CC BY
23
Section 4. Machinery construction
DOI: http://dx.doi.org/10.20534/AJT-16-9.10-18-28
Vasenin Valery Ivanovich, Perm National Research Polytechnic University, Associate Professor, Candidate of technical sciences, Department of Materials, technologies and machinery design
E-mail: vaseninvaleriy@mail.ru Bogomyagkov Aleksey Vasilyevich,
Assistant
Investigation of the operation of a ring-shaped gating system
Abstract: The description of laboratory ring-shaped gating system is provided. Results of theoretical and experimental determination of flow speed and liquid flow rate, depending on the quantity of simultaneously working feeders are stated. It is shown that the Bernoulli equation is suitable for calculating the gating systems with variable flow rate (mass), which varies manifold in the collector with the flow distribution to feeders. It takes stock of three loss pressure: by friction on length, in local resistances and on changing of pressure. The calculation is done by method of successive approximations until the desired value of difference of losses of the pressure from the opposite sides from the zero point is achieved. The decision is the method of successive approximations given the difference the loss head with opposition side from the zero point. A good agreement between the calculated and experimental data is obtained.
Keywords: pouring basin, sprue, collector, feeder, pressure, zero point, resistance coefficient, flow coefficient, stream velocity, fluid consumption.
Introduction ter in the basin is 103,5mm. Longitudinal axes of the col-L-shaped, branched, combined and step gating systems lector and feeders are in one horizontal plane. The level of (GS) were studied earlier. The difference between the cal- fluid H - the distance vertically from the section 1-1 in culated and experimental values ofvelocities, flow rates and the basin to the longitudinal axes of the collector and feed-pressures was several percent. The Bernoulli equation (BE) ers - was maintained by the constant way of undisrupted for the stream with variable flow rate (and mass) was used in adding of water to the basin and discharge of its surplus the calculations. Although, it is established [1, P. 10; 2, P. through a special crack in the basin: H = 0,363 m. The 205] for a fluid flow with constant flow rate (mass) given fluid flows from top from the feeders into the mould. Pi-the absence of the distribution of the stream to feeders, i. e. ezometers, glass pipes with the length of 370mm and infor GS with one feeder. It is not clear why BE works. And its ternal diameter of 4,5 mm, were installed in the sections of use in the calculation of GS with the fluid flow rate in the the collector 5-5, ... 16-16 to measure pressure. Piezom-collector (slag catcher) alternating from maximum to zero is eters bent by 90° (not shown in Fig. 1) were placed in the not proved theoretically. Thus, it appears practical to study, ex- section of the sprue 2-2, 3-3 and 4-4. The time of the perimentally and by way of calculations, the most complex fluid outflow from every feeder is 50-200s depending on GS - ring-shaped, in which there are 2 flows offluid and the the number of simultaneously operating feeders, and metal can be supplied to the feeder from two sides. the volume of water outflowed from every feeder is about Method of research 9 liters. These time and weight limitations ensured the de-The system (Fig. 1 and 2) consists of a pouring basin, viation from the mean value of the velocity ±0,005 m/s, sprue, collector and seven identical feeders I-VII. The in- not more. The outflow of fluid from every feeder was de-ner diameter of the basin is 272mm, and the height of wa- termined not less than 6 times.
Figure 1. Ring-shaped gating system
Figure 2. Scheme of a ring
Main body
First, let us calculate the characteristics of GS during the operation of only one feeder for the case, when the hydraulic system is disconnected in the section 16-16 (no ring). Let's establish the Bernoulli equation (BE) for the sections 1-1 and 17-17 of GS (let's consider that only feeder I is operating):
+ + h = ^ + aVl7 + h117, (1)
Y 2g y 2g
where p1 and p17 - pressures in the sections 1-1 and 1717, N/m 2 (equal to atmospheric pressure: p1 = p17 = pa ); a - coefficient of inequality of distribution of velocity along the section of the stream (Coriolis coefficient); accept a = 1,1 [2, p. 108]; g - acceleration of free fall;
-shaped gating system
g = 9,81 m/s 2; vx and v17 - velocities of the metal in the sections 1-1 and 17-17, m/s (due to a big difference in the areas of the basin Sj in the section 1-1 and of the feeder Sn in the section 17-17, one can accept Vj = 0 ); Y - specific weight of liquid metal, N/m 3; h117 - loss of pressure during the movement of fluid from section 1-1 to section 17-17, m. These losses of pressure are
(
hl-17 =
l
em j
v d ,
em
ZK +Z+xljmL d
em-l
k J
V
a-jem + 2g
.,2 (
a-
2g
l
(2)
Zn
V dn J
2g
where Zcm, ZK and Zn - coefficients of local resistances of the entry of the metal from basin into sprue, turn from the
2
2
V
7
sprue into collector and turn from the collector into feeder I; Z - coefficient of local resistance of the turn to 90° from section 5-5 to section 6-6 (without a change ofareas without the sections of the collector); A - coefficient of friction losses; - length (height) of the sprue, m; dm, dK and dn - hydraulic diameters of the sprue, collector and feeder I, m; vcm and v5 - velocity offluid in the sprue and in the collector in section 5-5, m/s; - distance from the sprue to the feeder I, m; ln - length of the feeder, m. Consumption in GS during the discharge from top is determined by the velocity of metal v17 in the output section 17-17 ofthe feeder I and area ofits cross section: Q = vl7Sn . Other velocities of fluid in the channels of GS are determined from the equation of flow continuity:
Q = v S = vcS = ,
^ cm cm 5 K 17 n f
(3)
where Sm, Sr - areas of the sections of the sprue and collector, m 2. Let us express all velocities ofmetal in (2) through velocity v17, using the equation of flow continuity (3):
h1-17(17) a
2g
Z +X —
cm j
d
( c ^
V cm J
(
L
ZK + Z + ^— d
( s >
_n_
S
i
+ Z„
d
. (4)
K JV"kJ
The expression in square brackets is specified as Z1-17(17) - it is a coefficient of resistance of the system from section 1-1 to section 17-17, reduced to the velocity of fluid in the section 17-17:
(
Z 1-17(17) (
Z +A —
cm j
d
( c ^
\ S I
V cm y
ZK +Z+A — d
( c ^
(5)
l
+ Z„ + A—.
Now, (1) can be written as:
H = av2n(l + Z1-17 (n))/2 g. (6)
And the coefficient of the flow rate of the system from section 1-1 to section 17-17, reduced to the velocity V17 ,
^1-17(17) =( + £-17(17) ) . (7)
The velocity
pgH /a . (8)
We find flow rate Q according to the expression (3). In the given GS, the length of the sprue = 0,2675 m, the length of every feeder ln = 0,0495 m, distance from the sprue to the first feeder lm- = 0,251 m. The diameters of the feeder, collector and sprue: dn = 0,00903 m, dK = d 5 =... = d12 = 0,01603 m, dT = 0,02408 m. Like in the works [3, 4], we accept that the coefficient of friction loss A = 0,03. The coefficient of local resistance of the discharge from the basin into the sprue depending on the radius of rounding of the entering edge is defined according to the reference book [5, p. 126]: Zm = 0,12 . The coefficients of local resistances are [6]: Z= 0,885, ZK = 0,396, Zn = 0,334 . The results of calculations according to correlations (5), (7), (8) and (3): Z1-17(17) = 0,683710 , ^1-17(17) = 0,770666 , v17 = 1,960978 m/s, Q17 = 125,5851141 -10"6 m 3/s.
During the calculation of the fluid outflow from the feeders II, III and IV, it should be taken into account that l „ = 0,370 m, l m = 0,489 m, l IV = 0,742 m, and
cm— l ' AJ"v cm-ll l ' AJ"v cm-IV ' AJ"v W11W
for the feeder IV - one more turn of fluid stream by 90° (without a change of the areas of the sections of the stream before and after the turn). The results of the calculations and experiments (in the denominator) are presented in Table 1.
Table 1. — Characteristics of GS during the operation of one feeder
2
2
v
17
2
2
n
+
n
n
Operating feeders Characteristics of the system
Z M , m/c 3KCU ' ' v , cm 3/c Q ЭKCп f ' Q *, %
1 2 3 4 5 6
I* 0,684 0,771 1,961 1,932 125,59 123,71 +1,5
I 0,631 0,783 1,992 1,976 127,60 126,51 +0,9
II* 0,706 0,766 1,948 1,925 124,76 123,26 +1,2
II 0,634 0,782 1,991 1,955 127,48 125,19 +1,8
III* 0,729 0,761 1,935 1,904 123,95 121,97 +1,6
III 0,637 0,782 1,989 2,010 127,37 128,71 -1,0
1 2 3 4 5 6
IV* 1,863 119,31
0,865 0,732 +2,3
1,822 116,66
IV 0,644 0,780 1,984 127,07 -2,0
2- 024 129.63
*) Qo =
Q - Q3
Q
■100
^ Hydraulic system is disconnected in the section 16-16.
When the feeder I is in the ring, the loss of the pressure in parallel pipelines 5-6 and 16-13-10-7 are not summed up, and they are equal to one another. The BE for sections 4-4 and 17-17 (on the way through sections 5-5 and 6-6):
,,2 ( V
С
P
--Г ОС
Y 2g
( l
Z +1
5(5) + ^ + Z
a-
2g
(9)
a
+ Pi
2g Y
The BE for sections 4-4 and 16-16 (through sections 16-16,13-13,10-10 and 7-7):
P±
Y
-a-
2g
b4-:
■16(16)
-A-
ст-1(16-Г)
-X
a
2g
d.
a
2g
In. Y
(10)
Here C5(5) — coefficient of resistance on the division of the stream in the sprue in the section 4-4 between sections 5-5 and 16-16, reduced to the velocity of metal in the section 5-5; C6(16) — coefficient of resistance on the division of the stream in the sprue in the section 4-4 between sections 5-5 and 16-16, reduced to the velocity of fluid in the section 16-16. We define these coefficients according to the following expression [5, p. 277]:
Z =[l + HV* / v) ] / ( / v)2 , (11)
where $ — coefficient depending on the rounding of the edges at the place of stream division; at big radius of rounding, 0 = 0,3; at zero radius of rounding, 0 = 1,5 ; for our GS, 0 = 1,5; v — velocity of the fluid before stream division, m/s; vd — velocity of the fluid in one of the channels after stream division, m/s.
Left parts of the expressions (9) and (10) are equal. We equate their right parts and, after transformations, we receive ( z = v5 / v16, lcm-K16-7) = 1,233 m):
z =
kd
Ь 4-
16(16)
-4,962548
Сад +1,354744
(12)
v S = v.S + v16S = z■ v16 ■ S + v16S =(z + l)v16S .
ст ст 5 к 16 к 16 к 16 к \ /16 к*
It is apparent that v5 > v16, see equations (9) and (10).
Let us assume that v5 = 1, lv16, i. e. z = 1,1. Then vl6/ vT = ST /(z +1) = 1,074550 . According to (11), we find that: C16(16) = 2,366058.
V16 = V5 1 Z , VCTSCT = (V5 + V16 К = (V5 + V5 =
= (1 +Uz^. And v5/vcm = Scm/(1 +1/z) = = 1,182005. According to (11), we define: Сад = 2,215750.
We put received values Z4 -5(5) and C16(16) in (12) and get: z = 1,432671. We set z = 1,1. We do the following approximation - z = 1,432671 - and repeat the calculation. After some approximations given the established z = 1,504590, we find according to the calculation z = 1,5045904 . The calculation of this relation can be finished because the difference between stablished and calculated values z is only 0,0000004 . We accept that z = v5 / v16 = 1,504590 . Herewith, v5 / vT = 1,355587 , Cscs) = 2,044183, v16 / vcm = 0,900968, C^) = 2,731917, v16 = 0,664633v 5.
Vct = V17Sn 1 , VJcr =(1 + 1/ Z) V 5SK = V17Sn , а
v5 = v13Sn /(1 +1 / z)SK . Coefficient of resistance of GS from section 1-1 to section 17-17, reduced to velocity v17 in the feeder I [(see dependences (2) and (9)],
i
Z 1-17(17) = Z a
V
d
S.
(
Сад +11JL + Z d
ст / V
V
S
+
1 ln d.
(1 +1/z )SK
We put in the known values, and receive: Z1-17(17) = 0,631241 , ^1_17(17) = 0,782964 , v17 = 1,992266 m/s, Q17 = 127,588871 -10"6 m 3/s.
As it can be seen, the closure of the ring around feeder I led to the reduction of resistance coefficient Z1-17(17) from 0,684 to 0,631. The appearance of a parallel collector led to the fall of fluid velocities in each line, reduction of friction losses and in local resistances, which caused the reduction Z1-17(17), growth M1-17(17), v17 and Q17 compared to the case, when feeder I was working at the disruption of fluid ring in the section 16-16.
When feeders II and III operate in GS (see Fig. 1), fluid from the sections 8-8 and 9-9 goes to feeder III, and to feeder II — only from the section 7-7, and not
2
к
2
2
2
fully. In the considered ring, there are two different streams: one anti-clockwise (16-13-9), the other — clockwise (5-7-8). The movement of fluid in the section 8-8 is from right to left. In this case, it is easily defined during the operation of two feeders. The streams meet at the entry to feeder III at the point A (Fig. 3), which is called the point of water division or zero point [2, p. 216217]. Mentally, we cut our ring along the intended line of water division and receive a net depicted in Fig. 3. Then, according to regular equations, we calculate the losses of pressure for the line 16-13-9 hu_9 and for the line 5-7-8 h5-8. After this, we compare two found losses of pressure. If hl6-9 = h5-8, we conclude that pressures in points A' and A" will be identical, which should be, because points A' and A" represent physically one point A. Consequently, having obtained the indicated equation, we can state that we established the correct values of flow rates of Q7, Q8, Q9, Q18 and Q19. If the specified equation is not obtained, one has to change the values of
these flow rates, and sometimes shift the point if water division, for instance, to point B — the point of fluid entry into feeder II (Fig. 3). Herewith, we refer to the 2nd, 3rd attempts making sure that the equation specified above had accurate precision.
Let us establish the BE for the sections 1-1 and 1818 (for the path through sections 2-2, 5-5, 7-7)
H =
(
Z +x1cm-
~ cm 1
d ,
cm J
I
v
a — + 2g
V
a
k y
2g
Z18 ++ 1 d„
(13)
a
2g
and for sections 1-1 and 19-19 (for the path through sections 2-2, 16-16,13-13, 9-9)
(
H =
(
Z
~ cm j
a
cm J
v
a — + 2*
(14)
Z4-16(16) + X+k
cm-111(16-9)
V
a-
2*
Z +1—+1
" a
a-
2*
Figure 3. Scheme for the calculation In the expression (13), Z18 — coefficient of resistance on the branching of a part of the stream from the collector into the feeder II with the output section 1818. Z8 — coefficient of resistance on the passage of fluid from section 7-7 into section 8-8 during the branching of a part of the stream from the collector into the feeder II. Coefficients of resistances determined by the branching of the stream from the collector into the feeder will be calculated according to the equations for three-way pieces [1, p. 112-115]. Coefficients ofresistance on the passage in the collector during the branching of a part of the stream into the feeder
Z np = 0,4 (1 - vnp / vK )2/( / vK)) (15)
and coefficient of resistance on the branching of a part of the stream into the feeder
during the operation of feeders II and III
C™ =[l + t{v„ / vK )2 ] / (vn / vK )2, (16)
where vK and vnp — velocities of metal in the collector before and after the branching of a part of the stream into the feeder, m/s; vn — velocity of fluid in the collector, m/s; t — coefficient. For our case, at Sn / SK = 0,317 t = 0,15 [7]. Coefficient Z„P is reduced to the velocity of passing stream vnp, and Zome — to the velocity in the feeder vn.
Let us write down apparent equations:
Q = vJt = Qs + Qi6 = vS + v16SK = Q18 + Q19 = vmsn + vS, Q8 = v gSK = Q7 - Q18 = v 7SK - vlsSn, Q19 = v A = Q8 + Q, =
= V8SK + V9SK .
Let us introduce the following notations: x = vl8 / v19, y = Q8 / Q7 = vgSK / v7SK = vs / v7 , z = Q5 / Qu =
= V5SK / V16SK = v5/ V16. And Q8 = yQ7, v7 = v8/ y,
2
2
18
2
2
K
O = O / z v = v / z v = v = v v = v = = v = v The flow rate of fluid in the system
Q = (V18 + V19 )Sn =(x ■ v 19 + V19 ) ■Sn = V19 (x + 1) ■Sn = v19Snp(19),
where Snp(19) =(1 + x)Sn - reduced to the velocity v19 -area of the feeders (considers the operation ofboth feeders). And vcm = vwSnp(w) / Smm . Similarly, we write down:
Q = (v18 + v19)S„ = (v18 + v18 / x) = v18 (1 +1 / x)Sn =
= v 18Snp(18), where Snp(18) = (1 +1/ x )Sn - reduced to the velocity v18 area of the feeders. And vcm = viSSnp(18) / S^. We also have:
Q = vJSm = (v5 + v16 )SK = (v5 + v5 / z) = v5 (1 +1 /z),
S_
S.
v = V
=v
p(18)
1 S_
=v
np(18)
(1 +1/z)SK 18 ST 1 +1/z SK 18 (1 +1/z)S/ Now, the equation (13) can be written down as:
v
H = a-^
2g
( S
np(18)
S
np(18)
+ Zi8 +A-T +1
ZLs)
l.
(1 +1/z )S
The expression in square brackets (except "1") is a coefficient of resistance from section 1-1 to section 18-18:
(
Z = Z + 2 m
S 1-18(18) rt cm ,
Vs.
np(18)
cm J \ \
+
r s.
np(18)
"k JV
I
(17)
+ Z18 ■
(1 +1/z )SKr
We accept (randomly): x = v 18/19 = 1, y = v8 /v7 = 0,4, z = v5 / v16 = 1,6. At x = 0,4 = 0,9, Z18 = 0,429714, see equations (15) and (16). For z = 1,6 according to (11), we find that C5(5) = 2,018579, and C16(16) = 2,827562. The results of calculations according to (17), (7), (8) and (3): Ci-i8(i8) = 1,178541 , Ml_lg(lg) = 0,677512 , v18 = 1,723946 m/s, Q18 = 110,405056 -10-6 m 3/s.
For the feeder III (line 1-16-19), the following correlations are apparent:
Q = VctSCT = V5SK + V16SK = (v5 + V16 =
= ( + vi6 ) = vi6 (z + 1)sK,
1 S„
S.
= v
np(19)
1 S„
z +1 S S z +1 S
k cm k
And the equation (14) will look as:
'( l ^ "
Z + À Cm "P^19) +
cm j
d
V cm y
=v
np(19)
(z + 1)Sk'
H = a —
2g
S
(
Z4-16(16) + 3C+A
+Z +1 d.
cm J l
cm-111(16-9)
(S
"p(19)
(z + 1)SK
+
In the square brackets (except "1") — coefficient of resistance of the system from section 1-1 to section 19-19 (for line 1-16-9):
(
L
Z = Z + 2
S 1-19(19) rt cm 1
V dcm J v
r s.
(
c
16(16)
+3Z+2
np(19)
S
cm \f
+
cm-111(16-9)
d.
np(19)
(z + 1)SK
+ Z +2—. d.
(18)
According to (18), (7), (8) and (3), we find:
Z_19(19) = 0,971933, M1-1,(19) = 0,712121, v19 = 1,812009 m/s, Q19 = 116,044835 -10-6 m 3/s.
The flow rate in the system Q = Q18 + Q19 = = 226,449891-10-6 m 3/s. vT = Q / Scm = 0,497244 m/s.
Q
q5=■
= 139,353779 -10-6 m 3/s, v5 = Q5/ SK =
1 +1/z = 0,690497 m/s.
Qi6 = 87,096112-10-6 m 3/s, v16 = Qu / Scm =
1 + z = 0,431561 m/s.
Q8 = Q5 - Qi8 = 28,948723 -10-6 m 3/s, v8 = Q8 / SK = = 0,143441 m/s.
In the ring-shaped hydraulic system, the losses of pressure hcm A from the sprue to point A, on the way through the sections 5-5, 6-6, 7-7 and 8-8, should be equal in losses of pressure hcm_A(16_9) from the sprue to point A on the way through the section 16-16, 1313 and 9-9. These losses ofpressure can be found according to the following equations:
h =
cm- A
z
d
4-5(5)
d.
a
2g
r o 1 d
(
h
'cm - A(16-9)
z 4-16(16)+3Z+^
Cm—111(16-9)
«71 >(19) 2g
a2~' (20) 2g
-16(16) - ^ - - ,
All values in (19) and (20) are known. v9 = v16, K-„.(16-9) = 0,995 m. We find that hcm _A(16_9) = 0,076692, the difference between them is hA = 0,020729 m.
Losses hcm_A more than hm_A(16_9), one should reduce the velocity of fluid on the way through the sections 5-5, 6-6, 7-7 and 8-8. We accept z = v5 / v16 = 1,4, and x and y remain same. We repeat the calculation and obtain: Ht -a = 0,087971 m,
hcm _A(16_9) = 0,087166
m,
hA = 0,000805 m, and y = v8 / v7 = 0,152813.
The results of the calculations at x = v18/19 = 1, y = v8 / v7 = 0,152813 and z = v5 / v16 = 1,4 : hcm_A = = 0,096260 m, hcm- A(16-9) = 0,090155 m, hA = 0,006104 m, x = 1,010657, y = 0,138314.
Acting in such way, we receive that x = v18/19 = = 1,031010, y = v8/v7 = 0,139022 and z = v5 / v16 = = 1,352501 h_A = 0,09340484 m, h^^ = = 0,09324289 m, hA = 4,91 -10-5 m.
2
2
2
d
S
cm y
\ cm J
k /
X
2
s
cm
2
2
2
2
Apparently, the difference hA can be reduced to any quired to ensure the working capacity of the proposed earlier established infinitely small value. It is clear that the method of GS calculation. The results of calculations and difference of pressures in 10-5 m is pointless. It was re- experiments (in the den°minat°r) are presented Table 2.
Table 2. - Characteristics of GS during the operation of several feeders
Operating feeders Characteristics of the system
£-19(19) ^1-19(19) v , m/s эксп ' V19 v , m/s эксп ' V18 v , m/s эксп ' V17 Ql9 3/ , cm 3/s эксп ' Q19 Q 3/ , cm 3/s Q ЭКСП ' Q, %
II, III* 1,530 0,629 1,600 1,626 1,514 1,547 102,46 104,13 199,36 203,21 -1,9
II, III 1,069 0,695 1.769 1.770 1,818 1,752 113,30 113,35 229,73 225,56 +1,8
I-III* 2,470 0,537 1,366 1,374 1,292 1,319 1,128 1,166 87,48 87,99 242,42 247,10 -1,9
I-III 1,646 0,615 1,564 1,525 1.557 1.558 1,516 1,455 100,17 97,66 295,76 290,62 +2,2
I-III, VII** 2,319 0,549 1,397 1,390 1,437 1,362 1,305 1,270 89,44 88,99 359,00 349,29 +1,8
I-III, V VI*** 2,998 0,500 1,273 1,256 1,258 1,209 1,118 1,098 81,50 80,42 390,57 384,96 +1,5
I-VII**** 4,775 0,416 1,059 1,004 0,945 0,954 0,811 0,830 67,81 64,31 427,68 422,62 +1,2
*Hydraulic system is disconnected in the section 16-16.
**) v23 = 1,466 m/s, v27 = 1,433 m/s
***) V3, = 1,292 m/s, v3m = 1,277 m/s, v22 = 1,158 m/s,
****) z = 4 904 Ь 1-20(20)
Ml_20(20) = 0,412, v20 = 1,047 м/с, v^ = 1,024 m/s During the operation of feeders I-III, the zero point will be located, apparently, in point B (Fig. 4). In this case, the fluid goes to feeder II from sections 7-7 and 8-8, to feeder I — only from sections 6-6, and not all fluid, and to feeder III — from section 9-9 (not all). Let us establish the BE for sections 1-1 and 17-17:
v2T = 1,171 m/s
(
H =
(
Z + 1 —
~ cm j
d ,
cm
a
2g
a-
2g
I.
(21)
V
Z17 +1
a
2g
for sections 1-1 and 18-18:
(
H =
(
Z
~ cm 1
d ,
cm J
a
Z7+Я-
v
2g
(
^z:5(5)+Z+^
v < ,
a
2 g
a
2g
I.
(22)
Z +1-^ +1 d.
a
2g
and for sections 1-1 and 19-19 (for thee way through sections 2-2, 16-16,13-13,9-9):
(
H =
V
Z +1—
~ cm 1
d ,
cm
a
2g
Сшб)+X +я
cm-111(16-9)
d
(
xa-
2g
l
Z19 +Я~Т + 1 d
a
(23)
2g
Let us indicate: x1 = v17 / v19, x2 = v18 / v19. Then,
V17 = x 1V19 , v18 = X2Vl9 , V19 = v17/ X, , V19 = v18/ x2,
V17 = x1V19 = V18x! / x2 , V18 = X2V19 = V17x2 / x! .
Let us introduce the following notations: y1 = Q7 / Qe = v7SK / v6SK = v7 / v6, y2 = Q8 / Q, =
= V8SK / V9SK = V 8 / V 9 , Z = Q5IQ16 = V5SK 1 VA = V5/V16 .
And v 6 = v7l yl, v 9 = vs / y2, v16 = vJz, v 5 = v6, V = v =... = v = v
9 K10 15 K16"
Let us assume that x1 = x2 = 1, yj = y2 = 0,4, and v5 = 1,6v 16, i. e. z = 1,6. The flow rate in the system
Q = VctSct = V 5SK + vJr. = zvJr. + vJr. =(z +And
the relation v16 / vT = ST /(z +1) = 0,867906. We calculate it according to (11): Z4-16a6) = 2,827562.
Similarly, we define:
Q = (v 5 + v16 )SK = (v 5 + v 5/ z ) =(1 +1/ z )v 5S5, Vs/ v™ = Scm / (1 +1/z) = 1,388649, ZU) = 2,018580.
At y1 = 0,4 Z7 = 0,9 — according to the correlation (15). vl7Sn =(1 -y,)v6SK, v17 /v6 =(1 -y,)SK /S„ = = 1,890788, Z17 = 0,429714 — according to the dependence (16).
For y 2 = 0,4 = 0,9, and Z19 = 0,429714. The flow rate of the fluid in the system
Q = VJt =(V17 + V18 + v 19 )S„ =
S_ =
V1 J
2
2
5
2
17
2
2
2
2
2
2
2
2
17
x, + x, + 1
= v
v 1-
S = v„S
17" np(17)'
where S
np(1l)
x l + x 2 +1 Snn — reduced to velocity v17 —
v
H = a-^
2g
the area of the feeders (considers the operation of all three feeders). And vcm = v17Snp(17) / Scm.
Also, Q = VcrScm = (v5 + v16 ) = ( + v5 / z ) =
/■ \ 1 S
= v5 (1 +1/z )SK, v5 = v6 = vT--T =
5V , 56 1 +1/z S
hcm d
S
np(17)
s
cm J V. cm J
+
d
* J
s
Snp(17)
+ Z17 +M /d +1 v(l +1/z)S n
The expression in square brackets (except "1") is
the coefficient of resistance of the system from section
1-1 to section 17-17 (for the line 1-5-17)
Z-
■>(17)
■17(17)
= v -
->(17)
1 s_
=v
np(17)
St 1 +1/z SK 17 (1 + 1/z)S/ And the correlation (21) will look as follows:
S
z +a—
J cm j
, d ,, ^ ,
\ cm / \ cm /
V
\ÎS f 1 ^
ZL(5)+Z+^k±
k /
(24)
np(17)
(1 +1/ z S
+ Z17 +Mn / dn.
Figure 4. Scheme for the calculation during the operation of feeders I, II and III
According to the correlations (24), (7), (8) and (3), we find: Zi—17(17) = 1,832576 , /u1_17(17) = 0,594168 , v17 = 1,511874 m/s, Q17 = 96,823562 -10-6 m 3/s.
Let us calculate the outflow of fluid from feeder II. The flow rate of the fluid in the system
v
2g
Q = VCTSCT = (( + V18 + V19 )Sn =
V 2
S. =
2
Z
hcm d
S.
np (18)
cm J \ cm J
Z° +Z+^
S
np(18)
(1 +1/ z )S
+Z +A—+1 nd
y,s„
p(18)
(1 +1/z )S
= Xi + x 2 + 1 S = S
= V18 Sn = V18Snp(18) ,
x
where S,
"P(18)
x + x + 1
—-2— Sn — reduced to velocity v18
x,
In the square brackets (except "1"), the coefficient of resistance of the system from section 1-1 to section 18-18 is written down:
( 7 y s „„,Y ( > i A
area of the feeders (considers the work of three feeders).
VT = VAP(18) 1ST.Wealsohave: Q = vcmScm = (v5 + v16 )St =
1 s
= (v5 + v5/ z) ^ = v 5 (1 + 1 / z) sk, V 5 = V6 = va
Z = Z + 2 lcm
S 1-18(18) ^ cm d
np(18)
S.
= v
-p(18)
1 S_
S
=v
-p(18)
1 +1/z SK
V 7 = ^V5 =
V cm J V cm J V
A2 z1 i \
np(18)
(1 + 1 /z)SK
Z7 +2L-J-d„
l
Z4-5(S) +Z+^ if
< J N
(25)
ln
+ Zn +2—. (1 +1/z)SK J dn
yS
np(18)
S^ 1 +1/z SK 18 (1 + 1/z)S/ = yV^T,—■ The dependence (22) will look as
(1 +1/z
follows:
Hie results of calculations according to (25), (7), (8) and (3): Zi—18(18) = 1,798513 , ^1_IS(IS) = 0,597773 , v18 = 1,521048 m/s, Q18 = 97,411043 -10-6 m 3/s.
For the feeder III, the flow rate of fluid in the system
2
X
X
2
X
K y
+
X
Q = VCTSCT = ( V17 + V18 + V19 ) Sn = (X1V19 + X2 V19 + V19 ) Sn = = V19 ( + X2 + = V19S„p(19)>
where Snp(19) = (x 1 + x2 + l)Sn — reduced to velocity v19 the area of the feeders. A vcm = vi9Snp{i9)l Scm
Q = = (v5 + v16 ) = (zv16 + v16) = v16 (z +1),
1 S_
S
V, = K = v„
■ = v
V 1C
np (19)
1S
S
■ = v
v ic
np(19)
Z +1 SK " Scm z +1 SK 19 (z + 1)S/ And the expression (23) will be written down as follows:
v
H =
2*
( i Y S Z •
~ cm i
d„
np(19)
S
+
cm J
lm
V cm J \
f l A
yd . iy . 1 cm-III(16-9)
Z 4-16(16) + -j-
V dk J
A2
f
X
S
np(19)
+ Z19 f- + 1 d
(z + 1)Sk
The correlation in square brackets (except "1") is the coefficient of resistance of the system from section 1-1 to section 19-19 (for the line 1-16-19):
Z1-19(19)
hcm d
np(19)
,, s
cm y \ cm
2 /
Z' + 3Z+A
cm-III(16-9)
S
np(19)
(z + 1)S
+ Z19 .
(26)
According to the equations (26), (7), (8) and (3), we define: Z™ = 1,659498. ^^ = 0,613197 , v19 = 1,560295 m/s, Q19 = 99,924513 -10-6 m 3/s.
The flow rate in the system ofthree feeders I, II and III Q = Q17 + Q18 + Q19 = 294,159118 ■ 10-6 m 3/s, and the velocity of fluid in the sprue vm = Q / Scm = 0,645921 m/s.
Q
q5=-
1 +1/z = 0,896958 m/s.
Q
= 181,020995 -10-6 m 3/s, v5 = Q5 / SK =
Qi6 =■
= 113,138122 ■ 10-6 m 3/s, v16 = Q16 / Scm =
1 + z
= 0,560599 m/s.
Q7 = Q5 - Q18 = 84,197434 ■ 10-6 m 3/s, v7 = Q7 / SK = = 0,417198 m/s.
Q8 = Q16 - Q19 = 13,213610-10-6 m 3/s, v 8 = Qsl SK = v 8 = QSISK = m/s.
In the ring-shaped hydraulic system, the losses of pressure hcm4>B from the sprue to point B on the way through sections 5-5, 6-6 and 7-7 should be equal to the losses of pressure hOT4B(16_9) from the sprue to point B on the way through sections 16-16,13-13, 9-9 and 8-8. These losses of pressure can be defined according to the following correlations:
h =
cm-B
C5(5) + z+^l-f-d„
„2 (
a
2g
l
Zv +1-T d
a-^,(27)
2g
cm-B(16-9)
Z 4-16(16) + 3Z
cm-III(16-9)
a
2g
(28)
+
Z +1— d.
a
2g
All values in (27) and (28) are known. v9 = v16, -9) = 0,995 m. We find that hm-B = 0,178106 ,
'"cm—III (16—9)
hm-B(16-9) = 0,133188 , the difference between them hB = 0,044918 m.
Losses hT-B are bigger than hcm_B(16_9), hence, one should reduce the velocity of fluid on the way through sections 5-5, 6-6, 7-7 and 8-8. We accept z = v5 / v16 = 1,4, and x1, x2, y1 and y2 remain unchanged. We repeat the calculation and receive: hm-B = 0,156807 m, hT-B (16-9) = 0,149202 m, hB = 0,007605 m, y1 = 0,426329, y2 =0,206438.
The results of calculations at x1 = 1, x2 = 1, y, = 0,426329, y 2 = 0,4 and z = 1,4: hcm_B = 0,155072 m, hT-B (16-9) = 0,148714 m, hB = 0,006358 m, y1 = 0,428153, y2 =0,9^05169.
At x1 = 1, x 2 = 1, y1 = 0,4, y2 = 0,206438 and z = 1,4, we have: hm-B = 0,159007 m, hcm-B(U-9) = 0,155007 m, hB = 0,004000 m, yl = 0,430048, y2 = 0,194150.
Acting in such way, we receive that, at Xj = 0.966830, x2 = 0.993380 , y1 = 0,440432 , y2 = 0,188701 and z = 1,393022 hcm_B = 0,155951 m, hcm-B{l6-9) = 0,155918 m, hB = 3,33 -10-5 m.
Acting similarly, we find the characteristics of GS during the operation of different numbers of feeders (Table 2).
Results of the research and their discussion
9695E8xperimental results differ from calculated ones by -2,0% to +2,2%, see Tables 1 and 2. The differences are small and it's difficult to make conclusions. In the 6w0h5o9l9e, one can consider that a good correspondence of theoretical and experimental data was obtained. And the Bernoulli equation established for the specific case — for 1t7h1e9s8ystem with one feeder, works in multi-feeder gating system, herewith, in the most complicated one — ring-shaped gating system.
Due to such small differences, there is a thought about a vicious circle when the data obtained in own experiments is used. Naturally, the coefficients of resistances on the turn in the collector by 90° and from the collector to the feeder and change ofareas of the sections ofthe stream before and after the turn were found for same GS. However, there is no vicious circle. Firstly, not a new, but known dependence — the Bernoulli equation, was used in the experiments on the definition of this coefficient during the operation of only one feeder (there was no stream division).
2
K
2
2
2
X
2
Secondly, to define the specified coefficient, independent experiments were conducted [6]. And, mainly, the coefficients of resistances in the hydraulics cannot be calculated and are defined experimentally. Only the resistance of a sharp expansion of the stream as well as with some allowances, the resistance of sharp narrowing and the resistance of turn by 90° without the change of areas of sections before and after the turn, are calculated theoretically. And our primary resistances — the turn in the collector by 90° and the turn from the collector into the feeder with the change of areas ofsections before and after the turn are defined only experimentally. As the coefficient of losses on friction A, the coefficients ofresistances on the division of the stream calculated according to (11) and passage and branching ofa part of the stream defined according to (15) and (16), are also obtained by way ofprocessing the results of experiments [1,5]. Since hydraulics is calculation and experimental science, one will have to use experimental data in theoretical researches.
Regardless the number of operating feeders, the Bernoulli equation is same — it is equation (1). Or, one can write down the BE for section 1-1 and any section of GS or any two sections (along the stream), although the flow rates of the fluid in these sections can differ by many times. I. e. we use the Bernoulli equation for the sections of the stream with different float rates and, unsurprisingly, experiments prove this, seemingly, absurd allowance. And the calculation of GS became possible at this expense. Without any additional principles. Only obvious: Q = Z Qi, where Qj — the flow rate of the fluid in i feeder. In any section of the hydraulic system, H acts in the form of a sum of velocity and piezometric pressures and losses of pressure.
Apart from two usual hydraulic losses — on friction along the length and in local resistances, the calculations take into account the losses on the change of pressure calculated according to the correlations (11), (15) and (16). The possibility to sum up losses on the change of pressure with the losses on friction along the length and in local resistances is not grounded theoretically. However, there is no experimental data contradicting this allowance.
The Bernoulli equation is established for the elementary spray of ideal («dry») fluid at the established movement strictly theoretically, without engagement of experimental data [2, P. 95-97]: h + p / y + v2 / 2g = const (along the spray), where h is the excess of the section over the plane of comparison. However, for the stream of real (viscous) fluid at the established movement, one has to implement the losses of pressure on friction and in local resistances and the coefficient of inequality of distribution of velocity on the section of the stream a [2, P. 108-111]. Herewith, to define losses on the friction, they find the experimental coefficient of losses A, and for the losses in local resistances — coefficients of local resistances Z . The coefficients A and Z depend on the velocity of the movement of the stream, coarseness of the surface of the pipe etc. I. e. the Bernoulli equation becomes calculation-experimental. And the expansion of its work on the streams with alternate flow rate of fluid with the use of experimental equations (11), (15) and (16) should not cause protest.
Let us note that the feeders «know» about each other, because switching on and switching off of only one feeder leads to the change of work of the entire hydraulic system (see Tables 1 and 2). Herewith, experimentally, the process of the fluid outflow is established very quickly, for 5-10s, even at sharp «imbalance» in the system, when, only feeders I, II and III operate. Apparently, there is something that is yet to be understood.
Conclusion
Thus, the ring-shaped gating system is researched theoretically and experimentally with the definition of velocities and flow rates of fluid in each feeder and in the entire system. At the calculation of such hydraulic system with changing flow rate of the fluid, the Bernoulli equation was used, although, it was established theoretically and checked in practice for the stream of fluid with constant flow rate. The calculation is done by method of successive approximations until the desired value of difference of losses of the pressure from the opposite sides from the zero point is achieved. A good agreement between the calculated and experimental data is obtained.
References:
1. Меерович И. Г., Мучник Г. Ф. Гидродинамика коллекторных систем. - М.: Наука, - 1986. - 144 с.
2. Чугаев Р. Р. Гидравлика. - М.: изд-во "Бастет", - 2008. - 672 с.
3. Токарев Ж. В. К вопросу о гидравлическом сопротивлении отдельных элементов незамкнутых литниковых систем//Улучшение технологии изготовления отливок. - Свердловск: изд-во УПИ, - 1966. - С. 32-40.
4. Jonekura Koji (et al.) Calculation of amount of flow in gating systems for some automotive castings//The Journal of the Japan Foundrymen's Society. - 1988. - Vol. 60. - № 8. - P. 326-331.
5. Идельчик И. Е. Справочник по гидравлическим сопротивлениям. - М.: Машиностроение, - 1992. - 672 с.
6. Васенин В. И., Васенин Д. В., Богомягков А. В., Шаров К. В. Исследование местных сопротивлений литниковой системы//Вестник ПНИПУ Машиностроение, материаловедение. - 2012. - Т. 14. - № 2. - С. 46-53.
7. Васенин В. И., Богомягков А. В., Шаров К. В. Исследования Ъ-образных литниковых системы//Вестник ПНИПУ. Машиностроение, материаловедение. - 2012. - Т. 14. - № 4. - С. 108-122.
DOI: http://dx.doi.org/10.20534/AJT-16-9.10-28-30
Kuliev Sabir,
Azerbaijan University of Architecture and Construction
Kazymov Musa, Azerbaijan State University of Oil and Industry E-mail: rahimova_mahluqa@mail.ru
Study of cracks formation in curved bars and rocks
Abstract: As it is know, different load-gripping devices, so called hooks, are used in the performance of loading and unloading works of mobile installations, cranes.
Years-long use of load hooks showed that the wrong selection of the material, production technology as well as violations of the exploitation regime lead to the appearance ofhollows, cracks, slag inclusions, which can cause the breakdown of hooks and emergency situations at production site. Hence, during the design of hooks, it is required to calculate its exact durability and crack-resistance.
Keywords: load-gripping devices, high curve bar, stress-intensity factor, cracks, buckling loads, normal fracture.
The work [1] presents the study of the stress condition A principle moment consists in the condition
of load-gripping devices, hooks, and defines total stress in of the limit equilibrium. The simplest variant of this
critical points B and C of cross-section (Figure 1). condition, based on physical ideas of Griffith, was for-
Having defined the maximal total stress in a criti- mulated by Irwin in case of a normal fracture [2; 3].
cal point of the cross-section of the bar during the cal- Irwin showed that the appearance of cracks in a brittle
culation of hook durability, it is required to determine or quasi-brittle body takes place when stress intensity
critical values of external load (i. e. the weight of a lifted at the top of the crack (k) reaches some (constant, for
load) at which cracks formation and local or complete this material) value. destruction of the body begin.
Figure1. Curved bar (hook)
