flow behind a shock wave in a cloud of particles is shown experimentally and substantiated ... Experiments were performed in a shock tube equipped for optical.
Combustion, Explosion, and Shock Waves, VoL 32, No. 2, 1996
INTERACTION OF A SHOCK WAVE WITH A CLOUD OF PARTICLES V. M. Boiko, V. P. Kiselev, S. P. Kiselev, A. N. Papyrin, S. V. Poplavskil, and V. M. Fomin
UDC 532.529
The present paper is devoted to experimental and theoretical investigation of shock-wave propagation in a mixture of a gas and solid particles with clearly defined boundaries of the two-phase region (cloud of particles). The effect of qualitative transformation of supersonic flow behind a shock wave in a cloud of particles is shown experimentally and substantiated theoretically for volume concentrations of the dispersed phase of 0.1-3%.
The hazard of dust explosions exists in a number of technological processes of production and processing of powder materials. A shock wave (SW) in gas-dust mixtures is an important factor of these explosions. Physical and mathematical modeling of this phenomenon requires sophisticated understanding of the relaxation processes behind the SW front at temperatures and pressures characteristic of gas-dust explosions. In particular, one of the main problems is the relaxation of phase velocities in the flow behind the SW front [1, 2]. Most experimental work in this field deals with subsonic flows. The phenomenon of velocity nonequilibrium in supersonic flows is substantia~y less studied for the case where the Mach number M12 of the relative motion of the gas and particles varies over a wide range during velocity relaxation. Extensive experimental data on the drag coefficient Ca for isolated particles are given in [3, 4]. Also, there are many correlations Cd = Cd(M12, Re) [3-8]. However, in flows with a high concentration of the dispersed phase there are deviations of Cd from the values characteristic of isolated particles. Thus, in low-velocity two-phase flows this effect becomes noticeable in sufficiently concentrated mixtures at volume particle concentration m2/> 5% [9], and in a supersonic nozzle the effects of "tightness" appear already at m2/> 1% [10]. Mathematical modeling of SW propagation in gas suspensions in a one-dimensional approximation has been carried out previously [11-13]. However, the complexity of the problem and inadequate representations of unsteady two-phase flows (in particular, of the behavior of the gas phase) behind the SW in a cloud of particles make necessary for further research along this line. The present work is devoted to experimental and theoretical investigation of SW propagation in a mixture of a gas and solid particles with well-defined boundaries of the two-phase region (cloud of particles). E X P E R I M E N T A L I N V E S T I G A T I O N OF T H E I N T E R A C T I O N OF SHOCK WAVES W I T H D I S P E R S E D - P H A S E CLOUDS 1.1. E x p e r i m e n t a l Technique. Experiments were performed in a shock tube equipped for optical visualization of shock-wave processes in two-phase media, registration of pressure profiles, and shock-wave velocity measurements (Fig. 1). The low-pressure and high-pressure chambers had lengths of 1.5 and 5 m, respectively, a channel cross-section of 52 x 52 mm. Helium at pressure p = 2.5-10 MPa was used as the
Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Fizika Goreniya i Vzryva, Vol. 32, No. 2, pp. 86-99, March-April, 1996. Original article submitted April 20, 1995; revision submitted July 31, 1995. 0010-5082/96/3202-0191 $15.00 © 1996 Plenum Publishing Corporation
191
r
f I J I L
Fig. 1. Experimental setup: continuous access camera (1), time-interval counter (2), shock tube (3), pressure gauges (4), device for injecting particles (5), windows (6), laser stroboscope (7).
propelling gas and oxygen at p = 0.01-0.1 MPa as the working gas. The SW Mach number varied within the range of M0 = 1.5-4.5. The gas parameters behind the SW were determined from the measured number M0 with allowance for temperature dependence of the adiabatic exponent [14, 15]. Powders of Plexiglas (d = 80-300/~m, p22 --- 1.2 g / c m 3) and bronze (d --- 80-130 #m, p22 = 8.6 g/cm 3) were injected into the channel using either an electromagnetic striker on the lower wall of the measuring section, which tossed the powder to a given height, or a vibrating mesh on the upper cover of the measuring unit, which produced a vertical flow of free-falling particles with a uniform concentration over the cross section. Changeable diaphragms on the mesh produced a cloud of required geometry and particle concentration. The dynamics of SW interaction with the particles was observed by high-speed methods of laser visualization in transmitted (multiframe laser shadow visualization) and dispersed (laser beam method) light. The exposure time (30 nsec), number of frames, and time intervals between frames were controlled by a laser stroboscope [16], and spatial separation of the frames was accomplished by a continuous access camera with a rotating mirror prism. Time intervals were adjusted with an accuracy of 0.1 /~sec. A synchronization scheme ensured precise control of the triggering sequence of the individual elements of the shock tube and diagnostic equipment. A generator producing a series of pulses was triggered by a pressure gauge placed in the observation region. This ensured correlation between light pulses and the moment of passage of the shock wave through the region under study with an accuracy of not worse than 1 /~sec. I n f l u e n c e o f P a r t i c l e C o n c e n t r a t i o n on P a r t i c l e - A c c e l e r a t i o n D y n a m i c s B e h i n d a S h o c k Wave. Let us consider experimental data on the dynamics of a particle cloud behind an SW. A series of photographs for two concentrations of Plexiglas particles are presented in Fig. 2. Even in the initial stage of motion the cloud "spreads" in the longitudinal and transverse directions and hence its average concentration decreases. One of the evident reasons for the growth of the cloud in the longitudinal direction is polydispersity. Small particles are accelerated more rapidly than coarse ones, and this leads to size separation of the particles along with a decrease in concentration. Transverse widening is partially due to collision of the particles, but, as will be shown later, more significant effects are connected with rearrangement of the gas flow. Figure 3 presents series of photographs obtained for particles of two materials (bronze and Plexiglas) at m2 ~< 0.1% and m2 > 1%. The trajectories of the leading boundary of the clouds of bronze and Plexiglas particles are shown in Figs. 4 and 5. One can see that the particle acceleration differs by almost a factor of two as the concentration changes from < 0.1% to ,-- 1%. In our opinion, the main reason for such a strong effect of concentration on the acceleration dynamics of particles in the cloud is due to a change in the wave structure formed at the particles in supersonic flow. Actually, in the initial stage of particle acceleration behind the SW at Mach numbers M0 > 2 the flow regime is supersonic. The photographs in Fig. 6 show that frontal shock waves are formed near each particle when 192
a
b
i?i,iii ~~;?:~ii!!iiiijsli!} i i ;i?i-i~ii]ii=:iii.~i~;~ ~F"~,
5",'~.
"
-
~".
....
Fig. 2. Interaction of a SW with the clouds of Plexiglas particles at different concentration of dispersed phase: m2 < 0.2 (a) and 3% (b); M0 = 4.5, p = 0.01 MPa, At = 20/~sec.
the particle concentration is low. With an increase in the concentration the shock waves interact with one another, overlap, and form a collective frontal shock wave. Before proceeding to consideration of the dynamics of the particles in the cloud we consider the acceleration of a single particle in supersonic flow behind the SW front. In a specially arranged experimental run with spherical bronze particles with d = 180 4-10 #m, which were specia~y selected with a microscope, we determined the aerodynamic drag coefficients. The range of variation in the Mach number 0.8 < M12 < 1.2 for the relative motion of the gas and particles corresponds to a change in the flow regime from supersonic to subsonic. Numerous experiments performed with particles of the same size showed that the spread of absolute values of Cd does not exceed 10%, while the mean value of Ca and the character of the dependence Ca = f(M12) agrees with those presented in Fig. 7. These data for a single spherical particle are described by the expression [6]
193
I
5mm
I
a
b
!ii: ii!i
....
C
1
2
2
Fig. 3. Acceleration of bronze (a and b) and Plexiglas (d, e) particles behind the SW: m2 ~< 0.1 (a and c) and > 1% (b and d); M0 = 2.8, p = 0.1 MPa, At = 100 #sec.
Ca(Re, M12)=
l+exp(M~2~7)
Re-- P-I~'~ - v21d
#
~ee+~e),
Mx2 - Iv~ -,,~1
(1)
~ = "Y~
c
This empirical formula will be used in numerical calculations. The obtained data suggest that the formation of a "collective" SW ahead of the cloud is responsible for the observed effect. As a consequence, the flow regime past the particles changes from supersonic to subsonic and Ca and the kinetic head of the gas decrease.
194
x, mm
x, mm
o3
12
"4
/
/
3O
o/
/ /
08
20.
2,., o~ °
04
0
50
100
/
i~
10
150 t, .usec
o °. L'~ 50
Fig. 4
/
100
150 t,~sec
Fig. 5
Fig. 4. Trajectories of Plexiglas particles (d = 80-130 #m, p22 = 8.6 g / c m 3) at m2 ~ 0.1 (3) and ~ 1% (4); M0 = 2.8, p = 0.1 MPa; 1 and 2) calculation for d = 130 #m. Fig. 5. Trajectories of bronze particles (d = 80-300/~m, P22 ~- 1.2 g/cm 3) at m2 ~ 0.1 (3) and ~ 3% (4); M0 = 2.8, p = 0.1 MPa; 1 and 2) calculation for d = 300 #m.
Fig. 6. Wave structure in supersonic flow behind the SW at rn2 ~< 0.1 (a) and m2 > 1% (b).
M A T H E M A T I C A L M O D E L I N G OF T H E I N T E R A C T I O N OF A SHOCK WAVE W I T H A CLOUD OF P A R T I C L E S C o n t i n u u m - D i s c r e t e M o d e l o f G a s - P a r t i c l e Flow. Let us consider a cloud of solid spherical particles with a SW running against them. It is required to determine the parameters of the gas and particles resulting from the interaction of the SW with the cloud. The motion of the particles is modeled by a noncollisional kinetic equation, and the gas is described by the averaged equations of a dusty gas. It is assumed that the particles can have dispersion in velocity and size. This model is thoroughly described in [17, 18] and applicable to the case where particle trajectories do not intersect in the flow region or particle colhsions are raze [Knudsen number Kn ~ (d/(6m2L) >t 1, L is the distance traveled by a particle in the cloud]. Directing the z-axis along the velocity D of the center of mass of the cloud of particles, following [17-19], we obtain the following system of equations: Of Of Of 0 0 0 0--t+V2~x+W2Oyy + ~v2 ( a z f ) + - ~ w 2 ( a , f ) + - ~ 2 ( q f ) = O , v2 = v 1 - D,
l-b, a= = a=
f = f ( t , x, y, v2, w2, r, T2),
w2 = 12, n =
/ fdVo,
a
1 ,=ay,
m2 =
'/
b = dD d~ , rSfdVo,
D-
f P22vlfdVo f P22f dVo ,
aVo = clv2dw~draT2,
195
.10-4
Re
Cd 0.7
C
~
2.0
0.5 ° .,,~ ,,. ~ ~ "
0.4
o:9
''~ *''"
'
i'.o
-1.5
'
Ill
tw-hi:
0.6
1'.1
~+
X1
Fig. 8
Fig. 7
X2
r4
$C
Fig. 7. Drag coefficient of a particle versus Mach number M12. Fig. 8. Calculation domain (one-dimensional case): f~l is the region filled with the gas; fig. is the gas-particle mixture; x+ is the SW front coordinate; xz is the leading edge of the cloud; x2 is the trailing edge of the cloud; 71-74 are the calculation domain boundaries. q = 2rArNu T1 - T2,
ml + rn2 = 1,
i
Csmp
3 ( Re# \p22d2)'t~dkRe'Mz2)'
r = 4
el = c~Tz,
=
1
ax
Vl - v2
7"
--
4 3 m , = ~ r r P22,
F
Op
Pl = P l l m l ,
G =
plVlWl
vl-D,
Wl
=
10p
7"
/922 0 y '
'
wll,
A1 ,
=
['Oml
plVlWl pl w2 + pro1
T
,
¢=
pa wl A a
el + prnl + -v 2- + , w~ p~ 2 Oral
A2 = v1¢1 + wa¢2 -t-pl,---~ - + va ~ COral / Vl -- V2 ¢1 = --P ~ + rnp ~ f dY,
Pr = cp# A'
(2)
p = (7--1)P11¢1,
pa va A1 =
Wl - w2
N u = 2 + 0.6Re°'SPr °'33,
ply21 -]- p m l
=
plWl
vl
1
a~ =
( lvl)( lwl
Oqo OF OG -0--~-+ -~x-x+ ff-y + ¢ = 0,
plY1
1
/922 OX'
(gml~
0 ¢1 ¢2
A2
P 2 2 ---- c o n s t
+ wl Oy ] -
px¢3,
Oml / wl - w2 f dV, ¢2 = - p -~y + rnp T
The superscript 1 stands for the parameters of the gas and the particles in the laboratory coordinate system; pll and p22 are the true densities of the gas and the particles; c~ and cs are the heat capacities of the gas and the particles;/z and X are the viscosity coefficient and the heat conductivity of the gas; v and w are the velocities atong the x and y axes; the subscripts 1 and 2 correspond to the parameters of the gas and the particles. System (1) and (2) is used below to calculate subsonic and supersonic flows of a gas with particles. This model does not allow one to calculate the frontal shock for each particle, but the presence of a Mach SW is taken into account in the dependence Ca(Ma2). This enables one to describe correctly the averaged supersonic motion of a particle cloud in a gas.
196
Calculation A l g o r i t h m . System (1) and (2) is solved numerically using an algorithm that consists in the following. A rectangular Eulerian grid with steps 2hz and 2hy along the x and y axes is constructed in the plane z, y. The equations for the gas are written on the Eulerian grid using an explicit finite-difference scheme of third-order accuracy [20, 21]. The noncollisional kinetic equation is solved in Lagrangian variables. At t = 0 the region occupied by the particles is split into rectangular Lagrangian particle ceils 2hx and 2h v in size, so that all particles within each ith cell have the same velocities v°i and w°i, temperature T°i, and radius ri; therefore at t = 0 the distribution function in the ith cell is written as
fo = ~fio~(V . v°i)~i(w2 .
.w°i)*(r8.
ri)*(T2 .
T°i),
Vi°
4h~h v,
where Ni is the number of particles in the ith cell and * is the delta function. Let us show that the number of particles in the ith cell will remain constant. We choose an individual volume within the phase space of the particles v(t). The condition of conservation of the total number of particles N in the individual volume has the form N
[ fdVvdVdrs = const,
(3)
where v(t) is formed by the same particles whose trajectories are determined from the equations
dx dt
v2,
dy dt
w2,
dv2 dt
dw2 dt - av'
az,
dT2 dt
dr s dr.
q'
o.
(4) System (4) is the characteristic system for the kinetic equation [the first equation in (2)]. The last equation allows for the absence of particle fragmentation and coagulation. Differentiating (3) with respect to time and taking into account the Ostrogradskii-Gauss theorem and system (4), we easily obtain the noncollisional kinetic equation in system (1) and (2). The individual volume v(t) coincides with the volume of the moving cell, and hence the number of particles in the ith cell remains constant. The distribution function in the ith cell at moment t = is be given by the formula f ? = v - Wig(v2 - v~i )~(w2 _ w~i )6(rs _ ri)$(T2-T~),
Vin = 4 h x h y ,
--
v~i -~ V~i 1 + TanTlzt '
x~i = x~i-1 + ~'v~i-1 + r2a~i-1/2,
-
n--1
W~i = W~ i 1 + Tay i ,
Y~i = Y~i--1 + rw~i--1 + ~'2a~i-1/2,
(5)
T~ = T~ -1 + ~'qT,
where ~- = t n - t n-l is the time step. As follows from (5), the Lagrangian cells move relative to the Eulerian grid with velocities v~i and w~i. The distribution function fj~ in the j t h Eulerian cell has the form m *~Ni
f; = ~
n
Vi--if- 8(v2 - v2i)8(w2 - w~i)~(rs - ri)6(T2 - T~),
• . ~ = ~yJj '
~f=S~3,
~=Sj,
(6)
S i = 4 h x h ~,
where ~ is the volume fraction occupied by the ith cell in the j t h Eulerian cell. Summation in (6) goes over m cells intersecting the j t h Eulerian cell. We use (6) to find the mean value CO
oO
= f ] QSdv d r .
(7)
--oo 0
Substituting (6) into (7), in the j t h Eulerian cell, we obtain
E Q~*~N~IV~ R
n
.
(Q) =
tg
(8) E
/
t
197
A~v.lO 3
1
o 60
80
100
120
d,'/Jm
Fig. 9. The function of size distribution of particles.
Formula (8) makes it possible to find the mean parameters of the particles in an arbitrary Eulerian cell. The gas parameters in the particle cells were found by means of linear interpolation. Conceptually this method of numerical solution of the kinetic equation is close to the method of particles in the dynamics of a low-density plasma [22]. However, there are substantial differences. In the present paper, the equations of a dusty gas are solved along with the kinetic equation; in [22], with Maxwell's equations. A similar method for calculating gas-particle flows in the steady-state case was proposed by Crowe [23]. O n e - D i m e n s i o n a l C a l c u l a t i o n o f t h e I n t e r a c t i o n of an S W w i t h a C l o u d o f P a r t i c l e s . Let us calculate the two-phase flow pattern and study the dynamics of the leading edge of a cloud of spherical particles behind an SW. The uniform cloud overlaps the cross-section of the flat channel (Fig. 8). In region g~l (without particles) the gas parameters are calculated from a system of Euler equations (ideal gas) by a third-order-accuracy method. System (1) and (2) was solved in the region f~2 by the numerical method described above. In Eqs. (1) and (2) we assumed that dO~dr = 0 (laboratory coordinate system). The initial conditions were specified in the form 0 2 pllMo Pll=P°l, p=p0, vl=wl=0, x > x +, Pl] = 1 - h + hM02'' p=p°((l+h)M~-h), Me = Do~co,
x < x +,
vl=(1-h)co(M0-1/M0), m2 = m °,
h = 7 -"1y + l '
wl=0,
(9)
fo = R(rs)~(w2)~(v2) - - in region ~/2,
where M0 is the SW Mach number, R(rs) = ( 6 m ° / r d ~ ) g ( r s - ds/2) for a monodisperse cloud of particles with diameter ds. We take wz = 0 as the boundary conditions for the gas at the boundaries V1 and 73, and impose the Hugoniot conditions on 72 and the condition of equality to zero of the gas-dynamic functions on 74 (see Fig. 8). The condition of mirror reflection is specified for the particles at the boundaries ~/1 and 73, while the condition of particle absorption is specified at 72 and 74- The values of the parameters of the gas and particles in the calculations were the same as in the experiments described above. The calculated trajectories of the left boundary x(t) of clouds of monodisperse bronze (d --- 130/~m) and Plexiglas (d = 300/~m) particles are shown in Figs. 4 (curves 1 and 2) and 5 (curves 1 and 2), respectively. There is good agreement between the calculation and experiment results. The size distribution of bronze particles for a polydisperse cloud is given in Fig. 9. Here A ¢ is the volume concentration of particles with diameter dl (in the range Ad), where A ¢ = m °. In the calculations we took eight fractions of from dl = 60/~m to d2 = 130 # m with increment Ad = 10 #m. It was found that the effect of polydispersity on the flow pattern is weak. Figures 10a and 10b presents dependences for the pressure p and the Mach number of the flow Ml(x) for a dense cloud. It is evident that a reflected ("collective") SW forms ahead of the cloud. A rarefaction wave forms inside the cloud. The gas is accelerated inside the wave and the flow becomes supersonic near the right boundary of the cloud. The amplitude of the passing SW decreases as compared with that of the incident wave, since a portion of the gas energy is expended on acceleration of the cloud.
198
p, MP~
,'-r:"k~
.
'~
b
M1
1.35
1.3
0.'* 0
t
I
t
I
1
1
2
3
4
5 x , mm
|
C
M1 1 2 3
1.1
1.30
0.9
t .25
0.7
1.20
0.5 0
1
2
3
4
5 x , mm
|
I
|
I
t
2
3
4
5
6 x, mm
I
Fig. 10. Pressure profiles p ( x ) (a) and flow Mach numbers MI(z) (b and c) in cloud of Plexiglas particles at times 50, 100, and 150 #sec "(curves 1, 2, and 3, respectively); rn2 -3- 10 -2 (a and b) and 10 -3 (c) (vertical curves show the cloud boundaries).
r3 £11 8W
r4
71
x
Fig. 11. Calculation domain (plane case), the same notation as in Fig. 8.
Figure 10c shows a curve of MI(x) for interaction of an SW with a cloud of Plexiglas particles with a low volume concentration of particles. It is seen that the flow in the cloud is supersonic and that no reflected "collective" SW arises. The appearance of small perturbations ahead of the left boundary is associated with the artificial viscosity. T w o - D i m e n s i o n a l C a l c u l a t i o n of t h e F l o w D u r i n g t h e I n t e r a c t i o n o f an S W w i t h a C l o u d of P a r t i c l e s . Let a SW run against a cloud of spherical Plexiglas particles in the region f/2 bounded along x and Y (Fig. 11). Let us consider the plane case: the motion of the gas and particles is described by system (1) and (2). At t = 0 the cloud is of a rectangular form. The particles are monodisperse (d = 100 #m), p22 = 1.2 g/cm 3, T ° = 300 K, v2 = w2 = 0, m ° = 10 -2, M0 = 3; the gas parameters ahead of the SW front: p° 1 = 1.3 kg/m 3, T O = 300 K. We specify Wl - ~ 0 at the boundary 3'1, the Hugoniot conditions at 72, and the condition of equality to zero of the gas-dynamic functions at ~3 and %. The condition of mirror reflection is specified for the particles on the boundary "71, while the condition of particle absorption is specified on %, %, and "Y4, Figure 12 shows isobars for times of 80 and 160 #sec. It is evident that a reflected SW is formed ahead of the particle cloud. A constant negative pressure gradient (rarefaction wave) is established inside the cloud. As the cloud is accelerated, the force of interaction between the gas and the particles is diminished f12 "~ Cg(vl - v2). A rarefaction wave, which attenuates the reflected SW, propagates upstream from the cloud. The pressure behind the SW is determined from the discontinuity decay that arises at the moment of arrival of the rarefaction wave at the reflected SW front. This process is unsteady. The time of arrival of the rarefaction wave at a certain point of the SW depends on its position, i.e., it increases with an increase in the transverse coordinate. As a result, the pressure behind the SW and hence the SW propagation velocity at the periphery is greater than near the axis (the half-space boundary), and the reflected SW front begins to straighten, while the SW itself separates from the cloud, which is clearly seen in Fig. 12.
199
a
3
4
b
5
4
5
6
7
8
x , cm
Fig. 12. Isobars (atm) for times 80 (a) and 160 #sec (b); dashed line shows the cloud boundary (laboratory coordinate system).
y,
cm
1.0
0.5
7.0
7:s
od
Fig. 13. Isolines of volume concentration rn2(x , y ) with increment A m 2 ----5 - 10 - 3 for time 160 #sec (laboratory coordinate system).
Volume-concentration isolines ra2(x, Y) are presented in Fig. 13. One can see that, because of the flow nonuniformity behind the SW, the particle cloud expands upward and extends downstream. In this case a region of elevated concentration is formed on the windward side of the cloud. The field of particle velocities in the center-of-mass system of the cloud is demonstrated in Fig. 14. It is seen that the particle trajectories are concentrated in the neighborhood of the leading cloud boundary, i.e., boundary caustics occurs as shown in Fig. 15. DISCUSSION OF THE RESULTS
Two mechanisms of generation of a "collective" SW ahead of a cloud of particles are currently known. According to [24], a common leading SW is formed in the case of joining of transonic zones behind the shock waves at individual particles. Then the condition of its generation is determined from the relation 1/d
