q = v S = (v5 + v16 )s =
^ cm cm \ 5 16 / k
= (v 5 + v 5/Z ) = v 5 (1 + 1/Z ) ,
Q16 = Q/(1 + Z) , V 5 = V 7 , V11 = V16,
v = v.S JS = v,,,S JS = v.aS n,,aJS ,
cm 17 np(17) cm 18 np(18) cm 19 np(19) cm '
K =■
= v
' 1
zv S
cm cm
(1+z )SK zS
zS
=v
' 1 '
=v
' 1i
,p (17)
( + z )
zS
¿°np (19)
(1+z)S, 19 (1+z S • Acting as abovementioned, we define the characteristics of GS in the course of work of the feeders I, II and III. Similarly we calculate GS with the feeders I-IV and I-V. The zero point in these systems is in the feeder III. The results are given the table 2.
Research results and their discussion The difference between the experimental and calculation results is from — 2.7 % to +0.5 %. The difference is not big, and it is difficult to make any conclusions. In general, we may believe that a good conformity of theoretical and experimental data is obtained. And Bernoulli's equation, adduced for the particular case — for the system with one feeder, also works in the gating system with many feeders and in the most complex one — P-shaped system, in which there is not only a split of the stream in the collector into parts, but also a confluence of streams in the collector, and there are also operating feeders between the rings with the feeders.
Let us note the influence of circularity of a hydraulic circuit on the characteristics of GS. In the case of work of 1, 2, 3, 4 and 5 feeders, the flow in a closed-circuit system, as opposed to an open-circuit system, is higher by 3.7, 16.7, 31.8, 43.7 and 51.2 % respectively — due the work of the second parallel collector (see the table 1 and 2).
Regardless of the number of working feeders, the calculation is based on Bernoulli's equation in the form of the formula (1). Initially we randomly assume speed ratios in feeders and collector after split or confluence of streams, and determine the fluid flow in each feeder and all system. Then we divide the system into two half-rings. The pressure losses in half-rings must be equal to each other. We try to achieve it by increasing or decreasing the fluid speed in separate lines of GS. This difference of pressure losses in half-rings may be adduced to any pre-assigned infinitely small value.
Pressure losses, fluid speed and flow in each feeder can be calculated when fluid flows in two parallel hydraulic lines (but not against flow of the stream). Although the fluid speeds in different parts of the collector can differ from each other by many times. And the difference of the feeder characteristics in the case of its calculation by different hydraulic lines can also be adduced to any pre-assigned infinitely small value. This unexpected result requires surely a further verification and discussion.
So, we use Bernoulli's equation for stream sections with different flow and for different hydraulic lines in the same system, and, surprisingly enough, the experiments confirm this seemingly absurd assumption. And due to this it became possible to calculate the P-shaped ring GS. Without any additional principles. Only evident features:
n
Q = T Q,
i=1
where Qj — the fluid flow in I - feeder. In any section of hydraulic system the pressure H consists of velocity and piezometric pressures and pressure losses.
In calculations apart from 2 conventional hydraulic losses — for friction along the length and in local resistances, we consider the losses for the pressure change, which are calculated using the formulas (11), (15) and (16) — in split of the stream, and using the formulas (21) and (22) — in confluence of the streams. The possibility of summing the losses for pressure change and the losses for friction along the length and in local resistances is theoretically not substantiated. However, now we do not have experimental data that contradict this assumption.
Bernoulli's equation for elementary filament of ideal ("dry") liquid in stable motion was derived strictly theoretically, without use of experimental data [2, 95-97]: h + p / y + v2 /2g = const (along the filament), where h — excess of the section above the plane of reference. However, for the flow of real (viscous) fluid in a stable motion we have to introduce the pressure losses for friction and in local resistances and the coefficient of speed distribution irregularity along the section of the stream [2, 108-111]. And in order to determine the friction losses, we calculate the loss coefficient X through experiments, and to determine the losses for local resistances — we calculate the local resistance coefficients Z . The coefficients X and Z depend on the speed of stream flow, roughness of pipeline inner surface and other aspects. So, Bernoulli's equation becomes a calculation-experimental one. And the extension of its work field into streams with variable fluid flow with the use of experimental formulas (11), (15), (16), (21) and (22) should not draw protest.
Let's note that the feeders "know" about each other, so switching on or off at least one of the feeder causes the work restructuring of the entire hydraulic system. Moreover, the process of liquid outflow is established experimentally very quickly, within 5-10 seconds, even in the case of serious malfunction of the system, when, for example, only feeders I and II work.
The question arises about conformity between the results of experiments with water and liquid metals. We may give the following answer to it. In 1946 E. Z. Rabi-novich proved [13; 14] that «the movement of liquid metal in the region of turbulent flow does not have any peculiarities in comparison with ordinary motion of "normal liquids"; the hydraulic resistances in the studied motion of liquid metal follow the common laws of hydraulics». In the article [15], published in 1958, E. Z. Rabinovich came to the following conclusion: "Now we may regard it as the established fact, that in turbulent conditions, the local resistance coefficients depend only on the form and structural peculiarities of resistances; the influence of viscosity starts to manifest itself only in the region of laminar flow, where these coefficients are also regarded as a function of the Reynolds number. Thus, for record of local pressure losses in calculation of gating systems we may proceed from resistance coefficients, which are usually recommended for this purpose. The mechanism of molten metal movement changes significantly, if the initial pouring temperature is not sufficient".
The experiments, which were conducted for study of the gating channels, confirm applicability of the laws of hydraulics to liquid aluminum and titanium alloys and cast iron [16-18]. In the review of the works on hydrodynamics of molten metals [19, 340-341], including the works on foundry hydraulics, it was formulated the following: "In the whole region of turbulent flow, the molten metals act like normal, Newtonian liquids. A considerable change of the flow coefficient is observed only when temperatures are close to the solidification temperature, i. e. in the region where the molten mass is presented by Newtonian liquid". According to the
book [20, 57]: "A frequent calibration of orifice meters on water, mercury, tin and other molten metals also confirms the absence of differences in the local resistance coefficients during the flow of water and molten metal".
In the works [21; 22] the hydrodynamics of filling of a foundry mould with aluminum alloy AK12 was investigated theoretically and experimentally. In the articles [23-25] we studied the filling of gating system with alloy AK12 with the same dimensions as in the filling of a system with water. And thus far we have not detected any contradictions, using water instead of liquid metals in experimental studies of gating systems.
Considering the previously investigated GS, which are listed in the introduction, the possibility of using Bernoulli's equation to the stream sections with different flow may be regarded as proven, i. e. for calculations of gating systems with many feeders. However, theoretically it is not substantiated.
Conclusion
Thus, the most complex gating system (P-shaped) was theoretically and experimentally investigated for the first time — with identification of speeds and flow of the fluid in each feeder and all system. In the system there is not only a ramification of some part of the stream from collector to collector, but also confluence of fluid streams from two collectors, and some feeders are located outside the rings with feeders. In the calculation of such hydraulic system with variable fluid flow we used Bernoulli's equation, although it is theoretically derived and practically tested for the fluid flow with a constant flow, that is, for GS with one feeder. For calculation the GS is split into two half-rings. The calculation is performed by successive approximations until the predetermined value of the pressure loss divergence in half-rings reaches the zero point. Pressure losses, speed and flow of fluid in each feeder are calculated during flow of fluid in two parallel hydraulic lines (but not against the flow of the stream). And the difference of the feeder characteristics in calculation by different hydraulic lines may be adduced to any pre-determined infinitely small value. A good conformity of experimental and calculation results is obtained.
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