CC BY
7
UDC 532.527
R. R. Usmanova, G. E. Zaikov ASSESSMENT OF A CRITICAL TRAJECTORY OF THE MOVEMENT OF A PARTICLE TAKING INTO ACCOUNT REGIME DESIGN DATA OF THE DEVICE
Keywords: separation factor; movement equation; geometrical complex; regime complex; trajectory ofparticles.
The method of forecasting of efficiency of gas purification on a basis is developed analysis of hydrodynamics of gasdisperse streams. Calculation of trajectories of the movement ofparticles is given, which will allow to find conditions of their catching and to define influence of essential factors on gas purification process. As the defining parameter the dimensionless Are complex is established (a separation factor), including an assessment of a critical trajectory of the movement of a particle considering regime design data of the device.
Ключевые слова: фактор сепарации; уравнение движения; геометрический комплекс; режимный комплекс; траектория
частиц.
Разработан метод прогнозирования эффективности газоочистки на основе анализа гидродинамики газодисперсных потоков. Приведен расчёт траекторий движения частиц, который позволит найти условия их улавливания и определить влияние существенных факторов на процесс газоочистки. В качестве определяющего параметра установлен безразмерный комплекс Ар (фактор сепарации), включающий оценку критической траектории движения частицы с учетом режимно-конструктивных параметров аппарата.
1. Statement of the problem, the assumptions
To studying vortex devices in many papers, extensive experimental data. However, many important problems of analysis and design of vortex devices have not yet found a systematic review.
Existing research in this area show a strong sensitivity of the output characteristics of the regime and design of the device. This indicates a qualitatively different flow hydrodynamics at different values of routine-design parameters.
Thus, it is important to consider the efficiency of fluid flow and vortex devices, receipt and compilation dependencies between regime-design parameters of the machine. Creating effective designs is the actual problem.
2. Derive the equation of motion of a particle
One of the most common devices for dust cleaning equipment considered centrifugal machines. Due to their widespread use simplicity of design, reliability, and low capital cost.
Consider the mechanism of dust-gas cleaning scrubber is something dynamic [1]. Capture dust scrubber is based on the use of centrifugal force. Dust particle flows at high velocity tangentially enters the cylindrical part of the body and makes a downward spiral.The centrifugal force caused by the rotational motion flow, dust particles are moved to the sides of the device (Fig. 1).
When moving in a rotating curved gas flow dust under the influence of centrifugal force and resistance.
Analysis of the swirling dust and gas flow in the scrubber will be carried out under the following assumptions:
1. Gas considered ideal incompressible fluid and, therefore, its potential movement.
2. Gas flow is axisymmetric and stationary.
3. Circumferential component of the velocity of the gas changes in law
ww = const -4r .
Fig. 1 - The trajectory of the particles in a dynamic scrubber
This law is observed in the experiments [2, 3], will provide a simple solution that is convenient for the quantitative analysis of the particles.
3. Particle does not change its shape over time and diameter, it does not happen any crushing or coagulation. Deviation of the particle shape on the field is taken into account the coefficient K.
4. Wrapping a strong flow of gas is viscous in nature. Turbulent fluctuations of gas is not taken into account, which is consistent with the conclusion of [4]: turbulent diffusion of the particles has no significant impact on the process of dust removal.
5. Not considered force Zhukovsky, Archimedes, severity, since these forces by orders of magnitude smaller than the drag force and centrifugal [5, 6, 3].
6. Concentration dust is small, so we can not consider the interaction of the particles
7. Neglect the uneven distribution of the axial projection of the radial velocity of the gas, which is in
accordance with the data of [7]. Axial component of the velocity of the particles changes little on the tube radius.
The rotation of the purified stream scrubber creates a field of inertial forces, which leads to the separation of a mixture of gases and particles. Therefore, to calculate the trajectories of the particles it is necessary to know their equations of motion and aerodynamics of the gas flow. In accordance with the assumption of a low concentration of dust particles on the influence of the gas flow can be neglected. Consequently, we can consider the motion of a single particle in the velocity field of the gas flow. Therefore, the task to determine the trajectories of particles in the scrubber is decomposed into two parts:
- Determination of the velocity field of the gas
flow;
- Integration of the equations of motion of a particle for a calculation of the velocity field of the gas.
The assumption of axial symmetry of the problem (with the exception of the mouth) allows for consideration of the motion of the particles using a cylindrical coordinate system.
The greatest difficulty is to capture fine dust, for which the strength of the resistance with sufficient accuracy is given by Stokes. By increasing the ratio of dust cleaning machine grows [8], so the calculation parameters scrubber with low dust content (by assumption) guarantees a minimum efficiency.
3. Derive the equation of motion of a particle
To calculate the trajectories of the particles need to know their equations of motion. Such a problem for some particular case is solved by the author [9].
We introduce a system of coordinates OXYZ. Its axis is directed along the OZ axis of symmetry scrubber (Figure 2). Law of motion of dust particles in the fixed coordinate system OXYZ can be written as follows:
/77-
dvp Л
= Ft
st
(1)
where m - mass of the particle; dvp - velocity of the
particle; Fst - aerodynamic force.
For the calculations necessary to present the vector equation (1) motion in scalar form. Position of the particle will be given by its cylindrical coordinates (r; 9; z). Velocity of a particle is defined by three components: Up- tangential, Vp - radial and Wp - axial velocity.
Fig. 2 - The velocity vector of the particle
We take a coordinate system O'X'Y'Z', let O'X' passes axis through the particle itself, and the axis O'Z' lies on the axis OZ. Adopted reference system moves forward along the axis OZ Wp speed and rotates around an angular velocity
01
(2)
The equation of motion of a particle of mass 1 3
/77 = ^ xPpdp coordinate system O'X'Y'Z' becomes:
/77-
dVp dt
_y —♦ —*•
- m an + m rP 0 + rp0
+ /77
-—► ;_^
0 ■ V0 + 2/77 Vp ■ 0
or
d7r
(3):
— -Ft — & n +
/77
_^ 1_^ _
rp 0 + 0 ■ rp 0 + 2 Vp ■ 0
dt
where a0 - translational acceleration vector of the reference frame; dvp - velocity of the particle; rp - the
radius vector of the particle;
to unevenness of rotation;
^ 0
acceleration; 2
\?p-A
_
0 ■ rp 0
_ _ - _
- acceleration due
- the centrifugal
- Coriolis acceleration.
The first term on the right-hand side of equation (3) is the force acting c gas flow on the particle, and is given by Stokes
^ — 3 Wgdp
V9~VP
(4)
/ug - dynamic viscosity of the gas. The second term (3) is defined as
dWn
dWn
-e7 —-
dt dt Convert the remaining terms:
rp
d 0 dt
1 dUp Up
Гр dt
о Vx 2 x
Г d * __
P dt . (
Up
ey —
V ГР
dUp + UpV^
v
dt
-1 0 U 2 _ p _ 1—► — U 2 — p — -* U 2 p
0 ■ rp ■0. rx ez ■ ex ■ ez. rp ez ■ ev — — ex rp
Vp 0
—2lv
2 Ux
— 2lV
ex< ■ 0
G «^-1 * & — —
X Z
- 2-
UpVp
eM,ev,ez- vectors of the reference frame and used the fact that rp =ex -rH-vx= Vp
+
e
z
Г
e
y
Г
Г
Substituting these expressions in the equation of motion (3),
m-
dv r _t_
dt
■ = Fst - mao + m
+ m
rp -ю
+m
ю
rp'ю
rp'ю
+ 2m
vP 'ю
or
dvp 1 ^ -T +
—7T = — Fst - a0 + dt m
We write this equation in the projections on the axes of the coordinate system O'X'Y'Z'
-_* - - -_- -
1 —► t —► + 2 -> —*■
rp 'ю + ю - rp 'ю vP 'ю
- - - - - _
dVr~
dt
u2
m
stx'
0 = — F,
m
sty
0=mF~--
dUp dt
dWn
upvp
dt
dVn
dt dU
■ = —F.
dt
dW„
m m
stx
P =—F -
- 'sty
.up
ГР
UPVP
(5)
dt
= —Fc
m
stz
We have the equation of motion of a particle in a rotating gas flow projected on the axis of the cylindrical coordinate system.
Substituting (2) and (4) in (5) we obtain the system of equations of motion of the particle:
dVn
dt dUr
18ц
Ppd I
Ui
dt
dWn
^-((J -U
U P>
P4d4 18ц
rn
(6)
dt
Ppd I
(Wg-Wp)
4. Output relationship between the geometrical and operational parameters
Formal analysis of relationships that define the motion of gas and solids in the scrubber. The analysis shows that the strict observance of similarity of movements in the devices of different sizes requires the preservation of four dimensionless complexes, such as
wD
w
vS
Red = — =^-,Res= —.
SPl ■[
v Dg' ' Dp1 v
Not all of these systems are affecting the motion of dust. Experimentally found that the influence of the Froude number Fr negligible [9] and can be neglected. It is also clear that the effect of the Reynolds number for
large values it is also insignificant. However, maintaining unchanged the remaining two complexes, still introduces significant difficulties in modeling devices.
On the other hand, there is no need for strict observance of the similarity in the trajectory of the particle in the apparatus. What is important is the end result - providing the necessary efficiency unit. To estimate the parameters that characterize the removal of particles of a given diameter, consider the approximate solution of the problem of the motion of a solid particle in a scrubber. A complete solution for a special case considered in the literature [10], this solution can be used to obtain the dependence of the simplified model of the flow.
For the three coordinates - radial, tangential and vertical equations of motion of a particle at a constant resistance can be written in the following form:
dw wl , .
— -— = -a(wr-vr)
dwz ~dt
= -a(wz -vz)
dw„
V -, wv i \
f + - a(w -Vy)
dt ■ r
where a - factor resistance to motion of a particle, divided by its mass.
M
a=- 2 , KpS2
K - factor, which takes into account the effect of particle shape (take K = 2).
The axial component of the velocity of gas and particles are the same, as follows from the equations of motion by neglecting gravity. Indeed, if the
= a{yvz -vz) , then takingwz - vz = Aw.
we
dt
get dlAWz =-aAw?. AH/, =Aw7ne~dt.
6 dt z z zo
If initially
wz = i/z (AWZ = 0);AWt =01Wt = const projection speed:
w2 = const -Jr .
Valid law W^(r) may differ markedly from the accepted, but this is not essential. In this case, it only makes us enter into the calculation of average
(W2 ^ 2 r
V y
Under these simplifications, the first of the equations of motion is solved in quadrature. Indeed, for now
dwr
wz
dt 'r then with the obvious boundary condition t = 0, vr = 0, we have
f 2 ^ Wçp
r
V У
av (l - e-dt).
The time during which the flow passes from the blade to the swirler exit from the apparatus as well
+
wr = —
А =•
On the other hand, knowing the law of the radial velocity, we can find the time during which the particle travels a distance of r1 (the maximal distance from the wall) to the vessel wall (r2).
w„
r2 - rx = I wrdt =
л - /1 = -
ar„
ч +a
1-e
-dt
еЛ-1
a
Substituting in this equation is the limiting value of t1 = tt. we get:
/ \ , ( n . i \
arav\r2 -1) / , 1
Vav
w7 a
epS wz -1
or
И
2 2 /12 - r22 W7 Л
1 2 - z > 1+
Kp82 21/V /
Ф av
Kpö2 wz
И
/
pSLK w, -1
(7)
The presented approach is based on the known dependence and model of the flow, it's different in a number of studies approach is only in the details. However, further to allocate two sets, one of which characterizes the geometry of the device, and the other -operating data. The use of these systems simplifies the calculation and, most importantly, takes into account the influence of some key factors to the desired gas velocity and height of the apparatus WZav.
Dependence structure (7) shows the feasibility of introducing two sets, one of which
2pS2l/VtK
characterizes the effect of the flow regime and the particle diameter, and the other is a geometric characteristic of the device.
A = yV (l -12) (8)
In (8) through r1 marked relative internal radius apparatus:
h = V2
and p1 - the average angle of the flow at the exit of the guide apparatus
t = £ Then (7) takes the simple form
A >f(A)
where
fe>=ii + ¿2 И -1) (9)
Expressed graphically in Figure 3, this dependence allows to determine the minimum value of the mode parameter Amin for the scrubber with geometrical parameters of Ar. And must take A >Amm
AOO° eooO a20O0 a6OQ°
Operational parameters Ap
Fig. 3 - The relationship between the geometrical and operational complexes
One of the important consequences of the resulting function is the relationship between the diameter of the dust particles and the axial velocity of the gas.
With this therefore
machine, with data Ag, Amin = const and
wz§ = const
This means that reducing the particle size axis (expenditure), the rate should be increased according to the dependence
w -,
w
zo
S
2
Unfortunately, a significant increase Wz permitted as this may lead to the capture of dust from the walls and ash. You can also change the twist angle p and the height of the flow system, without changing the axial velocity. If the reduction of the particle diameter 5 or increasing the size of the unit has increased the value of A, it must be modified accordingly Ag (using a graph Figure 3), and the new value of Ar to find an angle p:
(A, ^
ctgß ctgß.
о
9
go
Conclusions
1. Creating a mathematical model of the motion of a particle of dust in the swirling flow allowed us to estimate the influence of various factors on the collection efficiency of dust in the offices of the centrifugal type, and create a methodology to assess the effectiveness of scrubber.
2. Identified settlement complexes, one of which characterizes the geometry of the scrubber and the other operational parameters. The use of these systems and simplifies the calculation takes into account the influence of several key factors.
3. The developed method can be used in the calculation and design of gas cleaning devices, as constituent relations define the relationship between the technological characteristics of the dust collectors and their geometrical and operational parameters
References
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10. V.E. Mizonov, V. Blaschek, R. Colin, A. Greeks Tohti, 1994.28, 3, 277-280.; 11. A.T. Litvinov, Journal of Applied Chemistry, 1971.44, 6, 1221-1231.
© 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, [email protected]; G. E. Zaikov - DSc. Professor of the Chair Plastics Technology Kazan National Research Technological University in Kazan, Tatarstan, Russia, [email protected].
© Р. Р. Усманова - канд. техн. наук, доц. каф. СМ Уфимского госуд. авиационного технич. ун-та, [email protected]; Г. Е. Заиков - д-р хим. наук, проф. каф. ТПМ КНИТУ, [email protected].
