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The influence of particle size distributions on the velocity and the attenuation of sound in gas-solid suspensions
Chmical
Engiiening
Science,1976,Vol. 31, pp. 15-21. PcrgamonPress. Printed in Great Britain
THE INFLUENCE OF PARTICLE SIZE DISTRIBUTIONS ON THE VELOCITY AND THE ATTENUATION OF SOUND IN GAS-SOLID SUSPENSIONS W. GREGOR, J. RAASCH and H. RUMPF Institut fur Mechanische Verfahrenstechnii der Universitit Karlsruhe, D-75 Karlsruhe, West Germany (Received 26 May 1975;accepted 4 June 1975) Abskxt-On the basis of mass- and momentum-balances equations are derived for the velocity and the attenuation of sound depending on the volume fraction of the gas, the density ratio of the two phases, the gas viscosity, the slip velocity between gas and particles, the sound frequency, and on the normal&d volume-distribution-function q,(x) of the particle size. Specially very small particles have a big influence which is shown quantitatively with an assumed power-function for the volume-distribution q&).
INTRODUCTION
The sound velocity in a gas-solid suspension is less than in the pure gas phase and the sound attenuation in a suspension is higher. Both effects are caused by the different densities and compressibilities of the two phases. Generally for predicting the velocity of sound in a suspension, the wellknown Laplace equation111 is applied by using an average density [2,3]. Based on a flux model Rumpf and Gregor [4-61 have recently developed a theory which enables to calculate the velocity of sound in a gas containing small homogeneously distributed solid particles of uniform size. In the following this theory is extended for particles of variable sizes and arbitrary shapes.
In accordance with Fig. lb the mass balance for the gas yields (a - u)epf = (a - u - Au)(pr + Ap,)(r t Ae) or
a~~.A(r.pf)=e~~.Au.
(3)
If p(x). dx is defined as the number of particles with sizes between x and x t dx per unit volume, the total number of particles per unit volume is then determined by: OD cp(x)dx I0
N=
THEVELOClTYOFSOUNDINA
(4)
GAS.SOLID SUSPENSION
A suspension containing a gas phase F and a solid phase S is considered. The unique velocity of the gas is given by o; the velocity w of the solid particles depends on the particle size x, thus it is given by w(x). The relative velocity between the two phases is: u,,,(x) = 0 - w(x).
and the volume fraction of particles is given by: l--E=
(5)
with V,(x) = Volume of one particle of size x. A change of the porosity E causes a change of the distribution function p(x):
(1)
The compressive gas has the density pf, the density of the incompressible particles is ps. The gas volume related to the total volume is called the porosity E.A small pressure wave propagating with an absolute velocity a through this suspension is changing the velocities, densities and the porosity as shown in Fig. la. If the reference frame moves with the same velocity a as the advancing pressure wave, a steady state is reached (Fig. lb). Since the pressure wave passes through the continuous gas phase, the velocity of sound a,, in a suspension can be defined as the velocity of the advancing pressure wave relative to the motion of the gas: a,, = a - 1).
m Vs(x) . cp(x)dx I0
0 -he=
I Cl
Vs(x) . Acp(x)dx.
(6)
According to Fig. lb the mass balance of the particles with sizes between x and x tdx gives: (a - w(x)). ~8. V.(x). cp(x)dx = (a-w(x)-Aw(x)).ps.
V,(x).(cp(x)+Ap(x))dx
or Acp(x)= dx).
(2)
afAtw;,;x, s
15
l-2
(7)
16
W. GRI!OORet ai.
1
af
[---I
p+Ap
I
L
I
Phase F
i-L
I
pl+Ap~
PhaseS
; s’
1
E+AC
L---l 8
(12)
which is known as Laplace’s equation. In deriving it isentropy has been assumed. This equation is here applied to a gas-solid suspension, although a pressure wave in a suspension will certainly not propagate completely isentropicly. However, it is assumed that eqn (12) is still a good approximation in this case. Replacing the differentials in (12) with the corresponding increments, and using the eqns (3), (6), (7), (10) and (1l), the following expression for the velocity of sound af, in a suspension is obtained:
P
+
“+d
=eiJm
p, A
(0) m
r--l
P+AP
I
q3b)
a_.
Aw(x)
(l+ u,,(x)/u,,)’ ,
E
P
dx
Au .dx
a-v-Av L=4
I &
O-4
0-w
Phase S
I I E
I
l+AC
p,tAp,
(13)
!%seF
L--l 6
If the slip velocity (eqn (1)) between all particles and the gas is small compared with afS,then.eqn (13) simplifies to:
PI A
(b)
Fig. 1. Propagating pressure wave with (a) fixed reference frame: and (b) moving reference frame (steady state).
(14)
If q3(x) is defined as the normalised volume distributionfunction of the particle size, then the mass balance can also be written as:
as it has been already used in [7]. If only particles of one unique size are present, eqn (13) results in:
ni,. q3(x) dx = (ars + u,,,(x)). ps. V&Z). q(x) dx (8) where &fSis equal to the total mass flow per unit area:
I& = ps.
Im(a~s+t’dxN.
V,(x).cp(x)dx
0
eqns. (5), (8) and (9) yield:
(1-E)%(X) (p(x)=(af,+u,.,(x)).
V,(x)'
The total momentum balance gives:
+I_c .-.- af, Aw E af, + ursl Au ps af, + urea Aw E+(l--E).pl.-.af, Au 1
(9)
as it has been already shown in [4] and [5]. An unknown value in (13) is the relative acceleration Aw(x)/Au. For determining this term a momentum balance is made for the separate gas phase. The drag force between both phases W, related to the total cross(10) sectional area A has to be considered (Fig. 2). As explained in [5] and [6] the total drag force Wfs must be reduced by a force K, which is necessary to establish the slip velocity between solid particles and gas.
Ap = a&p, - (qs -Au)‘@ t Ac)(p, + Ap,)
W,s-K E. Ap = a&p, - (afs - Au)‘(e t Ae)(p, + Ap,) + A (16)
m
t
J I cl
ps(qs+ urn,(x))*. L's(x). p(x) dx
The momentum balance for the particles with sizes between x and x t dx gives:
OD
-
o ~,(a,‘ t u,,,(x)-Aw(x))*.
Vs(x) . (dx)
Vs. p . Acp(x)dx = Vs. ps. (a/s+
t Aq(x)) dx.
(13)
(11)
The velocity of sound in the pure gas af is described by 111:
ur&))*. cp(x)dx
- Vsp.(a,, +u~,,(x)--w(x))~((P(x~+~(P(~))~ _dWfs-K
7(x).
(17)
17
The influence of particlesizedistributionsin gas-solidsuspensions ____c
r----
1
I I a-v-Av -
I
I
I I
Gas - Phase
With the eqns (3), (6), (7), (lo), (12), (17)-(20), an integral equation of Fredholm type and 2nd kind[8] for Aw(x)/Ao is obtained with the solution:
I
, I
o-v
I
I I I
m
Al =E.
;_
(F)
I
m+b
In case of Stokes’ law eqn (Mb) is valid. XS,is the Stokes diameter of a particle of size x. According to Fig. 2, Al is equal to 4of the wavelength:
K/A
---
I
I
Aw(x) -=&+ A0
c
G(x)+
($* J 2
1 W&A
-
P(x)
.
dx
1
) (21)
+ A o_w_Aw
-
F+x1
W,JA --
-
-
(22)
-
-
I
* I-~-AS
Solid - Phase
I
6) I I___---__A
I I I
P(x)=;.
K/A -
(23)
Q(x)+G(x)
3 clu. V,l -.4T V.XN
(244
;9 *z17rlPt
(25a)
G(x) = when Stokes’ law is applied.
When the settling velocity wI of a particle141 is introduced, then the parameter G can also be written as: CW
W-4
+----Al---_-cl Fig. 2. Propagating pressure wave with separate reference frames for the two phases.
If the slip velocity between all particles and the gas is small compared with afS,then the eqns (21H23) simplify to:
In writing down eqns (16) and (17) it has been assumed that wall friction, if present, is not changed by the pressure wave. This means, that eqns (16) and (17) are only precisely true, if the slip velocity between the two phases is not caused by wall friction but by field forces. In accordance with [5] and [6] the drag force d [(W, IQ/A](x) of the particles with sizes between x and x + dx canbe obtained from
Aw(x) PC--_. Au
1
G(x)+
;+ G(x)
(26)
;c.(x).~~.~N*.~,.,(x).(Av-Aw(x)).A~.I~(x).~~ dv(x)
3?nl,.x,.$
. (Au -Aw(x))
Equation (Ha) is the general form. c, is the drag coefficient and xN is the diameter of a sphere having the same drag force as a particle of size x. The Reynolds number which determines c, is given by: Re = u,dx) . xN. pr/qf. CES Vol. 31.No. 1-B
(18a)
=
(19)
.A/. q(x). dx.
(W
which is identically equal with the result in [5] and [6] in case of particles with one unique size. The velocity of sound can be predicted completely by using the eqns (13) and (21)-(25). In case of small slip velocities, the eqns (14), (25) and (26) are valid. The equations yield:
W. GREGORet al.
18
(27)
with
THE EFFECT OF A PARTICLE-SIZEPOWER
St is the Stokes number with the definition:
DR7I’RIRIJTION FOR THE VELOCITY OF SOUND
St =
J(IIpr* > O.XSl
THE ATTENUATION
(28)
OF SOUNDIN A GAS43OLID SUSPENSlON
The total theory of the attenuation of sound in a gas containing solid particles of one unique size is described in [9]. In the following the attenuation of sound in a suspension with particles of variable sizes is confined to the so-called viscous attenuation caused by the friction between the solid particles and the gas phase. In accordance with the equations obtained in 191the intensity of the sonic wave is given by: Z(d) = ars i epfAu* m
I
t;P"
0
V,(x). (p(x). Aw’(x) . sin’ ot. dx . (29)
The decrease of the intensity Z with the length I can be determined by the product of the drag force and the relative velocity between particles and gas:
az -=-
O” ~(x).3?n7rx,t.(Ao-Aw(x))2.sin20t.dx
I
al
o
(30) if Stokes’ law is applied for the drag force. With the definition of the mean viscous attenuation coefficient [9]: 2n azlal
I.
“’
Id
l
l.+_l-c.f& pr . c
‘Xsr
=x
is valid for the particle diameter.
PARTICLESIZE DISTRIRDTION
FOR TEE ATTENUATIONOF SOUND
i_bm. (F)k}*(32)
The factor Aw(x)/Au can be taken from eqns (25) and (26). x. is the diameter of a sphere having the same volume as the particle with the size x. If all particles are of spherical shape, then x.
(34)
The influence of a particle size distribution on the attenuation of sound, which is given by eqn (32) together with eqns (21)-(26), is still more complicated than its influence on the velocity of sound. It can be stated, that in both cases the influence of q,(x) cannot be described by any statistical moment. Only if for all particle sizes Aw(x) =Au is valid, the attenuation coefficient, 6, can depend on the two moments M2.3and M-2.) as it is shown in [7].
q,(x)*(yr.dx
l-
* xm-‘.
Further on the particles are supposed to have spherical shape, so that eqn (33) is valid. For different exponents rn the results of the integrals in eqn (27) are shown in Table 1. The dependence of the sound velocity on the porosity E, the density ratio ps/p, and the Stokes number can be seen from Fig. 3 in case of one unique particle size x. For a power distribution with an exponent m = 1 for the particle sizes, the result is shown in Fig. 4. In Fig. 5 the porosity and the density ratio are kept constant (c = 099, p,/p, = 2000). On a line of constant Stokes number, the different particle size distributions have the same maximum particle size xmax,but the amount of small particles increases with decreasing value of m and thus the sound velocity decreases. Small particles always tend to follow the movements of the surrounding gas (Aw(x)/Au + 1 for x + 0) and their big density causes this decrease of the sound velocity[4]. According to Fig. 5, the particle distribution causes a considerable effect of decrease in the sound velocity only in a certain frequency range. Experimental data have not been available yet for checking this effect.
(31)
f%.(l-y)l.q,(x).dx o xv_ ”
*3(x) = 2
THEEFFECTOF A d(ot)
the following expression is obtained:
G” =,,.Ilrlpr.&.
The influence of a particle size distribution q,(x) on the velocity of sound is described by the eqns (13), (14) and (21H28). For demonstrating these results, a power distribution for q3(x) is assumed:
(33)
LIMITS OF THE TERORY
The equations derived above, predicting the velocity and the attenuation of sound in a gas-solid suspension, have been obtained on several simplifying assumptions. (1) A slip velocity between the two phases has been introduced which can be caused either by wall friction or by field forces. However, it has been assumed that the forces giving rise to the slip velocity are not changed by the pressure wave. Therefore the above equations are
The influence of particlesizedistributionsin gas-solid suspensions Table 1. Integrals of eqn (27)for different exponentsin eqn (34) O3G(x). s,(x) Io G(x)+ P,/Pf dr
S. G
m-3
nd,
G max
pf -2
= 18 . St”_
+
- 3.._,:+3
(G,,.
‘1 f/Pi = 18 . ~ 2.74 . x2,,
m-03
Fig. 3. Velocity of sound in a gas containing solid particles of one unique size.
only precisely true in case the slip velocity is the result of field forces. (2) Laplace’s equation for the velocity of sound in a pure gas has been used, although a pressure wave in a suspension does not propagate completely isentropicly. (3) All thermic effects upon the velocity of sound have been neglected. Soo[lO] mentions that these effects are extremely small compared with viscous effects. Experimental results for monosized particle suspensions fit in
very well with theoretical values obtained when only accounting for viscous effects [S]. (4) The total attenuation coefficient in a suspension is given by the sum of the attenuation coefficients caused by scattering, by irreversible heat transfer between the two phases, and by friction between the two phases. The last one has been considered in the present paper, the other two attenuation coefficients need a much more difficult analysis in case of polysized particle-suspensions, but
W. GREGOR et al.
Fig. 4. Velocity of sound in a gas containing solid particles with a size-distribution described by a power-function with an exponent m = 1.
I.0 0.6
W
Fig. 5. Velocity of sound in a gas containing solid particles with size-distributions obeying power-functions with variable exponents m.
The influenceof particlesizedistributionsin gas-solid suspensions
w ws x xN
velocity of the particles settling velocity particle size adequate diameter of a sphere with same drag force XS, Stokes diameter x, adequate diameter of a sphere with same volume
they are not at all negligible. As shown in [9] for monosized particles the viscous attenuation coefficient dominates only in a certain range of the Stokes number. These points restrict possible applications of the above equations. However, since the theoretical results agree very well with experimental values in case of monosized particles, no vast deviations are expected in case of polysized particles. NOTATION
intersectional area absolute velocity of sound propagation sound velocity in the pure gas sound velocity in the gas-solid suspension drag coefficient parameter gravity intensity force length-coordinate mass flow of solid particles per unit area exponent number of particles per unit volume pressure functions normalised volume-distribution function Reynolds number Stokes number time volume of one particle, equal to T. x:/6 velocity of the gas slip velocity between gas and particles friction force
21
Greek
symbols a” E Ilr
Pf Ps V w =25Tv
mean viscous attenuation coefficient porosity dynamic viscosity of the gas density of the gas density of the particles frequency of the sound angular frequency
REFERENCES
[l] Gerthsen Chr., Physik p. 114,lOth Edn. Springer, New York, 1969. [2] Fischer M., Thesis, University of Karlsrahe 1%7. [3] Deich M. E., Fillipow G. A. and Stekolshchikov E. V., Teploenergetika 1964 1133. [4] Rumpf H. and Gregor W., Chemie-Ing.-Techn. 197314924. [5] Gregor W. and Rumpf H., Chemie-Ing.-Techn 19744 155. [6] Gregor W. andRumpf H., Inr. J. MulriphaseFlow 19751753. [A Rumpf H., Gregor W. and Al-Taweel A., Preprints for GVC/AIChE-Joint Meeting Miinchen 1974,III, DS-2. [8] Bronstein I. and Semendjajew K., Taschenbuch der Mathematik, p. 528,3rd Edn. Harri Deutsch, Frankfurt, 1%2. [9] Gregor W. and Rumpf H., Powder Technology (in preparation). [lo] Soo S. L., Fluid Dynamics of Multiphase Systems, Blaisdell, Waltham, Mass., 1967.
Engiiening
Science,1976,Vol. 31, pp. 15-21. PcrgamonPress. Printed in Great Britain
THE INFLUENCE OF PARTICLE SIZE DISTRIBUTIONS ON THE VELOCITY AND THE ATTENUATION OF SOUND IN GAS-SOLID SUSPENSIONS W. GREGOR, J. RAASCH and H. RUMPF Institut fur Mechanische Verfahrenstechnii der Universitit Karlsruhe, D-75 Karlsruhe, West Germany (Received 26 May 1975;accepted 4 June 1975) Abskxt-On the basis of mass- and momentum-balances equations are derived for the velocity and the attenuation of sound depending on the volume fraction of the gas, the density ratio of the two phases, the gas viscosity, the slip velocity between gas and particles, the sound frequency, and on the normal&d volume-distribution-function q,(x) of the particle size. Specially very small particles have a big influence which is shown quantitatively with an assumed power-function for the volume-distribution q&).
INTRODUCTION
The sound velocity in a gas-solid suspension is less than in the pure gas phase and the sound attenuation in a suspension is higher. Both effects are caused by the different densities and compressibilities of the two phases. Generally for predicting the velocity of sound in a suspension, the wellknown Laplace equation111 is applied by using an average density [2,3]. Based on a flux model Rumpf and Gregor [4-61 have recently developed a theory which enables to calculate the velocity of sound in a gas containing small homogeneously distributed solid particles of uniform size. In the following this theory is extended for particles of variable sizes and arbitrary shapes.
In accordance with Fig. lb the mass balance for the gas yields (a - u)epf = (a - u - Au)(pr + Ap,)(r t Ae) or
a~~.A(r.pf)=e~~.Au.
(3)
If p(x). dx is defined as the number of particles with sizes between x and x t dx per unit volume, the total number of particles per unit volume is then determined by: OD cp(x)dx I0
N=
THEVELOClTYOFSOUNDINA
(4)
GAS.SOLID SUSPENSION
A suspension containing a gas phase F and a solid phase S is considered. The unique velocity of the gas is given by o; the velocity w of the solid particles depends on the particle size x, thus it is given by w(x). The relative velocity between the two phases is: u,,,(x) = 0 - w(x).
and the volume fraction of particles is given by: l--E=
(5)
with V,(x) = Volume of one particle of size x. A change of the porosity E causes a change of the distribution function p(x):
(1)
The compressive gas has the density pf, the density of the incompressible particles is ps. The gas volume related to the total volume is called the porosity E.A small pressure wave propagating with an absolute velocity a through this suspension is changing the velocities, densities and the porosity as shown in Fig. la. If the reference frame moves with the same velocity a as the advancing pressure wave, a steady state is reached (Fig. lb). Since the pressure wave passes through the continuous gas phase, the velocity of sound a,, in a suspension can be defined as the velocity of the advancing pressure wave relative to the motion of the gas: a,, = a - 1).
m Vs(x) . cp(x)dx I0
0 -he=
I Cl
Vs(x) . Acp(x)dx.
(6)
According to Fig. lb the mass balance of the particles with sizes between x and x tdx gives: (a - w(x)). ~8. V.(x). cp(x)dx = (a-w(x)-Aw(x)).ps.
V,(x).(cp(x)+Ap(x))dx
or Acp(x)= dx).
(2)
afAtw;,;x, s
15
l-2
(7)
16
W. GRI!OORet ai.
1
af
[---I
p+Ap
I
L
I
Phase F
i-L
I
pl+Ap~
PhaseS
; s’
1
E+AC
L---l 8
(12)
which is known as Laplace’s equation. In deriving it isentropy has been assumed. This equation is here applied to a gas-solid suspension, although a pressure wave in a suspension will certainly not propagate completely isentropicly. However, it is assumed that eqn (12) is still a good approximation in this case. Replacing the differentials in (12) with the corresponding increments, and using the eqns (3), (6), (7), (10) and (1l), the following expression for the velocity of sound af, in a suspension is obtained:
P
+
“+d
=eiJm
p, A
(0) m
r--l
P+AP
I
q3b)
a_.
Aw(x)
(l+ u,,(x)/u,,)’ ,
E
P
dx
Au .dx
a-v-Av L=4
I &
O-4
0-w
Phase S
I I E
I
l+AC
p,tAp,
(13)
!%seF
L--l 6
If the slip velocity (eqn (1)) between all particles and the gas is small compared with afS,then.eqn (13) simplifies to:
PI A
(b)
Fig. 1. Propagating pressure wave with (a) fixed reference frame: and (b) moving reference frame (steady state).
(14)
If q3(x) is defined as the normalised volume distributionfunction of the particle size, then the mass balance can also be written as:
as it has been already used in [7]. If only particles of one unique size are present, eqn (13) results in:
ni,. q3(x) dx = (ars + u,,,(x)). ps. V&Z). q(x) dx (8) where &fSis equal to the total mass flow per unit area:
I& = ps.
Im(a~s+t’dxN.
V,(x).cp(x)dx
0
eqns. (5), (8) and (9) yield:
(1-E)%(X) (p(x)=(af,+u,.,(x)).
V,(x)'
The total momentum balance gives:
+I_c .-.- af, Aw E af, + ursl Au ps af, + urea Aw E+(l--E).pl.-.af, Au 1
(9)
as it has been already shown in [4] and [5]. An unknown value in (13) is the relative acceleration Aw(x)/Au. For determining this term a momentum balance is made for the separate gas phase. The drag force between both phases W, related to the total cross(10) sectional area A has to be considered (Fig. 2). As explained in [5] and [6] the total drag force Wfs must be reduced by a force K, which is necessary to establish the slip velocity between solid particles and gas.
Ap = a&p, - (qs -Au)‘@ t Ac)(p, + Ap,)
W,s-K E. Ap = a&p, - (afs - Au)‘(e t Ae)(p, + Ap,) + A (16)
m
t
J I cl
ps(qs+ urn,(x))*. L's(x). p(x) dx
The momentum balance for the particles with sizes between x and x t dx gives:
OD
-
o ~,(a,‘ t u,,,(x)-Aw(x))*.
Vs(x) . (dx)
Vs. p . Acp(x)dx = Vs. ps. (a/s+
t Aq(x)) dx.
(13)
(11)
The velocity of sound in the pure gas af is described by 111:
ur&))*. cp(x)dx
- Vsp.(a,, +u~,,(x)--w(x))~((P(x~+~(P(~))~ _dWfs-K
7(x).
(17)
17
The influence of particlesizedistributionsin gas-solidsuspensions ____c
r----
1
I I a-v-Av -
I
I
I I
Gas - Phase
With the eqns (3), (6), (7), (lo), (12), (17)-(20), an integral equation of Fredholm type and 2nd kind[8] for Aw(x)/Ao is obtained with the solution:
I
, I
o-v
I
I I I
m
Al =E.
;_
(F)
I
m+b
In case of Stokes’ law eqn (Mb) is valid. XS,is the Stokes diameter of a particle of size x. According to Fig. 2, Al is equal to 4of the wavelength:
K/A
---
I
I
Aw(x) -=&+ A0
c
G(x)+
($* J 2
1 W&A
-
P(x)
.
dx
1
) (21)
+ A o_w_Aw
-
F+x1
W,JA --
-
-
(22)
-
-
I
* I-~-AS
Solid - Phase
I
6) I I___---__A
I I I
P(x)=;.
K/A -
(23)
Q(x)+G(x)
3 clu. V,l -.4T V.XN
(244
;9 *z17rlPt
(25a)
G(x) = when Stokes’ law is applied.
When the settling velocity wI of a particle141 is introduced, then the parameter G can also be written as: CW
W-4
+----Al---_-cl Fig. 2. Propagating pressure wave with separate reference frames for the two phases.
If the slip velocity between all particles and the gas is small compared with afS,then the eqns (21H23) simplify to:
In writing down eqns (16) and (17) it has been assumed that wall friction, if present, is not changed by the pressure wave. This means, that eqns (16) and (17) are only precisely true, if the slip velocity between the two phases is not caused by wall friction but by field forces. In accordance with [5] and [6] the drag force d [(W, IQ/A](x) of the particles with sizes between x and x + dx canbe obtained from
Aw(x) PC--_. Au
1
G(x)+
;+ G(x)
(26)
;c.(x).~~.~N*.~,.,(x).(Av-Aw(x)).A~.I~(x).~~ dv(x)
3?nl,.x,.$
. (Au -Aw(x))
Equation (Ha) is the general form. c, is the drag coefficient and xN is the diameter of a sphere having the same drag force as a particle of size x. The Reynolds number which determines c, is given by: Re = u,dx) . xN. pr/qf. CES Vol. 31.No. 1-B
(18a)
=
(19)
.A/. q(x). dx.
(W
which is identically equal with the result in [5] and [6] in case of particles with one unique size. The velocity of sound can be predicted completely by using the eqns (13) and (21)-(25). In case of small slip velocities, the eqns (14), (25) and (26) are valid. The equations yield:
W. GREGORet al.
18
(27)
with
THE EFFECT OF A PARTICLE-SIZEPOWER
St is the Stokes number with the definition:
DR7I’RIRIJTION FOR THE VELOCITY OF SOUND
St =
J(IIpr* > O.XSl
THE ATTENUATION
(28)
OF SOUNDIN A GAS43OLID SUSPENSlON
The total theory of the attenuation of sound in a gas containing solid particles of one unique size is described in [9]. In the following the attenuation of sound in a suspension with particles of variable sizes is confined to the so-called viscous attenuation caused by the friction between the solid particles and the gas phase. In accordance with the equations obtained in 191the intensity of the sonic wave is given by: Z(d) = ars i epfAu* m
I
t;P"
0
V,(x). (p(x). Aw’(x) . sin’ ot. dx . (29)
The decrease of the intensity Z with the length I can be determined by the product of the drag force and the relative velocity between particles and gas:
az -=-
O” ~(x).3?n7rx,t.(Ao-Aw(x))2.sin20t.dx
I
al
o
(30) if Stokes’ law is applied for the drag force. With the definition of the mean viscous attenuation coefficient [9]: 2n azlal
I.
“’
Id
l
l.+_l-c.f& pr . c
‘Xsr
=x
is valid for the particle diameter.
PARTICLESIZE DISTRIRDTION
FOR TEE ATTENUATIONOF SOUND
i_bm. (F)k}*(32)
The factor Aw(x)/Au can be taken from eqns (25) and (26). x. is the diameter of a sphere having the same volume as the particle with the size x. If all particles are of spherical shape, then x.
(34)
The influence of a particle size distribution on the attenuation of sound, which is given by eqn (32) together with eqns (21)-(26), is still more complicated than its influence on the velocity of sound. It can be stated, that in both cases the influence of q,(x) cannot be described by any statistical moment. Only if for all particle sizes Aw(x) =Au is valid, the attenuation coefficient, 6, can depend on the two moments M2.3and M-2.) as it is shown in [7].
q,(x)*(yr.dx
l-
* xm-‘.
Further on the particles are supposed to have spherical shape, so that eqn (33) is valid. For different exponents rn the results of the integrals in eqn (27) are shown in Table 1. The dependence of the sound velocity on the porosity E, the density ratio ps/p, and the Stokes number can be seen from Fig. 3 in case of one unique particle size x. For a power distribution with an exponent m = 1 for the particle sizes, the result is shown in Fig. 4. In Fig. 5 the porosity and the density ratio are kept constant (c = 099, p,/p, = 2000). On a line of constant Stokes number, the different particle size distributions have the same maximum particle size xmax,but the amount of small particles increases with decreasing value of m and thus the sound velocity decreases. Small particles always tend to follow the movements of the surrounding gas (Aw(x)/Au + 1 for x + 0) and their big density causes this decrease of the sound velocity[4]. According to Fig. 5, the particle distribution causes a considerable effect of decrease in the sound velocity only in a certain frequency range. Experimental data have not been available yet for checking this effect.
(31)
f%.(l-y)l.q,(x).dx o xv_ ”
*3(x) = 2
THEEFFECTOF A d(ot)
the following expression is obtained:
G” =,,.Ilrlpr.&.
The influence of a particle size distribution q,(x) on the velocity of sound is described by the eqns (13), (14) and (21H28). For demonstrating these results, a power distribution for q3(x) is assumed:
(33)
LIMITS OF THE TERORY
The equations derived above, predicting the velocity and the attenuation of sound in a gas-solid suspension, have been obtained on several simplifying assumptions. (1) A slip velocity between the two phases has been introduced which can be caused either by wall friction or by field forces. However, it has been assumed that the forces giving rise to the slip velocity are not changed by the pressure wave. Therefore the above equations are
The influence of particlesizedistributionsin gas-solid suspensions Table 1. Integrals of eqn (27)for different exponentsin eqn (34) O3G(x). s,(x) Io G(x)+ P,/Pf dr
S. G
m-3
nd,
G max
pf -2
= 18 . St”_
+
- 3.._,:+3
(G,,.
‘1 f/Pi = 18 . ~ 2.74 . x2,,
m-03
Fig. 3. Velocity of sound in a gas containing solid particles of one unique size.
only precisely true in case the slip velocity is the result of field forces. (2) Laplace’s equation for the velocity of sound in a pure gas has been used, although a pressure wave in a suspension does not propagate completely isentropicly. (3) All thermic effects upon the velocity of sound have been neglected. Soo[lO] mentions that these effects are extremely small compared with viscous effects. Experimental results for monosized particle suspensions fit in
very well with theoretical values obtained when only accounting for viscous effects [S]. (4) The total attenuation coefficient in a suspension is given by the sum of the attenuation coefficients caused by scattering, by irreversible heat transfer between the two phases, and by friction between the two phases. The last one has been considered in the present paper, the other two attenuation coefficients need a much more difficult analysis in case of polysized particle-suspensions, but
W. GREGOR et al.
Fig. 4. Velocity of sound in a gas containing solid particles with a size-distribution described by a power-function with an exponent m = 1.
I.0 0.6
W
Fig. 5. Velocity of sound in a gas containing solid particles with size-distributions obeying power-functions with variable exponents m.
The influenceof particlesizedistributionsin gas-solid suspensions
w ws x xN
velocity of the particles settling velocity particle size adequate diameter of a sphere with same drag force XS, Stokes diameter x, adequate diameter of a sphere with same volume
they are not at all negligible. As shown in [9] for monosized particles the viscous attenuation coefficient dominates only in a certain range of the Stokes number. These points restrict possible applications of the above equations. However, since the theoretical results agree very well with experimental values in case of monosized particles, no vast deviations are expected in case of polysized particles. NOTATION
intersectional area absolute velocity of sound propagation sound velocity in the pure gas sound velocity in the gas-solid suspension drag coefficient parameter gravity intensity force length-coordinate mass flow of solid particles per unit area exponent number of particles per unit volume pressure functions normalised volume-distribution function Reynolds number Stokes number time volume of one particle, equal to T. x:/6 velocity of the gas slip velocity between gas and particles friction force
21
Greek
symbols a” E Ilr
Pf Ps V w =25Tv
mean viscous attenuation coefficient porosity dynamic viscosity of the gas density of the gas density of the particles frequency of the sound angular frequency
REFERENCES
[l] Gerthsen Chr., Physik p. 114,lOth Edn. Springer, New York, 1969. [2] Fischer M., Thesis, University of Karlsrahe 1%7. [3] Deich M. E., Fillipow G. A. and Stekolshchikov E. V., Teploenergetika 1964 1133. [4] Rumpf H. and Gregor W., Chemie-Ing.-Techn. 197314924. [5] Gregor W. and Rumpf H., Chemie-Ing.-Techn 19744 155. [6] Gregor W. andRumpf H., Inr. J. MulriphaseFlow 19751753. [A Rumpf H., Gregor W. and Al-Taweel A., Preprints for GVC/AIChE-Joint Meeting Miinchen 1974,III, DS-2. [8] Bronstein I. and Semendjajew K., Taschenbuch der Mathematik, p. 528,3rd Edn. Harri Deutsch, Frankfurt, 1%2. [9] Gregor W. and Rumpf H., Powder Technology (in preparation). [lo] Soo S. L., Fluid Dynamics of Multiphase Systems, Blaisdell, Waltham, Mass., 1967.