CC BY
17
UDC 532.527
R. R. Usmanova, G. E. Zaikov
THE CALCULATION OF THE OPTIMAL POSITION OF THE NOZZLE FOR SUPPLYING IRRIGATION LIQUID
Keywords: An irrigation spray; Criterion of defining the optimum; Fractional coagulation; Optimum location; The conic dissector.
Is set and experimentally verified the criterion that determines the optimal position of the sprinkler. Determined the position nozzle for supplying irrigation liquid for the best crushing of water, a uniform distribution over the cross section of its system and high efficiency gas cleaning.
Ключевые слова: конический рассекатель; траектория движения; орошающая жидкость; факел распыла; дробление
жидкости.
Установлен и экспериментально подтвержден критерий, определяющий оптимальное положение оросителя. Определено положение патрубка для подачи орошающей жидкости, обеспечивающее наилучшее дробление воды, равномерное распределение ее по сечению аппарата и высокую эффективность газоочистки.
1. Current state of a problem
Spraying of a liquid is widely applied in modern technics. It is carried out, in particular, in chemical and the food-processing industry at extraction of firm substances from liquids, at drying, at any interactings between liquids and gases, and also in a number of other processes (mash crushing in the aluminium industry, cooling of gases by the sprayed liquid in a number of apparatuses etc.) [1-7]. So widespread application of spraying speaks that in all these processes decrease of sizes of drops sharply increases a surface-area factor and, hence, reduces a time of a leakage of process that allows to reduce gabarits of apparatuses considerably. Besides, spraying secures with the big uniformity of distribution of a liquid and its best interacting with the reacting medium. Numerous experimental researches show that the streams of a liquid outflowing from a hole on medium of gas, pulse [8, 9]. Under certain conditions the liquid pulsation fade-ins along a stream and leads to its disintegration on drops. Character of intermittent motion depends on the nozzle form from which the stream, liquid index of turbulence in flush, physical properties of a liquid and gas and their relative speed outflows.
Problem about disintegration of streams by means of consideration of stability of the given current of a liquid. Mathematical research of stability of motion in relation to perturbations can be solved by means of the motion equations. With that end in view on stationary main current the non-stationary perturbation is imposed so that resulting motion fulfilled to the motion equations. At an outflow velocity having practical interest, gravity effects on liquid motion can to be considered. In this case on a liquid stream forces of viscosity, a superficial tension and a seepage force act.
2. Development of motion spherical drops of liquid in a gas stream
At optimum location of a sprinkler the best crushing of a liquid and overlapping by a spray of an irrigation water of all cross-section of the apparatus will be secured.
Let us observe motion of drops of a liquid in the twirled gas stream at optimum location of a sprinkler &0.
For simplification of the solution of a problem we will accept following assumptions: we will consider that the drop is in the form of a sphere and is absolutely a solid. Actually the drop form is not strictly spherical and in the course of motion in it originate the deformation caused by force of resistance of gas and counteraction of this deformation, called by a liquid surface tension force [2].
Motion spherical drops of liquid in a gas stream is presented by the vector differential equation in an aspect:
^Mu+m«G' (i)
where m0, D0 - weight and diameter of a drop; pg - gas density; V, U - vectors of a relative and absolute traverse speed of a drop; /U/ - the module of a vector of relative speed; G - a vector of acceleration of a gravity; yc - coefficient of resistance to corpuscle motion, function
vr
It is defined or on an experimental curve, or under empirical formulas [7].
vr - kinematic viscosity of gas.
Having increased both sides of an Eq. (1) on
6
2
n-vr pr
Let us gain
pD0 dV = ^^ + 1_G (2)
PVr
dt 4 2
PVr
index
I/ = U,/ +U„! =-- = ———/ + ->—-/ (3)
x У
■ ■ ■ WD0 WrD0 WVD0 W = WJ + WJ =-0 = + (4)
'x' 1 " У
Ù.ÙJ + =
X, , v y.
vr vr
/ j. V J. V J.
Let us spread out the Eq. (2) on components on X-axes and y
prvr2 dt 4
PiD о
dvy
WC\Ug (7)
prvr- ui 4 v, prvr-
Let us restructure the left parts of the Eqs. (6) and (7) for what we will increase and we will divide a derivative in the first equation on dx, in the another on dy, then:
1
Prvr
2 -V
PiDо3
dvx _ 3 D, dx 4 vr-dvu
-Vc\U\Ux (8)
у
Prvr
У
dx
3 D,
Vc\U\uy +
PiDо
(9)
index
T _
Prvr
Prvrt .x_ Prx .Y _ РгУ .
■•ff
PlD о
PlD о
P/D0 '
P-^ff y-\vc\u\
Prvr 4
And in view of earlier accepted designations, from the Eqs. (6) and (7) we will gain the dimensionless equations:
i v
dT
duv .
—— - -yU v + P dT y
and from Eqs. (8) and (9) accordingly
. .. dux ■
U „ —— - -yU x
dx
dVv У _-/,u +P
(10)
(11)
(12) (13)
r dy
By means of the gained Eqs. (10-13) it is possible to define speed of a drop at any moment and to design a path of its motion in a gas stream, moving with a variable speed if these equations to present in final differences.
So, from Eqs. (10) and (11) it is had:
Шу _AUX
U„
Uy 4U,,
)
and from Eqs. (12) and (13) accordingly
and
AY УАйУ
4ùy - P
(14)
(15)
(15a)
The analysis of dimensionless quantities Eqs. (10-13) shows that compared motions of drops in geometrically similar sprinklers will be similar, if
prl
PlDо
■ _ idem,
UD,
P/Dl
Prv]
— _ idem,
g _ idem,
(16)
(17)
(18)
H*- - idem, Uy
here l - length of an irrigation pipe.
In this case mechanical trajectories of volumes of a liquid also will be similar [4]. The time of motion of a drop is subject to the condition:
PVrt/
T _■
1
(19)
Applying the gained expressions to observed motion of a stream of an irrigation water in a sprinkler and neglecting stream weight, we will gain as a first approximation:
Prdx P/Dо Ud
- idem,
_ idem,
Ux
'opt
_ idem,
(2о)
(21)
(22)
here dx - diameter of the conic dissector in the field of a sprinkler bevel, m; dl - diameter of a hole of a sprinkler, m; mopt - speed of gas in the field of an irrigation, mps; vt - a projection of speed of an irrigation water to a perpendicular to a gas traffic route, mps.
In this case we accept that all irrigation water was completely converted to flat radial streams.
Thus performance of conditions of similitude Eqs. (20-22) for compared streams of an irrigation water is necessary, but insufficient. These conditions do not consider liquid supply in the device in the form of the flat radial streams, secured by the conic dissector.
On Figure 1 the way of corpuscles of a surface of a cone is represented
In polar coordinates R-! and &. By an arrow the flow core direction of rotation is shown. The curve 1 represents cone boundary line, a curve 2 a corpuscle way. In case of the flat dissector the curve 2 represents the valid way of a corpuscle of the liquid arriving on a tangent on a round of a cone and outflowing in its apex.
R/R°
Fig. 1 - The trajectory of the drops on the surface of the conical divider
3
2
2
3
7о
Analysing process of separation of a liquid in a gas stream, the author [1]. Specifies that at drop disintegration acts following forces:
p,u2D0;CT0O; n,uD0pru2Dl;
The condition of disintegration of a drop under the influence of a gas stream can be presented in the form of following criteria:
a ■ = idem (23)
p,u 2D0
and
ft
■ = idem
n (24)
pruDo
Passing round the had dependences to a case of disintegration of continuous streams of a liquid under the influence of a gas stream as a first approximation we will have:
p,u 2d,
= idem
and
ft
= idem
(25)
(26)
PrUdi
Using the had dependences Eqs. (25) and (26) it is possible to tell that compared motions of streams of the irrigation water which is getting out a sprinkler in geometrically similar apparatuses with axial input of an irrigation water will be similar, if Prdx
P/d,
= idem
Ud,
=
opt
= idem
a
Pl ®optdl ft
■ = idem
=
(27)
(28)
(29)
(30)
(31)
PiaoPtdi
The analysis of these criteria shows that they are the connected magnitudes since at maintenance of the best crushing of a liquid and full overlapping of cross-section of the apparatus by an irrigation spray, change of one of them involves by all means change and others.
3. Sampling of criterion optimum a sprinkler rule
Let us assume that defining criterion 0O will be any other criterion merging had criteria all here.
Therefore, product of four of them, namely Eqs. (27), (28), (29), and (31) gives not that other, as the relation of a Reynolds number of a gas stream in the field of a dispensing of an irrigation water and an irrigation water stream, at an exit from the sprinkler hole, increased by a dimensionless quantity, equal
yj1+ vil(oopt 2
d.h.
d x® opt
-a/1+ b,<Dopt 2 = idem (32) V p,v,di v
For the devised build of a sprinkler for supply of an irrigation water magnitude
(1
/1+ {,l(Dopt 2 = /j (7?! ; Therefore expression (32) can be presented in an
aspect
Re, Re,
Tj (771 )= idem
(33)
here mi - magnitude of a specific irrigation, equal to the relation of the charge of a liquid to the volume flow rate of gas taken under a condition on an entry in a scrubber, l/m3.
Having denoted the relation of numbers Rer and Re; through □, d.h.
/77 = ■
Re,
aoptdxv!
(34)
Re7 vidtvr Expression (33) can be written down in an aspect
/77 = Л
1
/77,
were
/2
1
=
idem
(35)
(36)
/771 f1 ("1 )
For the chosen build of a sprinkler for supply of an irrigation water and at use of any one liquid with a = const the criterion Eq. (30) will assume the following air:
'opt
■ = idem,
(37)
were
GCI =_
a
P/d,
Having increased and having divided expression
(37) on vi, we will have
G0 • )2 = idem (3 8) V/ ®opt
The analysis of expression (38) shows that the relation v/oiopt ~ mi (by definition), and vl~ mi at a>opt -const the criterion Eq. (30) under the conditions specified above also will be proportional to a specific irrigation.
Research of criteria Eqs. (30) and (35) shows that as in the capacity of an irrigation water water first from them is more the general is usually used and consequently there are all bases to assume that as a first approximation it (U) will be defining at a finding of optimum location of feeding into of a liquid through a sprinkler, answering to the best crushing of an irrigation water and more its uniform distribution on apparatus cross-section at preset values mi and a2. Here a2 - speed of a gas stream in observed cross-section of a scrubber, mps.
Having defined experimentally dependence of criterion of optimum location of a connecting pipe of supply of an irrigation water (U) from a specific irrigation, it will be possible to find by means of this dependence an optimum place of feeding into of a liquid
a
and at others, distinct from experimental, parametres of gas and fluid-flow streams.
So, for the conic dissector observed inprocess (Fig. 2) it is had:
dx - d 2 \ + 2®Jg a;
and
- 4Q1
(39)
"opt
K22 1 + 2©„&f 2
(4о)
where Ql - a second gas rate at parametres on an entry in the apparatus; al - An angle of disclosing of a cone of the conic dissector.
10
Gas
g j'-l p
71
/
ев
ш
Fig. 2 - The scheme to the determination of the position of the sprinkler
Substituting the had dependences dx and &opt in the formula (34) and solving rather ©0, we will have
©о _ d
1
„X f
ъf
)Э/УГmd si
-1) (41)
where 9 - an angle between a heading of outflow of a liquid from a sprinkler and a traffic route of a gas stream (Fig. 3).
In that case when Q1 it is expressed through a2, and Q1 through vh the formula (41) will assume the following air:
1 ^2 _ u (42)
© о _
2tgf v,d,vryxm
Where y1 and y2 - relative density of gas at temperature and an inlet pressure and an exit from the apparatus, accordingly, kg/m3.
Optimum location of a sprinkler &0 can be defined also or from expression (43)
¿0 _ ¿1 _ yd 2
©1
4 ¿1
Or from expression (44)
(43)
©1 _
1
(
f
1 --
Q^v/d/
Qjvrm^si n^
(44)
where L1 - total length of an irrigation connecting pipe, m; d1 - diameter of a sprinkler, m; L0 - distance from the conic dissector to an irrigation connecting pipe, m; y -the relation of magnitude L0 to d1
Fig. 3 - Circuit for determining the geometric parameters
Research of experimental data [7-9] about character of a deviation various to diameter and an outflow velocity of streams of water under the influence of a gas stream, moving with various speed, shows that inferred above the formula and criterion scores for a finding of optimum location of a sprinkler O0 for liquid supply can be passed round, apparently, only for the zone of an irrigation restricted on a size. Definition of the maximum zone of the irrigation served by an axial sprinkler, at the moment a theoretical way is not obviously possible and, therefore, it should be made experimentally. Quite probablly that for industrial scrubbers at which the apparatus cross-section will be more than the maximum spray of an irrigation formed by one connecting pipe, uniform water delivery on cross-section and provision of full shutdown of a plane by a liquid can be had only at installation of several sprinklers. For scrubber set the optimum value ©i, in the range of criteria ©1 = (0,06-0,10).
Conclusions
1. By comparison of the forces causing convergence of corpuscles it is shown that capture of trapped corpuscles of a dust by water drops in a scrubber is carried out at the expense of inertia for corpuscles d > l mic and diffusions for corpuscles d < l mic.
2. At water delivery in the central sprinkler, a place of its feeding into essential impact makes on extent of a dispensing of water, its distribution on apparatus cross-section, creation of a flat spray of an irrigation and efficiency of trapping of a dust. There is an optimum location of a sprinkler 0O at which for set □ and m2 in the apparatus the best crushing of water, more its uniform distribution on cross-section of the apparatus and the highest efficiency of a dust separation is secured.
3. On the basis of the equation of motion of volume of a liquid in gas medium the criterion defining
optimum location of a sprinkler is found and
experimentally confirmed.
References
1. Tate R. W. and Marschall, W. R.; Atomization by centrifugal pressure nozzles. Chem. Eng. Progress. 1953, 49(5).
2. Turner, G. M. and Mоutоn, R. W. Drop size distribution from spray nozzles. Chem. Eng. Progress. 1953, 49(4).
3. Vasilevsky, M. B.; and Zykov, Е. G.; Methods of raise of efficiency of systems of dust removal of gases with group cyclonic apparatuses in small power engineering. Ind. Power Eng. 2004, 9, 54-57, (in Russian).
4. Vereschagin, L.V., Semerchan I. А. and Sekojan S. C. To a Question on Disintegration of a High-Speed Water Stream. Works of Academy of Sciences USSR. 1959.
5. Idelchik I. E., Alexanders, V. P., Kogan, E. I. Research of direct-flow cyclone separators of system of an ash collection of a state district power station. Heat Power Eng. 1968, 8, 4548, (in Russian).
6. Kutateladze S. S., Gyroscopes E. P. and Terekhov V.I. Aerodynamics in the Restricted Vortex Flows. Novosibirsk: Academy of Sciences of the USSR; 1987.
7. Kutateladze S. S. and Styrikovich, M. А; Hydraulics of Gaso-Liquid Systems. M.:Power Publishing House; 1958.
8. Vitman, L. A. Spraying of a viscous fluid by injectors not centrifugal type. In: The Collector of Scientific Works. L.: Chemistry; 1953.
9. Vitman L. A. About calculation of length of a continuous part at disintegration of a stream of a liquid. In: The Collector Questions of a Convective. M.: State Power Publishing House; 1961.
© R. R. Usmanova - She is currently Associate Professor of the Chair of Strength of Materials at the Ufa State Technical University of Aviation in Ufa, Bashkortostan, Russia, regina.uu2012@yandex.ru; G. E. Zaikov - DSc. Professor of the Chair Plastics Technology Kazan National Research Technological University, chembio@sky.chph.ras.ru.
© Р. Р. Усманова - канд. техн. наук, доц. каф. СМ Уфимского госуд. авиационного технич. ун-та, regina.uu2012@yandex.ru; Г. Е. Заиков - д-р хим. наук, проф. каф. ТПМ КНИТУ, chembio@sky.chph.ras.ru.
