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Computation of mean trajectories of charged aerosol particles in turbulent jets
Journalof
ELECTROSTATICS ELSEVIER
Journal of Electrostatics40&41 (1997) 503-508
Computation of mean trajectories of charged aerosol particles in turbulent jets "I.P. Vereshchagin, b V.G. Stepanyanz, "A.V. Orlov, "A.G. Temnikov "Moscow Power Engineering Institute (Technical University), Moscow, KrasnokaTarmennaya st. 14, Russia b Ministry of Science and Technology, Moscow, Tverskaya st. 11, Russia
1. I n t r o d u c t i o n Application of the artificially charged aerosol flows for discharge studying and in technology is combined with their formation conditions investigation and with the studying of the charged particles motion in a turbulent jet. It allows to determine the time from the jet charging beginning which is required for the inducing in any point of space the electric field strength from the charged pan of the flow which is necessary in the technological processes or for the discharge phenomena appearence. The charged turbulent jet formation computation task is nonstationary because the charged pan boundaries and the space charged distribution change with time. Because of the jet charge is concentrated on the aerosol particles, the flow in the considering nonstationaty task can be described by the "big panicle" model [1]. The flu~_afi_'onsoccurring in the turbulent flow can be taken into account by the approximation characteristics [2, 3]. Using the lasts, the mean trajectories of the additive particles which specify the charged aerosol behaviour are found. Self-coordinated task of the charged aerosol particles motion in high-speed gas-dynamical turbulent flow and in the electric field created by the charged pan of jet with continuously changed boundaries and charge density distributions has been solved.
2. Computation of the mean trajectories of the charged aerosol particles in the submerged turbulent jet The process of the noncharged aerosol jet propagation in time was quite detailly considered in [3] where the turbulent jet boundary contours under its development in time were obtained. The automodel profiles of the velocities and the additives concentrations in the jet were used. However, it is impossible to use this method for the setf-coordineted task solving of the charged parts formation dynamics of jet becJ_~_~the location and the dimJbution of the space charge and the force action on the aerosol panicles are changed with jet charged pan development. So, the different approach in the formation dynamics task solving of the charged pan of the axisymmetricai jet was considered. The model of "big particles" was used. It was supposed 0304-3886/97/$17.00 0 Elsevier ScienceB.V. All rights reserved. PII S0304-3886(97)00094-6
504
I.P. Vereshchagin et al./Journal o f Electrostatics 40&41 (1997) 503 -508
that the main charge in jet was transported by the charged aerosol drops. It was taken into account that the latters move under the influence of forces acting on them. Equation of motion of the charged particle is written as: m (dV/dt) = ZF
(1)
where m - mass of the aerosol particle; V - its velocity vector; ZF - sum of the forces acting on the panicle. In gasdynamical turbulent jet the forces acting on the charged spherical aerosol particles are: a) the Stokes force of flow resistance Fs F s = - 6 x B K h a ( V - U)
(2)
b) the electric force FE ICE = q E
(3)
where a, q - the radius and the charge of the drop respectively; U - the gasdynamical flow velocity; g - the viscosity coefficient for air; E - the electric field in the point where drop is situated in that moment; Kh - Cunningham correcter factor (was used for particles less than the micron dimension). The gravity force acting on the aerosol particle is not taken into account because it is very less in comparison with others for the aerosol particles less than 1 pan. Equation of motion (1) has been solved analytically on every time step. That has allowed to coordinate the solution with the electric field calculation equation from the space charge of the forming charged part of the jet. Equation of motion of the aerosol particle in the axial and radial directions had the following form: z" = 5~Fz/ m = wz z' = v~0 + z" At = Vz z = z 0+ Vz0 At + 0.5 wz Ate x" = ZFx / m = wx x' = v~0 + x" At = vx x = xo + v~0 At + 0.5 w~ At2
(4)
where v~0, v~o - the aerosol particle velocity on the previous time step into the axial and radial directions; wz, wx - the aerosol particle accelerations on that time step; ZFz, ZFx = the acting in both directions forces; At - time step. Thus, the number of the following conditions was put to the base on the formation computation method of the forming charged part of the turbulent jet. 1. The beginning part of the main section of the gas flow doesn't depend on the appenrence of the electric forces. This condition is applied for aerosol particles with small sizes (0.1 - 5.0 I~m diameter) which motion adds a very small perturbation to the main gas flow. As the experimental investigations [4 - 6] showed, the velocity approximation characteristics of the
LI? keshchagin
et al. /Journal
of Electrostatics 40&41 (1997) 503-508
505
gas-dynamical flow inside the jet boundaries determined by the equation suggested for the uncharged axisymmetrical gas-dynamical flows in 12, 31 can be used for the charged jet representation:
where R= z tg(cr) - transversal radius of the jet; a - a half angle of the turbulent jet; z and x axial and radial coordinate, respectively: u, - axial component of the flow velocity. And,
u,(z,O)
=
12.4
mu0
or0 (6)
Z
where ~0 - the flow velocity in the nozzle section; r. - nozzle radius, To solve the task of the charged part formation of the ax&symmetrical turbulent jet the approximation of the transverse velocity component ux of the gas-dynamical flow has been used [5]:
X
u, = u, -
Z
=
-yI;’
.x[l_(z.l;(.))ll]
Application of the distributions [2] and (5, 7) has allowed to compute in time the development of the uncharged submerged jet. Calculation results showed a good correlation with data published in [3]. 2. Conditions of the jet main section beginning for which the gas-dynamical parameters are well known were considered by the initial conditions under the formation of the charged turbulent flow. 3. The lift force for aerosol particles were not taken into account. Under the computation of the electric field forces the reflection with respect to a plane from which the jet has flown out was taken into account. 4. Axi-symmetrical jet was considered, so computation was carried out only in one of the planes passing through the jet axis. 5. At time t = 0 the charging and the charge progress in a disc state begin. The certain number n of drops was selected in the disc on the different distances from a centre, and what’s more the first drop was on the initial cross-section boundary R, of the main part of jet, but the n-drop was on the jet axis. Others (i-drops) was situated among them. For every selected drop the equation system which is determined its movement in axial and cross directions was set up. That system has been consequantly solved on every time step. 6. Electric field computation of the forming charged part of the turbulent jet was carried out on the method suggested by the authors in [6]. Presented computation algorithm was realized in program allowing to compute the mean trajectories of aerosol particles in charged and uncharged turbulent aerosol jets, and to compute electric fields in the charged part of jet and near it.
506
I.P Vereshchagin et al. /Journal
ofElectrostatics 40&41 (1997) 503-508
The mean trajectories calculation of the charged particles in the charged turbulent flows was carried out for parameters of the charged aerosol generator used in experiments [4-61: water steam was used for aerosol production; flow velocity in the nozzle section LQ = 400 m/s; the outlet current of generator I = 100 PA; the nozzle diameter Q = 6 mm; the aerosol particle radius a = 0.4 urn; the charge of aerosol particle q = 4 10-r’ C; the average space charge density in the beginning cross-section of the main part ofjet pm = 2.3 lOA C/m3. Development dynamics of the charged part of the turbulent jet within different time gaps after the charging beginning and comparing of the development dynamics of the charged and uncharged jets showed that the length of the forming charged and uncharged parts of jet is the same, but the charged part of jet with time after the charging beginning expands in radial direction sufficiently more than uncharged one ( Fig. 1 ), where X*= 2x/&; z* = 2z/Q.
Figure 1. Shapes of the uncharged and charged parts of the aerosol turbulent jets _________ _ uncharged turbulent jet; ______________ _ charged turbulent jet; -------- gas-dynamical boundary; 1 - 21 ms; 2 - 105 ms. After tens of milliseconds the part of charged aerosol begins to go out of the gas-dynamical flow boundaries. The mean trajectories of the charged aerosol particles which were in the charging beginning on the different distances ri from the jet axis in the beginning of the main part have been computed and are shown in Fig. 2 ( ri = ri / R,,, ; R, - the beginning crosssection of the main part of the turbulent jet: R,,, = 6 Q ). Computation allowed to determine how much of charged aerosol entering to the main part of turbulent jet in time moment t = 0 goes out of gas-dynamical flow boundaries through the
LP, Vereshchaginet aL/Journal of Electrostatics40&41 (1997) 503-508
507
certain time after the jet charging beginning. On the base of the computational experiments for different parameters of the charged aerosol generator and aerosol particles the dependance of leaving time of the charged aerosol particle out of gas-dynamical jet boundaries versus its location in the charging beginning was detected and experimentally proved [5]:
r tt=.2"In
ri
r,'l
F2L
ri
•
J
(8)
where the constant ¢ can be found using the pararaeters of the charged aerosol partide and the charged turbulent jet:
T
12.4
[16.:¢2
. 6 o . ra
(9)
Pm " q For the experimental condition [4 - 6] z = 0.027 s.
7O
60
3
/ s
fO 4,0 JO .2O fO
Figure 2. Mean trajectories of the charged aerosol particles. ....... the gas-dynamical boundary, 1 - ri" = 1; 2 - ri* = 0.75; 3 - ri* = 0.5; 4 - ri* = 0.25. Thus, it can be estimated how the outlet current of aerosol generator influences on mean trajectories of the charged aerosol particles. It is necessary to note that even for the aerosol particles situated on the outer mrfa~ of the charged part of jet the majority of the charged aerosol particles remalnes within gas-dynamical flow limits under the charged part propagation
508
1.17. Fereshchagin et al./Journal of Electrostatics 40&41 (1997) 503- 508
up to 400 - 500 nozzle calibres. For that time the charged part of the turbulent jet reaches dimensions which are sufficient for the creating the discharge phenomena near it.
3. Conclusions 1. Computation method of the mean trajectories of the charged aerosol particles in turbulent jets was developed. 2. It is shown that the formation dynamics of the charged part of submerged charged aerosol jet is different from uncharged one by the bigger expansion into radial direction under the same length conservation. 3. The dependence of the leaving time of the charged aerosol particles out of the gasdynamical jet boundaries versus their location in the charging beginning was found. 4. It is shown that the jet propagation up to 400 - 500 nozzle calibres the majority of charged aerosol remains within the gas-dynamical boundaries of submerged jet.
References [1] Ilyin V.P. Computational methods of the dectrophysics problems solving. Moscow, Nauka, 1985 (in Russian). [2] Abramovich G.N. Applied gas-dynamics. Moscow, Nauka, 1991 (in Russian). [3] Abramovich G.N. Theory of the turbulent jets. Moscow, Nauka, 1984 (in Russian). [4] Makalsky L.M., Odov A.V., Temnikov A.G. Changing of the space charge density in the charged aerosol flows. Moscow International Aerosol Symposium, 1994. [5] Tenmikov A.G. Dynamics of the formation of the turbulent charged aerosol flow. Vestnik MPEI, N 1, 1996 (in Russian). [6] Vereshchagin I.P., Koshelev M.A., Makalsky L.M., Temnikov A~G. Determination of electric fields induoed by a charged aerosol cloud. 9th International Symposium on HVE, Cn'az, Austria, 1995.
ELECTROSTATICS ELSEVIER
Journal of Electrostatics40&41 (1997) 503-508
Computation of mean trajectories of charged aerosol particles in turbulent jets "I.P. Vereshchagin, b V.G. Stepanyanz, "A.V. Orlov, "A.G. Temnikov "Moscow Power Engineering Institute (Technical University), Moscow, KrasnokaTarmennaya st. 14, Russia b Ministry of Science and Technology, Moscow, Tverskaya st. 11, Russia
1. I n t r o d u c t i o n Application of the artificially charged aerosol flows for discharge studying and in technology is combined with their formation conditions investigation and with the studying of the charged particles motion in a turbulent jet. It allows to determine the time from the jet charging beginning which is required for the inducing in any point of space the electric field strength from the charged pan of the flow which is necessary in the technological processes or for the discharge phenomena appearence. The charged turbulent jet formation computation task is nonstationary because the charged pan boundaries and the space charged distribution change with time. Because of the jet charge is concentrated on the aerosol particles, the flow in the considering nonstationaty task can be described by the "big panicle" model [1]. The flu~_afi_'onsoccurring in the turbulent flow can be taken into account by the approximation characteristics [2, 3]. Using the lasts, the mean trajectories of the additive particles which specify the charged aerosol behaviour are found. Self-coordinated task of the charged aerosol particles motion in high-speed gas-dynamical turbulent flow and in the electric field created by the charged pan of jet with continuously changed boundaries and charge density distributions has been solved.
2. Computation of the mean trajectories of the charged aerosol particles in the submerged turbulent jet The process of the noncharged aerosol jet propagation in time was quite detailly considered in [3] where the turbulent jet boundary contours under its development in time were obtained. The automodel profiles of the velocities and the additives concentrations in the jet were used. However, it is impossible to use this method for the setf-coordineted task solving of the charged parts formation dynamics of jet becJ_~_~the location and the dimJbution of the space charge and the force action on the aerosol panicles are changed with jet charged pan development. So, the different approach in the formation dynamics task solving of the charged pan of the axisymmetricai jet was considered. The model of "big particles" was used. It was supposed 0304-3886/97/$17.00 0 Elsevier ScienceB.V. All rights reserved. PII S0304-3886(97)00094-6
504
I.P. Vereshchagin et al./Journal o f Electrostatics 40&41 (1997) 503 -508
that the main charge in jet was transported by the charged aerosol drops. It was taken into account that the latters move under the influence of forces acting on them. Equation of motion of the charged particle is written as: m (dV/dt) = ZF
(1)
where m - mass of the aerosol particle; V - its velocity vector; ZF - sum of the forces acting on the panicle. In gasdynamical turbulent jet the forces acting on the charged spherical aerosol particles are: a) the Stokes force of flow resistance Fs F s = - 6 x B K h a ( V - U)
(2)
b) the electric force FE ICE = q E
(3)
where a, q - the radius and the charge of the drop respectively; U - the gasdynamical flow velocity; g - the viscosity coefficient for air; E - the electric field in the point where drop is situated in that moment; Kh - Cunningham correcter factor (was used for particles less than the micron dimension). The gravity force acting on the aerosol particle is not taken into account because it is very less in comparison with others for the aerosol particles less than 1 pan. Equation of motion (1) has been solved analytically on every time step. That has allowed to coordinate the solution with the electric field calculation equation from the space charge of the forming charged part of the jet. Equation of motion of the aerosol particle in the axial and radial directions had the following form: z" = 5~Fz/ m = wz z' = v~0 + z" At = Vz z = z 0+ Vz0 At + 0.5 wz Ate x" = ZFx / m = wx x' = v~0 + x" At = vx x = xo + v~0 At + 0.5 w~ At2
(4)
where v~0, v~o - the aerosol particle velocity on the previous time step into the axial and radial directions; wz, wx - the aerosol particle accelerations on that time step; ZFz, ZFx = the acting in both directions forces; At - time step. Thus, the number of the following conditions was put to the base on the formation computation method of the forming charged part of the turbulent jet. 1. The beginning part of the main section of the gas flow doesn't depend on the appenrence of the electric forces. This condition is applied for aerosol particles with small sizes (0.1 - 5.0 I~m diameter) which motion adds a very small perturbation to the main gas flow. As the experimental investigations [4 - 6] showed, the velocity approximation characteristics of the
LI? keshchagin
et al. /Journal
of Electrostatics 40&41 (1997) 503-508
505
gas-dynamical flow inside the jet boundaries determined by the equation suggested for the uncharged axisymmetrical gas-dynamical flows in 12, 31 can be used for the charged jet representation:
where R= z tg(cr) - transversal radius of the jet; a - a half angle of the turbulent jet; z and x axial and radial coordinate, respectively: u, - axial component of the flow velocity. And,
u,(z,O)
=
12.4
mu0
or0 (6)
Z
where ~0 - the flow velocity in the nozzle section; r. - nozzle radius, To solve the task of the charged part formation of the ax&symmetrical turbulent jet the approximation of the transverse velocity component ux of the gas-dynamical flow has been used [5]:
X
u, = u, -
Z
=
-yI;’
.x[l_(z.l;(.))ll]
Application of the distributions [2] and (5, 7) has allowed to compute in time the development of the uncharged submerged jet. Calculation results showed a good correlation with data published in [3]. 2. Conditions of the jet main section beginning for which the gas-dynamical parameters are well known were considered by the initial conditions under the formation of the charged turbulent flow. 3. The lift force for aerosol particles were not taken into account. Under the computation of the electric field forces the reflection with respect to a plane from which the jet has flown out was taken into account. 4. Axi-symmetrical jet was considered, so computation was carried out only in one of the planes passing through the jet axis. 5. At time t = 0 the charging and the charge progress in a disc state begin. The certain number n of drops was selected in the disc on the different distances from a centre, and what’s more the first drop was on the initial cross-section boundary R, of the main part of jet, but the n-drop was on the jet axis. Others (i-drops) was situated among them. For every selected drop the equation system which is determined its movement in axial and cross directions was set up. That system has been consequantly solved on every time step. 6. Electric field computation of the forming charged part of the turbulent jet was carried out on the method suggested by the authors in [6]. Presented computation algorithm was realized in program allowing to compute the mean trajectories of aerosol particles in charged and uncharged turbulent aerosol jets, and to compute electric fields in the charged part of jet and near it.
506
I.P Vereshchagin et al. /Journal
ofElectrostatics 40&41 (1997) 503-508
The mean trajectories calculation of the charged particles in the charged turbulent flows was carried out for parameters of the charged aerosol generator used in experiments [4-61: water steam was used for aerosol production; flow velocity in the nozzle section LQ = 400 m/s; the outlet current of generator I = 100 PA; the nozzle diameter Q = 6 mm; the aerosol particle radius a = 0.4 urn; the charge of aerosol particle q = 4 10-r’ C; the average space charge density in the beginning cross-section of the main part ofjet pm = 2.3 lOA C/m3. Development dynamics of the charged part of the turbulent jet within different time gaps after the charging beginning and comparing of the development dynamics of the charged and uncharged jets showed that the length of the forming charged and uncharged parts of jet is the same, but the charged part of jet with time after the charging beginning expands in radial direction sufficiently more than uncharged one ( Fig. 1 ), where X*= 2x/&; z* = 2z/Q.
Figure 1. Shapes of the uncharged and charged parts of the aerosol turbulent jets _________ _ uncharged turbulent jet; ______________ _ charged turbulent jet; -------- gas-dynamical boundary; 1 - 21 ms; 2 - 105 ms. After tens of milliseconds the part of charged aerosol begins to go out of the gas-dynamical flow boundaries. The mean trajectories of the charged aerosol particles which were in the charging beginning on the different distances ri from the jet axis in the beginning of the main part have been computed and are shown in Fig. 2 ( ri = ri / R,,, ; R, - the beginning crosssection of the main part of the turbulent jet: R,,, = 6 Q ). Computation allowed to determine how much of charged aerosol entering to the main part of turbulent jet in time moment t = 0 goes out of gas-dynamical flow boundaries through the
LP, Vereshchaginet aL/Journal of Electrostatics40&41 (1997) 503-508
507
certain time after the jet charging beginning. On the base of the computational experiments for different parameters of the charged aerosol generator and aerosol particles the dependance of leaving time of the charged aerosol particle out of gas-dynamical jet boundaries versus its location in the charging beginning was detected and experimentally proved [5]:
r tt=.2"In
ri
r,'l
F2L
ri
•
J
(8)
where the constant ¢ can be found using the pararaeters of the charged aerosol partide and the charged turbulent jet:
T
12.4
[16.:¢2
. 6 o . ra
(9)
Pm " q For the experimental condition [4 - 6] z = 0.027 s.
7O
60
3
/ s
fO 4,0 JO .2O fO
Figure 2. Mean trajectories of the charged aerosol particles. ....... the gas-dynamical boundary, 1 - ri" = 1; 2 - ri* = 0.75; 3 - ri* = 0.5; 4 - ri* = 0.25. Thus, it can be estimated how the outlet current of aerosol generator influences on mean trajectories of the charged aerosol particles. It is necessary to note that even for the aerosol particles situated on the outer mrfa~ of the charged part of jet the majority of the charged aerosol particles remalnes within gas-dynamical flow limits under the charged part propagation
508
1.17. Fereshchagin et al./Journal of Electrostatics 40&41 (1997) 503- 508
up to 400 - 500 nozzle calibres. For that time the charged part of the turbulent jet reaches dimensions which are sufficient for the creating the discharge phenomena near it.
3. Conclusions 1. Computation method of the mean trajectories of the charged aerosol particles in turbulent jets was developed. 2. It is shown that the formation dynamics of the charged part of submerged charged aerosol jet is different from uncharged one by the bigger expansion into radial direction under the same length conservation. 3. The dependence of the leaving time of the charged aerosol particles out of the gasdynamical jet boundaries versus their location in the charging beginning was found. 4. It is shown that the jet propagation up to 400 - 500 nozzle calibres the majority of charged aerosol remains within the gas-dynamical boundaries of submerged jet.
References [1] Ilyin V.P. Computational methods of the dectrophysics problems solving. Moscow, Nauka, 1985 (in Russian). [2] Abramovich G.N. Applied gas-dynamics. Moscow, Nauka, 1991 (in Russian). [3] Abramovich G.N. Theory of the turbulent jets. Moscow, Nauka, 1984 (in Russian). [4] Makalsky L.M., Odov A.V., Temnikov A.G. Changing of the space charge density in the charged aerosol flows. Moscow International Aerosol Symposium, 1994. [5] Tenmikov A.G. Dynamics of the formation of the turbulent charged aerosol flow. Vestnik MPEI, N 1, 1996 (in Russian). [6] Vereshchagin I.P., Koshelev M.A., Makalsky L.M., Temnikov A~G. Determination of electric fields induoed by a charged aerosol cloud. 9th International Symposium on HVE, Cn'az, Austria, 1995.