- Email: [email protected]
Estimation of the maximum stable drop sizes, coalescence frequencies and the size distributions in isotropic turbulent dispersions
Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295 www.elsevier.com/locate/colsurfa
Estimation of the maximum stable drop sizes, coalescence frequencies and the size distributions in isotropic turbulent dispersions Selim Ceylan, Gudret Kelbaliyev, Kadim Ceylan Department of Chemical Engineering, Inonu University, 44069 Malatya, Turkey Received 13 February 2002; accepted 3 July 2002
Abstract The process of coalescence or breakup of drops in turbulent flow is of importance in many technical applications. A new size distribution takes place due to the coalescence or the breakup of the drops during the motion of a dispersed system. Based on the experimental data given in the literature, some new empirical relationships are developed in this paper to evaluate the maximum stable drop sizes, the coalescence frequencies and the drop size distribution in an isotropic turbulent flow. The relationships are developed essentially in terms of the particle Reynolds number or of the physical properties of the system. The Focker /Planck equation is used to estimate the particle size distribution. The model predictions are compared with the experimental data given in the literature. The results indicated that the predicted values and the experimental data are in a good agreement. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Coalescence; Drop; Dispersions; Size distribution
1. Introduction Coalescence and breakup phenomena of drops in isotropic turbulent flow are essential for many chemical or physical process applications such as multi-phase reactions, liquid/liquid extraction, distillation, gas absorption, emulsion and suspension polymerization etc. The tendency of a drop
Corresponding author. Tel.: /90-422-341-0039; fax: /90422-341-0046 E-mail address: [email protected] (K. Ceylan).
for the coalescence or the breakup is essentially determined by its size. Numerous studies on the subject are given in the literature and they show in general that the size of the drops (a), the physical properties of the system and hydrodynamical properties of the flow are the main parameters that affect these phenomena [1 /12]. The first step in the coalescence of a drop is to approach to a close proximity of another drop. Once the drop collide with the other, a liquid film, or a lamella is formed between the two-neighboring drops and the contact lasts some time (contact time). Generally, the drops with a size comparable
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 3 2 6 - 6
286
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
to the length scale of turbulence (l ) may collide [1]. The collision occur, in general, when l /l0 which is observed in the fully developed turbulent flow, where l0 is the Kolmogoroff scale of turbulence and defined as l0 /(n3c /o R )1/4, nc is the kinematic viscosity of the medium, o R is the dissipation energy per unit of mass. Majority of particle transportation essentially takes place due to the turbulent diffusion in the turbulent flow. In the case of viscous liquids or in viscous sublayer, on the other hand, l B/l0 and molecular diffusion is dominant. The liquid film separating the collided drops resists the relative motion of the pair of the drops. During this stage, the forces from the contiguous turbulent flow can separate the drops; consequently not all the collisions will lead to coalescence. The second step in the coalescence is the drainage of the liquid film between the drop pairs [2,4,6]. If the drops are not separated away, the coalescence event occur when the liquid film drained and ruptured so that the two drops become one. The amount of the unified drops increase in general with the increasing collision frequency [6,9,12]. The minimum and maximum stable drop sizes (amin and amax) are the important parameters that closely affect the coalescence and breakup phenomena. The drops with a size larger than a characteristic maximum stable drop size (aS), have a tendency to breakup, but smaller drops show a tendency to coalesce. Several approximation formulas are given in the literature for estimation of the maximum stable drop size in dispersed systems. Most of these models are based on the model of Hinze [7,13]. One of the widely used expressions for the maximum stable drop size in a dilute dispersion is [10], 0:6 s aS 0:725 o R 0:4 rc where s is surface tension drop and rc is density of the dispersing medium. A correlation of the stable drop size in terms of Weber number is given as [14]: Wec 0:6 s0:6 aS o 0:4 (1a) R 2 (r2c rd )0:2 where Wec, is critical Weber number and rd is
density of the drop liquid. In the case of the flow of the dispersed system, it is suggested [15]:
aS (Wec )0:6
s0:6 0:2 0:6 0:2 (r0:3 c rd hc )
D1:5 U 1:1
(1b)
where hc is viscosity of the medium, D is the diameter of channel, U is the average flow velocity of the dispersed system. Similarly, considering the properties both the medium and drop, the following approximation formula is proposed [16]: aS rc U 2 hc U 1=2 h U 0:7 38 10:7 d (2) s s s
where hd is the viscosity of the drop liquid. Various other models have been proposed taking into account the effect of the dispersed phase properties on the maximum stable drop sizes [11,17,18]. Above given equations and several others proposed in the literature, suggest that the maximum stable drop size depends essentially on the properties of the drop and the dispersing medium and on the velocity of the flow. The size distribution in a dispersed system has a significant effect on equipment performance. Most of the studies regarding size distribution, in the literature, are for air /water systems, but the studies regarding two liquids are rather limited. Some of these studies are summarized recently by Simmons and Azzopardi [12]. In this paper, some new empirical relationships are developed to estimate the maximum stable sizes, the collision frequencies and the particle size distribution in the isotropic turbulent flow of a dispersed system. The relationships are developed in terms of the particle Reynolds number and of the physical properties of the system. The solutions of the Focker/Planck equation are used to estimate the size distribution. The Marquardt algorithm is used for the approximation of the coefficients of the proposed relationships. The validity of the developed relationships is confirmed with experimental data from the literature.
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
287
2. Development of empirical relationships for estimation of the maximum staple drop size, collision frequency and the particle size distribution in the turbulent flow of dispersed systems
may be inserted into Eq. (5) to obtain: 1=2 6=7 U s 4=7 oR aS 0:619g 12=7 1=3 2=3 nc rc rd
2.1. The maximum stable drop sizes in the coalescence processes
For a very low Rea range, i.e. Rea B/1, CD may be estimated as CD /24/Rea [22 /24]. This statement may be inserted into Eq. (5) to obtain: 1:5 U s aS 0:192g 3 o 1 (8) R 1=3 2=3 nc rc rd
The theoretical approach to estimate stable drop sizes is essentially based on the force balance between the surface energies and the dynamic pressure losses inside the drops [1,11]. Based on the literature data, it may be written: CD
rc V 2 4s aS 2
(3)
where CD is the drag coefficient. An internal circulation occurs during the motion of drops and this phenomenon affect the drag factor. A high scale of turbulence pulsation may not be effective but the small fluctuation velocities (V ) are effective on the coalescence of drops. The fluctuation velocity may be stated as [1], 1=3 rd V g oR l (4) rc where g is an empirical constant. For a stable large drop, together with the assumption of l :/aS, Eq. (3) may be rearranged to obtain the maximum stable drop size: 0:6 0:6 8 s aS g 1:2 o 0:4 (5) R 1=3 2=3 CD rc rd Since the value of CD depends essentially on the particle Reynolds number, Rea , (Rea /aU /nc), Eq. (5) may be modified depending on the ranges of Rea . In the range of 2 /103 5/Rea 5/5 /105, the drag factor for the drops or bubbles may be assumed constant as CD :/8/3 [19], then Eq. (5) is simplifies to: 0:6 s aS 1:93g 1:2 1=3 2=3 o 0:4 (6) R rc rd In the range of 2 5/Rea 5/103, CD may be estimated as CD /14/Re0.5 a [20,21]. This statement
(7)
Above equations are useful to estimate the maximum stable sizes of the drops under different conditions. As an example, a comparison of the predictions from Eq. (7) with the experimental data from literature [15] is given in Table 1. It is seen that the predicted values are in a very good agreement with the experimental data. 2.2. Collision frequencies of drops in turbulent flow Coalescence is indeed a very complex phenomenon that involves many interacting subprocesses. Several models are proposed in the literature to estimate the coalescence rate in turbulent dispersions. A detailed discussion of these models is given by Tobin et al. [7]. It is generally accepted that coalescence phenomena occur when a B/aS and it is essentially controlled by the collision frequency. For the collision of a drop to another one, the drops must be transported by some ways, such as molecular or turbulent diffusion, to the proximity of another drop. In order to visualize the collision phenomena; let us consider an imaginary sphere, which may be called as collision field, with a radius of R :/a as represented in Fig. 1. All the drops passing through this field may collide and have the tendency of coalescence. Therefore, the collision frequency is proportional with the surface of the field and with the flux of the drop, i.e. v 4pR2 DTP
@N @r
j
(9a)
r R
where v is the collision frequency, DTP is the eddy diffusion coefficient of drops, and N is the number of the drops per unit of volume, r is the variable radius. DTP may be estimated as [25],
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
288
Table 1 A comparison of the predicted maximum stable drop sizes with the experimental data from the literature (the experimental data from [15]) Experiment number
U (m/s)
Rea
o R (W/kg)
aS (m /103)
aS (Eq. (6))a (m /103)
1 2 3 4
3.50 3.98 4.59 6.04
1000 820 670 604
6.5 13.3 32.6 69.3
4.3 3.1 2.2 1.5
4.5 3.1 2.0 1.5
a
nc /10 6 m2/s, s /72 /10 3 N/m, rc /1.29 kg/m3, rd /1000 kg/m3, g /10.6.
two cases and it may be represented as [1]:
Fig. 1. Schematic representation of the diffusion a droplet into the collision field to collide.
DTP mR DT ;
(9b)
where DT is the turbulent diffusion coefficient of the medium, mR is the transport degree of particles by turbulent flow so that 0B/mR 5/1. The number of drops that passes through this field is also a function of the time, therefore, it may be represented by, @N 1 @ @N r2 DTP (10) @t r2 @r @r where, t is time. The boundary conditions for the solution of this equation are: t 0; rR; then N :N0 t0; r R; then N 0 t0; r 0 ; then N N0
(11a) (11b) (11c)
where N0 is number of drops per unit of volume at the beginning. The turbulent diffusion coefficient of the medium is related to the characteristics of flow and therefore, it may be stated in terms of the micro scale of the turbulence. As mentioned above, l /l0, is for the fully developed turbulent flow, but l B/l0 is for the viscous flow. The dependence of the diffusion coefficient on the micro scale of the turbulence is different for the
for ll0 ;
then
for lBl0 ;
then
DT a(o R l)1=3 l 1=2 o DT a R l2 nc
(12a) (12b)
where a , is an empirical constant. Since the effect of the system parameters on DT are not the same, then the solution of Eq. (10) is different for the two cases. In general, the scale of the turbulence may be assumed as l :/r for the both cases. Then, Eq. (10) may be rewritten after substitution 12a into it: @N a @ @N 1=3 mR o R r10=3 (13) @t r2 @r @r Considering the boundary conditions given in Eqs. (11a), (11b) and (11c), a solution of Eq. (13) may be given as: 1=3 X r 2 N(r; t) Cn J2 mn (14) e mn t R n 1 where Cn is the coefficients of the series and given as: R
g N J (m (r=R) 2 0 2
Cn
n
1=3
)r dr
0
R2
J12 (mn )
where J1 and J2 are the first and second order Bessel functions, respectively; mn is the eigenvalues and defined as: 1=3 o m a 1=2 mn qn R R 3R2=3 where qn are the constants and may be determined from the solution of the equation of J2(qn ) /0.
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
The roots for this equation are: q1 /5.05, q2 / 8.45, q3 /11.8, etc. These q values indicate that Eq. (14) is a rapidly converging series, and therefore, using the fist term is satisfactory for the practical purposes. The first terms for the eigenvalues and the coefficients are; 1=3 o R mR a 1=2 m1 5:05 (15a) 3R2=3 and C1
2N0 J3 (qn )
(15b)
qn J12 (qn )
Using the properties of the Bessel functions and the boundary conditions given above, it may be written that J3(qn ) / /J1(qn ), J1(5.05) / /033, and therefore, it may be evaluated that C1 / 1.2N0. Hence, (Eq. (14)) may be written in a simplified form as; 1=3 1=3 r o exp 8:5 R N(t; r) 1:2N0 J2 5:05 mR at R R2 @N @r
j
0:67N0 R 1 r R
1=3 o exp 8:5 R mR at R2
(16)
Inserting this statement into Eqs. (9a) and (9b) together with 12a and with an assumption that the drops are equal-sized with R /a , gives: 1=3 1=3 o o v 16mR a R 8 0 exp 8:5 R mR at a2 a2 (17a) where 80 is volumetric fraction of the drops, 80 / N0pa3/6. This equation may be modified by replacing t for t, 1=3 o t v 16mR 8 0 a R (17b) exp T1 a2 where t is the coalescence time, T1 is a parameter defined as T1 /0.117(a2/o R )1/3/mR a . The exponential term in (17b) is similar to those given in the literature [10] for the coalescence efficiency (gE) as gE :/exp( /t/T ). Therefore, a comparison with literature data suggests that above defined T1 may be related to the contact time. The collision frequency at the beginning may be estimated from
(17a) using t /0: 1=3 o v 168 0 mR a R a2
289
(17c)
A similar equation is given in the literature [26] and a comparison suggests that mR a /0.72. Eqs. (17a) and (17b) suggest that the collision frequency, v , is independent of viscosity of the medium in the fully developed turbulent flow (l /l0). The substitution of 12b into Eq. (10) enables to investigate the situation for the viscous flow, l B/ l0. In this case, the solution of Eq. (10) is simpler and may be given as: 3 1=2 R o N(r; t) N0 1 t (18) exp a R r nc Therefore, it may be written for r /R ; 1=2 @N o 3N0 R 1 exp a R t @r r R nc
j
(19)
Using the same procedure given above, it may be obtained that the collision frequency in Eqs. (9a) and (9b) may be written as, 1=2 1=2 o o v 728 0 amR R exp a R t (20a) nc nc For viscous flow it may be assumed that mR :/1. As discussed above, (20a) may be modified by replacing t for t , 1=2 o t v 728 0 a R exp (20b) T2 nc where T2 is a parameter defined as T2 /1/a (nc/ o R )1/2. A comparison with literature data suggests that above defined T2 may be related to the contact time for the viscous systems and it is different from that for the turbulent case. The collision frequency at the beginning may be estimated from 20a using t /0: 1=2 o v 728 0 a R (20c) nc This equation suggests that, in the viscous flow, the collision frequency is inversely proportional with the kinematic viscosity of the medium, i.e.
290
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
v /nc 1/2. For the collision frequency of differentsized drops, the following equation is given in the literature [5], 1=2 o v 1:294N0 (ai aj )3 R (21) nc The above results indicate that the collision frequency is directionally proportional with the concentration of drops for the both cases. For the equal sized drops, Eq. (21) is identical with 20c for the value of a /0.275.
roughness on the stable drop sizes, collision frequencies and coalescence efficiency. A comparison of Eqs. (6) /(8) with Eq. (24) suggests that the pipe roughness affects the maximum stable drop sizes. Depending on the ranges of particle Reynolds number, this effect may be summarized as follows: 2 103 5Rea 52 105 0:4 D 0:15 aS j U3 25Rea 5103
2.3. Effect of pipe roughness on the maximum stable sizes and on the coalescence of drops
oR
f (j)U 3 2D
(22)
where f(j) is the friction factor function, j is the relative pipe roughness defined as j /o /D , o is the size of the roughness. Since the roughness affects the dissipation energy, above equations suggest that the roughness also affects stable drop sizes, collision frequencies and coalescence efficiency. In general, the dependence of the friction factor, f, on Re and on the relative pipe roughness may be stated as, f c(Re; j)
(23)
where Re is Reynolds number for the flow in pipe and defined as Re /UD /nc. Several empirical equations are given in the literature [22 /25] to represent the Eq. (23). Based on the literature data [25] and using Eq. (22) it may be written for fully turbulent flow that, oR
U3 2D
c(j3=8 )
aS j 0:22
Rea 5 1 aS j
The dispersed systems are generally transported through circular pipes most of which includes roughness to some extent. The pipe roughness affects all the transport phenomena such as the heat, mass or momentum transfer. Similarly, the roughness also affects the dissipation energy as shown below [15]:
(25a)
3=8
D U3
D
4=7
U3
(25b) (25c)
Eqs. (25a), (25b) and (25c) suggest that aS / j k where k is an empirical constant. In other words, the maximum stable sizes of the drops decrease with increasing roughness. Similarly, the effects of the pipe roughness on the collision frequency may qualitatively be estimated by a comparison of Eqs. (17a), (17b), (20a), (20b) and (20c) with Eq. (24). The results suggest that: U ll0 ; then v j1=8 (26a) D1=3 3 1=2 U lBl0 ; then v j3=16 (26b) D Eqs. (26a) and (26b) shows that v /jk . In other words, the collision frequencies of the drops increase with increasing pipe roughness. An increase in the collision frequency result in increase in the coalescence rate. It is suggested in the literature [10] that coales0.89 cence efficiency gE /e oR . Using this relationship and 23d it may be written that ln gE / /j1/3 This relation suggest that the coalescence efficiency decreases with increasing roughness. 2.4. Modeling of the drop size distribution in turbulent dispersions
(24)
Eq. (24) enables to estimate the effect of the
After starting the motion, a new size distribution takes place in a dispersed system due to the
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
coalescence and breakup of the drops. Prediction of drop size distribution in such systems requires use of a distribution function. In the size distribution, generally fraction of drops having certain sizes, or the cumulative volume fraction, or the number density (concentration of the drops having certain volume) is investigated. The Rosin /Ramler distribution or upper limit lognormal distribution (ULLN) functions are used by several researchers to represent this distribution [12,16]. Recently, Ruiz et al. [27] showed that the drop size distribution in a stirred tank has a bell shaped form and it may be represented by the lognormal distribution function. In this study, the size distribution in the turbulent dispersion is evaluated by using the Focker /Planck equation. Accordingly, a size-time dependent distribution function may be represented as [28,29], @P(a; t) @ @ [f (a)P(a; t)] @t @a @a @P(a; t) B(a; t) @a
(27)
P (a ,t) is the distribution function of the drops depending on size and time, f(a) is the size variation function, B (a ,t ) is the stochastic diffusion coefficient. The variation the mass of a drop, due to the coalescence or the breakup, with time may be stated as, dm dt
4pR2 DTP
@N @r
j
dt
1 3
(17a), (17b), (20a), (20b) and (20c) the definition the f(a ) also varies depending on the frequency statement. As an example, the solution is discussed below for the turbulent case, i.e. l /l0. A comparison of Eq. (28c) with (17c) indicates n 1 that kR /(16/3)80mR a (o 1/3 ). If it is assumed R /aS as B (a,t) /B is a constant, then it may be possible to obtain various solutions for Eq. (27) by inserting 28c into it. In fact, any value may be chosen for the rate order, n , in 28c, but, the analytical solutions may be obtainable for only certain values of n. Some typical solutions for the selected n values are discussed below: a) If n /2/3, then from 28c f(a ) /kR , then a solution to Eq. (27) may be given as, P(a; t) P0 (a)e
k2R t=B
a p ffiffiffiffiffiffiffi ffi erf 4Bt
(29)
where P0(a ) /P (a ,0). b) If n /5/3, then from 28c f(a) /kR a . In this case, the process is dependent on particle sizes, therefore, the maximum stable sizes of the drops should be taken into account. In the case of small drops, i.e. a B/aS, the coalescence is the governing phenomena and it may be convenient to define the function of f (a ) as f(a) /kR (aS /a). Then the solution to Eq. (27) may be given as, 2
m vm
(28a)
r R
where m is the instantaneous mass of a drop and may be estimated as m /prda3/6. Then Eq. (28a) may be modified as: da
291
va f (a)
(28b)
For the nth order coalescence kinetics, the above equation may be rewritten as: da 1 an v n 1 kR an 2=3 f (a) dt 3 aS
(28c)
where kR is the rate constant of the coalescence. Since the collision frequency statement vary depending on the micro scale of the turbulence Eqs.
P(a; t) (aS a) 1=2 e kR (aS a) =4B X m2n kR t Cn e c m2n n
1 1 k (a a)2 ; ; R S 4 4 4B
(30)
where mn is an eigenvalue and Cn is the coefficients of the series, c [m2n /1/4; 1/ 4,kR (aS /a )2/4B ] is a hypergeometric function. In the case of large drops, i.e. a/aS, the break-up is the governing phenomena and it may be convenient to define the function of f(a) as f(a ) /Kp(a /aS), where Kp is the rate constant of the breakup. Under these conditions, the solution of Eq. (27) may be given as,
292
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295 2
P(a; t) e kR (a aS ) =4B X Cn e nKp t Hn (aS a) n
sffiffiffiffiffiffi Kp
(31)
4B
where Hn is the Hermite function. The coefficient of Cn is given as, sffiffiffiffiffiffi 1 Kp da P0 (a)Hn (aS a) Cn pffiffiffi 2n n! p 4B
g
c)
If n / /1/3, then from 28c f (a ) /kR /a . If the coalescence and breakup process occurs simultaneously it may be convenient to defined the function of f(a ) as f(a) /kR /a /Kpa . In this case, a solution of Eq. (27) may be given as, 2
2
P(b; t) bu e (Kp aS =2B)b X Kp a2S 2 2Kp nt b e Cn Lwn 2B n
(32)
where Cn is
u
u1
Cn y
g P (b)L ((K a =2B)b )db 0
0 (u1) (u 1) 2 2
2
w n
2 p S
2
G(n (u 1)=2)n!
where b /a /aS, u /kR /B, y /kR /Kpa2S, w / (u /1)/2, Lw n (b ) is the Laguerre function, G (n ) is the Gamma function. For a special case, if t 0/ in Eq. (32) then the solution assumes a simple form as: k a2 P (b) C0 bu exp R S b2 (33) 2B
where C0 /2(u /2y)((u1)/2). Eq. (33) suggests that the final size distribution (at t 0/) is not related to the beginning distribution. For a steady-state regime, Eq. (33) is similar to the Rosen /Ramler distribution function. However, for u /1 it is similar to the Rayleigh distribution, and for u /
2 it is similar to the Maxwell distribution function. The shape of the distribution function describes the extent of the heterogeneity of the particle size in the distribution. The average drop size at t may be estimated as: sffiffiffiffiffiffi 2B k (34) G 1 R a aP (a)da a Kp 2B
g 0
3. Results and discussion The literature data and above-developed mathematical statements indicate that the maximum stable drop sizes in an isotropic turbulent flow depends essentially on the particle Reynolds number, the collision frequencies and on the scale of the turbulence. The collision frequency increases with the increasing drop concentration and dissipation energies. The results suggest that if l /l0, the collision frequency is not affected from the viscosity of the medium but it is affected if l B/l0. The pipe roughness also affects the maximum stable sizes, the collision frequencies and the collision efficiency. The particle size distribution in a dispersed system, at any time, is different from the starting distribution due to the coalescence and breakup phenomena. Both the distribution functions and most of the experimental data indicates that the particle size distribution is unimodal (i.e. have one maximum in the distribution). However, some experimental data indicate that the particle size distribution may be multimodal (i.e. two or more maximums in the size distribution). For example the data of Tobin et al. [9] for the system of benzene /carbontetrachloride (5%) in water indicate that the particle size distribution is unimodal. However, the data for the system of hexane / carbontetrachloride (5%) in water show bimodal distribution. Most of the distribution functions given used in the literature are not able to represent the multimodal size distribution. It is shown here that the solution of the Focker/Planck equation also may be used for estimation of the unimodal or multimodal size distribution. The
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
typical solutions for this equation are given by Eqs. (29) /(33). These solutions may be simplified considering the characteristics of the system. For example, the solution given by Eq. (33) may be simplified as, C2 2 (35a) a f2 P(a; t) C1 a exp f1 where C1 and C2 are constants, f1 and f2 are time functions. The constants in this equation may be determined from experimental data. For this purpose, the data of Tobin et al [9] for the system of benzene /carbontetrachloride (5%) in water are used. The obtained equation is, 0:0395 P(a; t) 0:0085a2 exp 0:2 a f (35b) f where f is a time function defined as f /1/ 0.1t0.5, t is the time as minute. A plot of Eqs. (35a) and (35b) is given in Fig. 2 for t /0 and for t /30 min. Some experimental data from the literature [9] are also represented in the figure for comparison. The results indicate that Eqs. (35a) and (35b) is well suited with the data for the unimodal distribution. However, this equation does not represent data for the system of hexane/carbontetrachloride system, because this system shows a bimodal distribution. Based on the experimental result, Tobin et al. [9] commented
293
that coalescence of the smallest drops is very small, as consequence of which such droplets linger for long times, leading to bimodality. However, they did not give the physical reasons for that why the coalescence of these smallest drops is small. Indeed, it is expected that the probability of collision of the small drops is higher since the diffusion rates are higher. Therefore, the explanation of the bimodality, in terms of collision frequencies may not be satisfactory. We suggest here that the interaction energies between the drops may be effective in the coalescence and breakup phenomena and therefore, in the shape of the distribution function. At very small concentrations (8 /1) the interaction between the drops may be negligible since the drops are far away from each other. The probability of the collision of the drops under these conditions is very low and the coalescence may not occur. If the drop concentration is appreciable (i.e. 8 /0.3) then the interaction between the drops may be appreciable and the effectiveness of collision may be affected from this. The total interaction energy (UT) may be stated as the sum of the attraction energy (Ua) and the repulsion energy (Ur). Assuming that all the drops diffused into the collision field hit a drop, then the statement of collision frequency given in Eqs. (9a) and (9b) may be modified as: v 4pR2 DTP
@N @r
j
h
@UT
r R
@r
j
(36) rBR
where h is a parameter depending drop size and the diffusion resistance. For the small drops (Stokes’ drops) h is defined as h /(4pr2N )/ (6phca ). The solution of Eq. (27), using Eq. (36), may be given as: P(a; t) e (hUT (a))=B X 2 e mn Bt C1n n
C2n Fig. 2. Unimodal size distribution according to Eqs. (35a) and (35b). Experimental data are from Tobin et al. [9]. ': t /0 min, m: /30 min.
h
g exp B
@ 2 UT (a) UT (a) m2n da @a2
(37)
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
294
Considering the characteristics of this series, Eq. (37), may be simplified as, C h P(a; t) C1 a2 exp 2 a f2 UT (38) B f1 The coefficients of this equation should be determined from the experimental data. The attraction and the repulsion energies of the drops may be represented as [2]; hUa B
A
(39a)
a2
hUr X ( 1)n An a2n exp( mn an1 ) B n 0
(39b)
where A , An and mn are the empirical constants. Based on the literature data we suggest here that if Ua /Ur then h 10:8 UT ; (40a) B a2 but if Ua and Ur are comparable and then hUT B
10:8 a2
2
5:55e 0:3a 0:004a2 e 0:0006a
(40b)
Using the data of Tobin et al. [9] for the system of hexane/carbontetrachloride (5%) in water, the approximated equation is: 0:0395 P(a; t) 0:006a2 exp 0:2 a 0:7f f h UT (41) B where f /1/0.25t0.5. The shape of the distribution function defined by Eq. (41) depends on the definition of the interaction function of (h/B )UT. If the function defined by 40a is used, then a unimodal distribution is obtained just similar to that given by 35b. This result suggests that the effect of the attraction on the distribution is negligible. If the function defined by 40b is used, then a bimodal distribution is obtained. A plot of Eq. (41) together with 40b is given in Fig. 3 for t / 1 and t /60 min. Some experimental data from the literature [9] are also represented in the figure for comparison. The results indicate that Eq. (41) a
Fig. 3. Bimodal size distribution according to Eq. (41). Experimental data are from Tobin et al. [9]. ': t /1 min, m: /60 min.
relatively good fit is obtained for the bimodal distribution. A notable feature of the experimental data and above developed relationships is that the peak in the number density do not shifts appreciably with increasing time. The number density lower with time due to the coalescence phenomena. Several of small drops may combine to form one, and therefore, the decrease in the number density of small particles higher than the increase in the density of the large particles. The model prediction is also in accord with this situation since only a slight cross-over is predicted at large size particles.
4. Conclusions Both the theory and experimental data indicate that the maximum stable size, the coalescence frequency, and the drop size distribution in turbulent dispersions are closely related to the concentration of the drops and on the rheologic properties of the medium. The characteristics of turbulence and pipe roughness are also effective on these phenomena. Depending on the properties of the dispersed system, the size distribution in a dispersed system may be unimodal or multimodal. The results suggest that the effective interaction energies among the drops affect the shape of the
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
size distribution. The modified solutions of the Focker/Planck equation may be used to represent these distributions.
References [1] V.G. Levich, Physicochemical Hyrodynamics, PrenticeHall, Englewood Cliffs, Canada, 1962. [2] A. Scheludko, Adv. Colloid Interphase Sci. 1 (1967) 391. [3] C.A. Coulaloglu, L.L. Tavlarides, Chem. Eng. Sci. 32 (1977) 1289. [4] C. Malderlli, R.K. Jain, I.B. Ivanow, E. Ruckenstein, J. Colloid Interphase Sci. 78 (1980) 118. [5] K. Higashitani, K. Yamanchi, Y. Matsuno, J. Chem. Eng. Jpn. 18 (1983) 299. [6] E. Ruckenstein, A. Sharma, J. Colloid Interpahes Sci. 119 (1987) 1. [7] T. Tobin, R. Muralidhar, D. Ramkrishna, Chem. Eng. Sci. 45 (1990) 3491. [8] B. Yang, S. Hartland, Chem. Eng. Sci. 46 (1991) 1437. [9] K. Saneev, R. Kumar, K.S. Ghandi, Chem. Eng. Sci. 18 (1993) 2025. [10] S. Liu, D. Li, Chem. Eng. Sci. 54 (1999) 5667. [11] J. Baldyga, W. Podgorska, Can. J. Chem. Eng. 76 (1998) 456. [12] M.J.H. Simmons, B.J. Azzopardi, Int. J. Multiphase Flow 27 (2001) 843. [13] J.O. Hinze, AIChE J. 1 (1955) 289.
295
[14] R.P. Hesketh, A.W. Etchells, T.W. Russel, AIChE J. 33 (1987) 663. [15] R.P. Hesketh, A.W. Etchells, T.W. Russel, Chem. Eng. Sci. 46 (1991) 1. [16] A. Panagiota, G.F. Hewitt, Chem. Eng. Sci. 55 (2000) 3133. [17] R.V. Calabrese, T.P.K. Chang, P.T. Dang, AIChE J. 32 (Part I) (1986) 657. [18] J.S. Lagissety, P.K. Das, R. Kumar, K.S. Gandhi, Chem. Eng. Sci. 41 (1986) 65. [19] S.L. Soo, Fluid Dynamics of Multiphase Systems, Blaisdell Publishing, London, 1970. [20] P. Reist, Aerosol Science and Technology, McGraw Hill, New York, 1992. [21] F. Raymond, J.M. Rosant, Chem. Eng. Sci. 55 (2000) 943. [22] L.F. Moody, Friction factors for pipe flow, Trans. ASME 66 (1944) 671. [23] R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, second ed., Wiley, New York, 1978. [24] K. Ceylan, A. Altunbas, G. Kelbaliyev, Powder Technology 119 (2001) 175. [25] A. Altunbas, G. Kelbaliyev, K. Ceylan, J. Aerosol Sci. 33 (2002) 1075. [26] G. Kelbaliyev, Theor. Fundam. Chem. Technol. 26 (1992) 390. [27] M.C. Ruiz, P. Lermanda, R. Padilla, Hydrometallurgy 63 (2002) 65. [28] C.W. Gardiner, in: H. Haken (Ed.), Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, vol. 13, Springer, New York, 1985. [29] G. Kelbaliyev, K. Ceylan, Powder Technol. 115 (2001) 84.
Estimation of the maximum stable drop sizes, coalescence frequencies and the size distributions in isotropic turbulent dispersions Selim Ceylan, Gudret Kelbaliyev, Kadim Ceylan Department of Chemical Engineering, Inonu University, 44069 Malatya, Turkey Received 13 February 2002; accepted 3 July 2002
Abstract The process of coalescence or breakup of drops in turbulent flow is of importance in many technical applications. A new size distribution takes place due to the coalescence or the breakup of the drops during the motion of a dispersed system. Based on the experimental data given in the literature, some new empirical relationships are developed in this paper to evaluate the maximum stable drop sizes, the coalescence frequencies and the drop size distribution in an isotropic turbulent flow. The relationships are developed essentially in terms of the particle Reynolds number or of the physical properties of the system. The Focker /Planck equation is used to estimate the particle size distribution. The model predictions are compared with the experimental data given in the literature. The results indicated that the predicted values and the experimental data are in a good agreement. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Coalescence; Drop; Dispersions; Size distribution
1. Introduction Coalescence and breakup phenomena of drops in isotropic turbulent flow are essential for many chemical or physical process applications such as multi-phase reactions, liquid/liquid extraction, distillation, gas absorption, emulsion and suspension polymerization etc. The tendency of a drop
Corresponding author. Tel.: /90-422-341-0039; fax: /90422-341-0046 E-mail address: [email protected] (K. Ceylan).
for the coalescence or the breakup is essentially determined by its size. Numerous studies on the subject are given in the literature and they show in general that the size of the drops (a), the physical properties of the system and hydrodynamical properties of the flow are the main parameters that affect these phenomena [1 /12]. The first step in the coalescence of a drop is to approach to a close proximity of another drop. Once the drop collide with the other, a liquid film, or a lamella is formed between the two-neighboring drops and the contact lasts some time (contact time). Generally, the drops with a size comparable
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 3 2 6 - 6
286
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
to the length scale of turbulence (l ) may collide [1]. The collision occur, in general, when l /l0 which is observed in the fully developed turbulent flow, where l0 is the Kolmogoroff scale of turbulence and defined as l0 /(n3c /o R )1/4, nc is the kinematic viscosity of the medium, o R is the dissipation energy per unit of mass. Majority of particle transportation essentially takes place due to the turbulent diffusion in the turbulent flow. In the case of viscous liquids or in viscous sublayer, on the other hand, l B/l0 and molecular diffusion is dominant. The liquid film separating the collided drops resists the relative motion of the pair of the drops. During this stage, the forces from the contiguous turbulent flow can separate the drops; consequently not all the collisions will lead to coalescence. The second step in the coalescence is the drainage of the liquid film between the drop pairs [2,4,6]. If the drops are not separated away, the coalescence event occur when the liquid film drained and ruptured so that the two drops become one. The amount of the unified drops increase in general with the increasing collision frequency [6,9,12]. The minimum and maximum stable drop sizes (amin and amax) are the important parameters that closely affect the coalescence and breakup phenomena. The drops with a size larger than a characteristic maximum stable drop size (aS), have a tendency to breakup, but smaller drops show a tendency to coalesce. Several approximation formulas are given in the literature for estimation of the maximum stable drop size in dispersed systems. Most of these models are based on the model of Hinze [7,13]. One of the widely used expressions for the maximum stable drop size in a dilute dispersion is [10], 0:6 s aS 0:725 o R 0:4 rc where s is surface tension drop and rc is density of the dispersing medium. A correlation of the stable drop size in terms of Weber number is given as [14]: Wec 0:6 s0:6 aS o 0:4 (1a) R 2 (r2c rd )0:2 where Wec, is critical Weber number and rd is
density of the drop liquid. In the case of the flow of the dispersed system, it is suggested [15]:
aS (Wec )0:6
s0:6 0:2 0:6 0:2 (r0:3 c rd hc )
D1:5 U 1:1
(1b)
where hc is viscosity of the medium, D is the diameter of channel, U is the average flow velocity of the dispersed system. Similarly, considering the properties both the medium and drop, the following approximation formula is proposed [16]: aS rc U 2 hc U 1=2 h U 0:7 38 10:7 d (2) s s s
where hd is the viscosity of the drop liquid. Various other models have been proposed taking into account the effect of the dispersed phase properties on the maximum stable drop sizes [11,17,18]. Above given equations and several others proposed in the literature, suggest that the maximum stable drop size depends essentially on the properties of the drop and the dispersing medium and on the velocity of the flow. The size distribution in a dispersed system has a significant effect on equipment performance. Most of the studies regarding size distribution, in the literature, are for air /water systems, but the studies regarding two liquids are rather limited. Some of these studies are summarized recently by Simmons and Azzopardi [12]. In this paper, some new empirical relationships are developed to estimate the maximum stable sizes, the collision frequencies and the particle size distribution in the isotropic turbulent flow of a dispersed system. The relationships are developed in terms of the particle Reynolds number and of the physical properties of the system. The solutions of the Focker/Planck equation are used to estimate the size distribution. The Marquardt algorithm is used for the approximation of the coefficients of the proposed relationships. The validity of the developed relationships is confirmed with experimental data from the literature.
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
287
2. Development of empirical relationships for estimation of the maximum staple drop size, collision frequency and the particle size distribution in the turbulent flow of dispersed systems
may be inserted into Eq. (5) to obtain: 1=2 6=7 U s 4=7 oR aS 0:619g 12=7 1=3 2=3 nc rc rd
2.1. The maximum stable drop sizes in the coalescence processes
For a very low Rea range, i.e. Rea B/1, CD may be estimated as CD /24/Rea [22 /24]. This statement may be inserted into Eq. (5) to obtain: 1:5 U s aS 0:192g 3 o 1 (8) R 1=3 2=3 nc rc rd
The theoretical approach to estimate stable drop sizes is essentially based on the force balance between the surface energies and the dynamic pressure losses inside the drops [1,11]. Based on the literature data, it may be written: CD
rc V 2 4s aS 2
(3)
where CD is the drag coefficient. An internal circulation occurs during the motion of drops and this phenomenon affect the drag factor. A high scale of turbulence pulsation may not be effective but the small fluctuation velocities (V ) are effective on the coalescence of drops. The fluctuation velocity may be stated as [1], 1=3 rd V g oR l (4) rc where g is an empirical constant. For a stable large drop, together with the assumption of l :/aS, Eq. (3) may be rearranged to obtain the maximum stable drop size: 0:6 0:6 8 s aS g 1:2 o 0:4 (5) R 1=3 2=3 CD rc rd Since the value of CD depends essentially on the particle Reynolds number, Rea , (Rea /aU /nc), Eq. (5) may be modified depending on the ranges of Rea . In the range of 2 /103 5/Rea 5/5 /105, the drag factor for the drops or bubbles may be assumed constant as CD :/8/3 [19], then Eq. (5) is simplifies to: 0:6 s aS 1:93g 1:2 1=3 2=3 o 0:4 (6) R rc rd In the range of 2 5/Rea 5/103, CD may be estimated as CD /14/Re0.5 a [20,21]. This statement
(7)
Above equations are useful to estimate the maximum stable sizes of the drops under different conditions. As an example, a comparison of the predictions from Eq. (7) with the experimental data from literature [15] is given in Table 1. It is seen that the predicted values are in a very good agreement with the experimental data. 2.2. Collision frequencies of drops in turbulent flow Coalescence is indeed a very complex phenomenon that involves many interacting subprocesses. Several models are proposed in the literature to estimate the coalescence rate in turbulent dispersions. A detailed discussion of these models is given by Tobin et al. [7]. It is generally accepted that coalescence phenomena occur when a B/aS and it is essentially controlled by the collision frequency. For the collision of a drop to another one, the drops must be transported by some ways, such as molecular or turbulent diffusion, to the proximity of another drop. In order to visualize the collision phenomena; let us consider an imaginary sphere, which may be called as collision field, with a radius of R :/a as represented in Fig. 1. All the drops passing through this field may collide and have the tendency of coalescence. Therefore, the collision frequency is proportional with the surface of the field and with the flux of the drop, i.e. v 4pR2 DTP
@N @r
j
(9a)
r R
where v is the collision frequency, DTP is the eddy diffusion coefficient of drops, and N is the number of the drops per unit of volume, r is the variable radius. DTP may be estimated as [25],
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
288
Table 1 A comparison of the predicted maximum stable drop sizes with the experimental data from the literature (the experimental data from [15]) Experiment number
U (m/s)
Rea
o R (W/kg)
aS (m /103)
aS (Eq. (6))a (m /103)
1 2 3 4
3.50 3.98 4.59 6.04
1000 820 670 604
6.5 13.3 32.6 69.3
4.3 3.1 2.2 1.5
4.5 3.1 2.0 1.5
a
nc /10 6 m2/s, s /72 /10 3 N/m, rc /1.29 kg/m3, rd /1000 kg/m3, g /10.6.
two cases and it may be represented as [1]:
Fig. 1. Schematic representation of the diffusion a droplet into the collision field to collide.
DTP mR DT ;
(9b)
where DT is the turbulent diffusion coefficient of the medium, mR is the transport degree of particles by turbulent flow so that 0B/mR 5/1. The number of drops that passes through this field is also a function of the time, therefore, it may be represented by, @N 1 @ @N r2 DTP (10) @t r2 @r @r where, t is time. The boundary conditions for the solution of this equation are: t 0; rR; then N :N0 t0; r R; then N 0 t0; r 0 ; then N N0
(11a) (11b) (11c)
where N0 is number of drops per unit of volume at the beginning. The turbulent diffusion coefficient of the medium is related to the characteristics of flow and therefore, it may be stated in terms of the micro scale of the turbulence. As mentioned above, l /l0, is for the fully developed turbulent flow, but l B/l0 is for the viscous flow. The dependence of the diffusion coefficient on the micro scale of the turbulence is different for the
for ll0 ;
then
for lBl0 ;
then
DT a(o R l)1=3 l 1=2 o DT a R l2 nc
(12a) (12b)
where a , is an empirical constant. Since the effect of the system parameters on DT are not the same, then the solution of Eq. (10) is different for the two cases. In general, the scale of the turbulence may be assumed as l :/r for the both cases. Then, Eq. (10) may be rewritten after substitution 12a into it: @N a @ @N 1=3 mR o R r10=3 (13) @t r2 @r @r Considering the boundary conditions given in Eqs. (11a), (11b) and (11c), a solution of Eq. (13) may be given as: 1=3 X r 2 N(r; t) Cn J2 mn (14) e mn t R n 1 where Cn is the coefficients of the series and given as: R
g N J (m (r=R) 2 0 2
Cn
n
1=3
)r dr
0
R2
J12 (mn )
where J1 and J2 are the first and second order Bessel functions, respectively; mn is the eigenvalues and defined as: 1=3 o m a 1=2 mn qn R R 3R2=3 where qn are the constants and may be determined from the solution of the equation of J2(qn ) /0.
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
The roots for this equation are: q1 /5.05, q2 / 8.45, q3 /11.8, etc. These q values indicate that Eq. (14) is a rapidly converging series, and therefore, using the fist term is satisfactory for the practical purposes. The first terms for the eigenvalues and the coefficients are; 1=3 o R mR a 1=2 m1 5:05 (15a) 3R2=3 and C1
2N0 J3 (qn )
(15b)
qn J12 (qn )
Using the properties of the Bessel functions and the boundary conditions given above, it may be written that J3(qn ) / /J1(qn ), J1(5.05) / /033, and therefore, it may be evaluated that C1 / 1.2N0. Hence, (Eq. (14)) may be written in a simplified form as; 1=3 1=3 r o exp 8:5 R N(t; r) 1:2N0 J2 5:05 mR at R R2 @N @r
j
0:67N0 R 1 r R
1=3 o exp 8:5 R mR at R2
(16)
Inserting this statement into Eqs. (9a) and (9b) together with 12a and with an assumption that the drops are equal-sized with R /a , gives: 1=3 1=3 o o v 16mR a R 8 0 exp 8:5 R mR at a2 a2 (17a) where 80 is volumetric fraction of the drops, 80 / N0pa3/6. This equation may be modified by replacing t for t, 1=3 o t v 16mR 8 0 a R (17b) exp T1 a2 where t is the coalescence time, T1 is a parameter defined as T1 /0.117(a2/o R )1/3/mR a . The exponential term in (17b) is similar to those given in the literature [10] for the coalescence efficiency (gE) as gE :/exp( /t/T ). Therefore, a comparison with literature data suggests that above defined T1 may be related to the contact time. The collision frequency at the beginning may be estimated from
(17a) using t /0: 1=3 o v 168 0 mR a R a2
289
(17c)
A similar equation is given in the literature [26] and a comparison suggests that mR a /0.72. Eqs. (17a) and (17b) suggest that the collision frequency, v , is independent of viscosity of the medium in the fully developed turbulent flow (l /l0). The substitution of 12b into Eq. (10) enables to investigate the situation for the viscous flow, l B/ l0. In this case, the solution of Eq. (10) is simpler and may be given as: 3 1=2 R o N(r; t) N0 1 t (18) exp a R r nc Therefore, it may be written for r /R ; 1=2 @N o 3N0 R 1 exp a R t @r r R nc
j
(19)
Using the same procedure given above, it may be obtained that the collision frequency in Eqs. (9a) and (9b) may be written as, 1=2 1=2 o o v 728 0 amR R exp a R t (20a) nc nc For viscous flow it may be assumed that mR :/1. As discussed above, (20a) may be modified by replacing t for t , 1=2 o t v 728 0 a R exp (20b) T2 nc where T2 is a parameter defined as T2 /1/a (nc/ o R )1/2. A comparison with literature data suggests that above defined T2 may be related to the contact time for the viscous systems and it is different from that for the turbulent case. The collision frequency at the beginning may be estimated from 20a using t /0: 1=2 o v 728 0 a R (20c) nc This equation suggests that, in the viscous flow, the collision frequency is inversely proportional with the kinematic viscosity of the medium, i.e.
290
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
v /nc 1/2. For the collision frequency of differentsized drops, the following equation is given in the literature [5], 1=2 o v 1:294N0 (ai aj )3 R (21) nc The above results indicate that the collision frequency is directionally proportional with the concentration of drops for the both cases. For the equal sized drops, Eq. (21) is identical with 20c for the value of a /0.275.
roughness on the stable drop sizes, collision frequencies and coalescence efficiency. A comparison of Eqs. (6) /(8) with Eq. (24) suggests that the pipe roughness affects the maximum stable drop sizes. Depending on the ranges of particle Reynolds number, this effect may be summarized as follows: 2 103 5Rea 52 105 0:4 D 0:15 aS j U3 25Rea 5103
2.3. Effect of pipe roughness on the maximum stable sizes and on the coalescence of drops
oR
f (j)U 3 2D
(22)
where f(j) is the friction factor function, j is the relative pipe roughness defined as j /o /D , o is the size of the roughness. Since the roughness affects the dissipation energy, above equations suggest that the roughness also affects stable drop sizes, collision frequencies and coalescence efficiency. In general, the dependence of the friction factor, f, on Re and on the relative pipe roughness may be stated as, f c(Re; j)
(23)
where Re is Reynolds number for the flow in pipe and defined as Re /UD /nc. Several empirical equations are given in the literature [22 /25] to represent the Eq. (23). Based on the literature data [25] and using Eq. (22) it may be written for fully turbulent flow that, oR
U3 2D
c(j3=8 )
aS j 0:22
Rea 5 1 aS j
The dispersed systems are generally transported through circular pipes most of which includes roughness to some extent. The pipe roughness affects all the transport phenomena such as the heat, mass or momentum transfer. Similarly, the roughness also affects the dissipation energy as shown below [15]:
(25a)
3=8
D U3
D
4=7
U3
(25b) (25c)
Eqs. (25a), (25b) and (25c) suggest that aS / j k where k is an empirical constant. In other words, the maximum stable sizes of the drops decrease with increasing roughness. Similarly, the effects of the pipe roughness on the collision frequency may qualitatively be estimated by a comparison of Eqs. (17a), (17b), (20a), (20b) and (20c) with Eq. (24). The results suggest that: U ll0 ; then v j1=8 (26a) D1=3 3 1=2 U lBl0 ; then v j3=16 (26b) D Eqs. (26a) and (26b) shows that v /jk . In other words, the collision frequencies of the drops increase with increasing pipe roughness. An increase in the collision frequency result in increase in the coalescence rate. It is suggested in the literature [10] that coales0.89 cence efficiency gE /e oR . Using this relationship and 23d it may be written that ln gE / /j1/3 This relation suggest that the coalescence efficiency decreases with increasing roughness. 2.4. Modeling of the drop size distribution in turbulent dispersions
(24)
Eq. (24) enables to estimate the effect of the
After starting the motion, a new size distribution takes place in a dispersed system due to the
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
coalescence and breakup of the drops. Prediction of drop size distribution in such systems requires use of a distribution function. In the size distribution, generally fraction of drops having certain sizes, or the cumulative volume fraction, or the number density (concentration of the drops having certain volume) is investigated. The Rosin /Ramler distribution or upper limit lognormal distribution (ULLN) functions are used by several researchers to represent this distribution [12,16]. Recently, Ruiz et al. [27] showed that the drop size distribution in a stirred tank has a bell shaped form and it may be represented by the lognormal distribution function. In this study, the size distribution in the turbulent dispersion is evaluated by using the Focker /Planck equation. Accordingly, a size-time dependent distribution function may be represented as [28,29], @P(a; t) @ @ [f (a)P(a; t)] @t @a @a @P(a; t) B(a; t) @a
(27)
P (a ,t) is the distribution function of the drops depending on size and time, f(a) is the size variation function, B (a ,t ) is the stochastic diffusion coefficient. The variation the mass of a drop, due to the coalescence or the breakup, with time may be stated as, dm dt
4pR2 DTP
@N @r
j
dt
1 3
(17a), (17b), (20a), (20b) and (20c) the definition the f(a ) also varies depending on the frequency statement. As an example, the solution is discussed below for the turbulent case, i.e. l /l0. A comparison of Eq. (28c) with (17c) indicates n 1 that kR /(16/3)80mR a (o 1/3 ). If it is assumed R /aS as B (a,t) /B is a constant, then it may be possible to obtain various solutions for Eq. (27) by inserting 28c into it. In fact, any value may be chosen for the rate order, n , in 28c, but, the analytical solutions may be obtainable for only certain values of n. Some typical solutions for the selected n values are discussed below: a) If n /2/3, then from 28c f(a ) /kR , then a solution to Eq. (27) may be given as, P(a; t) P0 (a)e
k2R t=B
a p ffiffiffiffiffiffiffi ffi erf 4Bt
(29)
where P0(a ) /P (a ,0). b) If n /5/3, then from 28c f(a) /kR a . In this case, the process is dependent on particle sizes, therefore, the maximum stable sizes of the drops should be taken into account. In the case of small drops, i.e. a B/aS, the coalescence is the governing phenomena and it may be convenient to define the function of f (a ) as f(a) /kR (aS /a). Then the solution to Eq. (27) may be given as, 2
m vm
(28a)
r R
where m is the instantaneous mass of a drop and may be estimated as m /prda3/6. Then Eq. (28a) may be modified as: da
291
va f (a)
(28b)
For the nth order coalescence kinetics, the above equation may be rewritten as: da 1 an v n 1 kR an 2=3 f (a) dt 3 aS
(28c)
where kR is the rate constant of the coalescence. Since the collision frequency statement vary depending on the micro scale of the turbulence Eqs.
P(a; t) (aS a) 1=2 e kR (aS a) =4B X m2n kR t Cn e c m2n n
1 1 k (a a)2 ; ; R S 4 4 4B
(30)
where mn is an eigenvalue and Cn is the coefficients of the series, c [m2n /1/4; 1/ 4,kR (aS /a )2/4B ] is a hypergeometric function. In the case of large drops, i.e. a/aS, the break-up is the governing phenomena and it may be convenient to define the function of f(a) as f(a ) /Kp(a /aS), where Kp is the rate constant of the breakup. Under these conditions, the solution of Eq. (27) may be given as,
292
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295 2
P(a; t) e kR (a aS ) =4B X Cn e nKp t Hn (aS a) n
sffiffiffiffiffiffi Kp
(31)
4B
where Hn is the Hermite function. The coefficient of Cn is given as, sffiffiffiffiffiffi 1 Kp da P0 (a)Hn (aS a) Cn pffiffiffi 2n n! p 4B
g
c)
If n / /1/3, then from 28c f (a ) /kR /a . If the coalescence and breakup process occurs simultaneously it may be convenient to defined the function of f(a ) as f(a) /kR /a /Kpa . In this case, a solution of Eq. (27) may be given as, 2
2
P(b; t) bu e (Kp aS =2B)b X Kp a2S 2 2Kp nt b e Cn Lwn 2B n
(32)
where Cn is
u
u1
Cn y
g P (b)L ((K a =2B)b )db 0
0 (u1) (u 1) 2 2
2
w n
2 p S
2
G(n (u 1)=2)n!
where b /a /aS, u /kR /B, y /kR /Kpa2S, w / (u /1)/2, Lw n (b ) is the Laguerre function, G (n ) is the Gamma function. For a special case, if t 0/ in Eq. (32) then the solution assumes a simple form as: k a2 P (b) C0 bu exp R S b2 (33) 2B
where C0 /2(u /2y)((u1)/2). Eq. (33) suggests that the final size distribution (at t 0/) is not related to the beginning distribution. For a steady-state regime, Eq. (33) is similar to the Rosen /Ramler distribution function. However, for u /1 it is similar to the Rayleigh distribution, and for u /
2 it is similar to the Maxwell distribution function. The shape of the distribution function describes the extent of the heterogeneity of the particle size in the distribution. The average drop size at t may be estimated as: sffiffiffiffiffiffi 2B k (34) G 1 R a aP (a)da a Kp 2B
g 0
3. Results and discussion The literature data and above-developed mathematical statements indicate that the maximum stable drop sizes in an isotropic turbulent flow depends essentially on the particle Reynolds number, the collision frequencies and on the scale of the turbulence. The collision frequency increases with the increasing drop concentration and dissipation energies. The results suggest that if l /l0, the collision frequency is not affected from the viscosity of the medium but it is affected if l B/l0. The pipe roughness also affects the maximum stable sizes, the collision frequencies and the collision efficiency. The particle size distribution in a dispersed system, at any time, is different from the starting distribution due to the coalescence and breakup phenomena. Both the distribution functions and most of the experimental data indicates that the particle size distribution is unimodal (i.e. have one maximum in the distribution). However, some experimental data indicate that the particle size distribution may be multimodal (i.e. two or more maximums in the size distribution). For example the data of Tobin et al. [9] for the system of benzene /carbontetrachloride (5%) in water indicate that the particle size distribution is unimodal. However, the data for the system of hexane / carbontetrachloride (5%) in water show bimodal distribution. Most of the distribution functions given used in the literature are not able to represent the multimodal size distribution. It is shown here that the solution of the Focker/Planck equation also may be used for estimation of the unimodal or multimodal size distribution. The
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
typical solutions for this equation are given by Eqs. (29) /(33). These solutions may be simplified considering the characteristics of the system. For example, the solution given by Eq. (33) may be simplified as, C2 2 (35a) a f2 P(a; t) C1 a exp f1 where C1 and C2 are constants, f1 and f2 are time functions. The constants in this equation may be determined from experimental data. For this purpose, the data of Tobin et al [9] for the system of benzene /carbontetrachloride (5%) in water are used. The obtained equation is, 0:0395 P(a; t) 0:0085a2 exp 0:2 a f (35b) f where f is a time function defined as f /1/ 0.1t0.5, t is the time as minute. A plot of Eqs. (35a) and (35b) is given in Fig. 2 for t /0 and for t /30 min. Some experimental data from the literature [9] are also represented in the figure for comparison. The results indicate that Eqs. (35a) and (35b) is well suited with the data for the unimodal distribution. However, this equation does not represent data for the system of hexane/carbontetrachloride system, because this system shows a bimodal distribution. Based on the experimental result, Tobin et al. [9] commented
293
that coalescence of the smallest drops is very small, as consequence of which such droplets linger for long times, leading to bimodality. However, they did not give the physical reasons for that why the coalescence of these smallest drops is small. Indeed, it is expected that the probability of collision of the small drops is higher since the diffusion rates are higher. Therefore, the explanation of the bimodality, in terms of collision frequencies may not be satisfactory. We suggest here that the interaction energies between the drops may be effective in the coalescence and breakup phenomena and therefore, in the shape of the distribution function. At very small concentrations (8 /1) the interaction between the drops may be negligible since the drops are far away from each other. The probability of the collision of the drops under these conditions is very low and the coalescence may not occur. If the drop concentration is appreciable (i.e. 8 /0.3) then the interaction between the drops may be appreciable and the effectiveness of collision may be affected from this. The total interaction energy (UT) may be stated as the sum of the attraction energy (Ua) and the repulsion energy (Ur). Assuming that all the drops diffused into the collision field hit a drop, then the statement of collision frequency given in Eqs. (9a) and (9b) may be modified as: v 4pR2 DTP
@N @r
j
h
@UT
r R
@r
j
(36) rBR
where h is a parameter depending drop size and the diffusion resistance. For the small drops (Stokes’ drops) h is defined as h /(4pr2N )/ (6phca ). The solution of Eq. (27), using Eq. (36), may be given as: P(a; t) e (hUT (a))=B X 2 e mn Bt C1n n
C2n Fig. 2. Unimodal size distribution according to Eqs. (35a) and (35b). Experimental data are from Tobin et al. [9]. ': t /0 min, m: /30 min.
h
g exp B
@ 2 UT (a) UT (a) m2n da @a2
(37)
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
294
Considering the characteristics of this series, Eq. (37), may be simplified as, C h P(a; t) C1 a2 exp 2 a f2 UT (38) B f1 The coefficients of this equation should be determined from the experimental data. The attraction and the repulsion energies of the drops may be represented as [2]; hUa B
A
(39a)
a2
hUr X ( 1)n An a2n exp( mn an1 ) B n 0
(39b)
where A , An and mn are the empirical constants. Based on the literature data we suggest here that if Ua /Ur then h 10:8 UT ; (40a) B a2 but if Ua and Ur are comparable and then hUT B
10:8 a2
2
5:55e 0:3a 0:004a2 e 0:0006a
(40b)
Using the data of Tobin et al. [9] for the system of hexane/carbontetrachloride (5%) in water, the approximated equation is: 0:0395 P(a; t) 0:006a2 exp 0:2 a 0:7f f h UT (41) B where f /1/0.25t0.5. The shape of the distribution function defined by Eq. (41) depends on the definition of the interaction function of (h/B )UT. If the function defined by 40a is used, then a unimodal distribution is obtained just similar to that given by 35b. This result suggests that the effect of the attraction on the distribution is negligible. If the function defined by 40b is used, then a bimodal distribution is obtained. A plot of Eq. (41) together with 40b is given in Fig. 3 for t / 1 and t /60 min. Some experimental data from the literature [9] are also represented in the figure for comparison. The results indicate that Eq. (41) a
Fig. 3. Bimodal size distribution according to Eq. (41). Experimental data are from Tobin et al. [9]. ': t /1 min, m: /60 min.
relatively good fit is obtained for the bimodal distribution. A notable feature of the experimental data and above developed relationships is that the peak in the number density do not shifts appreciably with increasing time. The number density lower with time due to the coalescence phenomena. Several of small drops may combine to form one, and therefore, the decrease in the number density of small particles higher than the increase in the density of the large particles. The model prediction is also in accord with this situation since only a slight cross-over is predicted at large size particles.
4. Conclusions Both the theory and experimental data indicate that the maximum stable size, the coalescence frequency, and the drop size distribution in turbulent dispersions are closely related to the concentration of the drops and on the rheologic properties of the medium. The characteristics of turbulence and pipe roughness are also effective on these phenomena. Depending on the properties of the dispersed system, the size distribution in a dispersed system may be unimodal or multimodal. The results suggest that the effective interaction energies among the drops affect the shape of the
S. Ceylan et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 212 (2003) 285 /295
size distribution. The modified solutions of the Focker/Planck equation may be used to represent these distributions.
References [1] V.G. Levich, Physicochemical Hyrodynamics, PrenticeHall, Englewood Cliffs, Canada, 1962. [2] A. Scheludko, Adv. Colloid Interphase Sci. 1 (1967) 391. [3] C.A. Coulaloglu, L.L. Tavlarides, Chem. Eng. Sci. 32 (1977) 1289. [4] C. Malderlli, R.K. Jain, I.B. Ivanow, E. Ruckenstein, J. Colloid Interphase Sci. 78 (1980) 118. [5] K. Higashitani, K. Yamanchi, Y. Matsuno, J. Chem. Eng. Jpn. 18 (1983) 299. [6] E. Ruckenstein, A. Sharma, J. Colloid Interpahes Sci. 119 (1987) 1. [7] T. Tobin, R. Muralidhar, D. Ramkrishna, Chem. Eng. Sci. 45 (1990) 3491. [8] B. Yang, S. Hartland, Chem. Eng. Sci. 46 (1991) 1437. [9] K. Saneev, R. Kumar, K.S. Ghandi, Chem. Eng. Sci. 18 (1993) 2025. [10] S. Liu, D. Li, Chem. Eng. Sci. 54 (1999) 5667. [11] J. Baldyga, W. Podgorska, Can. J. Chem. Eng. 76 (1998) 456. [12] M.J.H. Simmons, B.J. Azzopardi, Int. J. Multiphase Flow 27 (2001) 843. [13] J.O. Hinze, AIChE J. 1 (1955) 289.
295
[14] R.P. Hesketh, A.W. Etchells, T.W. Russel, AIChE J. 33 (1987) 663. [15] R.P. Hesketh, A.W. Etchells, T.W. Russel, Chem. Eng. Sci. 46 (1991) 1. [16] A. Panagiota, G.F. Hewitt, Chem. Eng. Sci. 55 (2000) 3133. [17] R.V. Calabrese, T.P.K. Chang, P.T. Dang, AIChE J. 32 (Part I) (1986) 657. [18] J.S. Lagissety, P.K. Das, R. Kumar, K.S. Gandhi, Chem. Eng. Sci. 41 (1986) 65. [19] S.L. Soo, Fluid Dynamics of Multiphase Systems, Blaisdell Publishing, London, 1970. [20] P. Reist, Aerosol Science and Technology, McGraw Hill, New York, 1992. [21] F. Raymond, J.M. Rosant, Chem. Eng. Sci. 55 (2000) 943. [22] L.F. Moody, Friction factors for pipe flow, Trans. ASME 66 (1944) 671. [23] R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, second ed., Wiley, New York, 1978. [24] K. Ceylan, A. Altunbas, G. Kelbaliyev, Powder Technology 119 (2001) 175. [25] A. Altunbas, G. Kelbaliyev, K. Ceylan, J. Aerosol Sci. 33 (2002) 1075. [26] G. Kelbaliyev, Theor. Fundam. Chem. Technol. 26 (1992) 390. [27] M.C. Ruiz, P. Lermanda, R. Padilla, Hydrometallurgy 63 (2002) 65. [28] C.W. Gardiner, in: H. Haken (Ed.), Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, vol. 13, Springer, New York, 1985. [29] G. Kelbaliyev, K. Ceylan, Powder Technol. 115 (2001) 84.