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On the turbulent pipe flow of gas-solids suspensions
Aerosol Science, 1972, Vol, 3, pp. 157 to 165. Pergamon Press. Printed in Great Britain.
ON T H E T U R B U L E N T PIPE FLOW OF GAS-SOLIDS SUSPENSIONS M. NAGARAJAN'~ Mechanical Engineering Department, Imperial College of Science and Technology, London, S.W.7 (Received 11 August 197 I)
Abstract--Fully developed axisymmetric flow of gas-solids suspension in the presence of gravitational and electrostatic fields is considered with a view to predicting tile particle slip velocity and concentration profiles. The analytical model presupposes the existence of drift velocity (NAGA~JAN et al., 1971) and statistical properties for the solids phase. No particles are absorbed at the wall. By extending Kohnogoroff's scheme to the solids-phase turbulence it is shown that the particles are more evenly distributed in the regions away from the pipe wall and that the particle slip-velocity distribution is governed by a mean slip velocity for the flow. An expression relating the mean slip velocity to the particle diameter has been derived. The results are in close agreement with published data on gas-solids suspension flow.
NOMENCLATURE a C, c CD d e Ej
Fj 1 g p Po Uj, uj
Uav u* V i, v i
W Wm a, 3 e r, ep ec, ~s
r p p,,
/~p ~o co,,
pipe radius volume fraction of particles, its fluctuation particle drag co-efficient particle diameter Kolmogoroff energy scale electrostatic force components fluid-particle interaction force components Kolmogoroff length scale acceleration due to gravity pressure nett hydrostatic pressure fluid velocity components, Uo = axial velocity average fluid velocity over tube radius two-phase friction velocity particle velocity components, fluctuations slip velocity of solids constant value of slip under uniform flow conditions dimensionless parameters .momentum diffusivities eddy disperson co-efficients kinematic viscosity fluid density density of particle-fluid mixture particle density mass fraction of particles constant value of mass fraction under uniform flow conditions shear stress INTRODUCTION
THE PHENOMENA o f slip b e t w e e n p h a s e s is a c h a r a c t e r i s t i c o f t w o - p h a s e flows a n d this a s s u m e s s i g n i f i c a n t p r o p o r t i o n s in g a s - s o l i d s system. A s s o c i a t e d w i t h slip velocity is tNow at the Indian Institute of Technology, Powai, Bombay 76, India. 157
158
M. NAGARAJAN
distribution of concentration of solids, their interdependence being well recognised. In view of their fundamental importance in the transport characteristics of two-phase systems there have been attempts (Soo, 1962; WAKSTFJN, 1966; NAGARAJAN, 1971) to predict the slip velocity and concentration profiles. It is customary in such studies of suspension flows, to ignore the particle interaction effects except in the flows with very high concentrations of solids. Particles which strike the wall are assumed to be reflected back into the flow. The equations governing the flow of suspensions are written for the individual phases as follows: d ,%,[il-cWJ
= o
, l~
- - ~ c v s) = o
C
C
C).\'j
~X j
~-- [p( I - C)U i U j] = --- [( 1 - C)(iris - - pc}ij)f] -b-p ( [
~2~
--
C)g i - p F~
?.x l"(p p C k~ Vi) . . . .Cx . . ~CVru - pOu),, - p ,,Cg, + pF~ + pE~ .
~3)
i4}
Adding the equations (3) and (4) we get the equation of motion for the mixture which, incidentally, defines the stress of the medium, ( v u - p 6 u), assumed continuous and differentiable;
[p( J - c ) u , u , + p,, c ~;.vii = ~;~-[,is- po,s] + p..o, + pr.,
i 5t
where the mixture density, p,. is given by p., =- p ( 1 - C ) + C p p .
TURBULENT
~6)
F L O W OF S U S P E N S I O N S
Following the Reynolds procedure, we let a quantity of either phase, Q, be split into a time-independent average and a fluctuating part, as Q = 0.+q- It is understood that the average is taken over time intervals which are large compared to the time scales of motions of the particles (and hence of the fluid) and that ergodicity of all statistical quantities is confirmed. For the axisymmetric case considered g = gJfi: and E----- Ei6~,. Besides the compressibility effects in the fluid phase, all triple or higher order correlations containing the concentration fluctuations will be ignored. To simplify the notation, where there could be no confusion, the bars denoting the averages will be omitted. In the cylindrical ~.oordinates (r, 0, :) the continuity equations become
UI"
C
In gas-solids flows the co-volume, C, is very small even for appreciable loading and can be
On tile turbulent pipe flow of gas-solids suspensions
159
ignored in comparison with unity. Integrating the continuity equations we get the expressions for the drift Velocities, Ur = cu'S and
CVr =-~,.
(9)
The equation of motion in the z-direction is obtained from (5) as =--r(z,:-pu, u _ - p p C ~ ) cr
= r - - ( p o + zgp,,) . 3-_.
"
(10)
Here Po = Po(z) is the wall static pressure reckoned from r = a, z = 0, and is obtained by integrating the radial momentum balance.
P
po(-I = P +(v~ +~,~,)+o~(v/+~,)+ f °E r d r + P
o
fi
'
[(U~ +r°V~')+(~'-ffg)+°~(e2'-rJ~)]dr
=0
(11)
F
where w = Cpp/p is the mass fraction of particles. For a fully developed flow the coefficient of r in the right-hand side expression of (10) is a constant and we integrate the equation to obtain r dPo z,=-p(u,u._+ o~ vrv=) = -- - 2a dz where Po is the total hydrostatic pressure of the mixture given by Po = p o + p , , g z .
(12)
In terms of the momentum diffusion coefficients ~3U. g' I,'. ( v + ~S) ~ + og~p =-'or = - (r /a)u*: where and
(13)
v, ~y = molecular, eddy viscosity of fluid ep = particle momentum diffusivity u* = friction velocity for the suspension.
For the solids phase we obtain from (4): 8r(rPp ~ )
= 7Cr~:~o r - rC~-zz,( p + p p g z ) + rp .F., .
(14)
The viscous and pressure gradient effects can be neglected in comparison with the fluid particle interaction forces and the above equation simplifies to = (rpp
eCi~E.,v=)= rp(F..--ogg).
(15)
/1 I.
In effect, this amounts to defining F_. through the drag equation (15) whence F~ may be taken to include the effect of viscosity and pressure gradient previously neglected. The drag force per unit volume of the mixture can be given in terms of drag coefficient, CD, for a single particle. F. = w(1½/d). (p/pp). C.I'V:. t W.I .
(16)
160
M. NAGARAJAN
The drag coefficient of the suspended particle can be given in general by a power law of the form C o = K,,Re~"
0 < n < I
~I 7)
where Rep is the Reynolds number of the particle related to that of the fluid by , !~)
Rep = R e . ( d / 2 a ) . ( W . . / U , , , , ) R e = U,,,,.d;t).
The value of the exponent n = 1 for Stokes law regine, n = 0 for Newtons law regime and for the transition regime, INGEBO (1956) gives n = 0.84. The parametric constant K can be conveniently defined to include the effects due to the presence of other particles. TURBULENCE ENERGY BALANCE AND KOLMOGOROFF SCHEME The turbulent energy balance is constructed by the conventional technique (HINzE. 1959) along the mean motion, Urn, defined by the mass conservation equation, =~-(p,,, u,,,,,? = =- p ( U j + o ~ . ~ )
c.\-j
ux i
= o.
~i9)
As usual, triple correlations involving concentration fluctuations and simultaneous fluctuations of both particle and fluid velocities, are neglected. The scalar equation {summation indicated by repeated indices) after rearrangement reads
8-v'il C'iTwhere
.l/j_?. i = &\./ Dj+sx;~(Umir;j)-~,-B-P-
Q
,20)
OJ = -- [u'u" u' + °J'''ju'u' + ep ( ' ' +°~''')
B = O)'tl{m)i(gi--
fi)
~3U~ (1 +oY)P = ( u r % + c o ' u j V,,.)=- .... ,(~Xj
.Xj
We note that the structure of the equation is identical with that of a clear fluid and almost all of its terms are analogous to those of a pure fluid. The left-hand side represents, in the order of their appearance, convection of turbulent kinetic energy by mean motion and the terms on the right denote the diffusion of kinetic energy, work done by turbulent viscous stresses, dissipation of turbulent kinetic energy, work done by external body forces in suspending the solids and the production of turbulent energy from mean velocity gradients by mixing stresses. In the absence of axial velocity gradients the transport of kinetic energy by convection, advection and diffusion is negligible compared to its production and dissipation. The gravity term which could be very significant even for small concentrations, vanishe.~ due to the assumption of axisymmetry. The viscous stresses being small compared
On the turbulent pipe flow of gas-solids suspensions
161
to Reynolds stresses, we may disregard the work done by fluctuating viscous stresses. Thus we are left with only the terms for production and dissipation which are in near balance everywhere. Further we need to retain only those terms involving the velocity components in the flow direction in the absence of any appreciable flow in the radial direction. The final form of the equation after rearrangement, reads
For the determination of the coefficient ~p, ~¢, ~, of turbulent exchange and of the dissipation e, of turbulent energy, we propose to extend the KOLMOGOROFF (1942) scheme for pure fluids to the present situation. To pave the way we shall first consider the case of suspension flow with identical particles of size d uniformly distributed over the cross section of flow. For this "'mean" flow (denoted by subscript m) OUJc3r = c3V:/c3r and the ratio of eddy diffusivities,
(Pv C vr v.lp u,u:),,
= 3
(22a)
is a constant for the flow, being dependent only on the particle size and density. From the equation (13) for the motion of suspension we obtain for the regions away from the pipe wail: s:(1 +,8)
~U.
. - = --.
Cr
i" a
u * . 2,
(226)
which is a linear distribution of shear stress as for a pure fluid. The turbulence energy balance (21) reduces to Z i Z j p - 1 "C~jO(U i + O)Vi)l'OX J = . (1
+ e),~ r) 0 .U= .
(23)
UF
The dissipation term on the left shows that the effect due to the presence of solids as being proportional to their concentration, which is in agreement with LEvlcn's (1967) findings from his studies on the behaviour of suspended particles in isotropic turbulent fluid. The assumed uniformity of concentration distribution implies no mean motion in the radial direction so that the expression for the turbulent kinetic energy (20) becomes q,, = ½ Y i u ~
1+o~,,---- / . II l Ili/I
We observe that the total kinetic energy receives a contribution from the dispersed phase in proportion to their concentration to off-set the reduction in the turbulence level of the pure fluid, again in proportion to concentration, caused by the inertial mass centres of the particles with imperfect response to energy containing eddies. Thus we may possibly conclude that the total kinetic energy 02 is not very different from e = ½/7d7i of the pure fluid. In the absence of strong radial accelerations like gravity we may suppose that turbulent quantities of the suspension flow can still be described by e. For dilute suspensions the length scale of turbulence, l, is little affected by the presence of solids for the same reason. Following Kolmogoroff hypothesis and employing e and I for velocity and length scale respectively, we will assume:
162
M. NAGARAJAN 8t = l e ~
8p = % l e ~
e~ = ~ , l e ~
and
ec = ~ c l e ~
~, = z~e;l -~
!24)
where the ~'s are non-dimensional constant parmeaters. We note in passing that the various constants in the above scheme can be considered universal only in a restricted sense. Employing these newly defined quantities we now give the equations governing the motion of suspension in the non-dimensional form. For want of suitable parameters, all lengths and velocities have been non-dimensionalised respectively by a, the pipe radius and u*, the friction velocity for the flow. In what follows, prime denotes differentiation with respect to r, the non-dimensional radius. (25)
le~(U' +urtp V') = r
d
" e ~V )' = rF.. d- -p ( r . %
(26)
e = (IU'/.=,) 2.(1 + R i )
!27)
R i = (U') - 2. [(~c U' + as coV') Vco'+ (ac + coa,)Eco' + %(0~ V') 2] .
i 2S )
The last written equation defines the parameter R i an analogue of Richardson number tbr density stratified flows, reprc'~enting the added effects due to solids. As has already been pointed out, R i is proportional to the concentration, whence we have Ri,, = %eo~
and
Ri = (%co,,).to .
~29)
From particle diffusivity measurements made by See (1960) we infer that ~, ~ 10--' and we can safely neglect the product epco in the equation (25) thus effecting a considerable simplification o f the equations (27) and (26) which become respectively, I"
e = -il+Ri) ~
~30)
dr---5 (r2co V'/U') = CO(W~W.~)2 - ~
( 31 )
and d
where the m ~ n slip velocity IV,. has been obtained from (15) and (16) using the mean flow conditions (22), as
__W" = C 5.3~, v.o
p, Rd' (u',. yl.
LF~:~-2~. ~
@-2/a
+,,, .. , ~ \~/
'
~;:>
Equating (28) and (29) we obtain the equation for determining the concentration profile for the cases e~ -----% and ac ,~ e, respectively,
(~,/co.~,).(v/w).[u'+(l+co)}e/vlco' w' (-0-r-+vV)"
J~- = W
(~,/~,). (v! w). L(-6T_~]/-~j~ - w
(33a) ~33b)
While deriving the above equation we have made use of the approximation o~,.U'-~,) V ' : co,,W', since co _ co,. and U' ~ 1 in the turbulent core. The equations (31) and(33) together with the boundary conditions for the flow solve for the concentration and slip velocity
On the turbulent pipe flow of gas-solids suspensions
163
profiles in terms of the fluid velocity U(r) and the diffusivity ratios % and :~,. The ratio % may be obtained from a knowledge of shear stress distribution (25) while particle diffusivity measurements are required to evaluate %. DISCUSSION The equations (33a and b) for the concentration profile merit consideration. The expression on the left hand side has two large factors (V/W) and (~s/co=%). The latter especially, is very large for flows with not too high a concentration. We also find that the field forces do not make any significant contribution anywhere excepting the regions close to the wall, as indicated by the large dividing factor, V. The result is a concentration profile very much flatter than the slip velocity profile. We find experimental corroboration in the measurements taken by REDDY and PEI (1969) and Soo (1962). The later experimenters, notably BOYCE (1970) have taken uniform distribution of concentration for granted and the equations (33a and b) provide a rational basis for such an assumption. By making use of this result we can simplify the equation (31) for slip velocity profile to read
,1 / :dv'~
(A)
=
An approximate solution of this equation for the case n = 1 drag and W = 14Ioat U = Uo is W = W,,,+(Wo- W~,).exp [ ( U o -
viz Stokes law for particle
U)/14~]
(B)
o E
o.
= to
/
n = I0
----o--
Eauation
-----
Exptl.---dala, Boyce (1970)
.(22)
'
--
10
I0
Porticle
size,
I0
~m
F1G. I. Slip velocity with varying particle sizes for two-phase flow through circular pipe. BOYCE(] 970). Reynolds number
0"9
Pipe diameter Particle material
2"75 in. Silicadust (2-60 pro) Glass beads (100, 200, 840 t,m)
x
104 to 6"3 × 104
164
M. NAGAP.AJAN
Evidently the slip velocity is strongly dependent on 14/,,and gives a steq~r profile for smaller values of W,,. The slip velocity profiles measured by Soo (1962) and RF.~DY and PEI (1969) provide experimental verification. The slip velocity profiles obtained by them are reproduced in Figs. 2 and 3. The magnitude of the mean slip velocity may be taken as being nearly the same as its value at the centre of a large pipe where the concentration distribution is uniform over most part of the turbulent core. Particle velocity measurements made by BOYCE(1970) I'0
.
.
.
.
.
.
.
.
.
II
0.8 velocity ,~
/ /
::~ 0 . 6
/
o
// /
-x 0.4
/
/
/
//---"Solid phase velocity
/ W m ~ I 0 z, Uo
0.2
o
0.5
I
1.5
~adius,
2
25
in.
FIo. 2. Velocity distribution for air and particles. Data of Soo et al. (1964). Particles 2-75 lb MgO, 35 ~m nominal. zo
!
Air ~elocity profile
esentati~e ~artiele velocity prohle
Oisfarlcg I*r¢~l wall. Y c m
FIo. 3. Particle velocity obtained by double-flash storoo-photography. REDDYand PEt (t969).
Reynolds number Particle size Loading ratio
55000 200/.~m 0.02
over a wide range of particle size and flow Reynolds numbers reveal that the mean slip velocity is least affected by flow Reynolds number or concentration and is related mainly to particle size. An inspection of the expression (32) for mean slip velocity points to the same for the terms within the square bracket is almost independent of the concentration
On the turbulent pipc flow of gas-solids suspensions
165
and only weakly dependent on Reynolds n u m b e r since u*,. ~ R e -~" (Blasius Law). Finally, a justification for the present macroscopic a p p r o a c h to the problem m a y be found in BOYCE'S (ibid 1970) excellent correlation o f pressure drop data for gas-solids suspension flows by defining the flow Reynolds n u m b e r as if the two-phase system were a h o m o g e n e o u s fluid.
REFERENCES BORSHC~EVS~, I. YU. T. (1964) Dust Cooling Conference, London. BoYcE, M. P. (19"/0) J. Bas. Engng, Series D, 495. BUEVICH,IU. A. (1968) P M M 32, (1), 95. HINZE, J. O. (1959) Turbulence, McGraw Hill, New York. INGEBO,R. (1956) N.A.C.A. Tech. Note No. 3762. LEVlCH, V. G. (196"/) Doklady 12, (6), 546. NAGAR~OAN,M. (19"/1)`/. Aerosol Sci. 2, 1.5. REDDY, K. V. S. and PEI, D. C. T. (1969) Ind. Engng. Chert1. Fundls. Edn. 8, (3), 490. Soo, S. L. (1960),/. Bas. Engng, Series D, 82, 609. Soo, S. L. (1962) Ind. Engng Chem. Fundls. Edn., 1 (1), 33. Soo, S. L. (1967) Fhtid Dynamics of Multiphase systems, Blaisdelt. VASlL~V, O. P. (1969) 13th Congr. LA.H.R., Tokyo. WAKSTE~,C. (1966) Motion of small particles suspended in turbulent airflow in a vertical pipe. Penn. Univ WAKs'~rEIN,C. (19"/0)3". Aerosol Sei. 1, 69.
ON T H E T U R B U L E N T PIPE FLOW OF GAS-SOLIDS SUSPENSIONS M. NAGARAJAN'~ Mechanical Engineering Department, Imperial College of Science and Technology, London, S.W.7 (Received 11 August 197 I)
Abstract--Fully developed axisymmetric flow of gas-solids suspension in the presence of gravitational and electrostatic fields is considered with a view to predicting tile particle slip velocity and concentration profiles. The analytical model presupposes the existence of drift velocity (NAGA~JAN et al., 1971) and statistical properties for the solids phase. No particles are absorbed at the wall. By extending Kohnogoroff's scheme to the solids-phase turbulence it is shown that the particles are more evenly distributed in the regions away from the pipe wall and that the particle slip-velocity distribution is governed by a mean slip velocity for the flow. An expression relating the mean slip velocity to the particle diameter has been derived. The results are in close agreement with published data on gas-solids suspension flow.
NOMENCLATURE a C, c CD d e Ej
Fj 1 g p Po Uj, uj
Uav u* V i, v i
W Wm a, 3 e r, ep ec, ~s
r p p,,
/~p ~o co,,
pipe radius volume fraction of particles, its fluctuation particle drag co-efficient particle diameter Kolmogoroff energy scale electrostatic force components fluid-particle interaction force components Kolmogoroff length scale acceleration due to gravity pressure nett hydrostatic pressure fluid velocity components, Uo = axial velocity average fluid velocity over tube radius two-phase friction velocity particle velocity components, fluctuations slip velocity of solids constant value of slip under uniform flow conditions dimensionless parameters .momentum diffusivities eddy disperson co-efficients kinematic viscosity fluid density density of particle-fluid mixture particle density mass fraction of particles constant value of mass fraction under uniform flow conditions shear stress INTRODUCTION
THE PHENOMENA o f slip b e t w e e n p h a s e s is a c h a r a c t e r i s t i c o f t w o - p h a s e flows a n d this a s s u m e s s i g n i f i c a n t p r o p o r t i o n s in g a s - s o l i d s system. A s s o c i a t e d w i t h slip velocity is tNow at the Indian Institute of Technology, Powai, Bombay 76, India. 157
158
M. NAGARAJAN
distribution of concentration of solids, their interdependence being well recognised. In view of their fundamental importance in the transport characteristics of two-phase systems there have been attempts (Soo, 1962; WAKSTFJN, 1966; NAGARAJAN, 1971) to predict the slip velocity and concentration profiles. It is customary in such studies of suspension flows, to ignore the particle interaction effects except in the flows with very high concentrations of solids. Particles which strike the wall are assumed to be reflected back into the flow. The equations governing the flow of suspensions are written for the individual phases as follows: d ,%,[il-cWJ
= o
, l~
- - ~ c v s) = o
C
C
C).\'j
~X j
~-- [p( I - C)U i U j] = --- [( 1 - C)(iris - - pc}ij)f] -b-p ( [
~2~
--
C)g i - p F~
?.x l"(p p C k~ Vi) . . . .Cx . . ~CVru - pOu),, - p ,,Cg, + pF~ + pE~ .
~3)
i4}
Adding the equations (3) and (4) we get the equation of motion for the mixture which, incidentally, defines the stress of the medium, ( v u - p 6 u), assumed continuous and differentiable;
[p( J - c ) u , u , + p,, c ~;.vii = ~;~-[,is- po,s] + p..o, + pr.,
i 5t
where the mixture density, p,. is given by p., =- p ( 1 - C ) + C p p .
TURBULENT
~6)
F L O W OF S U S P E N S I O N S
Following the Reynolds procedure, we let a quantity of either phase, Q, be split into a time-independent average and a fluctuating part, as Q = 0.+q- It is understood that the average is taken over time intervals which are large compared to the time scales of motions of the particles (and hence of the fluid) and that ergodicity of all statistical quantities is confirmed. For the axisymmetric case considered g = gJfi: and E----- Ei6~,. Besides the compressibility effects in the fluid phase, all triple or higher order correlations containing the concentration fluctuations will be ignored. To simplify the notation, where there could be no confusion, the bars denoting the averages will be omitted. In the cylindrical ~.oordinates (r, 0, :) the continuity equations become
UI"
C
In gas-solids flows the co-volume, C, is very small even for appreciable loading and can be
On tile turbulent pipe flow of gas-solids suspensions
159
ignored in comparison with unity. Integrating the continuity equations we get the expressions for the drift Velocities, Ur = cu'S and
CVr =-~,.
(9)
The equation of motion in the z-direction is obtained from (5) as =--r(z,:-pu, u _ - p p C ~ ) cr
= r - - ( p o + zgp,,) . 3-_.
"
(10)
Here Po = Po(z) is the wall static pressure reckoned from r = a, z = 0, and is obtained by integrating the radial momentum balance.
P
po(-I = P +(v~ +~,~,)+o~(v/+~,)+ f °E r d r + P
o
fi
'
[(U~ +r°V~')+(~'-ffg)+°~(e2'-rJ~)]dr
=0
(11)
F
where w = Cpp/p is the mass fraction of particles. For a fully developed flow the coefficient of r in the right-hand side expression of (10) is a constant and we integrate the equation to obtain r dPo z,=-p(u,u._+ o~ vrv=) = -- - 2a dz where Po is the total hydrostatic pressure of the mixture given by Po = p o + p , , g z .
(12)
In terms of the momentum diffusion coefficients ~3U. g' I,'. ( v + ~S) ~ + og~p =-'or = - (r /a)u*: where and
(13)
v, ~y = molecular, eddy viscosity of fluid ep = particle momentum diffusivity u* = friction velocity for the suspension.
For the solids phase we obtain from (4): 8r(rPp ~ )
= 7Cr~:~o r - rC~-zz,( p + p p g z ) + rp .F., .
(14)
The viscous and pressure gradient effects can be neglected in comparison with the fluid particle interaction forces and the above equation simplifies to = (rpp
eCi~E.,v=)= rp(F..--ogg).
(15)
/1 I.
In effect, this amounts to defining F_. through the drag equation (15) whence F~ may be taken to include the effect of viscosity and pressure gradient previously neglected. The drag force per unit volume of the mixture can be given in terms of drag coefficient, CD, for a single particle. F. = w(1½/d). (p/pp). C.I'V:. t W.I .
(16)
160
M. NAGARAJAN
The drag coefficient of the suspended particle can be given in general by a power law of the form C o = K,,Re~"
0 < n < I
~I 7)
where Rep is the Reynolds number of the particle related to that of the fluid by , !~)
Rep = R e . ( d / 2 a ) . ( W . . / U , , , , ) R e = U,,,,.d;t).
The value of the exponent n = 1 for Stokes law regine, n = 0 for Newtons law regime and for the transition regime, INGEBO (1956) gives n = 0.84. The parametric constant K can be conveniently defined to include the effects due to the presence of other particles. TURBULENCE ENERGY BALANCE AND KOLMOGOROFF SCHEME The turbulent energy balance is constructed by the conventional technique (HINzE. 1959) along the mean motion, Urn, defined by the mass conservation equation, =~-(p,,, u,,,,,? = =- p ( U j + o ~ . ~ )
c.\-j
ux i
= o.
~i9)
As usual, triple correlations involving concentration fluctuations and simultaneous fluctuations of both particle and fluid velocities, are neglected. The scalar equation {summation indicated by repeated indices) after rearrangement reads
8-v'il C'iTwhere
.l/j_?. i = &\./ Dj+sx;~(Umir;j)-~,-B-P-
Q
,20)
OJ = -- [u'u" u' + °J'''ju'u' + ep ( ' ' +°~''')
B = O)'tl{m)i(gi--
fi)
~3U~ (1 +oY)P = ( u r % + c o ' u j V,,.)=- .... ,(~Xj
.Xj
We note that the structure of the equation is identical with that of a clear fluid and almost all of its terms are analogous to those of a pure fluid. The left-hand side represents, in the order of their appearance, convection of turbulent kinetic energy by mean motion and the terms on the right denote the diffusion of kinetic energy, work done by turbulent viscous stresses, dissipation of turbulent kinetic energy, work done by external body forces in suspending the solids and the production of turbulent energy from mean velocity gradients by mixing stresses. In the absence of axial velocity gradients the transport of kinetic energy by convection, advection and diffusion is negligible compared to its production and dissipation. The gravity term which could be very significant even for small concentrations, vanishe.~ due to the assumption of axisymmetry. The viscous stresses being small compared
On the turbulent pipe flow of gas-solids suspensions
161
to Reynolds stresses, we may disregard the work done by fluctuating viscous stresses. Thus we are left with only the terms for production and dissipation which are in near balance everywhere. Further we need to retain only those terms involving the velocity components in the flow direction in the absence of any appreciable flow in the radial direction. The final form of the equation after rearrangement, reads
For the determination of the coefficient ~p, ~¢, ~, of turbulent exchange and of the dissipation e, of turbulent energy, we propose to extend the KOLMOGOROFF (1942) scheme for pure fluids to the present situation. To pave the way we shall first consider the case of suspension flow with identical particles of size d uniformly distributed over the cross section of flow. For this "'mean" flow (denoted by subscript m) OUJc3r = c3V:/c3r and the ratio of eddy diffusivities,
(Pv C vr v.lp u,u:),,
= 3
(22a)
is a constant for the flow, being dependent only on the particle size and density. From the equation (13) for the motion of suspension we obtain for the regions away from the pipe wail: s:(1 +,8)
~U.
. - = --.
Cr
i" a
u * . 2,
(226)
which is a linear distribution of shear stress as for a pure fluid. The turbulence energy balance (21) reduces to Z i Z j p - 1 "C~jO(U i + O)Vi)l'OX J = . (1
+ e),~ r) 0 .U= .
(23)
UF
The dissipation term on the left shows that the effect due to the presence of solids as being proportional to their concentration, which is in agreement with LEvlcn's (1967) findings from his studies on the behaviour of suspended particles in isotropic turbulent fluid. The assumed uniformity of concentration distribution implies no mean motion in the radial direction so that the expression for the turbulent kinetic energy (20) becomes q,, = ½ Y i u ~
1+o~,,---- / . II l Ili/I
We observe that the total kinetic energy receives a contribution from the dispersed phase in proportion to their concentration to off-set the reduction in the turbulence level of the pure fluid, again in proportion to concentration, caused by the inertial mass centres of the particles with imperfect response to energy containing eddies. Thus we may possibly conclude that the total kinetic energy 02 is not very different from e = ½/7d7i of the pure fluid. In the absence of strong radial accelerations like gravity we may suppose that turbulent quantities of the suspension flow can still be described by e. For dilute suspensions the length scale of turbulence, l, is little affected by the presence of solids for the same reason. Following Kolmogoroff hypothesis and employing e and I for velocity and length scale respectively, we will assume:
162
M. NAGARAJAN 8t = l e ~
8p = % l e ~
e~ = ~ , l e ~
and
ec = ~ c l e ~
~, = z~e;l -~
!24)
where the ~'s are non-dimensional constant parmeaters. We note in passing that the various constants in the above scheme can be considered universal only in a restricted sense. Employing these newly defined quantities we now give the equations governing the motion of suspension in the non-dimensional form. For want of suitable parameters, all lengths and velocities have been non-dimensionalised respectively by a, the pipe radius and u*, the friction velocity for the flow. In what follows, prime denotes differentiation with respect to r, the non-dimensional radius. (25)
le~(U' +urtp V') = r
d
" e ~V )' = rF.. d- -p ( r . %
(26)
e = (IU'/.=,) 2.(1 + R i )
!27)
R i = (U') - 2. [(~c U' + as coV') Vco'+ (ac + coa,)Eco' + %(0~ V') 2] .
i 2S )
The last written equation defines the parameter R i an analogue of Richardson number tbr density stratified flows, reprc'~enting the added effects due to solids. As has already been pointed out, R i is proportional to the concentration, whence we have Ri,, = %eo~
and
Ri = (%co,,).to .
~29)
From particle diffusivity measurements made by See (1960) we infer that ~, ~ 10--' and we can safely neglect the product epco in the equation (25) thus effecting a considerable simplification o f the equations (27) and (26) which become respectively, I"
e = -il+Ri) ~
~30)
dr---5 (r2co V'/U') = CO(W~W.~)2 - ~
( 31 )
and d
where the m ~ n slip velocity IV,. has been obtained from (15) and (16) using the mean flow conditions (22), as
__W" = C 5.3~, v.o
p, Rd' (u',. yl.
LF~:~-2~. ~
@-2/a
+,,, .. , ~ \~/
'
~;:>
Equating (28) and (29) we obtain the equation for determining the concentration profile for the cases e~ -----% and ac ,~ e, respectively,
(~,/co.~,).(v/w).[u'+(l+co)}e/vlco' w' (-0-r-+vV)"
J~- = W
(~,/~,). (v! w). L(-6T_~]/-~j~ - w
(33a) ~33b)
While deriving the above equation we have made use of the approximation o~,.U'-~,) V ' : co,,W', since co _ co,. and U' ~ 1 in the turbulent core. The equations (31) and(33) together with the boundary conditions for the flow solve for the concentration and slip velocity
On the turbulent pipe flow of gas-solids suspensions
163
profiles in terms of the fluid velocity U(r) and the diffusivity ratios % and :~,. The ratio % may be obtained from a knowledge of shear stress distribution (25) while particle diffusivity measurements are required to evaluate %. DISCUSSION The equations (33a and b) for the concentration profile merit consideration. The expression on the left hand side has two large factors (V/W) and (~s/co=%). The latter especially, is very large for flows with not too high a concentration. We also find that the field forces do not make any significant contribution anywhere excepting the regions close to the wall, as indicated by the large dividing factor, V. The result is a concentration profile very much flatter than the slip velocity profile. We find experimental corroboration in the measurements taken by REDDY and PEI (1969) and Soo (1962). The later experimenters, notably BOYCE (1970) have taken uniform distribution of concentration for granted and the equations (33a and b) provide a rational basis for such an assumption. By making use of this result we can simplify the equation (31) for slip velocity profile to read
,1 / :dv'~
(A)
=
An approximate solution of this equation for the case n = 1 drag and W = 14Ioat U = Uo is W = W,,,+(Wo- W~,).exp [ ( U o -
viz Stokes law for particle
U)/14~]
(B)
o E
o.
= to
/
n = I0
----o--
Eauation
-----
Exptl.---dala, Boyce (1970)
.(22)
'
--
10
I0
Porticle
size,
I0
~m
F1G. I. Slip velocity with varying particle sizes for two-phase flow through circular pipe. BOYCE(] 970). Reynolds number
0"9
Pipe diameter Particle material
2"75 in. Silicadust (2-60 pro) Glass beads (100, 200, 840 t,m)
x
104 to 6"3 × 104
164
M. NAGAP.AJAN
Evidently the slip velocity is strongly dependent on 14/,,and gives a steq~r profile for smaller values of W,,. The slip velocity profiles measured by Soo (1962) and RF.~DY and PEI (1969) provide experimental verification. The slip velocity profiles obtained by them are reproduced in Figs. 2 and 3. The magnitude of the mean slip velocity may be taken as being nearly the same as its value at the centre of a large pipe where the concentration distribution is uniform over most part of the turbulent core. Particle velocity measurements made by BOYCE(1970) I'0
.
.
.
.
.
.
.
.
.
II
0.8 velocity ,~
/ /
::~ 0 . 6
/
o
// /
-x 0.4
/
/
/
//---"Solid phase velocity
/ W m ~ I 0 z, Uo
0.2
o
0.5
I
1.5
~adius,
2
25
in.
FIo. 2. Velocity distribution for air and particles. Data of Soo et al. (1964). Particles 2-75 lb MgO, 35 ~m nominal. zo
!
Air ~elocity profile
esentati~e ~artiele velocity prohle
Oisfarlcg I*r¢~l wall. Y c m
FIo. 3. Particle velocity obtained by double-flash storoo-photography. REDDYand PEt (t969).
Reynolds number Particle size Loading ratio
55000 200/.~m 0.02
over a wide range of particle size and flow Reynolds numbers reveal that the mean slip velocity is least affected by flow Reynolds number or concentration and is related mainly to particle size. An inspection of the expression (32) for mean slip velocity points to the same for the terms within the square bracket is almost independent of the concentration
On the turbulent pipc flow of gas-solids suspensions
165
and only weakly dependent on Reynolds n u m b e r since u*,. ~ R e -~" (Blasius Law). Finally, a justification for the present macroscopic a p p r o a c h to the problem m a y be found in BOYCE'S (ibid 1970) excellent correlation o f pressure drop data for gas-solids suspension flows by defining the flow Reynolds n u m b e r as if the two-phase system were a h o m o g e n e o u s fluid.
REFERENCES BORSHC~EVS~, I. YU. T. (1964) Dust Cooling Conference, London. BoYcE, M. P. (19"/0) J. Bas. Engng, Series D, 495. BUEVICH,IU. A. (1968) P M M 32, (1), 95. HINZE, J. O. (1959) Turbulence, McGraw Hill, New York. INGEBO,R. (1956) N.A.C.A. Tech. Note No. 3762. LEVlCH, V. G. (196"/) Doklady 12, (6), 546. NAGAR~OAN,M. (19"/1)`/. Aerosol Sci. 2, 1.5. REDDY, K. V. S. and PEI, D. C. T. (1969) Ind. Engng. Chert1. Fundls. Edn. 8, (3), 490. Soo, S. L. (1960),/. Bas. Engng, Series D, 82, 609. Soo, S. L. (1962) Ind. Engng Chem. Fundls. Edn., 1 (1), 33. Soo, S. L. (1967) Fhtid Dynamics of Multiphase systems, Blaisdelt. VASlL~V, O. P. (1969) 13th Congr. LA.H.R., Tokyo. WAKSTE~,C. (1966) Motion of small particles suspended in turbulent airflow in a vertical pipe. Penn. Univ WAKs'~rEIN,C. (19"/0)3". Aerosol Sei. 1, 69.